L e c tu re N o te s in C o m p u te r S c ie n c e E d ite d b y G . G o o s , J . H a rtm a n is a n d J . v a n L e e u w e n 1 8 0 3 3 B e r lin H e id e lb e rg N e w Y o rk B a rc e lo n a H o n g K o n g L o n d o n M ila n P a r is S in g a p o re T o k y o S te f a n o C a g n o n i e t a l. ( E d s .) R e a l-W o rld A p p lic a tio n s o f E v o lu tio n a ry C o m p u tin g E v o W o rk sh o p s 2 0 0 0 : E v o IA S P , E v o S C O N D I, E v o T e l, E v o S T IM , E v o R o b , a n d E v o F lig h t E d in b u rg h , S c o tla n d , U K , A p ril 1 7 , 2 0 0 0 P ro c e e d in g s 13 S e rie s E d ito rs G e r h a r d G o o s , K a r ls r u h e U n iv e r s ity , G e r m a n y J u r is H a r tm a n is , C o rn e ll U n iv e r s ity , N Y , U S A J a n v a n L e e u w e n , U tr e c h t U n iv e r s ity , T h e N e th e r la n d s M a in V o lu m e E d ito r S te fa n o C a g n o n i U n iv e r s ity o f P a r m a D e p a rtm e n t o f C o m p u te r E n g in e e rin g P a rc o d e lle S c ie n z e 1 8 1 /a , 4 3 1 0 0 P a rm a , Ita ly E -m a il: c a g n o n i@ c e .u n ip r.it C a ta lo g in g -in -P u b lic a tio n d a ta a p p lie d fo r D ie D e u ts c h e B ib lio th e k - C IP -E in h e its a u fn a h m e R e E v 2 0 B a S p a l w o rld a p p lic a tio n s o W o rk sh o p s 2 0 0 0 : E 0 0 . S te fa n o C a g n o n i rc e lo n a ; H o n g K o n g rin g e r, 2 0 0 0 (L e c tu re n o te s in c o m IS B N 3 -5 4 0 -6 7 3 5 3 -9 o f e v o lu v o IA S P . . . (e d .). ; L o n d o tio . . . - B n ; n a ry c o m , E d in b u e rlin ; H M ila n ; p u rg h e id P a r tin g : p ro c e , S c o tla n d , e lb e rg ; N e is ; S in g a p o e d in g s U K , A w Y o rk re ; T o / p ril 1 7 , ; k y o : p u te r s c ie n c e ; V o l. 1 8 0 3 ) C R S u b j e c t C l a s s i fi c a t i o n ( 1 9 9 8 ) : C . 2 , I . 4 , F . 3 , I . 2 , G . 2 , F . 2 , J . 2 , J . 1 , D . 1 IS S N 0 3 0 2 -9 7 4 3 IS B N 3 -5 4 0 -6 7 3 5 3 -9 S p rin g e r-V e rla g B e rlin H e id e lb e rg N e w Y o rk T h is w o rk is s u b je c c o n c e r n e d , s p e c i fi c a re p ro d u c tio n o n m ic o r p a rts th e re o f is p e in its c u rre n t v e rs io n lia b le fo r p ro s e c u tio t to c o p y rig h t. A ll rig h ts a re re s e rv e d , w h e th e r th e w h o le o r p a rt o f th e m a te ria l is lly th e rig h ts o f tra n s la tio n , re p rin tin g , re -u s e o f illu s tra tio n s , re c ita tio n , b ro a d c a s tin g , r o fi l m s o r i n a n y o t h e r w a y , a n d s t o r a g e i n d a t a b a n k s . D u p l i c a t i o n o f t h i s p u b l i c a t i o n rm itte d o n ly u n d e r th e p ro v is io n s o f th e G e rm a n C o p y rig h t L a w o f S e p te m b e r 9 , 1 9 6 5 , , a n d p e r m is s io n f o r u s e m u s t a lw a y s b e o b ta in e d f r o m S p r in g e r- V e r la g . V io la tio n s a r e n u n d e r th e G e rm a n C o p y rig h t L a w . S p rin g e r-V e rla g is a c o m p a n y in th e B e rte ls m a n n S p rin g e r p u b lis h in g g ro u p . c S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0  P rin te d in G e rm a n y T y p e s e ttin g : C a m e ra -re a d y b y a u th o r P rin te d o n a c id -fre e p a p e r S P IN : 1 0 7 2 0 1 7 3 0 6 /3 1 4 2 5 4 3 2 1 0 i r a si gla ti fi l d f lti a r p ta ti ( ) pr id s a la lt l s, i spir d t t r f a t ra ls l ti a d g ti i rita ,t pr l s li g, a i la r i g, a d pti ia ti i a r al - rl d a ppl ia ti s. spit s a rl it iti s a t , t at a da t d a t t i ti f p t rs, a d a tt r f r a ld fi iti f , a d i t 6 s, t q stf r r a l - rl d a ppl ia ti s f l ga i t l at s. dra a ti i r a s i p t r p rf r a si t l a std a d ft 2 t t r ga ris t a p siti f d a pr ss: t iq s a r a d r a ppl ia l,sti l a ti gt gr t fit r sti t ir st d ,a d a l l i g, i tr , p rf l pa ra dig s t d is d. I pa ra l ll it t r tia lr s l ts, t r ffi l ds t i is i ga ppl id is i r a si gda da ,a l g it t pl it fa ppl ia ti s a d a ppl ia ti d a i s.I pa rti l a r,i d stria l lr l a tfi l ds,s a s sig a l a d i a g pr ssi g, p t r isi , pa tt r r g iti , i d stria l tr l , tl ia ti , s d l i g a d ti ta l i g, a d a r spa gi ri g a r pl i g t iq s t s l pl r a l - rl d pr l s. is l ta i s t pr di gs f r s ps 2 : si r sps r al - rl d a ppl ia ti s f l d rr tl pril 7 t 2 i di rg . r s ps a r : I 2 ,t s d rp a r sp lti a r p ta ti i I a g a lsis a d ig a l r ssi g, I2 ,t first r p a rs p lti a r p ta ti i st s, tr la d ri s i I d str , l2 ,t s d rp a rs p lti a r l ia ti s, ti 2 , t first r p a rs p lti a r d l i ga d i ta l i g, 2 ,t t ird rp a rs p lti a r tis, a d l ig t2 ,t first r p a rs p lti a r r a tis. r s ps 2 as l di ti it t t r a r rp a ts: r 2 ,t rp a fr ti r gra i g, l d pril t a d 6 t , a d I 2 ,t ird I t r a ti a l fr la l st s: fr ilg t a rd a r , l d fr pril7 t t t. l dl i t ta t rs ft it r a ti a l pr gra itt s ft si r s ps i i ld d t r s a r rs a d r s a r rs i t sp ifi fi l ds fit r sta d s r d t ig q a l it ft pa p rs. pril2 a t fa a g i, i a rd l i, i, rg it , a id r , a rti a t s, a rt, ir a a i, g rt . . rs, a t r, a d r . ga rt r r s ps 2 I I - ha ir - ha ir O I ha ir l - ha ir l - ha ir l - ha ir I ha ir ha ir l ight ha ir a lha ir l ia tins ha ir r s ps 2 r t fa n a gn ni( ni rsit f a rma Ita l) i ar l i( ni rsit f irmingha m K) n i( ni rsit f l a sg K) rg mith ( ni rsit f a st ngl ia K) a i rn ( ni rsit f a ing K) a rtin O a t s ( ritish l m pl K) mma a rt( ni rsit f in rgh K) ir a a n i( il a n lt hni Ita l) g rt . . rs ( h th rl a n s) n a ht r ( a pir ni rsit K) rn . ga rt ( th a nk ni rsit K) r r a na gitis a mi is hn lgia l a tina lInstit tin f h ssa lniki r i a nni rni ni rsit f a rma Ita l a rja n a n n kk r h th rl an s l ista ir rmita g a pir ni rsit K l fga ng a n ha f ni rsit f rtm n rma n an al l. r as st rn s r ni rsit n ra na rini il a n lt hni Ita l a him . rn a imlr hr slr rma n l rt r ggi ni rsit f a ia Ita l t fa n a gn ni ni rsit f a rma Ita l ri l a ha kra rt a tina l ngin ring a ra t r K l a l a ri g ni rsit f irmingha m K ar lm tti il a n lt hni Ita l is rr ia ni rsit f is n rt ga l a s n a i a ni rsit f ihiga n ar rig r ni rsit f r ss l s l gi m ha ng . a n r t K g st n i n i n ni rsit h th rl an s tr lming ni rsit f h ffi l K a ri lr a n it rl an O s a r . a r ia ni rsit f ra na a pa in rg r¨ l rma n a rk r n lr ni rsit fZ a gr r a tia Inma n a r ni rsit f ss K r hil s a n s ni rsit f ss K ra n is rr ra ni rsit f ra na a pa in a sa ki ira f ji mp ta tina l l ing a ra t r a nil a r K Kn nt ni rsit f l a sg K ik a k i ni rsit f ss K a ri K pp n ra nh fr I K rma n hn K a ta nf r ni rsit Ka h n a n a tina l ni rsit f inga p r inga p r n i ni rsit f l a sg K ln tt n I I ra n an r a r nima t a ra n O ra i igl in ni rsit f a pls Ita l l ia n il lr ni rsit f irmingha m K t fa n l fi a tina l s a r h n ilIta l t r r in ha l m rs ni rsit n n a ht r a pir ni rsit K i ar l i ni rsit f irmingha m K im mith ni rsit fth st f ngl an K a thia s l a n n a ra l inga m kia K t r a nn l l s pl K rt mith ni rsit fth st f ngl an K Ka h n a n a tina l ni rsit f inga p r inga p r n r a . . tta ma n i n tia Ita l n rr l l ni rsit f rk K a ns- iha l igt I rma n . nr i rp l ni rsit K in a ni rsit f irmingha m K l iZ a la l a rit- a tt ni rsit K p sr s a pir ni rsit K t th t rk f s l ln in ltina r mp ting a pa n s p cil rp s I g n ltin ith l bl rd . m lin . . st r . . rn l n . r t r sc pic isin f r n id . . . r . rin n . . st r n tic ng . . ing.. 2 lst ring l g rith ................................... n . . ng 22 c n Int rpr t tin sing . r h . . kls n 34 s s in ign l p c s............................. 44 inding l f rs s: h l tr igh ch ppr ch.................... . . r . rkins . . r m . h ilr . . rt r . . ng . . rgh s . . m nski n . . l h 4 lt il ts nd tr n tic r gr p t tin ltin r . . rrir ntic . . b t sing r hl r .............. ltin r s rs nd c l i tin f r n id b tb ns f n tic r gr ing............................................................ . rlss n . rin n . r hl 6 n th c lbil it f n tic l g rith s t r rg - c l tr lctin ................................................................. . sr n . . rt 77 bining ltin r , nn ctinist, nd lssific tin l g rith s f r h p n lsis ........................................... . . sin n . O . ngs 7 p ri nt l t r in tin f r s phil br nic rdin t s b n tic l g rith s, th i pl th d, nd h ir brid .............. . . ir . . im kin . init n . sm n 7 p rs . n tic l g rith . i . . ith n c l rch f r . . ng ling b hp r bl s.. 7 istrib t d rning ntr lf r ffic ign l s.......................... . . . rs n . ll n . ils i ris r dictin b . hn n . i r ing tr l l 7 r l t rks...... 27 r jct r ntr l lr t rk nd Its sign t tin thr gh ltin r p ting................................................ . h ng n . i 3 ltin r p t tin nd nl in r r gr ingin l ti- d l b st ntr l sign .................................................. . lss n . r¨ l 47 nch rking st- ssign nt ch sfr l ti- b jcti ltin r l g rith s................................................ . k lkis n . i t tic nth sis f th th p lg nd r t rs f r ntr l lr f r hr - g lnt ith i - c nd l sing n tic r gr ing................................................... . . . . n . . l n . . nn tt 6 t tic sign f l ti ribl ntr l st i ltin r p t tin............................................................ . . n . . n . . hr 7 lp nt f r r nsf r r h r l d l s f r il p r t r r dictin ................................................. . . ng . ng . . tt ll . i h rs n . im ns n n . . s t . . tic l id tin f r t c lInt rf c s scrib d in rn . n r n . q illr lti ri d l ing f /I t rk r ffic f r Intr sin ......... 2 t ctin .. 2 4 l ti d l rf r nc r fils n th d pti istrib t d tb s n g nt r bl ................................................... 224 . O t s . rn n . r r t c l nstr ctin sing . h r ls n . km n n tic rch chniq s................. 23 s rs r dictin f . rn . . rs n r nic tin . . q ir nts f r igh- p d ng . n r n rchit ct r f r l ti- g nt . ll n . ils irc its............... 247 . i lnt rning st s....... 2 p rs n b lnc . . . r r st ring st .................................. 26 7 r . in l n . . grt st tic In stig tin f rf r nc n obsh p ch d l ing r bl s ............................................................... . rt n . ss 27 7 n nt l g rith ith hr n l tin lf r t l rdin ss r bl s..................................................... . rkl n . i n rf 2 7 n tic pr s nt tin nd n lst r r ss r f r b h p ch d l ing r bl s.......................................... . k . iji . i shi n . O k m r 2 7 pti ising n ltin r l g rith f r ch d l ing................... . rq h rt . hish lm n . ht r 3 7 p rs n- in ltin f ntr lf r r- gg d b t sing n tic r gr ing........................................................... . n rss n . nss n . r hl n . rin 3 pti i d l l isin r ltin r ft r . lm 327 l f- d pti . ll n . t tin in rst b t t t nt n r tin b th ......................................... ntr l lrs.............................. sing brid ltin r - * ppr ch f r rning cti h irs............................................................... . tt n . . r 33 347 ig p rs p r is d . . rl ltin r th ds in r d n . n r n . ir ic sign pti i tin . 3 7 n ltin r l g rith f r rg c l t ring r bl s ith ppl ic tin t irl in r ch d l ing................................. . rhi ri n . t n k 36 7 sign, I pl nt tin, nd ppl ic tin f lf r pti l ircr ft sitining.............................................................. 3 2 . f lgrf . rnk . i h n rgr n . t l n rg r d .................................................. 3 S p e c ia l P u r p o s e I m a g e C o n v o lu tio n w ith E v o lv a b le H a r d w a r e a J o e D u m o u lin , J a m e s A . F o s te r a N e w L ig h t In d u s trie s , L C e n te r fo r S e c u re a n d D c M ic ro e le c tro n ic s R e s e M o sc o w ID d D e p t. o f C o m p u te r S c ie e D e p t. o f E le c tric a l a n d b e m a i l : j o e @ b r e s g a l . c o m , b ,c ,d c ,e , Ja m e s F . F re n z e l , S te v e M c G re w a td ., S p o k a n e , W A e p e n d a b le S o ftw a re , U . Id a h o , M o s c o w , ID a r c h a n d C o m m u n ic a tio n s I n s t., U . I d a h o , n c e , U . Id a h o , M o sc o w , ID C o m p u te r E n g in e e rin g , U . Id a h o , M o s c o w , ID f o s t e r @ c s . u i d a h o . e d u , s t e v e m @ i e a . c o m j f f @ m r c . u i d a h o . e d u , A b str a c t. I n th is p a p e r , w e in v e s tig a te a u n iq u e m e th o d o f in v e n tin g lin e a r e d g e e n h a n c e m e n t o p e ra to r s u s in g e v o lu tio n a n d re c o n fig u r a b le h a r d w a r e . W e s h o w th a t th e te c h n iq u e is m o tiv a te d b y th e d e s ir e f o r a to ta lly a u to m a te d o b je c t re c o g n itio n s y s te m . W e s h o w th a t a n im p o r ta n t s te p in a u to m a tin g o b je c t r e c o g n itio n is to p ro v id e fle x ib le m e a n s to s m o o th im a g e s , m a k in g f e a tu r e s m o r e o b v io u s a n d r e d u c in g in te r fe re n c e . N e x t w e d e m o n s tra te a te c h n iq u e fo r b u ild in g a n e d g e e n h a n c e m e n t o p e r a to r u s in g e v o lu tio n a r y m e th o d s , im p le m e n tin g a n d te s tin g e a c h g e n e r a tio n u s in g th e X ilin x 6 2 0 0 f a m ily F P G A . F in a lly , w e p r e s e n t th e r e s u lts a n d c o n c lu d e b y m e n tio n in g s o m e a r e a s o f fu rth e r in v e s tig a tio n . I n tr o d u c tio n Im a g e e d g e e n h a n c e m e n t is a n im p o rta n t p a rt o f m o d e rn c o m p u te riz e d o b je c t re c o g n itio n m e th o d s . E d g e e n h a n c e m e n t ty p ic a lly re q u ire s c o n v o lu tio n o p e ra to rs th a t p ro d u c e w e ig h te d a v e ra g e tra n s fo rm a tio n s o n in d iv id u a l p ix e ls o f s o m e s o u rc e im a g e . E d g e e n h a n c e m e n t o p e ra to rs te n d to b e d e v e lo p e d fo r p a rtic u la r ty p e s o f p ro b le m s a n d la rg e r o p e ra to rs te n d to b e fo r v e ry s p e c ific p u rp o s e s . L a rg e irre g u la r p ix e l p a tte rn s re q u ire ra th e r la rg e a n d irre g u la r e d g e e n h a n c e m e n t o p e ra to rs . T h e s m o o th in g p ro c e s s a s s is ts a n e d g e d e te c tio n a lg o rith m to d is tin g u is h b e tw e e n b a c k g ro u n d n o is e a n d a c tu a l o b je c ts in a n im a g e . U n fo rtu n a te ly , s o ftw a re im p le m e n ta tio n s o f la rg e c o n v o lu tio n s a re e x tre m e ly s lo w . C o n s e q u e n tly , c o n v o lu tio n s o f a n y re s p e c ta b le s iz e a re ty p ic a lly im p le m e n te d in s p e c ia liz e d im a g e p ro c e s s in g h a rd w a re . A n o th e r p ro b le m w ith la rg e c o n v o lu tio n s is th a t it is o fte n v e ry d iffic u lt to " d is c o v e r" g o o d o n e s . T h e p ro c e s s o f c re a tin g th e s e c o n v o lu tio n o p e ra to rs re q u ire s tria l a n d e rro r, a n d e x p e rie n c e . R e c e n tly , a t N e w L ig h t In d u s trie s , w e n e e d e d s p e c ia l p u rp o s e c o n v o lu tio n o p e ra to rs fo r u s e in lo w -c o s t, c o m m e rc ia l o ff-th e -s h e lf h a rd w a re d e v ic e s . T h e s e c o n v o lu tio n s n e e d e d to b e v e ry e ffic ie n t a n d im p le m e n ta b le in re c o n fig u ra b le h a rd w a re . U s in g g e n e tic a lg o rith m s a n d X ilin x X C 6 0 0 0 te c h n o lo g y , w e im p le m e n te d a n e v o lu tio n a ry s y s te m th a t d e s ig n e d c o n v o lu tio n o p e ra to rs fo r o u r a p p lic a tio n . T h e g e n e tic a lg o rith m e v a lu a te d o p e ra to rs b y c o m p a rin g th e S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 − 1 1 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 J . D u m o u lin e t a l. c o n v o lu tio n o f o n e o rig in a l im a g e w a b le to p ro d u c e . T h e s e e x p e rim e n ts d e m o n s tra te d e n h a n c in g c o n v o lu tio n k e rn e ls u s in g d e s c rib e s o u r re s u lts a n d s h o w s a n a lg o rith m . F in a lly , th is p a p e r p r a p p lic a tio n s . ith s e v e ra l p re p a re d im a g e s th a t th e c o n v o lu tio n s h o u ld b e b o e v e v e se th th e o lu tio o lv e d n ts s v ia b n a ry c o n v o m e ility , a n d te c h n iq u e o lu tio n th p o te n tia l s o m e lim s a n d p re a t w a s u fu tu re ita tio n p a re d s e d in re se a r s , o f c re a tin g e d g e im a g e s. T h is p a p e r a n e d g e d e te c tin g c h d ire c tio n s a n d M o tiv a tio n C o m p u te riz e d im a g e p ro c e s s in g p ro v id e s m a n y o p p o rtu n itie s fo r im p le m e n tin g e v o lu tio n a ry p ro b le m -s o lv in g te c h n iq u e s . O u r p ro b le m is to id e n tify im p o rta n t fe a tu re s in a d iffra c tio n p a tte rn th a t c h a n g e s o v e r tim e . T h is is a n a lo g o u s to id e n tify in g a n d tra c k in g m u ltip le o b je c ts m o v in g th ro u g h a s e rie s o f fra m e s . C o n s id e r a c a m e ra a im e d a t a m o v in g o b je c t o r a n o b je c t illu m in a te d b y a m o v in g lig h t s o u rc e . T h e p ro b le m is to id e n tify th e o b je c t a s it m o v e s th ro u g h a s e rie s o f s till fra m e s . S in c e w e n e e d e d c o m p a c t, in e x p e n s iv e , e ffic ie n t h a rd w a re to p e rfo rm th e id e n tific a tio n , w e c h o s e to im p le m e n t o u r s o lu tio n in re c o n fig u ra b le s y s te m s u s in g X C 6 0 0 0 F P G A s fro m X ilin x . O u r ta rg e t im a g e s w e re a s e rie s o f d iffra c tio n p a tte rn s re c o rd e d o v e r tim e w ith a m o v in g lig h t s o u rc e . T h e d iffra c tio n p a tte rn s a re o p tic a lly d e riv e d F o u rie r T ra n s fo rm s o f fe a tu re s o n th e s u rfa c e o f a n o b je c t. T h e fra m e s a re c a p tu re d in 8 -b it in g ra y -s c a le a n d th e o b je c ts to b e re c o g n iz e d a re g e o m e tric " b lo b s " th a t b o th m o v e a n d c h a n g e s h a p e fro m fra m e to fra m e . W e m u s t id e n tify e a c h o b je c t a s it m o v e s a n d c h a n g e s s h a p e o v e r a s e rie s o f fra m e s . W e c h o s e a s im p le e d g e -d e te c tio n a lg o rith m [8 ] a s th e m e th o d fo r e x tra c tin g th e b o rd e r. A k e y s te p in id e n tify in g o b je c t b o rd e rs in a p a rtic u la r fra m e is a p p ly in g a la rg e c o n v o lu tio n o p e ra to r to th e im a g e . W e u s e d g e n e tic m e th o d s to d e v e lo p a p p ro p ria te c o n v o lu tio n o p e ra to rs fo r o u r tra in in g s e t. T h e fo llo w in g s e c tio n s e x p la in th e a p p ro a c h a n d th e re s u lts o f o u r in itia l e x p e rim e n ts . B a c k g r o u n d I m a g e P r o c e s s in g C o n s id e r a tio n s T h e lite ra tu re o n im a g e p ro c e s s in g c o n ta in s m a n y e x a m p le s a n d te c h n iq u e s fo r p e rfo rm in g im a g e e n h a n c e m e n t a n d im a g e s e g m e n ta tio n . G e n e ra lly , a s o n e m ig h t e x p e c t w ith a n y c o m p u ta tio n a l ta s k , th e re a re tra d e o ffs b e tw e e n s p e e d a n d a c c u ra c y in th e s e g m e n ta tio n p ro c e s s . W e u s e d o p tic a lly g e n e ra te d F o u rie r T ra n s fo rm s o f s u rfa c e fe a tu re s fo r th is e x p e rim e n t. Id e n tific a tio n o f fe a tu re s in th e fre q u e n c y s p a c e o f th e s e im a g e s a m o u n te d to id e n tify in g th e o b je c t b e in g s c a n n e d . W e n e e d e d to id e n tify n o t ju s t th e fe a tu re s o n m a n y im a g e s , b u t th e p a th o f e a c h fe a tu re th ro u g h m u ltip le c o n s e c u tiv e fra m e s . W e in v e s tig a te d s o m e w e ll-k n o w n m e th o d s fo r im a g e d e te c tio n th a t w e c o u ld u s e to fa c ilita te s e g m e n ta tio n a n d o b je c t tra c k in g . F o r th e e d g e d e te c tio n m e th o d , w e lo o k e d a t th re e a lg o rith m s : M a rr-H ild re th [8 ], th e In fin ite S y m m e tric E x p o n e n tia l F ilte r (IS E F ) [9 ], a n d m o rp h o lo g ic a l b o u n d a ry e x tra c tio n [1 0 ]. M a rr-H ild re th e d g e d e te c to rs u s e a s m o o th in g a lg o rith m , u s u a lly a c o n v o lu tio n w ith a G a u s s ia n fu n c tio n , fo llo w e d b y a z e ro -c ro s s in g b in a riz a tio n o f th e im a g e . W e c h o s e th is m e th o d b e c a u s e it is s im p le , fa s t, a n d e a s y to im p le m e n t in v a rie d h a rd w a re a n d s o ftw a re . T h e IS E F is a v e ry h ig h q u a lity e d g e d e te c to r o n im a g e s w ith h ig h fre q u e n c y u n ifo rm n o is e . T h e IS E F a ls o c o m p u te s z e ro c ro s s in g s to fin d e d g e s , b u t it p e rfo rm s a m u c h m o re c o m p le x s m o o th in g u s in g a b a n d -lim ite d L a p la c ia n o p e ra to r. T h e IS E F a lg o rith m w e im p le m e n te d [1 1 ] u s e d a n e d g e fo llo w in g te c h n iq u e th a t c o n s u m e d m e m o ry a n d p ro b a b ly S p e c ia l P u rp o s e Im a g e C o n v o lu tio n w ith E v o lv a b le H a rd w a re 3 c o u ld h a v e b e e n tim e -o p tim iz e d a s w e ll. W e fo u n d th is a lg o rith m to b e v e ry e ffe c tiv e , b u t m u c h s lo w e r th a n M a rr-H ild re th . S in c e th e la tte r w a s a d e q u a te fo r o u r ta s k , w e d id n o t u s e th e IS E F . M o rp h o lo g ic a l B o u n d a ry E x tra c tio n u s e s s o m e s im p le s e t o p e ra tio n s to o u tlin e e d g e s o n th e b o u n d a rie s o f o b je c ts in a n im a g e . In th e s im p le s t c a s e , th e e d g e e n h a n c e m e n t re s u lts fro m e ro d in g a c o p y o f th e im a g e u s in g a s im p le k e rn e l a n d th e n s u b tra c tin g th e re s u ltin g im a g e fro m th e o rig in a l im a g e . A n im a g e c o n ta in in g h ig h fre q u e n c y n o is e re q u ire s a n a d d itio n a l o p e n in g s te p to c le a n th e fra m e b e fo re th e b o u n d a ry e x tra c tio n w ill w o rk c o rre c tly . T h is a lg o rith m w a s n o t a s e ffe c tiv e fo r e x tra c tin g im a g e s fro m o u r im a g e s . T h e b o u n d a rie s o f d iffe re n t fe a tu re s in th e im a g e te n d to b le e d to g e th e r if th e e ro s io n k e rn e ls a re n o t o f th e p ro p e r s iz e fo r th e b o u n d a ry . If th e b o u n d a ry is d iffe re n t in d iffe re n t p a rts o f th e im a g e it m a y b e n e c e s s a ry to e ro d e d iffe re n t p a rts o f th e im a g e w ith d iffe re n t k e rn e ls to g e t a n a c c u ra te b o u n d a ry . W e a ls o in v e s tig a te d s o m e th re s h o ld in g m e th o d s , b u t fo u n d th e m to b e v e ry lim ite d . O n e m e th o d w e d id n o t e x p lo re w h ic h m ig h t p ro v e e ffe c tiv e is E d g e L e v e l T h re s h o ld in g , th o u g h th is m e th o d p ro v e s to b e s lo w re la tiv e to th e m e th o d s w e c h o s e to in v e s tig a te . W e d e c id e d to u s e th e M a rr-H ild re th m e th o d in itia lly , b e c a u s e it w a s b y fa r th e fa s te s t m e th o d g iv e n o u r e n v iro n m e n t. T h e IS E F te c h n iq u e p ro v e d m o re a c c u ra te in o b je c t s e g m e n ta tio n , b u t th e im p ro v e m e n t w a s n o t w a rra n te d g iv e n th a t th e p ro c e s s w a s s ig n ific a n tly s lo w e r th a n M a rr-H ilre th . T h e M o rp h o lo g ic a l te c h n iq u e w e in v e s tig a te d w o rk e d a d e q u a te ly o n ly w ith a g re a t d e a l o f “ h a n d -tw e a k in g ” th e p a ra m e te rs . T h is a c tu a lly m a d e th e m o rp h o lo g ic a l b o u n d a ry e x tra c tio n m e th o d a c a n d id a te fo r la te r e x p e rim e n ts w ith g e n e tic m e th o d s . F o r th e c u rre n t e x p e rim e n t, th o u g h , w e fo u n d it d iffic u lt to c o n tro l th e re s u lts w h e n w e trie d to a u to m a te th e a lg o rith m . C o n v o lu tio n C o n v o lu tio n o p e ra to rs a re c o m m o n ly u s e d in e d g e e n h a n c e m e n t te c h n iq u e s to in c re a s e c o n tra s t o r e m p h a s iz e fe a tu re s w ith p a rtic u la r s h a p e s o n d ig itiz e d im a g e s . " E d g e s " a re u s u a lly ra p id c h a n g e s in p ix e l v a lu e s . O n e w a y to th in k a b o u t a c o n v o lu tio n is a s a d iffe re n tia l o p e ra to r th a t m e a s u re s th e ra te o f c h a n g e in s o m e d ire c tio n a lo n g th e im a g e . C o n v o lu tio n o p e ra to rs fo r im a g e p ro c e s s in g a re u s u a lly e x p re s s e d a s o d d -v a lu e d s q u a re m a tric e s . T h e o p e ra tio n is a p p lie d a c ro s s th e p ix e l fie ld b y m u ltip ly in g in d iv id u a l p ix e l v a lu e s b y th e m a trix e le m e n ts a n d th e n a d d in g th e re s u lts to g e t a n e w p ix e l v a lu e . T h e n e w v a lu e re p la c e s th e v a lu e fo r th e c e n te r p ix e l in th e fie ld . T h e o p e ra tio n is a p p lie d re p e a te d ly a c ro s s th e p ic tu re u n til th e w h o le p ix e l fie ld h a s b e e n tra n s fo rm e d . S im p le c o n v o lu tio n s a re o fte n 3 × 3 o r 5 × 5 m a tric e s . O n e s im p le a n d s tra ig h tfo rw a rd c o n v o lu tio n is th e " d ire c tio n le s s " o r L a p la c ia n o p e ra to r. T h e L a p a c ia n o p e ra to r is d ire c tly = ∂ 2 + ∂ 2 a n a lo g o u s to th e L a p la c ia n o f d iffe re n tia l a n a ly s is : ∇ 2 L a p la c ia n c a n b e e x p re s s e d d is c re te ly a s a 3 × 3 m a trix − 1 0  0  − − 4  1  0 − 1 0 T h is is o n ly o n e c o m m o n e x a m p le o f a c o n v o lu tio n b y g e n e tic m e th o d s la te r in th is p a p e r. W h e n th e c o n v o lu tio n is a p p lie d to a 3 × 3 a re a o n v a lu e fo r th e c e n te r p ix e l in a tra n s fo rm e d im a g e . T h a s fo llo w s :   1   m a trix . W e w ill s h o w e x a m p le s c re a te d ∂ y 2 ∂ x 2 U s in g th e d e fin itio n o f a d iffe re n c e , f ′ ≈ ( f ( x ) − f ( x − h ) ) h , a n d le ttin g h = 1 , th e a p ix e l fie ld , th e o p e ra to r g iv e s a n e w e c o n v o lu tio n is th e n m o v e d a c ro s s a n d 4 J . D u m o u lin e t a l. d o w n th e p ix e l fie ld u n til a ll o f d iffe re n t te c h n iq u e s fo r o m a trix . In o u r e x p e rim e n ts , w T h e c o n v o lu tio n o p e ra tio n re s u lts . W e p re s e n t a little e o n e h a s a s e c tio n o f a g ra y p o rtio n o f a n im a g e . 2 1 0 1 5 1 2 0 2 8 1 0 F se c t th e p a d ig u r e io n s o c o n v o d e d . A 3 6 4 6 2 g r n in c ia n t a y g n h a n v e rte d . A t th e e d g e s , th e re a re a n u m b e r c a n n o t b e m a p p e d to th e c e n te r o f a 3 × 3 th e b o rd e rs o f th e p ix e l fie ld s . a n p ro d u c e c o m p le x a n d o fte n in s c ru ta b le a p p lie d to a s m a ll a rra y b e lo w . S u p p o s e a t in F ig u re 1 . T h is c o rre s p o n d s to s o m e 0 2 4 4 o f a e g in a p la lu tio 0 3 5 1 2 1 a . A s e c tio n f th e im a g e , b lu tio n w ith L ls o a n y c o n v o th e p ix e ls h a v e b e e n c o p e ra tin g o n p ix e ls th a t e ig n o re d th e v a lu e s a t , th o u g h q u ite s im p le , c x a m p le o f c o n v o lu tio n s c a le im a g e s u c h a s th 0 0 0 1 0 2 s c a le w ith o f im t su m 6 0 0 im th e a g e s to a g e h ig 1 a a n 0 . T h h lig . N e g a e h t o t tiv 0 1 9 1 7 0 0 0 0 c o e d e e 0 0 n v o se c th a t n u m 0 lu t tio th b e io n n . e e r is 0 9 0 0 0 o p e r a te s o n in c r e m e n ta l F ig u r e 1 b . T h e r e s u lt o f d g e s o f th e im a g e a r e 0 se t to z e r o . C o n v o lu tio n is d e fin e d a s th e s u m o f th e e le m e n t-b y -e le m e n t p ro d u c ts o f tw o m a tric e s . S o in th e c a s e o f th e L a p la c ia n o p e ra to r, a n d a n N × N m a trix , th e c o n v o lu tio n d e s c rib e s a n e w m a trix w h o s e e le m e n ts a re d e fin e d a s : A ∗ ∆ i, j A fte r m o v in g th is c o n v o lu tio n o p e ra to r a c c o n tin u e s w ith th e n e x t 3 × 3 s e c tio n o f th e m a s e c tio n o f th e m a trix a s s o c ia te d w ith lo c a l m a x T h e L a p la c ia n is o n ly o n e o f m a n y w id e ly u g iv e n a p a rtic u la r s h a p e to re c o g n iz e a n d a c o n v o lu tio n k e rn e ls th a t w ill b e m o s t e ffe c tiv e th e d e s ire d s h a p e o r s c a le . W h e n s h a p e s a re la e ffe c tiv e (o r e ffic ie n t) c o n v o lu tio n k e rn e l to re = ∑ 1 p = − 1 ro s s th e s trix . N o te im a th a t a se d c o n v o p a rtic u la r in e n h a n c rg e o r irre c o g n iz e th ∑ 1 q = − 1 a i+ p , j+ q ∆ p ,q h a d e d a re a o f th e im a g e a b h o w th e c o n v o lu tio n " e n h a n re n o t o n e d g e s. lu tio n o p e ra to rs (o r k e rn e ls ). s c a le , th e re is a p a rtic u la r in g th e im a g e to re c o g n iz e fe g u la r, it c a n b e d iffic u lt to d e e sh a p e . o v e , o n e c e s" o n e U s u a lly , c la s s o f a tu re s o f v e lo p a n E v o lu tio n a r y C o m p u tin g w ith X C 6 0 0 0 F P G A s W e p e rfo rm e d th e im a g e e x tra c tio n a n d o rig in a l im a g e p ro c e s s in g te s tin g u s in g a M a tro x G e n e s is im a g e g ra b b e r, w h ic h in c lu d e s a T M S 3 2 0 C 8 0 w ith p a ra lle l fix e d -p o in t M A C s a n d a 3 2 -b it d a ta b u s , a llo w in g u s to v e ry h a n d ily e v a lu a te th e d iffe re n t e d g e d e te c tio n m e th o d s d is c u s s e d a b o v e . A t th e tim e o f its p u rc h a s e , th e d e v ic e a p p ro a c h e d $ 8 ,0 0 0 U S , fa r fro m th e C O T S c rite ria th a t w e w e re lo o k in g fo r. F o r th is re a s o n , a n d to fa c ilita te th e e v o lu tio n a ry a s p e c t o f o u r p ro je c t, w e c h o s e th e V C C F a t H O T P C I c a rd fo r o u r d e v e lo p m e n t s y s te m . It w a s le s s th a n h a lf th e c o s t w h ile p ro v id in g th e n e c e s s a ry c o m p u tin g p o w e r. D e v e lo p in g p ro g ra m s fo r F P G A s u s in g e v o lu tio n a ry m e th o d s re q u ire s s o m e a tte n tio n to th e s p e c ia l c o n s tra in ts o f th e re c o n fig u ra b le h a rd w a re . F P G A -b a s e d d e s ig n s , e v o lu tio n a ry o r n o t, a re p h y s ic a lly c o n s tra in e d b y g e o m e try a n d fu n c tio n u n it c a p a b ilitie s . F u n c tio n u n its w ith in a n y F P G A p r o v id e b a s ic b in a r y lo g ic a l o p e r a tio n s ( e .g ., O R , A N D , X O R ) , b u t b e y o n d th is , th e re is a w id e v a ria tio n in th e p a rtic u la r fu n c tio n s th a t c a n b e im p le m e n te d w ith in a s in g le fu n c tio n u n it o n th e F P G A . S im ila rly , th e g e o m e try o f th e F P G A a s a w h o le h a s a n e ffe c t o n d e s ig n . T h e w a y in w h ic h in d iv id u a l fu n c tio n u n its a re c o n n e c te d to o th e r fu n c tio n u n its a n d to th e e d g e s o f th e d e v ic e a ls o c o n s tra in s th e ro u tin g o f c irc u itry in th e F P G A . W e h a v e c h o s e n th e X ilin x X C 6 0 0 0 s e rie s o f F P G A s fo r o u r re se a rc h fo r th e fo llo w in g re a s o n s : • T h e re is a s ig n ific a n t b o d y o f re s e a rc h o n e v o lu tio n a ry h a rd w a re d e s ig n b u ild in g a ro u n d th is c la s s o f F P G A d e v ic e s . • S p e c ia l P u rp o s e Im a g e C o n v o lu tio n w ith E v o lv a b le H a rd w a re 5 T h e c h ip d e s ig n is o p e n a n d c o n fig u ra tio n fo rm a ts a re a c c e s s ib le a t th e h a rd w a re le v e l. F u n c tio n u n it ro u tin g is e x tre m e ly fle x ib le . • T h e d e v ic e h a s a w e ll-d e fin e d P C I b u s in te rfa c e s ta n d a rd fro m th e m a n u fa c tu re r; a n d th is m a k e s th e d e v ic e u s e fu l fo r e x p e rim e n tin g w ith d e s k to p c o m p u te r in te rfa c e s in a c o n s is te n t m a n n e r w ith o th e r re s e a rc h e rs a n d m a n u fa c tu re rs . • T h e m a n u fa c tu re r h a s d e v e lo p e d A P I s e ts th a t m a k e d e v ic e p ro g ra m m in g e a s ily a c c e s s ib le to th e s o ftw a re d e v e lo p e r u s in g a d e s k to p P C . O th e r A P I s e ts a re a v a ila b le fo r d iffe re n t d e v e lo p m e n t e n v iro n m e n ts . T h e X C 6 0 0 0 g iv e s u s th e fle x ib ility to e x p e rim e n t w ith d iffe re n t e v o lu tio n a ry m e th o d s a n d to ra p id ly b u ild a n d u tiliz e n e w to o ls fo r o u r re s e a rc h . M a n y re s e a rc h e rs d e s c rib e th e X C 6 2 0 0 a rc h ite c tu re a s it a p p lie s to o u r e x p e rim e n t. In p a rtic u la r, s e e [3 ], [4 ] a n d [5 ]. M a n y p o s s ib le a p p ro a c h e s e x is t fo r e v o lu tio n a ry d e s ig n u s in g F P G A s . W e w ill d is c u s s re le v a n t a p p ro a c h e s to o n e fe a tu re re c o g n itio n s u b s y s te m th a t h a v e e m e rg e d fro m o u r p re s e n t re s e a rc h . W h a t d is tin g u is h e s d iffe re n t e v o lu tio n a ry d e s ig n a p p ro a c h e s is th e re p re s e n ta tio n a n d e v a lu a tio n o f tria l s o lu tio n s . • E v o lu tio n a r y H a r d w a r e D e s ig n R e p r e se n ta tio n T h e k e y to p ro b le m s o lv in g w ith g e n e tic a lg o rith m s is to fin d a s u th e p ro b le m . A g o o d re p re s e n ta tio n m u s t e n c o m p a s s th e fu ll ra n g e o f m u s t e n a b le th e g e n e tic a lg o rith m to g e n e ra te n e w tria l s o lu tio n s p ro b a b ility o f re ta in in g b e n e fic ia l fe a tu re s o f o ld tria l s o lu tio n s . M u c h a n g e tria l s o lu tio n s in w a y s th a t le a d to re la tiv e ly s m a ll fitn e s s c h a n o p e ra to rs s h o u ld te n d to a c c u m u la te u s e fu l fe a tu re s o f tw o o r m o re tr tria l s o lu tio n . W e in v e s tig a te d tw o d iffe re n t re p re s e n ta tio n s o f F P G m e th o d u s e s a c o d e d re p re s e n ta tio n o f p o s s ib le fu n c tio n s a n d a c p o s s ib le ro u tin g s . ita b le re p re s e n ta tio n o f p o s s ib le s o lu tio n s , a n d w ith a re la tiv e ly h ig h ta tio n o p e ra to rs s h o u ld g e s , a n d re c o m b in a tio n ia l s o lu tio n s in to a n e w A c o n fig u ra tio n s . O n e o d e d re p re s e n ta tio n o f F u n c tio n s 0 C o n s C o n s X A N X O R X X O IN V B U F 1 2 3 4 5 6 ta n ta n D Y R t 0 t 1 Y Y R o u tin g : 1 2 3 4 N o rth E a st W e st S o u th F ig u r e 2 : C o d e d R e p r e s e n ta tio n I m p le m e n te d a s a L o o k -U p T a b le . 6 J . D u m o u lin e t a l. C o d e d re p re s e n ta tio n s a re c o m m o n (s e e [4 ], [6 ]) in e v o lu tio n a ry p ro g ra m m in g , p a rtly b e c a u s e th e y a re re la tiv e ly e a s y to d e s c rib e . In o u r re p re s e n ta tio n , o u r c h ro m o s o m e is a 2 D a rra y o f in te g e rs th a t d e fin e d a c o n v o lu tio n k e rn e l n u m e ric a lly . T h e fitn e s s e v a lu a tio n s te p c o n v e rts th e k e rn e l in to a s e t o f b it v a lu e s a n d th e n c o n v e rts th e b it v a lu e s in to a c o n fig u ra tio n . T h e c o n fig u ra tio n p e rfo rm s th e c o n v o lu tio n b u t a llo w s u s to h a rd -c o d e (o r, m o re a c c u ra te ly , h a rd -w ire ) th e c o n v o lu tio n k e rn e l in to th e c irc u it d e s c rip tio n . T h is re d u c e s th e c irc u it s iz e a n d s p e e d s u p th e e v a lu a tio n o f th e c o n v o lu tio n . T h e p rim a ry a d v a n ta g e o f th is re p re s e n ta tio n is th a t it is a b s tra c te d fro m th e p a rtic u la r F P G A a rc h ite c tu re , s o th a t it c a n b e re w ritte n to c o n fig u re d iffe re n t d e v ic e s (a s lo n g a s o n ly c o m m o n fu n c tio n u n it/ro u tin g c o n fig u ra tio n s a re a llo w e d ). T h e p rim a ry d is a d v a n ta g e s o f th is re p re s e n ta tio n a re : • O n ly c o m m o n F P G A fu n c tio n c o n fig u ra tio n s a re a llo w e d , s o s p e c ia l fe a tu re s o f c e rta in F P G A s ( i.e ., s e p a r a b le c lo c k in g , r e g is te r p r o te c tio n in th e X C 6 0 0 0 ) w ill n o t b e a v a ila b le to th e e v o lu tio n a ry p ro c e s s . N o te th a t th is c a n b e a n a d v a n ta g e in s o m e c irc u m s ta n c e s , s u c h a s w h e n it is d e s ira b le to c o n s tra in th e F P G A to c lo c k e d c irc u its . • T h e a d d e d le v e l o f a b s tra c tio n (b e y o n d a b it-le v e l re p re s e n ta tio n ) a d d s a le v e l o f c o m p le x ity to th e te s tin g p ro c e s s . W e a ls o e v a lu a te d in trin s ic e v o lu tio n , in w h ic h w e d ire c tly u s e th e u n d e rly in g s tru c tu re o f th e X C 6 0 0 0 in th e fu n c tio n u n it e n c o d in g . (F o r e x a m p le s o f in trin s ic e v o lu tio n , s e e [3 ].) In th is re p re s e n ta tio n w e p a y p a rtic u la r a tte n tio n to th e a d d re s s /d a ta s tru c tu re o f th e F P G A c o n fig u ra tio n . P ro g ra m m in g th e F P G A a t th is le v e l is a n a lo g o u s to p ro g ra m m in g a s ta n d a rd C P U u s in g n a tiv e m a c h in e c o d e . T h e c o m p le x ity o f c re a tin g w o rk in g p ro g ra m s a t th is le v e l is a v o id e d in th e e v o lu tio n a ry p ro c e s s . F P G A c o n fig u ra tio n s e v o lv e to a p p ro a c h th e p ro b le m re p re s e n te d in th e fitn e s s s ta g e . B y te 0 B y te 1 B y te 2 O u tp u t R o u tin g F u n c tio n C o n fig / In p u t F u n c tio n C o n fig / In p u t F ig u r e 3 : F u n c tio n U n it C o n fig u r a tio n B y te s A c irc u it c o n fig u ra tio n is re p re s e n te d a s a s trin g o f b y te s th a t to g e th e r d e fin e th e c o n fig u ra tio n o f a fu n c tio n u n it o n th e X C 6 0 0 0 . T h e c o n fig u ra tio n o f a s in g le fu n c tio n u n it c o m p ris e s th re e b y te s th a t d e fin e th e lo c a l ro u tin g a n d th e lo g ic im p le m e n te d in th e fu n c tio n u n it. O u r c h ro m o s o m e is a tw o -d im e n s io n a l a rra y o f th e s e th re e -b y te u n its . E a c h e le m e n t o f th e a rra y re p re s e n ts a fu n c tio n u n it. T h e in d e x o f e a c h e le m e n t re p re s e n ts th e re la tiv e p o s itio n o f th e fu n c tio n u n it w ith in th e F P G A , th e re b y a llo w in g u s to c o n s tru c t a n a d d re s s fo r e a c h e le m e n t o f th e c o n fig u ra tio n . F o r d e ta ils a b o u t w h a t p a rtic u la r v a lu e s in d ic a te , s e e [1 ]. T h e p rim a ry a d v a n ta g e s o f th is re p re s e n ta tio n a re : • T h e re p re s e n ta tio n m a y b e d ire c tly in te rp re te d a s a " p ro g ra m " o f th e F P G A . V e ry little p re lim in a ry p ro c e s s in g is n e e d e d to c re a te a lo a d a b le c o n fig u ra tio n . • A ll p o s s ib le fu n c tio n s a re a llo w e d in p rin c ip le , b u t th e y c a n b e re s tric te d a t w ill b y m a s k in g th e a p p ro p ria te b its o f e a c h fu n c tio n u n it c o n fig u ra tio n . • • T h e p rim T h e re su u n d e rs ta T h e c o n m e th o d m a n u fa c a ry d is a d v a n ta g ltin g e v o lv e d c o n d . fig u ra tio n is a lw fo r o n e m a n u tu re r’s F P G A . e s o f a n in trin s ic re p re s e n ta tio n a re : n fig u ra tio n c a n b e v e ry d iffic u lt fo r a h u m a n F P G A p ro g ra m m e r to a y s s p e c ific to a p a rtic u la r F P G A . A p ro g ra m g e n e ra te d v ia th is fa c tu re r’s F P G A c a n n o t b e e x p e c te d to ru n o n a d iffe re n t S p e c ia l P u rp o s e Im a g e C o n v o lu tio n w ith E v o lv a b le H a rd w a re T h e first re p re a s o n s . U s in g d e v e lo p m e n t. e x p e rim e n ts to o th e r m e th o d s s im p le a n d a d e re s e n ta tio n d isc u s s e d a b o v e is s u p e rio r to th e firs t m e th o d , w e w ill n o t b e tie d to a n y It is , h o w e v e r, m u c h m o re tim e -c o n s u m d a te h a v e c o n c e n tra te d o n th e s e c o n d r fo r c re a tin g F P G A c o n fig u ra tio n s (S e e [6 q u a te fo r th e ta s k . th e s e c o n d fo r p a rtic u la r F P G in g to im p le m e p re s e n ta tio n . ] in p a rtic u la r) a n u m A fo r e n t th T h e re , b u t o b e r o th e p e fir a re , u r m f p u rp s t, o f e th 7 ra c tic a l o se s o f so o u r c o u rse , o d s a re F itn e s s E v a lu a tio n a n d C r o s s o v e r T h e e v a lu a tio n o f e a c h s o lu tio n c re a te d b y th e g e n e tic a lg o rith m is a n o th e r k e y a s p e c t o f e v o lu tio n a ry d e s ig n o f p ro g ra m s fo r o u r e x p e rim e n t. W e te s t e a c h tria l s o lu tio n c re a te d b y o u r g e n e tic a lg o rith m , a n d u s e th e te s t re s u lts to a s s ig n a fitn e s s v a lu e to th e tria l s o lu tio n . G e n e ra lly , th e o b je c tiv e o f th e d e s ig n p ro c e s s is to d e v e lo p a d e s ig n th a t m e e ts c e rta in p e rfo rm a n c e c rite ria . In e v o lu tio n a ry d e s ig n , h o w e v e r, it is im p o rta n t to u s e fitn e s s v a lu e s to g u id e e v o lu tio n e ffic ie n tly to w a rd s th e fin a l o b je c tiv e . T h e b a s ic flo w o f o u r F P G A e v o lu tio n a lg o rith m is : 1 . L o a d a f ile th a t r e p r e s e n ts a ll th e s ta tic c o n f ig u r a tio n p o r tio n s o f th e p r o g r a m ( e .g ., I /O R e g is te rs , C o u n te rs , a n d o th e r c o n tro l s e c tio n s ) 2 . C re a te a ra n d o m p o p u la tio n o f tria l s o lu tio n s . 3 . F o r e a c h tria l s o lu tio n , e v a lu a te a s fo llo w s : a . C o n v e rt th e tria l s o lu tio n to a lo a d a b le c o n fig u ra tio n . b . L o a d th e c o n fig u ra tio n . c . L o a d th e in p u t re g is te rs . d . W a it fo r s o m e fix e d a m o u n t o f tim e . e . R e a d th e o u tp u t re g is te r. f. C o m p a re th e re g is te r w ith a c a lc u la te d re s u lt b a s e d o n th e in p u t to g e t th e e rro r. g . A s s ig n th e fitn e s s o f th e tria l s o lu tio n b a s e d o n s te p f. 4 . S e le c t tria l s o lu tio n s a c c o rd in g to fitn e s s to g e n e ra te a n e w p o p u la tio n . 5 . G e n e ra te n e w tria l s o lu tio n s b y a p p ly in g g e n e tic o p e ra to rs (m u ta tio n a n d re c o m b in a tio n ) to th e s e le c te d tria l s o lu tio n s . 6 . A p p ly s te p s 3 th ro u g h 6 re p e a te d ly u n til a s a tis fa c to ry s o lu tio n e m e rg e s . In s te p g a b o v e , c h a n g e s c a n b e m a d e to e m p h a s iz e o r d e -e m p h a s iz e c e rta in a s p e c ts o f th e c a lc u la te d e rro r a t d iffe re n t s ta g e s in th e e v o lu tio n s o th a t, fo r e x a m p le , b a s ic fu n c tio n s a re e v o lv e d a n d o p tim iz e d b e fo re th e y a re c o m b in e d in to m o re c o m p le x s tru c tu re s . F itn e s s c a lc u la tio n c a n in c lu d e w e ig h tin g th e e rro r to g iv e m o re in flu e n c e to s o lu tio n s th a t m e e t c e rta in c o n s tra in ts o r s u b – c rite ria . T h is is e x tre m e ly im p o rta n t in th e c a s e o f o u r a p p lic a tio n s in c e w e w ill b e c o m p a rin g v e c to r v a lu e s re p re s e n tin g p a tte rn s o f fe a tu re s id e n tifie d b y th e F P G A to re fe re n c e v e c to rs c a lc u la te d b y o th e r m e a n s . In o rd e r to m a in ta in th e in te g rity o f w o rk in g p o rtio n s o f e a c h g e n o m e , w e d e v e lo p e d a c ro s s o v e r o p e ra to r th a t p re s e rv e s re c ta n g u la r p o rtio n s o f th e g e n o m e d u rin g c ro s s o v e r. B y d e fa u lt, th e 2 D in te g e r (3 D B in a ry ) g e n o m e in G a il w ill p e rfo rm b y te -le v e l c ro s s o v e r. T h is h a s th e e ffe c t o f d e s tro y in g lo c a l o p tim a l a re a s in th e g e n o m e fro m g e n e ra tio n to g e n e ra tio n . W e trie d to p re s e rv e lo c a l o p tim a l b e h a v io r in th e g e n o m e b y tre a tin g e a c h g e n o m e a s a 2 D a rra y a n d e a c h c ro s s o v e r c o m p o n e n t a s a 2 D s u b -a rra y o f th e g e n o m e . F o r e x a m p le , g iv e n tw o 9 × 9 g e n o m e s to c ro s s o v e r, th e c ro s s -o v e r a lg o rith m w o rk s a s fo llo w s : 1 . R a n d o m ly s e le c t a s ta rtin g ro w a n d s ta rtin g c o lu m n . T h is w ill b e th e u p p e r le ft c o rn e r o f th e c ro s s o v e r a rra y . 2 . R a n d o m ly s e le c t th e n u m b e r o f ro w s a n d c o lu m n s to c ro s s o v e r. T h is d e fin e s th e s iz e o f th e c ro sso v e r a rra y . 8 J . D u m o u lin e t a l. 3 . S e o f 4 . E x 5 . T h to L a le c t tw o g e n o m e s to c ro s s o v e r a n d re m o v e th e c a lc u la te d s iz e , fro m e a c h o f th e g e n o m c h a n g e th e se s u b -a rra y s b e tw e e n th e g e n o m e e v o lv e d c o n v o lu tio n s w e re c re a te d b y g e " re p ro d u c e " a n im a g e th a t w a s e n h a n c e d w p la c ia n ) in s p e c ia l p u rp o s e im a g e p ro c e s s in a su b -a rr e s. e s. n e tic a lly ith a g e n g h a rd w a a y fro m th e a p p ro p ria te p o s itio n a n d d e v e lo p in g a c o n v o lu tio n to a tte m p t e ra l-p u rp o s e c o n v o lu tio n (a p s e u d o re . W e e v a lu a te d fitn e s s a s fo llo w s : 1 . G e t 2 . C o n 3 . C o m (S e e re su 4 . C o m a g e n o m e . v o lv e th e o r p a re th e re F ig u re 4 b ltin g im a g e p a re fitn e s s ig in a l im a g e w ith th e g e n o m e . s u ltin g im a g e w ith th e im a g e c e lo w ) to g e t th e fitn e s s v a lu e . T w ith e a c h p ix e l in th e o rig in a l im e s p re s e rv in g th e lo w e st v a lu e (h o n v o lv e d w ith th e 9 × 9 p s e u d o -L a p la c ia n h e e v a lu a tio n c o m p a re s e a c h p ix e l in th e a g e . ig h e s t fitn e s s ) in d iv id u a ls . R e s u lts T h e e x p e rim e n ts G A L IB 2 .4 .2 a n d th 9 × 9 2 -d im e n s io n a l p ro b a b ility o f 2 0 % , th e G A p ro d u c e d v e A s w e se e in th e " te rrib le " re s u ltin g fitn e s s e v a lu a tio n . B re w a rd fitn e s s v a lu e s h o w n in F ig u re 6 . W e p re se n t so m p ro d u c e d b e lo w . T h F ig u re 4 . w e p e rfo rm e d w e re c a rrie d o u t u s in g a g e n e tic a lg o rith m d e fin e d u s in g e X IL IN X A P I lib ra ry fo r th e 6 2 0 0 p a rts . T h e G A w a s s e t u p to e v o lv e a in te g e r g e n o m e . W e ra n th e G A w ith p o p u la tio n s iz e o f 5 0 , c ro s s o v e r a n d m u ta tio n p ro b a b ility o f 5 % . W e n o tic e d th a t, u n d e r th e s e c o n d itio n s , ry s ta b le re s u lts a fte r a p p ro x im a te ly 1 0 0 g e n e ra tio n s . p ic tu re s b e lo w , lo w fitn e s s m e m b e rs o f th e p o p u la tio n a c tu a lly p ro d u c e d im a g e s . W e d e te rm in e d th a t th is w a s d u e to n u m e ric o v e rflo w d u rin g e c a u se o f th is th e fitn e s s e v a lu a tio n w a s re -e x a m in e d a n d w e d e c id e d to s c lo s e to h a lf th e m a x im u m v a lu e o f a lo n g in te g e r. O n e " g o o d " re s u lt is e e x a m p le s o f e v o lv e d c o n v o lu tio n s a n d th e e n h a n c e d im a g e s th e y e G A c re a te d a n d e v a lu a te d th e s e im a g e s u s in g th e tra in in g s e t im a g e in  − 1   0  0   0  − 1   0   0  0   − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 8 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0        − 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0       − 1  F ig u r e 4 : A n o r ig in a l d iffr a c tio n p a tte r n fr o m th e tr a in in g s e t (le ft), a p s e u d o L a p la c ia n o p e r a to r (r ig h t) a n d th e c o n v o lv e d im a g e u s e d fo r c o m p a r is o n (c e n te r ). E v o lv e d c o n v o lu tio n o p e ra to rs p ro d u c e s o m e w id e ly v a ry in g re s u lts . B e lo w a re a fe w e x a m p le s o f e v o lv e d o p e ra to rs a n d th e im a g e s th e y p ro d u c e d fro m th e o rig in a l im a g e in F ig u re 4 . S p e c ia l P u rp o s e Im a g e C o n v o lu tio n w ith E v o lv a b le H a rd w a re  1 1 0   − 6 6  4   − 2 3  − 3 9   − 3 2   1 1 7 − 1 1 5   − 2 0 − 5 7 8 3 − 7 1 − 5 7 − 3 1 1 2 7 4 3 3 1 0 7 2 2 9 1 − 6 2 − 4 1 3 5 − 5 1 5 0 − 3 9 − 1 9 − 1 6 9 − 1 5 1 1 9 6 1 2 0 0 − 6 4 1 1 0 6 0 6 6 − 6 3 3 4 0 0 0 5 7 3 8 5 − 1 1 9 8 0 E v o lv e d c o n v o lu tio n w ith fitn e s s = 1 2 6 4 4 . − 6 1 − 1 9 − 1 0 0 1 1 9 1 2 3 − 4 0 5 9 − 9 8 − 9 5 a n d − 3 6 1 1 1 − 4 8 − 6 9 − 9 4 − 6 1 1 8 4 6 − 3 0 9 2 − 1 6 − 6 9 − 1 1 7 6 9 − 9 9 8 2 − 1 2 3 − 6 e n h a n c e d − 8 0   − 1 3  − 3 0   1 6  4 9   2 7   4 4  − 3 6   7 1   1 2 7  8 6  2  − 4 7  − 5 4 4 5 im a g e E v o lv e d c o n v o lu tio n a n d e n h a n c e d w ith fitn e s s = 2 5 2 0 4 0 . 6 − 4 6 − 1 1 0 − 9 3 − 1 2 0 − 2 8 − 7 1 − 1 2 6 − 2 2 − 2 2 3 4 1 0 4 7 7 − 9 1 3 − 2 3  2 3   − 7 5  − 7 6 8 − 6 8 − 7 1 0 4 5 9  2 2 9 6 − 1 2 4 1 5 − 3 1 2 3    − 6 6 − 4 1 0 0 − 3 9 1 2 7 1 1 5 3 1 − 1 1 2 7 1 − 7 6 1 1 5 − 9 2 − 1 1 4 7 6 − 1 0 1 1 6 − 1 2 6 − 4 9 − 7 5 1 2 6 − 3 0 1 − 1 0 − 5 9 − 5 2 1 0 5 − 1 0 7 5 6 − 1 1 8 − 2 − 6 9 5 9 − 2 6 − 3 5 9 − 3 8   1 1 6  − 1 9   6 4  − 4 9   1 0 5   − 7 0  7 2   4 9  im a g e F ig u r e 5 : T w o H ig h F itn e s s R e s u lts In d is tr in tro T h is th e " h ig h -fitn ib u tio n o f p ix e d u c e d a h ig h -o s u g g e s ts th a t la e ss l v rd e rg e " c o n v o lu tio n a lu e s th ro u g h r ra n d o m n e ss r c o n v o lu tio n o p e th e a t th a rra y ra to rs, d a rk a e le v e s m a y w e re a s l o f p ro v c a n o f th e id e   110   70     109   − 66     − 92     − 8   − 113     48   105  se e th e o c o n v m o re − 47 − 93 98 − 12 105 − 107 95 126 − 67 w h a t a p p e a rs to b e a ra n d o m rig in a l fra m e . T h e s e o p e ra to rs o lu tio n a rra y s iz e (9 × 9 p ix e ls ). e ffe c tiv e m a tc h in g . − 32 − 71 − 120 58 − 119 − 112 71 116 80 37 − 109 101 103 − 68 − 38 113 − 15 82 − 31 − 15 84 58 − 85 37 70 15 99 35 − 122 127 101 − 52 98 116 − 84 44 − 128 103 − 55 − 52 − 57 30 − 86 − 61 88 4 − 86 58 − 57 − 72 − 43 33 − 50 5 0   88  − 106  − 54 85   − 37  − 81 39   − 96 F ig u r e 6 : E v o lv e d c o n v o lu tio n a n d e n h a n c e d im a g e w ith fitn e s s = 2 0 1 8 8 5 1 8 9 7 . A w a s F ig u e d g e lo w g e n e re 4 , d e te fitn e s s re s u ra te d in th y e t it p ro v c to r to fin d lt th a t p r e 1 0 0 th id e s v e ry fe a tu re s o v g e g in id e d n e ra o o d th e b e tte r m a tc h in g tio n . It m a tc h e s d e ta il e x tra c tio n s a m p le fra m e (s h a p p e a rs a lm o s t a n d w a o w n in in F ig u r n e g a tiv e s in fa c t F ig u re 4 e 6 a ly w u se d ) a n d b o v ith in o th e . T h is th e te m a M a rr e rs p ro o p e ra p la te -H ild r d u c e d to r in e th in 1 0 J . D u m o u lin e t a l. tim e -s e q u e n c e in th e s a m e m a n n e r a s o u r e x a m p le . N o te th a t th e fitn e s s is " c lo s e " to h a lf th e m a x im u m v a lu e o f a n u n s ig n e d 6 4 -b it in te g e r. C o n c lu s io n s W e h a v e a n u m b e r o f c o n c lu s io n s a n d p o s s ib le p a th s o f c o n tin u e d re s e a rc h fro m th e s e in itia l re s u lts . • I m p r o v e d F itn e s s E v a lu a tio n . W e n e e d a fitn e s s e v a lu a tio n m e th o d o lo g y m o re a p p ro p ria te fo r th e v e ry la rg e n u m b e r o f c a lc u la tio n s re q u ire d to c o m p a re th e s e re s u lts p ix e l-b y -p ix e l. O n e a p p ro a c h w o u ld a v e ra g e lo c a l re s u lts b e fo re d o in g th e s u m o f s q u a re s . • L a r g e r C o n v o lu tio n K e r n e ls . S in c e w e a re w o rk in g w ith im a g e s o f 6 4 0 × 4 8 0 p ix e ls , w e h a v e th e p o te n tia l to s e e b e tte r re s u lts w ith la rg e r c o n v o lu tio n k e rn e ls o r u s in g m u ltip le k e rn e ls a n d lin e a rly c o m b in in g th e re s u ltin g im a g e s . P la n s a re n o w u n d e r w a y to 1 0 0 × 1 0 0 p ix e l k e rn e ls u s in g a c lu s te r c o m p u te r. • T h e U tility o f th e F P G A v s . C lu s te r e d S y s te m s . T h e F P G A is v e ry u s e fu l fo r fitn e s s e v a lu a tio n a n d e x e c u tio n o f s m a ll k e rn e ls b e c a u s e th e c o n v o lu tio n m a th e m a tic s c a n b e d e s ig n e d in to th e c irc u it. T h e X C 6 2 0 0 in p a rtic u la r is v e ry u s e fu l w h e n w e c o n v o lv e a fra m e u s in g m u ltip le k e rn e ls . F o r la rg e c o n v o lu tio n k e rn e ls , h o w e v e r, m u ltip le F P G A s a re n e e d e d . T h is is b e c a u s e th e d a ta b u s o f th e F P G A b e c o m e s a lim ita tio n w h e n e v a lu a tin g la rg e s e ts o f n u m b e rs . W e w ill b e e x p lo rin g th e u s e o f m a trix m a th o p e ra tio n s b u ilt in to m o d e r n C O T S m ic r o p r o c e s s o r s ( e .g ., M M X , 3 D n o w ) to o p tim iz e c a lc u la tio n tim e f o r m a trix o p e ra tio n s . W e h o p e to p ro d u c e a p e rfo rm a n c e c o m p a ris o n e v a lu a tin g C O T S te c h n iq u e s fo r in te g e r m a trix c a lc u la tio n s in th e fu tu re . • A p p lic a b ility to M o tio n D e te c tio n . F o r s o m e v e ry ru d im e n ta ry ty p e s o f o b je c ts , o u r te c h n iq u e s c o u ld p ro v e u s e fu l fo r id e n tify in g " c u s to m " k e rn e ls th a t c o rre s p o n d to p a rtic u la r o b je c ts . F o r e x a m p le , a ro b o tic v e h ic le c o u ld u s e th is m e th o d to is o la te u n re c o g n iz e d o b je c ts a n d s a v e a lib ra ry o f d is c o v e re d te m p la te s fo r fu tu re re fe re n c e . W e h a v e n o c o n c lu s iv e re s u lts a lo n g th is lin e , o n ly th e s u g g e s tio n o f v a lu e . • O v e r a ll V ia b ility . T h e m e th o d o lo g y d e s c rib e d in th is p a p e r p ro v e d u s e fu l fo r o u r p a rtic u la r a p p lic a tio n . W e h o p e to e x te n d th e m e th o d o lo g y to p ro v id e u s e fu l g e n e ra tio n o f a u to m a tic c o n v o lu tio n k e rn e ls fo r o th e r g e n e ra l a p p lic a tio n s . A c k n o w le d g e m e n ts T h is w o rk w a s fu n d e d b y B M D O . J a m e s A . F o s te r w a s a ls o p a rtia lly fu n d e d b y D O D /O S T . B ib lio g r a p h y a n d R e fe r e n c e s P C C R G [1 ] X IL IN X X C 6 2 0 0 F ie ld P ro g ra m m a b le G a te A rra y s , A p ril 2 4 , 1 9 9 7 , X IL IN X , L td [2 ] R . M u rg a i, R . B ra y to n , A S a n g io v a n n i-V in c e n te lli; L o g ic S y n th e s is fo r F ie ld ro g ra m m a b le G a te A rra y s " 1 9 9 5 , K lu w e r A c a d e m ic P u b lis h e rs [3 ] A . T h o m p s o n , I. H a rv e y a n d P . H u s b a n d s ; U n c o n s tra in e d E v o lu tio n a n d H a rd o n s e q u e n c e s C S R P 3 9 7 , (in T o w a rd s E v o lv a b le H a rd w a re , S p rin g e r-V e rla g L e c tu re N o te s in o m p u te r S c ie n c e , 1 9 9 6 ) [4 ] J . K o z a , S . B a d e , F . B e n n e tt III, M . K e a n e , J . H u tc h in g s , D . A n d re ; R a p id ly e c o n fig u ra b le F ie ld -P ro g ra m m a b le G a te A rra y s fo r A c c e le ra tin g F itn e s s E v a lu a tio n in e n e tic P ro g ra m m in g , P u b lis h e d in K o z a , J o h n R . (e d ito r). L a te B re a k in g P a p e rs a t th e S p e c ia l P u rp o s e Im a g e C o n v o lu tio n w ith E v o lv a b le H a rd w a re 1 1 G e n e tic P ro g ra m m in g 1 9 9 7 C o n fe re n c e , S ta n fo rd U n iv e rs ity , J u ly 1 3 -1 6 , 1 9 9 7 . S ta n fo rd , C A : S ta n fo rd U n iv e rsity B o o k s to re . P a g e s 1 2 1 - 1 3 1 . [5 ] D . M o n ta n a , R . P o p p , S u ra j Iy e r, a n d G . V id a v e r; E v o lv a w a re : G e n e tic P ro g ra m m in g fo r O p tim a l D e s ig n o f H a rd w a re -B a s e d A lg o rith m s , 1 9 9 8 , B B N T e c h n o lo g ie s , P ro c . In t. C o n f. o n G e n e tic P ro g ra m m in g . [6 ] J . M ille r, P . T h o m s o n ; E v o lv in g D ig ita l E le c tro n ic C irc u its fo r R e a l-V a lu e d F u n c tio n G e n e ra tio n u s in g a G e n e tic A lg o rith m , 1 9 9 8 , N a p ie r U n iv e rs ity , P ro c . In t. C o n f. o n G e n e tic P ro g ra m m in g . [7 ] L . P a g ie , P . H o g e w e g ; E v o lu tio n a ry C o n se q u e n c e s o f C o e v o lv in g T a rg e ts, E v o lu tio n a ry C o m p u ta tio n 5 (4 ):4 0 1 -4 1 8 , 1 9 9 8 . [8 ] D . M a rr, E . H ild re th ; T h e o ry o f E d g e D e te c tio n , P ro c e e d in g s o f th e R o y a l S o c ie ty o f L o n d o n , S e rie s B , V o l. 2 0 7 , p p .1 8 7 -2 1 7 , 1 9 8 0 . [9 ] J . S h e n , S .C a s ta n ; A n O p tim a l L in e a r O p e ra to r fo r S te p E d g e D e te c tio n , C o m p u te r V is io n , G ra p h ic s , a n d Im a g e P ro c e s s in g : G ra p h ic a l M o d e ls a n d U n d e rs ta n d in g , V o l.5 4 , 2 :p p . 1 1 2 -1 3 3 , 1 9 9 2 . [1 0 ] J . S e rra ; Im a g e A n a ly s is a n d M a th e m a tic a l M o rp h o lo g y , A c a d e m ic P re s s , 1 9 8 8 . [1 1 ] J .R . P a rk e r; A lg o rith m s fo r Im a g e P ro c e s s in g a n d C o m p u te r V is io n , J o h n W ile y a n d S o n s, 1 9 9 7 . S te r e o s c o p ic V is io n fo r a H u m a n o id R o b o t U s in g P r o g r a m m in g C h r is to p h e r T .M . G r a a e , P e te r a n d M a ts N o rd a h l C o m p le x S y s te m s G ro u p , In s titu te o f P h y s ic a l T h e o ry , C h a lm e rs U n iv e rs ity o f T e c h n o lo g y , S -4 1 2 9 6 G ö te b o rg , S w e d e n In th is w e in tro d u c e a n e w a d a p tiv e s te re o s c o p ic V is io n . W e u s e g e n e tic p ro g ra m m in g , w h e re th e in p u t to th e in d iv id u a ls is ra w d a ta fro m s te re o im a g e -p a irs a c q u ire d b y tw o C C D th e in d iv id u a ls is th e d is p a rity m a p , w h ic h is tra n s fo r u s in g tria n g u la tio n . T h e u s e d g e n e tic in d iv id u a ls , a n d th e re b y h ig h P e rfo rm T h e e v o lv e d in d iv id u a ls h a v e a n 1 .5 w h ic h is e q u iv a le n t to a n u n c e rta in ty o f T h is w o rk is b y a p p lic a tio n s to th e c o n tro m a n o id ro b o ts T h e H u m a n o id a t C h a lm e rs . fro m th e 1 a n d T h e o u tp u t m e d to a 3 D m a p o f e n g in e e v o lv e s a n c e o n w e a k d is p a rity -e rro r o f 1 0 % o f th e tru e l o f a u to n o m o u s M o tiv a tio n M a n is th e S ta n d a rd fo r to o ls h u m a m a n s o f h u T T h e T h u m th a t w ith V m a n a ll in te ra c tio n s in o u r w o rld w h e re m o s t e n v iro n m e n ts , a n d m a c h in e s a re a d a p te d to th e a b ilitie s , m o tio n c a p a b ilitie s a n d g e o m e try o f n s . W a lk in g ro b o ts h a v e a v e ry la rg e p o te n tia l in e n v iro n m e n ts c re a te d fo r a s w e ll a s in m o re n a tu ra 1 T h e la rg e s t p o te n tia l is a s s o c ia te d w ith ro b o ts h u m a n o id ro b o ts . It c o u ld b e m o re m a n -lik e d im e n s io n s w a lk in g o n tw o le g s to c o n tro l v a rio u s m a c h in e s b y th e s e ro b o ts th a n to re b u ild a ll m a c h in e s fo r C o m p u te r c o n tro l h is w o rk is p a rt o f th e H u m a n o id a t C h a lm e rs U n iv e rs ity o f T e c h n o lo g y . a s e rie s o f h u m a n o id e x p e rim e n ts , a ll o f w h ic h w ill b e p rim a rily b y e v o lu tio n a ry a d a p tiv e m e th o d s . h e fin a l g o a l o f th e re s e a rc h is to b u ild a h u m a n -s iz e d ro b o t b a s e d o n a a n S k e le to n to e n s u re a u th e n tic ity . T h e o f th e is a s e c o n d -g e n e ra tio n p ro to ty p e o f a s m a ll h u m a n o id is b e in g d e v e lo p e d E L V IS a h e ig h t o f 6 0 c m . is io n is th e m o s t im p o rta n t o f o u r fiv e A s a n e x a m p le a o f th e h u is d e v o te d to b u t so m e d e v o te o v e r h a lf o f th e ir A c c o rd in g to a n e w S t. L o u is . e s tim a te b y n e u ro s c ie n tis t D a v id V a n E s s e n o f W a s h in g to n U n iv e rs ity in S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 2 − 2 1 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 S te re o s c o p ic V is io n fo r a H u m a n o id R o b o t 1 3 1 4 C .T .M . G r a a e , P . N o r d in , a n d M . N o r d a h l S te re o s c o p ic V is io n fo r a H u m a n o id R o b o t 1 5 1 6 C .T .M . G r a a e , P . N o r d in , a n d M . N o r d a h l S te re o s c o p ic V is io n fo r a H u m a n o id R o b o t 1 7 1 8 C .T .M . G r a a e , P . N o r d in , a n d M . N o r d a h l S te re o s c o p ic V is io n fo r a H u m a n o id R o b o t 1 9 2 0 C .T .M . G r a a e , P . N o r d in , a n d M . N o r d a h l S te re o s c o p ic V is io n fo r a H u m a n o id R o b o t 2 1 s r en , c s r g H , a nd Z Z Y o n gr pa rt t f lctrica l gi ri ga d lctr ics i rsit f i rp l , i rp l , 6 3 3 , .K. q.h.wu@liv.ac.uk pa rt t f lctr ic gi ri g i rsit, 5 6 , . . ia c. ispa p rpr s tsa l g tic clst ri ga l g rit c i i g a g tic a l g rit ( ) it t cl a ssica l a rd - a s clst ri g a l g rit ( ). Itpr c ss s pa rtiti a tric s ra t r t a s ts f c t r p its a d t s pr id s a i pl ta ti sc f r t g tic p ra t r -r c i a ti . r c pa ris fp rf r a c it t r isti g clst ri g a l g rit s, a gra -l li a g q a tia ti pr l is c sid r d. p ri ta lr s l ts s t a tt pr p s d al g rit c rg s r q ickl t t gl a l pti a d t s pr id s a tt r a t ft dil a i ic t tra diti a lclst ri g al g rit s a r a sil tra pp d i lca l pti a a d t g tic a ppr a c is ti c s i g. r c lsterin etho ds pl a a ita lro l e in e pl o ra to r da ta a na lsis. In the el d o fpa ttern reco nitio n [ ], the co n entio na l- ea ns clsterin a lo rith s ( s) ha e been idela ppl ied. ro a dlspea in , s ca n be cl a ssi ed into r s (H ) a nd s ( ) clsterin a lo rith s. H a rd clsterin dea l s ith a ssinin ea ch o b jectpo intto e a ctl o ne o fthe clsters, herea s f clsterin e tends this co nceptto a sso cia te ea ch o b jectpo intto ea cho fthe clsters itha ea s re o fbel o n in ness. he bel o n in nesso fo b ject po intsto the ha rd/ f clstersist pica l lrepresented b a e bership a tri ca l l ed a ha rd/ f pa rtitio n,respectiel.In enera l , s a i a t ndin the o pti a lpa rtitio n a nd o pti iin a clsterin o b jectie f nctio n sin ca l c lsba sed etho ds. H o e er, clsterin o b jectie f nctio ns a re hihl no n-l inea r a nd l ti- o da lf nctio ns. s a co nseq ence, b hil l -cl i bin , s ca n be ea sil tra pped into l o ca le tre a a sso cia ted ith a r pa rtitio n (i.e., a pa rtitio n itho ne o r o re e pt ro s, ea nin tha tfe er tha n clsters ere o bta ined in the na lpa rtitio n). o reo er,the a re a l so sini ca ntlsensitie to the initia lco nditio ns. a to a chie e bo tha o ida nce o fl o ca le tre a a nd ini a lsensitiit to initia l ia tio n is to se sto cha stic o pti ia tio n a ppro a ches, s ch a s r r s ( s). n is inspired b o r a nic e o ltio n a nd ha s been idel bel ie ed to be a n effectie l o ba lo pti ia tio n a lo rith . In [3] [6 ], enetica l l ided a ppro a ches ere de ned fo r the o pti a lclsterin pro bl e s a nd S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 2 − 3 3 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 A F a s te r G e n e tic C lu s te rin g A lg o rith m 2 3 e peri ents ere perfo r ed ith different da ta sets. es l ts sho ed tha t a n ca n a el io ra te the diffic l t o fcho o sin a n initia l ia tio n fo r the s a nd pro ides a ia bl e a to a o id l o ca le tre a . H o e er, a s sta ted in [3], a n clsterin a ppro a ch ta es p to t o o rders o f a nit de o re ti e tha n H / clsterin a lo rith s. s a re itera tie sche es o pera tin o n a po p l a tio n o fso -ca l l ed indiid al s. a ch indiid a lin the po p l a tio n co rrespo nds to a ia bl e so ltio n to the pro bl e in ha nd. In pre io s o r o n enetica l l ided clsterin ([4] [6 ]), ea ch indiid a lin the po p l a tio n ha s been desined to representa a tri o f clster centers. l tho h nder t o sets o fclsterin criteria the a ppin bet een a pa rtitio n a nd the co rrespo ndin seto fclster centers is e clsie, the o a lo fa t pica lclsterin a ppro a ch is essentia l l the o pti a lpa rtitio n ra ther tha n the po sitio ns o fclster centers. In this pa per, a no el enetica l l ided clsterin a lo rith is de ned, hich ses pa rtitio n a trices a s the indiid al s in po p l a tio n. his h brid a lo rith co bines a enetic a lo rith ( ) iththe cl a ssica lha rd - ea ns clsterin a lo rith (H ) a nd is therefo re ter ed a s a enetic ha rd - ea ns clsterin a lo rith ( H ). H ha s been a ppl ied to a ra -l e eli a e q a ntia tio n pro bl e . o pa red ith H a nd a ppro a ches, H co n er es fa ster a nd a la s to a n no nde enera te pa rtitio n hich is o r is nea rb the l o ba lo pti . r r o nsider a seto f ecto rs X = ... to be clstered into ro ps o fl ie da ta . a ch ℜ is a fea t re ecto r co nsistin o f rea l - a led ea s re ents describin the fea t res o fthe o b jectrepresented b . H a rd o r f clsters o fthe o b jectpo ints ca n be represented b a ha rd/ f e bership a tri ca l l ed a ha rd/ f pa rtitio n. he seto fa l l no nde enera te ha rd pa rtitio n a trices is deno ted b a nd de ned a s = = ℜ ; ; i () here is a ha rd pa rtitio n a tri. he n ber o fpo ssibl e ’ s, i.e. the n ber o f a s o fclsterin o b jects into no ne pt ro ps,is a tirl in n ber o fthe seco nd ind ien b (see [7 ]) ( H !) (− ) (2) . − he clsterin o b jectie f nctio n fo r ha rd - ea ns (H f nctio n ( )= · (v ) ) pa rtitio n is the (3) 2 4 L . M e n g , Q .H . W u , a n d Z .Z . Y o n g here = [v v ...v ] is a a tri o fpro to t pe pa ra eters (clster centers) v ℜ ; a nd (v ) is the cl idea n dista nce bet een the th fea t re ecto r a nd the ith clster pro to t pe v , hich is o f the fo r ( ) = ( − ) ( − ). his o b jectie f nctio n describes the a cc l a ted sq a red erro r hen repl a cin ea ch fea t re ecto r b the center o f clster to hich it bel o n s. H ence, itis a ct a l l a ea s re o fdisto rtio n. ini iin the clsterin o b jectie f nctio n ith respectto l ea ds to the fo l l o in sets o fclsterin criteria . o r ea ch clster i(i [ ]), rstl, (4) v = i.e.,the clstercenters a re po sitio ned a tthe center o f a ss o fthe fea t re ecto rs bel o n in to the clster. he seco nd set o fcriteria tha t ini ies the H f nctio n sta tes tha t a n o b jectie po int sho l d be a sso cia ted ith the cl o sest clster center: ifi= in [ (v )]; = () o ther ise. In eq a tio ns( ) ( ),i= 2 ... deno testhe inde a clster center a nd = 2 ... deno tes the inde n H a do ptsbo thsetso fthese clsterin criteria a nd f nctio n ( ) b al terna tiel pda tin a nd sin 3 c s r g n bero fa clstero r ber o fa n o b jectpo int. ini iesthe o b jectie eq a tio ns(4) a nd ( ). gr ta rtin ith a n initia lco nditio n, a e o les a po p l a tio n to a rds s ccessiel better re io ns o fthe sea rch spa ce b ea ns o fra ndo ied pro cesses o f r , ,a nd s . he ien o pti ia tio n pro bl e de nes a n en iro n enttha tdel iersa q a l it info r a tio n ( ss s) fo rne sea rch po ints, a nd the sel ectio n fa o rs tho se indiid a l s o fhiher q a l it to repro d ce o re o ften tha n o rse indiid a l s. he reco bina tio n echa nis s a l l o s fo r iin o fpa renta linfo r a tio n hil e pa ssin itto their o ffsprin ,a nd ta tio n intro d ces inno a tio n into the po p l a tio n. In brief, a sea rch fo r the o pti a lindiid a lis t pica l l i pl e ented a s fo l l o s. . enera te a n initia lpo p l a tio n; 2. a la te the tness a le o fea chindiid a lin the c rrentpo p l a tio n; 3. el ectpa irs o fpa rents; 4. enera te o ffsprin o fthe sel ected pa rents ia reco bina tio n a nd ta tio n; . epl a ce the pa rents ith their o ffsprin a nd crea te a ne enera tio n; 6. H al tthe pro cess ifa ter ina tio n criterio n is et. ther ise,pro ceed to step 3. o red ce the sea rchspa ce dra stica l l, e intro d ce H into the t pica l i pl e enta tio n pro cess. rin e er enera tio n, ith tnesse a la tio n A F a s te r G e n e tic C lu s te rin g A lg o rith m 2 5 ta in pl a ce,a sin l eH rea l l o ca tio n step is a ppl ied to ea chindiid a lin the po p l a tio n. cco rdin to the pa rtitio n a tri represented b a n indiid a l , the seto fclster centers a re l o ca ted sin eq a tio n (4) a nd then the pa rtitio n a tri is pda ted b eq a tio n ( ),i.e.b a ssinin ea cho b jectpo intto itscl o sest clster. fter a l lthe indiid a lpa rtitio ns a re pda ted sin H ,the three ba sic enetic o pera tio ns - sel ectio n, reco bina tio n a nd ta tio n - sta rt. In o rder to a ppla enetic a ppro a chto a ien pro bl e ,a n ber o ff nda enta liss es stbe a ddressed in a d a nce. he a re so ltio n representa tio n, tness f nctio n, crea tio n o fthe initia lpo p l a tio n a nd a s cceedin ne enera tio n, i pl e enta tio n sche es o f enetic o pera to rs, ter ina tio n criterio n, a nd the pa ra etersettin s. he resto fthis sectio n describes ea cho fthese iss es in deta il . . t pr s t t s entio ned in sectio n , ea chindiid a lin the po p l a tio n is a ha rd pa rtitio n a tri . In ha rd clsterin , a n o b jectpo intbel o n s to the cl o sest clstere clsiel. here iso nla do n a n co l n o fa ha rd pa rtitio n a tri. H ereo f, it is po ssibl e to si pif a ha rd pa rtitio n a tri into a n di ensio na l ecto r u ith the ith el e entdescribes hich ro the l ies do n the ithco l n o fthe o riina l . he po ssibl e a les o fthe el e ents o fu ra n e fro to . his si pl i ca tio n is a do pted in the pro po sed H . .2 t ss ct o al l o co pa riso n o fperfo r a nce ith the cl a ssica lH a s el la s a n e istin enetic clsterin a ppro a ch[3]),H f nctio n ( ) is sed here a s the o b jectie f nctio n to be ini ied. nd the a tri is ca l c l a ted ithrespectto sin eq a tio n (4). o l a r elred ce the cha nce o f H beco in st c a ta de enera te pa rtitio n, e ha e ta en the n ber o fe pt clsters into co nsidera tio n. n a le o bta ined sin H f nctio n is sca l ed ith a pena l t fa cto r. ifferent fro the o ne in [3], o r o b jectie f nctio n is rede ned a s fo l l o s: ( )= ( ) ( + ) (6 ) here is the to ta ln ber o fclsters a nd deno tes the n ber o f e pt clsters a nd is e a la ted ia co ntin the a l l - ro s in . he o a l o fa clsterin a ppro a chisto ini ie the o b jectie f nctio n hil ea fa o rs tterindiid a l s. o co pro ise, e se the in erse o fa n indiid a l ’ s sca l ed H f nctio n a le a sits tness a le.In a dditio n,a l inea r tnesssca l in echa nis [ ] ha s been intro d ced to a inta in rea so na bl e sel ectio n press re. . t t o nsider tha t o o d cho ice fo r sta rtin co n ra tio ns sho l d be free o f o ert bia ses. o r the ha rd pa rtitio n ecto rs u in the initia lpo p l a tio n, ea ch el e ent 2 6 L . M e n g , Q .H . W u , a n d Z .Z . Y o n g p a re n t 0 1 : th e w o rs t m e m b e r 1 k o f c lu s te r i : : : l : : p a re n t _ 1 2 3 4 8 : i : i : 4 i’ : : 0 4 6 1 7 8 : j 2 : 1 2 5 : j : : : k : : : l 3 : : o ffs p rin g 0 1 : k : : : l : : 2 3 4 8 : i’ : i : i’ : : i. . ra p ica ld scripti ft r c s pa rtiti . t clst r fpa r t ’ _ 2 th e ra n d o m ly c h o s e n m e m b e r fro m c lu s te r j _ 1 m o v e d to c lu s te r i’ i a ti stra t g itis a ppl id t t is setto a ra ndo l enera ted n ber in the ra n e o f[ ]. do in so , e a ct a l l ra ndo l pa rtitio n the o b jectpo ints to initia lclsters. .4 tc p r t rs In e er enera tio n, a sel ect pa rents fro the c rrent po p l a tio n. heo retica l l, the pro ba bil it o fo ne indiid a lbein sel ected is pro po rtio na lto its tness a le rel a tie to the o thers’ tness a les. fter bein sel ected, pa rents a re a ted to ie birth to their o ffsprin . ffsprin a re enera ted ia the o pera tio ns o freco bina tio n a nd ta tio n. . ct . s to the sel ectio n o pera to r,the s s rs s sche e is a ppl ied [9 ]. a sed o n the theo retica la nd e pirica la na lsis, a er co nclded [9 ] tha tthis sche e is a n o pti a lseq entia lsa pl in sche e hich, fo r the rstti e, a ssin o ffsprin a cco rdin to the theo retica lspeci ca tio ns. . c t . bra nd-ne reco bina tio n stra te ha sbeen desined fo r the clsterin pro bl e s. Itrea l l o ca tes the o rst e ber in ea chclster o fa pa rent’ spa rtitio n a cco rdin to its a tin pa rtner’ spa rtitio n. he o rst e ber o fa certa in clsteristhe fa rthesto b jectpo intto the centero ftha tclster,a o n tho se bel o n in to it. a rtic l a rl, fo r the ith clster o fpa rent’ s pa rtitio n u , reco bina tio n is ca rried o ta s fo l l o s: A F a s te r G e n e tic C lu s te rin g A lg o rith m 2 7 . nd the o rst e ber o fthe ith clster o fu (s ppo se it’ s ); 2. a cco rdin to the a tin pa rtner’ s pa rtitio n ecto r u , nd o t to hich clster this o rst e ber bel o n s (s ppo se it’ s the th clster o fu ); 3. stil la cco rdin to u , ra ndo l cho o se a no ther o b ject po int fro the th clster o fu (s ppo se itha ppens to be ); 4. ba c to u , chec o t to hich clster o f u the cho sen o b ject po int bel o n s a nd si pl rea l l o ca te to tha tclster. hil e stil l ho l din the ra ndo nesspro pert,thisreco bina tio n is el l ided. In step 2, the a tin pa rtner’ s pa rtitio n u is referred to a ns er the fo l l o in q estio n: hicho b jectpo ints sho l d sha re the sa e clster iththe o rst e ber o fa certa in clster o fpa rtitio n u ? ince there is a tl ea sto ne (itsel f) hil e a be o re tha n o ne o b jectpo ints a a il a bl e, step 3 ra ndo l cho o se o ne o f the . ina l l,in step 4,the o rst e ber o fpa rtitio n u ’ s ithclster is rea l l oca ted to a ne clster s chtha titbel o n s to the sa e clster a s this ra ndo l cho sen o b jectpo int. o r better ndersta ndin , these i pl e enta tio n steps a re described ra phica l lin i re . he reco bina tio n pro cessisa ppl ied to ea chpa rento fthe a tin pa irclster b clster. ha tis to sa , ea ch clster is a dj sted independentl o fthe o thers. c. t t . fter e er reco bina tio n, ta tio n is i po sed o n ea ch el eento fthe ne lco nstr cted pa rtitio n ith a ta tio n pro ba bil it p . ta tio n sets the cho sen el e ents to a ra ndo l enera ted inte er ra n in fro to . . r t f rt r enetic clsterin a lo rith is a stea d -sta te , hich repl a ces o nl a fra ctio n o f the po p l a tio n ea ch enera tio n. he o tia tio n o f intro d cin a stea d -sta te is to eep a o o d ba l a nce bet een e pl o ita tio n o f the best re io ns fo nd so fa r a nd co ntin ed e pl o ra tio n fo r po tentia l lbetter pa o ffa rea . H o e er,a stea d -sta te il l increa se the a ria nce a l o n the ro thc r es o findiid a l s [ ]. o red ce the a ria nce,a irst-In- irst- t( I ) del etio n [ ] is e pl o ed a nd th s per itthe se o fthis stea d -sta te ith s a l l er po p l a tio ns. ith I del etio n the po p l a tio n is si pl a rst-in- rst-o t q e e ith ne indiid a l s a dded to o ne end a nd del eted indiid a l s re o ed fro the o ther end. o reo er,el itis is i pl e ented to eep the best e ber o fthe po p l a tio n. .6 r tr tt gs he enetic clsterin a ppro a chis co p ta tio na le pensie. he do ina ntco st is tha to fca l c l a tin the tness a le fo r ea ch indiid a lin the po p l a tio n fo r e er enera tio n. his co st is a f nctio n o f , , a nd , i.e., the n ber o f fea t res, n ber o fo b jectpo ints, a nd n ber o fcl a sses to clster the o b ject 2 8 L . M e n g , Q .H . W u , a n d Z .Z . Y o n g po ints. n increa se in a n o fthese pa ra etersres l ts in a l inea r increa se in ti e per enera tio n. ince the to ta l co p ta tio na l ti e isa l so pro po rtio na l to the po p l a tio n sie, s al lpo p l a tio ns a re preferred. hro ho tthe e peri ents ith H , e sed a po p l a tio n sie o f3 . In e er enera tio n,6 % indiid a l s o fthe po p l a tio n il lnder o the three ba sic enetic o pera tio nsto enera te o ffsprin . he pro ba bil itieso fs cceedin reco bina tio n a nd ta tio n a re .9 a nd . ,respectiel[ ]. hese pa ra eter settin s ha e been fo nd to o ffer bestres l ts. he ter ina tio n criterio n sed here is the n ber o f enera tio ns. he a ppro a chsto ps hen the req ired enera tio n is crea ted. p r s s clsterin is o ften a ppl ied to i a e pro cessin a nd i a es a re rea l - o rl d do a ins o f sini ca nt co pl e it in ter s o f n ber o f o b ject po ints to be clstered a nd n ber o f cl a sses, a ra l e eli a e q a ntia tio n pro bl e is co nsidered. In this a ppl ica tio n ca se, the 2 6 2 6 bl a c -a nd- hite ena i a e is rstl diided e enl into s a l lbl o c s o f4 4piel s. hen the ra l e el s o fthe piel s in ea ch o fthese bl o c co po se a ecto r s ch tha tthere a re 49 6 i a e ecto rs o f 6 fea t res ( ra l e el s). he o a lo fthis i a e q a ntia tio n pro bl e is to clster these 49 6 i a e ecto rs into 2 6 cl a sses. H ence, = 49 6 , = 2 6 ,a nd = 6 . o this i a e q a ntia tio n pro bl e , the cl a ssica lH , a present enetica l l ided clsterin a lo rith [3],a nd o r o n H ha e been a ppl ied independentl. nd the o b jectpo intspro cessed b a l l o fthese a lo rith s ere the 49 6 i a e ecto rs co nsistin o f 6 pielra l e el s. he cl a ssica lH a s i pl e ented a s fo l l o s: . a ndo l pa rtitio n the i a e ecto rs into initia lclsters. a l c l a te the co rrespo ndin seto fclster centers sin eq a tio n (4). 2. q a tio n ( ) is a ppl ied s ch tha ta cco rdin to its 6 pielra l e el s, ea ch i a e ecto r is a ssined to the cl o sestclster center. ea n hil e, the cl idea n dista nce bet een ea ch i a e ecto r a nd its cl o sestclster center is s ed to the a cc l a ted sq a red erro r . 3. q a tio n (4) is a ppl ied. a chclster center is pda ted a s the center o f a ss o fa l lthe i a e ecto rs bel o n in to it. teps 2 a nd 3 a re a ppl ied a l terna tiel ntilthe rel a tie difference in the a cc l a ted sq a red erro r o ft o s ccessie itera tio ns is l ess tha n = . − (i.e. a bs( ) ). t ep 2 i s repea t ed sin the l a stset rr rr o fclster centers. he res l tin a cc l a ted sq a red erro r a nd n ber o f e pt clsters a re reco rded. H al l . desined the a ppro a chfo l l o in the t pica li pl e enta tio n steps o fa a nd the H as al so intro d ced in ea ch enera tio n. he three enetic o pera to rs ere to rna entsel ectio n,t o -po intreco bina tio n a nd A F a s te r G e n e tic C lu s te rin g A lg o rith m 2 9 ra ndo ta tio n. In , indiid a l s o fthe po p l a tio ns ere sets o fclster centers instea d o fpa rtitio n a trices. o r co pa riso n, the a ppro a ch ha s been repea ted. H o e er, there a re so e differences bet een o r e peri ents a nd theirs a nd these a re hihl ihted a s fo l l o s: . hil e the ha d sed -fo l d to rna entsel ectio n ith = 2, e ha e sed sto cha stic niersa lsa pl in sche e, a s in o r o n H H . 2. hil e the ha d sed a enera tio na l ith a n el itiststra te o fpa ssin the t o ttestindiid a l s to the ne t enera tio n, e ha e sed a stea d sta te ith enera tio n a p o f .6 thro ho t. l itis is i pl e ented to eep the ttestindiid a l . 3. hil e the ha d sed a bina r ra co de representa tio n fo r the indiid a l s, e ha e sed rea la le representa tio n. ie tho se sed thro ho tthe e peri ents fo r o r enetic clsterin a lo rith , e ha e cho sen the sa e pa ra eter settin s in a l lthe repea ted a ppro a ches. he ere a po p l a tio n sie o f3 , a reco bina tio n pro ba bil it o f .9 , a nd a ta tio n pro ba bil it o f . . cepttha tbetter so ltio n ca n be o bta ined ith l a r er po p l a tio n sie, these a le o ffer the best perfo ra nce [3]. er a ppro a chsto pped a tthe 4th enera tio n. s s ests o n clsterin i a e ena b the cl a ssica lH , the a ppro a ch, a nd o r H a re nderta en respectiel. he e ea s resfo rco pa rin a lo rith perfo r a ncesa re the ea n sq a red erro r( ) a nd n bero fe pt clsters, , a sso cia ted ith a pa rtitio n a tri. he ea n sq a red erro r is the a le o fthe H f nctio n a era ed b the n ber o fi a e piel s( = (2 6 2 6 )), hichindica tes the disto rtio n bet een the q a ntied i a e a nd the o riina li a e. rin ea ch r n, the pa rtitio n ith the l o esta cc l a ted sq a red erro r is tra ced a nd the a nd a les a sso cia ted ith the o pti a l pa rtitio n na l l fo nd a re reco rded. he reco rded a leso f a nd a re a era ed o erthe to ta l n bero fr ns, i.e. ,4 a nd 4 fo r H , a nd H respectiel. he a era e a les a nd the sta nda rd de ia tio ns o f a nd a re repo rted in a bl e . he a nd a les a sso cia ted iththe er bestpa rtitio n e er o bta ined b ea ch a lo rith a re a l so inclded. i re 2 sho sthe histo ra o fthe reco rded a nd a les fo rH , a nd H r ns, respectiel. he histo ra is a sta tistica lrepresenta tio n o fthe distrib tio n o fl o ca lo pti a lo bta ined b the a lo rith . ddin res l ts o f o re r ns did no tcha n e the distrib tio n f rther. ro a bl e a nd i re 2, the fo l l o in o bser a tio ns a re dra n: . U sin H , a bro a d distrib tio n o fl o ca lo pti a lis o bser ed. he sta nda rd de ia tio ns o f a nd a re ch hiher tha n tho se o fthe enetic 3 0 L . M e n g , Q .H . W u , a n d Z .Z . Y o n g . s l ts f r tria l s it t r a ra g st. d . f ft st a ra g st. d . f ft st clst ri g a l g rit s 3 .3 7 .5 35 7 7 .7 27 5 5.36 35 3. 2 .26 7 25. 7 3 7 6 . 357 7 4. 5 4. 3. 5 6 .5 7 . 4 6 5 a ppro a ches. his indica tes tha tcl a ssica lH is sensitie to the initia l distrib tio n o fclster centers a nd ea sil tra pped in l o ca lo pti a , hil e, o n the o ther ha nd, sho s the ia bil it o fa enetic a ppro a chto o erco e these pro bl e s. 2. o th enetic ided clsterin a lo rith s o tperfo r the cl a ssica lH in the sense tha tthe end p in pa rtitio nsa sso cia ted ith chl o erdisto rtio n a nd sini ca ntfe er e pt clsters. a rtitio ns o fsi il ar a les ere repea tedl fo nd b a nd H . his indica tes tha tthese res l ts a re indeed nea rlo pti a l . 3. In no ca se did H res l tin a de enera te pa rtitio n. n a era e, the best a le o bta ined b H is sl ihtl l o er tha n tha to bta ined b . o is fo r the er best e er fo nd. rin ea ch enetica l l ided clsterin a ppro a ch sin either or H , the a le a nd e pt clster n ber o f the best pa rtitio n fo nd p to a nd incldin ea ch s ccessie enera tio n a re reco rded a nd then a era ed o er the to ta ln ber o fr ns. i re 3 sho s the res l ta nta era e a les o f a nd ith respectto the enera tio n n ber fo r bo th a nd H . cco rdin to the t o c r es, e see tha t, fo r bo th enetic a ppro a ches, the initia lco n er ence ra tes a re er hih a nd a s the enera tio ns pro ress co n er ence ra te decrea ses ra pidl. H o e er,in the ea rl ier enera tio ns, H co n er es ch fa ster tha n a nd q ic l rea ch the desired reio n here the no nde enera te pa rtitio n a trices reside. H ca n nd a n a le l o ertha n the l o este erfo nd b H a fterthe th enera tio n, hil e do es ita fter enera tio n . s sta ted,fo r clsterin pro bl e s the tness e a la tio n a te er enera tio n is ti e co ns in . l tho h so o n ca tches p a nd fro enera tio n the difference in a les bet een t o enetic clsterin a lo rith s is o itta bl e, in the specia lca ses here the speed a s el la s perfo r a nce is req ired H a pro ide a ch fa ster a to nd a n a ccepta bl e so ltio n. rther o re, a s H a ss res l o ca lo pti a l it a nd d e to its hil l -cl i bin etho d co n er es ch fa ster tha n a n enetic a ppro a ch, instea d o f a itin fo r the enetic a ppro a ches rea ch a n e a cto pti a l so ltio n, e a sto p enetic sea rch a fter a necessa r n ber o f enera tio ns a nd se H to nd the co rrespo ndin l o ca lo pti . A F a s te r G e n e tic C lu s te rin g A lg o rith m 6 3 1 c s In this pa per a no elenetic clsterin a lo rith is pro po sed, hich co bines a enetic a lo rith ( ) ith the cl a ssica lha rd - ea ns clsterin a lo rith (H ). U nl ie o ther clsterin a lo rith s, H pro cesses pa rtitio n a trices ra ther tha n sets o fcenter po ints a nd th s a l l o s a ne i pl e enta tio n sche e fo r the enetic o pera to r -reco bina tio n. o r co pa riso n o fperfo r a nce ith o ther present clsterin a lo rith s, e peri ents o n a ra -l e eli a e q a ntia tio n pro bl e ha e been co nd cted. he res l ts sho tha t H co n er es ch q ic er to the l o ba lo pti a nd pro ides a ia bl e a to so le the dil e a here the cl a ssica lH is fo nd ea sil ca htin l o ca l o pti a a nd a enetic a ppro a chreq ires l a r e ti e co ns ptio n. fr c s . . 2. 3. 4. 5. 6. 7. . . . . . a d . . al, r g ii ri i s. ddis - sl , a di g, a ssa c s tts, 7 4. . c d rs, g tic l d- a i a g q a tia ti a l g rit ,” r g ii ., l . 7 , pp. 547 -556 , 6. . . al l , I. . rta d . . d k, lst ri g it a g tica l l g id d pti i d a ppr a c ,” r s. i r m i , l . 3, . 2, pp. 3- 2, . . c d rs, g tic - a s clst ri g a l g rit a ppl id t c lr i a g q a tia ti ,” r g i., l . 3 , . 6 , pp. 5 - 6 6 , 7. . Kl a , clst ri g it lti a r a l g rit s,” i r . h r grss, l . 2, pp. 3 2-323, 7. . . a a d . . rt, lst ri g it lti a r stra t gis,” r g i., l . 27 , . 2, pp. 32 -32 , 4. . ra it a d I. . t g , ds., f hmi i s, . . pa rt t f rc , a ti a l r a f ta da rds ppl id a t a tica l ris. 55, 6 4. . . l d rg, i grihms i rh imi i hi ri g. ddis - sl l is i g pa ,I c, . . . a k r, d ci g ia s a d i ffici c i t s lcti a l g rit ,” r . . f. i grihms, pp. 4-2 , 7. K. . ga d . a r a , ra ti ga ps r isit d,” i s f i grihms 2 , . itl ( d.), pp. -2 . a il , : rga Ka f a , 3. . . c ra d l p a d . K. l , a ic pa ra t r c di g f r g tic al g rit s,” hi r i g, l . , . , pp. -2 , 2. 3 2 L . M e n g , Q .H . W u , a n d Z .Z . Y o n g 7 no. of occurences among 100 runs no. of occurences among 100 runs 3 2.5 2 1.5 1 0.5 0 60 80 100 120 140 160 6 5 4 3 2 1 0 -50 0 50 100 no. of empty clusters 150 200 0 50 100 no. of empty clusters 150 200 0 50 100 no. of empty clusters 150 200 MSE (a ) 12 no. of occurences among 40 runs no. of occurences among 40 runs 3 2.5 2 1.5 1 0.5 0 60 80 100 120 140 10 8 6 4 2 0 -50 160 MSE 4 40 3.5 35 no. of occurences among 40 runs no. of occurences among 40 runs () 3 2.5 2 1.5 1 0.5 0 60 80 100 120 140 160 MSE 30 25 20 15 10 5 0 -50 (c) i. . istri ti ft a sq a r it t pti a l pa rtiti sfi a l lf d a d , r sp cti l. rr r a d pt clst r clst ri gt a i ag r a ss cia t d it , A F a s te r G e n e tic C lu s te rin g A lg o rith m empty cluster number of the best partition 600 MSE of the best partition 500 400 300 200 100 0 0 10 20 30 generation 40 50 40 30 20 10 0 0 50 3 3 10 20 30 generation 40 50 10 20 30 generation 40 50 (a ) empty cluster number of the best partition 600 MSE of the best partition 500 400 300 200 100 0 0 10 20 30 generation 40 50 25 20 15 10 5 0 0 () i. . r rg c pr p rtis f a d it r sp ctt t g ra ti Scene Interpretation using Semantic Nets and Evolutionary Computation? D. Prabhu1, B. P. Buckles2 , and F. E. Petry2 i2 Technologies, 1603 LBJ Freeway, Suite 780, Dallas TX 75234, USA and Department of Electrical Engineering & Computer Science, Tulane University, New Orleans, LA 70118, USA 1 2 Abstract. The tness function used in a GA must be measurable over the representation of the solution by means of a computable function. Often, the tness is an estimation of the nearness to an ideal solution or the distance from a default solution. In image scene interpretation, the solution takes the form of a set of labels corresponding to the components of an image and its tness is dicult to conceptualize in terms of distance from a default or nearness to an ideal. Here we describe a model in which a semantic net is used to capture the salient properties of an ideal labeling. Instantiating the nodes of the semantic net with the labels from a candidate solution a chromosome provides a basis for estimating a logical distance from a norm. This domain-independent model can be applied to a broad range of scene-based image analysis tasks. 1 Introduction We describe how a genetic algorithm GA can be employed to classifylabel objects in a scene for which no prior truth data exists. Relationships among the objects in a typical scene from the domain of discourse are encapsulated within a semantic net. The method was validated using a test suite of images captured by satellites. Mainly, these included the infrared band of North Atlantic scenes and two bands of AVHRR data depicting regions of the Western U.S. The objective of the North Atlantic image analysis was to identify currents such as the Gulf Stream and eddies. The objective of using AVHRR images was to detect and identify clouds by type. 2 Background Classication or labeling of segments is the focus of this paper. Labeling a segment of an image is a particularly di cult subtask because there must be an automatic method of assigning a gure of merit to a candidate solution. A very ? This work was supported in part by a grant from NASAGoddard Space Flight Center, NAG5-8570 and in part by DoD EPSCoR and the State of Louisiana under grant F49620-98-1-0351. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 4 − 4 3 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 S c e n e In te rp re ta tio n 3 5 general and computationally reasonable method for dening tness function for image labeling is described that is based upon developing a semantic net for a typical scene from the domain. The classes of the semantic net are dened by the labels to be assigned to the segments. The predicates are relationships that exist among objects bearing the corresponding labels. Given a candidate solution a labeling for an image, a measure is described that estimates the conformance of the labeling to the semantic net. 3 Methodology In this section, the formal approach and extensions to our prior ad hoc model using semantic nets. 1 is developed. A semantic network is a structure used to represent knowledge as a combination of nodes interconnected with arcs. 3.1 Description of Scene Properties with Semantic Nets Let CL be the set of possible classication labels or classes specic to the application. CL = fc1  c2  : : :  cm g 1 Let S be the set of segments components in the given image: S = fs1 s2  : : :  sn g 2 Each of these components is assumed to be completely characterized by T , the set of features specic to the application. Let, T = ft1  t2  : : :  th g 3 For any given image component, each of these features, ti , takes values from some corresponding domain Di . Assume that every statement about such application-specic knowledge that is useful for the classication task can be described by using at most kmax number of components drawn from the image. First, consider only those statements which involve exactly k for some 1  k  kmax  components of the image. Let Ck be the set of all k-tuples built from the indices of the elements of CL. i.e., Ck = f hj1  : : :  jk i j 1  j1  : : :  jk  m g 4 Every image component s 2 S can be an instance of only one of the m classes drawn from the set CL. Denote this by ISA, an instance function which maps each image component to the index of a class. ISA : S ,! f1 : : :  mg 5 Thus, given a k-tuple of image components segments, ISA can be used to generate a corresponding k-tuple hj1  : : :  jk i 2 Ck . Let Fk be a set of feature-value 3 6 D . P r a b h u , B .P . B u c k le s , a n d F .E . P e tr y comparator functions such that each function f 2 Fk maps a k-tuple of feature values drawn from the domain Di corresponding to some feature ti 2 T , to an absolute comparison value. Fk = f f j f : Dik ,! R g 6 Let Pk be the set of predicates such that each p 2 Pk is a fuzzy predicate mapping a given absolute comparison value to a fuzzy truth-value in the continuous range of 0 1 . Pk = f p j p : R ,! 0 1 g 7 Predicates based on fuzzy logic are more appropriate than binary predicates for this task in view of the heuristic knowledge employed. In addition, null 2 Pk , and nullx = 0 for all x. For every k-tuple of image components, every feature-value based relationship inherited from the corresponding k-tuple of classes hj1  : : :  jk i 2 Ck among its elements can be quantied by using some specic feature-value comparator function and a corresponding predicate. For a given k-tuple of classes, hj1  : : :  jk i 2 Ck , there may exist multiple relationships among the elements of the k-tuple of class-instances. Therefore, we dene a mapping, R1k such that every given tuple of classes denotes a specic subset of the available predicates, i.e., R1k : Ck ,! 2Pk 8 We also dene another functional mapping R2k such that every predicate is mapped to a feature comparator function. i.e., R2k : Pk ,! Fk 9 Thus, every relationship involving k components can be uniquely denoted by the tuple hhj1  : : :  jk i pi where, hj1  : : :  jk i 2 Ck and p 2 R1k hj1  : : :  jk i . Consider a k-tuple of image componentssegments that is instantiated by virtue of some ISA mapping to be a tuple of class-instances corresponding to hj1  : : :  jk i. Every relationship hhj1  : : :  jk i pi inherited by the image components can be quantied as follows: Suppose that f = R2k p and that f makes use of the feature ti 2 T using k-tuple of values from the domain Di . Now, this relationship can be quantied by the composition pf x1  : : :  xk where x1  : : :  xk 2 Di are the feature values of the image segments. The set of all such relationships constitutes a semantic net, SNk . In other words, SNk = f hhj1  : : :  jk i pi j hj1  : : :  jk i 2 Ck ^ p 2 R1k hj1  : : :  jk i g 10 This formal model for representing and utilizing the knowledge relating k elements can be summarized by FNk , a tness net. FNk = hCL S T ISA Ck  Pk  R1k  Fk  R2k  SNk  Wk  E i 11 where, Wk is a set of weights, one for each predicate p 2 Pk , and E is a tness function described in section 3.3. S c e n e In te rp re ta tio n 3 7 This model can now be generalized to represent all possible statements from the knowledge base, i.e., for all values of k, 1 k kmax . Such a complete model is given by: FN = hCL S T ISA C  P  R1 F  R2 SN  W E i 12 where, S S P = S1 k kmax Pk C = S 1 k kmax Ck R1 = S1 k kmax R1k F = S 1 k kmax Fk R2 = S1 k kmax R2k SN = 1 k kmax SNk W = 1 k kmax Wk To illustrate the notions discussed above, consider a simple domain such as a chair shown in Figure 1. Here we have, CL = fs b a lg, where the symbols de- s4 s7 s1 s2 s6 s8 s3 s5 Fig. 1. A chair and its segments note, and index, the classes Seat", Back", Arm", and Leg" respectively. Also, S = fs1 s2 s3 s4 s5 s6 s7 s8g from Figure 1. There are two measuresfeatures for each segment: angle of rotation from the horizontal, t1 , and surface area, t2 . While relationships of any degree except zero are permitted, for the sake of simplicity we assume only binary relationships for this example, i.e., we consider only the case of k = 2 and C = C2 . The set of feature comparator functions is given by F = F2 = ff1 f2 g. f1 takes the angle-measures of two segments as arguments and computes their relative orientation. f2 computes the dierence in the surface area of the given argument pair of segments. Further, let the set of predicates be P = P2 = fperpendicular-to parallel-to area-greater-than nullg. The predicates perpendicular-to and parallel-to return 1 if the segments are mutually perpendicular and parallel respectively. Otherwise, they return 0. Similarly, the predicate area-greater-than returns 1, if the dierence computed by f2 is positive and 0 otherwise. Obviously, fuzzy predicates can be used instead to assign values in the range 0 1. The mapping from class tuples to predicates, R1 = R12, is simplied in this case since no class pair has more than one relationship. This is graphically shown in the top-half of Figure 2. The functional mapping, R2 = R22 , can be constructed easily since predicates perpendicular-to 3 8 D . P r a b h u , B .P . B u c k le s , a n d F .E . P e tr y and parallel-to use the function f1 and the predicate area-greater-than uses the function f2 . In this simple example, the semantic net SN = SN2 closely correα > α > L e g e n d : p e r p e n d ic u la r -to : B a c k S 1 Fig. 2. a chair. S e a t S 2 S 3 L e g S 4 S 5 A rm S 6 α > S 7 : p a r a lle l-to a r e a -g r e a te r -th a n S 8 A sample semantic net and a candidate ISA mapping for a scene consisting of sponds to R12 and the Figure 2 shows the useful parts of the semantic net for a typical chair using the three predicates. Figure 2 also shows a particular instantiation, i.e., an ISA mapping, of the segments from the scene shown in Figure 1. This mapping of fhs1,si hs2,ai hs3,li hs4,bi hs5,li hs6,li hs7,ai hs8,lig results in the correct classication of the segments. Also, it is obvious that any other assignment of labels to the segments would result in a lower consistency evaluation. Formally, the semantic net shown in Figure 2 is given by SN = f hhb ai i hhb li nulli hhb si ?i hhb bi nulli hhs ai i hhS Li ?i hhs si nulli hhl ai ?i hhl li ==i hha ai nulli g where, the symbols ", ?", and ==" denote the predicates area-greater-than, perpendicular-to , and parallel-to respectively. 3.2 Representing Candidate Solutions in GAs A candidate solution for the classication task takes the form of a vector of indices hj1  : : :  j  : : :  j i, containing one element for each segment s 2 S . This vector of indices represents a possible ISA mapping for the segments in the set S . In other words, ISAs  = j where, 1  i  n and 1  j  m. For example, the vector hs a l b l l a li represents the labeling shown in Figure 2. Similarly, the vector hl a s b a l l li represents another candidate solution, albeit of inferior quality. i n i i i i S c e n e In te rp re ta tio n 3 9 3.3 Computing Fitness Using Semantic Net Description Fitness is a quantitative measure of the consistency of an ISA relationship that a candidate solution represents. The procedure described below may appear to be computationally expensive. However, we have found that, in practice, there are very few relationships involving more than two componentssegments and that a large number of high-order relationships are reduced to null predicates. Further, all the predicates can be precomputed for the segments in a given image and the repeated tness computations need only do the summation of the various predicate values using a table look-up. E= X X Xw max k k pf x1  : : :  x c p 13 k =1 hh k i i2 k h k i s SN where hc i abbreviates hj1  : : :  j i 2 C and hs i represents any k-tuple of image segments instantiated to hj1  : : :  j i via the ISA mapping. Also, f = R2 p , w 2 W is the weight corresponding to predicate p, and x1  : : :  x 2 D are the feature values of the image segments in hs i corresponding to some feature t 2 T , depending on the feature comparator function f . For domains in which knowledge is unevenly distributed in the semantic net, practice may dictate that normalization over the set of predicates for each class or class-tuple be performed. To illustrate tness, examine the best solution hs a l b l l a li taken from Figure 2. Since there are four and two instances of the classes l and a respectively, we need to sum twenty-three predicate values, ignoring the null predicate. Assuming unit weights for all predicates, on summation, the tness value for the ideal solution can be seen to be twenty three, since all the predicates have a value of 1 in this case. In contrast, consider the obviously sub-optimal solution hl a s b a l l li. Here, only ten out of the twenty three predicate values have a value of 1 resulting in a tness value of ten. k k k k k k k k i k i 4 Experiments and Analysis The goal in the oceanographic problem is to label the mesoscale features of the North Atlantic from satellite images using a known set of classes. The classes in this case are Gulf Stream North Wall"n , Gulf Stream South Wall"s , Warm Eddy"w , Cold Eddy"c , and Other"o , i.e., CL = fn s w c og. We use edge-segmented images of the region in the infrared band 10.3-11.3 m . An infrared satellite image and its companion segmented image are shown in Figures 3 and 4. For the image shown, we have, S = fs1 s2 : : : s35g. We use only two measures for each segment in the image  its position and length. They are computed by using coordinates of the centroid based on the mass of the segment and the two end-points. The set of predicates P and the set of functions F are informally shown in Table 1. The semantic net SN for the domain is shown in Table 2. The mappings R1 and R2 are implicit in these tables. We use unit weights for all predicates. 4 0 D . P r a b h u , B .P . B u c k le s , a n d F .E . P e tr y Fig. 3. Table 1. Original Infrared Image of the Gulf Stream Description of predicates and functions for oceanic labeling Predicate Is-North-Ofi,j Function Comments If AvgLati AvgLatj , Segment i is north of segment j = 1 Otherwise, = 0. Is-Neari,j exp,  X  X is distance between segments i and j . Is-Not-Neari,j 1 , Is-Neari j  Fuzzy complement of Is-Near. Is-North-Of-And min f Is-North-Ofi j  Segment i is north of segment j -Fifty-Km-Fromi,j exp,  jX , 50j g and is 50 km from it. Arcs-Of-Circle is estimated based on Arcs-Of-Circle-And min f Arcs-Of-Circlei j  intersection of cords from segments. -Less-Than-Hundred Less-Than-Hundred- Second predicate is computed as -Km-Distanti,j Km-Distanti j  g = 1, if X  100, and = exp,  jX , 100j, otherwise ij ij ij ij ij Also, it is noted that the predicates are computed a priori for all the segment pairs and stored in a lookup table. Candidate solutions are represented as vectors of labels. For the image shown in Figure 4, any label vector h 1 2 2f g for all 35 i such that c c :::c ci n s w c o S c e n e In te rp re ta tio n Table 2. A semantic net for oceanic 4 1 segment labeling Class tuple Predicate Name h i Is-Neari,j h i Arcs-Of-Circle-And -Less-Than-Hundred-Km-Distanti,j h i Is-North-Ofi,j h i Is-Neari,j h i Is-Neari,j h i Is-North-Of-And-Fifty-Km-Fromi,j h i Is-Neari,j h i Is-North-Ofi,j h i Is-Neari,j h i Is-Neari,j h i Arcs-Of-Circle-And -Less-Than-Hundred-Km-Distanti,j h i Is-Not-Neari,j h i Is-Not-Neari,j h i Is-Not-Neari,j h i Is-Not-Neari,j h i Is-Neari,j Other tuples null Legend: w = Warm Eddy c = Cold Eddy o = Other n = North Wall of Gulf Stream s = South Wall of Gulf Stream w w w w w n w n n n n s s s s c s c c c c c o w o n o s o c o o i, constitutes a feasible candidate solution. Such label vectors are encoded as bit strings suitable for GA search. Table 3 shows the parameters used for the GA runs. Each run with these settings was repeated 10 times, each starting with a dierent initial random population. The accuracy of the best solution generated by the GA in each run with respect to the fairly dicult image shown in Figures 3 and 4 is compared with that of manual labeling and is listed in Table 4. Figure 4 shows the best labeling obtained over all the runs. 5 Conclusions Here we describe a domain-independent framework for labeling image segments for scene interpretation. This approach is based on the abstract representation of a typical scene from the domain of discourse. The abstraction form, i.e., semantic network, permits encoding the descriptions of relationships of arbitrary degree among the instances of scene objects. A GA is used in searching the space of candidate solutions for the best labeling. Fitness of a candidate solution is 4 2 D . P r a b h u , B .P . B u c k le s , a n d F .E . P e tr y Table 3. Parameters of GA runs for oceanic labeling Description Population size Number of generations Selection operator Value 200 200 Proportional selection using stochastic remainder sampling with replacement Crossover operator Uniform crossover allele level Probability of crossover 0.600 Mutation operator Bit mutation Probability of mutation 0.005 Table 4. Accuracy of GA-generated oceanic Run 1 2 3 4 5 Accuracy 80 57 66 83 83 labeling Run Accuracy 6 63 7 71 8 77 9 71 10 69 estimated by evaluating the conformance of the solution to the relationships depicted in the semantic net. References 1. C. A. Ankenbrandt, B. P. Buckles, and F. E. Petry, Scene recognition using genetic algorithms with semantic nets", Pattern Recognition Letters, vol. 11, no. 4, pp. 285293, 1990. 2. B. P. Buckles and F. E. Petry, Eds., Genetic Algorithms, IEEE Computer Society Press, 1992. 3. B. Bhanu, S. Lee, and J. Ming, Self-optimizing image segmentation system using a genetic algorithm", in Proceedings of the Fourth International Conference on Genetic Algorithms, R.K. Belew and L.B. Booker, Eds., San Mateo, CA, 1991, pp. 362369, Morgan Kaufmann. 4. S. M. Bhandarkar and H. Zhang, Image segmentation using evolutionary computation", IEEE Trans. on Evolutionary Computation, vol. 3, no. 1, pp. 121, apr 1999. 5. R. Tonjes, S. Growe, J. Buckner, and C.-E. Liedtke, Knowledge-based interpretation of remote sensing images using semantic nets", Photogrammetric Engineering & Remote Sensing, vol. 65, no. 7, pp. 811821, jul 1999. 6. J. Bala, K. DeJong, and P. Pachowicz, Using genetic algorithms to improve the performance of classication rules produced by symbolic inductive methods", in S c e n e In te rp re ta tio n Fig. 4. 4 3 Best Labeling of the Gulf Stream found by the GA Proceedings of 6th International Symposium Methodologies for Intelligent Systems ISMIS'91, Z. W. Ras and M. Zemankova, Eds., Charlotte, NC, 16-19 Oct 1991, pp. 286 295, Springer-Verlag, Berlin, Germany. 7. S. Truve, Using a genetic algorithm to solve constraint satisfaction problems generated by an image interpreter", in Theory and Applications of Image Analysis. Selected Papers from the 7th Scandinavian Conference, Aalborg, Denmark, P. Johansen and S. Olsen, Eds. Aug, 13-16 1991, pp. 133 147, World Scientic. 8. A. Hill and C. J. Taylor, Model-based image interpretation using genetic algorithms", Image and Vision Computing, vol. 10, no. 5, pp. 295 300, Jun 1992. 9. D. B. Fogel, Evolutionary programming for voice feature analysis", in Proceedings of 23rd Asilomar Conference on Signals, Systems, and Computers, oct 1989, pp. 381 383. Evolutionary Wavelet Bases in Signal Spaces Adelino R. Ferreira da Silva Universidade Nova de Lisboa, Dept. de Eng. Electrotecnica, 2825 Monte de Caparica, Portugal afs@mail.fct.unl.pt Abstract. We introduce a test environment based on the optimization of signals approximated in function spaces in order to compare the performance of di erent evolutionary algorithms. An evolutionary algorithm to optimize signal representations by adaptively choosing a basis depending on the signal is presented. We show how evolutionary algorithms can be exploited to search larger waveform dictionaries for best basis selection than those considered in current standard approaches. 1 Introduction In order to facilitate an empirical comparison of the performance of dierent evolutionary algorithms a test environment must be provided. Traditionally, sets of test functions with specic topological properties, commonly known as tness landscapes, have been proposed by several authors to be used in performance benchmarking. In particular, the De Jong's test function set has been a standard for genetic algorithm benchmarks since 1975. In most cases, the optimization objective is formulated as a global function minimization problem. In this paper, we depart from this view by considering the optimization of functions approximated in function spaces. Series expansions of continuous-time signals go back at least to Fourier's original expansion of periodic functions. A basis is a set of linearly independent functions that can be used to produce all admissible functions f t . The idea of representing a signal as a sum of elementary basis functions, or equivalently to nd orthogonal bases for certain function spaces, is very powerful. However, classic approaches have limitations, in particular there are no "good" local Fourier series that have both time and frequency localization. An alternative is the construction of wavelet bases, which use scaling instead of modulation in order to obtain an orthonormal basis for L2 R . An entropy-based algorithm for best basis selection has been proposed in the literature 6. Under the specic conditions of its application, the standard best basis SBB algorithm nds the optimum basis decomposition according to a specied cost functional. We show that this algorithm can be used to benchmark evolutionary algorithms. A second objective of this paper, is to show how evolutionary algorithms can be exploited to search larger waveform dictionaries for best basis selection than those considered in current standard approaches. We extend the scope of S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 4 4 − 5 3 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 E v o lu tio n a ry W a v e le t B a s e s in S ig n a l S p a c e s 4 5 the SBB algorithm by searching larger waveform dictionaries in order to nd better adaptive signal representations. In Sect. 3, we present an evolutionary algorithm for best basis selection. Adapted waveform analysis uses libraries of orthonormal basis and an ecient functional to match a basis to a given signal or family of signals. Two often used libraries are wavelet-packets and localized trigonometric functions, since they support the expansion of the waveforms in orthonormal basis whose elements have good time-frequency localization properties. These libraries constitute huge collections of basis from which we can pick and choose the best matching basis. Flexible decompositions are important for representing time-frequency atoms whose time-frequency localizations vary widely. In this article, we propose the use of evolutionary algorithms 10 as the main searching tool for best basis selection. The proposed approach generates a population of solutions based on basis expansions of multi-lter, time-shifted, wavelet-packet libraries. An evolutionary algorithm operates on the population to evolve the best solution according to a given objective function. Libraries of bases represent the population from which we want to select the best-t individuals. This optimization approach permits more exibility in searching for best basis representations than traditional approaches. 2 Wavelet Spaces In this section, we brie y review the framework of wavelet basis selection. Let x 2 R be an input signal and let 0 0 = R represent the signal space and B0 0 = e1  : : :  e  be the standard basis of R . Wavelet packets split this original space into two mutually orthogonal subspaces smoothly and recursively, i.e., n n   n n  =  +1 2   +1 2 +1 1 for j = 0 1 : : :  J k = 0 : : :  2 , 1, and J  log2 n is the maximum i level of recursions specied by the user. Here, we have n = dim  = n=2 . The wavelet jk j  k j  k j  j j: j packet transforms, recursively split the frequency domain via the so-called conjugate quadrature lters. These splits naturally generate a set of subspaces with the binary tree structure. Let  = j k be an index to specify a node i.e., a subspace with its basis set of this binary tree. The index j species the depth of the binary tree this is an index of the width of frequency bands for wavelet packets. The index k species the location of the frequency bands for wavelet packets. Let =  be such a collection of subspaces and let B = fB g be the corresponding set of basis vectors where B = ! 1  : : :  ! j  is a set of basis vectors that spans  . Each basis vector in B is called a time-frequency atom, and the whole set B is referred to as a time-frequency dictionary or a dictionary of orthonormal bases. These dictionaries contain many orthonormal bases. If the depth of the tree is J , each dictionary contains more than 22 J , dierent bases. An important question is how to select from a large number of bases in the dictionary a basis which performs "best" for one's task. In order to measure the performance of each basis, we need a measure of eciency or tness of a basis for     n  1 4 6 A .R . F e r r e ir a d a S ilv a the task at hand. For this purpose, several so-called information cost functionals have been proposed. A commonly used information cost functional is entropy. The entropy of a vector = f  g is dened by, d H  = , d Xj d k k k k2 j dk d 2 log2 j k kk j2 2 d d : 2 This cost functional was used as the objective function to drive the evolutionary optimization approach outlined in Sect. 3. The goal of the optimization approach is to nd an optimal, or quasi-optimal in some sense basis representation for a given dataset. The above considerations are the fundamentals behind the SBB algorithm 6 . However, it is possible to extend the library of bases in which the best representations are searched for, by introducing additional degrees of freedom that adjust the time-localization of the basis functions 2 . One such extension is the shift-invariant wavelet packet transform. Actually, one well-known disadvantage of the discrete wavelet and wavelet packet transforms is the lack of shift invariance. The added dimension in the case of shift-invariant decompositions is a relative shift, between a given parent-node and its relative children nodes. Shifted versions of these transforms for a given input signal, represent new bases to be added to the library of bases, which may further improve our ability to nd the "best" adapted basis. These modications of the wavelet transform and wavelet packet decompositions lead to orthonormal best-basis representations which are shift-invariant and characterized by lower information cost functionals 5 . Wavelet packet trees may be extended as joint wavelet packet trees to prot from enlarged libraries of bases, thus increasing our chances of getting truly adapted waveform representations. However, enlarged search spaces entail combinatorial explosion problems. We rely on evolutionary optimization approaches to guide us on the search process. 3 Evolutionary Formulation The two major major steps in applying any heuristic search algorithm to a particular problem are the specication of the representation and the evaluation tness function. When dening an evolutionary algorithm one needs to choose its components, such as mutation and recombination that suit the representation, selection mechanisms for selecting parents and survivors, and an initial population. Each of these components have parameter values which determine whether the algorithm will nd an near-optimum solution and whether it will nd a solution eciently. 3.1 Representation In the work reported here, a variable length integer sequence is used as the basic genotype. The objective is to evolve wavelet decomposition trees through the E v o lu tio n a ry W a v e le t B a s e s in S ig n a l S p a c e s 4 7 evolution of genetic sequences. The technique used to initialize the population is based on generating an initial random integer sequence, according to the values of the allele sets specied for the individual genes. The initial genotype sequence which codies the wavelet tree matches the breadth-rst BF sequence required to generate a complete binary tree, up to a pre-specied maximum depth. We refer to these sequences as tree-mapped sequences. A well-built decomposition tree for wavelet analysis purposes, is generated by imposing appropriate constraints to the genotype sequence as specied in Sect. 3.2. The imposition of the constraints yields variable length code sequences after resizing. An alphabet A = f0 1 2g is used to codify the wavelet tree nodes according to their types as specied in Sect. 3.3, thus enabling us to map any tree structure into a code sequence. The mapping of a code sequence to a complete BF tree traversal yields an initial sequence with length = 2 , 1, for a tree of depth . The length is also the number of nodes in a complete binary tree of depth L. When coding a complete binary tree using a complete BF sequence the last level of terminal nodes is redundant. Therefore, we have used a codication based on resized complete BF sequences to code genetic sequences. The chromossomes are constructed as follows. The rst gene assumes integer values 0 2 F, where F is the set of possible lter types used in the implementation, as explained in Sect. 3.3. The remaining genes are used to codify the wavelet decomposition tree. L l L l g 3.2 Constraints There are several methods for generating trees which can be used to initialize the population. The full, grow and ramped half-and-half methods of tree generation were introduced in the eld of genetic programming 9. These methods are based on tree depth. The ramped half-and-half method is the most commonly used method of generating random parse trees because of its relative higher probability of generating subtrees of varying depth and size. However, these methods do not produce a uniform sampling of the search space. In this work, we use constrained genetic sequences for genome initialization. Two types of constraint operators are used to guarantee that valid tree-mapped genetic sequences are generated: 1 top-down operator, and 2 bottom-up operator. In addition, by applying these operators we look for a uniform sampling of the tree search space. In terms of binary tree data structures, the top-down constraint guarantees that if a node has null code = 0 then its two sons 0 and 1 must have null code 0 = 0 and 1 = 0. The bottom-up constraint guarantees that if at least one of the sons 0 and 1 of a node has non-null code, then the parent 0 must have non-null code 6= 0. These constraint operators are biased in opposite ways. Starting from a uniform random code sequence, the bottom-up constraint operator constructs valid genetic sequences which are biased towards complete, full-depth trees. By he same token, the top-down constraint operator constructs valid genetic sequence which is biased towards null, minimum depth trees. To get a more uniform sampling of the sequence space, for sequences of maximum = 2 , 1, we use the following initialization procedure: ti ci ci ti ti ci s ti ti ci L ti ti 0.0 0.1 0.2 0.3 0.4 0.5 A .R . F e r r e ir a d a S ilv a 0.000 0.002 0.004 4 8 0 50 100 150 200 250 0 InteriorNodes 2 4 6 8 Depth Fig. 1. Histogram for the distribution of interior nodes in generated sequences left panel histogram for the distribution of trees with specied depth right panel 1. initialize sequence to terminal code value = 0 2 A , 2. get a random value 2 1 , 3. initialize subsequence 1 = 1 2 with random interior code values, = f1 2g 2 A , 4. randomly select one of the two constraint operators, bottom-up or top-down, to apply to . s ci r s s ci r s  s By resizing pruning constrained code sequences we allow for genetic sequences of variable length, hence tree representations of variable depth. The left panel in Fig. 1 shows a histogram for the number of interior nodes generated by the initialization procedure, for a maximum specied depth = 8 of the equivalent tree, and 3000 stochastic genetic sequences. The right panel in Fig. 1 presents a histogram for the distribution of trees with specied depth, generated from the same stochastic samples with resizing. L 3.3 Specication The approach of organizing libraries of bases as a tree has been extended to construct a joint tree, to guide the process of generating shifted wavelet packet transforms. Libraries of bases represent the population from which we want to select the best-t individuals. In the current formulation, the genotype sequence G allows for three optimization parameters: best lter, best wavelet packet basis and best shifted basis. The genetic representation is used to create an initial population and evolve potential solutions to the optimization problem. The genotype is made up of the genes which guide the discrete wavelet decomposition of each waveform, in accordance with the joint tree representation. A cost functional is then applied to the wavelet coecients, and its value is used to derive the tness of the individual. In terms of entropy, the optimization problem amounts to evolve a minimum-entropy genotype. Therefore, the best individual is the one with minimum evolved entropy in a given library space. Since we were able to formulate the three subtasks, wavelet packet decomposition, shifted wavelet transform, and wavelet lter to be applied in a common data structure, the original multiple optimization problem can be solved in terms of a single aggregate E v o lu tio n a ry W a v e le t B a s e s in S ig n a l S p a c e s 4 9 functional. The rst gene 0 in G is responsible for the optimization of the lter used in the decomposition. We have used in the implementation 16 possible types of dierent lters, thus 0 = f0 15g. The lters considered in the implementation were the Haar lter, the Daubechies lters D4, D6 and D8, and several biorthogonal lters commonly used in image analysis as implemented in 7. In particular, the lter set included the 7 9 spline lter referred to in 1, the 7 9, 11 13, 10 6, 3 5, 6 2, and 3 9 lters dened in 11, the 7 9 "FBI-ngerprint" lter, and the 10-tap lter listed in 3. The analysis phase of the discrete shift wavelet packet transform is codied in the genetic sequence G . The collection of wavelet packets comprises a library of functions with a binary tree structure. To obtain the wavelet packet analysis of a function, or data set in the discrete case, we rst nd its coecient sequence in the root subspace, then follow the branches of the wavelet packet coecient tree to nd the expansion in the descendent subspaces. Assigning to each tree node a wavelet split value 2 f0 1 2g we may enumerate all possible binary tree structures. The value w = 1 references unshifted interior nodes, i.e., nodes with left and right children subtrees associated with unshifted decompositions. The value w = 2 references time shifted interior nodes. The value w = 0 references the leaves. g g ::: = = = = = = si  = =  s s 4 s Spaces of Test Signals A well-known implementation of the SBB algorithm which can be used for comparison purposes is contained in the WaveLab package 4. We reference by evolutionary best basis EBB, the evolutionary formulation presented in Sect. 3 for best basis selection using multilter, time shifted wavelet packet bases. By canonical best basis CBB, we mean an algorithm which is able to reproduce approximate the results of the SBB algorithm, using optimization methodologies for best basis selection dierent from those conceived for the SBB algorithm. In this sense, we may map the EBB algorithm into a CBB algorithm. The SBB algorithm is based on building an information cost tree in order to minimize some cost measure on the transformed coecients 6. The evolutionary algorithm proposed in Sect. 3 was applied to a set of test signals and the results compared with the results produced by SBB, based on the entropy minimization criterion. For evaluation purposes, we will use the test signals depicted in gure 2. Two of these signals are articial signals. The other two signals are built from collected data. The signal HypChirps includes two hyperbolic chirps. The signal MishMash includes a quadratic chirp, a linear chirp, and a sine, as used in the WaveLab package. The signal Seismic is distributed throughout the seismic industry as a test dataset. Finally, the signal Sunspots represent the monthly sunspot numbers. The following basic parameters have been used in the steady-state evolutionary algorithm 8, 12: 1 population size: 50 2 crossover probability: 0.95 3 mutation probability: 0.02 4 replacement percentage: 0.6. 0 200 400 600 800 -3 -2 -1 0 1 2 3 MishMash -1.5 -0.5 0.5 1.5 A .R . F e r r e ir a d a S ilv a HypChirps 5 0 1000 0 200 200 400 600 600 800 1000 800 0 50 100 150 200 sunspots 0.0 0.5 1.0 0 400 Index -1.0 seismic Index 1000 0 Index 500 1000 1500 2000 Index Fig. 2. Test signals 4.1 Standard Wavelet Spaces To reproduce the application conditions of the SBB algorithm, the EBB approach was restricted to handle unshifted wavelet packet decompositions and use a specic lter. Hence, the CBB algorithm is a restricted version of the EBB algorithm, which is used to reproduce the SBB results using a dierent methodology. Both the EBB and the CBB algorithms are evolutionary. The minimum entropy values associated with the best basis selected by the SBB WaveLab algorithm for these signals, using the Daubechies D8 lter and L = 9 decomposition levels, are reported in Table 1. The SBB entropy values are also depicted by the dashed lines in Fig. 3. The CBB algorithm was applied to the same set of test signals to evolve the best basis using the same entropy cost functional. Fig. 3 shows the evolution of the median minimum entropy median best value with the number of generations for each of the test signals. The entropy values represent the median over 30 runs of the CBB algorithm. The median entropy values after ngen = 80 generations are shown in Table 1 as well. The optimal entropy values and the rate of convergence can be used as benchmarks to compare the performance of dierent evolutionary algorithms, or simply to tune the value of the control parameters. Table 1. Comparative median minimum entropy values and reconstruction errors Median Minimum Entropy L=9 Reconst. Error SBB CBB EBB EBB D8 D8, ngen=80 ngen=80 ngen=80 HypChirps 3.7908 3.7908 2.7707 1.0 e-6 MishMash 4.4805 4.5162 3.5910 2.6 e-6 Seismic 2.6022 2.6025 1.9735 2.7 e-7 Sunspots 3.0059 3.0059 2.7749 3.9 e-8 Signal E v o lu tio n a ry W a v e le t B a s e s in S ig n a l S p a c e s 40 60 80 100 Median minimum entropy 3.0 40 60 80 100 120 0 20 100 120 40 60 80 100 120 Evolution of the median minimum entropy by the CBB algorithm 3.2 5.0 4.5 EBB WaveLab EBB WaveLab 3.5 4.0 3.6 Median minimum entropy 4.0 MishMash 2.8 40 60 80 100 120 60 80 60 80 100 120 120 3.00 EBB WaveLab 2.90 2.8 2.4 40 100 3.10 sunspots Generation Fig. 4. 40 seismic 2.0 20 20 Generation EBB WaveLab 0 0 Generation 2.80 20 3.2 0 Median minimum entropy Median minimum entropy 80 Generation HypChirps Median minimum entropy 60 CBB WaveLab Generation Fig. 3. 40 sunspots 2.8 Median minimum entropy 20 seismic 2.6 20 4.9 0 Generation CBB WaveLab 0 4.7 120 Generation 3.02 3.04 3.06 3.08 20 3.2 0 CBB WaveLab 4.5 CBB WaveLab 5.1 MishMash Median minimum entropy 3.80 3.85 3.90 3.95 4.00 Median minimum entropy HypChirps 5 1 0 20 40 60 80 100 120 Generation Evolution of the median minimum entropy by the EBB algorithm 5 2 A .R . F e r r e ir a d a S ilv a HypChirps MishMash seismic sunspots Fig. 5. Evolved trees by the EBB algorithm 4.2 Enlarged Wavelet Spaces To show the application of the EBB technique we have applied it to evolve basis for the optimization of three parameters: wavelet packet decomposition, shiftable decomposition and lter to use. Fig. 4 presents the evolution of the minimum entropy values generated by the EBB algorithm for the signals in the test set. The values are the median values over 30 runs of the EBB algorithm. The dashed horizontal line in Fig. 4 is the value of the best basis entropy generated by SBB. We notice that the EBB algorithm is able to greatly reduce the minimum entropy value used to assess best basis adaptability, compared to both the SBB and the CBB algorithms. Table 1 references the numeric median values for the EBB minimum entropy after ngen = 80 generations. The most selected lters among the best evolved lters for each test signal, were the following: Brislawn 10-tap lter for signals HypChirps and MishMash, Villasenor 3 9 lter for signal Seismic, and Villasenor 6 2 lter for signal Sunspots. Fig. 5 depicts typical best evolved trees for a sample run of the EBB algorithm. Darker lines represent shifted wavelet packet transforms. Thinner lines represent unshifted transforms. Another important evaluation factor is the reconstruction error. Given a signal f , we reconstruct an approximate signal f^ from the transformed coecients by applying the inverse shifted wavelet packet transform, and calculating the l2 error between these two signals k f , f^ k2 . Table 1, presents the numeric median E v o lu tio n a ry W a v e le t B a s e s in S ig n a l S p a c e s 5 3 values of the reconstruction errors for each of the test signals, using the EBB algorithm for 80 generations over 30 runs. 5 Conclusion The approximation of signals in functions spaces was used to introduce a test environment aimed at the comparative performance of dierent evolutionary algorithms. We have considered entropy as the optimization tness criterion to be used. However, other cost functions may prove useful to extend the range of the test environment. In particular, measures directed to the optimization of multiobjective criteria may be incorporated in the proposed framework. On the other hand, the proposed test environment may be easily extended to incorporate two dimensional signal spaces. In terms of signal processing, well-adapted signal expansions are important, for instance, in signal compression. For orthonormal basis and additive cost measures, the standard algorithm for best basis selection is e cient. However, with the introduction of overcomplete waveform dictionaries the algorithm has increasing di culty in nding well-adapted signal representations. The proposed evolutionary approach oers more exibility in searching for well-adapted signal representations than standard approaches. References 1. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies. Image coding using wavelet transform. IEEE Trans. on Image Process., 12, April 1992. 2. G. Beylkin. On the representation of operators in bases of compactly supported wavelets. Society for Industrial and Applied Mathematics, 66:1716 1740, December 1992. 3. C. M. Brislawn. Two-dimensional symmetric wavelet transform tutorial program. Technical report, Los Alamos National Laboratory, December 1992. 4. J. Buckheit and D. L. Donoho. Wavelab and reproducible research. Technical report, Department of Statistics, Stanford University, 1995. 5. I. Cohen, S. Raz, and D. Malah. Orthonormal shift-invariant wavelet packet decomposition and representation. Signal Processing, 573:251 270, March 1997. 6. R. R. Coifman and M. V. Wickerhauser. Entropy based methods for best basis selection. IEEE Trans. on Inf. Theory, 382:719 746, 1992. 7. G. Davis. Baseline Wavelet Transform Coder Constrution Kit. Mathematics Department, Dartmouth College, January 1997. 8. D. E. Goldberg. Genetic Algorithms in Search, Optimization, and machine learning and Filter Banks. Addison-Wesley, Reading, Massachusetts, 1989. 9. John R. Koza. Genetic Programming - On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, MA, 1992. 10. Z. Michalewicz. Genetic algorithms + data structures = evolution programs. Articial Intelligence. Springer-Verlag, New York, 1992. 11. J. Villasenor, B. Belzer, and J. Liao. Wavelet lter evaluation for image compression. IEEE Trans. on Image Process., 48:1053 1060, August 1995. 12. M. Wall. GAlib: A C++ Library of Genetic Algorithm Components. Mechanical Engineering Department, Massachusetts Institute of Technology, August 1996. Finding Golf Courses: The Ultra High Tech Approach Neal R. Harvey, Simon Perkins, Steven P. Brumby, James Theiler, Reid B. Porter, A. Cody Young, Anil K. Varghese, John J. Szymanski and Jeffrey J. Bloch Space and Remote Sensing Sciences Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract. The search for a suitable golf course is a very important issue in the travel plans of any modern manager. Modern management is also infamous for its penchant for high-tech gadgetry. Here we combine these two facets of modern management life. We aim to provide the cuttingedge manager with a method of finding golf courses from space! In this paper, we present Genie: a hybrid evolutionary algorithm-based system that tackles the general problem of finding features of interest in multi-spectral remotely-sensed images, including, but not limited to, golf courses. Using this system we are able to successfully locate golf courses in 10-channel satellite images of several desirable US locations. 1 Introduction There exist huge volumes of remotely-sensed multi-spectral data from an everincreasing number of earth-observing satellites. Exploitation of this data requires the extraction of features of interest. In performing this task, there is a need for suitable analysis tools. Creating and developing individual algorithms for specific feature-detection tasks is important, yet extremely expensive, often requiring a significant investment of time by highly skilled analysts. To this end we have been developing a system for the automatic generation of useful feature-detection algorithms using an evolutionary approach. The beauty of an evolutionary approach is its flexibility: if we can derive a fitness measure for a particular problem, then it might be possible to solve that problem. Many varied problems have been successfully solved using evolutionary computation, including: optimization of dynamic routing in telecommunications networks [1], optimizing image processing filter parameters for archive film restoration [2], designing protein sequences with desired structures [3] and many others. When taking an evolutionary approach, a critical issue is how one should represent candidate solutions in order that they may be effectively manipulated. We use a genetic programming (GP) method of representation of solutions, due to the fact that each individual will represent a possible image processing algorithm. GP has previously been applied to image-processing problems, including: S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 5 4 - 6 4 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 F in d in g G o lf C o u rs e s 5 5 edge detection [4], face recognition [5], image segmentation [6], image compression [7] and feature extraction in remote sensing images [8–10]. The work of Daida et al. Brumby et al. and Theiler et al. is of particular relevance since it demonstrates that GP can be employed to successfully evolve algorithms for real tasks in remote-sensing applications. 2 System Overview We call our feature detection system “Genie” (GENetic Image Exploitation) [9, 10] Genie employs a classic evolutionary paradigm: a population of individuals is maintained and each individual is assessed and assigned a fitness value. The fitness of an individual is based on an objective measure of its performance in its environment. After fitness determination, the evolutionary operators of selection, crossover and mutation are applied to the population and the entire process of fitness evaluation, selection, crossover and mutation is iterated until some stopping condition is satisfied. 2.1 Training Data The environment for each individual in the population consists of a set of training data. This training data consists of a data “cube” of multi-spectral data together with some user-defined data defining “ground-truth”. Ground-truth, in this context, is not what is traditionally referred to as ground-truth (this being in-situ data collected at, or as close as possible to, the time the image was taken). Here, ground-truth refers to what might normally be referred to as “analyst-supplied interpretation” or “training data”. This training data for our system is provided by a human analyst, using a Java-based tool called Aladdin. Through Aladdin, the user can view a multi-spectral image in a variety of ways, and can “mark up” training data by “painting’ directly on the image using the mouse. Training data is ternary-valued with the possible values being “true”, “false”, and “unknown”. True defines areas where the analyst is confident that the feature of interest does exist. False defines areas where the analyst is confident that the feature of interest does not exist. Fig. 1 shows a screen capture of an example session. Here the analyst has marked out golf courses as of interest. 2.2 Encoding Individuals Each individual chromosome in the population consists of a fixed-length string of genes. Each gene in Genie corresponds to a primitive image processing operation, and so the whole chromosome describes an algorithm consisting of a sequence of primitive image processing steps. Genes and Chromosomes A single gene consists of an operator name, plus a variable number of input arguments, specifying where input is to come from; 5 6 N .R . H a r v e y e t a l. Fig. 1. GUI for Training Data Mark-Up. Note that Aladdin relies heavily on color, which does not show up well in this image. The light colored patches in the center-right and upper-right parts of the image are two golf courses that have been marked up as “true”. Most of the rest of the image has been marked up as “false”, except for a small region around the golf courses which has been left as “unknown”. output arguments, specifying where output is to be written to; and operator parameters, modifying how the operator works. Different operators require different numbers of parameters. The operators used in Genie take one or more distinct image planes as input, and generally produce a single image plane as output. Input can be taken from any data planes in the training data image cube. Output is written to one of a small number of scratch planes — temporary workspaces where an image plane can be stored. Genes can also take input from scratch planes, but only if that scratch plane has been written to by another gene positioned earlier in the chromosome sequence. The image processing algorithm that a given chromosome represents can be thought of as a directed acyclic graph where the non-terminal nodes are primitive image processing operations, and the terminal nodes are individual image planes extracted from the multi-spectral image used as input. The scratch planes are the ‘glue’ that combines together primitive operations into image processing pipelines. Traditional GP ([11]) uses a variable sized (within limits) tree representation for algorithms. Our representation differs in that it allows for reuse of values computed by sub-trees since many nodes can access the same scratch plane, i.e. the resulting algorithm is a graph rather than a tree. It also differs in that the total number of nodes is fixed (although not all of these may be actually used in the final graph), and crossover is carried out directly on the linear representation. We have restricted our “gene pool” to a set of useful primitive image processing operators. These include spectral, spatial, logical and thresholding operators. Table 1 outlines these operators. For details regarding Laws textural operators, the interested reader is referred to [12, 13]. The set of morphological operators is restricted to function-set processing morphological operators, i.e. gray-scale morphological operators having a flat structuring element. The sizes and shapes of the structuring elements used by F in d in g G o lf C o u rs e s 5 7 Table 1. Image Processing Operators in the Gene Pool Code ADDP SUBP ADDS SUBS MULTP DIVP MULTS DIVS SQR SQRT LINSCL LINCOMB SOBEL PREWITT AND OR CL LAWB LAWD LAWF LAWH Operator Description Add Planes Subtract Planes Add Scalar Subtract Scalar Multiply Planes Divide Planes Multiply by Scalar Divide by Scalar Square Square Root Linear Scale Linear Combination Sobel Gradient Prewitt Gradient And Planes Or Planes Clip Low Laws Textural Operator S3T Laws Textural Operator E3T Laws Textural Operator L3T Laws Textural Operator S3T × L3 × E3 × S3 × S3 Code Operator Description MEAN VARIANCE SKEWNESS KURTOSIS MEDIAN SD EROD DIL OPEN CLOS OPCL CLOP OPREC CLREC HDOME HBASIN CH LAWC LAWE LAWG Local Mean Local Variance Local Skewness Local Kurtosis Local Median Local Standard Deviation Erosion Dilation Opening Closing Open-Closing Close-Opening Open with Reconstruction Close with Reconstruction H-Dome H-Basin Clip High Laws Textural Operator L3T × E3 Laws Textural Operator S3T × E3 Laws Textural Operator E3T × S3 these operators is also restricted to a pre-defined set of primitive shapes, which includes, square, circle, diamond, horizontal cross and diagonal cross, and horizontal, diagonal and vertical lines. The shape and size of the structuring element are defined by operator parameters. Other local neighborhood/windowing operators such as mean, median, etc. specify their kernels/windows in a similar way. The spectral operators have been chosen to permit weighted sums, differences and ratios of data and/or scratch planes. We use a notation for genes that is most easily illustrated by an example: the gene [ADDP rD0 rS1 wS2] applies pixel-by-pixel addition to two input planes, read from data plane 0 and from scratch plane 1, and writes its output to scratch plane 2. Any additional required operator parameters are listed after the input and output arguments. Note that although all chromosomes have the same fixed number of genes, the effective size of the resulting algorithm graph may be smaller than this. For instance, an operator may write to a scratch plane that is then overwritten by another gene before anything reads from it. Genie performs an analysis of chromosome graphs when they are created and only carries out those processing steps that actually affect the final result. Therefore, in some respects, we could refer to the fixed length of the chromosome as a “maximum” length. 2.3 Backends Complete classification requires that we end up with a single binary-valued output plane from the algorithm. It would be possible to treat, say, the contents of 5 8 N .R . H a r v e y e t a l. scratch plane 0 after running the chromosome algorithm, as the final output from the algorithm (thresholding would be required to obtain a binary result). However, we have found it to be of great advantage to perform the final classification using a non-evolutionary algorithm. To do this, we first select a subset of the scratch planes and data planes to be answer planes. Typically in our experiments this subset consists of just the scratch planes. We then use the provided training data and the contents of the answer planes to derive the Fisher Discriminant, which is the linear combination of the answer planes that maximizes the mean separation in spectral terms between those pixels marked up as “true” and those pixels marked up as “false”, normalized by the “total variance” in the projection defined by the linear combination. See [14] for details of how this discriminant works. The output of the discriminant-finding phase is a gray-scale image. This is then reduced to a binary image by using Brent’s method [15] to find the threshold value that minimizes the total number of misclassifications (false positives plus false negatives) on the training data. 2.4 Fitness Evaluation The fitness of a candidate solution is given by the degree of agreement between the final binary output plane and the training data. This degree of agreement is determined by the Hamming distance between the final binary output of the algorithm and the training data, with only pixels marked as true or false contributing towards the metric. The Hamming distance is then normalized so that a perfect score is 1000. To put this in a more formal/mathematical context. Let H be the Hamming distance between the final binary output of the algorithm and the training data, with only pixels marked as true or false contributing towards the metric, let N be the number of classified pixels in the training image (i.e. pixels marked as either “true” or “false”) and let F be the fitness of the candidate solution. F = (1 − (H/N )) × 1000 2.5 (1) Software Implementation The genetic algorithm code has been implemented in object-oriented Perl. This provides a convenient environment for the string manipulations required by the evolutionary operations and simple access to the underlying operating system (Linux). Chromosome fitness evaluation is the computationally intensive part of the evolutionary process and for that reason we currently use RSI’s IDL language and image processing environment. Within IDL, individual genes correspond to single primitive image operators, which are coded as IDL procedures, with a chromosome representation being coded as an IDL batch executable. In the present implementation, an IDL session is opened at the start of a run and communicates with the Perl code via a two-way unix pipe. This pipe is a low-bandwidth connection. It is only the IDL session that needs to access the input and training F in d in g G o lf C o u rs e s 5 9 data (possibly hundreds of Megabytes), which requires a high-bandwidth connection. The Aladdin training data mark-up tool was written in Java. Fig. 2 shows the software architecture of the system. Fig. 2. Software Architecture of the System Described. Note that the feature depicted on the right of this diagram represents the input data, training data and scratch planes 3 Why Golf Courses? The usefulness of devising algorithms for the detection of golf courses may not, at first, seem apparent (except to a manager, perhaps!). However, due to the nature of golf courses and their characteristics in remotely-sensed data, they are of great use in testing automatic feature-detection systems, such as described here. They possess distinctive spectral and spatial characteristics and it is the ability of feature-detection algorithms to utilize both these “domains” that we seek to test. It is also useful that there exists a great deal of “ground truth” data available: a great many golf courses, for the benefit of low-tech managers, are marked on maps. In addition, golf courses usually possess a well-known, particular type of vegetation and it is rare to find information regarding specific vegetation types on maps. Fig. 3 (a) shows a map of NASA’s Moffet Field Air Base, clearly showing the position of a golf course. Fig. 3 (b) shows a false col- (a) (b) Fig. 3. (a) Map of NASA’s Moffet Field Air Base, showing a golf course (available at http://george.arc.nasa.gov/jf/mfa/thesite2.html) (b) Image from remotely-sensed data of NASA’s Moffet Field Air Base 6 0 N .R . H a r v e y e t a l. our image of some remotely sensed data of the same region. The airfield and golf course are clearly visible. 4 Remotely-Sensed Data The remotely-sensed images referred to in this paper are 10-channel simulated MTI data, produced from 224-channel AVIRIS data, each channel having 614 × 512 pixels. The images displayed are false-color images (which have then been converted to gray-scale in the printing process). The color mappings used are the same for all images shown (an exception being Fig. 1 where the false-color image has had a red and green overlay, corresponding to “false” and “true” pixels, as marked by the human analyst). The particular color mappings used here involve averaging bands A and B for the blue component, bands C and D for the green component and bands E and F for the red component. In addition, the images have been contrast enhanced. The choice of color mappings was arbitrary, in that it was a personal decision made by the analyst, made in order to best “highlight” the feature of interest, from his/her perspective and thus enable him/her to provide the best possible training data. This choice of colormappings, together with a contrast-enhancement tool, are important and very useful features of Aladdin. Table 2 provides details about MTI data. Table 2. MTI Band Characteristics Band Wavelength (µm) Color SNR Ground Sample Distance A B C D E F G H I O J K L M N 0.45-0.52 0.52-0.60 0.62-0.68 0.76-0.86 0.86-0.89 0.91-0.97 0.99-1.04 1.36-1.39 1.55-1.75 2.08-2.35 3.50-4.10 4.87-5.07 8.00-8.40 8.40-8.85 10.2-10.7 blue/green green/yellow red NIR NIR NIR SWIR SWIR SWIR SWIR MWIR MWIR LWIR LWIR LWIR 120 120 120 120 500 300 600 4 700 600 250 500 800 1000 1200 5m 5m 5m 5m 20m 20m 20m 20m 20m 20m 20m 20m 20m 20m 20m Figs. 3(a), 4(a) and 5(a) are data taken over an area of NASA’s Moffet Field Air Base in California, USA. Fig. 3(a) is a sub-set of the data shown in Fig. 4(a). Figs. 3(a) and 5(a) are non-adjacent regions of the original data. These sub-sets of the data contain a lot of different features, but, of course, have a common feature of interest: golf courses. F in d in g G o lf C o u rs e s 5 6 1 Searching for Golf Courses We reserve the data described above (Fig. 3(a)) for testing an evolved golfcourse finder algorithm and set the system the task of finding a golf course on some other data. This data, showing the “truth” as marked out by an analyst, is shown in Fig. 1. The golf course area has been marked as “true” and most of the remaining data has been marked as “false”. The system was run for 400 generations, with a population of 100 chromosomes, each having a fixed length of 20 genes. At the end of the run the best individual had a fitness of 966 (a perfect score would be 1000). This fitness score actually translates into a detection rate of 0.9326 and a false alarm rate of 0.00018. The results of applying the best overall algorithm found during the run to the data used in the training run are shown in Fig. 4. (a) (b) Fig. 4. (a) Image of training data (b) Result of applying algorithm found to training data It can be seen that the algorithm has been able to successfully detect the golf course and has not detected any of the other features within the image. In order to test the robustness of the algorithm found, it was applied to outof-training-sample data, as described previously, and shown in Fig. 3 (b). The results are shown in Fig. 5. It should be noted that the data shown in Fig. 5 covers a greater area than shown by the map in Fig. 3 (a). It can be seen that the algorithm has successfully found the golf course shown on the map. It can also be seen that the algorithm has detected other golf courses. On closer examination of the data, it would appear that further golf courses do, in fact, exist at those locations. It can also be seen that the algorithm has not found any spurious features. The “short” (redundant genes stripped out) version of the chromosome found is detailed below. [LAWG rD2 wS0] [OPREC rD3 wS3 5 1] [ADDP rS0 rS3 wS1] [ADDP rS1 rD6 wS1] [LAWE rD6 wS4] [LAWG rD6 wS0] [OPCL rS4 wS3 1 1] [DIL rS1 wS1 1 0] [OPREC rS1 wS1 5 0] [MEDIAN rS1 wS2 1] [LAWH rD2 wS4] 6 2 N .R . H a r v e y e t a l. (a) (b) Fig. 5. (a) Image of out-of-training-sample data (b) Result of applying algorithm found to out-of-training-sample data A graphical representation of the algorithm found is shown in Fig. 6. Note that the circles at the top of the graph indicate the data planes input to the algorithm (in this case only 3 data planes out of a possible 10 have been selected), the 5 circles in the center represent the scratch planes and the circle at the bottom represents the final, binary output of the overall algorithm. The operations above the line of scratch planes represent that part of the overall algorithm incorporated in the chromosome. The operations below the line of scratch planes represent the optimal linear combination of scratch planes and intelligent thresholding parts of the overall algorithm. It is interesting to have some kind of objective measure of the algorithm’s performance on the out-of-training-sample data. To this end an analyst marked up training data (i.e. true and false) for this data, with respect to the golf courses present. This enabled determination of a fitness for the algorithm on this data as well as detection and false alarm rates. The fitness of the algorithm was 926.6, the detection rate was 0.8532 and false-alarm rate was 3.000E-05. 6 Comparison with Other Techniques In order to compare the feature-extraction technique described here to a more conventional technique, we used the Fisher discriminant, combined with the intelligent thresholding, as described previously, to try and extract the golf courses in the images shown/described. This approach is based purely on spectral information. On application to the data used in the training run (Fig. 4(a)), this “traditional” approach produced a result having a fitness of 757.228 (with respect to the training data/analyst-supplied interpretation), which translates into a detection rate of 0.5159 and a false-alarm rate of 0.00141. On application to the out-of-training-sample data, the result had a fitness of 872.323, which translates into a detection rate of 0.7477 and false-alarm rate of 0.00305. Both of these results are significantly below the performance of the results produced by the Genie system described here. F in d in g G o lf C o u rs e s 7 6 3 Conclusions A system for the automatic generation of remote-sensing feature detection algorithms has been described. This system differs from previously described systems in that it combines a hybrid system of evolutionary techniques and more traditional optimization methods. It’s effectiveness in searching for useful algorithms has been shown, together with the robustness of the algorithms discovered. It has also been shown to significantly out-perform more traditional, purely-spectral approaches. D 3 D 2 D 6 O p e n R e c . la w G A d d P la n e s A d d P la n e s D ila te la w E O p e n R e c . O p e n -C lo s e M e d ia n la w H S 1 S 4 - 3 .9 1 8 E - 7 M u ltip ly S c a la r la w G S 2 S 3 1 .1 7 2 E - 5 M u ltip ly S c a la r 3 .0 2 3 E - 6 M u ltip ly S c a la r M u ltip ly S c a la r S 0 6 .0 4 9 E - 6 M u ltip ly S c a la r A d d P la n e s - 1 .8 1 7 A d d S c a la r 0 .6 8 2 T h re s h o ld O /P Fig. 6. Graphical representation of algorithm found 5 .7 1 9 E - 7 6 4 N .R . H a r v e y e t a l. References 1. Cox, L.A., Jr., Davis, L., Qiu, Y.: Dynamic anticipatory routing in circuit-switched telecommunications networks, in Handbook of Genetic Algorithms, L. Davis, ed., pp. 124-143, Van Nostrand Reinhold, New York, 1991. 2. Harvey, N.R., Marshall, S.: GA Optimization of Spatio-Temporal Grey-Scale Soft Morphological Filters with Applications in Archive Film Restoration. In: Poli, R., Voigt, H.-M., Cagnoni, S., Corne, D., Smith, G.D., Fogarty, T.C. (eds.): Evolutionary Image Analysis, Signal Processing and Telecommunications (1999) pp. 31–45 3. Dandekar, T., Argos, P.: Potential of genetic algorithms in protein folding and protein engineering simulations, Protein Engineering 5(7), pp. 637-645, 1992. 4. Harris, C., Buxton, B.: Evolving edge detectors, Research Note RN/96/3, University College London, Dept. of Computer Science, London, 1996. 5. Teller, A., Veloso, M.: A controlled experiment: Evolution for learning difficult image classification, in 7th Portuguese Conference on Artificial Intelligence, Volume 990 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1995. 6. Poli, R., Cagoni, S.: Genetic programming with user-driven selection: Experiments on the evolution of algorithms for image enhancement, in Genetic Programming 1997: Proceedings of the 2nd Annual Conference, J. R. Koza, et al., editors, Morgan Kaufmann, San Francisco 1997. 7. Nordin, P., Banzhaf, W.: Programmatic compression of images and sound, in Genetic Programming 1997: Proceedings of the 2nd Annual Conference, J. R. 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Image Understanding Workshop, Nov. 1979, pp. 47–51. 13. Pietikainen, M., Rosenfeld, A., Davis, L.S.: Experiments with Texture Classification using Averages of Local Pattern Matches IEEE Trans. on Systems, Man and Cybernetics, Vol. SMC-13, No. 3, May/June 1983, pp. 421–426. 14. Bishop, C.M.: Neural Networks for Pattern Recognition, pp. 105–112, Oxford University Press, 1995. 15. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd Edition, Cambridge University Press, 1992, pp. 402–405.. L c t M s G ika rd Ka rl sso , o pl r H tc Pr r t r o rdi a d R t a ts o rda l st s ro p, a l rs i rsit o f c o l o , -42 9 6 , ö t o r , d , a il : o rdi ,tf @ f .c a l rs.s c l i ar s st as s d to so l t pro l o f so d l o ca l ia tio fo r a a to o o s a o id ro o t, it t o icro p o s a s a rs. o d t r i t a lto t so d so rc , a old pro ra as s d i a l o o p o r a st r o sa plstr a , r t tic pro ra ts t l a t stsa plpa irplsf d a ckfro t pr io s it ra tio a s i p t. pr cisio o ft o l d pro ra s a s d p d t o t p ri ta ls t p. o r a sa to o t a fro a fi d dista c t s al lst rro r a s 8 . ltti t dista c to t sa so rc a r t rro r a s 23 . o r a a o ic a t a r i dista c s t rro r a s p to 4 . tr ct p rpo s o ft is pa p r is to i stiga t so d l o ca l ia tio si g . s sis it d d to b s d i a a o id ro bo t q ipp d it t o icro p o s. dl o ca l ia tio is p rfo r d b a s a l lbi a r a c i co d pro gra o a pro c ss d st r o str a o fsa pld so d. i c t s st is it d d fo r a a o id ro bo t,so d gr o fsi il a rit to a so d l o ca l ia tio isd sir d (s [ ] fo r a o t r st d o ft is pro bl i a ro bo tco t t). ca s o ft l i it d po r o -bo a rd t ro bo tt co p ta tio a lr q ir ts d to b ii i d s st ds to g ra l i fro a l i it d s to ftra i i gda ta . s st a l so ds to b a blto l o ca l i a diff r tki ds o fso ds, so ds fro a l l dir ctio si t o rio ta l pl a ,so ds it diff r tit sit,so ds fro a l ldista c s,so ds c o s a r pr s t,a d so ds it ba ckgro d o is pr s t. s a co pa riso , rst gi a s o rt d scriptio o ft a a dito r s st a d its so d l o ca l ia tio ca pa bil itis. t o . a i r s s a bil it to l o ca l i so ds is a i po rta tpa rto ft a dito r s st a d as b ss tia lto o r s r ia l . o d pro pa ga t s t ro g a di a s a l o git di a l a . its a to t a rdr itpa ss s t o t r a r,ca l ld t pi a , a d t a dito r ca a l . l o git di a l a i t a ir is tra sfrr d S . 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H pro id sa o t r po t tia lc fo r so d l o ca l ia tio . or l o ca l ia tio i t o rio ta lpl a I a d II a r a jo r c s fo r a s, il t H is a c fo r l o ca l ia tio i t rtica lpl a s. I S o u n d L o c a liz a tio n fo r a H u m a n o id R o b o t 6 7 a st I is s d fo r fr q cis s a l lrt a 2 kH , ilt II is s d fo r fr q cis a bo 2 kH . s rs l ts a gr fa irl l l it t a rg ts a bo . o r d ta il so a so d l o ca l ia tio a d its a cc ra c ca b fo d i [2]. I t o rio ta lpl a , a s a k a rro ro f4 to d p di go t dir ctio to t so d so rc . o ds stra ig ta a d o f s a r a sistto l o ca l i , ilitis a rd r to l o ca l i so ds i t dia pl a . Itis o r diffic l t to l o ca l i so ds ifo is d a fo o a r. titis po ssibl to s o l t H as c . 2 p r t s tp o icro p o s r s d to r co rd so ds i st r o fro diff r tdir ctio s i t o rio ta lpl a . o r so d l o ca l ia tio to b a s si pla d ffici ta s po ssibl o l d pro gra st a tco l dl o ca l i so d si go la ra sa pld so d sig a la s i p t. pro gra a s it ra t d it f dba ck o ft o tp t fro t pr io s it ra tio a d t tpa ir o fso d sa pls a s i p t. 2. c r i s s o d r co rdi gs i st r o r ad it t o icro p o s pl ac d i t d a d o ft ro bo t. sig a lfro t icro p o s a s pr a pl i d, a d sa pld a d r co rd d i st r o b a so d ca rd. l lr co rdi gs r a d i a o ffic it ba ckgro d o is a i l fro co p t r fa s. c o ic o fsa pl i g fr q c a d r so ltio l i its ic c s ca b s d fo r so d l o ca l ia tio , si c t sa pl i g la ds to a discr tia tio rro r. is c o ic a l so d t r i s t fr q c ra g . a rd a r r stricts s to o r 6 bits r pr s ta tio . discr tia tio rro r is a t o st a l ft r so ltio i ti a d sa pl a l . o ca l c l at t I a d II a diff r c b t sig a l s is i o l d, so t a i a lrro r is t r so ltio i ti a d sa pl al . s o fI fo r so d l o ca l ia tio r q ir s ig ti r so ltio ; t sa plr so ltio is lss i po rta t. Ift so d a rri s fro i it t I i r is d p ds o t a glto t so d a ppro i a t la s = t dista c b t t a rs, is t so d l o cit a d is t a glto t so d so rc . rro r is t gi b : = o o bta i a rro r o f fo r a so d fro stra ig ta a d t sa pl i gti − stb a ss o rta s46 s,co rr spo di gto a sa pl i gfr q c o f22 kH . rq cis p to 5 H ca b s d to ca l c l at I fo r a s, ilt l i itis 9 H fo r a ro bo t it 9 c b t its a rs. II o t o t r a d is a i ls siti to t sa plr so ltio . I pra ctic ,fr q cis a bo 7 H ca b s d to ca l c l a t II fo r a s, ilt l i itfo r t ro bo t is 3 H . o s t H a l a rg ra g o ffr q cis stb pr s ta d a ig r so ltio i t sa pl a l is d d. 6 8 R . K a rls s o n , P . N o rd in , a n d M . N o rd a h l a ki g t s fa cts ito a cco t, a 6 kH sa pl i g fr q c a s d cid d po , ba s d o a co pro is b t r d ci g t discr tia tio rro r, a d r d ci gt a o to fi p tda ta fo r t g tic pro gra . ra lso ds r r co rd d si gdiff r t p ri ta ls t ps. a si gi g o ic a s r co rd d it a pt a d o ft ro bo t(a c bic a l i bo , o p i t to p a d ba ck). o s ts o f 6 H sa to o t a s r r co rd d it diff r tpi a . i co a ds spo k b a a o ic r al so r co rd d. s r co rdi gs a r r fr d to a s r co rdi gs , a I a d II a d o a dsi t r sto ft pa p r. sa pls r r co rd d si g 6 bits. I t s t p fo r r co rdi g t a d a s pt. pi a r ad o f pl a stic, a d r si il a r to a pi a . o a dito r ca a l s r pr s t. al a si gi ga to a sr co rd d fro 6 ldistrib t d dir ctio s i t o rio ta lpl a . o r t r co rdi g a I t ad as l ld it i s l a ti g a t ria l . dito r ca a l s it a dia t r o f7 a d a d pt o f r a d o to fpl a stic. o r t s al lpi a . 6 H sa to o t a as tra s itt d fro a l o dsp a k r .2 t rs fro t a d. o ds r r co rd d fro 6 dir ctio s q a l l spa c d i t o rio ta lpl a . I t r co rdi g o a ds t ad as l ld it i s l a ti g a t ria l , a d t ro bo t a s q ipp d it a rs a d o f o d l i gcl a a d o d ld a ft r a a rs. a dito r ca a l s r a ppro i a t l 2 l o g. a ccid t, t a rca a l sdiff r d so a ti idt . o ds r r co rd d it t d a d pl ac d i diff r tpo sitio s i t ro o .I a c r co rdi gs r a d fro 6 l distrib t d dir ctio s i t o rio ta lpl a . so d so rc a s pl a c d it r o , t o , t r o r fo r t rs fro t d a d. r co rd d so ds r a a l o ic gii g t co a ds a l k, sta d, fo r a rd, ba ck, sto p, rig t, lfta d gra b. I t r co rdi g a II t sa a d a s i r co rdi g o a ds a s s d ( c pt t a t t a r ca a l s r ad q al l id ). a to o t a s it a fr q c of 6 H r r co rd d it t a d pl a c d a t t r diff r t l o ca tio s i t ro o , ,2 a d 3 fro t so d so rc . t so ds fro 6 dir ctio s l distrib t d i t o rio ta lpl a r r co rd d. I t o t r t o po sitio s so ds fro i dir ctio s a s r co rd d. 2.2 ic pr ra i rs ca ia i pro gra s o l d s d a ra sa pld so d sig a la s i p t. pro gra a s it ra t d it f dba ck o ft o tp t fro t pr io s it ra tio a d t t pa ir o f so d sa pls a s i p t. i diid a lpro gra ca b i d a s a f ctio o s i p ts a d o tp ts a r a rra s. i p tco sists o ft o o ris, t pr io s r a lo tp ts, t o sa pls a d a co sta t. o tp t co sists o ft t o o ris a d t o r a lo tp ts. o ris r pr s tt o tp tfro t i diid a l s d a s i p ti t tit ra tio . ’ r a lo tp ts’a r t o tp ts s d to a la t t a glto t so d so rc . s a r s d a s f dba cki t sa a as t o ris. sa pls a r a pa ir o fi p tsa pls fro t lfta d rig tso d c a l . S o u n d L o c a liz a tio n fo r a H u m a n o id R o b o t 6 9 I itia l l t i p ts i fo r o f o ris a d pr io s r a lo tp ts a r s t to ro . ri g s bs q tit ra tio s, o ris a d pr io s r a lo tp ts, a r s d a s f dba ck. rstti a i diid a lis r itis gi t rstpa ir o f sa pls i t t ss ca s a s i p t, a d a pa ir o fsa pls is t s ppl id i a c it ra tio . ca d scrib o t i diid a l s a r s d i ps do -co d , s gr . sa p sa p ss as ss as r ) ) ) ) a sa p sa p sa p a a a s o fa i ps do -co d . ss as ss as s s r sa p s ) ) ) r pr sr a p s i diid a lto d t r i t dir ctio to a so d so rc is s o t o r a lo tp ts ’ o t ’a d ’ o t2’a r it rpr t d a s t -a d co o rdi a t i a ca rt sia co o rdi a t s st , a d t a gl is ca l c l a t d fro t is i fo r a tio . is r pr s ta tio is s gg st d b t g o tr o ft pro bl (a a g l a r r pr s ta tio as al so trid). b r o fit ra tio s d d ca b sti a t d fro t p sics o ft pro bl . o s t I t i diid a l ds to s a tla sta b ro fsa pls co rr spo di gto t ti d l a . a i a lti d l a is a ppro i a t l t ti itta k s t so d to tra lt dista c b t t a rs, a ppro i a t l .26 s fo r a 9 c ro bo t a d. is co rr spo ds to 4.2 sa pls fo r a sa pl ig fr q c of 6 H . o s t II t i diid a l ds to s a tla sto a l gt . o r a so d o f4 H t is co rr spo ds to 4 sa pls. 4 H is l l b l o t fr q ciso f ig stit sit i t a sp a ki g o ic . i c ig r fr q cis a r o stit r sti gfo r t H o o r sa pls t a to a la t t II a r l ik l to b d d. 2. s s l i ar s st o li g bi a r a c i co d a s s d [3]. s st o p ra t s it a fo r i diid a lto r a t s lctio si g st a d sta t . d s r s d, to g t r it a a rit tic f ctio s t. 7 0 R . K a rls s o n , P . N o rd in , a n d M . N o rd a h l t ss ca s co sists o fa b r o fpa irs o fsa pls. s r c os ra do l fro a l o g r a rra o fsa pls fro a r co rdi g co rr spo di g to a sp ci c dir ctio to t so d so rc . l ldir ctio s r gi q a l ig t i t tra i i g s t. t ss ca s s r c o s a c ti a i diid a l a s a la t d, to i pro g ra l ia tio . t sts t a s c o s ra do lfro t l a rg r s to fsa pls i t sa a a s t tra i i gs t. a la tio a s do o t ss ca s s c o s t sa a a s t o s s d fo r tra i i g. al ida tio s t a s s d to c ck o l lt g tic pro gra g ra l i s to a diff r ts tt a t tra i i gs t. rro r o f a i diid a l a s ca l c l a t d a s diff r c , t s o rt st a a ro d t circl, b t t co rr cta gla d t a glca l c l a t d fro t o tp to ft g tic pro gra . t ss a s d d b t fo l l o i g pr ssio :  i = − i r is t b r o f t ss ca s s i t tra i i gs ta d t i d spo ds to a sp ci c t ss ca s . 3 co rr - s ts s cc ss o ft o l d pro gra a rid sig i ca tl it t p ri ta l s t p. rs l ts fro p ri ts I to ar s o i g r s 2 to 7 . ig r s 2 to 5s o t t ss o ft b sti diid a l a la t d o t tra i i g,t sta d al ida tio s ts. al so s o t a ra g a d dia t ssfo rt po p l a tio asa o l,a d t pro gra l gt o ft b stg tic pro gra a d t a ra g l gt o ft g tic pro gra s i t po p l a tio . i dia gra s i g r 6 a d g r 7 s o t a glca l c l a t d a ft r a c it ra tio d ri g t c tio o ft g tic pro gra i p ri ts I a d .I gr 6 o gra p is s o i a c dia gra a d i g r 7 t r gra p s co rr spo di g to so ds co i g fro diff r tdista c s a r s o . I t i dia gra s t d sir d a s r ra g s fro -9 i t rstto + 9 i t l a st it a spa ci g o f22.5 . a la tio s r do o a t sts tt a t a s id tica lto t tra i i gs t. a bl s o s t pa ra t rs t a t r a rid b t p ri ts. fo l l o i gpa ra t rs r id tica li a l l p ri ts: () g tic pro gra s r tra i d o so ds co i gfro t dir ctio s = , 22 5 , 45 , 6 7 5 a d 9 . (2) l lsa pls r a pl i d i al l p ri ts i s c a a t a t t a i a lsa pla pl it d a s id tica li a l l t ss ca s s. ft r t is a pl i ca tio a s co d a pl i ca tio a s p rfo r d i a l l p ri ts c ptfo r p ri t . l a sta pl i ca tio a s c o s ra do l b t . a d fo r a c t ss ca s ,to a o id t a tt g tic pro gra la r d to r co g i a c r co rd d so d l. S o u n d L o c a liz a tio n fo r a H u m a n o id R o b o t p. a ta fo r: rr. rr. r. ra i i st Va l . ra i . st I a I a I 2 - 7 (9 ) (9 ) II a a a I 4 34 - 6 2 (27 ) ( 8) (9 ) III a I a I a 8 6 (9 ) ( 8) (9 ) IV a I a I a 3 (9 ) (9 ) ( 8) V a II a II a I 23 24 - 28 (27 ) (27 ) (9 ) a rr. Va l . - tti s: l o o ps 2 4 3 4 5 4 - 56 4 38 -5 7 1 . 42 - 6 3 [36 , 45] 27 45 - 53 [36 , 45] 32 7 (45) a . ta blis a s a r o fpa ra t r s tti gs a d r s l ts fo r p rits Ito . o l t o to fo r s o t so d ls s d a s tra i i g,t sta d al ida tio s ts, a d t b r o fso d ls s d. I co l to s t t ss o ft b sti diid a lca l c l a t d o t tra i i g,t sta d a l ida tio s ts ar s o . fo l l o i gco l s o st b r o fit ra tio s o ft g tic pro gra s. I p ri t I a d t b r o fit ra tio s a s c o s ra do l i t gi it r a l . b rs i t l a stco l so t b r o fg ra tio s a c p ri t a s r . () po p l a tio co sist d o f4 i diid a l s diid d ito 4 d s. () 9 % cro sso r pro ba bil it a s s d. pro ba bil it fo r ta tio s to o cc r a s t sa . pro ba bil it fo r o o l o go s cro sso r a s 4 % a d t r a s o igra tio b t d s. ra t o fcro sso r b t d s as % . () o l co sta ta a il a blto t g tic pro gra s a s -43 6 o i diid a l a s a l l o d to b l o g r t a 2 b t s (32 bl o cks). 2 . (6 ) I p ri tIa d IIt t ss a s a la t d o t t ss ca s s fro a c r co rd d so d l. I t o t r p ri ts t a t b r as . t t ss a s a la t d o t t sts ta d tra i i g s tt t t ss ca s s r ta k fro a c r co rd d so d li t t sta d a l ida tio s ts. o r a ls cc ss r o bta i d t tra i i g s t r fro co rdi g . rro r fo r so d l o ca l ia tio o t tra i i gs t a s rs al lr t a s 2 . is r s l t a s o bta i d al li diid a l s r a la t d o a ctl t sa t ss ca s s. rro r o f 42 a s o bta i d t t ss ca s s r c a gd b t ac a la tio o fa i diid a l . is i dica t s t a tt g tica l l o l d pro gra o rtra i d a d o ri d t t ss ca s s i st a d o fso li gt r a lpro bl o fso d l o ca l ia tio . 7 2 R . K a rls s o n , P . N o rd in , a n d M . N o rd a h l 130 length (bytes) 120 110 100 90 80 70 60 0 50 100 150 200 generations 250 300 350 400 0 50 100 150 200 generations 250 300 350 400 100 error (degrees) 80 60 40 20 0 s l ts fro p ri tI. pp r dia ra s o s t l t i t s o ft sti diid a l(so l id) a d t a ra l t (da s d). l o r dia ra s o s t fit ss o ft sti diid a lo t tra i i s t(so l id) a d t t sts t(do tt d), t a ra fit ss (da s d) a d t dia fit ss (da s do t). 140 length (bytes) 120 100 80 60 40 0 20 40 60 80 generations 100 120 140 160 0 20 40 60 80 generations 100 120 140 160 error (degrees) 100 80 60 40 20 t t t t s l ts fro p ri tII. pp r dia ra s o s t l t i ts o f sti diid a l(so l id) a d t a ra l t (do tt d). l o r dia ra s o s fit ss o ft sti diid a lo t tra i i s t(so l id), t t sts t(do tt d) a d al ida tio s t(da s d l o r). Ita l so s o s t a ra fit ss (da s d pp r) a d dia fit ss (da s do t). S o u n d L o c a liz a tio n fo r a H u m a n o id R o b o t 7 3 130 length (bytes) 120 110 100 90 80 70 0 50 100 150 200 generations 250 300 350 400 0 50 100 150 200 generations 250 300 350 400 120 error (degrees) 100 80 60 40 20 0 t t t t s l ts fro p ri tIII. pp r dia ra s o s t l t i ts o f sti diid a l(so l id) a d t a ra l t (do tt d). l o r dia ra s o s fit ss o ft sti diid a lo t tra i i s t(so l id), t t sts t(do tt d) a d al ida tio s t(da s d l o r). Ita l so s o s t a ra fit ss (da s d pp r) a d dia fit ss (da s do t). 140 length (bytes) 120 100 80 60 40 0 50 100 150 generations 200 250 300 0 50 100 150 generations 200 250 300 error (degrees) 80 60 40 20 0 t t t t s l ts fro p ri tIV. pp r dia ra s o s t l t i ts o f sti diid a l(so l id) a d t a ra l t (do tt d). l o r dia ra s o s fit ss o ft sti diid a lo t tra i i s t(so l id), t t sts t(do tt d) a d al ida tio s t(da s d l o r). Ita l so s o s t a ra fit ss (da s d pp r) a d dia fit ss (da s do t). 7 4 R . K a rls s o n , P . N o rd in , a n d M . N o rd a h l 0 -50 -100 -150 100 0 50 -20 0 -40 -50 0 20 40 60 20 0 -100 -60 0 20 40 60 -80 20 60 0 40 -20 20 0 20 40 60 0 20 40 60 0 20 40 60 -20 -40 -60 0 20 40 60 -40 80 100 60 0 40 -100 0 20 40 60 0 100 80 60 20 0 20 40 60 -200 40 0 20 40 60 20 6 s l ts fo r p ri tIV. a c o ft dia ra s s o s t a l ca l c l atd fro t o tp ts o ft sti diid a lo o fit ss ca s fro t t sts t. a l is ca l c l a t d a ft r a c c tio o ft i diid a li t l o o p c ptfo r t first. t sts t a s id tica lto t tra i i s t. I a c o ft i dia ra s t o rio ta l l i s o t co rr cta s r. 50 0 -50 -100 0 20 40 60 50 100 100 50 50 0 0 -50 -50 -100 0 20 40 60 -100 50 100 0 0 50 -50 -50 0 -100 0 20 40 60 100 -100 0 20 40 60 80 20 40 60 0 20 40 60 0 20 40 60 100 60 50 -50 0 80 40 0 -50 60 20 0 20 40 60 0 0 20 40 60 40 7 s l ts fo r p ri tIV. a c o ft dia ra s s o s t a l ca l c l atd fro t o tp ts o ft sti diid a lo t r fit ss ca s s fro t t sts t. t r fit ss ca s s co rr spo d fro so ds fro t sa dir ctio tfro diff r t dista c s. dista c s r (so l id),2 (do tt d) a d 3 (da s do t). a l is ca l c l a t d a ft r a c c tio o f t i diid a li t l oop c pt fo r t first. I a c dia ra , t t sts t r id tica lto t tra i i s t. I a c o ft i dia ra s t o rio ta ll i s o s t co rr cta s r. S o u n d L o c a liz a tio n fo r a H u m a n o id R o b o t 7 5 sc ss rstr a rk t a t ca tio s o l d b o bs r d co pa ri g t r s l ts fro t diff r t p ri ts, si c t diff r t tra i i g s ts r r co rd d it diff r to t r a rs. rs l ts o f[4] s o t a tift s a p o ft a pi a is c a g d o g , o r a bil it to l o ca l i so ds is dra a tica l lr d c d. t s sta rt it p ri t . fa ct t a t t t ss s fo r t b st i diid a l a la t d o t tra i i ga d t sts ts a r a lo st q a l ls al l(s ta bl ), a st a tt g tic pro gra g ra l i sto a l lso dsi t tra i i g s t. is is a l so t ca s fo r t o t r p ri ts c ptfo r p ri tII, a s ca b s i g r s 2 to 5. I ta bl o ca s t a tt o l d pro gra i p ri t do s o t g ra l i to a a sp a ki g o ic ( o a ds) tra i d o sa to o t a s. is is o ts rprisi g si c itis a rd fo r t g tic pro gra to l o ca l i a sa to o t a so d co i gfro diff r tdir ctio s a d dista c s, s g r 7 . is g r a l so s o s t a tt g tic pro gra do s o tg ra l i t at l lto so ds co i g fro diff r tdista c s. diffic l t is pro ba bl d to c o s fro al l s a d f r it r i t ro o . ra i i go a a o ic a s a l so diffic l t, s g r 3. b st o l d pro gra a s a t ss o f4 . t ss ca b co pa r d to t a to bta i d fro a pro gra t a ta la s gi s t co sta to tp t a l = . t t ss is 5 , ic s o s t a tso la r i g a s o cc rr d. I gr 6 o ca s t a tt a glfro o i diid a la ris p rio dica l l a la t d o o t ss ca s , ilco rgi g to t rig ta s r. p rio d is t sa a s t a to ft sa to o t a t i diid a lis a lt d o . is i dica t s t a tt g tic pro gra s s t II i so a . is p rio dic pa tt r a s lss cla r t g tic pro gra s r o l d o so d co i g fro diff r tdista c s,s g r 7. r a so fo rt is a b t a tt g tic pro gra s stla r to co p it c o s, ic a k s it o r diffic l tto s t II . r a so t a titis po ssiblto o l a s al l a c i co d pro gra t a t ca it rpr ta pro c ss d str a o fsa pld st r o so d it o ti s rtio o fdo a i k o ldg o r o t r str ct r . o l d pro gra ca l o ca l i dir ctio o fa so d it a r i gs cc ss. a rti cia lso d t i diid a l s a si il a r p rfo r a c to a s as r d as a g l a r r so ltio , b t c r s a rc r a i s fo r t is p rfo r a c to o l d d r o r r al istic s tti gs a d co ditio s. c ts is r s a rc a s s ppo rt d b l op t. K, t dis o a rd fo r c ol o gica l 7 6 R . K a rls s o n , P . N o rd in , a n d M . N o rd a h l r c s . Iri . . s i i i i i r r i s r i , t sis, I , a rid 9 9 3. 2. l a rt . i ri h s h h si s i i , I r ss, 9 9 7 . 3. a a f ., t.a l . i r r i , dp kt. rl a a d or a a f a l is rs, H id r a d a ra sico , 9 9 8. 4. H o f a . ., t.a l ., la r i so d l o ca l ia tio it a rs, r rsi , 5, pp. 47 -42 , 9 9 8. t c iit f t V r L r -Sc dr a s osr a d tic tr . a ra si a rit s cti urt r an s a rc nt r f r rtificia lInt l l ig nc 676 Ka is rsl a ut rn, r a n moser@dfki.de pa rt nt f put r cinc a nd ut a tin India n Institut f cinc a nga lr - 56 2, India mnm@csa.iisc.ernet.in . a tur lctin is a r pr ising pti isa tin stra t g f r a tt rn c gnitin s st s. ut, a s a n -c plt ta sk, itis tr l difficul tt ca rr ut. a ststudis t r f r r ra t r l i it d in it r t ca rdina l it ft f a tur spa c r t nu r fpa tt rns util is d t a ss ss t f a tur su s tp rf r a nc . is stud a in s t sca l a il it f istri ut d n tic l g rit s t r l a rg -sca l a tur lctin. s d a in fa ppl ica tin, a cl a ssifica tin s st f r ptica l a ra ct rs is c s n. s st is ta ilr d t cl a ssif a nd- ritt n digits, in ling7 6 ina r f a tur s. u t t a stn ss ft in stiga t d pr l , t is stud f r s a st p int n r a ls in a tur lctin f r cl a ssifica tin. pr s nt a s t f cust isa tins f s t a t pr id f r a n a ppl ica tin f kn n c nc pts t a tur lctin pr l s f pra ctica l int r st. l i ita tins f s in t d a in f a tur lctin a r unr a ld a nd i pr nts a r sugg st d. id l us d stra t g t a cc lra t t pti isa tin pr c ss, ra ining t a pl ing, a s s r d t fa ilin t is d a in fa ppl ica tin. p ri nts n uns n a l ida tin da ta sugg st t a t istri ut d s a r ca pa l fr ducingt pr l c pl it significa ntl. r sul ts s t a tt cl a ssifica tin a ccura c ca n a inta in d ilr ducing t f a tur spa c ca rdina l it a ut5 % . n tic l g rit s a r d nstra t d t sca l l l t r l a rg -sca lpr l sin a tur lctin. tr cti r c td l op ti I fo r a tio c ol o g a s ro ug ta o uta tr do us flo o fi fo r a tio . o pa is,o rga isa tio s a d i diidua l s ar l it ra l l g tti gdro d i a flo o d o f a sur ts. a tt r co g itio ,a d i pa rticul a rpa tt r cl a ssi ca tio ,o ff rspro isi g a s to a rds a o r r d a o f d al i g it i fo r a tio . I o rd r to i pro t pro c sso fcl a ssi ca tio , a tur lctio a a ppl id to i cr a s t t ro ug puto fp rti ti fo r a tio . Itdo s o to lr duc t a o u to f S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 7 7 − 8 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 7 8 A . M o s e r a n d M .N . M u r ty da ta ta k ito a cco u tfo ra I fo r a tio ro c ssi gta sk, ut a a l so i pro t o ra l lpro c ss a ccura c . Ho r pro isi g t s o s r a tio s a , t r a r s rio us c a l lgs to t: a tur lctio is k o to a -co plt pro l . is fa ct a k s it tr l difficul tto a i l a rg -sca l,r a l - o rl d do a i s. ic t a rl studis o f a tur lctio t o ds i t s tis, t o tio o f a t“ l a rg -sca l” a ctua l l a s a s c a g d. rst p ri ts d al t it ra t r sm llpro l s o fso t s o ffa tur s. it t i cr a s o f co puta tio a lpo r, studis o flr s l ta sks ( - 5 fa tur s) ca po ssi l. o da , do a i s it o r t a 5 fa tur s ca sa id to trul c al l gi g;su s qu tl, suc pro l s s a l l ca l ld r lr s l. r a r a sica l l t o a s o fca rr i g o ut a a tur lctio pro c ss: il rm sa d r rm s [ ]. fo r r cl a ss o fa ppro a c s r l is o g ra lsta tistica lpro p rtis o ft pro l do a i a d a stra cts fro pa rticul a r cl a ssi r o d l s. ra pp r t o ds o t o t r a d o pti is t p rfo r a c o fa co cr t cl a ssi r. r a s il t r t o ds t pica l ld l i r or g ra lr sul ts, ra pp r t o ds a r co sid r d to il d a tt r cl a ssi ca tio p rfo r a c fo r sp ci c ta sks [2]. us, a i i ga ta ig cl a ssi ra ccura c , ra pp r t o dss to t o r pro isi gca dida t s. utt a sta ss ss tti fo r cl a ssi r p rfo r a c ltsuc a ppro a c sa pp a r i fa si lfo r r a l - o rl d a ppl ica tio s a tt rstgl a c .H o r,a d a c s i t r a l o f istri ut d rti cia lI t l l ig c pro id a s to a ppl ra pp r t o ds to c a l l gi gr a l - o rl d pro l s. fo l l o i gpa g s il ld scri a succ ssfula ppl ica tio o f istri ut d tic l go rit s -a ra pp r t o d -to a r l a rg -sca lpro l i a tur lctio fo r a s st . irst, t g ra lcusto isa tio o f s fo r a tur lctio is o utl i d. Itis fo l l o d a d scriptio o fo ur ta il o ri go f s fo r r l a rg -sca l a tur lctio . p ri ta ls ctio d o stra t s t us ful ss o ft s tup. i a l l, t r sul ts a r su a ris d a d po ssi l t sio s a r sugg st d. tic rit st ct tr s s a pro d to ca pa lo fd a l i g it -co plt pro l s i a rio us l ds. idlck a d l a sk sugg st d t ir a ppl ica tio to a tur lctio fo r cl a ssi ca tio [3]. t r ra pp r (a d il t r) t o ds a a ppl id i a rio us studis [ ]. ut sp cia l l i gfa c d it r d a di gl a rg sca lta sks, s a pp a r to t stc o ic du to t ir i r tpa ra l ll is a d o i a ld a d fo r a ckgro u d k o ldg . . si i p s I t ir ilsto pa p r, idlck a d l a sk us d a i pl to g ra t pro isi gfa tur su s ts. a i o ft irstud a sto r duc t fa tur spa c ca rdi a l it a d a tt sa ti to k p t pr dicti a ccura c a cc pta l ig . O n th e S c a la b ility o f G e n e tic A lg o rith m s 7 9 o a ppl s, t t o r s a rc rs “ co r” t pa tt r s it a i a r a sk t ro ug ic t cl a ssi r p rc i s t sa pls - t fa tur su s t. ’’ i dica t s t pr s c a d a ’’t a s c o fa fa tur . c ro o so is o ta i d tru ca ti g t ro s o ft fa tur a sk. idlck a d l a sk ’ s o d las succ ssful l a do pt d a d r d a o t r r s a rc rs. co o l a ppl id t ss fu ctio fo r a fa tur su s t co sists o f a co i a tio o fcl a ssi r a ccura c a d su s tco pl it : ( )= () ( )− I t is fo r ul a, d o t s a a ccura c sti a t fo r t i sta tia t d cl a ssi r, a d ( ) is a a sur fo r t co pl it o ft fa tur s t- usua l lt u r o futil is d fa tur s. urt r o r , il ds t fa tur spa c ca rdi a l it, a d isa pu is tfa cto rto ig t ul tiplo jcti so ft t ssfu ctio . u r o ffa tur s us d a su s tis it d d to la d t a l go rit to r gio s o fs a l lco pl it. . i is ri u s s tup a s d scri d a o o rks pr tt l lfo r fa tur spa c s o fco pa ra ti l s a l lca rdi a l it. ut i g co fro t d it r l a rg -sca ldo a i s, t a sta ss ss tti fo r fa tur su s ts pro i its t is a ppro a c . u c t a l pl o d istri ut d s fo r a tur tra ctio i o rd r to stud ig r-o rd r pro l s [4] . us d a icro ra i istri ut d to a cc lra t t al ua tio pro c ss, il di gup to l i a r sp dups. U si gsuc co tio a l icro ra i s, a fa tur su s tis ra t d s di gitto a r o t a l ua to r. su s t’ s pr dicti a ccura c is sti a t d t r ru i ga co plt cl a ssi r. r sul ts a r t r tur d to a a st r o d a d us d to guid t . c tl, istri ut d ri l s r sugg st d i o rd r to a cc lra t t al ua tio pro c ss fo r cl a ssi r o pti isa tio s i furt r [5]. us o ft is al ua tio sc fa cil ita t s fo r a stud o f r l a rg sca ldo a i s i a tt r co g itio . is o d lca cia l l a ppl id to a tur lctio . . s u is ro to da ’ s p rsp cti , pa ststudis a ppli g lctio r l i it d i a rio us a s: a jo rit o fr s a rc rsi stiga t d do t a tis l o fa tur s [3,6 – ]. tic l go rit s to a i s o fra t rs a l lco ug unc t a l . ca l lt ir c ntri utin rth r s a rch tin a d l a ssifi ca ti si g tic l grith s , t in fa ctca rr tra cti pr c ss. a tur pl it, a tur uta lcat r 8 0 A . M o s e r a n d M .N . M u r ty tudis o rki go l a rg sca la d r l a rg sca ldo a i sr l id o a ra t r s al lu ro ftra i i gda ta [9 – ],([ 2]). r- tti gt d cist us r quit l ik l. a l ida tio a s usua l l o tr po rt d. o pa ra ti studis o f tic l go rit s a d o t r t o ds r it r as d o s al l -sca lpro l s [ , 3, 4], o r t circu sta c s fo r s do o ta pp a r to cia l[ 2]. ais i su s qu t s ctio s il ld scri a s to stud r l a rg -sca l do a tur lctio ila o idi gt a o l i ita tio s. 3 r s r r c is a ppli g s to do a i s o fl a rg co pl it i a tur lctio , t a jo r c a l l g is t o r li gti co pl it o ft pro c ss. . r s i o ug t ti r quir ts fo r t t ss a l ua tio ca dra a tica l ld cr a s d irtu o f istri ut d ( rtica l ) s, t dura tio o ft a ss ss t pro c ss r a i s t t r a t i gl i ita tio o ft s a rc pro c ss. t us a s to i tgo o d so l utio s i f r c cls a s co pa r d to o t r l ds o fa ppl ica tio . crucia ltra d o ff is to k p t al ua tio pro c ss a s t si a s r quir d - a d a s co cis a s po ssi l. . I p cs s si I o ur stud , s ra l a sur ts r ta k i o rd r to co p up it t s r quir ts. o sto ft s a sur ts a r po rt d fo r ;itis t ir ca r fulco i a tio t a tfa cil ita t s fo r a succ ssfula ppl ica tio s o f s to r l a rg -sca l a tur lctio . mi r rs [ 5]pro id a ffici t a s to fo rc co rg c o f t s a rc pro c ss. is is c ssa r a s t ti r quir ts to a ss ss t i diidua l s pr tt fro co rgi gi a “ a tura l ” a . ft ra a turi gp a s o fa o uto - u dr d g ra tio s,t s lctio pr ssur a s i cr a s d i o ur s tup i o rd r to la d t a l go rit to a - c ssa rill o ca l -o pti u . is a sa c i d r duci gt uta tio ra t a d i cr a si g cro sso r ra t a d i fo r a tio it rc a g t i diidua l s. i d scri s furt r d ta il s o t pa ra t r s tti gs. I o rd r to a c t i itia l isa tio pro c ss, t ra do g ra to r a s a ipul a t d a cco rdi gto a pl icit ii lis i i s. I its usua li sta tia tio ,t i itia lpo pul a tio o ft is o ta i d to ssi ga fa ir co i fo r r i diidua la d r fa tur . I diidua l si r l a rg -sca la ppl ica tio s il lr fl ctt is c o ic i t a tt il l s ttld a ro u d t 5 % l l o fco pl it. al go rit i gfo rc d to co rg fa st o ul d t us a a po o r i itia lco ra g o ft s a rc spa c . O n th e S c a la b ility o f G e n e tic A lg o rith m s U si g a I itia l isa tio ia s, t pro a il it fo r a ’’to o ccur i t po pul a tio is c o s a cco rdi gto ( )= ( )= ( − + o ) 8 1 i itia l o o I t s qua tio s, t I itia l isa tio ia s ra g s fro to - it r li a rl o r o tia l l. o i d o t s t u r o fi diidua l si t po pul a tio a d it u r o fi diidua l ( i o i ); ca a djust d usi gt fr pa ra t r n. s t co pl it o ft i diidua l s ca p ct d to r fl ctt is ia s,t s a rc spa c co ra g a l a rg d t is a sur t. pa rt fro t sta da rd uta tio o p ra to r, s r s m i as a ppl id. ro ppi g t a ssu ptio o f qua l uta tio pro a il itis a d ( d o ti gt pro a il it o f’ i’to c a g d ito ’ j’ ) fa cil ita t s fo r a pl o ra tio o fa ro a d r ra g o ft s a rc spa c . t di g t s to fusua l l a ppl id t ss fu ctio a s itro duc d i t fo r r s ctio , l mi l is m i s r a ppl id: ( )= ( )− ( () ) − () op a s t a t t is a sur t o ul d co ura g t a l go rit to pl o r r gio s o fs a l lco pl it pr fra l. o a o id o sitio a l ia s, i mi l rss r[ 6 ] a s us d: i diidua l s a r co i d to ssi ga ia s d co i fo r r g . sid s t is a sur t, a d a ppi gs r t st d to co ura g t pro ductio o f i diidua l si l o r a d upp r ra g s o fco pl it o ft s a rc spa c . l so sugg st d i [ 6 ] a d id l us d i a tur lctio studis, ra ii g t a pl ig ( ) a s trid i o rd r to a cc lra t t p rfo r a c a ss ss t. id a is to ra do ls lcta su s to ft cl a ssi r tra i i g da ta fo r sti a ti gt cl a ssi ca tio a ccura c o ft su s ts. s du to t r -sa pl i gt t ss a sur c a g s i t co a rs o ft s a rc pro c ss, i diidua l s i rit d fro fo r r g ra tio s a to r - al ua t d. I t l it ra tur , a s r po rt d to sp d up t ss a l ua tio it o ut itro duci gto o uc o f o is ito t s a rc pro c ss p ri ts o al ida t t us ful ss o ft custo isa tio s d scri d a o ,t sts r co duct d,ta ckl i ga r l a rg sca lta ski a tur lctio . ti -co su i g p ri ts r co duct d a tt r a s a rc t r fo r rti cia lI t l l ig c , Ka is rsl a ut r , r a . 8 2 A . M o s e r a n d M .N . M u r ty . up I o ur stud i stiga t d a a tur ur cl a ssi r a s ta il o r d to ca t go ris lctio pro l fo r a a d ritt digits. s st . s la r i gda ta co sist d o f i a r pa tt r s it 24 32 pi l s, diid d ito t cl a ss s o f qua lca rdi a l it. da ta s t a s us d s ra l ti s fo r i t co u it, o str c tl ra ka s a d urt [ 7 ] a d a ra d i[ ]. I o rd r to i stiga t t us ful ss o f s i t is r l a rg -sca ldo a i , t la r i gda ta a s ra do lspl itito a 5 % tra i i gs t,a 3 % t sts t,a d a 2 % al ida tio s ta tt gi i go f a c ru . sa pls r o r a l is d to qua lsi i a pr pro c ssi g st p, usi g a stra ig tfo r a rd sca l i g pro c ss ic pr s r d t sp ct a tio o ft pa tt r s. ssi r o ra t a s to ffa tur s,t t sts t a s ca t go ris d a cco rdi g to t tra i i g s t usi g a - a r st- ig o ur l a ssi r. ra lstudis pro d t is o d lto a ro ust, ffici t a d l l -p rfo r i g a ppro a c fo r -a ppl ica tio s[ 9 ]. o o la ista c usi gt o p ra to rfa cil ita t d fo r a it-pa ra l ll a ppi go ft pa tt r s. o ug co sid ra l ffo rt a d sp t to a cc lra t t cl a ssi ca tio pro c ss,t ti r quir d to a ss ssa si glfa tur su s ta o u t d to o - u dr d- ft s co ds o a U U o rksta tio . o il l ustra t t i pa cts o ft is gur : i t po pul a tio to co sisto fo u dr d i diidua l sa d t fa tur spa c ca rdi a l it to 7 6 ,a si t pro l i stiga t d r,a a usti s a rc usi g a si gl a c i o ul d r quir a o ut a rs! uri g t o pti isa tio pro c ss, t al ida tio s t a s ful l k pta sid . It a s us d to judg t g ra l it o ft r sul ts a ft r t t r i a t d. u to t a str so urc r quir ts,co pa ra ti studis it o t r a tur lctio t o ds r o tco duct d. H o r, t ca r ful p ri ta ls tup it a s pa ra t a l ida tio s tfa cil ita t s fo r a o jcti a sur to ft t o d’ s p rfo r a c . istri ut d rtica l a s ru o t irt U o rksta tio s o f tp - , -2 a d U . Its a da pti l o ad al a ci gpro id d fo r a ffici t us o f t t ro g o us ul ti-us r iro t. o r furt r d ta il s a o utt co c pto f rtica l s t r a d r is r frr d to [5] a d [2 ]. po pul a tio o fo - u dr d i diidua l s a s a ita i d. o s lctt pa r ts o ft g ra tio ,a co i d d t r i istic a d ra do stra t g a s us d: sti diidua l s r co pid dir ctl ito t tg ra tio , a d t r a i i g pl ac s i t a ti g po o l r l ld usi g a fo rtu l a s d o sca ld t ss a l u s. is sca l i g a s do suc t a tt diff r c s i cl a ssi ca tio a ccura c r a pl i d. i i a l t ss a l u a s a dd d to p r itfo r a s lctio o fi diidua l s it l o pr dicti a ccura c . fo l l o ig O n th e S c a la b ility o f G e n e tic A lg o rith m s qua tio s o s o t t ss a l u to t a ra g t ss a d t ( )= ( ) − (2 ¯− t ss 8 3 ( ) o fa i diidua lp is sca ld, gi a i a l t ss i t po pul a tio : )+ t ss if t r is a l su a ris s t o st i po rta t pa ra t r s tti gs i t co a rs o fa t pica lru . d o ts t u r o fg ra tio s pa ss d, a d cro ss ia s i dica t s t pro a il it fo r t i o ia lcro sso rto c o o s a g fro pa r t o fo r c il do . g -5 ( lra ti 5 -75 (gr i g 7 5( a t ri g (c . rg c a ra t r in itn ss uta tin a t cr ss r a t cr ss ia s in itn ss uta tin a t cr ss r a t cr ss ia s in itn ss uta tin a t cr ss r a t cr ss ia s in itn ss uta tin a t cr ss r a t cr ss ia s al .4 2-3% 7 5% % .2 % % 7 5% .2 .5% % 6 % ff ct d ra t s lctin pr ssur f cus n plra tin d ra t c r s int ra ctin l g n int ra ctin incr a s d s lctin pr ssur plra tin a nd c ina tin d ra t g n int ra ctin f cus n a turing incr a s d c r s int ra ctin incr a s d g n int ra ctin ig s lctin pr ssur % nf rc c n rg nc % ig c r s int ra ctin 5% ig g n int ra ctin . na ic p ra t rs su s si r ii pi ra i i g t a pl i g( ) as us d a r s a rc rs i t l d o f a tur lctio to co p up it t o r l i gti r quir ts. is t c iqu a s as l ltrid fo r t purpo s s o ft is stud . tra i i gs ts a r -sa pld a tt % a d3 % l l . i r s o st ol utio o ft st t ss a l u s o t t stda ta , t a tis t da ta us d i t o pti isa tio pro c ss its l f. tra i i gs t a s r -sa pld i 3 % ra tio s. o t t a tdu to t is r -sa pl i gt t ss fu ctio a ris fro g ra tio to g ra tio . sca s i t gur , o pro gr ssi t r so ft p rfo r a c crit rio o t t stda ta is o ta i d. a sur ts a ki g us o ft al ida tio da ta tur d o utto o r disa stro us. o is itro duc d to t s a rc pro c ss s o d to to o a stfo r t is ig -di sio a lda ta s ti co ju ctio it t ig u r o ftra i i gi sta c s. us, as o t furt rl co sid r d i t is stud . 8 4 A . M o s e r a n d M .N . M u r ty 91.5 91 Accuracy 90.5 90 89.5 89 88.5 88 0 20 i. . 40 60 80 100 Generation number 120 stfitn ss a l u s using ra ining 140 160 180 t a pl ing rc p c r U si gt sta da rd o p ra to rs,t as o s r d to tra c i diidua l s a rt 5 % l l o fco pl it pr fra l. is a l ad to t p cul ia ritis o ft pro l do a i ,suc a s ig o - o o to o usit o ft t ss fu ctio , ig i s sitiit o ft l a ssi r to c a g s i fa tur spa c ca rdi a l it,t l i it d a turi gp a s a d s a rp p rfo r a c p a ks i ra g s o fl o co pl it. o gt a sur ts sugg st d i t fo r rs ctio ,a s c ro o us uta tio tur d o utto t o st ff cti a sto ro a d t a i d r gio . i co rpo ra tio o fa pu is tfa cto r ito t t ss fu ctio a l o as o t ca pa lo fguidi g t a l go rit to o t r r gio s o fit r st. us o f a ppi gs i suppl t o fso o ft cro sso r o p ra tio s a s o s r d to la d to pr a tur co rg c : al go rit go tstuck i su o pti a lr gio s it pr dicti a ccura c l o t ful lfa tur s t. c ii urt r p ri ts usi gt ful ltra i i gs tfo r a ccura c sti a tio r co duct d. r sul ts s o t a t s sca l l lto do a i s o fl a rg co pl it i a tur lctio : a s o fd a ic o p ra to rs, t co rg d it i a o utt o - u dr d g ra tio s. u r o futil is d fa tur s co ul d r duc d a o ut5 % ilpr s r i gt pr dicti a ccura c o ft cl a ssi r. i r 2 s o st d l op to ft p rfo r a c o t u s al ida tio pa tt r s: ft r r a c i ga c rta i d gr o f a turit,t ca l cul a t d fa tur su - O n th e S c a la b ility o f G e n e tic A lg o rith m s 8 5 s ts d l i r cl a ssi ca tio a ccura cis co pa ra lto o r tt r t a t ful l fa tur s t. is r a rka lr sul tco ul d r pro duc d r l ia li t co s cuti ru s. 93.2 Calculated feature set 93 Accuracy (%) 92.8 92.6 92.4 Full feature set 92.2 92 91.8 91.6 0 20 40 50 60 80 100 120 Generation number 140 150 160 180 200 i. . a l ida tin p rf r a nc r t k is r po rta i d a ti stiga ti gt us ful ss o f tic l go rit s fo r r l a rg -sca l a tur lctio . ra lpa rtia l l al r a d r po rt d a sur ts r co i d i o rd r to co p it t co pl pro l . o l i ita tio s o f s fo r r l a rg -sca l a tur lctio pro l s r o sr da d a sto o rco t r sugg st d. p ri tss o d t a t s sca l l lto do a i s o fl a rg co pl it i a tur lctio . id l a ppl id a ppro i a tio t o d, ra i i g t a pl i g, a s o s rd to fa ilgro ssli o ur a ppl ica tio . o l l o i gt a sic id a o ft is a ppro a c , o r suita lcusto isa tio s s o ul d trid i futur studis. I st a d o fra do lr -sa pl i gt tra i i gs t,a ro to t p lctio pro c ss a ca rrid o utsi ul ta o usl to a tur lctio . a ra d ia ppl id t is id a succ ssful l to s a l l -sca l a tt r co g itio pro l s [ ]. is co i a tio a pp a rs to pro isi gfo r r l a rg -sca lta sks, a s l l . utur o rk il la i a ta stud o ffurt r a tur lctio t o ds i r l a rg -sca ldo a i s. pro l o f a tur r a tio il l a ddr ss d i o rd r to a c t s a rc pro c ss i furt r. 8 6 A . M o s e r a n d M .N . M u r ty fr c s . 2. 3. 4. 5. 6. 7. . . . . 2. 3. 4. 5. 6. 7. . . 2 . . a s a nd . iu. a tur s lctin f r cl a ssifica tin. I t ll ig t a ta a lsis, 7. . n, . K a i, a nd K. fl g r. Irr l a ntf a tur s a nd t su s ts lctin pr l . rc di gs fth I t r a ti a l fr c a chi a r i g, , 4. . idlckia nd . kl a nsk . n t n g n tic a l g rit s f r l a rg -sca lf a tur s lctin. a tt r c g iti tt rs, :335–347 , . unc , d a n, i, a i ia - un, . l a nd, a nd . n d . urt r r s a rc n f a tur s lctin a nd cl a ssifica tin using g n tic a l g rit s. rc di gs fth 5th I t r a ti a l fr c f tic l grith s, 3. . s r. distri ut d rtica lg n tic a l g rit f r f a tur s lctin. ifth I t r a ti a l fr c c t a lsis a d c g iti s a rch r , . . lt ing r. a tur s lctin g n tic a l g rit s. II rt ri s, 36 , 3. . ra ka s a nd . . urt. a tur s lctin t i pr c cl a ssifica tin a ccura c usinga g n tic a l g rit . r a l fth I dia I stit t f ci c , 7. . K. a in a nd . Z ngk r. a tur s lctin: a l ua tin, a ppl ica tin a nd s a l l sa plp rf r a nc . I ra sa cti s a tt r a lsis a d a chi I t l l ig c , (2), 7. . . it , . . ga rt,a nd I. . ns n. n tic f a tur s lctin f r cl ust ringa nd cl a ssifica tin. rc di gs fth I llq i tic l grith s i I a g rc ssi g& isi ; I igst 4/ 3, 4. . u rra - a l c d a nd . itl . n tic s a rc f r f a tur s lctin: c pa ris n t n a nd I . rc di gs fth si tic lgrith s, . . a nga nd . na a r. a tur su s ts lctin usinga g n tic a l g rit . a t r tra cti str cti a d lcti a ta i i g rs cti , . . . rri, . udil , . a t f, a nd . Kittlr. pa ra ti stud ft c niqu s frl a rg -sca lf a tur r ductin. a tt r c g iti i ra ctic I , 4. I. . I a a nd . a fa i. n prica lc pa ris n t n gl a la nd gr d -l ik s a rc f r f a tur s lctin. rc di gs fth lrida I s a rch si , 4. . I. a ng a nd . . ipp a nn. sing g n tic a l g rit s t i pr pa tt rn cl a ssifica tin p rf r a nc . d a c s i ra lI f r a ti rc ssi g, 3, . . a sl , . . ul l , a nd . . a rtin. n ri fg n tic a l g rit s;pa rt 2: s a rc t pics. i rsit ti g, 5(4):5 –6 , 3. . Z . ril l , . . r n, a nd . . a rtin. a stg n tic s lctin ff a tur s f r n ura ln t rk cl a ssifi rs. I ra sa cti s f ra l t rks, 3(2), 2. . ra ka s a nd . . urt. r ingsu spa c pa tt rn r c gnitin t ds a nd t ir n ura l -n t rk d l s. I ra sa cti s ra l t rks, ( ), 7. . . a ra d i. a tt r rs ta ti a d rt t lcti i lassifi ca ti . a st r sis, pa rt nt f put r cinc a nd ut a tin,India n Institut f cinc , a nga lr , . . lstr , . K istin n, a nd . ja . ura la nd sta tistica lcl a ssifi rs -ta n a nd t ca s studis. I ra sa cti s ra l t rks, ( ), 7. . s r. istri ut d g n tic a l g rit s f r f a tur s lctin. i l a h sis i rsit f a is rsl a tr r a , . ii ti ssi c ti r rit sis cti ist sfr p a ul . o sin pa rt nto f o put r cinc a rdiff ni rsit K P.L.Rosin@cs.cf.ac.uk a nd H nr . o ng sa cho o lo f o puting h ffi l d H al l a ni rsit K H.Nyongesa@shu.ac.uk c . his pa p r pr s nts a n in stiga tin int th cl a ssifica tin f a difficul tda ta s tc nta ining l a rg intra -cl a ss a ria il it utl int rcl a ss a ria il it. ta nda rd cl a ssifi rs a r a k a nd fa ilt a chi sa tisfa ct r r sul ts h r, itis pr p s d tha ta c m ina tin fsuch a k cl a ssifi rs ca n impr ra l lp rf rma nc . h pa p r a l s intr duc s a n l l utina r a ppr a ch t fu rulg n ra tin f r cl a ssifica tin pr lms. I tr cti his pa p r d scrib s a s ris o f p ri nts in ta c l inga difficul tcl a ssi ca tio n pro bl . h da ta co nsists o f a rio us b a ns a nd s ds, a pls o f hich a r sho n in gur . l tho ugh so o f th o b jcts a r l a rg r tha n o th rs ( .g. a lo nds co pa r d to lntil s) a r int r st d in cl a ssif ing th ba s d o n th ir sha p a l o n itho utusinginfo r a tio n a bo utth ir si . his co rr spo nds to th situa tio n h r th dista nc b t n th o b jcts a nd th ca ra is no t d, a nd so th ir a ppa r nti a g d si s o ul d a r . h difficul t o fth ta s l is in th r l a ti l s a l lint r-cl a ss diff r nc in sha p a nd th high intra cl a ss diff r nc s. In o th r o rds,a l lth o b jcts l o o si il a r, a pp a ringro ughl l l iptica l . l tho ughth sha p so fso o b jcts( .g.a lo nds) a r fa irlco nsist nt o th rs a r co nsid ra bl( .g. co rn rn l s). h ba sisfo rcl a ssif ingth o b jcts il l b a s to fsha p pro p rtis a sur d fro th ir sil ho u tt s. inc si info r a tio n is to b disca rd d th pro p rtis n d to b in a ria ntto sca l ing. i is , in a ria nc to po sitio n a nd o rinta tio n S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 8 7 − 9 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 8 8 P .L . R o s in a n d H .O . N y o n g e s a cha ng s is n c ssa r . urth r o r ,it a b us fulto incl ud in a ria nc to a dditio na ltra nsfo r a tio ns o f th sha p . o r insta nc , if th d t r ining sha p fa cto r o fa cl a ss is its si il a rit to a n l l ips th n th a sp ctra tio a b irr l a nt. h co put r isio n l it ra tur pro id s a a rit o fsha p a sur s [ ]. s lctio n o fth s , in co bina tio n ith so n sha p pro p rtis d l op d b o sin [2], ha b n a ppl id to g n ra t a s to f 7 a sur nts o f a ch sa pl. h ca n b diid d into subgro upsa cco rdingto th irpro p rtisa nd/ o r al go rith s: s –fo ur a ttribut s in a ria ntto ro ta tio n,tra nsl a tio n,a nd sca l ing (in a ria ntund r si il a rit tra nsfo r a tio ns). s 2 – thr a ttribut s in a ria ntto ro ta tio n, tra nsl a tio n, sca l ing, a nd s (in a ria ntund r a ffin tra nsfo r a tio ns). s r – fo ur sta nda rd a ttribut s – cc ntricit, circul a rit, co pa ctn ss, a nd co n it (in a ria ntund r si il a rit tra nsfo r a tio ns). r – thr a sur nts o f l l ipticit, r cta ngul a rit, a nd tria ngul a rit (in a ria ntund r a ffin tra nsfo r a tio ns, si il a rit tra nsfo r a tio ns a nd str tchinga l o ngth a s, a nd a ffin tra nsfo r a tio ns r sp cti l). r 2 –thr a l t rna ti a sur nts o f l l ipticit, r cta ngul a rit, a nd tria ngul a rit. i. . a mpls fda ta ; r s c nta in: ( ) a l m nds, (2) chickp a s, (3) c ff a ns, (4) lntil s, (5) p a nuts, (6 ) c rn k rn l s, (7 ) pumpkin s ds, ( ) ra isins, ( ) sunfl r s ds. In this pa p r ha in stiga t d fo ur diff r nt cl a ssi ca tio n t chniqu s, a nd co bin d th in a n a tt ptto i pro o ra l lcl a ssi ca tio n. o i plnta tio ns o fd cisio n tr s a r us d. h rst tho d is th l l no n 4.5 C o m b in in g E v o lu tio n a ry , C o n n e c tio n is t, a n d F u z z y C la s s ific a tio n A lg o rith m s 8 9 a chin la rning tho d d l op d b uinl a n [3]. 4.5induc s tr s tha tpa rtitio n fa tur spa c into quia lnc cl a ss s usinga is-pa ra l llh p rpl a n s ( .g. in 2 this o ul d co nsisto fho rio nta la nd rtica ll in s). h s co nd a ppro a ch is [4] hich is a g n ra l isa tio n in tha t, ra th r tha n ch c ingth a l u o fa singla ttribut a t a chno d ,itt sts a l in a r co bina tio n o fa ttribut s. a tur spa c isco ns qu ntlpa rtitio n d b o bl iqu h p rpl a n s. h third tho d isa n ns blo fn ura ln t o r s tha ta r tra in d o n diff r ntfa tur s pa tt rns, incl udingo n n t o r tha tinco rpo ra t s a l lpa tt rns. ina l l, ha a l so o l d fu ruls using o l utio na r pro gra ing. 2 rif r i f ssi c ti cti ist cisi r ura l n t o r sa r co nn ctio nists st stha ta r id lus d fo rr gr ssio n a nd functio n a ppro i a tio n. h co pris inputno d s, hich a r us d to pro id tra iningda ta to th n t o r a nd o utputno d s us d to r pr s ntth d p nd nt a ria bls. Input- utput a pping is a chi d b a djusting co nn ctio n ights b t n th inputa nd o utputno d s,but o r usua l l,thro ugha n int r dia t l a r o fno d s. his cha ra ct ristic ca n b o di d fo r cl a ssi ca tio n pro bl s b sp cif ing d sira bln t o r o utputs to b bina r a l u s. ura ln t o r s a r rl ia bl us d in cl a ssi ca tio n o fco pl da ta . u cl a ssi ca tio n isa rul-ba s d a ppro a chin hichI - H rulsa r us d to ca t go ris da ta . h ruls r l a t g n ra l is d o r i pr cis gro upings o finput da ta ,a nd th d cisio n o fa gi n rulr pr s nts a d gr o fb l o ngingto a gi n o utput cl a ss. his t p o fcl a ssi r is pa rticul a rl us ful h n itis n c ssa r to pro id int rpr ta bil it to th cl a ssi ca tio n s st in th fo r o fl inguistic ruls. H o r, th pro c ss o fcr a ting cl a ssi ca tio n ruls is o ft n difficul ta nd ti -co nsu ing. ra l studis ha a tt pt d to co p iththis pro bl using la rninga l go rith s,a nd in pa rticul a r th us o fn ura ln t o r s. cisio n tr s a r a l lsta bl ish d t chniqu ba s d o n structuringth da ta , a cco rdingto info r a tio n th o r , into utua l l cl usi crisp r gio ns. l a ssi ca tio n is g n ra l l p rfo r d b sta rting a tth tr ’ s ro o tno d a nd ch c ing th a l u o fa singla ttribut , a nd d p ndingo n its a l u s th a ppro pria t l in is fo l l o d. his pro c ss is r p a t d a tsucc dingno d s untila la fis r a ch d a nd a cl a ssi ca tio n is a ssign d. n ra ting th o pti a ld cisio n tr is g n ral l -ha rd, a nd th r fo r a sub-o pti a ltr is induc d inst a d, usinggr d o r ra ndo is d hil lcl i bing a l go rith s fo r insta nc . h r sul ting tr is o ft n prun d; subtr s a r r pl a c d b la fno d s ifthis r duc s th p ct d rro r ra t s. his r sul ts in po t ntia l ls al lr a nd o r a ccura t tr s. 3 ti r ssi c ti ol utio na r t chniqu s ha no tb n pr io usla ppl id to fu cl a ssi ca tio n pro bl s. h cl o s str l a t d o r studis us g n tic a l go rith s to o pti is 9 0 P .L . R o s in a n d H .O . N y o n g e s a fu cl a ssi ca tio n rulsa nd th ir b rship functio ns. n disa d a nta g ith this a ppro a chis tha titis in a ria bln c ssa r to pr -sp cif th structur o fth ruls, hich o ft n r sul ts in sub-o pti a lcl a ssi ca tio n. In this pa p r, ha pro po s d a n t chniqu in hich fu cl a ssi ca tio n ruls o fa rbitra r si a nd structur ca n b g n ra t d using g n tic pro gra ing. his is d sira bl fo r co pl pro bl s, ith l a rg nu b rs o finputfa tur s, fo r hich itis no t fa sibl to fo r ul a t th structur o fruls a nua l l. urth r o r , ith such l a rg nu b rs o f fa tur s it is usua l l th ca s tha t c rta in fa tur s a r no t signi ca nt in cl a ssi ca tio n o fdiff r nt o utput cl a ss s. H nc , in this ca s , ca n sa tha tg n tic pro gra ingis us d fo r unco nstra in d ruldisco r a nd o pti isa tio n. n tic pro gra ingis a n o l utio na r t chniqu in principlto H o l l a nd’ s g n tic a l go rith s. h a in diff r nc s a r ,( ) th structur o fa g n tic pro gra is a tr , (2) th no d s o fth tr s a r functio ns (o r t r ina l s), hich na bls th tr s to int rpr t d a s pro gra s a nd (3) th si o f a ch tr in a po pul a tio n is a ria bl, unl i o stg n tic a l go rith s h r a l lindiidua l s ar th sa si . th r is , sta nda rd o p ra to rs a ppl ica bl to g n tic a l go rith s a r us d in g n tic pro gra ing. In this stud , th no r a l is d input spa c a s pa rtio n d into thr fu b rship functio ns, r a nd s [5]. t , Z , b th fu b rship functio ns. a ssu d si pl rul co nstructs co pris d o f t o inputs a nd o n o utput. h no n-t r ina lno d s o fth tr s r pr s ntth s si plruls, hich a r co bin d to fo r th co pl cl a ssi ca tio n rul. a ch si pl rul is a l ua t d b a tching its inputs a ga instth fu a nt c d nts a nd th o utputis o bta in d usinga n o p ra to r,na l I . hus,a no d pr ss d a s Z ( , ) is int rpr t d a s: I ( ) H I ( ) hr is th d gr o fb l o ngingo f to th fu b rship functio n Z . his t p o frulco nstructis pr fra blto dir ctco bina tio n o fth inputpa ra t rs b ca us ita ssists in int rpr ta bil it o fth cl a ssi ca tio n s st . h r a r nin diff r ntfu ruls hich ca n b fo r d fro th co bina tio n o fth thr b rship functio ns. h stud us d so urc co d (l il gp) d l o p d a t ichiga n ta t nirsit [6 ]. his a s us d to o l a s to fco pl fu cl a ssi ca tio n ruls, o n fo r a ch o utputcl a ss o fda ta . h functio n s tis co pris d o fth si pl ruls d scrib d a bo , hilth t r ina ls tco pris d ra ndo co nsta nts a nd co o n a rith tic o p ra to rs. h tn ss o fth tr s o n th ir a l ua t d o utputs r d t r in d a ga instta rg ts o f . fo r th co rr ctda ta cl a ss a nd . o th r is . s a n a pl, th fo l l o ingis a s a l lpo rtio n o fa co pl rul: (ZP (NP a b) (ZN d c) ) Itca n b int rpr t d a s: I d is Z I a is c is b is H H t p = t p2 = I ( I ( ) ) C o m b in in g E v o lu tio n a ry , C o n n e c tio n is t, a n d F u z z y C la s s ific a tio n A lg o rith m s 9 1 I t p is Z t p2 is H o ut= I ( ) t p t p2 ti c s ca us n ura ln t o r s a r tra in d o n l i it d sa pl s ts o f r pr s nta ti da ta th r a r a la s signi ca nt rro rs in th g n ra l isa tio n o fco pl functio ns. n a to o rco this pro bl is to tra in ul tipln ura ln t o r s o n ind p nd ntda ta s ts a nd th n us o tingsch s to d t r in a n o ra l lcl a ssi ca tio n [7 ]. h t o po pul a r sch s a r co o nl no n a s ns bla nd o dul a r n t o r s. In ns bls o r co itt s r dunda ntn t o r s a r tra in d to ca rr o utth sa ta s , ith o tingsch s b inga ppl id to d t r in a n o ra l lcl a ssi ca tio n. n th o th r ha nd, itis po intlss to tra in id ntica ln ura l n t o r sa nd co nsid ra tio n is thus o ft n gi n to usingdiff r ntto po l o gis,da ta s ts o r tra ining a l go rith s. In o dul a r n t o r s th cl a ssi ca tio n pro bl is d co po s d into subta s s. o r a pl,n ura ln t o r s a r tra in d to r spo nd to o n cl a ss o r a gro up o fcl a ss s. h o utputo fa cl a ssi ca tio n n t o r ca n b us d to indica t th d gr to hichth inputfa tur s a r a tch d to th diff r ntcl a ss s. h r fo r a si pl a ppro a chto n t o r co bina tio n isto su th a ctia tio n l l so fco rr spo nding o utputno d s. r n ntto this sch is to sca lth o utputl l s ithin a n t o r suchtha tth su to o n . hisa l l o sth co ntributio nsa cro ssn t o r s to b o r co pa ra bl. n al t rna ti a ppro a chis ba s d o n th co nfusio n a tri,a ta blco nta ining ntris , hich indica t th fr qu nc tha t da ta sa pls fro cl a ss r l a b ld a s cl a ss . uch a ta bl is us fulfo r a na lsing th p rfo r a nc o f a cl a ssi r. ur a ppro a ch is ba s d o n th cl a ssi ca tio n a ccura cis fo r a ch cl a ss gi n b h co ntributio ns o f a ch cl a ssi r a r ight d b th s p ct d a ccura cis. n a plo fa co nfusio n a tri is sho n in a bl . Itca n b s n tha tth cl a ssi r is ca pa blo fco nsist ntl co rr ctl cl a ssif ing a l linsta nc s o fcl a ss , buto nl 7 5% o fcl a ss2. p ri ts h p ri nts co pa r d th p rfo r a nc o ffo ur cl a ssi rs o n th sha p cl a ssi ca tio n pro bl . h tra ining a nd t sting da ta bo th co nsisto f 3 sa pls a ch co nta ining s nt n co ntinuo us a ttribut s o f sha p . o l l o ing sta nda rd pra ctic th da ta a s no r a l is d prio r to pr s nta tio n to th n ura la nd fu cl a ssi rs, h r a s this a s no tn c ssa r fo r th d cisio n tr s. h rsts to f p ri nts co nc ntra t d o n a l ua ting a chindiidua lcl a ssi r. a plso fth iro utputs a r sho n in igur s 2 to 4. h irp rfo r a nc s 9 2 P .L . R o s in a n d H .O . N y o n g e s a 2 3 4 56 7 2 3 4 5 6 7 % ccura c . 3 2 5 3 2 23 3 3 2 7 5. 3.3 7 .6 . 32 25. 6 4.2 2 33 7 .7 3 7 6 .5 . nfusin ma tri a r sho n in a bl2. h bo tto thr ntris co rr spo nd to a l t rna ti a ns to d riinga cl a ssi ca tio n ba s d o n n ura lo utputno d a ctia tio n l l s. h s a r sta nda rd inn r-ta -a l l( ),a nd usingth t o d cisio n tr s ( a nd 4.5). Itis o fint r stto no t tha tth si pl inn r-ta -a l lp rfo r s b tt r tha n th o r co pl d cisio n tr s in co bining th o utputs o fth n ura l n t o r s. h s co nd s to f p ri nts co pa r d a rio us o ting sch s a ppl id to n ura ln t o r ns bls. h n ura ln t o r s r tra in d o n th subgro ups o fsha p pro p rtis,a nd a n a dditio na ln ura ln t o r a s tra in d o n th co bin d s nt n pro p rtis. h o tingsch s ith r s lcto nl th a iu a ctia tio n l l ithin a n t o r ( inn r-ta -a l l ),o r a l t rna ti la l lth a ctia tio ns l l s a r us d. h l l s a rstb sca ld, a nd ight d b th co nfusio n a tri,a nd a r th n su d o r th n t o r s. h r sul tin a bl3 sho s tha tb tt r p rfo r a nc is a chi d h n th n t o r ns blincl ud s th n t o r tra in d ith a l lpro p rtis. urth r o r , th us o fth co nfusio n a tri g n ra l l i pro s p rfo r a nc . H o r, th o dul a r a ppro a ch s pro id l ittli pro nto r th ba sic singln t o r cl a si ca tio n r sul t. h na ls t o f p ri nts in stiga t d h bridisa tio n o fthr cl a ssi rs, na l,n ura ln t o r ,fu a nd 4.5,usingth co nfusio n a tri tho d. his do s no t r l o n po t ntia l l inco nsura t l i l iho o d a l u s tha t a b pro duc d b diff r ntcl a ssi rs. s fro a bl4tha tfurth r ga ins in p rfo r a nc ha b n a chi d,indica tingtha tth n ura l n t o r a nd d cisio n tr pro id us fulco pl i nta r info r a tio n. 6 c si s his pa p r ha s pr s nt d a co pa riso n o fdiff r ntcl a ssi ca tio n t chniqu s o n a difficul tsha p a na lsis pro bl . i il a r to o th r r po rts in th l it ra tur [ ,9 ] itha s b n sho n th r is no signi ca ntdiff r nc s b t n th indiidua lt chniqu so n o urcl a ssi ca tio n pro bl .H o r, ha sho n tha ti pro nts ca n b a chi d thro ugh diff r ntco bina tio ns o fth s t chniqu s [ ]. C o m b in in g E v o lu tio n a ry , C o n n e c tio n is t, a n d F u z z y C la s s ific a tio n A lg o rith m s 1 T I 1 9 3 1 1 1 1 1 11 1 1 7 3 7 7 3 3 7 2 73 3 252 3 2 32 5 2 5 25 2 3 5 22 5 2 2 5 5 2 3 22 8 3 4 55 3 5 38 5 4 5 6 5 26 5 6 88 55 4 5 4 68 48 6 6 6 6 86 8 6 684 4444 9 8 8 4 8 8 68 6 7 2 2 9 9 5 4 9 6 9 99 6 6 99 6 8 9 89 9 89 E F i. . n a mpl fth pa rtitining ff a tur spa c f r th a n cl a ssifica tin ta sk th d cisin tr using nl t pr p rtis. h h ri nta la is pr p rt is l l ipticit, a nd th rtica la is pr p rt is tria ngul a rit. H % 4.5 u + + + 4.5 53 54 44 57 52 5 . ccura c f r singlcl a ssifi rs 9 4 P .L . R o s in a n d H .O . N y o n g e s a Rule 4: property 16 <= 0.225411 -> class 6 Rule 7: property property property property property property property property property -> class 8 1 > 63.7441 1 <= 65.4694 4 > 0.161205 4 <= 0.179735 11 <= 0.967232 13 > 0.710997 15 <= 0.061688 16 > 0.225411 16 <= 0.329981 Rule 8: property 1 > 65.4694 property 4 <= 0.179735 property 15 <= 0.061688 -> class 6 i. . t pica ls t fruls g n ra t d 4.5 (ZP (NP b (PN (- (PN a 0.83860) (NP (ZP (- (- (ZN d c) (NZ -0.02504 d)) (PZ a f)) h) (PZ e h))) g)) (ZP a (ZP (- (ZN d c) (NP d (ZN d c))) (ZP (NP f b) (ZN d c))))) i. . p rtin fa c mpl fu rul in pr fi n ta tin. pp r ca s ltt rs d n t fu ruls, a nd l r ca s ltt rs d n t inputsha p pr p rt a l u s. C o m b in in g E v o lu tio n a ry , C o n n e c tio n is t, a n d F u z z y C la s s ific a tio n A lg o rith m s % u gr ups u gr ups + r thing 3 46 4 46 55 5 44 55 4 56 42 4 43 5 5 56 5 5 H um um um um , um fa ctia tins , um fa ctia tins ith sca l ing , um f , um fa ctia tins, ight d , um fa ctia tins ith sca l ing, ight d fa ctia tins fa ctia tins, ith sca l ing fa ctia tins, ight d fa ctia tins, ith sca l ing, ight d . ccura c f r 9 5 tingsch m s H % + + 4.5 u u + 4.5 + 4.5+ .H rid 63 5 57 6 l a ssifi rs n a ppro a ch to g n ra t fu cl a si ca tio n ruls usingg n tic pro gra ing,o n o th r ha nd,ha s no td o nstra t d sa tisfa cto r r sul t. his a b du to th l i it d pa rtitio ning o fth fu input spa c into o nl thr b rship functio ns i pl nt d in th stud a nd furth r o r il lin stiga t th po t ntia lo fthis tho d. fr c s . 2. 3. 4. 5. 6. 7. . nka , . H l a a c, a nd . l. I g r ssi g lsis hi isi . ha pma n a nd H a l l , 3. . . sin. a suringsha p : l l ipticit,r cta ngul a rit,a nd tria ngul a rit. s b itt f r bli ti . . . uinl a n. 4.5 rgr s f r hi r i g. rga n Ka ufma nn, 3. .K. urth , . Ka sif, a nd . a l rg. st m f r inductin f l iqu d cisin tr s. r l f rtifi i lI t llig s rh, 2: –33, 4. . . ink ns a nd H . . ng sa . n tic a l g rithms f r fu c ntr l- pa rti. I r trl h r l., 42: 6 – 7 6 , 5. . Z ngk r a nd . unch. htt / / isl . s. s . / / s ft r/ lil-g. 6. rl t rs. pring r- rl a g, . . . ha rk , dit r. bi i g rtifi i l . 9 6 P .L . R o s in a n d H .O . N y o n g e s a . H . h n, . untin, . h , . utja hj, . mm r,a nd . l. p rtpr dictin,s m l ic la rning,a nd n ura ln t rks: n p rim nt n gr h und ra cing. I rt, :2 –27 , 4. . . ul hl l a nd, . . H i rt, . . H a dda d,a nd . a rsl . c mpa ris n fcl a ssifica tin in a rtificia lint l l ig nc ,inductin rsus n ura ln t rks. h tri s I t llig t b rt r st s, 3 : 7 – 2 , 5. . .Xu, . Kr a k, a nd . . u n. th ds f c m ining mul tipl cl a ssifi rs a nd th ir a ppl ica tin t ha nd riting r c gnitin. I r s. st s b r ti s, 22:4 –435, 2. Experimental Determination of Drosophila Embryonic Coordinates by Genetic Algorithms the Simplex Method and Their Hybrid Alexander V Spirov Dmitry L Timakin John Reinitz and David Kosman The Sechenov Institute of Evolutionary Physiology and Biochemistryflflfl Thorez AveflflStflPetersburgflflflflflflflflRussiafl Deptflof Automation and Control SystemsflPolytechnic Universityflflfl Polytechnic StflStflPetersburgflflflflflflflflRussia Deptflof Biochemistry and Molecular BiologyflBox flflflfl MtflSinai Medical Schoolfl One Gustave LflLevy PlaceflNew YorkflNY flflflflfl USA Abstract Modern largeflscale flfunctional genomicsfl projects are inconfl ceivable without the automated processing and computerflaided analysis of imagesfl The project we are engaged in is aimed at the construction of heuristic models of segment determination in the fruit fly s s flThe current emphasis in our work is the automated transfl formation of gene expression data in confocally scanned images into an electronic database of expressionfl We have developed and tested profl grams which use genetic algorithms for the elastic deformation of such imagesfl In additionfl genetic algorithms and the simplex methodfl both separately and in concertfl were used for experimental determination of s embryonic curvilinear coordinatesflComparative tests demonfl strate that the hybrid approach performs bestflThe intrinsic curvilinear coordinates of the embryo found by our optimization procedures appear to be well approximated by lines of isoconcentration of a known morfl phogenflBicoidfl Introduction Computer Aided Analysis of Biological Images The ongoing revolution in molecular genetics has progressed from the large scale automated characterization of genomic sequence to the characterization of the biological function of the genome These investigations mark the beginning of the era of functional genomics A key feature of genomic scale approaches is the automated treatment of large amounts of data Both current and future work in the eld is impossible without the automated processing and computer aided analysis of images in connection with updating interactive electronic image databases A key aspect of such processing involves the segmentation of individual im ages and the registration of serial images Many problems involving the recog nition classi cation segmentation and registration of images can be formulated as optimization problems These optimization problems are typically di cult S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 9 7 − 1 0 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 9 8 A .V . S p ir o v e t a l. involving multiple minima and a complex search space topologyflContemporary approaches based on evolutionary computations are a promising avenue for the solution of such problems flEvoIASPflflflfl Here we describe a new method for the determination of intrinsic biological coordinates in embryos of the fruit fly Drosophila melanogaster by means of genetic algorithms flGAsflflGAsflthe simplex methodfland a hybrid of both were applied to the problemfland we flnd that hybrid methods perform the bestflOur results indicate that these coordinates may be determined by a morphogenetic gradient of the protein Bicoidfla result of some biological interestfl Stripe Straightening Search of Intrinsic Coordinates of Early Embryo Early in the development the fruit fly embryo is shaped roughly like a hollow prolate ellipsoidflcomposed of a shell of nuclei which are not separated by cell membranesfl Deviations from the elipsoidal shape reveal the future polarity of the animalfls bodyflThe more pointed end on the long axis makes anterior flheadfl structuresfland the rounder end posterior fltailfl structuresflFrom a lateral flsidefl perspectiveflone long edge of the embryo is flat and will will make dorsal flflbackflfl structuresflwhile the other long edge is rounded and makes ventral flflundersideflfl structuresflIn this paper we follow the standard biological convention and show embryos with anterior to the left and flif a lateral viewfl dorsal up flFigflflflfl Figfl fl shows that so called pairflrule stripes flearly markers of the future segmental pattern flflflfl are not parallel and straightfl but have a crescentfllike formflThe curvature of the stripes is highest at the terminifland minimal at the central partflEach stripe specifles an anteriorflposterior flAflPfllocationfland these stripes can be regarded as contours in an intrinsic coordinate system that is being created by the embryo itselffl Another set of embryonic determinants exists for the dorsoflventral flDflVfl axisflIf the image is smoothly transformed such that the curvilinear coordinates are plotted orthogonallyflthe stripes appear straightflso the determination of these coordinates can be viewed as a flstripe straighteningfl procedureflOur task is to understand and characterize this curvilinear coordinate system as it relates to the AflP axisfl Two coupled objectives of this study arefl flfl To characterize the intrinsic embryonic curvilinear coordinatesfl flfl To use carefully characterized and tested computational procedures for the purpose of automatically processing large numbers of imagesfl Methods and Approaches The work reported here is part of a large scale project to construct a model of segment determination in the fruit fly D melanogaster based on coarseflgrained chemical kinetic equations flflflfl The acquisition and mapping of gene expression data at a heretofore unprecedented level of precision is an integral part of this E x p e rim e n ta l D e te rm in a tio n o f D ro s o p h ila E m b ry o n ic C o o rd in a te s 9 9 Fig Image of early flblastoderm stagefl fly embryo with crescentfllike stripesfl in Cartesian physical coordinatesfl This is a confocally scanned image of an embryo stained by indirect fluorescence flimmunostaining with polyclonal antisera against the segmentation proteinflflEach small dot is an individual nucleusfl projectflThe current emphasis in our work is on the automated transformation of gene expression data in confocally scanned images into an electronic database of expressionfl Images of Drosophila Genes Expression Transformations of embryonic coordinates begin with data expressed in terms of the average fluorescence level flproportional to gene expression levelfl of each nucleusflwhere segmentation proteins exert their biological functionflThis data was obtained as followsfl Antibodies for flfl protein products of segmentation genes were raised and over flflfl images were prepared and scanned flflflfl These images were computationally treated by means of the Khoros package flflflfl Embryos were rotated and cropped automatically such that the physical long axis of the embryo was parallel with the x axis and the short physical axis with the y axisfl Nextfl the images were segmented flflfl Kosman et alfl in preparationflfl About flflflfl segmented and identifled nuclei are obtained from each imageflEach nucleus is labeled numericallyfl and the x and y coordinates of its centroid are foundfl together with the average fluorescence level over that nucleusfl The segmented 1 0 0 A .V . S p ir o v e t a l. data takes the form of tables in ASCII text formatflThe result is the conversion of an image to a set of numerical data which is then suitable for further processingfl Stripe straightening algorithm In Figflfl the crescentfllike pairflrule stripes of an embryo in near saggital projecfl tion are shownflWe assume that the center of a pairflrule stripe follows a curve of constant AflP positionfl The origin of the image coordinate system is at the top leftflwith image coordinates for width w increasing to the right and height h increasing downfl Our goal here is to flnd the true AflP and DflV coordinates on the imageflWe approximate the true coordinate system by a Taylor series as followsflWe denote the true AflP coordinate by x fl and the true DflV coordinate by yfl fl We pick the origin of flfl x yfl fl and the origin of new image coordinates flx yfl so that they are the same and as close as possiblefl We note that there is an AflP position at which a stripe is exactly vertical on its whole lengthflThe center of that stripe deflnes x fl fl flflwhich is the yfl fl axisfl Each pairflrule stripe other than the one at xfl fl fl is curvedfland we imagine the x fl axis to intersect each of the stripes at the point where it is exactly verticalfl Now we pick new image coordinates x and y such that they have the same origin and orientation as the flfl x yfl fl coordinatesflthat is x fl w − w y fl −h − h flflfl We now turn our attention to xflFor fl nowflwe can assume that yfl fl yflEven if we donflt do thatfl two important things will be true about the relationship between flx yfl and flfl x yflflflflfl fl The y and yfl axes are coextensivefland flflfl The loci yfl fl const are orthogonal to the y and yfl axes as they cross y fl yfl fl flflBoth of these important points follow from the existence of the vertical stripeflWe would like to write x fl in terms of x and yflso that x fl fl f flx yfl flflfl We expand in a Taylor series to third order around the originflThat gives x fl fl f flfl flfl fl x fl fl flfl flfl x fl flfl flfl xy fl flfl flfl fl flfl flfl x fl flfl flfl x y fl flfl flfl fl flfl flfl y y x xy flflfl Now consider the terms and what they meanflf flfl flfl fl fl by deflnitionflWe picked flfl x yfl fl such that at the origin fl fl and fl flflFor pure y terms we can say more than thatflThe fact that the y and the yfl axes are coextensive means fl fl fl as wellfl Thus far we have shown that f flfl yfl fl fl yfl and so that flve of the ten terms of the Taylor expansion vanishfl E x p e rim e n ta l D e te rm in a tio n o f D ro s o p h ila E m b ry o n ic C o o rd in a te s The unit vector e in the x fl direction is proportional to flso 1 0 1 measures the change in length of e as we move along the x axisflThis means that fl flfl flThis term can be thought of as the rate of change in size of Now consider the unit vector e fl along the yflaxisflAlong the yflaxis where x fl x fl fl flfl fl fl yflso that derivatives of this quantity with respect to y vanishfland hence this term of the series vanishesflThis has eliminated all but three terms from the seriesflso now we write the flrst order model of image transformation as x fl fl x fl Axy fl Bx y fl Cx flflfl All of these terms have a clear interpretationfl The xy term is the main onefl it gives quadratic DflV curvature that increases with distance from the xflaxisfl The x y term gives residual DflV asymmetry and the x term gives residual AflP asymmetryfl Lastlyfl if one expresses the above equation in terms of w and hfl expansion will bring back lower order terms in h and w when expanding x fl fl w − w fl Aflw − w flfl−h − h fl fl Bflw − w fl fl−h − h flfl Cflw − w fl flflfl in terms of w and hfl We tested this flflst order model and found that in more then half of cases it is insufl cient for straightening stripesfl We expanded the model empiricallyfl with the result that an empirical extension of the flflst order model is given by x fl fl Aflw − w flfl−h − h fl fl Bflw − w fl fl−h − h flfl Cflw − w fl fl−h − h fl fl Dflw − w flfl−h − h fl flflfl We can treat of these additional fourth order members as followsflCx y is a correction term for parabolic splayflwhile Dxy serves to correct DflV asymmetryfl In generalflthe situation is typical of a polynomial approximation problemfl there is one polynomial that is best but there are a number of distinct ones that can approximate it very wellfl Preliminary calculations have shown that the best outcome is achieved with an independent deformation of the anterior and posterior half of an embryoflIn summary it requires the determination of fl parameters of a deformation plus an evaluation of values w flh flh fl Genetic Algorithms Technique Simplex Method and Their Hybrid GA Search The optimization problem of flnding the coefl cient values for proper elastic transformations was initially implemented with GAsfl We have reduced the problem to the determination of factors AflBflC and D of equation flfl We use the following cost functionflEach embryofls image under consideration was subdivided into a series of longitudinal stripsflThen each strip is subdivided into boxes and the mean value of the product flEVEN SKIPPED proteinfl is 1 0 2 A .V . S p ir o v e t a l. calculated for each boxflEach row of means gives the local proflle of even skipped gene expression along each stripflThe cost function is computed by comparing each proflle and summing the squares of diflerences between the stripsflThe task of the GA is to minimize this cost functionfl Following the classical GA algorithmflthe program generates a population of floatingflpoint chromosomesfl Initial chromosomes are randomly generatedfl Affl ter that the program evaluates every chromosome as described abovefl thenfl according to the truncation strategyflthe average score is calculatedflCopies of chromosomes with above average scores replace all chromosomes with a score less than averagefl On the next step a predetermined proportion of the chromosome population undergoes mutationflso that one of the coefl cients gets a small incrementflThis cycle is repeatedflall chromosomes are consecutively evaluatedflthe average score is calculated and the winnersfloflspring substitutes for the losers in the process of reproductionfluntil an accepted level of stripe straightening is achievedfl Simplex Search We also solved the optimization problem by the downhill simplex method in multidimensions of Nelder and Mead flflflflThe method requires only function evaluationflnot derivativesflThis is an important speed advantage over gradient methodsfl since calculation of the gradients requires many more evaluations of the cost functionfl A simplex is the geometric flgure in N dimensions of N fl fl vertices and all their interconnecting line segmentsflThe NelderflMead method starts with such a set of N fl fl points deflning an initial simplexflThe downhill simplex method operates by moving the point of the simplex where the function is largest through the opposite face of the simplex to a lower pointfland so on until it reaches the vicinity of an extremumfl Hybrid Procedure Initial experience indicated that that the simplex method is fast but does not give high quality answersflwhile GAs give excellent answers but are slowflWe noted that both multiple simplex runs and GA search perform numerous evaluations for many random points in search spaceflIf we use small increments as mutations we will perform practically the same search by using GAs or the simplex methodfl If sofl we could use a set of chromosomes from the GA technique as a starting simplex for NelderflMead optimizationfl In the hybrid algorithmflwe implement a simple evolution strategy with floatingflpoint chromosomes with small mutational incrementsflSelection and reproduction are performed as described aboveflIn additionflfrom the very beginning the program links pointers to mutant oflspring so as to achieve complete lists of N fl fl pointers on N fl fl relativesfl These flclans of mutantsfl are ready for simplex procedurefl Following the completion of the flrst list of N fl fl pointers the program starts to perform not only mutationfl selectionflreproduction proceduresflbut also the simplex procefl dure for the lowest scoring members of flcompletefl clansflThe more clans achieve completion the more species undergo simplex procedureflIn summaryflGAs must E x p e rim e n ta l D e te rm in a tio n o f D ro s o p h ila E m b ry o n ic C o o rd in a te s 1 0 3 provide search of global optima together with local onesflwhile simplex provides fast downhill movingfl Results and Discussion Search Space Features for Stripe Straightening Problem The above described task of image elastic deformations turned out to be a diffl flcult numerical problemflThis is caused flrst of all by the unusual geometry of search spacefl Figfl fl gives a picture of its features through crossections of the search space for one typical embryo under considerationfl Fig Search space features for one of crossections flA fl Dfl for typical imagefl This is surface plot where vertical flZfl axis is evaluation oneflwhile X and Y are A and D coefl cients of expression flfl As we can see this cross section includes two groovesflone of which is deeper than anotherflThe sharp rectangular walls in Figflfl are caused by penalty confl ditionsflThe omission of the penalties gives a smoother surface with one groovefl which corresponds to the deeper groove flnot shownflflIn turnflpenalties are abfl solutely needed to avoid highly nonlinear folding of an image instead of smooth deformationsfl The bottom of both grooves have several local minimaflAs a result the simplex search gives in serial searches tens of such local extremafl GA search is more 1 0 4 A .V . S p ir o v e t a l. eflective and it flnds the best local extremum on the bottom of the deepest grooveflHowever to jump from the shallow to the deeper groove is still a difl cult task for GA search as wellflTo overcome this we use large population sizes or a series of runs to achieve the best solutionfl The Results of Stripe Straightening After completion of the stripe straightening procedure with flfl coefl cients flw and two sets of Afl Bfl Cfl D and h fl h fl for about two hundred images from the stages when all seven crescent stripes are visible we could compare found coefl cient setsfl These coefl cient sets show considerable diversityfl so that we fail to elucidate a general formula of appropriate elastic deformations to achieve satisfactory stripe straighteningflHoweverflthe resulting transformation of coorfl dinates are very similar for most of the imagesfl Typical example is shown in Figflflfl Fig Typical example of curvilinear coordinates found by our optimization procefl duresfl On the contraryflcomparison of coordinate curves for anterior and posterior halves of embryos reveal small but quite evident diflerences flCfflFigflfl and Figflflflfl A biological subject of interest is the source of the pairflrule stripesflcurvaturefl It is known that in Drosophila segmentation the maternally expressed protein BICOID forms an anteriorflposterior morphogenetic gradient in the egg which controls all following segmentation events flflflfl It is interesting that contour lines of a flDflconcentration map of the BICOID gradient closely coincide with curvifl linear coordinates determined by our methodflThe full biological implications of this observation will be reported elsewherefl E x p e rim e n ta l D e te rm in a tio n o f D ro s o p h ila E m b ry o n ic C o o rd in a te s 1 0 5 E ectivity and Cost of GAs Simplex and Hybrid Techniques In a table fl the results of the comparative tests on elastic transformation of a typical image by means of the simplexflmethodflGAsfland GAs with simplex flthe hybrid techniquefl are presentedflTo flnd parameters of optimization for a simplex and GAs giving the most eflective optimizationfl careful tuning of the limits in variation of a mutational increment and the range of variability of the initial population was carried outfl The hybrid method was tested at the same values of parametersflas GA techniqueflThe result of testing was compared according to the time required for calculation and according to the standard deviation of the resultfl On each tested procedure flflfl independent runs were carried outfl Inspection of the table reveals that the simplexflmethod is fastestfl but also the least preciseflThe GA technique is the most preciseflbut also requires flfl times as much computingflOur results indicate that the hybrid technique is approximately twice faster than GAs at the same accuracyfl Table Comparison of eflectiveness of three approaches on calculation time flin a summarized amount of evaluationsfl and on a divergence flin a standard deviation valfl uesfl Method Simplex Method GA Technique Hybrid Technique Time in evaluations Standard Deviation flflflflfl flflflflflflflfl flflflflflfl flflflflflflfl flflflflflfl flflflflflflfl The problems we encountered with our optimization task arefl flfl An abundance of local minima very close to the global minimumflThe simfl plex can get stuck in very smallflscale holesfl even with starting conditions exceptionally close to a known solutionflWe need to allow the optimizer to move from a position that is very close to the global minimum towards the global minimumfl It seems that this last stage is the difl cult part for the NealderflMead simplex methodfl It is possible to produce good scores with a number of distinct end points in polynomial parameter spaceflsuggesting that the problem is probably overflspecifledfl flfl This is the polynomial approximation problemflThere will be one polynomial that is the best for the stripe straightening but there are a number of distinct ones that can approximate it very wellfl As to the flrst itemfl the successful approach to a solution of this problem is to employ hybrid techniquesfl Genetic algorithms alone are usually slow in optimization problemsfl since they are too coarseflgrained to obtain a solution quicklyfl On the other handfl downhill algorithms are usually fast flin terms of processor cyclesflflif they are close to the solutionflbut tend to get stuck in local minimaflCombining both kind of algorithms manages to avoid local minimafland flnds solutions accuratelyfl 1 0 6 A .V . S p ir o v e t a l. Conclusions In the task of optimization of parameters of elastic transformation a simplexfl method is fastestflbut also the least preciseflgiving the greatest divergenceflThe GA technique is the most preciseflbut also requires at least flfl times more timefl The hybrid technique is twice faster than GAs at the same accuracyfl The intrinsic curvilinear coordinates of an embryo found by our procedures of optimization appears to be approximated by contour lines of a map of a gradient of the morphogen bicoidfl It is in the good agreement with known ideas about a governing role of this gradient in consequent processes of segmentation of an early fly embryofl Acknowledgements This work is supported by INTASflgrant No flflflflflflflflflRussian Foundation for Basic Researchesfl grant No flflflflflflflflflflflfl USA National Institutes of Healthfl grant ROflflRRflflflflflfland CRDFflgrant No RBflflflflflflAflSflwishes to thank Timfl othy Bowler for stimulating discussions and KingflWai Chu for help with profl grammingfl References flfl AkamflMflfl The molecular basis for metameric pattern in the Drosophila embryofl Development flflflflflfl flflflflfl flfl KosmanflDfland ReinitzflJflflRapid preparation of a panel of polyclonal antibodies to Drosophila segmentation proteinsflDevelopmentflGenesfland Evolution flflflflflfl flflflflflflflfl flfl KosmanflDflflReinitzflJfland Sharp DflHflflflflflflflAutomated assay of gene expression at cellular resolutionfl In AltmanflRflflDunkerflKflflHunterflLfland KleinflTfleditorsflProfl ceedings of the flflflfl Paciflc Symposium on Biocomputingfl pages flflflflfl Singaporefl World Scientiflc Pressfl flfl LanderflEflSflflThe new genomicsflGlobal view of biologyflScience flflflflflfl flflflfl flfl PressflWflHflfl FlanneryflBflPflfl TeukolskyflSflAfl and VetterlingflWflTflfl flflflflfl Numerical Recipes in CflThe Art of Scientiflc ComputingflCambridgefl Cambridge University Pressfl flfl Rasure Jfland Young MflflAn open environment for image processing software develfl opmentfl Infl flflflfl SPIEflISET Symposium on Electronic Imagingfl Vflflflflfl of SPIE ProcessingsflSPIEflflflflflfl flfl ReinitzflJflfl KosmanflDflfl VanarioflAlonsoflCflEfl SharpflDflflStripe forming architecture of the gap gene systemflDevelopmental Genetics flflflflflfl flflflflflfl flfl SanchezflCflfl LachaizeflCflfl JanodyflFflflet alflflGrasping at molecular interactions and genetic networks in Drosophila melanogaster using FlyNetsflan Internet databasefl Nucleic Acids Research flflflflflfl flflflflflfl A Genetic Algorithm with Local Search for Solving Job Problems  ½    ¾    ½ ½            ¾                  !"# $                   % &    &              '     (    )                    &    * +"), -   (    ( .(      &*  + .                       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Yong                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Distributed Learning Control of Trac Signals Y. J. Cao, N. Ireson, L. Bull and R. Miles Intelligent Computer Systems Centre Faculty of Computer Studies and Mathematics University of the West of England, Bristol, BS16 1QY, UK Abstract. This paper presents a distributed learning control strategy for trac signals. The strategy uses a fully distributed architecture in which there is eectively only one low level of control. Such strategy is aimed at incorporating computational intelligence techniques into the control system to increase the response time of the controller. The idea is implemented by employing learning classi er systems and TCP IP based communication server, which supports the communication service in the control system. Simulation results in a simpli ed trac network show that the control strategy can determine useful control rules within the dynamic trac environment, and thus improve the trac conditions. 1 Introduction Trac control in large cities is a dicult and non-trivial optimization problem. Most of the existing automated urban trac control systems, such as TRANSYT1, SCATS2, LVA3 and SCOOT4, have a centralized structure, i.e. information gathering and processing, as well as control computations, are carried out in a centralized manner, in which case eciency is decreased due to the large volume and the heterogeneous character of information 5. To achieve global optimality, hierarchical control algorithms are generally employed. However, these algorithms have a slow speed of reaction and it has been recognized that incorporating some computational intelligence into lower levels can remove some burdens of algorithm calculation and decision making from higher levels 6. Recently, there is a growing body of work concerned with the use of evolutionary computing techniques for the control of trac signals. Montana and Czerwinski 7 proposed a mechanism to control the whole network of junctions using genetic programming 8. They evolved mobile creatures" represented as rooted trees which return true or false, based on whether or not the creature wished to alter the trac signal it has just examined. Cao et al has developed an intelligent local trac junction controller using learning classier systems and fuzzy logic 9 and showed that the local controller can determine useful junction control rules within the dynamic environment. Mikami and Kakazu used S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 1 7 − 1 2 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 1 1 8 Y .J . C a o e t a l. a combination of local learning by a stochastic reinforcement learning method with a global search via a genetic algorithm. The reinforcement learning was intended to optimize the trac ow around each crossroad, while the genetic algorithm was intended to introduce a global optimization criterion to each of the local learning processes 12 . Escazut and Fogarty proposed an approach to generate a rule for each junction using classi er systems in biologically inspired con gurations 13 . This paper is devoted to developing a distributed learning control strategy for trac signals. A fully distributed architecture has been developed in which each subsystem is solely responsible for one aspect of the system and where a coherent global control plan emerges from the interactions of the subsystems no hierarchical structure is included. Such an approach is aimed at increasing the speed of response of the local controller to changes in the environment. To do this, we have developed an agent-alike controller, which is implemented by employing learning classi er systems 14, 15 and TCPIP based communication server, supporting the communication in the control system. Simulation results in a simpli ed trac network show that the control strategy can determine useful control rules within the dynamic trac environment, and therefore improve the trac conditions. 2 An Agent-alike Controller Optimization of a group of trac signals over an area is a large and multiagent type real-time planning problem without precise reference model given. To do this planning, each signal should learn not only to acquire its control plans individually through reinforcement learning but also to cooperate with each other. This requires communication between the agents. If each signal simultaneously communicates with each other and controls its phases according to the change of the global trac ow, the total volume of the area will be well optimised. However, to provide the ecient communication is a dicult task, caused by the inecient accounting of interactions between subsystems in decentralized case and the complex communication structure in the hierarchical case. In this work, we developed an agent-alike controller, consisting of a learning classi er system and a communication server, as shown in Figure 1. Rule-based controller as classi er systems lie midway between neural network and symbolic processing systems that can combine the bene ts of both. The limitation of specifying a single classi er system is that while it may work for a simple controller, the method does not scale up to complex control systems. Work done so far has addressed this problem by dividing a complex system into its simplest, physical sub-systems, specifying rule-based controller for each of these and thus creating a multi-agent system. For trac control problem, we associate an agent to each junction of the trac network. So, the whole control strategy developed in this work contains a number of distributed, communicating agents, where each agent has a classi er D is trib u te d L e a rn in g C o n tro l o f T ra ffic S ig n a ls 1 1 9 A g e n t C la s s ifie r S y s te m C o m m u n ic a tio n S e rv e r Figure 1: Structure of the agent-alike controller system providing the control strategy and a communication server which is used to connect the agent to the user interface, the application and to other agents. The two elements of the agent, classi er system and communication server, are separate since as messages are passed around the agent network, the communication server acts independently of the classi er system to route the message to its neighbours. Another reason for keeping the communication server distinct from the classi er system is that communication is likely to be implementation speci c even in the test applications. Thus it is necessary when specifying the communication server to consider the general requirements of setting up and maintaining the communication in a distributed learning system rather than those in a speci c software and hardware implementation. 2.1 Classier systems A classi er system is a learning system in which a set population of conditionaction rules called classiers compete to control the system and gain credit based on the system's receipt of reinforcement from the environment. A classier's cumulative credit, termed strength, determines its in uence in the control competition and in an evolutionary process using a genetic algorithm in which new, plausibly better, classi ers are generated from strong existing ones, and weak classi ers are discarded. A classi er c is a condition-action pair c = condition : action  with the interpretation of the following decision rule: if a current observed state matches the condition, then execute the action. The condition is a string of characters from the ternary alphabet f 0 1  g, where  acts as a wildcard allowing generalization. The action is represented by a binary string and both conditions and actions are initialized randomly. The real-valued strength of a classi er is estimated in terms of rewards obtained according to a payo function. Action selection is implemented by a competition mechanism, where a strength 1 2 0 Y .J . C a o e t a l. C o m m u n ic a tio n C o n fig u ra tio n P a ra m e te rs C o m m u n ic a tio n S e rv e r 2 . C re a te S e rv e r 3 . C o n n e c t (C o m m u n ic a tio n S e rv e r ) C o n n e c to r 4 . R e q u e s t C o n n e c tio n 1 . C re a te M 6 . A d d C o n n e c tio n C o n n e c to r M 5 . R e q u e s t C o n n e c tio n U s e r I n te r fa c e , A p p lic a tio n o r A g e n t N e ig h b o u r o n ito r o n ito r 7 . A c c e p t C o n n e c tio n C o m m u n ic a tio n S e rv ic e 8 . In itia lis e S tre a m s C o n n e c tio n Figure 2: Initialisation structure of communication server proportionate selection method is usually used. To modify classier strengths, the given credit assignment algorithm is used, e.g. the Bucket brigade 14 . To create new classiers a standard GA is applied, with three basic genetic operators: selection, crossover and mutation. The GA is invoked periodically and each time it replaces low strength classiers with the o spring of the selected tter ones the reader is referred to 14 for full details . 2.2 Communication Server The communication server provides the service for each agent connecting to the user interface, application and other agents. All these channels might involve two-way communication. During the initialisation the agents open a communication channel and await a connection message. The channel is tested to ensure the communication is setup correctly as although the conguration parameters have been previously checked for consistency, the parameters may be inconsistent with the physical communication process, also this process might be faulty. Although the term socket is used in the specication as the medium to connect communication channels in implementation other methods can be used, such as calls to remote objects, when using RMI or DCOM. The basis of the communication initialisation and run-time processes are not a ected. The creation of the communication object and binding in a remote registry on a given hostname and port replaces the creation of a server socket and calls to the remote object replace read and write calls to the sockets. Note that it is possible for the communication server to create separate processes to listen on the communication channel for messages, this allows the agent to be reactive to external messages. The communication with neighbours requires a single channel for incoming messages, and separate channels from sending to each neighbor except if the messages are broadcast on sent via a proxy . The initialisation of the communication server, shown in Figure 2, involves the following steps: D is trib u te d L e a rn in g C o n tro l o f T ra ffic S ig n a ls 1 2 1 1. The Communication Server object creates the specic Communication Services Application, User Interface or Neighbourhood as specied by the conguration. 2. The Communication Server object create a monitor which maintains the list of current connections 3. The Communication Server object passes the Communication Service object and connection conguration information to the Connector object which, for connection with the User Interface and Application and incoming channel from the neighbouring agents, opens a Server Socket on the specied port and waits for a request to connect. For the outgoing channel to the neighbouring agents the Connector object intermittently requests a connection to the neighbours specied port.. 4. The User Interface, Application or Neighbouring Agent sends a request to connect. 5. The request to connect is accepted by the neighbour's server socket. 6. The Connector sends the Communication Service object and open socket to the Communication Monitor. 7. The Communication Monitor object tests the communication channel, if the test succeeds the Communication Service is passed to the Connection object, otherwise the socket is closed and the failure reported. 8. The Connection object starts the thread to handle the connection and passes the input and output streams to the Communication Service object. 3 How to Control Trac Signals To control a trac network, we associate an agent to each junction of the trac network. The agents are initialised according to the trac network conguration and user-specied parameters. For the simulated 2 2 trac network, shown in Figure 3, four agents, i.e., agents I, II, III, and IV, associating with junctions I, II, III, and IV, are need to provide comprehensive control of the network. Agent I has the neighbouring agents II and III, and agent II has the neighbouring agents I and IV, etc. The communication server in each agent provides the control actions of its neighbouring agents, and these information is used to construct control rules for its junction. The classier system employed is a version of Wilson's zeroth-level" system ZCS 15. ZCS is a Michigan-style classier system, without internal memory. In order to avoid the genetic algorithm manipulating unnecessarily long rules, we extend the binary string representation in ZCS to a more general representation, which uses 0 to L L  10 for each variable bit position instead of the binary code. This reduces the string length signicantly and appears to benet multiple variable problems. For these hybrid strings, mutation in the GA is performed by changing an allele to a randomly determined number between 0 and L other than itself 16. 1 2 2 Y .J . C a o e t a l. I II III IV Figure 3: The simulated trac environment 3.1 Individuals The classiers have the representation shown in Figure 4. The condition part of each classier consists of six bits, which re ects the scalar level of queue length from each direction and the previous actions of the neighbouring agents. In this application, the scalar level of the queue length is set to 4, which ranges from 0 to 3, corresponding to the four linguistic variables, fzero small medium large g. The action part indicates the required state of the signal. For instance, for junction I, the rule 130201:1 says that if the queue from directions east and west are small 1 and zero 0, but the queue from directions south and north are large 3 and medium 2, and the previous neighbourhood junction controllers' actions are vertically red 0 junction II and green 1 junction III, then the trac light stays green vertically 1 for a xed period of time. 3.2 Evaluation of actions We assume that the junction controller can observe the performance around it, let the evaluated performance be P . Trac volume sensors are set at each of the intersections. They are able to count the numbers of the cars that come from all directions, pass through the intersection and stop at the intersection. In this study, the evaluation function we use to reward the individuals is the average queue at the specic junction. Let qi denote the queue length from direction i at the intersection i = 1 2 3 4, then the evaluation function is: 1 P4 q . We thus attempted to minimize this measure. Let us identify f = i=1 i 4 the k-th cycle by a subscript k, then fk for the cycle k is calculated by observing the sensor from the beginning of the k-th cycle to the end of this cycle. Thus, the evaluated performance of the action performed at the k-th cycle is computed as Pk = fk,1 , fk . Specically, if Pk  0, the matched classiers containing the performed action should be rewarded, otherwise penalized. D is trib u te d L e a rn in g C o n tro l o f T ra ffic S ig n a ls A c tio n C o n d itio n T ra ffic c o n d itio n fro m e a st T ra ffic c o n d itio n fro m n o rth T ra ffic c o n d itio n fro m s o u th 1 2 3 T ra ffic c o n d itio n fro m w e st T ra ffic lig h t s ta te N e ig h b o u rh o o d a c tio n s Figure 4: Structure of the classier system rules 3.3 Reinforcement learning After the controller has produced an action, the environment judges the output, and accordingly, gives payo in the following manner: Rewards: The objective of the local signal controller is to minimize the average queue length, . We have found the performance-based rewards are helpful in the environments we used in our experiments. The reward P function we used was = 14 4=1 100 , 4 +  , ,1 3 , where denotes the queue length of the th direction at the th cycle. Punishments: We use punishments i.e., negative rewards . We found the the use of appropriate punishments results in improved performance in a xed number of cycles , at least in the environments used in our experiments. We also found that large punishments could lead to instability of the classi ers and slow convergences of the rules. The appropriate punishments should be determined by trial tests. fi r qki q qki ki i qk i i k 4 Simulation Results For the trac network shown in Figure 3, we developed a simpli ed trac simulator, which is similar to the one used in 12. The simulator is composed of four four-armed junctions and squared roads. Each end of a road is assumed to be connected to external trac, and cars are assumed to arrive at those roads according to a Poisson distribution. Each intersection has two complementary" signals: when the horizontal signal is red, the vertical signal is green and vice versa. Each of the cars attempts to attain the same maximum speed. When a car passes an intersection, it changes its direction according to the probability associated with that intersection. Speci cally, let , = 1 2 3, be the next directions for a car, that is, f g = f right, forward, left g. At each of the intersections, the probabilities f g are previously given, where corresponds to di i   di pdi pdi 1 2 4 Y .J . C a o e t a l. 7.5 7 Traffic Speed 6.5 6 5.5 5 4.5 0 1000 2000 a 3000 4000 5000 6000 Time Steps 7000 8000 9000 10000 9000 10000 Number of Cars = 30 6.5 6 5.5 Traffic Speed 5 4.5 4 3.5 3 2.5 0 1000 2000 b 3000 4000 5000 6000 Time Steps 7000 8000 Number of Cars = 60 the probability of selecting an action d for the car passing through the intersection. Roads are not endless, thus only a limited number of cars is allowed to be on the road at a given time. If a car reaches the end of the road, then the car is simply removed from the simulation, and another car is generated, entering on a randomly selected road. For comparison purpose, two types of control strategies are employed: random control strategy and the developed distributed learning control DLC strategy. The random control strategy determines the tra c light's state 0 or 1 randomly at 50 of probability whilst distributed learning control DLC strategy determines the tra c light's state according to the action of the winning classier of the agent. The parameters used for the DLC were as follows: i Population size: 100 Mutation probability: 0.05 Crossover probability: 0.85 Selection method: Roulette wheel selection D is trib u te d L e a rn in g C o n tro l o f T ra ffic S ig n a ls 1 2 5 5.5 5 4.5 Traffic Speed 4 3.5 3 2.5 2 1.5 0 1000 2000 c 3000 4000 5000 6000 Time Steps 7000 8000 9000 10000 Number of Cars = 90 Figure 5: Comparison performance of the control strategies As the major task is to test whether the proposed DLC can learn some good rules in the trac network, experiments were carried out for three di erent types of trac conditions. In these simulations, the mean arrival rates for the cars are the same but the number of cars in the area is limited to 30, 60, and 90, corresponding to a sparse, medium, and crowded trac condition. In all cases, the DLC strategy is found to learn how to reduce the average queue length and improve the trac speed in the network. For example, Figure 5 shows the average performances of the random control strategy and DLC strategy respectively over 10 runs in all cases, where the solid line represents DLC strategy and the dotted line represents random control strategy. It can be seen that the DLC strategy consistently learns and improves the trac speed over 10,000 iterations. 5 Conclusion and Future Work In this paper we have presented a distributed learning control strategy for trac signals. The simulation results on a simplied trac environment are encouraging since we have shown that the developed control strategy can learn to coordinate and determine useful control rules within a dynamic environment. This preliminary work needs, of course, a number of extensions. We are currently extending this work in a number of directions, particularly examining ways of improving the learning capability of classier systems and the performances in much more complicated trac network. 6 Acknowledgment This work was carried out as part of the ESPRIT Framework V Vintage project ESPRIT 25.569. 1 2 6 Y .J . C a o e t a l. References 1 Robertson, D. I.: TRANSYT A trac network study tool. Transport and Research Laboratory, Crowthorne, England 1969 2 Luk, J. Y., Sims, A. G. and Lowrie, P. R.: SCATS application and eld comparison with TRANSYT optimized xed time system. In Proc. IEE Int. Conf. Road Trac Signalling, London 1982 3 Lowrie, P. R.: The Sydney coordinated adaptive trac system. In Proc. IEE Int. Conf. Road Trac Signalling, London 1982 4 Hunt, P. B., Robertson, D. I., Bretherton, R. D. and Winston, R. I.: SCOOTA trac responsive method of co-ordinating trac signals. Transport and Research Laboratory, Crowthorne, England 1982 5 Scemama, G.: Trac control practices in urban areas. Ann. Rev. Report of the Natl Res. Inst. on Transport and Safety. Paris, France 1990 6 Al-Khalili, A. J.: Urban trac control  a general approach. IEEE Trans. on Syst. Man and Cyber. 15, 1985 260271 7 Montana, D. J. and Czerwinski, S.: Evolving control laws for a network of trac signals. Proc. of 1st Annual Conf. on Genetic Programming, 1996 333338 8 Koza, J. R: Genetic Programming. MIT Press, Cambridge, MA 1992 9 Cao, Y. J., Ireson, N. I., Bull, L. and Miles, R.: Design of Trac Junction Controller Using a Classier System and Fuzzy Logic. In Computational Intelligence: Theory and Applications, Reusch, B. ed , Lecture Notes in Computer Sciences, 1625, Springer Verlag, 1999 342353 10 Cao, Y. J. and Wu, Q. H.: An improved evolutionary programming approach to economic dispatch. International Journal of Engineering Intelligent Systems, 6, 2 , 1998 187194 11 Cao, Y. J. and Wu, Q. H.: Optimisation of control parameters in genetic algorithms: a stochastic approach. International Journal of Systems Science, 20, 2 , 1999 551559 12 Mikami, S. and Kakazu, K.: Genetic reinforcement learning for cooperative trafc signal control. Proceedings of the IEEE World Congress on Computational Intelligence, 1994 223229 13 Escazut, C. and Fogarty, T. C.: Coevolving classier systems to control trac signals. In Koza, J. R ed : Late breaking papers at the Genetic Programming 1997 Conference, Stanford University, 1997 5156 14 Holland, J. H.: Adaptation in Natural and Articial Systems. MIT Press, Cambridge, MA 1992 15 Wilson, S. W.: ZCS: A zeroth level classier system. Evolutionary Computation, 2, 1994 118 16 Cao, Y. J. and Wu, Q. H.: A mixed-variable evolutionary programming for optimisation of mechanical design. International Journal of Engineering Intelligent Systems, 7, 2 , 1999 7782 T im e S e r ie s P r e d ic tio n b y G r o w in g L a te r a l D e la y N e u r a l N e tw o r k s L ip to n C h a n a n d Y u n L i C e n tre fo r S y s te m s a n d C o n tro l, a n d D e p a rtm e n t o f E le c tric a l a n d E le c tro n ic s E n g in e e rin g , U n iv e r s ity o f G la s g o w , G la s g o w G 1 2 8 L T , U .K . E m a i l : L . C h a n @ e l e c . g l a . a c . u k A b s tr a c t. T im e -s e rie s p re d ic tio n a n d fo re c a s s c ie n c e a n d e c o n o m ic s . N e u ra l n e tw o rk s p ro b le m s . H o w e v e r, th e d e s ig n o f th e s e n e tw o u n d e rs ta n d in g to o b ta in u s e fu l re s u lts . In th is b a s e d in n o v a tiv e te c h n iq u e to g ro w n e tw o s im p lify th e ta s k o f tim e -s e rie s p re d ic tio n . A th is n e tw o rk is a ls o g iv e n to ta k e a d v a n ta g n e tw o rk is n o t re s tric te d to tim e -s e rie s p re d m o d e llin g d y n a m ic s y s te m s . 1 . tin g is m u c h u s e d in e n g in e e rin g , a re o fte n u s e d fo r th is ty p e o f rk s re q u ire s m u c h e x p e rie n c e a n d p a p e r, a n e v o lu tio n a ry c o m p u tin g rk a rc h ite c tu re is d e v e lo p e d to n e ffic ie n t tra in in g a lg o rith m fo r e o f th e n e tw o rk d e s ig n . T h is ic tio n a n d c a n a ls o b e u s e d fo r I n tr o d u c tio n D y n a m ic m o d e llin g a d d re s s e s th e m o d e llin g p ro b le m fro m d a ta o f a d y n a m ic s y s te m . A d y n a m ic s y s te m is a s y s te m w h ic h h a s in te rn a l s ta te s re p re s e n te d in a n a b s tra c t p h a s e o r tim e s p a c e . Its fu tu re s ta te a n d o u tp u ts d e p e n d s o n its c u rre n t s ta te . T h e y c a n b e m a th e m a tic a lly d e s c rib e d b y a n in itia l v a lu e p ro b le m [1 ]. A n e x a m p le o f d y n a m ic m o d e llin g is th e m o d e llin g o f tim e s e rie s d a ta , w h e re p re d ic tio n s h a s to b e m a d e o n th e fu tu re v a lu e s o f th e tim e s e rie s b a s e d o n c u rre n t v a lu e s o f th e s e rie s . T h is ty p e o f m o d e llin g trie s to c a p tu re th e g e o m e try a n d g e o m e tric a l in v a ria n ts o f a d y n a m ic s y s te m fro m p a s t o u tp u ts o f th e s y s te m [2 ]. T h e u s e o f p a s t o u tp u ts , d e la y c o -o rd in a te s , to m o d e l d y n a m ic s y s te m s c a n b e tra c e d b a c k a s fa r a s 1 9 2 7 to th e w o rk o f Y u le , w h o u s e d a u to -re g re s s io n (A R ) to c re a te a p re d ic tiv e m o d e l fo r s u n s p o t c y c le s [1 ], [3 ]. T h e m o s t p o p u la r m e th o d o f m o d e llin g tim e -s e rie s d a ta to d a y is th e s ta tis tic a l m e th o d o f B o x -J e n k in s . T h e B o x -J e n k in s m e th o d o lo g y s e a rc h fo r a n a d e q u a te m o d e l fro m A R , m o v in g a v e ra g e (M A ), a u to -re g re s s io n m o v in g a v e ra g e (A R M A ), a n d a u to re g re s s io n in te g ra te d m o v in g a v e ra g e (A R IM A ) [5 ]. T h is m o d e llin g is a th re e -s ta g e p ro c e s s : id e n tific a tio n , e s tim a tio n , a n d d ia g n o s tic s . T h e id e n tific a tio n in v o lv e s th e u s e o f s a m p le a u to c o rre la tio n fu n c tio n s (S A C F ) a n d s a m p le p a rtia l a u to c o rre la tio n fu n c tio n s (S P A C F ) to a n a ly s is th e lin e a r re la tio n s h ip s o f th e tim e s e rie s w ith its la g g e d im a g e s . T h e e s tim a tio n p ro c e s s in v o lv e s fin d in g a m o d e l, o n e o f A R , M A , A R M A , a n d A R IM A , w ith a th e o re tic a l A C F a n d P A C F s im ila r to th e S A C F a n d S P A C F o f th e tim e s e rie s . T h e th ird s ta g e , d ia g n o s tic s , in v o lv e s re s id u a l a n a ly s is , g o o d n e s s o f fit s ta tis tic s , a n d c ro s s -v a lid a tio n . T h e lim ita tio n o f th is m e th o d is th a t h u m a n d e c is io n a n d a s s o c ia te d e rro rs a re in h e re n t S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 2 7 − 1 3 8 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 1 2 8 L . C h a n a n d Y . L i in e v e ry s ta g e o f th e p ro c e s s [6 ]. A ls o th e S A C F a n d S P A C F fu n c tio n s m e a s u re o n ly lin e a r re la tio n s h ip s . N e u ra l n e tw o rk s h a v e b e e n s h o w n to b e u n iv e rs a l fu n c tio n a p p ro x im a to rs [4 ]. B y u s in g d e la y c o -o rd in a te s a s th e in p u ts o f a n e u ra l n e tw o rk , it c a n b e u s e d a s a n o n lin e a r a p p ro x im a to r fo r a d e la y d iffe re n tia l e q u a tio n . W ith a n a p p ro p ria te s tru c tu re a n d le a rn in g s tra te g y , th is d e la y d iffe re n tia l e q u a tio n c a n b e tu n e d to h a v e s im ila r b e h a v io u r to th e d y n a m ic s y s te m b e in g m o d e lle d . N e u ra l n e tw o rk s a re c o m m o n ly u s e d fo r tim e -s e rie s p re d ic tio n s a ls o . D iffe re n t n e tw o rk a rc h ite c tu re s h a v e b e e n e m p lo y e d to ta c k le th e p re d ic tio n o f tim e -s e rie s d a ta , fo r e x a m p le th e m u lti-la y e r p e rc e p tro n s [7 ], th e fin ite -d u ra tio n im p u ls e re s p o n s e (F IR ) n e tw o rk s [8 ], a n d th e re c u rre n t n e tw o rk s [9 ]. G o o d p re d ic tio n re s u lts h a v e b e e n o b ta in e d fro m s u c h n e tw o rk s . T h e d iffic u lty w ith th e s e a rc h ite c tu re s is th a t th e y a re s ta tic , fix e d b e fo re th e tra in in g is b e g u n . T h a t is , th e d e s ig n e r n e e d s to d e c id e o n th e n u m b e r o f d e la y c o -o rd in a te s to u s e , th e n u m b e r o f h id d e n n e u ro n s to h a v e , e tc . In th is p a p e r, a n o v e l n e tw o rk a rc h ite c tu re a n d tra in in g s tra te g y is p ro p o s e d fo r th e m o d e llin g o f d y n a m ic s y s te m s w h ic h a lle v ia te s th e d e s ig n e r o f m u c h o f th e s e d e c is io n s . T h e n e tw o rk s tru c tu re is n o t s ta tic a n d c h a n g e s d u rin g th e tra in in g p ro c e s s w h ic h m a k e s u s e o f e v o lu tio n a ry a lg o rith m s (E A ). A d e s c rip tio n o f th e n o v e l n e tw o rk a rc h ite c tu re is g iv e n in S e c tio n 2 . T h e e v o lu tio n a ry te c h n iq u e fo r g ro w in g a n d tra in in g th e n e tw o rk is d e s c rib e d in S e c tio n 3 . In S e c tio n 4 , re s u lts o f p re d ic tin g c h a o tic tim e s e rie s u s in g th is te c h n iq u e a n d n e tw o rk a re s h o w n . S e c tio n 5 d e m o n s tra te s th e a p p lic a tio n o f tim e s e rie s p re d ic tio n o n re a l d a ta , a n d s h o w s a m e th o d o f im p ro v in g p re d ic tio n . F in a lly , th e p a p e r is c o n c lu d e d in S e c tio n 6 . 2 . L a te r a l D e la y N e u r a l N e tw o r k A n e u ra l n e tw o rk re q u ire s m e m o ry to h a v e d y n a m ic b e h a v io u r [8 ]. T h is m e m o ry c a n b e d e la y e le m e n ts in th e a rc h ite c tu re o r th e u s e d e la y c o -o rd in a te in p u ts . T h e n u m b e r o f d e la y c o -o rd in a te in p u ts re la te s to th e e m b e d d in g d im e n s io n o f th e s y s te m , a n d th e n u m b e r o f h id d e n n e u ro n s is d ic ta te d b y n e c e s s a ry d e g re e s o f fre e d o m [1 0 ]. B o th o f th e s e h a v e to b e e s tim a te d fo r s ta tic n e tw o rk a rc h ite c tu re s . T h e p rin c ip le o f g ro w in g th e n e tw o rk s tru c tu re o n e h id d e n n e u ro n a t a tim e h a s s h o w n to b e a fa s t a n d e ffic ie n t m e th o d o f a p p ro x im a tin g a fu n c tio n w ith a n e u ra l n e tw o rk [1 1 ]. S e v e ra l s u c h in c re m e n ta l le a rn in g a lg o rith m s h a v e b e e n im p le m e n te d [1 1 ], [1 2 ]. It is d e s ira b le to u s e th e s e in c re m e n ta l a p p ro a c h e s to s p e e d th e le a rn in g p ro c e s s a n d re d u c e th e d im e n s io n a lity o f o p tim is a tio n . Y e t th e re a re tw o s tru c tu ra l p a ra m e te rs th a t n e e d s to b e e s tim a te d , n a m e ly , n u m b e rs o f d e la y s a n d h id d e n n e u ro n s . T h e d e s ig n o f th e la te ra l d e la y n e u ra l n e tw o rk (L D N N ) c o m b in e s th e s e tw o p a ra m e te rs in to o n e fo r th e im p le m e n ta tio n o f in c re m e n ta l le a rn in g a lg o rith m s . T h is a llo w s fo r th e in c re a s e d s p e e d a n d e ffic ie n c y o f m o d e llin g o f d y n a m ic s y s te m s . T h e c o m b in a tio n o f d e la y e le m e n ts a n d h id d e n n e u ro n s is s h o w n in F ig . 1 o f th e n e tw o rk a rc h ite c tu re . T h is a rc h ite c tu re is in c re m e n ta l w ith th e h id d e n n e u ro n s fo rm in g a o n e -w a y c h a in o f d e la y e d e le m e n ts . T h e s im p lic ity o f d e s ig n is fa c ilita te d b y n o t re q u irin g s tru c tu ra l d e c is io n s . Im p ro v in g th e p e rfo rm a n c e o f n e tw o rk is a c h ie v e d b y a d d in g a n e u ro n to th e la te ra l c h a in . T h e re a re v e ry fe w s y n a p tic w e ig h ts a s s o c ia te d w ith e a c h n e u ro n , th u s th e c o m p le x ity o f th e w e ig h t o p tim is a tio n is v e ry lo w . T im e S e rie s P re d ic tio n b y G ro w in g L a te ra l D e la y N e u ra l N e tw o rk s x 1 x 2 z -1 z -1 1 2 9 yˆ . . . . . . x m z -1 F ig . 1 . A rc h ite c tu re o f L D N N . T h e L D N N s y s te m in p u ts p re d ic tio n , th e th e o u tp u t is w h e re x n is th e 3 . 3 .1 . c a n a n d re is th e p th n d b e u s e d in d y n a m ic i t s o u t p u t yˆ b e i n g o n ly o n e in p u t w h ic re d ic te d fu tu re v a lu a ta in th e s e rie s . m o th e h th e o d e llin g w p re d ic te is th e c u f tim e -s e ith its in p d s y s te m rre n t v a lu r ie s , i.e ., u ts x ∈ ℜ m o u tp u t. I e o f th e tim x = [x n ] a b e in g th n tim e -s e e -s e rie s , n d xˆ n + 1 = e m rie s a n d yˆ , E v o lu tio n a r y N e tw o r k G r o w in g a n d T r a in in g E A -o n ly T r a in in g D u e to th e L D N N a rc h ite c tu re ’s s im p lic ity a n d in c re m e n ta l d e s ig n , tra in in g a lg o rith m s c a n b e d e v is e d to ta k e a d v a n ta g e o f th e s e p ro p e rtie s . E A s c a n b e u s e d to tra in th e n e tw o rk in c re m e n ta lly b y firs t o p tim is in g th e s y n a p tic w e ig h ts o f th e n e tw o r k w ith o n ly o n e h id d e n n e u r o n , i.e ., n o d e la y e le m e n ts y e t. T h e b e s t s o lu tio n fo u n d is th e n u s e d to “ h o t-s ta rt” s u b s e q u e n t E A o p tim is a tio n s o f th e n e tw o rk w ith a n a d d e d h id d e n n e u ro n . T h is c a n b e ite ra te d u n til th e d e s ire d a c c u ra c y is re a c h e d o r u n til o v e r-fittin g b e g in s to o c c u r. U s in g E A fo r tra in in g in th is w a y c a n p ro d u c e g o o d re s u lts , b u t th e d im e n s io n a lity o f th e o p tim is a tio n s p a c e in c re a s e s ra p id ly . If th e re is a w e ig h t a s s o c ia te d w ith e a c h s y n a p tic c o n n e c tio n a n d o n e fo r th e th re s h o ld o f e a c h n e u ro n , th e n th e d im e n s io n a lity in v o lv e d is (m + 3 )n − 1 w h e re m is th e n u m b e r o f in p u ts a n d n is th e n u m b e r o f h id d e n n e u ro n s. 1 3 0 L . C h a n a n d Y . L i 3 .2 . O r th o g o n a l T r a in in g w ith E A T h e d im e n s io n a lity o f th e o p tim is a tio n c a n b e g re a tly re d u c e d b o p tim is e d w e ig h ts o f th e p re v io u s n e tw o rk fix e d a n d o n ly o p tim is in g th e n e w ly a d d e d n e u ro n . A le a rn in g a lg o rith m b a s e d o n th is id e a is L D N N h e re . B e lic z y n s k i g a v e a n in c re m e n ta l a lg o rith m fo r tra in in g o n p e rc e p tro n s [1 1 ]. T h is a lg o rith m is s lig h tly m o d ifie d to a c c o m m o d a te L D N N s a n d la te ra l d e la y s ; th e p ro o fs c a n b e fo u n d in h is w o rk [1 1 ]. T h u s , firs t a s s u m e th a t th e tim e s e rie s to b e m o d e lle d is d e fin e d b y in p u t-o u tp u t p a irs : { ( x 1, f( a 1, x 1) ) , ( x 2, f( a 2, x 2) ) , … y k e e p in g th e th e w e ig h ts o f g iv e n fo r th is e -h id d e n -la y e r th e le a rn in g o f a fin ite s e t o f , (x N, f(a N, x N))} (1 ) w h e re f : ℜ → ℜ is th e s y s te m o u tp u t (th e n e x t d a ta ), x j is a n in p u t (th e c u rre n t c d a ta ), x j ∈ ℜ , a j is th e in te rn a l s ta te o f th e s y s te m , a j ∈ ℜ , j = 1 , … , N , a n d N is th e n u m b e r o f in p u t-o u tp u t p a irs . It fo llo w s th a t fo r tim e s e rie s , c ,1 x = f ( a j, x j) j+ 1 , N − 1 . D e fin e th e v e c to r X w h e re j = 1 , … o rd e r, a s g k( e la w h o rd aˆ , x y e d in e re n in g to j,k [  ) d e n o p u t to ≤ n m ax th e s tr j aˆ j,k , f ( a f (a 1, x 1 ), f (a 2 , x 2 ), T x 2 , x 3 , , x N +1 ] [ te th e fu n c tio th e h id d e n n e a n d n m ax is th u c tu re o f th e  0 =   g ℜ T a n d o u tp u t v e c to r F ( X ) = = o f in p u ts , i.e ., th e tim e s e r ie s d a ta in , x N] ∈ X = [ x 1, x 2, … L e t is th e d … , N , A c c (2 ) n o f u ro n e m a n e tw N N (3 ) , x N ) ]T ∈ ℜ N (4 ) a h id d e n n e u ro n , w h e re g k : ℜ → ℜ , aˆ j , k , fo r e v e ry n e u ro n k = 1 , … , n , a n d tim e j = 1 , x im u m n u m b e r o f h id d e n n e u ro n s a llo w e d . o rk 1 ,1 if j = 1 , o r k = 1 k − 1 ( aˆ , x j − 1 ,k − 1 j− 1 ) (5 ) o th e rw is e w h e re k = 1 , … , n , a n d j = 1 , … , N . T h e d e la y s o f th e n e tw o rk c o n s titu te s th e i n t e r n a l s t a t e o f t h e n e t w o r k , t h e r e b y t h e n e t w o r k s t a t e v e c t o r aˆ j a n d t h e s t a t e m a t r i x Aˆ n c a n b e d e fin e d a s fo llo w s aˆ = [ aˆ j Aˆ n j 1, , aˆ j,2 = [ aˆ 1 , aˆ 2 , , , a ˆ , a ˆ N j,n ]T ∈ ℜ ]T ∈ ℜ (6 ) n (7 ) N ,n D e f in e th e v e c to r G k( X ) w h ic h is c o m p o s e d o f th e o u tp u ts o f th e k fo r th e w h o le tim e s e rie s th h id d e n n e u ro n T im e S e rie s P re d ic tio n b y G ro w in g L a te ra l D e la y N e u ra l N e tw o rk s G k ( X ) = [ g = [ aˆ k ( aˆ 2 ,k + 1 , x 1 ), g , aˆ 3 , k + 1 ,  1 ,k k ( aˆ , aˆ  , x 2 ) , , g k ( aˆ N , k , x T ˆ + 1 , g k ( a N ,k , x N ) ] 2 ,k N ,k a n d d e fin e th e m a trix H A ls o le t W n n ( X ) = [G 1( X ),G ( X ), 2 , G ( X )] ∈ ℜ n N )] T ∈ ℜ 1 3 1 N (8 ) (9 ) N ,n d e n o te th e v e c to r o f w e ig h ts o f th e o u tp u t n e u ro n W , w n] T ∈ = [ w 1, w 2, … n ℜ n (1 0 ) w h e re w k is th e w e ig h t o f th e s y n a p tic c o n n e c tio n fro m n e u ro n k to th e o u tp u t n e u ro n , k = 1 , ,n . T h u s th e n e tw o r k ’ s p r e d ic tio n s , F n( X ) , o f th e tim e s e r ie s X is  F (X ) = H n (X )W n n ∈ ℜ N (1 1 ) N o w th e tra in in g e rro r in n e tw o rk p re d ic tio n c a n b e d e fin e d a s E n  F ( X )  F ( X ) − F ( X ) =  fo r n = 0 fo r 1 ≤ n ≤ n ( X ) n (1 2 ) m a x a n d th e m e a n s q u a re d tra in in g e rro r is e n ( X ) = F ro m E n ( X ) [1 1 ], E n ( X ) 1 E N n ( X ) 2 (1 3 ) is n o n -in c re a s in g , a n d th e m a x im u m o c c u rs w h e n th e n e w ly -a d d e d n e u ro n , g su p E g ∈ G n ( X ) G n + 1 ( X ) G n + 1 ( X ) T n + 1 ra te o f d e c re a s e in , is c h o s e n s u c h th a t (1 4 ) is a c h ie v e d . T h is is p r o v id e d th a t n m ax < N . I n p r a c tic e , N h a s to b e la r g e f o r tim e s e r ie s p r e d ic tio n a n d n m ax < < N . T h e q u a n tity in E q u a tio n 1 4 , is th e s c a la r p r o d u c t o f th e tw o v e c to rs , a n d b y m a x im is in g it, th e tw o v e c to rs a re m a d e q u a s i-p a ra lle l. T h e e rro r is a ls o o rth o g o n a l to th e o u tp u t o f e v e ry h id d e n n e u ro n in th e n e tw o rk . T h e re a re tw o o p tim is a tio n s to b e p e rfo rm e d fo r e a c h n e w n e u ro n b e in g a d d e d . T h e o p tim is a tio n o f th e n e w ly a d d e d h id d e n n e u r o n , g n, a n d W n, th e w e ig h ts o f th e o u tp u t n e u ro n . T h e s e a re o p tim is e d u s in g E A s . T h e d im e n s io n a lity o f th e s e s e a rc h a r e m + 1 f o r g n a n d n + 1 f o r W n, w h e r e m is th e n u m b e r o f in p u ts a n d n is th e to ta l n u m b e r o f h id d e n n e u ro n s . T h e w h o le o rth o g o n a l in c re m e n ta l tra in in g p ro c e s s is s u m m a ris e d in th e flo w c h a rt in F ig . 2 . T h e h i d d e n - l a y e r n e u r o n f u n c t i o n , g k ( aˆ j , k , x j) , c a n u s e d i f f e r e n t a c t i v a t i o n fu n c tio n s . N e ith e r o f th e tw o tra in in g a lg o rith m s d is c u s s e d in th is s e c tio n h a v e m a n y re s tric tio n s o n th e a c tiv a tio n fu n c tio n s u s e d fo r th e h id d e n n e u ro n s . F o r a ll th e tim e s e rie s p re d ic tio n s m a d e in th is w o rk , s ig m o id fu n c tio n s h a v e b e e n u s e d a s a c tiv a tio n fu n c tio n s fo r th e h id d e n n e u ro n s . A lin e a r fu n c tio n is c h o s e n fo r th e o u tp u t n e u ro n ’s 1 3 2 L . C h a n a n d Y . L i a c tiv a tio n . T h e u s e o f s ig m o b o u n d e d , th u s th e tim e -s e rie s T h e tra in in g a lg o rith m s in th e tra in in g p ro c e s s . T h u s th a n d th e tim e -s e rie s th a t c a n b id a c tiv b e in g m th is s e c e s y s te m e p re d ic a tio n f o d e lle tio n a r s th a t te d a re u n c tio n d m u st e s ta tic c a n b e s ta tio n s m e a ls o ; th e m o d a ry . a n s th a t th e n e tw o rk o u tp u t is b e b o u n d e d . n e tw o rk d o e s n o t c h a n g e a fte r e lle d a re a u to n o m o u s s y s te m s S T A R T n = 0 F in d g n + 1 g iv in g su p E g ∈ G U p d a t e s t a t e m a t r i x Aˆ n F in d W n + 1 n ( X ) fo r m in e w ∈ ℜ G n + 1 G n + 1 T n + 1 ( X ) ( X ) u s in g E A ( X ) u s in g E A n = n + 1 N o D e s ire d a c c u ra c y re a c h e d ? Y e s E N D F ig . 2 . F lo w c h a rt o f th e o rth o g o n a l in c re m e n ta l tra in in g p ro c e s s . T h e re s u lts s h o w n in F ig . 3 a re th e tim e s ta k e n to tra in a n a d d itio n a l n e u ro n b y th e tw o tra in in g a lg o rith m s a s th e n u m b e r o f n e u ro n s in c re a s e s . T h e s e tim e s a re th e a v e ra g e tim e s o f 5 n e tw o rk tra in in g s fo r th e p re d ic tio n o f th e M a c k a y -G la s s c h a o tic tim e -s e rie s . T h e a d d itio n a l tra in in g tim e ta k e n b y th e o rth o g o n a l tra in in g a lg o rith m , w ith re d u c e d c o m p le x ity , s ta y s n e a r c o n s ta n t – ris in g o n ly s lig h tly a s th e n u m b e r o f h id d e n n e u ro n s in c re a s e s . W h e re a s , w ith th e E A -o n ly tra in in g , th e e x tra tim e ta k e n p e r n e u ro n in c re a s e s e x p o n e n tia lly w ith th e a d d itio n o f h id d e n n e u ro n s . E x p e rim e n ta tio n s h o w s th a t th e E A -o n ly a lg o rith m o fte n p ro d u c e s v e ry g o o d p re d ic tio n a c c u ra c y w ith v e ry fe w n e u ro n s , w h ile th e o rth o g o n a l a lg o rith m m a y n e e d m a n y m o re n e u ro n s to a c h ie v e th e s a m e le v e l o f a c c u ra c y . D u e to th e lo w c o m p u ta tio n a l p o w e r re q u ire d b y th e o rth o g o n a l tra in in g a lg o rith m , th e re s t o f th e p re d ic tio n re s u lts in th is p a p e r a re o b ta in e d u s in g th is tra in in g m e th o d . T im e S e rie s P re d ic tio n b y G ro w in g L a te ra l D e la y N e u ra l N e tw o rk s A d d itio n a l tra in in g tim e p e r n e u ro n 1 0 0 0 8 0 0 T im e ta k e n (s ) 1 3 3 6 0 0 O rth o g o n a l tra in in g 4 0 0 E A tra in in g 2 0 0 0 2 3 4 H id d e n n e u ro n s 5 F ig . 3 . C o m p a ris o n o f tra in in g tim e s ta k e n b y th e tw o tra in in g a lg o rith m s w h e n a d d in g n e w n e u ro n s. 4 . P r e d ic tio n o f C h a o tic S e r ie s 4 .1 . L o g is tic M a p P re d ic tio n o f L o g is tic M a p 0 -0 .3 2 s te p a h e a d L o g (E rro r) -0 .6 -0 .9 1 s te p a h e a d -1 .2 -1 .5 -1 .8 1 2 3 4 5 F ig . 4 . R M S v a lid a tio n e rro rs o f L D N N lo g is tic m a p . 6 7 8 9 1 0 H id d e n n e u ro n s p re d ic tio n 1 s te p a n d 2 s te p s in to th e fu tu re o f th e T h e lo g is tic m a p is a c h a o tic s e rie s g iv e n b y th e e q u a tio n x U s in g th e o r m a p 1 s te p in to re s u ltin g v a lid a h id d e n n e u ro n s th a t th e tra in in m o d e l’s g e n e ra n e u ro n n e tw o rk n + 1 = 4 x n (1 − x n ) (1 5 ) th o g o n a l tra in in g a lg o rith m , a L D N N is tra in e d to p re d ic t th e lo g is tic th e fu tu re , a n d a n o th e r L D N N to p re d ic t 2 s te p s in to th e fu tu re . T h e tio n e rro rs o f p re d ic tio n , o f d a ta u n s e e n d u rin g tra in in g , fo r u p to 1 0 a re p lo tte d in F ig . 4 . T h e u s e o f in c re m e n ta l te c h n iq u e s g u a ra n te e s g e rro r m o n o to n ic a lly d e c re a s e s . T h e v a lid a tio n e rro rs s h o w th e lis a tio n c a p a b ility . T h e p re d ic tio n v a lid a tio n v a lu e s o f th e 1 0 h id d e n a re c o m p a re d w ith th e a c tu a l v a lu e s in F ig . 5 , s h o w in g a c lo s e m a tc h . 1 3 4 L . C h a n a n d Y . L i L o g is t ic M a p P r e d ic t io n s ( 1 s t e p a h e a d ) 1 .2 1 0 .8 0 .6 0 .4 0 .2 0 -0 .2 0 2 0 A c tu a l 1 .2 1 0 .8 0 .6 0 .4 0 .2 0 -0 .2 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 T im e P r e d ic t io n L o g is tic M a p P re d ic tio n s (2 s te p s a h e a d ) 0 2 0 A c tu a l 4 0 6 0 8 0 1 0 0 P re d ic tio n 1 2 0 1 4 0 T im e F ig . 5 . 1 s te p a n d 2 s te p s a h e a d lo g is tic m a p p re d ic tio n re s u lts w ith 1 0 h id d e n n e u ro n s . 4 .2 . M a c k a y -G la s s S e r ie s T h e M a c k a y -G la s s d iffe re n tia l e q u a tio n q u a s i-p e rio d ic . It is d e fin e d a s x = − b x t + a x 1 + x p ro d u c e s c h a o tic t − τ 1 0 t − τ tim e -s e rie s th a t a re (1 6 ) T h e d e c re a s e o f p re d ic tio n v a lid a tio n e rro rs w ith th e in c re a s e o f h id d e n n e u ro n s c a n b e s e e n in F ig . 6 fo r th e p re d ic tio n o f th e M a c k a y -G la s s s e rie s u p to 5 s te p s in to th e fu tu re . T h e p re d ic tio n v a lid a tio n re s u lts a re s h o w n in re la tio n to th e a c tu a l d a ta v a lu e s fo r 1 , 3 , a n d 5 s te p s a h e a d p re d ic tio n in F ig . 7 . It c a n b e s e e n th a t p re d ic tio n re s u lts a re v e ry c lo s e to th e a c tu a l d a ta v a lu e s , th o u g h th e p re d ic t re s u lts g e t m o re e rra tic th e fu rth e r in to th e fu rth e r th e p re d ic tio n is . A ll th e s e p re d ic tio n s a re m a d e w ith o n ly o n e in p u t to th e L D N N . It m a y b e p o s s ib le to im p ro v e o n th e s e re s u lts b y e x p lic itly e m b e d d in g o th e r d is ta n t d e la y c o o r d in a te s a s in p u ts to th e n e tw o r k , f o r e x a m p le x t– 5, x t– 15, e tc . T im e S e rie s P re d ic tio n b y G ro w in g L a te ra l D e la y N e u ra l N e tw o rk s P re d ic tio n o f M a c k a y -G la s s S e rie s 5 s te a h e a 4 s te a h e a 3 s te a h e a 2 s te a h e a 1 s te a h e a -0 .6 L o g (E rro r) -0 .9 -1 .2 -1 .5 -1 .8 -2 .1 1 2 3 4 5 6 7 8 9 1 0 1 3 5 p d p d p d p d p d H id d e n n e u ro n s F ig . 6 . R M S v a lid a tio n e rro rs o f p re d ic tio n s o f th e M a c k a y -G la s s s e rie s a t s e v e ra l s te p s in to th e fu tu re . M -G P re d ic tio n s (3 s te p a h e a d ) M -G P re d ic tio n s (1 s te p a h e a d ) 1 .5 1 .5 1 .2 1 .2 0 .9 0 .9 0 .6 0 .6 0 .3 0 .3 0 0 0 5 0 A c tu a l 1 0 0 1 5 0 2 0 0 2 5 0 P re d ic tio n 3 0 0 T im e 0 5 0 1 0 0 A c tu a l 1 5 0 2 0 0 2 5 0 P re d ic tio n 3 0 0 T im e M -G P re d ic tio n s (5 s te p a h e a d ) 1 .5 1 .2 0 .9 0 .6 0 .3 0 0 5 0 1 0 0 A c tu a l 1 5 0 2 0 0 P re d ic tio n 2 5 0 3 0 0 T im e F ig . 7 . 1 , 3 a n d 5 s te p s a h e a d M a c k a y -G la s s p re d ic tio n re s u lts w ith 1 0 h id d e n n e u ro n s . 5 . S u n u u s a p P r e d ic tio n o f S u n s p o t N u m b e r s n s p o ts n u m m b e rs fo rm e d to tra in p lic a tio n . T b e rs a re a c y c le a L D N h e d a ta in d ic a to o f a p p r N , d e m u se d a re rs fo o x im o n s y e a r th e le a te ly 1 tra tin g rly n u m v e l 1 y th e b e r o f s o la r e a rs. H u se o f s fro m 1 a c tiv ity in e re , y e a rly th is n e tw 8 5 1 to 1 9 9 th e S su n s o rk 8 [1 3 u n . T h e su n sp o t p o t n u m b e rs a re a n d its e a s e o f ]. 1 3 6 n e tra a b c o a d a n L . C h a n a n d Y . L i F ig . 8 s h o w s th e R M S e rro rs u ro n s in c re a s e s . U s in g in c re in in g e rro rs , a n d th e a n a ly s is ility o f th e L D N N m o d e l. F n tin u a lly in c re a s in g w h e n th e d itio n o f h id d e n n e u ro n s w ill m d in c re a s e o v e r-fittin g to th e tra o f p re d ic tio n v a lid a tio n a s th e m e n ta l tra in in g g u a ra n te e s th e o f v a lid a tio n e rro rs w ill s h o w ro m F ig . 8 , th e v a lid a tio n e rr n u m b e r o f h id d e n n e u ro n s re a o s t lik e ly d e c re a s e th e m o d e l’s g in in g d a ta . m b e r o f h id d e n c re a s e in R M S e g e n e ra lis a tio n a p p e a r to s ta rt d 6 -7 . F u rth e r ra lis a tio n a b ility P re d ic tio n o f Y e a rly S u n s p o t N u m b e rs 1 .7 5 1 .6 5 L o g (E rro r) n u d e th o rs c h e e n e 2 s te a h e a 1 s te a h e a 1 .5 5 1 .4 5 p d p d 1 .3 5 1 .2 5 1 2 3 4 5 6 7 8 9 1 0 H id d e n n e u ro n s F ig . 8 . R M S v a lid a tio n e rro rs fo r th e p re d ic tio n o f y e a rly s u n s p o t n u m b e rs . T o im p ro v e o n th e s e re s u lts , o n e c a n c h o o s e to u s e a d iffe re n t a c tiv a tio n fu n c tio n fo r s u b s e q u e n tly a d d e d h id d e n n e u ro n s o n th e d e te c tio n o f o v e r-fittin g . A lte rn a tiv e ly , e x p lic itly e m b e d d in g d e la y c o -o rd in a te s a s in p u ts to th e n e tw o rk c a n b e e m p lo y e d . T h e la tte r a p p ro a c h is u s e d in th is c a s e a n d th e re s u lts o f F ig . 8 c a n b e u s e d to s u g g e s t w h ic h d e la y c o -o rd in a te to e m b e d . S in c e o v e r-fittin g o c c u rs a t a b o u t th e a d d itio n o f th th th e 7 h id d e n n e u ro n a n d h e n c e th e b e s t n e tw o rk o u tp u ts d e p e n d o n o n ly u p to th e 7 th d e la y c o - o r d in a te , th e 8 d e la y c o - o r d in a te is c h o s e n , i.e ., xˆ xˆ w h e re a n = f1 (a n + 1 n + 2 = f2 (a n n , x n , x n − 7 , x n , x n − 7 (1 7 ) ) (1 8 ) ) is th e in te rn a l s ta te o f th e m o d e l, a n d x n is th e n v a lu e o f th e tim e s e rie s . P re d ic tio n o f Y e a rly S u n s p o t N u m b e rs 1 .7 5 1 .6 5 L o g (E rro r) th 2 s te p a h e a d 1 .5 5 1 s te p a h e a d 1 .4 5 1 .3 5 1 .2 5 1 2 3 4 5 6 7 8 9 1 0 H id d e n n e u ro n s F ig . 9 . R M S v a lid a tio n e rro rs w ith th e e x p lic it e m b e d d in g o f a n e x tra d e la y c o -o rd in a te . T im e S e rie s P re d ic tio n b y G ro w in g L a te ra l D e la y N e u ra l N e tw o rk s T h n e u ro d o e s a n d v re sp e e re s u ltin n s c a n b n o t o c c u r a lid a tio n c tiv e ly . g d e c re a s e s e e n in so so o n w re s u lts fo e in F ig ith r 1 R M . 9 th is a n d S v a lid a tio n w h e n u s in g a a rra n g e m e n t 2 s te p s a h e a d e rro n e a n d p re rs w ith x tra d e g e n e ra d ic tio n in la y lis s a c re a s in g n u m c o -o rd in a te a tio n is g o o d re s h o w n in 2 0 0 2 0 0 1 6 0 1 6 0 1 2 0 1 2 0 8 0 8 0 4 0 4 0 1 3 7 b e r o f h id d e . O v e r-fittin . T h e tra in in F ig . 1 0 a n d 1 n g g 1 0 0 0 2 0 4 0 6 0 A c tu a l 8 0 1 0 0 0 1 2 0 1 0 T ra in in g re s u lt 2 0 3 0 4 0 A c tu a l V a lid a t io n r e s u lt F ig . 1 0 . T ra in in g a n d v a lid a tio n re s u lts fo r 1 s te p a h e a d p re d ic tio n o f s u n s p o t n u m b e rs . 2 0 0 2 0 0 1 6 0 1 6 0 1 2 0 1 2 0 8 0 8 0 4 0 4 0 0 0 0 2 0 4 0 A c tu a l 6 0 8 0 1 0 0 1 2 0 T ra in in g re s u lt 0 1 0 2 0 3 0 A c tu a l V a lid a tio n re s u lt 4 0 F ig . 1 1 . T ra in in g a n d v a lid a tio n re s u lts fo r 2 s te p a h e a d p re d ic tio n o f s u n s p o t n u m b e rs . A 6 . C o n c lu s io n s n o v e l a rc h ite c tu re a n d g ro w in g te c h n iq u e a re p ro p o s im p lifie s th e d e s ig n p ro c e s s o f n e u ra l n e tw o rk s fo r tim e v o lu tio n a ry in c re m e n ta l tra in in g a lg o rith m p re s e n te d fo r s h o w s its e lf to b e a fa s t a n d e ffic ie n t m e th o d o f a p p ro x d y n a m ic s y s te m . T h e tra in in g a lg o rith m a llo w s fo r d a c tiv a tio n s fu n c tio n s to b e u s e d in th e n e tw o rk . It is p o s s ib s e d in th is p a p e r w h ic h e -s e rie s p re d ic tio n . T h e th is n e tw o rk a rc h ite c tu re im a tin g th e n e tw o rk to a iffe re n t c o m b in a tio n s o f le th a t a c h a n g e in h id d e n 1 3 8 L . C h a n a n d Y . L i n e c o In o f e x a s u ro n a c tiv a tio n fu n c tio n w ill in c re a s e th e ra te o f e rro r re d u c tio n , w h e n th is ra te is n v e rg in g . a d d itio n to th e p re d ic tio n o f tim e -s e rie s , th e n e tw o rk m a y b e u s e d fo r th e m o d e llin g d y n a m ic s y s te m s . Im p ro v e d re s u lts m a y b e o b ta in e d w ith th e a d d itio n o f o g e n o u s v a ria b le s o r e x p lic it e m b e d d in g o f d e la y c o -o rd in a te s . W ith s ta tic tra in in g g iv e n in th is p a p e r, a u to n o m o u s s y s te m s a n d s ta tio n a ry tim e -s e rie s c a n b e m o d e lle d . T h is m e th o d o lo g y is v a lid a te d b y th e a p p lic a tio n to a re a l e x a m p le , n a m e ly th e p re d ic tio n o f s u n s p o t n u m b e rs . Its e a s e a n d e ffic ie n c y is d e m o n s tra te d a lo n g w ith e x p lic it e m b e d d in g o f d e la y c o -o rd in a te s to im p ro v e re s u lts . F o r th e m o d e llin g o f n o n -a u to n o m o u s s y s te m s a n d n o n -s ta tio n a ry tim e -s e rie s , re a l-tim e a d a p ta tio n o f th is n e tw o rk m a y b e re q u ire d , w h ic h is a s u b je c t o f o n g o in g re s e a rc h h e re in C S C , U n iv e rs ity o f G la s g o w . 7 . R e fe r e n c e s 1 . M e is s J .: N o n lin e a r S c ie n c e F A Q , M a y 1 9 9 9 , V e r . 1 .3 .1 . I n te r n e t F A Q C o n s o r tiu m , h ttp ://w w w .f a q s .o r g /f a q s /s c i/n o n lin e a r - f a q /. O n lin e . 2 . P a c k a r d N .H ., e t a l.: G e o m e tr y f r o m a T im e S e r ie s . P h y s ic a l R e v ie w L e tte r s , V o l. 4 5 , N o . 9 . (1 9 8 0 ) 7 1 2 -7 1 6 3 . Y u le G .U .: P h ilo s o p h ic a l T r a n s a c tio n s o f th e R o y a l S o c ie ty o f L o n d o n A , V o l. 2 2 6 . ( 1 9 2 7 ) 2 6 7 4 . H o r n ik K ., S tin c h c o m b e M ., a n d W h ite H .: M u ltila y e r F e e d f o r w a r d N e tw o r k s a r e U n iv e r s a l A p p ro x im a to rs . N e u ra l N e tw o rk s , V o l. 2 . (1 9 8 9 ) 3 5 9 -3 6 6 5 . N e e r c h a l N .K .: T im e D o m a in , A u g u s t 1 9 9 9 . T im e S e r ie s T u to r : A n I n te r a c tiv e I n tr o d u c tio n to T im e S e r ie s A n a ly s is , h ttp ://m a th .u m b c .e d u /~ n a g a r a j/. O n lin e . 6 . E lk a te b M .M ., S o la im a n K ., a n d A l- T u r k i Y .: A c o m p a r a tiv e s tu d y o f m e d iu m - w e a th e r d e p e n d e n t lo a d fo re c a s tin g u s in g e n h a n c e d a rtific ia l/fu z z y n e u ra l n e tw o rk a n d s ta tis tic a l te c h n iq u e s . N e u ro c o m p u tin g , V o l. 2 3 . (1 9 9 8 ) 3 -1 3 7 . C o n w a y A .J ., e t a l.: A n e u r a l n e tw o r k p r e d ic tio n o f s o la r c y c le 2 3 . J o u r n a l o f G e o p h y s ic a l R e s e a rc h , V o l. 1 0 3 , N o . A 1 2 . (1 9 9 8 ) 2 9 7 3 3 -2 9 7 4 2 8 . H a y k in S .: N e u r a l N e tw o r k s . M a c m illa n . ( 1 9 9 4 ) 9 . M a ts u o k a M ., G o le a M ., a n d S a k a k ib a r a Y .: C o lu m n a r R e c u r r e n t N e u r a l N e tw o r k a n d T im e S e rie s A n a ly s is . F u jits u S c ie n tific & T e c h n ic a l J o u rn a l, V o l. 3 2 , N o . 2 . (1 9 9 6 ) 1 8 3 -1 9 1 1 0 .L o w e D ., a n d H a z a r ik a N .: C o m p le x ity m o d e llin g a n d s ta b ility c h a r a c te r is a tio n f o r lo n g te r m ite r a te d tim e s e r ie s p r e d ic tio n . I E E C o n f e r e n c e P u b lic a tio n , N o .4 4 0 . ( 1 9 9 7 ) 5 3 - 5 8 1 1 .B e lic z y n s k i B .: I n c r e m e n ta l A p p r o x im a tio n b y O n e - H id d e n - L a y e r N e u r a l N e tw o r k s : D is c re te F u n c tio n s R a p p ro c h e m e n t. IE E E In te rn a tio n a l S y m p o s iu m o n In d u s tria l E le c tr o n ic s V o l.1 . ( 1 9 9 6 ) 3 9 2 - 3 9 7 1 2 .F r itz k e B .: F a s t le a r n in g w ith in c r e m e n ta l R B F n e tw o r k s . N e u r a l P r o c e s s in g L e tte r s 1 . (1 9 9 4 ) 2 -5 1 3 .S u n s p o t N u m b e r s , O c to b e r 1 9 9 9 . S o la r - T e r r e s tr ia l P h y s ic s D iv is io n o f th e N a tio n a l G e o p h y s ic a l D a ta C e n te r , h ttp ://w w w .n g d c .n o a a .g o v /s tp /s tp .h tm l. O n lin e . T r a je c to r y C o n tr o lle r N e tw o r k a n d I ts D e s ig n A u to m a tio n th r o u g h E v o lu tio n a r y C o m p u tin g G re g o ry C h o n g a n d Y u n L i C e n t r e f o rS y s t e m s & C o n tr o l , D e p a r t m e n t o f E l e c t r o n i c s & E le c t r i c a l E n g i n e e r i n g U n i v e r s i t y o f G l sa g o w , G l a s g o w , G 1 2 8 L T , U K . g r e g c c y @ e l e c . g l a . a c . u k A b s t r a c t . C la s s i c a l c o n t r o l l e r s a r e h i g h l y p o p u l a r i n in d u s t r i a l a p p l i c a t io n s . H o w e v e r , m o s t c o n t r o l l e r s a r e t u n e d m a n u a l l y in a t r i a l a n d e r r o r p r o c e s s t h o u g h c o m p u t e r s im u l a t i o n . T h i s i s p a r t i c u l a r l y d i f f i c u l t w h e n t h e s y s te m t o b e c o n t r o l l e d is n o n l i n e a r . T o a d d r e s s t h is p r o b l e m a n d h e l p d e s i g n o f i n d u s t r i a l c o n t r o l l e r s f o r a w i d e r ra n g e o f o p e r a t i n g t ra j e c t o r y , t h i s p a p e r p r o p o s e s a t r a j e c t o r y c o n t r o l l e r n e t w o r k ( T C N ) te c h n i q u e b a s e d o n l i n e a r a p p r o x i m a t i o n m o d e l ( L A M ) t e c h n i q u e . I n a T C N, e a ch c on t r o l l e r c an b e o f a s i m p l e f o r m , w h ic h m a y b e ob ta in e d s tr a ig h tfo r w a r d ly v ia c la s s ic a l d e s ig n or e v o lu tio n a ry m e a n s . T o c o - o dr i n a t e t h e o v e r a l l c o n t r o l l e r p e r f o r m a n c e , t h e s c h e d u l i n g o f h t e T C N i s e v o l v e d t h r o u g h t h e e n t i r e o p e r a t i n g e n v e l o p e . S i n c e p la n t s t e p r e s p o n s e d a t a a r e o f te n r e a d i l y a v a i l a b l e i n e n g i n e e r i n g p r a c t i c e , th e de s ig n o f s u c h T C N is f u l l y a u t o m a t e d u s i n g a n e v o l u t i o n a r y a l g o r i t h m w i t h o u t t h e n e e d o f m o d e l i d e n t i f i c a t i o n . T h i s i s l il u s t r a t e d a n d v a l i d a t e d t h r o u g h a n o n l i n e a r c o n t r o l ex a m p l e . 1 I n tr o d u c tio n A d y n a m ic e n g in e e rin g s y s te m is u s u a lly n o n lin e a r a n d c o m p le x in p ra c tic e . P la n t d y n a m ic s m a y v a ry s ig n ific a n tly w ith c h a n g e s o f o p e ra tin g c o n d itio n s . T h e re fo re , th e u s e o f a s in g le n o m in a l lin e a r m o d e l u n d e r o n e o p e ra tin g c o n d itio n , a n d h e n c e c o n tro lle rs d e s ig n e d o u t o f s u c h a p la n t m o d e l, a re o fte n u n re lia b le a n d in a d e q u a te to re p re s e n t a p ra c tic a l s y s te m . T h e re c e n tly d e v e lo p e d lo c a l c o n tro lle r n e tw o rk te c h n iq u e s [5 ], h a v e p ro v id e d s o m e e ffe c tiv e s o lu tio n s to th e s e p ro b le m s , b u t th e y a re b a s e d o n lo c a lly lin e a ris e d m o d e ls . T o a d d re s s th e s e p ro b le m s m o re c o m p le te ly fo r a w id e r ra n g e o f o p e ra tin g tra je c to rie s a n d to m a k e u s e o f p la n t s te p -re s p o n s e d a ta th a t a re o fte n re a d ily a v a ila b le in e n g in e e rin g p ra c tic e , th is p a p e r p ro p o s e s a tra je c to ry c o n tro lle r n e tw o rk (T C N ) te c h n iq u e b a s e d o n lin e a r a p p ro x im a tio n m o d e l (L A M ) te c h n iq u e [2 ]. S u c h a L A M n e tw o rk is o b ta in a b le d ire c tly fro m p la n t s te p -re s p o n s e b y fittin g n o n lin e a r tra je c to rie s b e tw e e n tw o o p e ra tin g le v e ls . A s p re lim in a rie s to d e s ig n , th is m o d e llin g te c h n iq u e is o u tlin e d in S e c tio n 2 . In a T C N , e a c h c o n tro lle r c a n b e o f a s im p le fo rm , s u c h a s a p ro p o rtio n a l p lu s in te g ra l p lu s d e riv a tiv e (P ID ) c o n tro lle r, w h ic h m a y b e o b ta in e d s tra ig h tfo rw a rd ly v ia S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 3 9 − 1 4 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 1 4 0 G . C h o n g a n d Y . L i c la s s ic a l p e rfo rm a n e n v e lo p e . te c h n iq u e S e c tio n 5 . 2 d e s ig n c e , th e T h is is th ro u g h o r sc d e a e v o lu tio h e d u lin g ta ile d in n o n lin e a n a r o f S e c r c y m th e tio n o n tr e a n s T C N 3 . S o l e x . T o is e v e c tio n a m p le c o -o r o lv e d 4 illu . F in a d in a th r s tra lly , te o u g te s c o n th e o v e ra ll th e e n tire a n d v a lid a te s c lu s io n s a re h L in e a r A p p r o x im a tio n M o d e l fo r N o n lin e a r S y s te m H e re , th e L A M to a p T h e p la n t u s e d fo r th e m o d e ls liq u id -le v e l fo a ls o b e fo u n d in th e e q u a tio n s , th e s y s te m    =    h 2  a te a n o n lin e a r le is a tw in -ta n k c h e m ic a l a n d d to ry . B a s e d o n re is g iv e n b y : − sg n (h1 − h  h 1   p ro x im e x a m p u n d in la b o ra s tru c tu sg n (h1 − h  2 ) c1a A 1 2 ) c1a A 2 g h1 − h p la n c o u ia ry th e 2 − c 2a A s tra n lin T h e lli’s te d e a r sc m th ro u h y d ra a le d d a ss-b a  2 g h1 − h 1 t is illu p le d n o p la n ts . B e rn o u  Q  +  A   0  2 2 g (h 2 − H 2 ) 0  tro lle r ra tin g T C N w n in M o d e llin g g h a u lic o w n la n c  0  v i   0   0  1 c o n o p e th e d ra n e x a m s y s te m m o d e l e a n d f p le . th a t c a n lo w (1 ) T h e s y s t e m i n p u t i s t h e v o l t a g e a p p l i e d t o t h e p u m p , v i, a n d t h e s y s t e m o u t p u t i s t h e liq u id le v e l in ta n k 2 , h 2. T h e c o e ff ic ie n ts o f th e tw in ta n k a r e ta b u la te d in T a b le 1 . T h e n o n -lin e a rity o f th e p la n t m o d e l is c le a rly p lo tte d a s s h o w n in F ig . 1 . T a b le 1 . N o n lin e a r s y s te m p a ra m e te rs H e ig h t o f w a te r in ta n k 1 h 1(m ) H e ig h t o f w a te r in ta n k 2 h 2(m ) m in im u m H h e ig h t o f w a te r in ta n k = 0 .0 3 m 0 2 C ro s s s e c tio n a l a re a o f ta n k 1 & 2 A = 0 .0 1 m D is c h a rg e c o e ffic ie n t o f o rific e 1 c 1 = 0 .5 3 D is c h a rg e c o e ffic ie n t o f o rific e 2 c 2 = 0 .6 3 C ro s s s e c tio n a l a re a o f o rific e 1 a = 0 .0 0 0 0 3 9 6 m 2 1 C ro s s s e c tio n a l a re a o f o rific e 2 a = 0 .0 0 0 0 3 8 6 m 2 2 G ra v ita tio n a l c o n s ta n t g = 9 .8 1 m P e r-v o lt P u m p F lo w ra te Q F lo w ra te fro m D is c h a rg e ra te s -2 = 0 .0 0 0 0 0 7 (m Q 3 1 (m s ) Q (m 3 o s ) ta n k 1 to ta n k 2 i -1 -1 3 s -1 V -1 ) T ra je c to ry C o n tro lle r N e tw o rk 1 4 1 output (h2) versus input(vi ) 0.18 h2, Liquid level (m ) 0.16 0.14 Tank 2 0.12 0.1 0.08 0.06 0.04 0.02 0 0 2 4 6 8 Pump voltage, vi (volt) F ig . 1 . N o n -lin e a rity o f th e p la n t. h v o lta g e v i is a p p lie d to th e p u m p 2 is th e s te a d y s ta te liq u id le v e l o f ta n k 2 w h e n th e in p u t If th re e o p e ra tin g p o in ts o n th e n o n lin e b a s e d o n th e ra te o f c h a n g e o r e q u a lly d liq u id le v e l. T h e s im p le d iv is io n is u s e d L A M m o d e ls . T h e th re e o p e ra tin g p o in ts re s p o n s e s fro m th e L A M a t th e s e p o in ts a a r iv b a r re tra je c to ry id e d fro m e c a u se o f e 0 .0 5 m , 0 s h o w n in F a re th e th e .1 m ig . u se fu ll tra je a n d 2 . d , th o p e r c to ry 0 .1 5 e y a tin c a m . c a n b e c g ra n g e p a b ility G e n e ra te Step response from LAM Tank 2 Liquid level(m) 0.16 0.15m 0.14 0.12 0.1 0.1m 0.08 0.06 0.05m 0.04 0.02 0 0 200 400 600 Time (sec) F ig . 2 . S te p re s p o n s e s a t th re e o p e ra tin g p o in ts o f a L A M 800 n e tw o rk 1000 h o o f o f d s se th th te n e p e 1 4 2 G . C h o n g a n d Y . L i 3 E v o lv in g a T C N M a s im n o n n o n n y c o n tro l s y s te m d e s ig n m e th o d s a re b a s e d o n lin e a r s y s te m s a n a ly s is . p le P ID c o n tro l s y s te m is d e s ig n fo r a L A M . T o a p p ly th e T C N te c h n lin e a r p la n ts , c o n tro lle rs m u s t b e d e s ig n e d fo r th e e n tire L A M n e tw o rk lin e a r s y s te m . T h is is to p ro v id e a d e q u a te p e rfo rm a n c e a c ro s s th e o p e ra tin g e n v e lo p e s y s te m . T h re e P ID c o n tro lle rs in th e T C N a re s c h e d u le d o r s w itc h b e tw e e n s h o w n in F ig . 3 . D u rin g c o n tro lle r o p e ra tio n , a v a ria b le in d ic a tin g o p e ra tin g m o n ito re d a n d d iffe re n t c o n tro lle rs (o r c o n tro lle r p a ra m e te rs ) a re a c tiv a te d a c to th is s c h e d u lin g v a ria b le . In th is d e s ig n , th e p la n t o u tp u t y (t) is u s e d a s s c h v a ria b le to s c h e d u le th e o u tp u t o f th e c o n tro lle rs . H e re , a iq u e to o f th e o f th e th e m a s p o in t is c o rd in g e d u lin g S c h e d u lin g v a ria b le u C o n tro lle r1 r e 2 u 3 C o n tro lle r2 u 1 In te rp o la tin g C o n tro lle r O u tp u t u N o n -lin e a r p la n t A c tu a to r C o n tro lle r3 A n ti W in d -u p F ig . 3 . M u ltip le c o n tro lle rs b a s e d tra je c to ry c o n tro lle r n e tw o rk . T h e T C N u s e s a lin e a r in te rp o la tio n o r w e ig h tin g s c h e d u le a s s h o w n in F ig . 4 . Weight 100% Controller 1 P1 Controller 2 y(t) P2 F ig . 4 . A s im p le in te rp o la tio n s c h e d u le in fo rm in g a T C N Controller 3 P3 Operating Level y T ra je c to ry C o n tro lle r N e tw o rk 1 4 3 T h e r e f o r e , a t a n y o u t p u t l e v e l y ( t ) , t h e i n d i v i d u a l c o n t r o l l e r o u t p u t s u i( t ) a r e in te r p o la te d g iv in g a f in a l c o n tr o llin g o u tp u t u ( t) u s in g e q u a tio n ( 2 ) , w h e r e P 1= 0 .0 5 m , P 2= 0 .1 m a n d P 3= 0 .1 5 m .  P  P   i+ 1 i+ y (t) − P − y (t) × u i(t) + P i+ 1 − P 1 − P i u (t) =  u  u  i × u i+ 1 (t) if P i ≤ y (t) ≤ P i+ 1 i if y (t) < P i if y (t) > P n i n (2 )  w h e r e i= 1 ,… ,n -1 a n d n is th e to ta l n u m b e r o f lin e a r c o n tro lle r 4 H e re , in te rp o la tio n m a y a ls o a p p lie d to th e c o n tro lle r p a ra m e te rs K P , K i a n d K d. D e s ig n E x a m p le a n d V a lid a tio n 4 .1 G e n e r a t in g T r a j e c t o r y C o n t r o lle r s f r o m S te p R e sp o n se s In d iv id u a l P ID c o n tro lle rs fro m a s te p -re s p o n s e tra je c to ry to e a c h o f th e th re e o p e ra tin g p o in ts a re g e n e ra te d fro m th e P ID e a s y ™ d e s ig n a u to m a tio n p a c k a g e [4 ], a s s h o w n in F ig . 5 . F ig . 5 . D ire c t d e s ig n fro m p la n t re s p o n s e u s in g P ID e a s y ™ P ID e a s y ™ a n a ly s e s s te p re s p o n s e d a ta a n d g e n e ra te s a n a p p ro p ria te P ID c o n tro lle r fro m th e m . A t e a c h o p e ra tin g p o in t, fittin g th e s te p re s p o n s e g e n e ra te d fro m a L A M p ro d u c e s fa s t g e n e ra tio n o f P ID c o n tro lle r. T h e c lo s e d lo o p re s p o n s e s a t th e s e 1 4 4 G . C h o n g a n d Y . L i o p e ra tin g p o in ts a re p lo tte d o n th e s a m e g ra p h s h o w n in F ig . 6 . N o te th a t th e fa s t g e n e ra tio n o f P ID c o n tro lle rs u s in g lin e a r P ID e a s y ™ te c h n iq u e is te s te d a g a in s t th e n o n lin e a r p la n t. T h is re v e a ls th e n e e d o f n e tw o rk tu n in g . Performance of each individual trajectory PID controller 0.18 Tank 2 Liquid Level (m) 0.16 0.15m 0.14 0.125m 0.12 0.1 0.1m 0.08 0.075m 0.06 0.05m 0.04 0.02 0 0 200 400 600 800 1000 1200 Times(sec) F ig . 6 . P e rfo rm a n c e o f e a c h In d iv id u a l tra je c to ry P ID c o n tro lle r 4 .2 . N e tw o r k in g T h r o u g h E v o lu tio n A g e n e tic a lg o rith m (G p ro b le m s b y e m u la tin g u s in g th e e v o lu tio n a ry o p tim a l s o lu tio n s . O n e fitn e s s fu n c tio n n e e d s m in im is e d is th e s u m o p e ra tin g p o in ts w ith in A ) p ro v id e s g lo b a lly o p tim a l s o n a tu ra l e v o lu tio n . A p o p u la tio n o p e ra to rs o f c ro s s o v e r, m u ta tio a d v a n ta g e o f a G A fo r o p tim is n o t to b e d iffe re n tia b le . H e re , m a tio n o f a ll e rro rs a c ro s s th e a g iv e n tim e p e rio d m . J = H e re , sh o w n 0 .1 5 m p o in ts th e in . T . E re a re n = 5 re f F ig .7 . T h e 5 h e s e 5 p o in ts a c h re fe re n c e ∑ n lu tio n s to e n g in e e rin g d e s ig o f p o te n tia l s o lu tio n s e v o lv e n a n d s e le c tio n to a p p ro a c a tio n is th a t th e o b je c tiv e o th e o b je c tiv e fu n c tio n to b e n tire T C N a t n re fe re n c n s h r e e m ∑ e (t) re f = 1 t= 0 e re n c e le v e ls u s e d to e v a re fe re n c e le v e ls a re s e t c o v e r th e w h o le tra je c to is te s te d fo r a p e rio d o f m lu a te to 0 ry a n = 1 0 e (t)= | r (t)– y (t) |. (3 ) th e .0 5 m d in 0 0 s e rr , 0 c lu e c , o r tra c k in g p e rfo rm a n c e a s .0 7 5 m , 0 .1 m , 0 .1 2 5 m a n d d in g tw o u n s e e n o p e ra tin g w h e re (4 ) re p re s e n ts th e tra c k in g e rro r b e tw e e n th e c lo s e d -lo o p o u tp u t y (t) a n d th e c o m m a n d r (t). T ra je c to ry C o n tro lle r N e tw o rk Controller 1 Controller 2 1 4 5 Controller 3 100% =Reference points R1 R2 R3 R4 R5 Operating Level F ig . 7 . E v a lu a tio n p o in ts in th e o p e ra tin g e n v e lo p e . T o e v o p a ra m e s im u lta T h e c lo a p o p u d e p ic te th e n o n lv e T te rs o n e o u s s e d lo la tio n d in F lin e a r C N b a s e d o n th e fa s t g e n e ra te d P ID f th e th re e lin e a r c o n tro lle rs a n d th e ly , a t th e s e e n a n d u n s e e n o p e ra tin g p o p re s p o n s e s o f th e fin a lly e v o lv e d T C s iz e o f 5 0 a r e s h o w n in F ig .8 f o r ig .7 . I t c a n b e s e e n th a t th e lin e a r T C N c o n tro l p ro b le m . c o n tro lle rs o u t o f L A s c h e d u lin g w e ig h ts , a o in ts a lo n g in o p e ra tin g N a t th e e n d o f 5 0 g e n e a ll o f th e te s te d o p e ra p ro v id e d a n e x c e lle n t M , a ll th e re e v o lv e d e n v e lo p e . ra tio n w ith tin g le v e ls s o lu tio n to Closed loop responses of TCN 0.18 0.16 Tank 2 Liquid Level (m) 0.15m 0.14 0.125m 0.12 0.1 0.1m 0.08 0.075m 0.06 0.05m 0.04 0.02 0 0 200 400 600 800 1000 1200 Time(sec) F ig . 8 . C lo s e d lo o p re s p o n s e s o f th e T C N 0 .1 2 5 m a n d 0 .0 7 5 m . a t o p e ra tin g p o in ts in c lu d in g th e u n s e e n o n e s a t 1 4 6 G . C h o n g a n d Y . L i 5 D is c u s s io n a n d C o n c lu s io n T o a s s is t c o n tro l s y s te m d e s ig n fo r a w id e ra n g e o f o p e ra tin g e n v e lo p p la n ts , th is p a p e r h a s d e v e lo p e d a tra je c to ry c o n tro lle r n e tw o rk (T b a s e d o n lin e a r a p p ro x im a tio n m o d e l (L A M ) te c h n iq u e . T h e e x a m p le lin e a r T C N u s e d to c o n tro l a n o n lin e a r s y s te m p e rfo rm s w e ll in th e e e n v e lo p e . T h is o ffe rs p o te n tia l b e n e fits a n d s im p lic ity fo r c o n tro s y s te m s . T h e re s u lts s h o w th a t th e G A b a s e d a u to m a tic c o n tro lle r n e tw n o n lin e a r s y s te m s is p o s s ib le a n d u s e fu l. S u c h a n e tw o rk is e a s ily s a m p le d re s p o n s e d a ta . e fo r n o n lin e a r C N ) te c h n iq u e s h o w s th a t th e n tire o p e ra tin g l o f n o n lin e a r o rk d e s ig n fo r d e s ig n e d fro m R e fe r e n c e s 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . G .J . G r a y , D .J . M u r r a y S m ith , Y . L i, K .C . S h a r m a n , T . W e in b r e n n e r : N o n lin e a r m o d e l s tr u c tu r e id e n tif ic a tio n u s in g g e n e tic p r o g r a m m in g . C o n tr o l E n g in e e r in g P r a c tic e , V o l.6 , N o .1 1 . (1 9 9 8 ) 1 3 4 1 -1 3 5 2 Y . L i a n d K .C . T a n : L in e a r a p p ro x im a tio n m o d e l n e tw o r k a n d its f o r m a tio n v ia e v o lu tio n a ry c o m p u ta tio n , A c a d e m y P ro c e e d in g s in E n g in e e rin g S c ie n c e s (S A D H A N A ), In d ia n A c a d e m y o f S c ie n c e s , In v ite d p a p e r (1 9 9 9 ) D .E . G o ld b e r g : G e n e tic A lg o r ith m in S e a r c h , O p tim is a tio n a n d M a c h in e L e a r n in g , A d d is o n -W e s le y , R e a d in g (1 9 8 9 ) Y . L i, W . F e n g , K .C . T a n , X .K . Z h u , X . G u a n a n d K .H . A n g : P I D e a s y ™ a n d a u to m a te d g e n e ra tio n o f o p tim a l P ID c o n tro lle rs , T h e T h ird A s ia -P a c ific C o n fe re n c e o n M e a s u re m e n t a n d C o n tro l, D u n h u a n g , C h in a , P le n a ry p a p e r. (1 9 9 8 ) 2 9 -3 3 G .J . G r a y , Y . L i, D .J . M u r r a y - S m ith a n d K .C . S h a r m a n : S p e c if ic a tio n o f a c o n tr o l s y s te m fitn e s s fu n c tio n u s in g c o n s tra in ts fo r g e n e tic a lg o rith m b a s e d d e s ig n m e th o d s , P ro c . F irs t I E E /I E E E I n t. C o n f . o n G A in E n g . S y s t.: I n n o v a tio n s a n d A p p l., S h e f f ie ld . ( 1 9 9 5 ) 5 3 0 5 3 5 Y . F a th i: A lin e a r a p p ro x im a tio n m o d e l fo r th e p a ra m e te r d e s ig n p ro b le m , E u ro p e a n J o u r n a l O f O p e r a tio n a l R e s e a r c h , V o l.9 7 , N o .3 . ( 1 9 9 7 ) 5 6 1 - 5 7 0 K la tt a n d E n g e ll: G a in -s c h e d u lin g tra je c to ry c o n tro l o f a c o n tin u o u s s tirre d ta n k re a c to r, C o m p u te r s & C h e m ic a l E n g in e e r in g , V o l.2 2 , N o .4 - 5 . ( 1 9 9 8 ) 4 9 1 - 5 0 2 G . C o rrig a , A . G iu a , G . U s a i: A n im p lic it g a in -s c h e d u lin g c o n tro lle r fo r c ra n e s , IE E E T r a n s a c tio n s O n C o n tr o l S y s te m s T e c h n o lo g y , V o l.6 , N o .1 . ( 1 9 9 8 ) 1 5 - 2 0 E v o lu tio n a r y C o m p u ta tio n a n d N o n lin e a r P r o g r a m m in g i n M u l t i - mo d e l - ro b u s t C o n t r o l D e s i g n D o ro th e a K o lo s s a d o r o t h e a . k o l o s s a @ d a i m l e r c h r y s l e r . c o m G e o rg G rü b e l * g e o r g . g r u e b e l @ i e e e . o r g A b s tr a c t. A n a lg o rith m ic p a ra m e te r tu n in g m e th o d o lo g y fo r c o n tro lle r d e s ig n o f c o m p le x s y s te m s is n e e d e d . T h is m e th o d o lo g y s h o u ld o ffe r d e s ig n e rs a g re a t d e g re e o f fle x ib ility a n d g iv e in s ig h t in to th e p o te n tia ls o f th e c o n tro lle r s tru c tu re a n d th e c o n s e q u e n c e s o f th e d e s ig n d e c is io n s th a t a re m a d e . S u c h a m e th o d is p ro p o s e d h e re . F o r a n e x p lo ra to ry p h a s e a n e w p a re to -ra n k e d g e n e tic a lg o rith m is p ro p o s e d to g e n e ra te a n e v e n ly d is p e rs e d s e t o f n e a r o p tim a l, g lo b a l, s o lu tio n s . B y p a ir-w is e p re fe re n c e s ta te m e n ts o n d e s ig n a lte rn a tiv e s a lin e a r p ro g ra m is s e t u p a s a fo rm a l m e a n s fo r s e le c tin g th e s o lu tio n w ith b e s t o v e ra ll d e s ig n e r s a tis fa c tio n . In a fo llo w in g in te ra c tiv e d e s ig n p h a s e u s in g n o n lin e a r p ro g ra m m in g te c h n iq u e s w ith a p rio ri d e c is io n s o n a llo w e d q u a lity le v e ls , a b e s t tu n in g c o m p ro m is e in c o m p e tin g re q u ire m e n ts s a tis fa c tio n is s e a rc h e d fo r w h ile g u a ra n te e in g p a re to -o p tim a lity . In p a rtic u la r, th is tw o -p h a s e tu n in g a p p ro a c h a llo w s th e d e s ig n e r to b a la n c e n o m in a l c o n tro l p e rfo rm a n c e a n d m u lti-m o d e l c o n tro l ro b u s tn e s s . 1 I n tr o d u c tio n C o n tro l e n g in e e rin g w o rk is m a in ly o c c u p ie d w ith a d a p tin g a c o n tro l s y s te m a rc h ite c tu re w ith g iv e n c o n tro l la w s tru c tu re , s e n s o rs a n d a c tu a to rs , to n e e d s o f c h a n g e d p ro d u c t re q u ire m e n ts o r n e w p ro d u c t v e rs io n s . T h is is c a lle d ‘in c re m e n ta l d e s ig n ’. In p ra c tic e th is o c c u rs m u c h m o re o fte n th a n s ta rtin g c o n tro l s y s te m d e s ig n a fre s h . T h e e s s e n c e o f in c re m e n ta l d e s ig n is a d a p ta tio n b y tu n in g th e c o n tro l la w p a ra m e te rs , p a rtia l re p la c e m e n t o r a u g m e n ta tio n o f th e c o n tro l la w s tru c tu re b y d y n a m ic c o m p e n s a to rs , filte rs , a n d s ig n a l lim ite rs , a n d tu n in g th e o v e ra ll s tru c tu re in c o n c u rre n c e w ith th e b a s ic c o n tro l la w p a ra m e te rs . C o m m o n in d u s tria l p ra c tic e is h a rd w a re -in -th e -lo o p , m a n u a l, tu n in g c a lle d ‘c a lib ra tin g ’. H o w e v e r, s in c e th e re m a y b e v e ry m a n y p a ra m e te rs to b e tu n e d m a n u a l tu n in g is n o t e ffic ie n t n e ith e r in re q u ire d e n g in e e rin g c o s ts n o r in e x p lo itin g th e fu ll p o te n tia l o f th e c h o s e n s y s te m a rc h ite c tu re w ith re s p e c t to m u ltiv a ria te re q u ire m e n ts . T h e re fo re , ‘v irtu a l p ro d u c t e n g in e e rin g ’ b a s e d o n h ig h fid e lity s y s te m m o d e l s im u la tio n s is m o re a n d m o re b e c o m in g th e e n g in e e rin g life s ty le o f c h o ic e . C A C S D , i.e ., C o m p u te r A u to m a te d C o n tr o l S y s te m D e s ig n , is th e d is c ip lin e to p r o v id e th e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * E v o S c o n d i N o d e , D L R - In s titu te o f R o b o tic s a n d M e c h a tro n ic s , O b e rp fa ffe n h o fe n S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 4 7 − 1 5 7 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 1 4 8 D . K o lo s s a a n d G . G rü b e l p e rtin a u to m p a ra m p a rtic e n t a te d e te r u la r, te c h n sy m se a r C A C o lo b o c h S D g y in c o lic a l / n v ia n o n s u p p o rts n tro l u m e ric lin e a r th e d e synthesis model / rules G O paretooptimal algorith. tuning T Q D in g , m d m in p u ta m a k in g u s e o f re c y n a m ic s m o d e lin g g o r e v o lu tio n a ry tio n lo o p d e p ic te d in executable s/w specification G controller synthesis e n g in e e r a l s y s te p ro g ra m s ig n c o m G O K q u a lity fu n c tio n s 4 = ^PD[ /0  + ` plant (p) models commands / disturbances evaluation cases simulation analysis in ic In ]. G controller model C e n t a d v a n c e s a n d a lg o rith m c o m p u ta tio n . F ig u re 1 , c f. [1 characteristics extraction L, H requirements capture Q C D quality functions characteristics design directors T tuner parameters K controller parameters L, H quality scalings F ig . 1 . C A C S D g e n e ric c o m p u ta tio n lo o p to s u p p o rt in c re m e n ta l c o n tro l d e s ig n T h is C A C S D c o m p u ta tio n lo o p is g e n e ric in th a t it a llo w s to in c o rp o ra te a n y c o n tro lle r s tru c tu re to b e tu n e d . T u n in g p a ra m e te rs T m a y re la te to c o n tro lle r p a r a m e te r s e ith e r d ir e c tly , K = T , e .g ., in P I D c o n tr o l [ 2 ] , o r in d ir e c tly v ia a n a n a ly tic s y n th e s is m e th o d , K = f ( T , s y n th e s is m o d e l) , e .g ., in H ∞ [ 3 ] c o n tr o l, a s w e ll a s f u z z y c o n tro l, K = f(T , fu z z y ru le s ), w h e re in th e la tte r c a s e p o s s ib le tu n in g p a ra m e te rs a re th e s c a lin g fa c to rs o f th e m e m b e rs h ip fu n c tio n s a n d th e w e ig h tin g fa c to rs o f th e fu z z y c o n tr o l r u le s [ 4 ] . F u r th e r m o r e , it a llo w s to a p p ly a n y a n a ly s is m e th o d , e .g ., lin e a r m e th o d s in s ta te s p a c e a n d fre q u e n c y d o m a in a s w e ll a s n o n -lin e a r tim e s im u la tio n , to c o p e w ith n o n -c o m m e n s u ra b le c o n tro l q u a lity e v a lu a tio n s s im u lta n e o u s ly . T o a s s e s s re q u ire m e n ts s a tis fa c tio n , d e s ig n re q u ire m e n ts a re c a p tu re d fo rm a lly b y q u a lity fu n c tio n s . T h e fu z z y -ty p e in te rv a l fo rm u la tio n o f q u a lity fu n c tio n s , c f. S e c tio n 2 , a llo w s to d e a l w ith q u a lity le v e ls , e .g ., le v e ls o f ‘ g o o d ’ , ‘ a c c e p ta b le ’ , ‘ b a d ’ , re q u ire m e n t s a tis fa c tio n . T h e d a ta o b ta in e d b y e v a lu a tio n o f a ll th e q u a lity fu n c tio n s fe e d a tu n in g a lg o rith m to c o m p u te p a re to -o p tim a l tu n in g p a ra m e te r v a lu e s . F o r th is k in d o f d a ta -d riv e n tu n in g b o th e v o lu tio n a ry a lg o rith m s , c f. S e c tio n 3 , o r n o n -lin e a r p ro g ra m m in g a lg o rith m s , c f. S e c tio n 4 , c a n b e u s e d . P a r e to - o p tim a lity le n d s its e lf n o t to a u n iq u e s o lu tio n . T h e r e f o r e , in S e c tio n 3 .1 a n e w m u lti-o b je c tiv e g e n e tic a lg o rith m is p ro p o s e d , w h ic h h a s th e s p e c ia l p ro p e rty th a t it y ie ld s e v e n ly d is p e rs e d s o lu tio n s in o r n e a r to th e p a re to -o p tim a l s e t, th u s m a k in g b e s t u s e o f e v o lu tio n a ry c o m p u ta tio n to p ro d u c e a ric h s e t o f d e s ig n a lte rn a tiv e s . H a v in g m a n y a lte rn a tiv e s a v a ila b le to c h o o s e fro m , s e le c tio n o f th e b e s t c a n d id a te n e e d s to f o llo w a fo rm a l a p p ro a c h . T h is is d e a lt w ith in S e c tio n 3 .2 . N o n lin e a r p ro g ra m m in g fo rm u la tio n s in S e c tio n 4 a re u s e d to g e n e ra te d e d ic a te d p a re to -o p tim a l d e s ig n a lte rn a tiv e s e ith e r to a tta in o p tim a l d e s ig n e r s a tis fa c tio n o r to ite ra te q u a n tita tiv e c o m p ro m is e s in c o m p e tin g r e q u ir e m e n ts s a tis fa c tio n . E v o lu tio n a ry C o m p u ta tio n a n d N o n lin e a r P ro g ra m m in g T h is s u g g e s ts a tw o -p h a s e tu n in g p ro c e d u re to a c h ie ro b u s tn e s s , S e c tio n 5 . In p h a s e o n e a g lo b a l m u lti-o b je c tiv e d m o d e l in s ta n tia tio n is c a rrie d o u t u s in g e v o lu tio n a ry c o m p u in te ra c tiv e n o n lin e a r p ro g ra m m in g c o m p u ta tio n s a re a p p lie d c o n tro l b e h a v io r w ith o ff-n o m in a l b e h a v io r c h a ra c te riz e d b y p la n t m o d e l in s ta n tia tio n s . B y th is tu n in g o ff-n o m in a l b e h a v ‘a c c e p ta b le ’ w h ile n o m in a l b e h a v io r is to b e k e p t w ith in th e ‘ 2 v e m u lti-m o d e l c o e s ig n fo r a n o m in a l ta tio n , a n d in p h a s e to c o m p ro m is e n o m a n u m b e r o f o ff-n o m io r is to b e c o m e a t g o o d ’ q u a lity le v e l. 1 4 9 n tr p la tw in in le a o l n t o a l a l st R e q u ir e m e n ts C a p tu r e a n d S a tis fa c tio n A s s e s s m e n t F o r d e s ig n a s s e s s m e n t d e s ig n c h a ra c te ris tic s lik e s y s te m d a m p in g , s te a d y s ta te e rro r, g a in a n d p h a s e m a rg in s , o r m a x im u m c o n tro l ra te , h a v e to b e tra n s fo rm e d in to a q u a lity v a lu e w h ic h in d ic a te s th e d e g re e to w h ic h re q u ire m e n ts a re m e t. T w o k in d s o f m a th e m a tic a l fo rm u la tio n s a re c o m m o n ly in u s e : p o s itiv e d e fin ite ‘th e -s m a lle r-th e b e tte r ’ f u n c tio n a ls o f tim e a n d f r e q u e n c y r e s p o n s e s , e .g ., [ 2 ] , w h ic h o u g h t to b e m in im iz e d , a n d in e q u a litie s o n th e d e s ig n c h a ra c te ris tic s , w h ic h o u g h t to b e s a tis fie d a s c o n s tr a in ts , e .g ., [ 5 ] , [ 3 ] . A d v a n ta g e s o f th e tw o a p p ro a c h e s fo r q u a lity m o d e lin g c a n b e c o m b in e d b y th e s m a lle r-th e -b e tte r in te rv a l q u a lity fu n c tio n s , w h e re re q u ire m e n t s a tis fa c tio n is c o n s id e re d ‘g o o d ’ fo r o n e ra n g e w ith fu n c tio n v a lu e z e ro , ‘a c c e p ta b le ’ in a ra n g e w ith fu n c tio n v a lu e n o t g re a te r th a n o n e , a n d ‘b a d ’ o u ts id e a lim itin g ra n g e . S u c h a n in te rv a l q u a lity fu n c tio n , q (c ), is m a th e m a tic a lly d e fin e d o n th e d e s ig n c h a r a c t e r i s t i c s , c , b y t h e m a x - o p e r a t o r ( 1 ) w i t h f o u r i n t e r v a l v a l u e s EO < J O < J K < EK c o m p lia n t w ith ‘b a d ’, ‘a c c e p ta b le ’ a n d ‘g o o d ’ c h a ra c te ris tic s v a lu e s , c f. F ig u re 2 , T( F ) = m a x fu n o p e q u a se t c o m h a n { /( F ) , 0 , + ( F ) } , /( F ) = ( F − J O ) / ( EO − J O ) , EO < J O + ( F ) = ( F − J K ) / ( EK − J K ) , J O < J K < EK . (1 ) R e q u ire m e n t s a tis fa c tio n is a s s e s s e d a s th e b e tte r, th e s m a lle r a v a lu e th is q u a lity c tio n a s s u m e s . F u rth e rm o re , th e m a x -fo rm u la tio n fits to fu z z y lo g ic A N D ra tio n w ith m a x -o p e ra to r [7 ] to m a k e o v e ra ll ‘g o o d ’, ‘a c c e p ta b le ’, ‘b a d ’ s y s te m lity s ta te m e n ts in th e v e in o f fu z z y lo g ic . It a ls o a llo w s to c o m b in e a n e n u m e ra te d o f c o m m e n s u r a b l e q u a l i t y c h a r a c t e r i s t i c s FN , /N ( FN ) , + N ( FN ) , t o f o r m a p o u n d q u a lity f u n c tio n f o r , e .g ., ta k in g a c c o u n t o f a ll e ig e n v a lu e s c o n c u r r e n tly o r d l i n g a l l v a l u e s FN : = F ( W N ) o f a d i s c r e t i z e d t i m e r e s p o n s e a s a n e n t i t y i n re q u ire m e n ts c a p tu re fo r ro b u s t tra c k in g th u m b p rin t p e rfo rm a n c e [6 ]. W ith H := 0 , th e ‘g o o d ’ in te rv a l is o p e n to th e rig h t, s e e th e e x a m p le o f F ig u re 2 , w ith L := 0 it is o p e n to th e le ft. 1 5 0 D . K o lo s s a a n d G . G rü b e l T ζ 1 acceptable bad good L H b l = 0 .3 ζ g l = 0 .7 F ig . 2 . Q u a lity f u n c tio n f o r e ig e n v a lu e d a m p in g , w ith ζ > 0 .7 ‘ g o o d ’ a n d > 0 .3 ‘ a c c e p ta b le ’ 3 P a r e to -T u n in g b y G e n e tic A lg o r ith m s a n d D e s ig n P r e fe r e n c e C o n tro l s y s te m s p a ra m e te r tu n in g is a lw a y s a m u lti-o b je c tiv e p ro b le m w ith c o m p e tin g re q u ire m e n ts o f c o n tro l p e rfo rm a n c e v e rs u s c o n tro l e ffo rt a n d ro b u s tn e s s . H e n c e th e d e s ig n e r’s p rim e ta s k is to s e a rc h fo r a s u ita b le tra d e o ff w h ile g e n e ra tin g fe a s ib le s o lu tio n s . T h is s e a rc h o u g h t to b e c o n fin e d to th e s e t o f ‘b e s t a c h ie v a b le ’ c o m p ro m is e s o lu tio n s k n o w n a s p a re to -o p tim a l s o lu tio n s . G e n e ra lly , a d e s ig n a lte rn a tiv e a is s a id to b e p a re to p re fe rre d to a n a lte rn a tiv e a if a ll q u a lity m e a s u re s q o f a a re b e tte r (s m a lle r) th a n o r e q u a l to th o s e o f a , w ith a t le a s t o n e b e in g s tric tly b e tte r. T h u s , a p a re to -o p tim a l, o r n o n -d o m in a te d , s o lu tio n is o n e w h e re n o q u a lity m e a s u re c a n b e im p ro v e d w ith o u t c a u s in g d e g ra d a tio n o f a t le a s t o n e o th e r q u a lity m e a su re . T o m a k e a n in fo rm e d tra d e o ff d e c is io n , th e d e s ig n e r n e e d s a ric h s e t o f d e s ig n a lte rn a tiv e s a s w e ll a s fo rm a l m e th o d s to s u p p o rt a s y s te m a tic s e le c tio n p ro c e s s . S u c h a m e th o d o lo g y is p ro p o s e d h e re . A n e w p a re to -ra n k e d g e n e tic a lg o rith m g e n e ra te s a n e v e n ly d is p e rs e d s e t o f d e s ig n a lte rn a tiv e s o f n e a r o p tim a l s o lu tio n s , g iv in g th e d e s ig n e r a g lo b a l o v e rv ie w o f w h a t c a n b e a c c o m p lis h e d w ith th e u s e d c o n tro lle r s tru c tu re , a n d b y th e m e th o d o f p re fe re n c e -d ire c te d d e s ig n th a t s o lu tio n , w h ic h re s u lts in g re a te s t o v e ra ll d e s ig n e r s a tis fa c tio n , is s e le c te d . L L 3 .1 M M A N e w G e n e tic A lg o r ith m to G e n e r a te a n E v e n ly D is p e r s e d S e t o f S o lu tio n s A n a tu ra l w a y o f fin d in g th e p a re to s e t b y g e n e tic a lg o rith m s w a s p ro p o s e d b y F o n s e c a a n d F le m in g [8 ]. T h is a p p ro a c h m e a s u re s th e fitn e s s o f a n in d iv id u a l b y th e n u m b e r o f o th e r in d iv id u a ls th a t d o m in a te it in th e s e n s e o f p a re to p re fe re n c e . A c c o rd in g ly , a p o p u la tio n is ra n k e d , w h e re th e b e s t s o lu tio n s w ill b e th e n o n d o m in a te d o n e s . T h u s , th e n o n -d o m in a te d s o lu tio n s w ill a lw a y s b e m o s t lik e ly to b e s e le c te d , le a d in g to a c o n v e rg e n c e o f th e p o p u la tio n to th e p a re to s e t. F ig u re 3 g iv e s a n e x a m p le o f h o w a p o p u la tio n w o u ld b e ra n k e d w ith th is is th e g o a l. In th is a lg o rith m , w h e re m in im iz a tio n o f tw o q u a lity m e a s u re s C , C e x a m p le , th e re a re th re e n o n -d o m in a te d s o lu tio n s w h ic h a re ra n k e d w ith a z e ro ; a ll   E v o lu tio n a ry C o m p u ta tio n a n d N o n lin e a r P ro g ra m m in g o th e r s o lu tio n s a re d o m in a te d a n d th e ir ra n k s o lu tio n s a re b e tte r in th e p a re to s e n s e . is d e te rm in e d b y h o w m a n y 1 5 1 o th e r &       & F ig . 3 . P a re to ra n k in g a c c o rd in g to F o n s e c a a n d F le m in g [8 ] H o w e v e r, d u e to g e n e tic d rift, th e p o p u la tio n te n d s to c o n v e rg e to a s in g le p o in t o n th e p a re to s u rfa c e . S in c e it is th e a im o f a n e x p lo ra to ry d e s ig n p h a s e to p ro v id e th e d e s ig n e r w ith a ric h b a s is o f a lte rn a tiv e s fo r p o s s ib le tra d e o ff d e c is io n s , fin d in g o n ly o n e p a re to s o lu tio n is u n s a tis fa c to ry . T h u s , g e n e tic d rift s h o u ld b e a v o id e d , w h ic h c a n b e a c c o m p lis h e d b y m e th o d s lik e m u ltip le s u b -p o p u la tio n s o r p e n a liz in g o v e rc ro w d e d n e ig h b o rh o o d s (fitn e s s s h a rin g ) [3 ]. A v a ria n t o f fitn e s s s h a rin g is p ro p o s e d h e re fo r th e p u rp o s e o f fin d in g a n e v e n ly d is p e r s e d n e a r-p a re to -o p tim a l s e t o f s o lu tio n s . T h e id e a is to in c o rp o ra te th e p ro x im ity o f o th e r in d iv id u a ls in to th e fitn e s s o f o n e in d iv id u a l s u c h th a t in d iv id u a ls in re m o te re g io n s o f th e s e a rc h s p a c e w ill e n jo y a n a d v a n ta g e o v e r th o s e in m o re o v e rc ro w d e d re g io n s . S in c e s o m e in d iv id u a ls in th e c ro w d e d re g io n s s h o u ld re m a in , s o a s to n o t d is tra c t th e g e n e tic a lg o rith m fro m p ro m is in g re g io n s o f th e s e a rc h s p a c e , p e n a ltie s fo r c ro w d in g a re lim ite d to v a lu e s le s s th a n 1 . A s th e fitn e s s s c o re fro m p a re to ra n k in g is a n in te g e r, th is e n s u re s th a t th e in d iv id u a ls a re a lw a y s ra n k e d firs t b y p a re to d o m in a tio n b u t a m o n g in d iv id u a ls o f th e s a m e p a re to ra n k , th e in d iv id u a ls a re fo rc e d to s p re a d o u t e v e n ly . )/(c -c ) th e H a v in g m a p p e d th e n c r ite r ia v a lu e s to [ 0 ,1 ] b y c = ( c - c p e n a lty to a tta in a n e v e n ly d is p e rs e d s e t is c o m p u te d b y th e fo llo w in g fo rm u la : L S= (1 − δ ) ⋅ 2 2 ∑ FL , FXUUHQW − FL , FORVHVW ⋅ 1 L LPLQ Q. LPD[ LPLQ (2 ) L= 1 . . Q w h e r e δ i s c h o s e n s m a l l b u t g r e a t e r t h a n z e r o HJ δ = 0 1 FLM s i g n i f i e s t h e L c r i t e r i o n v a l u e o f t h e M i n d i v i d u a l , ’ FXUUHQW’ i s t h e i n d i v i d u a l f o r w h i c h t h e p e n a l t y i s c a l c u l a t e d a n d ’ FORVHVW’ i s t h e i n d i v i d u a l w h i c h i s t h e c l o s e s t p r e c e d i n g t h e c u r r e n t o n e in a lis t o f a ll in d iv id u a ls s o rte d b y p a re to -ra n k . T h is lis t is s o rte d in a s c e n d in g o rd e r, s o th a t th e b e s t in d iv id u a ls c o m e firs t, a n d b e tw e e n tw o in d iv id u a ls o f th e s a m e ra n k , th e p o s itio n in th e lis t is d e c id e d b y ra n d o m . T h u s , th e p e n a lty is a lw a y s s m a lle r th a n 1 a n d th e a d ju s te d fitn e s s v a lu e is a lw a y s s tric tly p o s itiv e , w h ic h is n e c e s s a ry fo r s o m e g e n e tic s e a rc h a lg o rith m s . T h is v a ria n t o f a m u ltio b je c tiv e g e n e tic a lg o rith m h a s b e e n im p le m e n te d b y m e a n s o f th e G e n e tic a n d E v o lu tio n a ry A lg o rith m T o o lb o x [9 ]. R e s u lts s h o w th a t th is WK WK 1 5 2 D . K o lo s s a a n d G . G rü b e l ra n k in g p ro c e d u re a s s u re s c o n v e rg e n c e to w a rd s th e e n tire p a re to s e t a s o p p o s e d to e ith e r c o n v e rg in g to ju s t a p a rt o f it o r fa v o rin g re m o te a re a s o f th e s e a rc h s p a c e w ith le s s th a n o p tim a l c rite ria v a lu e s . T h e e v o lu tio n a ry o p tim iz a tio n is c a rrie d o u t u n til e ith e r a c e rta in n u m b e r o f g e n e ra tio n s is re a c h e d o r a liv e lin e s s c rite rio n is m e t. B u t s in c e n o a n a ly tic a l c o n v e rg e n c e c rite rio n is u s e d th e re is n o g u a ra n te e fo r o b ta in in g e x a c t o p tim a l s o lu tio n s ra th e r th a n a tta in in g a n e a r p a re to -o p tim a l s e t. O n e e x a m p le o f h o w th is m u ltio b je c tiv e g e n e tic a lg o rith m fa re s w ith p e n a ltie s (2 ) c a n b e s e e n in F ig u re 4 . T h e o p tim iz a tio n re s u lts a re s h o w n to g e th e r w ith th e b o u n d a ry o f th e s e t o f a ll p o s s ib le s o lu tio n s . A s m e a s u re fo r th e d e g re e o f s p re a d a lo n g th e p a re to s u rfa c e th e a v e ra g e s ta n d a rd d e v ia tio n in th e o b ta in e d p a re to v a lu e s e t, c o m p u te d o v e r 1 0 0 0 ru n s a n d s c a le d to a m a x im u m o f 1 , is s h o w n in T a b le 1 . T a b le 1 . A v e ra g e s ta n d a rd d e v ia tio n in c o m p u te d p a re to v a lu e s e t W ith o u t P e n a ltie s 0 .1 4 9 3 A v e ra g e S ta n d a rd D e v ia tio n W ith P e n a ltie s 0 .2 4 1 7 All Generations 4000 : In itia l P o p u la tio n 3000 2000 1000 0 1 2 3 4 5 6 7 8 6 7 8 Final Generation 4000 3000 2000 1000 0 1 2 3 4 5 F ig . 4 . E x a m p le : c o n v e rg e n c e to e v e n ly d is p e rs e d p a re to -o p tim a l v a lu e s e t 3 .2 D e s ig n S e le c tio n fo r B e s t D e s ig n e r S a tis fa c tio n A fte r a n e d e s ig n e r s b y a n o rm th e s m a lle v e n ly a tis fa c a liz e d r th e v d is p e rs e d s e t o f s o lu tio n s is fo u n d , th a t s o lu tio n w h ic h g iv e s h ig h e s t tio n is to b e s e le c te d . C o m m o n ly , d e s ig n e r s a tis fa c tio n is m e a s u re d w e ig h te d -s u m v a lu e fu n c tio n v (q ) a n d a d e s ig n is ju d g e d th e b e tte r a lu e v (q ): Y = ∑ Z ⋅ T ( 7 ) , w i t h : ∑ ZN = 1 , ∀ N : ZN > 0 , TN ≥ 0 . N (3 ) E v o lu tio n a ry C o m p u ta tio n a n d N o n lin e a r P ro g ra m m in g T h e p ro b le m , c o m p lia n c e w ith m in im a l v a lu e v ( [1 0 ] c o p e s w ith n u m b e r o f p a ir-w If a d e s ig n e r m th is im p lie s h o w e v e r, is h o w to a ttrib u te d e s ig n e r’s in te n tio n s to fo rm a q ). T h e a p o s te rio ri a p p ro a c h o th is p ro b le m b y s o -c a lle d im is e p re fe re n c e s ta te m e n ts to b e a k e s a p re fe re n c e s ta te m e n t “ T ∑ Y( 7 ) = L ∑ Z ⋅ T (7) < N LN L N a p rio ri th e n u m e ric a l w e ig lly d e c id e o n th e ‘b e s t’ s o lu f p re fe re n c e -d ire c te d d e s ig n p re c is e v a lu e fu n c tio n s b a m a d e b y th e d e s ig n e r. u n i n g T L i s s u p e r i o r t o T M” t h Z ⋅ T (7) , N MN 1 5 3 h ts w N in tio n w ith s e le c tio n se d o n a e n b y (3 ) (4 ) M N w h ic h c a n b e re w ritte n a s ∑ ∑ Z ( T ( 7 ) − T ( 7) ) > 0 , N MN M LN L N W ith k n o w n q , q a d m is s ib le w e ig h ts w w h e th e r a c o m p a tib le w e ll. T h is is fo rm a lly LN , a se t . In [1 0 p re fe re d e c id e d MN N m in ∑ Z o f ] th n c e b y O − 1 ,N ( 72 * O − 1 (5 ) N p re fe re n c e s ta te m e is c o n s titu te s a n ‘im o rd e rin g e x is ts a m s o lv a b ility o f th e fo Z ( T2 * N ZN = 1 , ∀ N : ZN > 0 . ) − T2 * n ts p re o n g llo w , O N (5 ) c is e o th in g d e s c rib e s v a lu e fu n e r d e s ig n lin e a r p ro a su b sp a c e fo r c tio n ’ to c h e c k a lte rn a tiv e s a s g ra m : (6 ) ( 72 * ) O N V. W. ∑ Z ( T (7 ) − T (7)) > 0 N MN M LN L N : ∑ Z ( T2 * N O − 1 ,N ( 72 * O − 1 ) − T2 * , O N ( 72 * ) ) > 0 O N ∑ ZN = 1 , ∀ N : ZN > 0 , N , q w h e re fo r j = (2 * l-1 ) a n d i = (2 * l) th e q u a lity fu n c tio n v a lu e s q o f a p a ir o f a d d itio n a l in fe rio r a n d s u p e rio r d e s ig n a lte rn a tiv e s , re s p e c tiv e ly . B a s e d o n a fe w c o m p a tib le p re fe re n c e s ta te m e n ts (5 ) th is fo rm a lis m p a rtia lly o rd e r th e s e t o f d e s ig n a lte rn a tiv e s fo r s e le c tin g ‘th e b e s t’ o n e . T h e v a lu e fu n c tio n is a m e a n s to p ru n e in fe rio r p a th s d u rin g d e s ig n s p a c e s e re d u c in g th e c o m p le x ity o f th e s e le c tio n p ro c e s s . H a v in g fin a lly s e le c te d a v a ila b le d e s ig n a lte rn a tiv e , a c o m p a tib le s e t o f w e ig h ts c a n b e c o m p u te m in im iz in g th e d iffe re n c e o f th e v a lu e fu n c tio n o f th e c h o s e n a lte rn a tiv e to i.e . s u p e r io r , v a lu e o f d e s ig n e r s a tis f a c tio n , w h ic h is c h a r a c te r iz e d b y q = 0 .  ON  N ON a re th o s e a llo w s to im p re c is e a rc h th u s th e b e s t d b y (6 ), its id e a l, 1 5 4 4 D . K o lo s s a a n d G . G rü b e l P a r e to -T u n in g b y I n te r a c tiv e N o n lin e a r P r o g r a m m in g T h e g e n e tic a lg o rith m w ith p a re to p re fe rre d ra n k in g , a s p r o p o s e d in S e c tio n 3 .1 , fits w e ll to th e n a tu re o f e v o lu tio n a ry c o m p u ta tio n s in c e o n re tu rn to th e m a n y fu n c tio n e v a lu a tio n s th a t e v o lu tio n a ry c o m p u ta tio n re q u ire s it y ie ld s a ric h s e t o f s o lu tio n s e v e n ly d is p e rs e d in o r c lo s e to th e e n tire p a re to -o p tim a l s e t. F u rth e rm o re , g e n e tic a lg o rith m s c o p e w e ll w ith a la rg e n u m b e r o f p a ra m e te rs a n d w ith a la rg e s e a rc h s p a c e , w h ic h m a k e s th e m lik e ly to fin d th e g lo b a l in s te a d o f a lo c a l s o lu tio n in m u ltim o d a l p ro b le m s . T o g e th e r w ith fo rm a l d e c is io n s u p p o rt, a s d e a lt w ith in S e c tio n 3 .2 , th is is w e ll s u ite d fo r s e le c tin g a p o s te rio ri a d e s ig n c a n d id a te w ith b e s t d e s ig n e r s a tis fa c tio n a m o n g a n u m b e r o f g lo b a l d e s ig n a lte rn a tiv e s . O n th e o th e r s id e , n o n lin e a r p ro g ra m m in g a p p ro a c h e s to a tta in p a re to -o p tim a l s o lu tio n s a re b a s e d o n a n a n a ly tic a l o p tim a lity c o n d itio n , w h ic h m a k e s th e m v e ry e ffic ie n t to c o m p u te ju s t o n e , a p rio ri d e d ic a te d , p a re to -o p tim a l s o lu tio n in th e lo c a l n e ig h b o rh o o d o f w h e re th e a lg o rith m s g e ts s ta rte d . F u rth e rm o re , th e n e c e s s a ry K a ru s h -K u h n -T u c k e r o p tim a lity c o n d itio n s y ie ld a n u m e ric a l c o n v e rg e n c e c o n d itio n th a t a llo w s to a tta in a p a re to o p tim u m w ith h ig h a c c u ra c y . T h is m a k e s n o n lin e a r p ro g ra m m in g a lg o rith m s s u ita b le fo r ‘fin e tu n in g ’. If p a ra m e te riz e d in a d e c is io n in tu itiv e w a y , in te ra c tiv e , d e c la ra tiv e s e a rc h to a tta in a s p e c ific , ‘b e s t’, c o m p ro m is in g s o lu tio n fo r re q u ire m e n t s a tis fa c tio n u n d e r lo c a l d e s ig n c o n flic ts b e c o m e s fe a s ib le . N o n lin e a r P ro g ra m m in g c a n b e u s e d to c o m p u te p a re to -o p tim a l tu n in g v a lu e s T b y s o lv in g [1 1 ] th e n o rm a liz e d w e ig h te d -s u m m in im iz a tio n p ro b le m , c f. (3 ) m i n ∑ ZN TN , 7 V. W. ∑ ZN = 1 , ∀ N : ZN > 0 , T ≥ 0 , N N (7 ) o r th e m in -m a x o p tim iz a tio n p ro b le m { T ( 7) / G } m in m a x 7 L L L , V. W.  ≤ T ( 7 ) ≤ G , N = { L, M} . N M (8 ) O p tim iz a tio n (7 ) o r (8 ) is a s u ffic ie n t c o n d itio n to y ie ld a p a re to -o p tim a l s o lu tio n w ith p a ra m e te riz e d s p e c ific p ro p e rtie s . T h e re a re s ta n d a rd a lg o rith m s o f n o n lin e a r p r o g r a m m in g , lik e S Q P , to b e u s e d s ta r tin g w ith a ( g lo b a l) s o lu tio n b y S e c tio n 3 .1 . F o rm u la tio n (7 ) y ie ld s th e s o lu tio n o f o p tim a l d e s ig n e r s a tis fa c tio n w ith re s p e c t to a ttr ib u te d w e ig h ts a s f o u n d , e .g ., b y th e a p o s te r io r i d e c is io n p r o c e d u r e o f S e c tio n 3 .2 . A p rio ri s e le c tio n o f w e ig h ts is n o t d e c is io n in tu itiv e [1 1 ] to a tta in s p e c ific p ro p e rtie s . F o rm u la tio n (8 ) y ie ld s a p a re to -o p tim a l s o lu tio n d e p e n d in g o n p a ra m e te rs d . T h is fo rm u la tio n is w e ll s u ite d fo r c o m p ro m is in g c o m p e tin g re q u ire m e n ts b y u s in g th e s e p a ra m e te rs ite ra tiv e ly a s a p rio ri ‘d e s ig n d ir e c to r s ’ to b a la n c e re q u ire m e n ts s a tis fa c tio n w ith in a fe a s ib le s o lu tio n s e t. A d e c is io n in tu itiv e a p p ro a c h to c h o o s e d e s ig n d ire c to rs d is n o w p ro p o s e d , w h ic h is in s p ire d b y th e p e rfo rm a n c e v e c to r d e c is io n s y s te m a tic s d u e to K re is s e lm e ie r [1 2 ]. In itia liz a tio n S te p : S ta rt b y a g lo b a l s o lu tio n w ith d e s ig n e r s a tis fa c tio n a c c o rd in g to S e c tio n 3 a n d p a re to -o p tim iz e th is s o lu tio n b y s o lv in g th e u n c o n s tra in e d m in -m a x p ro b le m w ith d = { 1 } , j ∈ ∅ . T h is y ie ld s a b a la n c e d s o lu tio n w h e re a ll q u a lity fu n c tio n s g e t p a re to -m in im a l a n d th e v a lu e fu n c tio n fo r d e s ig n e r s a tis fa c tio n is fu rth e r im p ro v e d if th e s ta rt s o lu tio n is n o t y e t a n o p tim iz e d o n e . L E v o lu tio n a ry C o m p u ta tio n a n d N o n lin e a r P ro g ra m m in g 1 5 5 B y s o lv in g th e u n c o n s tra in e d m in -m a x p ro b le m , th e q u e s tio n fo r th e m a in c o n flic ts * in re q u ire m e n ts s a tis fa c tio n c a n b e a n s w e re d : If T = T is a m in im iz e r, th e n m a x { TL ( 7 * ) } = TF* 1 = TF* 2 = α * > 0 , (9 ) w h i c h m e a n s t h a t t h e v a l u e s o f q u a l i t y f u n c t i o n s TF1 , TF 2 , . . . b e l o n g i n g t o t h e m o s t c o m p e tin g re q u ire m e n ts a re e q u a l a n d th a t th e y h a v e th e la rg e s t v a lu e a m o n g a ll q u a lity fu n c tio n s . M o re o v e r, α ≤ 1 c h a ra c te riz e s a fe a s ib le (‘a c c e p ta b le ’) s o lu tio n , w h ic h g iv e s ro o m fo r c o m p ro m is in g th e m o s t c o m p e tin g re q u ire m e n ts w ith in th e s e t o f p a re to -o p tim a l a lte rn a tiv e s . Ite r a tiv e C o m p r o m is in g S te p s : S ta rtin g w ith a p a re to -o p tim a l s o lu tio n , s a tis fa c tio n o f c o m p e tin g re q u ire m e n ts c a n n o t b e im p ro v e d s im u lta n e o u s ly . T h is m e a n s th a t l o w e r i n g t h e v a l u e o f o n e q u a l i t y f u n c t i o n , q F 7 , c a n b e a c h i e v e d o n l y a t t h e e x p e n s e o f a h i g h e r v a l u e o f a n o t h e r , q F 7 , a n d v i c e v e r s a . D i f f e r e n t c o m p r o m i s e s o l u t i o n s q F v e r s u s q F c a n b e a c h i e v e d b y d i f f e r e n t c h o i c e s o f t h e d e s i g n d i r e c t o r s d in a n ite ra tiv e p ro c e d u re : W i t h a g i v e n p a r e t o - o p t i m a l s o l u t i o n 4 ( ν − 1 ) = { TL( ν − 1 ) , TF( ν − 1 ) } , f o r t h e n e x t i t e r a t i o n s t e p d e c i d e w h i c h o f t h e m o s t c o n f l i c t i n g q u a l i t y f u n c t i o n s { q F q F` s h a l l b e i m p r o v e d , s a y q F. T h i s c h o i c e m a y b e m a d e d e p e n d e n t o n t h e w e i g h t s t h a t a r e a s s o c ia te d to th e s e q u a lity fu n c tio n s v ia th e fo rm a lis m o f S e c tio n 3 . T h e n , c o n c a te n a te N ( ν ) ∈ { L, F1 } , a n d c h o o s e G N = 1 a n d G F 2 s u c h t h a t TF( ν 2 − 1 ) < G νF 2 (≤ 1 ). S o lv in g th e c o n s tr a in e d m in -m a x p ro b le m , m i n m a x { TN( Y ) ( 7 ) } , 7 N V. W. TF 2 ( 7 ) ≤ G F( 2Y ) , (1 0 ) th e a re to p ro re q n n a tta in s th e b e s t p o s s ib le s o lu tio n in th e s e n s e th a t a ll q u a lity fu n c tio n s o f in te re s t m in im iz e d u p to th e c o n s tra in t o f th e q u a n tifie d lim it o f d e g ra d a tio n o n e d e c la re s b e a c c e p ta b le fo r th e m a in c o n flic tin g q u a lity fu n c tio n . T h u s in a n ite ra tiv e c e d u re o n e c a n s e a rc h fo r a ‘b e s t’ c o m p ro m is e s a tis fa c tio n o f c o m p e tin g u ire m e n ts . Ite ra tiv e c o m p ro m is in g is b e s t c a rrie d o u t in a n in te r a c tiv e m o d e o f w o r k in g w h ic h e e d s fa s t a lg o rith m s to e x e c u te th e C A C S D c o m p u ta tio n lo o p o f F ig u re 1 . In d d itio n it n e e d s v is u a l d e c is io n s u p p o rt o n v a rio u s in fo rm a tio n le v e ls to b e s t g ra s p e s ig n p ro b le m c o m p le x ity . In p a rtic u la r, a g ra p h ic a l u s e r in te rfa c e [1 3 ] w ith a a ra lle l c o o rd in a te s d is p la y o f th e m a n y q u a lity fu n c tio n s , u s e d a s in te ra c tiv e s te e rin g id to d e te c t c o m p ro m is e c o n flic ts a n d to c h o o s e d e s ig n d ire c to rs a t ru n tim e , g re a tly n h a n c e s e n g in e e rin g p ro d u c tiv ity . A N D E C S _ M O P S is s u c h a n e n v iro n m e n t [1 4 ]. a d p a e 5 P a r e to -O p tim a l M u lti-m o d e l R o b u s tn e s s T u n in g F e e d b a c k c o n tro l s u ffe rs fro m p o te n tia l s ta b ility p ro b le m s , b u t if p ro p e rly d e s ig n e d fe e d b a c k re d u c e s p a ra m e te r s e n s itiv ity . T h e re fo re d e s ig n o f c o n tro lle rs , w h ic h a re s ta b ility a n d p e rfo rm a n c e ro b u s t w ith re s p e c t to o ff-n o m in a l o p e ra tio n , is o f p rim e 1 5 6 D . K o lo s s a a n d G . G rü b e l c o n c e rn . A n a ly tic a l ro b u s t-c o n tro l th e o ry , lik e µ -s y n th e s is , re lie s o n a n a ly tic a l s ta b ility c rite ria a n d p e rtin e n t (lin e a r) p la n t m o d e l a n d u n c e rta in ty d e s c rip tio n s . T h u s , it is re s tric te d to p ro b le m s w ith s p e c ific , c o m m e n s u ra b le , p e rfo rm a n c e m e a s u re s . A c o m p le te ly g e n e ra l a p p ro a c h to ro b u s t c o n tro l d e s ig n is s o -c a lle d m u lti-m o d e l d e s ig n a s im p lie d b y th e C A C S D tu n in g lo o p , c f. F ig u re 1 . It is a p p lic a b le to a n y k in d o f (n o n -lin e a r) p la n t m o d e ls a n d n o n -c o m m e n s u ra b le p e rfo rm a n c e m e a s u re s s in c e it re lie s o n ly o n th e d a ta o f th e p e rfo rm a n c e m e a s u re s a n d n o t o n th e ir a n a ly tic a l d e s c rip tio n . S tru c tu ra l in d e p e n d e n c e m a k e s th is k in d o f ro b u s t c o n tro l a p p ro a c h a p p lic a b le to a n y ty p e o f c o n tr o lle r , i.e ., P I D , o b s e r v e r f e e d b a c k , f u z z y c o n tr o l, e tc . T h e id e a o f ro b u s t m u lti-m o d e l d e s ig n is to s ta te th e d e s ig n p ro b le m fo r a n o m in a l p la n t m o d e l in s ta n tia tio n re fle c tin g n o m in a l o p e ra tio n c o n d itio n s a n d n o m in a l s y s te m p a ra m e te rs w ith in p e rtin e n t to le ra n c e b a n d s . T h e n , th e s a m e p ro b le m is s ta te d fo r a n u m b e r o f o f f - n o m in a l m o d e l in s ta n tia tio n s r e f le c tin g w o r s t c a s e p la n t b e h a v io r , e .g ., fa s t, lig h tly d a m p e d , a n d s lo w , o v e r-d a m p e d , b e h a v io r w ith in th e ra n g e o f a s s u m e d o p e ra tio n c o n d itio n s a n d p a ra m e te r u n c e rta in ty in te rv a ls . T h e q u a lity fu n c tio n s o f a ll th e s e fo rm u la tio n s a re c o n c a te n a te d to a s in g le m u ltio b je c tiv e p ro b le m fo r w h ic h a s a tis fy in g p a re to -o p tim a l s o lu tio n is to b e fo u n d . T h is a p p ro a c h is h ig h ly c o m p e titiv e in c o m p a ris o n to o th e r (a n a ly tic ) ro b u s t c o n tro l a p p ro a c h e s [1 5 ]. T h e s o lu tio n a p p ro a c h e s th a t a re d e a lt w ith in S e c tio n s 2 , 3 , 4 , a re p a rtic u la rly s u ita b le to b e c o m b in e d fo r th is ty p e o f m u lti-m o d e l, m u lti-o b je c tiv e , ro b u s t c o n tro l d e s ig n in fo rm o f a tw o -p h a s e d e s ig n p r o c e d u r e : In p h a s e 1 , o n ly a n o m in a l m o d e l in s ta n tia tio n is c o n s id e re d a n d in te rv a l q u a lity fu n c tio n s fo rm u la te d a c c o rd in g to S e c tio n 2 a re o p tim iz e d b y th e m u ltio b je c tiv e g e n e tic a lg o rith m o f S e c tio n 3 .1 to y ie ld a ric h s e t o f g lo b a l, p a re to -o p tim a l, d e s ig n a lte r n a tiv e s a s b a s is f o r d e s ig n e r p r e f e r e n c e s e le c tio n , S e c tio n 3 .2 . In p h a s e 2 , th e ‘b e s t’ n o m in a l p e rfo rm a n c e a c h ie v e d in p h a s e 1 is e m b e d d e d in ‘g o o d ’ in te rv a ls b y re -s c a lin g th e q u a lity le v e ls a s re q u ire d . T h is s o lu tio n is u s e d to s ta rt fu rth e r tu n in g u n d e r th e a s p e c t o f ro b u s tn e s s : T h e o ff-n o m in a l d e s ig n c a s e s a re a d d e d to th e n o m in a l c a s e a n d in te ra c tiv e n o n lin e a r p ro g ra m m in g ite ra tio n s a c c o rd in g to S e c tio n 4 a re s im u lta n e o u s ly a p p lie d to a ll d e s ig n c a s e s to ro b u s tify th e re s u lt o f th e firs t, n o m in a l, d e s ig n p h a s e . T h u s , th e u s e r is a llo w e d to m a k e q u a n tita tiv e tra d e -o ff d e c is io n s c o n c e rn in g n o m in a l v e rs u s ro b u s t p e rfo rm a n c e . In th e s e d e c is io n s o ff-n o m in a l c o n tro l b e h a v io r, c h a ra c te riz e d b y w o rs t-c a s e p la n t m o d e l in s ta n tia tio n s , s h o u ld b e c o m e a t le a s t ‘a c c e p ta b le ’, w h ile n o m in a l b e h a v io r is to b e k e p t w ith in a ‘g o o d ’ q u a lity le v e l. 6 C o n c lu s io n A p a ra m e te r tu n in g m e th o d o lo g y to s u p p o rt c o n tro l d e s ig n a u to m a tio n is d e s c rib e d . It u s e s a m u ltio b je c tiv e g e n e tic a lg o rith m w ith fitn e s s s h a rin g to fin d a ric h s e t o f g lo b a l s o lu tio n s e v e n ly d is p e rs e d in o r n e a r to th e p a re to -o p tim a l s e t, fro m w h ic h a d e s ig n c a n d id a te fo r b e s t d e s ig n e r s a tis fa c tio n is fo rm a lly s e le c te d v ia p a ir-w is e p re fe re n c e s ta te m e n ts . T h e n m in -m a x n o n lin e a r p ro g ra m m in g is a p p lie d fo r c o m p ro m is e tu n in g to a tta in a p a re to -o p tim a l s o lu tio n w ith b e s t tra d e o ffs in re q u ire m e n ts s a tis fa c tio n . A n o n -lin e in te ra c tiv e m o d e o f w o rk in g u s in g n o n lin e a r p ro g ra m m in g in th e c o m p ro m is in g p h a s e is s u p p o rte d b y a s y s te m a tic s fo r c h o o s in g ‘d e s ig n d ire c to rs ’ a s E v o lu tio n a ry C o m p u ta tio n a n d N o n lin e a r P ro g ra m m in g a llo T o g p h a n o n w a b le e th e r se a p p -c o m m u p p w ith ro a c e n s e r b o u n d s to re q u ire m e n h is w e ll s u u ra b le p e rfo lim it th e ts c a p tu re ite d to q u rm a n c e a n e x b y a n d p e n se o fu z z y tita tiv e ro b u s tn n e ty p m u e ss is w e in ltire q illin g to p te rv a l q u a m o d e l-ro b u ire m e n ts 1 5 7 a y in m a k in g tra d e o ffs . lity fu n c tio n s , th is tw o u s t c o n tro l d e s ig n w ith . R e fe r e n c e s 1 . G r ü b e l, G .: P e r s p e c tiv e s o f C A C S D : E m b e d d in g th e C o n tr o l S y s te m D e s ig n P r o c e s s in to a V irtu a l E n g in e e rin g E n v iro n m e n t. P ro c . IE E E In t. S y m p o s iu m o n C o m p u te r A id e d C o n tro l S y s te m D e s ig n , H a p u n a -B e a c h , H a w a ii (1 9 9 9 ) 2 9 7 -3 0 2 2 . F e n g , W ., L i, Y .: P e r f o r m a n c e I n d ic e s in E v o lu tio n a r y C A C S D A u to m a tio n w ith A p p lic a tio n to B a tc h P ID G e n e ra tio n . P ro c . IE E E In t. S y m p o s iu m o n C o m p u te r A id e d C o n tro l S y s te m D e s ig n , H a p u n a -B e a c h , H a w a ii (1 9 9 9 ) 4 8 6 -4 9 1 3 . C h ip p e r f ie ld , A .J ., D a k e v , N .V ., F le m in g , P .J ., W h id b o r n e , J .F .: M u ltio b je c tiv e R o b u s t C o n tro l U s in g E v o lu tio n a ry A lg o rith m s . P ro c . IE E In t. C o n f. In d u s tria l T e c h n o lo g y (1 9 9 6 ) 2 6 9 -2 7 3 4 . J o o s , H .- D ., S c h lo th a n e , M ., G r ü b e l, G .: M u lti- O b je c tiv e D e s ig n o f C o n tr o lle r s w ith F u z z y L o g ic . P ro c . IE E E /IF A C J o in t S y m p o s iu m o n C o m p u te r A id e d C o n tro l S y s te m D e s ig n , T u c so n , A Z (1 9 9 4 ) 7 5 -8 2 5 . Z a k ia n , V ., A l-N a ib , U .: D e s ig n o f D y n a m ic a l a n d C o n tr o l S y s te m s b y th e M e th o d o f In e q u a litie s . P ro c . In s titu te o f E le c tric a l E n g in e e rs , V o l. 1 2 0 , N o . 1 1 . (1 9 7 3 ) 1 4 2 1 -1 4 2 7 6 . T a n , K .C ., L e e , T .H ., K h o r , E .F .: C o n tr o l S y s te m D e s ig n A u to m a tio n w ith R o b u s t T r a c k in g T h u m b p rin t P e rfo rm a n c e U s in g a M u lti-O b je c tiv e E v o lu tio n a ry A lg o rith m . P ro c . IE E E S y m p o s iu m o n C o m p u te r-A id e d C o n tro l S y s te m D e s ig n , H a p u n a -B e a c h , H a w a ii (1 9 9 9 ) 4 9 8 -5 0 3 7 . K ie n itz , K .H .: C o n tro lle r D e s ig n U s in g F u z z y L o g ic – A C a s e S tu d y . A u to m a tic a , V o l. 2 9 , N o . 2 . (1 9 9 3 ) 5 4 9 -5 5 4 8 . F o n s e c a , C .M ., F le m in g , P .J .: A n O v e r v ie w o f E v o lu tio n a r y A lg o r ith m s in M u ltio b je c tiv e O p tim iz a tio n . E v o lu tio n a ry C o m p u tin g , V o l. 3 , N o . 1 . (1 9 9 5 ) 1 -1 6 9 . P o h lh e im , H .: G e n e tic a n d E v o lu tio n a ry A lg o rith m T o o lb o x fo r U s e w ith M a tla b D o c u m e n ta tio n . T e c h n ic a l R e p o rt, T e c h n ic a l U n iv e rs ity Ilm e n a u , (1 9 9 6 ) 1 0 . d 'A m b r o s i o , J . G . , B i r m i n g h a m , W . P . : P r e f e r e n c e - d i r e c t e d D e s i g n . A r t i f i c i a l I n t e l l i g e n c e f o r E n g in e e rin g D e s ig n , A n a ly s is a n d M a n u fa c tu rin g , V o l. 9 . (1 9 9 5 ) 2 1 9 -2 3 0 1 1 . M ie ttin e n , K .M .: N o n lin e a r M u ltio b je c tiv e O p tim iz a tio n , K lu w e r A c a d e m ic P u b lis h e r s , (1 9 9 8 ) 1 2 . K r e is s e lm e ie r , G ., S te in h a u s e r , R .: A p p lic a tio n o f V e c to r P e r f o r m a n c e O p tim iz a tio n to R o b u s t C o n tro l L o o p D e s ig n o f a F ig h te r A irc ra ft. In t. J o u rn a l C o n tro l, V o l. 3 7 , N o . 2 . (1 9 8 3 ) 2 5 1 -2 8 4 . 1 3 . F in s te r w a ld e r , R ., J o o s , H .- D ., V a r g a , A .: A G r a p h ic a l U s e r I n te r f a c e f o r F lig h t C o n tr o l D e v e lo p m e n t. P ro c . IE E E S y m p o s iu m o n C o m p u te r-A id e d C o n tro l S y s te m D e s ig n , H a p u n a -B e a c h , H a w a ii (1 9 9 9 ) 4 3 9 -4 4 4 1 4 . G r ü b e l, G ., F in s te r w a ld e r , R ., G r a m lic h , G ., J o o s , H .- D ., L e w a ld , S .: A N D E C S : A C o m p u ta tio n E n v iro n m e n t fo r C o n tro l A p p lic a tio n s o f O p tim iz a tio n . In : C o n tro l A p p lic a tio n s o f O p tim iz a tio n , R . B u lir s c h , D . K r a f t, e d s ., I n t. S e r ie s o f N u m e r ic a l M a th e m a tic s , V o l. 1 1 5 . B irk h ä u s e r V e rla g , B a s e l (1 9 9 4 ) 2 3 7 -2 5 4 1 5 . G r ü b e l, G .: A n o th e r V ie w o n th e D e s ig n C h a lle n g e A c h ie v e m e n ts . I n : R o b u s t F lig h t C o n tr o l – A D e s ig n C h a lle n g e , M a g n i, J .F ., B e n n a n i, S ., T e r lo u w , J .C ., E d s ., L e c tu r e N o te s in C o n tro l a n d In fo rm a tio n S c ie n c e s 2 2 4 , S p rin g e r V e rla g , B e rlin H e id e lb e rg N e w Y o rk (1 9 9 7 ) 6 0 3 -6 0 9 Benchmarking Cost-Assignment Schemes for Multi-objective Evolutionary Algorithms Konstantinos Koukoulakis, Dr Yun Li Department of Electronics and Electrical Engineering, University of Glasgow Abstract. Currently there exist various cost-assignment schemes that perform the necessary scalarization of the objective values when applied to a multi-objective optimization problem. Of course, the final decision depends highly on the nature of the problem but given the multiplicity of the schemes combined with the fact that what the user ultimately needs is a single compromise solution it is evident that elaborating the selection of the method is not a trivial task. This paper intends to address this problem by extending the benchmarks of optimality and reach time given in [1] to mutliobjective optimization problems. A number of existing cost-assignment schemes are evaluated using such benchmarks. 1. Introduction Having in mind the number of existing approaches to cost-assignment one could presume that the next step would be an appropriate choice. The concept of Pareto dominance has proven to be a great aid towards the formulation of the various schemes but further thinking reveals that what the user would like to have is simply a single compromise solution and not all of the solutions that form the Pareto-optimal set. As stated in [2], “although a Pareto-optimal solution should always be a better compromise solution than any solution it dominates, not all Pareto-optimal solutions may constitute acceptable compromise solutions”. Therefore, what is needed is a performance index that can be used for the evaluation of the suitability of each scheme in the context of a specific problem. To address this issue, two benchmarks used presented in Section 2. Section 3 describes a benchmark problem used for the comparison. Section 4 presents the evolutionary algorithm employed along with an outline of the various cost-assignment methods. Comparison is made between a number of evolutionary algorithms in Section 5. Conclusions are drawn in section 6. 2. The Benchmarks Two benchmarks used for the evaluation of the different approaches have been defined in [1]. In this section they are going to be briefly presented. S. Cagnoni et al. (Eds.): EvoWorkshops 2000, LNCS 1803, pp. 158-167, 2000.  Springer-Verlag Berlin Heidelberg 2000 Benchmarking Cost-Assignment Schemes 159 The first benchmark is called ‘optimality’. Suppose that we have a test function f(x): X F, where X Rn, F Rm, where R ⊆n represents the search space⊆in n dimensions, m ⊆ R represents the space of all the possible objective values, n is the number of parameters, m is the number of the objectives f ∈ F is the collection of the individual objective elements Also, consider the theoretical objective vector fo = { f(x) } that contains the objective values that can ultimately be reached. Finally, consider an objective reached as in Eq. 1. ⎛ ^ ⎞ ^ ^ f ⎜ xo ⎟ = f 0 , xo ∈ X ⎝ ⎠ (1) with ^xo, representing a corresponding solution found. The optimality measures how close an objective reached is to the theoretical objective vector and is calculated using the formula in Eq.2. ^ ⎛ ⎞ Optimality⎜ f o ⎟ = 1 − ⎝ ⎠α ^ fo − f o α _ ∈ [0,1] f− f (2) _ α - where f and f- are the upper and lower bounds of f respectively. Any norm can be used to evaluate the optimality of an objective and this paper uses the Euclidean metric (a=2) for this purpose. The above formula needs to be refined when the problem addressed is a nondominant or non-commensurate one since no such concept as ‘overall optimality’ can be assessed in a problem of this kind. Since this is the case for this paper, the ‘distance to demands’ method explained in [3], is used here. The second benchmark this short study uses is one that measures the convergence of the algorithm and is called ‘reach time’. The reach time is defined as the total number of function evaluations performed by the algorithm by which the optimality of the best individual first reaches b. Re ach _ time b = C b (3) For the purposes of the tests, b is set to 0.999, a certainly high value that may not always be reached by the algorithm. Because of that, a single algorithm terminates when either the set optimality threshold is reached or 20n generations of size 20nxm have been evolved. Those termination conditions are identical to the ones used in [1] with the latter one meaning that the algorithm is not supposed to perform worse than an O(n2) algorithm in terms of computational time. 160 K. Koukoulakis and Y. Li 3. The Problem A set of two objective functions (Fonsceca and Fleming, 1995) was chosen for the evaluation of the cost-schemes. The functions of Eq.4 and Eq.5 were chosen in an effort to produce as “standard” a Pareto optimal front as possible. 2 ⎛ n ⎛ 1 ⎞ ⎞⎟ f1 ( x) = 1 − exp⎜ − ∑ ⎜ xi − ⎟ ⎜ i =1 ⎝ n ⎠ ⎟⎠ (4) ⎝ 2 ⎛ n ⎛ 1 ⎞ ⎞⎟ ⎜ f 2 (x ) = 1 − exp − ∑ ⎜ xi + ⎟ ⎜ i =1 ⎝ n ⎠ ⎟⎠ ⎝ (5) Each individual consists of a real-valued vector of n parameters. For the purposes of this paper n was set to the value of 2, with each parameter coded in the interval [-2, 2). The individuals that form the Pareto-optimal set belong on the line shown in Fig. 1. Functions f1 and f2 are plotted for n=2 in figures 2 and 3 respectively. Fig. 1. Pareto optimal front Fig. 2. f1 plotted for n=2 Fig. 3. f2 plotted for n=2 Benchmarking Cost-Assignment Schemes 161 4.The Algorithm 4.1 Selection, Crossover and Mutation The evolutionary algorithm that was used is quite a simple and straightforward one. It uses a binary tournament selection scheme to form the mating pool of the individuals after, of course, the cost assignment procedure has taken place. Each individual in the mating pool then randomly mates with another one using arithmetic crossover since each individual consists of a real-valued vector. Arithmetic crossover, as described in [4], is a canonical intermediate recombination operator, which produces the i-th component of the offspring by averaging it with some weight as defined in Eq.6. xi/ = αx1i + (1 − α )x2 i (6) Next, the offspring are evaluated and then refined by the simulated annealing (SA) technique. The SA positive mutation cycles where conducted using a non-linear Boltzmann learning schedule as the one employed in [5]. For the purposes of mutation, the creep mutation operator (Davis, 1989) was employed. As suggested in [6], entrapment must be alleviated in the case of this operator when used with a bounded small random amount for mutation. As such is the case here, entrapment can be said to have been partially overcome by the probabilistic nature of the SA technique, which maintains a probability of retaining lesser-valued individuals. After SA, the parents are merged with the offspring the new population is formed with binary tournament selection. The parameter settings of the algorithm, most of which are suggested in [5], are listed in table 1. The optimality threshold was set to a high value as the tests were intended to prohibit the algorithm to reach it so that a clearer picture of each scheme’s behavior could be obtained. Table 1. Parameter settings Optimality threshold Number of parameters Weight vector Priority vector Goal vector Number of generations Population size Arithmetic crossover constant Creep mutation probability Tournament size Boltzmann constant Initial temperature Final temperature Initial annealing factor Transient constant 0.999 2 { 1, 1 } { 1, 0=top } { 0.0, 0.0 } 160 320 0.5 0.05 2 5E-06 1E05 1 0.3 16 162 K. Koukoulakis and Y. Li 4.2 The Cost-Assignment Schemes Each scheme is successively described below with a minimization problem assumed. 4.2.1 The Weighted Sum Method According to this approach, all of the objectives are weighted by positive coefficients defined by the user and are added together to obtain the cost. Φ : ℜn → ℜ m (7) f ( x ) = ∑ wk f k (x ) k =1 ,where Ö denotes the cost assignment scheme and x is the parameter vector. It must be noted that the same weights that are used here are also used to weight the objective vectors prior to the calculation of the norms of the optimality benchmark measure as suggested in [1]. 4.2.2 The Minimax Method This method tries to minimize the maximum weighted difference between the objectives and the goals, with the weights and the goals supplied by the user. Φ : ℜn → ℜ f (x ) − g k f ( x ) = max k k =1Km wk (8) 4.2.3 The Target Vector Method This approach minimizes the distance of the objective vector from the goal vector using a defined distance measure. Again the user supplies the goals. The Euclidean metric was used as the distance measure in this case. Φ : ℜn → ℜ f ( x ) = [ f ( x ) − g ]W −1 (9) α 4.2.4 The Lexicographic Method Here, the objectives are assigned distinct priorities and the selection proceeds with the comparison of the individuals with respect to the objective of the highest priority Benchmarking Cost-Assignment Schemes 163 with any ties resolved by a successive comparison with respect to the objective with the second-highest priority, until the lowest priority objective is reached. Φ : ℜn → {0,1, K µ − 1}, µ = pop _ size f ( xi ) = ∑ l ( f (x j )l < f ( xi )), µ = pop _ size µ (10) j =1 where l(condition) evaluates to unity if condition is true and f (x j )l < f ( x i ) ⇔ ∃ p ∈ {1K m } : ∀k ∈{p,L, m}, f k (x j ) ≤ f k (xi ) ∧ f p (x j ) < f p (xi ) (11) 4.2.5 Pareto Ranking (Goldberg’s Approach) According to the definition in [6] all non-dominated individuals are assigned a cost of one and then removed from contention with the next set of non-dominated individuals assigned a cost of two until the whole population has been ranked. Φ : ℜ n → {1, K µ }, µ = pop _ size 1 ⇐ not ( f (x j ) p < f ( xi )), ∀j ∈ {1K µ } f (xi ) = { φ ⇐ not ( f (x j ) p < f ( xi )), ∀ j ∈ {1K µ } (12) \ {l : Φ( f ( xl )) < φ } ,where the p< condition denotes partial domination of the individual j over the individual i and is true if and only if ∀k ∈ {1K m} f k (x j ) ≤ f k ( xi ) ∧ ∃k ∈{1K m} : f k (x j ) < f k ( xi ) (13) 4.2.6 Pareto Ranking (Fonseca and Fleming’s Approach) Proposed in 1993, this approach ranks an individual according to the number of individuals that dominate him. Φ : ℜ n → {0,1, K µ − 1}, µ = pop _ size µ ⎞ ⎛ f (xi ) = ∑l ⎜ f (x j ) p < f (xi )⎟, µ = pop _ size ⎠ j =1 ⎝ (14) 164 K. Koukoulakis and Y. Li 4.2.7 Pareto Ranking (With Goals and Priorities) This approach combines the pareto-optimality concept with goal and priority information. Equal priorities may be assigned to different objectives with both the priorities and the goals supplied by the user. Individuals are compared as in the lexicographic method but it is also affected from whether the individuals attain the goals set or not. Φ : ℜ n → {0,1, K µ − 1}, µ = pop _ size f ( xi ) = ∑ l ⎛⎜ f (x j ) p f ( xi ) ⎞⎟, µ = pop _ size g ⎠ j =1 ⎝ µ (15) ,where the condition within the brackets denotes preferrability of the j-th individual over the i-th individual and g is the preference vector, a vector that contains the goals of each objective grouped by priority As for the evaluation of the condition it is deemed too detailed to mention here but is fully described in [2]. 5. Comparison Results For each method, 10 experiments were carried out each with a random initial population, with an experiment terminating either when the optimality threshold has been reached or 400mn2 generations have been evolved. A discussion of the results obtained follows. 5.1 Pareto Front Sampling and Diversity A cost-assignment scheme is considered successful if it has managed to offer a diverse sample of the pareto-optimal front as quickly as possible. With this in mind, a short discussion for each scheme tested follows. A snapshot of the population in which the most optimal individual was found for each scheme can be seen at figures 4-10. Remember that the goals were set to 0.0 for both objectives. The weighted sum approach was unable to sample the concave region of the line, focusing entirely on the zero-cost line f2 = -f1. With identical, equally weighted goals, the minimax scheme failed to sample the pareto front. Nevertheless, with appropriate goal and weight settings it can prove successful, but surely less successful than the pareto-based approach with goals and priorities, which provides a better sampling in a quick and more efficient manner using only the goal information available. The target vector scheme has introduced better sampling diversity than the minimax approach in roughly the same time. The diversity is even better than the pareto-based approach with goals and priorities but the latter scheme is significantly quicker in providing its results. Benchmarking Cost-Assignment Schemes 165 Fonseca and Fleming’s approach along with Goldberg’s original one has indeed most quickly sampled a very good proportion of the pareto-optimal front with the former being better at that. It must be noted that they have performed better at that than the last scheme without using any information available. This leads us to the conclusion that in the case of unattainable goals both of these schemes can offer a better sample of the front than the last scheme in some applications. The last approach has performed very well using both the goal and priority information. It is interesting to compare it with the lexicographic cost-assignment scheme, which only uses priorities. It can clearly be seen that the latter scheme has driven the population to the minimisation of f2, which has a higher priority over f1. So, it can be said that the lexicographic method needs an aid for better results and the most obvious one is niching combined with mating restriction. 5.2 Optimality As far as optimality is concerned, as can be seen in Fig. 11, Goldberg’s and Fonseca and Fleming’s approaches have both quickly given optimal solutions without using the goal information and also have the added bonus of good diversity. Of course with attainable goals, the pareto-based approach with goals and priorities should be the quickest cost-assignment scheme to offer the most optimal solution. 5.3 Reach Time As no method managed to reach the high optimality threshold of 0.999 (max = 1.0), all of them had a reach time of 400mn2 = 51200. 6. Conclusions The cost-assignment scheme acts as the driving force of the algorithm. Performing the scalarization of the objectives, it is the determining factor of evaluation. The purpose of this paper was to expose the magnitude of its impact on (a) the quality of the sampling of the Pareto-optimal front and (b) on the speed at which this quality is achieved. The goals were deliberately set unattainable because they were intended to be a means to push the population towards the front. The optimality benchmark itself worked in accord with the goal settings so that it could act as an observer of the algorithm’s behaviour rather than a strict evaluator. It is thus concluded that for a given problem, the sampling of the Pareto-optimal front is generally easier achieved with the Pareto-based cost schemes. As for the rest of the schemes, it is thought that their usefulness can only be experienced with proper tuning of their associated parameters. As a further study, it would be interesting to test all of the schemes in the context of a harder problem, that is a problem with a more diverse front. Finally, it is believed that this testing should employ a wide range of weights so that the promising aggregating ‘target vector’ scheme can be examined more closely. 166 K. Koukoulakis and Y. Li Fig. 4. Pareto ranking (Fonseca and Fleming), Gen. 46 Fig. 5. Pareto ranking (Goldberg), Gen.41 Fig. 6. Lexicographic, Gen. 158 Fig. 7. Target vector, Gen. 156 Fig. 8. Minimax, Gen. 145 Fig. 9. Weighted sum, Gen. 67 Benchmarking Cost-Assignment Schemes Fig. 10. Pareto ranking (goals and priorities), Gen. 47 167 Fig. 11. Maximum optimality References 1.Benchmarks for testing evolutionary algorithms, The Third Asia-Pacific Conference on Measurement and Control, Dunhuang, China, 31 Aug. - 4 Sept 1998, 134-138. (W. Feng , T. Brune, L. Chan, M. Chowdhury, C.K. Kuek and Y. Li). 2.Back T., Fogel D. B. and Michalewicz Z., Handbook of Evolutionary Computation (New York, Oxford: Oxford University Press, Bristol, Philadelphia: Institute Of Physics Publishing, 1997) 3.Michalewicz Z., Genetic Algorithms + Data structures = Evolution programs. (Berlin: Springer-Verlag, 1992) 4.Michalewicz Z., Nazhiyath G. and Michalewicz M, A note on the usefulness of geometrical crossover for numerical optimization problems, Proc 5th Ann. Conf. on Evolutionary Programming ed L. J. Fogel, P. J. Angeline and T. Back (Cambridge, MA: MIT Press, 1996) 5.Tan K.C., Evolutionary methods for Modelling and Control of Linear and Nonlinear Systems, Ph.D. thesis (Department of Electronics and Electrical Engineering, University of Glasgow, 1997) 6.Goldberg D. E., Genetic algorithms in Search, Optimization and Machine Learning (Reading, MA: Addison-Wesley, 1989) 7.Davis L., Adapting operator probabilities in genetic algorithms, Proc 3rd Int. Conf. on GAs (Fairfax, VA, June 1989) ed J. D. Schaffer (San Mateo, CA: Morgan Kaufmann) pp 61-69 8.Fonseca C. M. and Fleming P. J., Multiobjective genetic algorithms made easy: selection sharing and mating restriction (First Int. Conf. on GAs in Eng. Systems: Innovations and Applications, Sheffield, UK, 1995) pp 45-52 A u to m a tic S y n th e s is o f B o th th e T o p o lo g y a n d P a r a m e te r s fo r a C o n tr o lle r fo r a T h r e e -L a g P la n t w ith a F i v e - S e c o n d D e l a y Us i n g G e n e t i c P r o g r a m m i n g J o h n R . K o z a S ta n fo rd U n iv e rs ity , S ta n fo rd , C a lifo rn ia k o z a @ s t a n f o r d . e d u M a r tin A . K e a n e E c o n o m e tr ic s I n c ., C h ic a g o , I llin o is m a k e a n e @ i x . n e t c o m . c o m J e sse n Y u G e n e tic P r o g r a m m in g I n c ., L o s A lto s , C a lif o r n ia j y u @ c s . s t a n f o r d . e d u W illia m M y d lo w e c G e n e tic P r o g r a m m in g I n c ., L o s A lto s , C a lif o r n ia m y d @ c s . s t a n f o r d . e d u F o r r e st H B e n n e tt III G e n e tic P ro g ra m m in g In c . (C u rre n tly , F X P a lo A lto L a b o ra to ry , P a lo A lto , C a lifo rn ia ) f o r r e s t @ e v o l u t e . c o m A b str a c t T h is p a p e r d e s c rib e s h o w th e p ro c e s s o f s y n th e s iz in g th e d e s ig n o f b o th th e to p o lo g y a n d th e n u m e ric a l p a ra m e te r v a lu e s (tu n in g ) fo r a c o n tro lle r c a n b e a u to m a te d b y u s in g g e n e tic p ro g ra m m in g . G e n e tic p ro g ra m m in g c a n b e u s e d to a u to m a tic a lly m a k e th e d e c is io n s c o n c e rn in g th e to ta l n u m b e r o f s ig n a l p ro c e s s in g b lo c k s to b e e m p lo y e d in a c o n tro lle r, th e ty p e o f e a c h b lo c k , th e to p o lo g ic a l in te rc o n n e c tio n s b e tw e e n th e b lo c k s , a n d th e v a lu e s o f a ll p a ra m e te rs fo r a ll b lo c k s re q u irin g p a ra m e te rs . In s y n th e s iz in g th e d e s ig n o f c o n tro lle rs , g e n e tic p ro g ra m m in g c a n s im u lta n e o u s ly o p tim iz e p re s p e c ifie d p e rfo rm a n c e m e tric s (s u c h a s m in im iz in g th e tim e re q u ire d to b rin g th e p la n t o u tp u t to th e d e s ire d v a lu e ), s a tis fy tim e -d o m a in c o n s tra in ts (s u c h a s o v e rs h o o t a n d d is tu rb a n c e re je c tio n ), a n d s a tis fy fre q u e n c y d o m a in c o n s tra in ts . E v o lu tio n a ry m e th o d s h a v e th e a d v a n ta g e o f n o t b e in g e n c u m b e re d b y p re c o n c e p tio n s th a t lim it its s e a rc h to w e ll-tra v e le d p a th s . G e n e tic p ro g ra m m in g is a p p lie d to a n illu s tra tiv e p ro b le m in v o lv in g th e d e s ig n o f a c o n tro lle r fo r a th re e -la g p la n t w ith a s ig n ific a n t (fiv e -s e c o n d ) tim e d e la y in th e e x te rn a l fe e d b a c k fro m th e p la n t to th e c o n tro lle r. A d e la y in th e fe e d b a c k m a k e s th e d e s ig n o f a n e ffe c tiv e c o n tro lle r e s p e c ia lly d iffic u lt. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 6 8 − 1 7 7 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 A u to m a tic S y n th e s is o f B o th th e T o p o lo g y a n d P a ra m e te rs 1 1 6 9 I n tr o d u c tio n T h e p ro c e s s o f c re a tin g (s y n th e s iz in g ) th e d e s ig n o f a c o n tro lle r e n ta ils m a k in g d e c is io n s c o n c e rn in g th e to ta l n u m b e r o f p ro c e s s in g b lo c k s to b e e m p lo y e d in th e c o n tr o lle r , th e ty p e o f e a c h s ig n a l p r o c e s s in g b lo c k ( e .g ., le a d , la g , g a in , in te g r a to r , d iffe re n tia to r, a d d e r, in v e rte r, s u b tra c to r, a n d m u ltip lie r), th e v a lu e s o f a ll p a ra m e te rs fo r a ll b lo c k s re q u irin g p a ra m e te rs , a n d th e to p o lo g ic a l in te rc o n n e c tio n s b e tw e e n th e s ig n a l p ro c e s s in g b lo c k s . T h e la tte r in c lu d e s th e q u e s tio n o f w h e th e r o r n o t to e m p lo y in te r n a l f e e d b a c k ( i.e ., f e e d b a c k in s id e th e c o n tr o lle r ) . T h e p ro b le m o f s y n th e s iz in g a c o n tro lle r to s a tis fy p re s p e c ifie d re q u ire m e n ts is s o m e tim e s s o lv a b le b y a n a ly tic te c h n iq u e s (o fte n o rie n te d to w a rd p ro d u c in g c o n v e n tio n a l P ID c o n tro lle rs ). H o w e v e r, a s B o y d a n d B a rra tt s ta te d in L in e a r C o n tr o lle r D e s ig n : L im its o f P e r fo r m a n c e (1 9 9 1 ), " T h e c h a lle n g e fo r c o n tro lle r d e s ig n is to p ro d u c tiv e ly u s e th e e n o rm o u s c o m p u tin g p o w e r a v a ila b le . M a n y c u rre n t m e th o d s o f c o m p u te r-a id e d c o n t r o l l e r d e s i g n s i m p l y a u t o m a t e p r o c e d u r e s d e v e l o p e d i n t h e 1 9 3 0 ’s t h r o u g h t h e 1 9 5 0 's … " T h is p a p e r d e s c rib e s h o w g e n e tic p ro g ra m m in g c a n b e u s e d to a u to m a tic a lly c re a te b o th th e to p o lo g y a n d th e n u m e r ic a l p a r a m e te r v a lu e s ( i.e ., th e tu n in g ) f o r a c o n tr o lle r d ire c tly fro m a h ig h -le v e l s ta te m e n t o f th e re q u ire m e n ts o f th e c o n tro lle r. G e n e tic p ro g ra m m in g c a n , if d e s ire d , s im u lta n e o u s ly o p tim iz e p re s p e c ifie d p e rfo rm a n c e m e tric s (s u c h a s m in im iz in g th e tim e re q u ire d to b rin g th e p la n t o u tp u t to th e d e s ire d v a lu e a s m e a s u re d b y , s a y , th e in te g ra l o f th e tim e -w e ig h te d a b s o lu te e rro r), s a tis fy tim e -d o m a in c o n s tra in ts (in v o lv in g , s a y , o v e rs h o o t a n d d is tu rb a n c e re je c tio n ), a n d s a tis fy fre q u e n c y d o m a in c o n s tra in ts . E v o lu tio n a ry m e th o d s h a v e th e a d v a n ta g e o f n o t b e in g e n c u m b e re d b y p re c o n c e p tio n s th a t lim it th e ir s e a rc h to w e ll-tra v e le d p a th s . S e c tio n 2 d e s c rib e s a n illu s tra tiv e p ro b le m o f c o n tro lle r s y n th e s is . S e c tio n 3 p ro v id e s g e n e ra l b a c k g ro u n d o n g e n e tic p ro g ra m m in g . S e c tio n 4 d e s c rib e s h o w g e n e tic p ro g ra m m in g is a p p lie d to c o n tro l p ro b le m s . S e c tio n 5 d e s c rib e s th e p re p a ra to ry s te p s n e c e s s a ry to a p p ly g e n e tic p ro g ra m m in g to th e illu s tra tiv e c o n tro l p ro b le m . S e c tio n 6 p re s e n ts th e re s u lts . 2 I llu s tr a tiv e P r o b le m T h e illu s tra tiv e p ro b le m e n ta ils c re a tio n o f b o th th e to p o lo g y a n d p a ra m e te r v a lu e s fo r a c o n tro lle r fo r a th re e -la g p la n t w ith a s ig n ific a n t (fiv e -s e c o n d ) tim e d e la y in th e e x te rn a l fe e d b a c k fro m th e p la n t o u tp u t to th e c o n tro lle r s u c h th a t p la n t o u tp u t re a c h e s th e le v e l o f th e re fe re n c e s ig n a l in m in im a l tim e (a s m e a s u re d b y th e in te g ra l o f th e tim e -w e ig h te d a b s o lu te e rro r), s u c h th a t th e o v e rs h o o t in re s p o n s e to a s te p in p u t is le s s th a n 2 % , a n d s u c h th a t th e c o n tro lle r is ro b u s t in th e fa c e o f d is tu rb a n c e (a d d e d in to th e c o n tro lle r o u tp u t). T h e d e la y in th e fe e d b a c k m a k e s th e d e s ig n o f a n e ffe c tiv e c o n tro lle r e s p e c ia lly d iffic u lt (A s tro m a n d H a g g lu n d 1 9 9 5 ). T h e tra n s fe r fu n c tio n o f th e p la n t is A G (s ) = K e − 5 s (1 + τ s ) 3 c o n tro lle r p re s e n te d in A s tro m a n d H a g g lu n d 1 9 9 5 (p a g e 2 2 5 ) d e liv e rs c re d ib le p e rfo rm a n c e o n th is p ro b le m fo r v a lu e s o f K = 1 a n d τ = 1 . 1 7 0 b y lim se v b y J .R . K o z a e t a l. T o m a k e th e th e c o n tro lle r ite d to th e ra n e ra l d iffe re n t A s tro m a n d H 3 p ro p re g e c o m a g g b le m s e n te b e tw b in a lu n d m o d in e e n tio n w a s re re a lis tic A s tro m a n -4 0 a n d + 4 s o f v a lu e s in te n d e d o , w e a d d e d d H a g g lu n d 0 v o lts . T h fo r K a n d n ly fo r K = a n a d d itio n a l 1 9 9 5 ) th a t th e e p la n t in th is τ (w h e re a s th e 1 a n d τ = 1 ). c o n s tra in t in p u t to th p a p e r o p e r c o n tro lle r (sa e p a te d e tis fie d la n t is s o v e r s ig n e d B a c k g r o u n d o n G e n e tic P r o g r a m m in g 4 G e n e tic p ro g ra m m in g is a n a u to m a tic te c h n iq u e fo r g e n e ra tin g c o m p u te r p ro g ra m s to s o lv e , o r a p p ro x im a te ly s o lv e , p ro b le m s . G e n e tic p ro g ra m m in g (K o z a 1 9 9 2 ; K o z a a n d R ic e 1 9 9 2 ) is a n e x te n s io n o f th e g e n e tic a lg o rith m (H o lla n d 1 9 7 5 ). G e n e tic p ro g ra m m in g is c a p a b le (K o z a 1 9 9 4 a , 1 9 9 4 b ) o f e v o lv in g re u s a b le , p a ra m e triz e d , h ie ra rc h ic a lly -c a lle d a u to m a tic a lly d e fin e d fu n c tio n s (A D F s ) s o th a t a n o v e ra ll p ro g ra m c o n s is ts o f a m a in re s u lt-p ro d u c in g b ra n c h a n d o n e o r m o re re u s a b le a n d p a ra m e te riz a b le a u to m a tic a lly d e fin e d fu n c tio n s (fu n c tio n -d e fin in g b ra n c h e s ). In a d d itio n , a rc h ite c tu re -a lte rin g o p e ra tio n s (K o z a , B e n n e tt, A n d re , a n d K e a n e 1 9 9 9 ; K o z a , B e n n e tt, A n d re , K e a n e , a n d B ra v e 1 9 9 9 ) e n a b le g e n e tic p ro g ra m m in g to a u to m a tic a lly d e te rm in e th e n u m b e r o f a u to m a tic a lly d e fin e d fu n c tio n s , th e n u m b e r o f a rg u m e n ts th a t e a c h p o s s e s s e s , a n d th e n a tu re o f th e h ie ra rc h ic a l re fe re n c e s , if a n y , a m o n g s u c h a u to m a tic a lly d e fin e d fu n c tio n s . G e n e tic p ro g ra m m in g o fte n c re a te s n o v e l d e s ig n s b e c a u s e it is a p ro b a b ilis tic p ro c e s s th a t is n o t e n c u m b e re d b y th e p re c o n c e p tio n s th a t o fte n c h a n n e l h u m a n th in k in g d o w n fa m ilia r p a th s . F o r e x a m p le , g e n e tic p ro g ra m m in g is c a p a b le o f s y n th e s iz in g th e d e s ig n o f b o th th e to p o lo g y a n d s iz in g fo r a w id e v a rie ty o f a n a lo g e l e c t r i c a l c i r c u i t s f r o m a h i g h - l e v e l s t a t e m e n t o f t h e c i r c u i t ’s d e s i r e d b e h a v i o r a n d c h a ra c te ris tic s (K o z a , B e n n e tt, A n d re , a n d K e a n e 1 9 9 9 ; K o z a , B e n n e tt, A n d re , K e a n e , a n d B ra v e 1 9 9 9 ). F iv e o f th e e v o lv e d a n a lo g c irc u its in th a t b o o k in frin g e o n p re v io u s ly is s u e d p a te n ts w h ile fiv e o th e rs d e liv e r th e s a m e fu n c tio n a lity a s p re v io u s ly p a te n te d in v e n tio n s in a n o v e l w a y . A d d itio n a l in fo rm a tio n o n c u rre n t re s e a rc h in g e n e tic p ro g ra m m in g c a n b e fo u n d in B a n z h a f, N o rd in , K e lle r, a n d F ra n c o n e 1 9 9 8 ; L a n g d o n 1 9 9 8 ; R y a n 1 9 9 9 ; K in n e a r 1 9 9 4 ; A n g e l i n e a n d K i n n e a r 1 9 9 6 ; S p e c t o r , L a n g d o n , O ’R e i l l y , a n d A n g e l i n e 1 9 9 9 ; K o z a , G o ld b e rg , F o g e l, a n d R io lo 1 9 9 6 ; K o z a , D e b , D o rig o , F o g e l, G a rz o n , Ib a , a n d R io lo 1 9 9 7 ; K o z a , B a n z h a f, C h e lla p illa , D e b , D o rig o , F o g e l, G a rz o n , G o ld b e rg , Ib a , a n d R io lo 1 9 9 8 ; B a n z h a f, P o li, S c h o e n a u e r, a n d F o g a rty 1 9 9 8 ; B a n z h a f, D a id a , E ib e n , G a rz o n , H o n a v a r, J a k ie la , a n d S m ith 1 9 9 9 ; P o li, N o rd in , L a n g d o n , a n d F o g a rty 1 9 9 9 ; a t w e b s ite s s u c h a s w w w . g e n e t i c - p r o g r a m m i n g . o r g ; a n d in th e G e n e tic P r o g r a m m in g a n d E v o lv a b le M a c h in e s jo u rn a l (fro m K lu w e r A c a d e m ic P u b lis h e rs ). G e n e tic P r o g r a m m in g a n d C o n tr o l C o n tro lle rs p ro c e s s in g o u tp u t(s ), a in s id e th e c b o th th e to b e tw e e n th g e rm a n e to c a n b e re p re s e n te d b y fu n c tio n s , in w h ic h e x n d in w h ic h c y c le s in o n tro lle r. G e n e tic p ro g p o lo g y a n d p a ra m e te r e p ro g ra m tre e s u s e d c o n tro lle rs . b lo c k d ia g ra m te rn a l p o in ts th e b lo c k d ia ra m m in g c a n v a lu e s fo r a in g e n e tic p s in w h ic h th e b lo c k s re p re r e p r e s e n t t h e c o n t r o l l e r ’s i n g ra m c o rre s p o n d to in te rn a b e e x te n d e d to th e p ro b le m c o n tro lle r b y e s ta b lis h in g ro g ra m m in g a n d th e b lo c k s e n t s ig n a l p u t(s ) a n d l fe e d b a c k o f c re a tin g a m a p p in g d ia g ra m s A u to m a tic S y n th e s is o f B o th th e T o p o lo g y a n d P a ra m e te rs 1 7 1 T h e n u m b e r o f re s u lt-p ro d u c in g b ra n c h e s in th e to -b e -e v o lv e d c o n tro lle r e q u a ls th e n u m b e r o f c o n tro l v a ria b le s th a t a re to b e p a s s e d fro m th e c o n tro lle r to th e p la n t. E a c h re s u lt-p ro d u c in g b ra n c h is a c o m p o s itio n o f th e fu n c tio n s a n d te rm in a ls fro m a re p e rto ire (b e lo w ) o f fu n c tio n s a n d te rm in a ls . P ro g ra m tre e s in th e p o p u la tio n d u rin g th e in itia l ra n d o m g e n e ra tio n (g e n e ra tio n 0 ) c o n s is t o n ly o f re s u lt-p ro d u c in g b ra n c h (e s ). A u to m a tic a lly d e fin e d fu n c tio n s a re in tro d u c e d in c re m e n ta lly (a n d s p a rin g ly ) in to th e p o p u la tio n o n s u b s e q u e n t g e n e ra tio n s b y m e a n s o f th e a rc h ite c tu re -a lte rin g o p e ra tio n s . E a c h a u to m a tic a lly d e fin e d fu n c tio n is a c o m p o s itio n o f th e fu n c tio n s a n d te rm in a ls a p p ro p ria te fo r c o n tro l p ro b le m s , re fe re n c e s to e x is tin g a u to m a tic a lly d e fin e d fu n c tio n s , a n d (p o s s ib ly ) d u m m y v a ria b le s (fo rm a l p a ra m e te rs ) th a t p e rm it p a ra m e te riz a tio n o f th e a u to m a tic a lly d e fin e d fu n c tio n . A u to m a tic a lly d e fin e d fu n c tio n s p ro v id e a m e c h a n is m fo r in te rn a l fe e d b a c k (re c u rs io n ) w ith in th e to -b e -e v o lv e d c o n tro lle r. A u to m a tic a lly d e fin e d fu n c tio n s a ls o p ro v id e a m e c h a n is m fo r re u s in g u s e fu l s u b s tru c tu re s . E a c h b ra n c h o f e a c h p ro g ra m tre e in th e in itia l ra n d o m p o p u la tio n is c re a te d in a c c o rd a n c e w ith a c o n s tra in e d s y n ta c tic s tru c tu re . E a c h g e n e tic o p e ra tio n e x e c u te d b y g e n e tic p ro g ra m m in g (c ro s s o v e r, m u ta tio n , re p ro d u c tio n , o r a rc h ite c tu re -a lte rin g o p e ra tio n ) p ro d u c e s o ffs p rin g th a t c o m p ly w ith th e c o n s tra in e d s y n ta c tic s tru c tu re . G e n e tic p ro g ra m m in g h a s re c e n tly b e e n u s e d to c re a te a c o n tro lle r fo r a p a rtic u la r tw o -la g p la n t a n d a th re e -la g p la n t (K o z a , K e a n e , Y u , B e n n e tt, a n d M y d lo w e c 2 0 0 0 ). B o th o f th e s e g e n e tic a lly e v o lv e d c o n tro lle rs o u tp e rfo rm e d th e c o n tro lle rs d e s ig n e d b y e x p e rts in th e fie ld o f c o n tro l u s in g th e c rite ria o rig in a lly s p e c ifie d b y th e e x p e rts . 5 P r e p a r a to r y S te p s S ix m a jo r p re p a ra d e te rm in e th e a rc h th e fu n c tio n s , (4 ) d a n d (6 ) c h o o s e th e 5 .1 S in c e th e re th e c o n tro lle o n e re s u lt-p (g e n e ra tio n g e n e ra tio n s , d e fin e d fu n c to ry s te ite c tu re e fin e th te rm in a p s o f e f tio a re re q u ire d th e p ro g ra m itn e s s m e a s u r n c rite rio n a n b e fo re tre e s , (2 e , (5 ) c h d m e th o P ro g ra m A rc h ite c tu re is o n e re s u lt-p ro d u c in g b ra n c h in th r a n d th is p ro b le m in v o lv e s a o n e -o u ro d u c in g b ra n c h . E a c h p ro g ra m tr 0 ) h a s n o a u to m a tic a lly d e fin e d a rc h ite c tu re -a lte rin g o p e ra tio n s m tio n s (u p to a m a x im u m o f fiv e p e r p a p p ly in g g e ) id e n tify th o o s e c o n tro d o f re s u lt d e p ro g ra m tp u t c o n tro e e in th e fu n c tio n s . a y in s e rt ro g ra m tre n e tic p ro g ra m m in g : (1 ) e te rm in a ls , (3 ) id e n tify l p a ra m e te rs fo r th e ru n , e s ig n a tio n . tre e fo r e a c h lle r, e a c h p ro g in itia l ra n d o m H o w e v e r, in a n d d e le te a e ). o u tp u t fro m ra m tre e h a s g e n e ra tio n su b se q u e n t u to m a tic a lly 5 .2 T e rm in a l S e t A c o n s tra in e d s y n ta c tic s tru c tu re p e rm its o n ly a s in g le p e rtu rb a b le n u m e ric a l v a lu e to a p p e a r a s th e a rg u m e n t fo r e s ta b lis h in g e a c h n u m e ric a l p a ra m e te r v a lu e fo r e a c h s ig n a l p ro c e s s in g b lo c k re q u irin g a p a ra m e te r v a lu e . T h e s e n u m e ric a l v a lu e s in itia lly r a n g e f r o m - 5 .0 to + 5 .0 . T h e s e n u m e r ic a l v a lu e s a r e p e r tu r b e d d u r in g th e r u n b y a G a u s s ia n m u ta tio n o p e ra tio n th a t o p e ra te s o n ly o n n u m e ric a l v a lu e s . N u m e ric a l c o n s ta n ts a re la te r in te rp re te d o n a lo g a rith m ic s c a le s o th a t th e y re p re s e n t v a lu e s in a ra n g e o f 1 0 o rd e rs o f m a g n itu d e (K o z a , B e n n e tt, A n d re , a n d K e a n e 1 9 9 9 ). T h e r e m a i n i n g t e r m i n a l s a r e t i m e - d o m a i n s i g n a l s . T h e t e r m i n a l s e t , T, f o r t h e re s u lt-p ro d u c in g b ra n c h a n d a n y a u to m a tic a lly d e fin e d fu n c tio n s (e x c e p t fo r th e p e rtu rb a b le n u m e ric a l v a lu e s m e n tio n e d a b o v e ) is 1 7 2 J .R . K o z a e t a l. T= { R E F C S p a c e (a lth o u g h K o z a , K e 5 .3 T h e te rm a u to F= E R E O N S d o e th e a n e , N C E T A N s n o m e a Y u , _ S I G N A L , C O N T R O L T _ 0 } . t p e rm it a d e ta ile d d e s n in g o f th e a b o v e te rm B e n n e tt, a n d M y d lo w e L E R _ O U T P U T , P L A N T _ O U T P U T , c rip tio n o f th e v a rio u s te rm in a ls u s e d h e re in in a ls s h o u ld b e c le a r fro m th e ir n a m e s ). S e e c 2 0 0 0 fo r d e ta ils . F u n c tio n S e t fu n c tio n s a re s ig n a l p ro c e s s in g fu n c tio n s th a t o p e ra te o n tim e -d o m a in s ig i n a l s i n T) . T h e f u n c t i o n s e t , F, f o r t h e r e s u l t - p r o d u c i n g b r a n c h m a tic a lly d e fin e d fu n c tio n s is { G A I N , I N V E R T E R , L E A D , L A G , L A G 2 , D I F F E R E N T I A L _ I N P U T _ I N T E G R A T O R , D I F F E R E N T I A T O R , A D D _ S I G N A L , S U B _ S I G N A L , A D D _ 3 _ S I G N A L , D E L A Y , A D F 0 , … A D F 4 } . A D F 0 , … , A D F 4 d e n o te a u to m a tic a lly d e fin e d fu n c tio n s a d d e d d u rin g th a rc h ite c tu re -a lte rin g o p e ra tio n s . T h e fu n c tio n a lity o f e a c h o f th e a b o v e s ig n a l p ro c e s s in g fu n c tio n s is s u g g th e ir n a m e s a n d is d e s c rib e d in d e ta il in K o z a , K e a n e , Y u , B e n n e tt, a n d M 2 0 0 0 . n a ls (th e a n d a n y , e ru n b y e s te d b y y d lo w e c 5 .4 F itn e s s G e n e tic p ro g ra m m in g is a p ro b a b ilis tic a lg o rith m th a t s e a rc h e s th e s p a c e o f c o m p o s itio n s o f th e a v a ila b le fu n c tio n s a n d te rm in a ls . T h e s e a rc h is g u id e d b y a fitn e s s m e a s u re . T h e fitn e s s m e a s u re is a m a th e m a tic a l im p le m e n ta tio n o f th e h ig h le v e l re q u ire m e n ts o f th e p ro b le m . T h e fitn e s s m e a s u re is c o u c h e d in te rm s o f “ w h a t n e e d s to b e d o n e ”  n o t “ h o w to d o it.” T h e fitn e s s m e a s u re m a y in c o rp o ra te a n y m e a s u ra b le , o b s e rv a b le , o r c a lc u la b le b e h a v io r o r c h a ra c te ris tic o r c o m b in a tio n o f b e h a v io rs o r c h a ra c te ris tic s . T h e fitn e s s m e a s u re fo r m o s t p ro b le m s o f c o n tro lle r d e s ig n is m u lti-o b je c tiv e in th e s e n s e th a t th e re a re s e v e ra l d iffe re n t (u s u a lly c o n flic tin g ) re q u ire m e n ts fo r th e c o n tro lle r. T h e f itn e s s o f e a c h in d iv id u a l is d e te r m in e d b y e x e c u tin g th e p r o g r a m tr e e ( i.e ., th e re s u lt-p ro d u c in g b ra n c h a n d a n y a u to m a tic a lly d e fin e d fu n c tio n s th a t m a y b e in v o k e d ) to p ro d u c e a n in te rc o n n e c te d s e q u e n c e o f s ig n a l p ro c e s s in g b lo c k s  th a t is , a b lo c k d ia g ra m fo r th e c o n tro lle r. A S P IC E n e tlis t is th e n c o n s tru c te d fro m th e b lo c k d ia g ra m . T h e S P IC E n e tlis t fo r th e re s u ltin g c o n tro lle r is w ra p p e d in s id e a n a p p ro p ria te s e t o f S P IC E c o m m a n d s . T h e c o n tro lle r is th e n s im u la te d u s in g o u r m o d if ie d v e r s io n o f th e S P I C E s im u la to r . T h e 2 1 7 ,0 0 0 - lin e S P I C E 3 s im u la to r (Q u a rle s , N e w to n , P e d e rs o n , a n d S a n g io v a n n i-V in c e n te lli 1 9 9 4 ) is a n in d u s tria ls tre n g th s im u la to r. It is ru n a s a s u b m o d u le w ith in o u r g e n e tic p ro g ra m m in g s y s te m . T h e S P IC E s im u la to r re tu rn s ta b u la r o u tp u t a n d o th e r in fo rm a tio n fro m w h ic h th e fitn e s s o f th e in d iv id u a l is th e n c o m p u te d . T h e fitn e s s o f a c o n tro lle r is m e a s u re d u s in g 1 3 e le m e n ts c o n s is tin g o f 1 2 tim e d o m a in -b a s e d e le m e n ts b a s e d o n a m o d ifie d in te g ra l o f tim e -w e ig h te d a b s o lu te e rro r (IT A E ) a n d o n e tim e -d o m a in -b a s e d e le m e n t m e a s u rin g d is tu rb a n c e re je c tio n . T h e f itn e s s o f a n in d iv id u a l c o n tr o lle r is th e s u m ( i.e ., lin e a r c o m b in a tio n ) o f th e d e trim e n ta l c o n trib u tio n s o f th e s e 1 3 e le m e n ts o f th e fitn e s s m e a s u re . T h e s m a lle r th e s u m , th e b e tte r. T h e firs t 1 2 e le m e n ts o f th e fitn e s s m e a s u re e v a lu a te h o w q u ic k ly th e c o n tro lle r c a u s e s t h e p l a n t t o r e a c h t h e r e f e r e n c e s i g n a l a n d t h e c o n t r o l l e r 's s u c c e s s i n a v o i d i n g A u to m a tic S y n th e s is o f B o th th e T o p o lo g y a n d P a ra m e te rs 1 7 3 o v e rs h o o t. T w o re fe re n c e s ig n a ls a re u s e d . T h e firs t re fe re n c e s ig n a l is a s te p fu n c tio n th a t ris e s fro m 0 to 1 v o lts a t t = 1 0 0 m illis e c o n d s w h ile th e s e c o n d ris e s fro m 0 to 1 m ic ro v o lts a t t = 1 0 0 m illis e c o n d s . T h e tw o s te p fu n c tio n s a re u s e d to d e a l w ith th e n o n -lin e a rity c a u s e d b y th e lim ite r. T w o v a lu e s o f th e tim e c o n s ta n t, τ , a re u s e d ( n a m e ly 0 .5 a n d 1 .0 ) . T h r e e v a lu e s o f K a r e u s e d , n a m e ly 0 .9 , 1 .0 , a n d 1 .1 . E x p o s in g g e n e tic p ro g ra m m in g to d iffe re n t c o m b in a tio n s o f v a lu e s o f s te p s iz e , K , a n d τ p ro d u c e s a ro b u s t c o n tro lle rs a n d a ls o p re v e n ts g e n e tic p ro g ra m m in g fro m e n g a g in g in p o le e lim in a tio n . F o r e a c h o f th e s e 1 2 fitn e s s c a s e s , a tra n s ie n t a n a ly s is is p e rfo rm e d in th e tim e d o m a in u s in g th e S P IC E s im u la to r. T a b le 1 s h o w s th e e le m e n ts o f th e fitn e s s m e a s u re in its le ft-m o s t fo u r c o lu m n s . T h e c o n trib u tio n to fitn e s s fo r e a c h o f th e s e 1 2 e le m e n ts o f th e fitn e s s m e a s u re is b a s e d o n th e in te g ra l o f tim e -w e ig h te d a b s o lu te e rro r (IT A E ) 3 6 ∫ (t − 5 ) e (t) A (e (t))B C d t . t= 5 B e c a u s e o f th e b u ilt-in fiv e -s e c o n d tim e d e la y , th e in te g ra tio n ru n s fro m tim e t = 5 s e c o n d s to t = 3 6 s e c o n d s . H e re e (t) is th e d iffe re n c e (e rro r) a t tim e t b e tw e e n th e d e la y e d p la n t o u tp u t a n d th e re fe re n c e s ig n a l. T h e in te g ra l o f tim e -w e ig h te d a b s o lu te e rro r p e n a liz e s d iffe re n c e s th a t o c c u r la te r m o re h e a v ily th a n d iffe re n c e s th a t o c c u r e a rlie r. W e m o d ifie d th e in te g ra l o f tim e -w e ig h te d a b s o lu te e rro r in fo u r w a y s . F irs t, w e u s e d a d is c re te a p p ro x im a tio n to th e in te g ra l b y c o n s id e rin g 1 2 0 3 0 0 -m illis e c o n d tim e s te p s b e tw e e n t = 5 to t = 3 6 s e c o n d s . S e c o n d , w e m u ltip lie d e a c h fitn e s s c a s e b y th e re c ip ro c a l o f th e a m p litu d e o f th e re fe re n c e s ig n a ls s o th a t b o th re fe re n c e s ig n a ls (1 m ic ro v o lt a n d 1 v o lt) a re e q u a lly in flu e n tia l. S p e c ific a lly , B is a fa c to r th a t is u s e d to n o rm a liz e th e c o n trib u tio n s a s s o c ia te d w ith th e tw o s te p fu n c tio n s . B m u ltip lie s th e d iffe re n c e e (t) a s s o c ia te d w ith th e 1 -v o lt s te p fu n c tio n b y 1 a n d m u ltip lie s th e 6 d iffe re n c e e (t) a s s o c ia te d w ith th e 1 -m ic ro v o lt s te p fu n c tio n b y 1 0 . T h ird , th e in te g ra l c o n ta in s a n a d d itio n a l w e ig h t, A , th a t v a rie s w ith e (t). T h e fu n c tio n A w e ig h ts a ll v a r ia tio n u p to 1 0 2 % o f th e r e f e r e n c e s ig n a l b y a f a c to r o f 1 .0 , a n d h e a v ily p e n a liz e s o v e r s h o o ts o v e r 2 % b y a f a c to r 1 0 .0 . F o u r th , th e in te g r a l c o n ta in s a s p e c ia l w e ig h t, C , w h ic h is 5 .0 f o r th e tw o f itn e s s c a s e s f o r w h ic h K = 1 a n d τ = 1 , a n d 1 .0 o th e r w is e . T h e 1 3 th e le m e n t o f th e fitn e s s m e a s u re is b a s e d o n d is tu rb a n c e re je c tio n . T h e p e n a lty is c o m p u te d b a s e d o n a tim e - d o m a in a n a ly s is f o r 3 6 .0 s e c o n d s . I n th is a n a ly s is , th e re fe re n c e s ig n a l is h e ld a t a v a lu e o f 0 . A d is tu rb a n c e s ig n a l c o n s is tin g o f a u n it s te p is a d d e d to th e C O N T R O L L E R _ O U T P U T a t tim e t = 0 a n d th e re s u ltin g d is tu rb e d s ig n a l is p ro v id e d a s in p u t to th e p la n t. T h e d e trim e n ta l c o n trib u tio n to fitn e s s is 5 0 0 /3 6 tim e s th e tim e re q u ire d to b rin g th e p la n t o u tp u t to w ith in 2 0 m illiv o lts o f th e r e f e r e n c e s ig n a l o f 0 v o lts ( i.e ., to r e d u c e th e e f f e c t to w ith in 2 % o f th e 1 -v o lt d is tu rb a n c e s ig n a l) a s s u m in g th a t th e p la n t s e ttle s to w ith in th is ra n g e w ith in 3 6 s e c o n d s . If th e p la n t d o e s n o t s e ttle to w ith in th is ra n g e w ith in 3 6 s e c o n d s , th e d e trim e n ta l c o n trib u tio n to fitn e s s is 5 0 0 p lu s th e a b s o lu te v a lu e o f th e p la n t o u tp u t in v o lts tim e s 5 0 0 . F o r e x a m p le , if th e e ffe c t o f th e d is tu rb a n c e w a s n e v e r re d u c e d b e lo w 1 v o lts , th e d e trim e n ta l c o n trib u tio n to fitn e s s w o u ld b e 1 0 0 0 . A c o n tro lle r th a t c a n n o t b e s im u la te d b y S P IC E is a s s ig n e d a h ig h p e n a lty v a lu e o f fitn e s s (1 0 8 ). 1 7 4 5 .5 T h e te rm 1 0 0 p a ra ra n g J .R . K o z a e t a l. C o n tro l P a ra m p o p u la tio n s iz e , M , w a in a ls ) w a s e s ta b lis h e d p o in ts w a s e s ta b lis h e m e te rs fo r c o n tro llin g e o f p ro b le m s (K o z a , B 5 .6 T e rm T h e ru n w a s m a n s u c c e s s iv e b e s t-o s in g le b e s t-s o -fa r e te rs s 5 0 0 ,0 0 0 . A m a x fo r e a c h re s u lt-p ro d fo r e a c h a u to m th e ru n s a re th e d e n n e tt, A n d re , a n d im u m s iz e o f 1 5 0 d u c in g b ra n c h a n a tic a lly d e fin e d e fa u lt v a lu e s th a t K e a n e 1 9 9 9 ). p o in ts d a m a fu n c tio w e a p (fu x im n . p ly n c tio n s a n d u m s iz e o f T h e o th e r to a b ro a d in a tio n u a lly m o n ito re d a n d m a n u a lly te rm in a te d w h e n th e fitn e s s o f m a n y f-g e n e ra tio n in d iv id u a ls a p p e a re d to h a v e re a c h e d a p la te a u . T h e in d iv id u a l is h a rv e s te d a n d d e s ig n a te d a s th e re s u lt o f th e ru n . 5 .7 P a ra lle l Im p le m e n ta tio n T h is p ro b le m w a s ru n o n a h o m e -b u ilt B e o w u lf-s ty le (S te rlin g , S a lm o n , B e c k e r, a n d S a v a re s e 1 9 9 9 ; B e n n e tt, K o z a , S h ip m a n , a n d S tiffe lm a n 1 9 9 9 ) p a ra lle l c lu s te r c o m p u te r s y s te m c o n s is tin g o f 1 ,0 0 0 3 5 0 M H z P e n tiu m II p ro c e sso rs (e a c h a c c o m p a n ie d b y 6 4 m e g a b y te s o f R A M ). T h e s y s te m h a s a 3 5 0 M H z P e n tiu m II c o m p u te r a s h o s t. T h e p ro c e s s in g n o d e s a re c o n n e c te d w ith a 1 0 0 m e g a b it-p e r-s e c o n d E th e rn e t. T h e p ro c e s s in g n o d e s a n d th e h o s t u s e th e L in u x o p e ra tin g s y s te m . T h e d is trib u te d g e n e tic a lg o rith m w ith u n s y n c h ro n iz e d g e n e ra tio n s a n d s e m i-is o la te d s u b p o p u la tio n s w a s u s e d w ith a s u b p o p u la tio n s iz e o f Q = 5 0 0 a t e a c h o f D = 1 ,0 0 0 d e m e s . T w o p ro c e s s o rs a re h o u s e d in e a c h o f th e 5 0 0 p h y s ic a l b o x e s o f th e s y s te m . A s e a c h p ro c e s s o r (a s y n c h ro n o u s ly ) c o m p le te s a g e n e ra tio n , fo u r b o a tlo a d s o f e m ig ra n ts fro m e a c h s u b p o p u la tio n (s e le c te d p ro b a b ilis tic a lly b a s e d o n fitn e s s ) a re d is p a tc h e d to e a c h o f th e fo u r to ro id a lly a d ja c e n t p ro c e s s o rs . T h e m ig ra tio n ra te is 2 % (b u t 1 0 % if th e to ro id a lly a d ja c e n t n o d e is in th e s a m e p h y s ic a l b o x ). 6 R e s u lts T h e b e s t in d iv id u a l in g e n e r a tio n 0 h a s a f itn e s s o f 1 9 2 6 .4 9 8 . T h e b e s t-o f-ru n c o n tro lle r e m e rg e d in g e n e ra tio n 1 2 9 (fig u re 1 ). T h is b e s t-o f-ru n c o n tr o lle r h a s a f itn e s s o f 5 2 2 .6 0 5 . T h e r e s u lt- p r o d u c in g b r a n c h o f th is b e s t- o f - r u n in d iv id u a l h a s 1 1 9 p o in ts (fu n c tio n s a n d te rm in a ls ) a n d 9 5 , 9 3 , a n d 7 0 p o in ts , re s p e c tiv e ly , in its th re e a u to m a tic a lly d e fin e d fu n c tio n s . N o te th a t g e n e tic p r o g r a m m in g e m p lo y e d a 4 .8 s e c o n d d e la y ( c o m p a r a b le to th e f iv e - s e c o n d p la n t d e la y ) in th e tra n s fe r fu n c tio n o f th e e v o lv e d p re -filte r. T h is b e s t-o f-ru n c o n tro lle r fro m g e n e ra tio n 1 2 9 h a s a b e tte r v a lu e o f fitn e s s fo r a s te p s iz e o f 1 v o lt, a n in te rn a l g a in , K , o f 1 .0 , a n d a tim e - c o n s ta n t, τ ,o f 1 .0 ( th e s p e c if ic c a s e c o n s id e r e d b y A s tr o m a n d H a g g lu n d 1 9 9 5 ). F ig u re 2 c o m p a re s th e tim e -d o m a in re s p o n s e to s te p in p u t o f th e b e s t-o f-ru n c o n tro lle r fro m g e n e ra tio n 1 2 9 (tria n g le s ) w ith th e c o n tro lle r in A s tro m a n d H a g g lu n d 1 9 9 5 ( s q u a r e s ) f o r a s te p s iz e o f 1 v o lt, a n in te r n a l g a in , K , o f 1 .0 , a n d a tim e - c o n s ta n t, τ ,o f 1 .0 . F ig u re 3 c o m p a re s th e d is tu rb a n c e re je c tio n o f th e b e s t-o f-ru n c o n tro lle r fro m g e n e ra tio n 1 2 9 (tria n g le s ) w ith th e c o n tro lle r in A s tro m a n d H a g g lu n d 1 9 9 5 (s q u a re s ) f o r a s te p s iz e o f 1 v o lt, a n in te r n a l g a in , K , o f 1 .0 , a n d a tim e - c o n s ta n t, τ ,o f 1 .0 . A u to m a tic S y n th e s is o f B o th th e T o p o lo g y a n d P a ra m e te rs Reference Signal 0 .718 (1 + 7.00 s + 16 .74 s 2 ) + 0.282 e 4.80531s (1 + 4 .308 s )(1 + 1 .00106 s ) 0 .1215 s (1 + 0.0317 s )(1 + 0 .01669 s ) + 1 7 5 Plant Output Control Variable Plant - (1 + 3 .96 s + 4.205 s 2 )(1 + 0 .238 s + 0.0837 s 2 ) F ig u re 1 B e s t-o f-ru n c o n tro lle r fro m T a b le th e A s tro C = 5 .0 . A s tro m a 1 c o a n A ll n d H m m p d H 1 2 a g g a re s th e fitn a g g lu n d 1 9 e n trie s a re lu n d 1 9 9 5 c g e n e ra tio n 1 2 9 fo r th re e -la g p la n t w ith fiv e -s e c o n d d e la y . e s s o f th e b e s t-o f-ru n c o n tro lle r fro m g e n e ra tio n 1 2 9 a n d 9 5 . T w o o f th e e n trie s a re d iv id e d b y th e s p e c ia l w e ig h t b e tte r fo r th e g e n e tic a lly e v o lv e d c o n tro lle r th a n fo r th e o n tro lle r. T a b le 1 F itn e s s o f tw o c o n tr o lle r s fo r th r e e -la g p la n t w ith B e s t-o f-ru T im e P la n t E le m e n t S te p g e n e ra tio n c o n s ta n in te rn a l s iz e 1 2 9 G a in , K (v o lts ) t, τ 0 1 0 .9 1 .0 1 3 .7 1 1 0 .9 0 .5 2 5 .6 2 1 1 .0 1 .0 3 4 .0 / 5 = 3 1 1 .0 0 .5 1 8 .6 4 1 1 .1 1 .0 4 .4 5 1 1 .1 0 .5 1 6 .3 6 1 0 -6 0 .9 1 .0 1 3 .2 7 1 0 -6 0 .9 0 .5 2 5 .5 8 1 0 -6 1 .0 1 .0 3 0 .7 / 5 = 9 1 0 -6 1 .0 0 .5 1 8 .5 1 0 1 0 -6 1 .1 1 .0 4 .3 1 1 1 0 -6 1 .1 0 .5 1 6 .2 D is tu rb a n c e 1 1 1 3 0 2 n fiv e -s e c o n d d e la y . A s tro m a n d H a g g lu n d c o n tro lle r 2 7 .4 3 8 .2 6 .8 2 2 .9 2 9 .3 2 5 .4 2 2 .7 2 7 .4 3 8 .2 6 .1 2 2 .9 2 9 .3 2 5 .4 2 2 .7 3 7 3 R e fe r e n c e s A n g e lin e , P e te r J . a n d K in n e a r, K e n n e th E . J r. (e d ito rs ). P r o g r a m m in g 2 . C a m b rid g e , M A : T h e M IT P re s s . A s tro m , K a rl J . a n d H a g g lu n d , T o re . 1 9 9 5 . P ID C o n tr o T u n in g . 2 n d E d itio n . R e s e a rc h T ria n g le P a rk , N C : In s tru B a n z h a f , W o lf g a n g , D a id a , J a s o n , E ib e n , A . E ., G a r z o n , J a k ie la , M a rk , a n d S m ith , R o b e rt E . (e d ito rs ). 1 9 9 9 . G th e G e n e tic a n d E v o lu tio n a r y C o m p u ta tio n C o n fe r e n c e , F lo r id a U S A . S a n F ra n c is c o , C A : M o rg a n K a u fm a n n . B a n z h a f , W o lf g a n g , N o r d in , P e te r , K e lle r , R o b e r t E ., a n d G e n e tic P r o g r a m m in g – A n In tr o d u c tio n . S a n F ra n c is c a n d H e id e lb e rg : d p u n k t. 1 9 9 6 . A d v a n c e s in G e n e tic lle r s : T m e n t S M a x H E C C O J u ly 1 3 h e o r y , D e s ig n , a n d o c ie ty o f A m e ric a . ., H o n a v a r , V a s a n t, -9 9 : P r o c e e d in g s o f -1 7 , 1 9 9 9 , O r la n d o , F ra n c o n e , F ra n k D . 1 9 9 8 . o , C A : M o rg a n K a u fm a n n 1 7 6 J .R . K o z a e t a l. B a n z h a f, W o lfg a n g , P o li, R ic c a rd o , S c h o e n a u e r, M a rc , a n d F o g a rty , T e re n c e C . 1 9 9 8 . G e n e tic P r o g r a m m in g : F ir s t E u r o p e a n W o r k s h o p . E u r o G P ’9 8 . P a r is , F r a n c e , A p r il 1 9 9 8 P r o c e e d in g s . P a r is , F r a n c e . A p r il l9 9 8 . L e c tu re N o te s in C o m p u te r S c ie n c e . V o lu m e 1 3 9 1 . B e rlin , G e rm a n y : S p rin g e r-V e rla g . B e n n e tt, F o r r e s t H I I I , K o z a , J o h n R ., S h ip m a n , J a m e s , a n d S tif f e lm a n , O s c a r . 1 9 9 9 . B u ild in g a p a r a lle l c o m p u te r s y s te m f o r $ 1 8 ,0 0 0 th a t p e r f o r m s a h a lf p e ta - f lo p p e r d a y . I n B a n z h a f , W o lf g a n g , D a id a , J a s o n , E ib e n , A . E ., G a r z o n , M a x H ., H o n a v a r , V a s a n t, J a k ie la , M a rk , a n d S m ith , R o b e rt E . (e d ito rs ). 1 9 9 9 . G E C C O -9 9 : P r o c e e d in g s o f th e G e n e tic a n d E v o lu tio n a r y C o m p u ta tio n C o n fe r e n c e , J u ly 1 3 -1 7 , 1 9 9 9 , O r la n d o , F lo r id a U S A . S a n F ra n c is c o , C A : M o rg a n K a u fm a n n . 1 4 8 4 - 1 4 9 0 . B o y d , S . P . a n d B a rra tt, C . H . 1 9 9 1 . L in e a r C o n tr o lle r D e s ig n : L im its o f P e r fo r m a n c e . E n g le w o o d C liffs , N J : P re n tic e H a ll. H o lla n d , J o h n H . 1 9 7 5 . A d a p ta tio n in N a tu r a l a n d A r tific ia l S y s te m s . A n n A rb o r, M I: U n iv e rs ity o f M ic h ig a n P re s s . K in n e a r, K e n n e th E . J r. (e d ito r). 1 9 9 4 . A d v a n c e s in G e n e tic P r o g r a m m in g . C a m b rid g e , M A : T h e M IT P re s s . K o z a , J o h n R . 1 9 9 2 . G e n e tic P r o g r a m m in g : O n th e P r o g r a m m in g o f C o m p u te r s b y M e a n s o f N a tu r a l S e le c tio n . C a m b rid g e , M A : M IT P re s s . K o z a , J o h n R . 1 9 9 4 a . G e n e tic P r o g r a m m in g II: A u to m a tic D is c o v e r y o f R e u s a b le P r o g r a m s . C a m b rid g e , M A : M IT P re s s . K o z a , J o h n R . 1 9 9 4 b . G e n e tic P r o g r a m m in g II V id e o ta p e : T h e N e x t G e n e r a tio n . C a m b rid g e , M A : M IT P re s s . K o z a , J o h n R ., B a n z h a f , W o lf g a n g , C h e lla p illa , K u m a r , D e b , K a ly a n m o y , D o r ig o , M a r c o , F o g e l, D a v id B ., G a r z o n , M a x H ., G o ld b e r g , D a v id E ., I b a , H ito s h i, a n d R io lo , R ic k . (e d ito rs ). 1 9 9 8 . G e n e tic P r o g r a m m in g 1 9 9 8 : P r o c e e d in g s o f th e T h ir d A n n u a l C o n fe r e n c e . S a n F ra n c is c o , C A : M o rg a n K a u fm a n n . K o z a , J o h n R ., B e n n e tt I I I , F o r r e s t H , A n d r e , D a v id , a n d K e a n e , M a r tin A . 1 9 9 9 . G e n e tic P r o g r a m m in g III: D a r w in ia n In v e n tio n a n d P r o b le m S o lv in g . S a n F ra n c is c o , C A : M o rg a n K a u fm a n n . F o rth c o m in g . K o z a , J o h n R ., B e n n e tt I I I , F o r r e s t H , A n d r e , D a v id , K e a n e , M a r tin A ., a n d B r a v e S c o tt. 1 9 9 9 . G e n e tic P r o g r a m m in g III V id e o ta p e : H u m a n -C o m p e titiv e M a c h in e In te llig e n c e . S a n F ra n c is c o , C A : M o rg a n K a u fm a n n . K o z a , J o h n R ., D e b , K a ly a n m o y , D o r ig o , M a r c o , F o g e l, D a v id B ., G a r z o n , M a x , I b a , H ito s h i, a n d R io lo , R . L . (e d ito rs ). 1 9 9 7 . G e n e tic P r o g r a m m in g 1 9 9 7 : P r o c e e d in g s o f th e S e c o n d A n n u a l C o n fe r e n c e S a n F ra n c is c o , C A : M o rg a n K a u fm a n n . K o z a , J o h n R ., G o ld b e r g , D a v id E ., F o g e l, D a v id B ., a n d R io lo , R ic k L . ( e d ito r s ) . 1 9 9 6 . G e n e tic P r o g r a m m in g 1 9 9 6 : P r o c e e d in g s o f th e F ir s t A n n u a l C o n fe r e n c e . C a m b rid g e , M A : M IT P re s s . K o z a , J o h n R ., K e a n e , M a r tin A ., Y u , J e s s e n , B e n n e tt, F o r r e s t H I I I , a n d M y d lo w e c , W illia m . 2 0 0 0 . A u to m a tic c re a tio n o f h u m a n -c o m p e titiv e p ro g ra m s a n d c o n tro lle rs b y m e a n s o f g e n e tic p ro g ra m m in g . G e n e tic P r o g r a m m in g a n d E v o lv a b le M a c h in e s . (1 ) 1 2 1 - 1 6 4 . K o z a , J o h n R ., a n d R ic e , J a m e s P . 1 9 9 2 . G e n e tic P r o g r a m m in g : T h e M o v ie . C a m b rid g e , M A : M IT P re s s . L a n g d o n , W illia m B . 1 9 9 8 . G e n e tic P r o g r a m m in g a n d D a ta S tr u c tu r e s : G e n e tic P r o g r a m m in g + D a ta S tr u c tu r e s = A u to m a tic P r o g r a m m in g ! A m s te rd a m : K lu w e r. A u to m a tic S y n th e s is o f B o th th e T o p o lo g y a n d P a ra m e te rs 1 7 7 P o li, R ic c a r d o , N o r d in , P e te r , L a n g d o n , W illia m B ., a n d F o g a r ty , T e r e n c e C . 1 9 9 9 . G e n e tic P r o g r a m m in g : S e c o n d E u r o p e a n W o r k s h o p . E u r o G P ’9 9 . P r o c e e d in g s . L e c tu re N o te s in C o m p u te r S c ie n c e . V o lu m e 1 5 9 8 . B e rlin : S p rin g e r-V e rla g . Q u a r le s , T h o m a s , N e w to n , A . R ., P e d e r s o n , D . O ., a n d S a n g io v a n n i- V in c e n te lli, A . 1 9 9 4 . S P I C E 3 V e r s io n 3 F 5 U s e r ’s M a n u a l . D e p a r tm e n t o f E le c tr ic a l E n g in e e r in g a n d C o m p u te r S c ie n c e , U n iv . o f C a lifo rn ia . B e rk e le y , C A . M a rc h 1 9 9 4 . R y a n , C o n o r. 1 9 9 9 . A u to m a tic R e -e n g in e e r in g o f S o ftw a r e U s in g G e n e tic P r o g r a m m in g . A m s te rd a m : K lu w e r A c a d e m ic P u b lis h e rs . S p e c t o r , L e e , L a n g d o n , W i l l i a m B . , O ’R e i l l y , U n a - M a y , a n d A n g e l i n e , P e t e r ( e d i t o r s ) . 1 9 9 9 . A d v a n c e s in G e n e tic P r o g r a m m in g 3 . C a m b rid g e , M A : M IT P re s s . S te r lin g , T h o m a s L ., S a lm o n , J o h n , B e c k e r , D . J ., a n d S a v a r e s e , D . F . 1 9 9 9 . H o w to B u ild a B e o w u lf: A G u id e to Im p le m e n ta tio n a n d A p p lic a tio n o f P C C lu s te r s . C a m b rid g e , M A : M IT P re s s . 1.2 1 Voltage 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 30 35 40 Time F ig u re 2 C o m p a ris o n fo r s te p in p u t. 1 0.8 Volt age 0.6 0.4 0.2 0 -0.2 0 5 10 15 20 25 30 35 Time F ig u re 3 C o m p a ris o n fo r d is tu rb a n c e re je c tio n . 40 A u to m a tic D e s ig n o f M u ltiv a r ia b le Q F T C o n tr o l S y s te m v ia E v o lu tio n a r y C o m p u ta tio n K . C . T a n , T . H . L e e , a n d E . F . K h o r D e p a rtm N a tio 1 0 K e n t R { e le ta n k c , e n t n a l U id g e e le le o f E le c tric a l n iv e rs ity o f C re s c e n t S in e th , e n g p 8 6 2 E n g in S in g a g a p o r 6 } @ n e e p o e 1 u s rin g re 1 9 2 6 0 .e d u .s g A b str a c t. T h is p a p e r p ro p o s e s a m u lti-o b je c tiv e e v o lu tio n a ry a u to m a te d d e s ig n m e th o d o lo g y fo r m u ltiv a ria b le Q F T c o n tro l s y s te m s . U n lik e e x is tin g m a n u a l o r c o n v e x o p tim is a tio n b a s e d Q F T d e s ig n a p p ro a c h e s , t h e ’i n t e l l i g e n t ’ e v o l u t i o n a r y t e c h n i q u e i s c a p a b l e o f a u t o m a t i c a l l y e v o lv in g b o th th e n o m in a l c o n tro lle r a n d p re -filte r s im u lta n e o u s ly to m e e t a ll p e rfo rm a n c e re q u ire m e n ts in Q F T , w ith o u t g o in g th ro u g h th e c o n s e rv a tiv e a n d s e q u e n tia l d e s ig n s ta g e s fo r e a c h o f th e m u ltiv a ria b le s u b -s y s te m s . In a d d itio n , it a v o id s th e n e e d o f m a n u a l Q F T b o u n d c o m p u ta tio n a n d tria l-a n d -e rro r lo o p -s h a p in g d e s ig n p ro c e d u re s , w h ic h is p a rtic u la rly u s e fu l fo r u n s ta b le o r n o n -m in im u m p h a s e p la n ts fo r w h ic h s ta b ilis in g c o n tro lle rs m a y b e d iffic u lt to b e s y n th e s is e d . E ffe c tiv e n e s s o f th e p ro p o s e d Q F T d e s ig n m e th o d o lo g y is v a lid a te d u p o n a b e n c h m a rk m u ltiv a ria b le s y s te m , w h ic h o ffe rs a s e t o f lo w -o rd e r P a re to o p tim a l c o n tro lle rs th a t s a tis fy a ll th e re q u ire d c lo s e d -lo o p p e rfo rm a n c e s u n d e r p ra c tic a l c o n s tra in ts . 1 I n tr o d u c tio n Q u a n tita tiv e F e e d b a c k T h e o ry (Q F T ) is w e ll-k n o w n a s a n e ffic ie n t fre q u e n c y d o m a in c o n tro lle r d e s ig n m e th o d o lo g y th a t u tilis e s N ic h o ls c h a rt to a c h ie v e a d e s ire d ro b u s t d e s ig n o v e r s p e c ifie d ra n g e s o f s tru c tu re d p la n t p a ra m e te r u n c e rta in tie s w ith a n d w ith o u t c o n tro l e ffe c to r fa ilu re s [1 -3 ]. T h e b a s ic id e a o f Q F T is to c o n v e rt d e s ig n s p e c ific a tio n o n c lo s e d -lo o p re s p o n s e a n d p la n t u n c e rta in ty in to ro b u s t s ta b ility a n d p e rfo rm a n c e b o u n d s o n o p e n -lo o p tra n s m is s io n o f th e n o m in a l s y s te m a s s h o w n in F ig . 1 . A fix e d s tru c tu re c o n tro lle r G (s ) a n d p re -filte r F (s ) is th e n s y n th e s iz e d u s in g g a in -p h a s e lo o p -s h a p in g te c h n iq u e s o th a t th e tw o -d e g re e -fre e d o m o u tp u t fe e d b a c k s y s te m is c o n tro lle d w ith in s p e c ific a tio n fo r a n y m e m b e r o f th e p la n t te m p la te s . F o r m u lti-in p u t m u lti-o u tp u t (M IM O ) s y s te m s , c o n v e n tio n a l Q F T m e th o d re q u ire s th e d e s ig n p ro c e s s to b e tu rn e d in to a s e q u e n c e o f m u lti-in p u t s in g le -o u tp u t (M IS O ) p ro b le m s b e fo re a n y Q F T d e s ig n p ro c e d u re c a n b e p e rfo rm e d [1 -3 ]. G iv in g a c o m b i n e d s o l u t i o n o f c o n t r o l l e r G ( s ) = d i a g [ g i( s ) ] a n d p r e - f i l t e r F ( s ) = [ f ij] , " i , j = 1 , 2 , … , m fo r a n m in p u t a n d m o u tp u t c o n tro l p ro b le m , th e s o lu tio n o f th e firs t s e t o f M IS O p ro b le m is th e firs t tra n s fe r fu n c tio n o f th e d ia g o n a l c o n tro lle r g 1(s ) a n d f11(s ). S. Cagnoni et al. (Eds.): EvoWorkshops 2000, LNCS 1803, pp. 178-194, 2000. ' Springer-Verlag Berlin Heidelberg 2000 Automatic Design of Multivariable QFT Control System T f1 f2 M in m h e n a (s) w 2 ( s ) , … IS O p u ts /o n e w se t o h ic h re s u , f2m (s ) a e q u iv a le n u tp u ts , w R (s) f M IS O lts to th n d e tc ., t lo o p s h ic h c a n p ro b le m is d e f e la tte r s o lu tio H e n c e , th is d e a n d its s iz e b e v e ry te d io u in e d b a s e d o n o f th e c o n s ig n m e th o d g ro w s e x p o s to th e d e s ig n th e p la n t, g 1(s ), f11(s ), f12( tro lle r g 2(s ) a n d p re -filte r le a d s to a n o v e ra ll d e s ig n n e n tia lly to th e m n u m b n e r [3 ]. 179 s ) ,… , f21(s ), o f m 2 e r o f Y (s) F (s) G (s) P (s) F ig . 1 . A ty p ic a l o u tp u t fe e d b a c k m u ltiv a ria b le c o n tro l s y s te m in Q F T A s p o in te d o u t in [4 ], th e re a re n o o th e r d e s ig n m e th o d s th a n th e m a n u a l tria la n d -e rro r p ro c e s s c o u ld b e e m p lo y e d to d e te rm in e th e s e s e rie s o f lo o p s . M o re o v e r, th is s e q u e n tia l M IS O d e s ig n p ro c e d u re m a y b e c o n s e rv a tiv e s in c e th e s o lu tio n o f a s e t o f M IS O is h ig h ly d e p e n d e n t a n d re s tric te d to th e fo rm e r s e ts o f M IS O s o lu tio n . T h e u s e r m a y th u s h a v e to re p e a t o r re -re s ta rt th e d e s ig n p ro c e d u re fro m th e firs t M IS O lo o p s y n th e s is , if th e re e x is t a n y s e t o f M IS O s o lu tio n s th a t is u n fe a s ib le d u e to im p ro p e r o r o v e rly d e s ig n o f th e p re v io u s s e ts o f M IS O lo o p s . B e s id e s , Q F T b o u n d s in N ic h o ls c h a rt fo r e a c h s p e c ific a tio n o f a ll fre q u e n c y p o in ts m u s t b e a c q u ire d b e fo re th e d e s ig n , w h ic h is o fte n a n e x h a u s tiv e tria l-a n d -e rro r p ro c e s s . T h e re a s o n is th a t, fo r e v e ry fre q u e n c y p o in t w ith s u ffic ie n tly s m a ll fre q u e n c y in te rv a l, th e te m p la te n e e d s to b e m a n u a lly s h ifte d u p o r d o w n o n th e N ic h o ls c h a rt u n til th e g a in v a ria tio n o f th e te m p la te is e q u a l to th e g a in v a ria tio n a llo w e d fo r a n y p a rtic u la r ro b u s t s p e c ific a tio n a t th a t fre q u e n c y . In a d d itio n , o n ly th e c o n tro lle r c a n b e s y n th e s iz e d v ia Q F T b o u n d c o m p u ta tio n u s in g th e c o n v e n tio n a l lo o p -s h a p in g m e th o d . A n o th e r in d e p e n d e n t d e s ig n ta s k h a s to b e a c c o m p lis h e d in o rd e r to o b ta in th e p re -filte r w ith in a tw o -s ta g e d e s ig n fra m e w o rk fo r e a c h s e t o f M IS O s o lu tio n . T h e a c q u is itio n o f a n o p tim a l Q F T c o n tro lle r is in fa c t a m u lti-o b je c tiv e m u ltim o d a l d e s ig n o p tim is a tio n p ro b le m th a t in v o lv e s s im u lta n e o u s ly d e te rm in in g m u ltip le c o n tro lle r a n d p re -filte r p a ra m e te rs to s a tis fy d iffe re n t c o m p e tin g p e rfo rm a n c e re q u ire m e n ts , s u c h a s s e n s itiv ity b o u n d s , c ro s s -c o u p lin g b o u n d s , ro b u s t m a rg in a n d e tc ., T o s o lv e th e s e p r o b le m s , a fe w a n a ly tic a l/m a th e m a tic s o r ie n te d o p tim is a tio n o r 'a u t o m a t i c d e s i g n ' t e c h n i q u e s h a v e r e c e n t l y b e e n i n v e s t i g a t e d a n d d e v e l o p e d [ 5 - 7 ] . T h e s e c o n v e x b a s e d o p tim is a tio n a p p ro a c h e s , h o w e v e r, im p o s e m a n y u n p ra c tic a l o r u n re a lis tic a s s u m p tio n s th a t o fte n le a d to v e ry c o n s e rv a tiv e d e s ig n s o r a re h a rd in fin d in g th e g lo b a l P a re to o p tim a l s o lu tio n s in th e m u lti-o b je c tiv e m u lti-d im e n s io n a l d e s ig n s p a c e . W ith th e v ie w o f ta c k lin g th e s e d ra w b a c k s a n d a u to m a tin g th e Q F T d e s i g n p r o c e d u r e , c o m p u t e r i s e d 'i n t e l l i g e n t ' t r i a l - a n d - e r r o r b a s e d m e t h o d o l o g y b a s e d o n e v o lu tio n a ry o p tim is a tio n h a s b e e n p ro p o s e d a n d s u c c e s s fu lly a p p lie d to in d u s tria l o r b e n c h m a rk a p p lic a tio n s [8 -1 0 ]. T h i s p a p e r f u r t h e r d e v e l o p s t h e m u l t i - o b j e c t i v e 'i n t e l l i g e n t ' a u t o m a t e d Q F T d e s ig n m e th o d o lo g y to M IM O c o n tro l s y s te m u s in g a h ig h p e rfo rm a n c e e v o lu tio n a ry a lg o rith m to o lb o x [1 1 ]. U n lik e e x is tin g m e th o d s , th e e v o lu tio n a ry Q F T d e s ig n a p p ro a c h is c a p a b le o f c o n c u rre n tly e v o lv in g th e c o n tro lle r a n d p re -filte r fo r th e e n tire 180 K.C. Tan, T.H. Lee, and E.F. Khor s e t o f M IS O s u b -s y s te m s to m e e t a ll p e rfo rm a n c e re q u ire m e n ts in Q F T , w ith o u t g o in g th ro u g h th e c o n s e rv a tiv e a n d s e q u e n tia l d e s ig n s ta g e s fo r e a c h o f th e M IS O s u b s y s te m s . B e s id e s , th e e v o lu tio n a ry to o lb o x is b u ilt w ith c o m p re h e n s iv e u s e r in te rfa c e a n d p o w e rfu l g ra p h ic a l d is p la y s fo r e a s y a s s e s s m e n t o f v a rio u s s im u la tio n re s u lts o r tra d e -o ffs a m o n g th e d iffe re n t d e s ig n s p e c ific a tio n s . T h e p a p e r is o rg a n iz e d a s fo llo w s : T h e v a rio u s Q F T d e s ig n s p e c ific a tio n s a n d th e ro le o f th e M O E A to o lb o x in th e m u ltiv a ria b le Q F T d e s ig n a re g iv e n in S e c tio n 2 . V a lid a tio n o f th e p ro p o s e d m e th o d o lo g y a g a in s t a b e n c h m a rk M IM O s y s te m is illu s tra te d in S e c tio n 3 . C o n c lu s io n s a re d ra w n in S e c tio n 4 . 2 . E v o lu tio n a r y A u to m a te d M u ltiv a r ia b le Q F T D e s ig n 2 .1 M u lt i-O b j e c t iv e A s m e n tio n e d in th e o b je c tiv e s n e e d to b e to th e c o n v e n tio n a l tw a re fo rm u la te d a s a m to c o n c u rre n tly d e s ig s a tis fy a ll th e re q u ire d Q F T D e s ig n S p e c ific a tio n s In tro d u c tio n , th e re a re a n u m b e r o f u s u a s a tis fie d c o n c u rre n tly in m u ltiv a ria b le Q F o -s ta g e lo o p -s h a p in g a p p ro a c h , th e s e p e rf u lti-o b je c tiv e d e s ig n o p tim is a tio n p ro b le m n th e n o m in a l c o n tro lle r G (s ) a n d p re -f s p e c ific a tio n s a s d e s c rib e d b e lo w : lly c o n flic tin g d e T d e s ig n s . In c o n o rm a n c e re q u ire m h e re . T h e a im is ilte r F (s ) in o rd e s ig n tra s t e n ts th u s r to (i) S ta b ility (R H S P ) T h e c o s t o f s ta b ility , R H S P , is in c lu d e d to e n s u re s ta b ility o f th e c lo s e d -lo o p s y s te m , w h ic h c o u ld b e e v a lu a te d b y s o lv in g th e ro o ts o f th e c h a ra c te ris tic p o ly n o m ia l. C le a rly , a s ta b le c lo s e d -lo o p s y s te m fo r a ll th e p la n t te m p la te s ¨ re q u ire s a z e ro v a lu e o f R H S P ,    P G F       > 0 , ∀ P i ∈ ℘ R H S P = N r  r e a l  p o l e  i (1 )   I + P i G    i  In o rd e r to e n s u re in te rn a l s ta b ility a n d to g u a ra n te e n o u n s ta b le p o le a n d n o n m in im u m p h a s e z e ro c a n c e lla tio n s , it is d e s ire d th a t a m in im u m p h a s e a n d s ta b le c o n tro lle r b e d e s ig n e d . T h is im p lie s th a t th e s e a rc h ra n g e fo r a p o ly n o m ia l c o e ffic ie n t s e t i s l i m i t e d t o e i t h e r t h e f i r s t o r t h e t h i r d ’q u a d r a n t ’, i . e . , a l l c o e f f i c i e n t s i n t h e n u m e ra to r o r d e n o m in a to r m u s t b e o f th e s a m e s ig n [1 0 ]. T h e p o le s a n d z e ro s o f th e c o n tro lle rs c a n b e c a lc u la te d e x p lic itly to a v o id R H P c a n c e lla tio n s o r a lte rn a tiv e ly , th e H o ro w itz m e th o d fo r Q F T d e s ig n o f u n s ta b le a n d n o n -m in im u m p h a s e p la n ts c a n b e u s e d , i.e ., Q F T b o u n d s fo r a n u n s ta b le /n o n -m in im u m p h a s e n o m in a l p la n t c a n b e tra n s la te d to th o s e fo r a s ta b le a n d m in im u m p h a s e p la n t, if n e c e s s a ry . ∑ (ii) R o b u s t U p p e r a n d L o w e r T r T h e c o s t o f u p p e r tra c k in g p e rfo lo o p tra n s fe r fu n c tio n , g iv e n b y u p p e r tra c k in g b o u n d a s s h o w n a t e a c h fre q u e n c y p o in t a s g iv e n a c k in g P e r fo r m a n c r m a n c e o f t h e i th d E R R U T (i,i), i s i n c l u in F ig . 2 . It is c o m b y , E R R U T (i,i) = ∑ e (E ia g o d e d p u te R R U n a l e to a d d a s T & le m d re th e E e n ss su R R L T ) t o f M IM O c lo s e d th e s p e c ific a tio n o f m o f a b s o lu te e rro r n e k = 1 ( i,i ) u t (ω k ) (2 ) Automatic Design of Multivariable QFT Control System w h e re n is th e to ta l n u m b e r o f in te b e tw e e n th e u p p e r b o u n d o f th e C L (i,i)U a n d t h e p r e - s p e c i f i e d u p p e r b o u n d o f th e c lo s e d -lo o p s y s te m b o u n d o r le s s th a n th e p re -s p e c ifie d e q u a l to z e ro a s illu s tra te d in F ig . 2 e a c h f r e q u e n c y wk r e p r e s e n t s t h e m re s te d fre q u ( i,i) e le m e n tra c k in g b o is g re a te r lo w e r tra c k , fo r w h ic h a g n itu d e o f 181 e n c y p o i n t s ; e ( i , i ) u t ( wk ) i s t h e d i f f e r e n c e t o f th e c lo s e d -lo o p tra n s fe r fu n c tio n u n d T ( i , i ) U a t f r e q u e n c y wk , i f t h e u p p e r th a n th e p re -s p e c ifie d u p p e r tra c k in g i n g b o u n d T ( i , i ) L ; o t h e r w i s e , e ( i , i ) u t ( wk ) i s th e le n g th o f th e v e rtic a l d o tte d lin e s a t e ( i , i ) u t ( wk ) . F i g . 2 . C o m p u t a t i o n o f u p p e r t r a c k i n g p e r f o r m a n c e f o r t h e i th d i a g o n a l e l e m e n t T h e c o s t f o r l o w e r t r a c k i n g p e r f o r m a n c e o f t h e i th d i a g o n a l e l e m e n t o f t h e c l o s e d - l o o p t r a n s f e r f u n c t i o n , g i v e n b y E R R L T (i,i), c a n b e d e f i n e d a s t h e s u m o f a b s o lu te e rro r a t e a c h fre q u e n c y p o in t, E R R L T ( i,i ) = ∑ n e ( i,i ) lt (ω k ) (3 ) k = 1 w h e r e n i s t h e n u m b e r o f f r e q u e n c y p o i n t s ; e (i,i)lt i s t h e b o u n d o f t h e c l o s e d - l o o p s y s t e m C L (i,i)L a n d t h e p r e - s p T (i,i)L , i f t h e l o w e r b o u n d o f t h e c l o s e d - l o o p s y s t e m i s u p p e r t r a c k i n g b o u n d T (i,i)U o r l e s s t h a n t h e p r e - s p e c i f i e O t h e r w i s e , e (i,i)lt i s e q u a l t o z e r o a s i l l u s t r a t e d i n F i g . 3 . d iffe r e c ifie g re a te d lo w e n c e b e tw e d lo w e r tra r th a n th e e r tra c k in g e n th e lo w e r c k in g b o u n d p re -s p e c ifie d b o u n d T (i,i)L ; F i g . 3 . C o m p u t a t i o n o f l o w e r t r a c k i n g p e r f o r m a n c e f o r t h e i th d i a g o n a l e l e m e n t 182 K.C. Tan, T.H. Lee, and E.F. Khor (iii) C r o s s -c o u p lin g P e r fo r m a n c e (E R R U C ) A p a rt fro m a d d re s s in g th e tra c k in g p e rfo rm a n c e o f d ia g o n a l e le m e n ts , it is a ls o e s s e n tia l to re d u c e th e c o u p lin g e ffe c t o f th e o ff-d ia g o n a l tra n s fe r fu n c tio n s fo r a ll th e p la n t te m p la te s in M IM O Q F T c o n tro l s y s te m d e s ig n . S in c e th e o b je c tiv e is to re d u c e th e g a in a n d b a n d w id th o f th e o ff-d ia g o n a l tra n s fe r fu n c tio n , o n ly th e u p p e r b o u n d s in th e fre q u e n c y re s p o n s e n e e d to b e p re s c rib e d [4 ]. T h e u p p e r b o u n d o f c o u p lin g e ffe c t r e p r e s e n t e d b y t r a n s f e r f u n c t i o n T (i,j)U f o r t h e o f f - d i a g o n a l t r a n s f e r f u n c t i o n s ( i , j ) w h e re i ž j c a n b e d e fin e d a c c o rd in g to th e a llo w a b le g a in K a n d th e b a n d w id th b e tw e e n w 1 a n d w 2, w h ic h is s h o w n in F ig . 4 a n d ta k e s th e fo rm o f 1 K ( s ) w 1 (4 ) T (i, j)U ( s ) =  1  1   s + 1   s + 1   w 1  w 2  T h e c o s t o f c ro s s -c o u p lin g e ffe c t fo r th e o ff-d ia g o n a l e le m e n ts o f M IM O c lo s e d -lo o p s y s t e m , g i v e n b y E R R U C (i,j), i s i n c l u d e t o a d d r e s s t h e s p e c i f i c a t i o n o f u p p e r c r o s s c o u p lin g b o u n d a n d is c o m p u te d a s th e s u m o f a b s o lu te e rro r a t e a c h fre q u e n c y p o in t, E R R U C w h e re n th e u p p e c o u p lin g p re -sp e c e (i,j))u c i s is th r b o b o u ifie d e q u a e (ω ( i, j) u c k ( i, j) = ∑ n e ( i, j) u c (ω k ) , fo r i žj (5 ) k = 1 e n u n d n d u p l to u m b e r o f fre q u e n c y p o in o f th e c lo s e d -lo o p s y s te T (i,j)U , i f t h e u p p e r b o u n d p e r c ro s s -c o u p lin g b o u n z e ro a s g iv e n b y ,  C L (i, j)U ( ω k ) − T (i, j)U ( ω k ) =  0  ) m t s ; e ( i , j ) u c ( wk ) a t wk C L (i,j)U a n d t h e o f th e c lo s e d -lo o p d T (i,j)U a s i l l u s t r a ,C L ( i, j)U (ω k is p re sy te d ) > T o th e r w is e th e -sp e s te m in ( i, j)U (ω d iffe c ifie is g F ig . k ) , re n d u re a 4 ; c e b e p p e r te r th O th e tw c r a n rw fo r i ≠ j e e n o ssth e is e , (6 ) F ig . 4 . C o m p u ta tio n o f u p p e r c ro s s -c o u p lin g p e rfo rm a n c e fo r o ff-d ia g o n a l e le m e n ts Automatic Design of Multivariable QFT Control System (iv ) R o b u s t M a r g in (R M ) P ra c tic a l c o n tro l a p p lic a tio n s o fte n in v o lv e n e g le c te d d y n a m ic s a t th e h ig h fre q u e n c y . B e c a u s e o f th e s e m m o d e l u s e d in c o n tro l s y s te m d e s ig n is o fte n in a c c u n e g le c t o r d u e to th e la c k o f u n d e rs ta n d in g o f th e p h y a d d re s s th e s e u n m o d e lle d u n c e rta in tie s , th e u n c e rta in m u l t i p l i c a t i v e p l a n t u n c e r t a i n t y , P ip ( s ) = P ( s ) { I + W iI ( u n c e rta in tie s o r is s in g d y n a m ic s , ra te e ith e r th ro u g s ic a l p ro c e s s [1 2 ] fe e d b a c k s y s te m − 1 s ) ∆ iI } a s s h o w n u n m o d e th e n o m h d e lib e . In o rd e w ith in v in F ig . 183 lle d in a l ra te r to e rse 5 is c o n s d ire d in th e Q F T d e s ig n . T h e ro b u s t m a rg in s p e c ific a tio n th a t a d d re s s e s th e c lo s e d -lo o p s ta b ility d u e to th e in v e rs e m u ltip lic a tiv e p la n t u n c e rta in ty fo r a n u n c e r t a i n t y w e i g h t i n g f u n c t i o n W iI c a n b e d e f i n e d a s [ 1 2 ] , R M IM 1 = I + L i( jω ) < W 1 ( jω ) iI , ∀ ω (7 ) w h e r e L i ( j ω ) i s t h e i th o p e n - l o o p t r a n s f e r f u n c t i o n w i t h t h e j th l o o p b e i n g c l o s e d i n a n M IM O s y s te m , w h ic h is s im p ly th e lo o p tra n s m is s io n P ( jω )G ( jω ) in a n S IS O s y s te m . W iI (s) ∆ P iI ip Y (s) G (s) P (s) F ig . 5 . F e e d b a c k s y s te m (v ) T h m a m a 6 , p la e x g w T h e o f S fro m n S e n s itiv ity R e je c tio n (R S ) ro b u s t s e n s itiv ity re je c tio im u m a m p litu d e o f th e re g u n itu d e . A g e n e ra l s tru c tu re h ic h d e p ic ts th e p a rtic u la r t o u tp u t. T h e m a th e m a tic a l w ith in v e rs e m u ltip lic a tiv e u n c e rta in ty n is to fin d a Q F T c o n tro lle r th a t m in im is e s la te d o u tp u t o v e r a ll p o s s ib le d is tu rb a n c e s o f b o u n to re p re s e n t th e d is tu rb a n c e re je c tio n is g iv e n in c a s e w h e re th e d is tu rb a n c e e n te rs th e s y s te m a t re p re s e n ta tio n is g iv e n b y , Y − 1 S = = {I + P (s )G (s )} D m a trix S (s ) is k n o w n a s th e d is tu r b a n c e re je c tio n . T h e m a x im u m s in g u la r v a d e te rm in e s th e d is tu rb a n c e a tte n u a tio n s in c e S is in fa c t th e c lo s e d -lo o p tra n d is tu rb a n c e D to th e p la n t o u tp u t Y . D (s) G (s) P (s) Y (s) W s (s) F ig . 6 . F o rm u la tio n o f a s e n s itiv ity re je c tio n p ro b le m th e d e d F ig . th e (8 ) lu e s sfe r 184 K.C. Tan, T.H. Lee, and E.F. Khor T h e d is tu rb a n c e a tte n u a tio n s p e c ific a tio n fo r th e c lo s e d -lo o p s y s te m w ritte n a s , σ (S ) ≤ W − 1 s ⇒ ∞ W s S ∞ m a y th u s b e < 1 (9 ) w h e re σ d e fin e s th e la rg e s t s in g u la r v a lu e a n d W s th e d e s ire d d is tu rb a n c e a tte n u a tio n fa c to r, w h ic h is a fu n c tio n o f fre q u e n c y to a llo w a d iffe re n t a tte n u a tio n fa c to r a t e a c h fre q u e n c y . (v i) H ig h F r T h e h ig h fre tra n s m is s io n n o is e a n d th a c tu a to r s a tu is g iv e n a s , e q u e n c y G a in R q u e n c y g a in p e L (s ) a t th e h ig e u n m o d e lle d ra tio n a n d in s ta o ll-o ff (H rfo rm a n c h fre q u e n h ig h -fre q b ility . T h F G ) e , H F c y in u e n c y e h ig h G , is o rd e d y n fre q in c lu r to a a m ic s u e n c y d e d to v o id th /h a rm o g a in o re d u e h ig n ic s f lo o c e th h -fre th a t p tra lim s rL ( s ) s → w h e re r is is to b e o p h ig h fre q u s tru c tu re g th e re la tim is e d e n c y g iv e n a s tiv e o rd e r o f L (s ). S in , th is p e rfo rm a n c e re q a in o f th e c o n tro lle r [9 ], b ( i,i) n s n G ( i,i) ( s ) = a ( i,i) m s m e g a in o f q u e n c y se m a y re su n s m is s io n lo o p n so r lt in L (s) (1 0 ) ∞ c e o n ly th e c o n tro lle r in th e lo o p tra n s m is s io n u ire m e n t is e q u iv a le n t to th e m in im iz a tio n o f o r th e m a g n itu d e o f b n/a m fo r a c o n tro lle r + b + a ( i,i) n − 1 ( i,i) m − 1 n − 1 s s m − 1 + Lb + La ( i,i ) 0 (1 1 ) ( i,i)0 w h e r e n a n d m is th e o r d e r o f th e n u m e r a to r a n d d e n o m in a to r fo r th e ( i,i) e le m e n t o f d ia g o n a l c o n tro lle r G (s ), re s p e c tiv e ly . 2 .2 E v o lu tio n a r y A lg o r ith m T o o lb o x a n d I ts R o le in Q F T D e s ig n A lth o u g h th e m u lti-o b je c tiv e o p tim is a tio n b a s e d Q F T d e s ig n m e th o d h a s th e m e rit o f a v o id in g c o n v e n tio n a l in d e p e n d e n t tw o -s ta g e c o n tro lle r s y n th e s is o r th e te d io u s s e q u e n tia l d e s ig n s to d e te rm in e th e s e rie s o f lo o p s fo r e a c h o f th e M IS O s u b -s y s te m , th e a p p ro a c h n e e d s to s e a rc h fo r m u ltip le o p tim is e d c o n tro lle r a n d p re -filte r c o e ffic ie n ts to s a tis fy a s e t o f n o n -c o m m e n s u ra b le a n d o fte n c o m p e tin g d e s ig n s p e c ific a tio n s . S u c h a n o p tim is a tio n p ro b le m is o fte n s e m i-in fin ite a n d g e n e ra lly n o t e v e ry w h e re d iffe re n tia b le [9 ]. It is th u s h a rd to b e s o lv e d v ia tra d itio n a l n u m e ric a l a p p ro a c h e s th a t o fte n re ly o n a d iffe re n tia b le p e rfo rm a n c e in d e x , w h ic h fo rm s th e m a jo r o b s ta c le fo r th e d e v e lo p m e n t o f a g e n e ra lis e d n u m e ric a l o p tim is a tio n p a c k a g e fo r Q F T c o n tro l a p p lic a tio n s . T h is p a p e r p ro p o s e s a n e v o lu tio n a ry a u to m a te d d e s ig n m e th o d o lo g y fo r th e m u lti-o b je c tiv e Q F T c o n tro l o p tim is a tio n p ro b le m . F ig . 7 s h o w s a g e n e ra l a rc h ite c tu re fo r th e c o m p u te r a id e d c o n tro l s y s te m d e s ig n (C A C S D ) a u to m a tio n o f M IM O Q F T c o n tro l s y s te m u s in g a m u lti-o b je c tiv e e v o lu tio n a ry a lg o rith m (M O E A ) to o lb o x . T h e d e s ig n c y c le a c c o m m o d a te s th re e d iffe re n t m o d u le s : th e in te ra c tiv e h u m a n d e c is io n -m a k in g m o d u le (c o n tro l e n g in e e r), th e o p tim is a tio n m o d u le (M O E A to o lb o x ) a n d th e Q F T c o n tro l m o d u le (s y s te m a n d s p e c ific a tio n s ). A c c o rd in g to th e s y s te m p e rfo rm a n c e re q u ire m e n ts a n d a -p r io r i k n o w le d g e o n th e p ro b le m o n -h a n d if a n y , c o n tro l e n g in e e rs m a y s p e c ify o r s e le c t th e d e s ire d Q F T s p e c ific a tio n s a s Automatic Design of Multivariable QFT Control System 185 d is c u s s e d in p re v io u s s e c tio n s to fo rm a m u lti-o b je c tiv e fu n c tio n , w h ic h n e e d n o t n e c e s s a ry b e c o n v e x o r d iffe re n tia b le . B a s e d o n th e s e d e s ig n s p e c ific a tio n s , re s p o n s e s o f th e c o n tro l s y s te m c o n s is ts o f th e s e t o f in p u t/o u tp u t s ig n a ls , th e p la n t te m p la te a s w e ll a s th e c a n d id a te c o n tro lle r G (s ) a n d p re -filte r F (s ) re c o m m e n d e d fro m th e o p tim is a tio n m o d u le a re s im u la te d a s to d e te rm in e th e d iffe re n t c o s t v a lu e s fo r e a c h d e s ig n s p e c ific a tio n in th e m u lti-o b je c tiv e fu n c tio n . A c c o rd in g to th e e v a lu a tio n re s u lts o f th e m u lti-o b je c tiv e fu n c tio n in th e c o n tro l m o d u le a n d th e d e s ig n g u id a n c e s u c h a s g o a l o r p rio rity in fo rm a tio n fro m th e d e c is io n -m a k in g m o d u le , th e o p tim is a tio n m o d u le (M O E A to o lb o x ) a u to m a te s th e Q F T d e s i g n p r o c e s s a n d i n t e l l i g e n t l y s e a r c h e s f o r t h e ’o p t i m a l ’ c o n t r o l l e r a n d p r e filte r p a ra m e te rs s im u lta n e o u s ly th a t b e s t s a tis fy th e s e t o f Q F T p e rfo rm a n c e s p e c ific a tio n s . O n -lin e o p tim is a tio n p ro g re s s a n d s im u la tio n re s u lts , s u c h a s th e d e s ig n tra d e -o ffs o r c o n v e rg e n c e a re d is p la y e d g ra p h ic a lly a n d fe e d b a c k to th e d e c is io n -m a k in g m o d u le . In th is w a y , th e o v e ra ll Q F T d e s ig n e n v iro n m e n t is s u p e rv is e d a n d m o n ito re d e ffe c tiv e ly , w h ic h h e lp s c o n tro l e n g in e e rs to m a k e a p p ro p ria te a c tio n s s u c h a s e x a m in in g th e c o m p e tin g d e s ig n tra d e -o ffs , a lte rin g th e d e s ig n s p e c ific a tio n s , a d ju s tin g g o a l s e ttin g s th a t a re to o s trin g e n t o r g e n e ro u s , o r e v e n m o d ify in g th e Q F T c o n tro l a n d s y s te m s tru c tu re if n e c e s s a ry . T h is m a n -m a c h in e in te ra c tiv e d e s ig n a n d o p tim is a tio n p ro c e s s m a y b e p ro c e e d e d u n til th e c o n tro l e n g in e e r is s a tis fie d w ith th e re q u ire d p e rfo rm a n c e s o r a fte r th e d e s ig n s p e c ific a tio n s h a v e b e e n m e t. S u c h a n e v o lu tio n a ry a u to m a te d a p p ro a c h a llo w s th e Q F T d e s ig n p ro b le m a s w e ll a s th e in te ra c tio n w ith o p tim is a tio n p ro c e s s to b e c lo s e ly lin k e d to th e e n v iro n m e n t o f th a t p a rtic u la r a p p lic a tio n . C o n tro l e n g in e e r, fo r m o s t o f th e p a rt, is n o t re q u ire d to d e a l w ith a n y d e ta ils th a t a re re la te d to th e o p tim is a tio n a lg o rith m o r to g o th ro u g h th e m a n u a l tria l-a n d -e rro r tw o -s ta g e a n d s e q u e n tia l d e s ig n a s a d o p te d in c o n v e n tio n a l Q F T d e s ig n m e th o d s . T h e M O E A to o lb o x [1 1 ] h a s b e e n d e v e lo p e d u n d e r th e M a tla b [1 3 ] p ro g ra m m in g e n v iro n m e n t, w h ic h is e ffe c tiv e fo r g lo b a l o p tim is a tio n a n d a s s e s s m e n t o f m u lti-o b je c tiv e d e s ig n tra d e -o ff s c e n a rio s , a id in g a t d e c is io n -m a k in g fo r a n o p tim a l s o lu tio n th a t b e s t m e e ts a ll d e s ig n s p e c ific a tio n s . It is a ls o c a p a b le o f h a n d lin g p ro b le m s w ith c o n s tra in ts a n d in c o rp o ra tin g a d v a n c e d g o a l a n d p rio rity in fo rm a tio n w ith lo g ic a l A N D /O R o p e ra tio n s fo r h ig h e r-d e c is io n s u p p o rt. B e s id e s , it is fu lly fu n c tio n e d w ith g ra p h ic a l u s e r in te rfa c e (G U I) a n d is re a d y fo r im m e d ia te u s e w ith m in im a l k n o w le d g e o n e v o lu tio n a ry c o m p u tin g o r M a tla b p ro g ra m m in g . T h e to o lb o x a ls o a llo w s th e d iffe re n t re p re s e n ta tio n o f s im u la tio n re s u lts in v a rio u s fo rm a ts , s u c h a s te x t file s o r g ra p h ic a l d is p la y s fo r th e p u rp o s e o f o n -lin e v ie w in g a n d a n a ly s is . W ith th e to o lb o x , d e s ig n e r m e re ly n e e d s to g iv e a m o d e l file re la tin g to h is /h e r p a rtic u la r o p tim is a tio n p ro b le m , a n d c o n fig u re s th e p ro b le m b a s e d o n a fe w s im p le G U I s e tu p s . F u rth e r d e s c rip tio n s o f th e to o lb o x a n d G U Is m a y b e re fe rre d to [1 1 ] o r th e tu to ria ls in th e to o lb o x , w h ic h is fre e ly a v a ila b le fo r d o w n lo a d in g a t h ttp ://w e b .s in g n e t.c o m .s g /~ k a y c h e n /m o e a .h tm . 186 K.C. Tan, T.H. Lee, and E.F. Khor Setting/modifing objective functions Design performance Decision-making Module Goals and priorities QFT De sign Spe c if ic at io ns Tracking bounds Coupling bounds Robust margin Sensitivity rejection High freq. gain Objective vector Multi-objective function evaluation Robust stability Multi-objective Optimisation Module Test signals, weighting functions Results Gragh ic al Displ ays QFT design parameters System response QFT Control Module R(s) F(s) G (s) Plant Templates Y (s) ... Fig. 7 . A general evolutionary design automated QFT control framework 3. A Be nc h mark M I M O QFT De sign Pro b l e m The benchmark MIMO QFT control problem given in [14] is studied in this section, which is shown in Fig. 1 with the MIMO uncertain plant sets given as, 3  0.5a º ª a «/ ( s ) / (s) » P ( s) « (12) » 8 » « 1 « / (s) » ¬/ ( s ) ¼ where / ( s ) s 2  0.03as  10 and a  [6, 8] . Apart from the few design speciciations studied by [14], additional performance requirements such as robust tracking and cross-coupling specifications are included here for wider consideration of the QFT design objectives, which subsequently adds to the design difficulty and complexity. The specification of high frequency gain [9, 10] is also incorporated to avoid any high-frequency sensor noise and unmodelled high-frequency dynamics/harmonics. The various closed-loop performance requirements for this MIMO QFT design are formulated as follows: (i) Robust Tracking Bounds for diagonal transfer functions: T (i ,i ) L (Z ) d CL(i ,i ) ( jZ ) d T(i,i )U (Z ) , for i = 1, 2 (13) Upper Tracking Model: T(1,1)U (Z ) T( 2,2)U (Z ) 1.9 u104 ( jZ )  6.4 u105 ( jZ )3  2.3 u102 ( jZ )2  1.9 u104 ( jZ )  6.4 u105 6.4 u103 ( jZ )  3.4 u105 ( jZ ) 3  1.5 u102 ( jZ ) 2  8 u103 ( jZ )  3.4 u105 Lower Tracking Model: (14a) (14b) Automatic Design of Multivariable QFT Control System 1 u10 6 T(1,1) L (Z ) ( jZ ) 3  3 u10 2 ( jZ ) 2  3 u10 4 ( jZ )  1 u10 6 2.5 u105 T(2,2) L (Z ) ( jZ )3  2.3 u102 ( jZ )2  1.5 u104 ( jZ )  2.5 u105 (ii) Robust Cross-Coupling Bounds for off-diagonal transfer functions: CL(i , j ) ( jZ ) d T(i , j )U (Z ) ,for i ž j, and i, j = 1,2 187 (15a) (15b) (16) where, T(1, 2 )U (Z ) 0 .0032 ( jZ ) >0.016 ( jZ )  1@ >0.016 ( jZ )  1@ 6.3 u 10 3 ( jZ ) >0.016 ( jZ )  1@ >0.016 ( jZ )  1@ T( 2,1)U (Z ) (iii) Robust Sensitivity Rejections for full matrix transfer functions: S i , j ( jZ )  ai , j ( jZ ) , for Z  10 (17a) (17b) (18) where, ai,j = 0.01w , for i = j; ai,j = 0.005w , for i ž j (iv) Robust Stability Margin: 1  1.8 1  Li ,i ( jZ ) ,for " i 1,2 , and Z ! 0 (19) The performance bounds of QFT are computed within a wide frequency range of 10-2 rad/s to 103 rad/s. Without loss of generality, the structure of the diagonal controller G (s) is chosen in the form of a general transfer function [9] as given by, 4 ¦b s m ¦a s n m G i ,i ( s) ,  bm , a n  ƒ  , for i = 1, 2 m 0 4 (20) n n 0 Note that the controller can also be designed by refining position of poles and zeros directly or by using other structures such as the realisable (non-ideal) PID structure if desired. The filter is fixed to a full matrix first-order transfer function as it is relevant to the tracking and cross-coupling bound in the frequency response. Since the resultant pre-filter must satisfy lim[ F ( s )] 1 for a step forcing function [9], the so 0 structure of pre-filter F(s) is chosen as a full matrix first-order transfer function as given by, 1  cn  ƒ  , for " i, j = 1, 2 (21) Fi , j ( s ) 2 n cn s 1 ¦ j n 188 K.C. Tan, T.H. Lee, and E.F. Khor A p a rt fro m m o st d e fa w ith a p o p u la tio n a n d g e n e v o lu tio n a ry Q F T d e s ig n p e rfo rm a n c e re q u ire m e n ts d e te rm in a tio n o f th e g o a l a p e rfo rm a n c e re q u ire m e n ts , c o m m itm e n t’ d e s ig n [9 ]. p e rfo rm a n c e s p e c ific a tio n s e v o lu tio n a ry o p tim is a tio n a u lt s e ttin g s , th e e v o lu tio e ra tio n s iz e o f 2 0 0 a n d o p tim is a tio n p ro c e s s , g m a y b e in c lu d e d o p tio n a n d p rio rity m a y b e a s u b it m a y b e u n n e c e s s a ry a n In p rin c ip le , a n y n u c a n b e a d d e d to th e p p ro a c h if n e c e s s a ry . n a ry to o lb o x h a s b e e n c o n fig u re 1 0 0 , re s p e c tiv e ly . T o g u id e th o a l a n d p rio rity fo r e a c h o f th lly a s s h o w n in F ig . 8 . A lth o u g je c tiv e m a tte r a n d d e p e n d s o n th d c a n b e ig n o re d fo r a ‘m in im u m m b e r o r c o m b in a tio n o f Q F d e s ig n u s in g th e m u lti-o b je c tiv d e e h e T e F ig . 8 . S e ttin g s o f th e M O E A to o lb o x fo r th e b e n c h m a rk Q F T d e s ig n p ro b le m A p o w e rfu l fe a tu re o f th e e v o lu tio n a ry Q F T d e s ig n is th a t it a llo w s o n -lin e e x a m in a tio n o f d iffe re n t tra d e -o ffs a m o n g th e m u ltip le c o n flic tin g s p e c ific a tio n s , m o d ific a tio n o f e x is tin g o b je c tiv e s a n d c o n s tra in ts , a n d z o o m in to a n y re g io n o f in te re s t b e fo re s e le c tin g o n e fin a l s e t o f c o n tro lle r a n d p re -filte r fo r re a l tim e im p le m e n ta tio n . T h e tra d e -o ff g ra p h o f th e re s u lta n t Q F T c o n tro l s y s te m is s h o w n in F ig . 9 , w h e re e a c h lin e re p re s e n tin g a s o lu tio n fo u n d b y th e e v o lu tio n a ry o p tim is a tio n . T h e c o s t o f o b je c tiv e s s u c h a s s ta b ility (R H S P ), ro b u s t tra c k in g a n d c ro s s c o u p lin g p e rfo rm a n c e s (E R R U T a n d E R R L T ) a re la b e lle d a s o b je c tiv e s 1 -7 , w h ic h a re a ll e q u a l to z e ro a s d e s ire d a c c o rd in g to th e g o a l s e ttin g s in F ig . 8 . T h e x -a x is s h o w s th e d e s ig n s p e c ific a tio n s , th e y -a x is s h o w s th e n o rm a lis e d c o s t fo r e a c h o b je c tiv e a n d th e c ro s s m a rk s h o w s th e d e s ire d g o a l s e ttin g fo r e a c h p e rfo rm a n c e re q u ire m e n t. C le a rly , tra d e o ffs b e tw e e n a d ja c e n t s p e c ific a tio n s re s u lts in th e c ro s s in g o f th e lin e s b e tw e e n th e m , w h e re a s c o n c u rre n t lin e s th a t d o n o t a c ro s s e a c h o th e r in d ic a tin g th e s p e c ific a tio n s d o n o t c o m p e te w ith o n e a n o th e r. F o r e x a m p le , th e ro b u s t s e n s itiv ity o b je c tiv e o f 1 2 (R S 2 1 ) a n d 1 3 (R S 2 2 ) a re n o t c o m p e tin g w ith e a c h o th e r, w h e re a s th e ro b u s t m a rg in o b je c tiv e 8 (R M 1 ) a n d 9 (R M 2 ) a p p e a r to c o m p e te h e a v ily , a s e x p e c te d . T h e in fo rm a tio n c o n ta in e d in th is tra d e -o ff g ra p h a ls o s u g g e s ts th a t lo w e r g o a l s e ttin g s fo r ro b u s t s e n s itiv ity (o b je c tiv e s 1 0 -1 3 ) a re p o s s ib le , w h ic h c a n b e fu rth e r o p tim is e d to a rriv e a t a n e v e n b e tte r ro b u s t p e rfo rm a n c e . Automatic Design of Multivariable QFT Control System 189 F ig . 9 . T ra d e -o ff g ra p h o f th e e v o lu tio n a ry d e s ig n e d Q F T c o n tro l s y s te m N o te th a t th e e v o lu tio n a ry Q F T d e s ig n a ls o a llo w s e n g in e e rs to d iv e rt th e e v o lu tio n to o th e r fo c u s e d tra d e -o ff re g io n o r to m o d ify a n y p re fe re n c e s o n th e c u rre n t s p e c ific a tio n s e ttin g s a fte r o b s e rv a tio n fo r a n u m b e r o f g e n e ra tio n s . F o r e x a m p l e , t h e d e s i g n e r c a n c h a n g e h i s p r e f e r e n c e a n d d e c i d e t o r e d u c e t h e 9 th g o a l s e ttin g fo r r o b u s t m a r g in ( R M 2 ) fr o m 1 .8 to 1 .3 . F ig . 1 0 illu s tr a te s th e b e h a v io u r o f th e e v o lu tio n u p o n th e m o d ific a tio n o f th is g o a l s e ttin g a fte r th e e v o lu tio n a ry Q F T d e s ig n in F ig . 9 . D u e to th e s u d d e n c h a n g e o f a tig h te r g o a l s e ttin g , in itia lly n o n e o f th e in d iv id u a ls m a n a g e to m e e t a ll th e re q u ire d s p e c ific a tio n s a s s h o w n in F ig . 1 0 (a ). A fte r c o n tin u in g th e e v o lu tio n fo r 2 g e n e ra tio n s , th e p o p u la tio n m o v e s to w a rd s s a tis fy in g th e o b je c tiv e o f R M 2 a s s h o w n in F ig . 1 0 (b ) a t th e p e rfo rm a n c e e x p e n s e o f o th e r o b je c tiv e s s in c e th e y a re h ig h ly c o rre la te d a n d c o m p e tin g to e a c h o th e r. T h e e v o lu tio n c o n tin u e s a n d a g a in le a d s to th e s a tis fa c tio n o f a ll th e re q u ire d g o a l s e ttin g s in c lu d in g th e s tric te r s e ttin g o f o b je c tiv e R M 2 a s s h o w n in F ig . 1 0 (c ). C le a rly , th is m a n -m a c h in e in te ra c tiv e d e s ig n a p p ro a c h h a s e n a b le d Q F T d e s ig n e rs to d iv e rt th e e v o lu tio n in to a n y in te re s te d tra d e -o ff re g io n s o r to m o d ify c e rta in s p e c ific a tio n s a n d p re fe re n c e s o n -lin e , w ith o u t th e n e e d o f re s ta rtin g th e e n tire d e s ig n p ro c e s s a s re q u ire d b y c o n v e n tio n a l Q F T d e s ig n m e th o d s . N o rm a lis e d c o s ts 1 0 .8 O n -lin e M o d ific a tio n 0 .6 0 .4 0 .2 0 2 4 6 8 1 0 1 2 1 4 O b je c tiv e s (a ) O n -lin e g o a l m o d ific a tio n o f ro b u s t m a rg in o b je c tiv e (R M 2 ) 190 K.C. Tan, T.H. Lee, and E.F. Khor 1 1 N o rm a liz e d c o s ts N o rm a liz e d c o s ts 0 .8 0 .6 0 .4 0 .2 0 .8 0 .6 0 .4 0 .2 0 0 2 4 6 8 1 0 1 2 2 1 4 4 6 8 1 0 1 2 1 4 O b je c tiv e s O b je c tiv e s (b ) A fte r 2 g e n e ra tio n s (c ) A fte r a n o th e r 2 g e n e ra tio n s F ig . 1 0 . E ffe c ts o f th e e v o lu tio n u p o n th e o n -lin e m o d ific a tio n o f g o a l s e ttin g F ig . 1 1 s h o w s th e ro b u s t tra c k in g p e rfo rm a n c e s in th e fre q u e n c y d o m a in fo r th e tw o d ia g o n a l e le m e n ts o f th e c lo s e d -lo o p s y s te m . It c a n b e s e e n th a t a ll th e fre q u e n c y re s p o n s e s o f C L U a n d C L L fo r b o th th e d ia g o n a l c h a n n e ls a re lo c a te d s u c c e s s fu lly w ith in th e ir re s p e c tiv e p re -s p e c ifie d tra c k in g b o u n d s o f T U a n d T L . B e s id e s , th e c o u p lin g e ffe c t fro m th e o ff-d ia g o n a l e le m e n ts o f th e c lo s e d -lo o p s y s te m fo r a ll th e p la n t te m p la te s h a s a ls o b e e n re d u c e d s a tis fa c to ry a n d s u c c e s s fu lly b o u n d e d b y th e u p p e r c o u p lin g b o u n d w ith m in im a l g a in a n d b a n d w id th o f th e o ff-d ia g o n a l tra n s fe r fu n c tio n s a s s h o w n in F ig . 1 2 . 1 0 1 0 T ( 1 ,1 )U 0 T 0 ( 2 ,2 ) U -1 0 C L -2 0 M a g n itu b e (d B ) M a g n itu b e (d B ) -1 0 ( 1 ,1 ) -3 0 T ( 1 ,1 ) L -4 0 -5 0 C L ( 2 ,2 ) T ( 2 ,2 ) L -3 0 -4 0 -5 0 -6 0 -7 0 -6 0 -7 0 -2 0 -8 0 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 F re q u e n c y (ra d /s e c ) ( a ) D ia g o n a l e le m e n t o f ( 1 ,1 ) 1 0 3 -9 0 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 F re q u e n c y (ra d /s e c ) ( b ) D ia g o n a l e le m e n t o f ( 2 ,2 ) F ig . 1 1 . T h e tra c k in g p e rfo rm a n c e in th e fre q u e n c y d o m a in 1 0 3 Automatic Design of Multivariable QFT Control System 0 -1 0 -1 0 -2 0 T ( 1 ,2 )U C L -4 0 ( 1 ,2 ) -5 0 -6 0 -7 0 -7 0 -8 0 -9 0 -1 0 0 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 ( 2 ,1 ) -6 0 -9 0 -2 C L -5 0 -8 0 1 0 ( 2 ,1 ) U -4 0 M a g n itu b e (d B ) -3 0 T -3 0 -2 0 M a g n itu b e (d B ) 191 3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 F re q u e n c y (ra d /s e c ) F re q u e n c y (ra d /s e c ) ( a ) O ff -d ia g o n a l e le m e n t o f ( 1 ,2 ) ( b ) O ff-d ia g o n a l e le m e n t o f ( 2 ,1 ) F ig . 1 2 . T h e c ro s s -c o u p lin g p e rfo rm a n c e in th e fre q u e n c y d o m a in F ig s . 1 3 a n d 1 la n t te m p la te s n e d c o n tro lle r rm a n c e s h a v e in g b o u n d s , a s th e p d e s ig p e rfo tra c k 4 sh o in th a n d p b e e n d e s ire w th e u n e tim e d re -filte r. s a tis fie d d . it s te p o m a in C le a rly su c c e tra fo , a ssf c k in r a ll th u lly g a n d c o u p lin g p e rfo rm ra n d o m s e le c te d s e t o f e tim e d o m a in tra c k in g a n d w ith in th e re q u ire a n c e v o a n d d p e s fo r a ll lu tio n a ry c o u p lin g re s c rib e d 1 .4 1 .4 T 1 .2 T ( 1 ,1 )U 1 .2 ( 2 ,2 ) U 1 1 0 .8 C L 0 .6 T 0 .4 M a g n itu d e M a g n itu d e 0 .8 ( 1 ,1 ) ( 1 ,1 )L C L ( 2 ,2 ) 0 .4 T 0 .2 0 .2 0 0 .6 ( 2 ,2 )L 0 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 T im e (s e c ) ( a ) D ia g o n a l e le m e n t o f ( 1 ,1 ) 0 .1 8 0 .2 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 T im e (S e c ) ( b ) D ia g o n a l e le m e n t o f ( 2 ,2 ) F ig . 1 3 . T h e tra c k in g re s p o n s e s o f th e d ia g o n a l e le m e n ts in th e c lo s e d -lo o p s y s te m 0 .2 192 K.C. Tan, T.H. Lee, and E.F. Khor 0 .1 5 0 .0 6 0 .1 0 .0 4 0 .0 5 0 .0 2 0 - 0 .0 5 M a g n itu d e M a g n itu d e 0 - 0 .1 - 0 .1 5 - 0 .2 C L - 0 .0 2 - 0 .0 4 - 0 .0 6 ( 1 ,2 ) - 0 .2 5 C L ( 2 ,1 ) - 0 .0 8 - 0 .3 - 0 .3 5 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 - 0 .1 0 .2 0 0 .0 2 0 .0 8 0 .0 4 0 .0 6 T im e (S e c ) 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 0 .2 T im e (s e c ) ( a ) O ff -d ia g o n a l e le m e n t o f ( 1 ,2 ) ( b ) O ff-d ia g o n a l e le m e n t o f ( 2 ,1 ) F ig . 1 4 . T h e c o u p lin g re s p o n s e s o f th e o ff-d ia g o n a l e le m e n ts in th e c lo s e d -lo o p s y s te m T o illu s tra te ro b u s tn e s s o f th e e v o lu tio n a ry d is tu rb a n c e re je c tio n , a u n it s te p d is tu rb a n c e s ig n a l O u tp u t d is tu rb a n c e re s p o n s e s fo r a ll th e fin a l P e le m e n t o f th e c lo s e d -lo o p tra n s fe r m a trix a re illu s s te p d is tu rb a n c e h a s b e e n s u c c e s s fu lly a tte n u a te d iffe re n t v a lu e s o f p a ra m e te r u n c e rta in tie s , a s s p e c ific a tio n o f ro b u s t s e n s itiv ity re je c tio n . . h it e e 0 .3 0 .2 0 .6 0 .1 M a g n itu d e M a g n itu d e n 0 .4 1 0 .8 0 .4 Y 0 .2 ( 1 ,1 ) D 0 - 0 .1 - 0 .2 - 0 .3 0 Y - 0 .4 - 0 .2 - 0 .4 d e s ig n e d Q F T c o n tro l s y s te m o w a s a p p lie d to th e M IM O s y s te m a re to o p tim a l c o n tro lle rs a t e a c tra te d in F ig . 1 5 . C le a rly , th e u n d to z e ro e v e n tu a lly fo r a ll th q u a n tifie d b y th e p e rfo rm a n c ( 1 ,2 ) D 0 - 0 .5 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 0 .2 T im e (s e c ) (a ) D is tu rb a n c e re s p o n s e o f e le m e n t (1 , 1 ) - 0 .6 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 0 .2 T im e (s e c ) (b ) D is tu rb a n c e re s p o n s e o f e le m e n t (1 , 2 ) Automatic Design of Multivariable QFT Control System 193 1 0 .1 5 0 .1 0 .0 5 - 0 .0 5 Y ( 2 ,1 ) ( 2 ,2 ) D 0 D - 0 .1 Y M a g n itu d e M a g n itu d e 0 .5 0 - 0 .1 5 - 0 .2 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 0 .2 - 0 .5 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 (c ) D is tu rb a n c e re s p o n s e o f e le m e n t (2 , 1 ) 0 .1 0 .1 2 0 .1 4 0 .1 6 0 .1 8 0 .2 T im e (s e c ) T im e (s e c ) (d ) D is tu rb a n c e re s p o n s e o f e le m e n t (2 , 2 ) F ig . 1 5 . O u tp u t re s p o n s e s fo r th e u n it s te p d is tu rb a n c e in th e tim e d o m a in 4 C o n c lu s io n T h is p a p e r h a s a n a ly s e d d iffic u ltie s in e x is tin g Q F T d e s ig n te c h n iq u e s fo r m u ltiv a ria b le c o n tro l s y s te m s . T o a d d re s s th e s e d e s ig n d e fic ie n c ie s , a n a u to m a te d m u ltiv a ria b le Q F T d e s ig n m e th o d o lo g y u s in g a h ig h p e rfo rm a n c e M O E A to o lb o x h a s b e e n p r o p o s e d . U n l i k e e x i s t i n g d e s i g n m e t h o d s , t h e ’i n t e l l i g e n t ’ d e s i g n a p p r o a c h i s c a p a b le o f a u to m a tic a lly e v o lv in g b o th n o m in a l c o n tro lle r a n d p re -filte r s im u lta n e o u s ly to m e e t a ll p e rfo rm a n c e re q u ire m e n ts in Q F T , w ith o u t g o in g th ro u g h th e s e q u e n tia l d e s ig n s ta g e s fo r e a c h o f th e m u ltiv a ria b le s u b -s y s te m s . B e s id e s , th e a p p ro a c h a ls o a v o id s th e n e e d o f m a n u a l Q F T b o u n d c o m p u ta tio n a n d tria l-a n d -e rro r lo o p -s h a p in g p ro c e d u re s a s re q u ire d b y c o n v e n tio n a l m e a n s . It is s h o w n th a t c o n tro l e n g in e e rs ’ e x p e rtis e s a s w e ll a s g o a l a n d p rio rity in fo rm a tio n c a n b e e a s ily in c lu d e d a n d m o d ifie d o n -lin e a c c o rd in g to th e e v o lv in g tra d e -o ffs , in s te a d o f re p e a tin g o r re s ta rtin g th e w h o le d e s ig n p ro c e s s . It is o b v io u s th a t th e p ro p o s e d e v o lu tio n a ry Q F T d e s ig n fra m e w o rk is fu lly e x p a n d a b le to o n -lin e d e s ig n o p tim is a tio n a n d im p le m e n ta tio n . T h is c a n b e re a lis e d e ith e r v ia th e h a rd a n d s o ftw a re s y s te m s s u c h a s d S P A C E [1 5 ] o r M IR C O S [1 6 ] fo r g ra p h ic a l p ro g ra m m in g a n d re a l-tim e o p e ra tio n to p ro v id e n e c e s s a ry lin k a g e s b e tw e e n th e to o lb o x a n d th e p h y s ic a l e n v iro n m e n ts . A p a rt fro m th e d e v e lo p m e n ts fo r o n -lin e a d a p ta tio n , th e m u lti-o b je c tiv e e v o lu tio n a ry Q F T d e s ig n p a ra d ig m is c u rre n tly b e in g e x te n d e d to ro b u s t c o n tro l o f n o n lin e a r s y s te m s a n d to in c o rp o ra te o th e r d e s ig n s p e c ific a tio n s s u c h a s e c o n o m ic a l c o s t c o n s id e ra tio n . P ro g re s s a n d re s u lts w ill b e re p o rte d in d u e c o u rs e . R e fe r e n c e s 1 . 2 . 3 . Y a n iv , O ., H o r o w itz , I .: A Q u a n tita tiv e d e s ig n m e th o d f o r M s y s te m h a v in g u n c e rta in p la n ts . In t. J . C o n tr o l, v o l. 4 3 , n o . 2 Y a n iv , O ., S c h w a r tz , B .: A C r ite r io n fo r lo o p s ta b ility in th o f M IM O fe e d b a c k s y s te m s . In t. J . C o n tr o l, v o l. 5 3 , n o . 3 , p H o u p is , C . H .: Q u a n tita tiv e fe e d b a c k th e o r y ( Q F T ) te c h n iq e d . (1 9 9 6 ). T h e C o n tr o l H a n d b o o k , C R C P re s s & IE E E P re s IM O lin e a r fe e d b a c k , p p . 4 0 1 -4 2 1 , 1 9 8 6 . e H o ro w itz S y n th e s is p . 5 2 7 -5 3 9 , 1 9 9 0 . u e . I n L e v in e , W . S ., s, p p . 7 0 1 -7 1 7 , 1 9 9 3 . 194 4 . 5 . 6 . 7 . 8 . 9 . 1 0 . 1 1 . 1 2 . 1 3 . 1 4 . 1 5 . 1 6 . K.C. Tan, T.H. Lee, and E.F. Khor S n e ll, S . A ., H e s s , R . A .: R o b u s t, d e c o u p le d , flig h t c o n tr o l d e s ig n w ith r a te s a tu ra tin g a c tu a to rs . C o n f. a n d E x h ib it. O n A IA A A tm o s p h e r ic F lig h t M e c h a n ic s , p p . 7 3 3 -7 4 5 , 1 9 9 7 . T h o m p s o n , D . F ., a n d N w o k a h , O . D . I .: A n a ly tic a l lo o p -s h a p in g m e th o d s in q u a n tita tiv e fe e d b a c k th e o ry , J . D y n a m ic S y s te m s , M e a s u r e m e n t a n d C o n tr o l, v o l. 1 1 6 , p p . 1 6 9 -1 7 7 , 1 9 9 4 . B r y a n t, G . F ., a n d H a lik ia s , G . D .: O p tim a l lo o p -s h a p in g fo r s y s te m s w ith la r g e p a ra m e te r u n c e rta in ty v ia lin e a r p ro g ra m m in g , In t. J . C o n tr o l, v o l. 6 2 , n o . 3 , p p . 5 5 7 -5 6 8 , 1 9 9 5 . C h a it, Y .: Q F T lo o p -s h a p in g a n d m in im is a tio n o f th e h ig h -f r e q u e n c y g a in v ia c o n v e x o p tim is a tio n , P r o c . S y m . Q u a n tita tiv e F e e d b a c k T h e o r y a n d o th e r F r e q . D o m a in M e th o d a n d A p p lic a tio n s , G la s g o w , S c o tla n d , p p . 1 3 -2 8 , 1 9 9 7 . C h e n , W . H ., B a lla n c e , D . J . L i, Y .: A u to m a tic lo o p -s h a p in g in Q F T u s in g g e n e tic a lg o r ith m s . P r o c . o f 3 rd A s ia - P a c ific C o n f. o n C o n t. & M e a s ., p p . 6 3 -6 7 , 1 9 9 8 . T a n , K . C ., L e e , T . H . K h o r , E . F .: C o n tr o l s y s te m d e s ig n a u to m a tio n w ith r o b u s t tra c k in g th u m b p rin t p e rfo rm a n c e u s in g a m u lti-o b je c tiv e e v o lu tio n a ry a lg o rith m " , I E E E I n t . C o n f . C o n t r o l A p p l . a n d S y s . D e s i g n , H a w a i i , 2 2 - 2 6 th A u g u s t , p p . 4 9 8 5 0 3 , 1 9 9 9 . C h e n , W . H ., B a lla n c e , D . J ., F e n g , W ., a n d L i, Y .: G e n e tic a lg o r ith m e n a b le d c o m p u te r-a u to m a te d d e s ig n o f Q F T c o n tro l s y s te m s , IE E E In t. C o n f. C o n tr o l A p p l . a n d S y s . D e s i g n , H a w a i i , 2 2 - 2 6 th A u g u s t , p p . 4 9 2 - 4 9 7 , 1 9 9 9 . T a n , K . C ., W a n g , Q . G ., L e e , T . H ., K h o o , T . T ., a n d K h o r , E . F .: A M u ltio b je c tiv e E v o lu tio n a r y A lg o r ith m T o o lb o x fo r M a tla b , ( h ttp ://v la b .e e .n u s .e d u .s g /~ k c ta n /m o e a .h tm ) , 1 9 9 9 . S k o g e s ta d , S ., P o s tle th w a ite , I .: M u ltiv a r ia b le F e e d b a c k C o n tr o l: A n a ly s is a n d D e s ig n . J o h n W ile y & S o n s L td , W e s t S u s s e x . E n g la n d , 1 9 9 6 . T h e M a th W o r k s , I n c .: U s in g M A T L A B , v e r s io n 5 , 1 9 9 8 . B o r g h e s a n i, C ., C h a it, Y . a n d Y a n iv , O .: Q u a n tita tiv e F e e d b a c k T h e o r y T o o lb o x U s e r M a n u a l, T h e M a th W o rk In c , 1 9 9 5 . H a n s e lm a n n , H .: A u to m o tiv e c o n tr o l: F r o m c o n c e p t to e x p e r im e n t to p r o d u c t, IE E E In t. C o n f. C o n tr . A p p l. a n d S y s . D e s ., D e a rb o rn , 1 9 9 6 . R e b e s c h ie ß , S .: M I R C O S - M ic r o c o n tr o lle r -b a s e d r e a l tim e c o n tr o l s y s te m to o lb o x fo r u s e w ith M a tla b /S im u lin k , IE E E In t. C o n f. C o n tr . A p p l. a n d S y s . D e s ig n , H a w a ii, U S A , p p . 2 6 7 -2 7 2 , 1 9 9 9 . p H t s r a ng ,H Z ng ,K I h i tta l l,Z r r s r r r p r t r r icti icha rdso n , i o nso n ,a nd H pa rtm nt f lctrica l ngin ringa nd lctr nics ni rsit f i rp l , i rp l , 6 3 , . . q.h.wu@liv.ac.uk lctric r s a rch Instit t ingh , ijing 5, . . hina ngin ring a nd chn lg h N a tina l rid mpa n pl c, . . c . his pa p r d scrib s a n th rma lm d l f il -imm rs d, f rc d-a ir c ld p r tra nsf rm rs a nd a m th d lg f r m d lc nstr ctin singint l l ig ntla rninga ppl id t n-sit m a s r m nts. h m d ld l i rs th a l fb tt m- ila nd t p- ilt mp ra t r s f r th rma lp rf rma nc pr dictin a nd n-l in m nit ring fp r tra nsf rmrs. h r s l ts bta in d sing th n th rma lm d la r c mpa r d ith th r s l ts fa tra ditina lth rma lm d la nd th r s l ts d ri d fr m a rtificia ln ra ln t rks. I tr cti n-l in o nito ringo fpo r tra nsfo r rso p ns th po ssi il it fo r t ndingth o p ra tingti o fpo rtra nsfo r rs,r d cingth risko f p nsi fa ilr s a nd pro idingpo t ntia lfo r cha ngingth a int na nc stra t g [ ] [3]. h s f l l if o fa tra nsfo r r is d t r in d in pa rt th a il it o fth tra nsfo r r to dissipa t th int rna l l g n ra t d h a t to its s rro ndings. o ns q ntl, th co pa riso n o fa ct a la nd pr dict d o p ra ting t p ra t r s ca n pro id a s nsiti a s r o ftra nsfo r r co nditio n a nd ightindica t a no r a lo p ra tio n. od l ing tra nsfo r r th r a ld na ics is r ga rd d a s o n o fth o sti po rta ntiss sa nd co nstr ctio n o fa n a cc ra t th r a l o d lisa n i po rta nta sp ct o ftra nsfo r rco nditio n o nito ring. h g n ra l la cc pt d tho ds [5][6 ],ca n s d to pr dict o n s o f c ss t p ra t r in a tra nsfo r r. H o r, th co n ntio na lca l c l a tio n o fint rna ltra nsfo r r t p ra t r is no to nl a co pl ica t d a nd diffic l tta sk ta l so la dsto a co ns r a ti sti a t a s d o n so a ss ptio ns o fth o p ra ting co nditio ns. Its a il it to pr dictth tra nsfo r r t p ra t r nd r r a l istic l o a dingco nditio ns is th r fo r so ha tl i it d. In this pa p r, t o diff r ntint l l ig ntla rning tho ds, g n tic a l go rith ( ) a nd a rti cia ln ra ln t o rk ( ), a r s d to co nstr ctth r a l o dl s fro th o n-sit as r nts. h st d sho s tha t int l l ig nt la rning tho ds ca n pr dicto nl in tra nsfo r r t p ra t r s in r a lti ith gr a t r S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 1 9 5 − 2 0 4 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 1 9 6 W .H . T a n g e t a l. a cc ra c tha n tha to ta in d singth tra ditio na l o d l s. h d l o p d th ra l o d lco l d s d a s th a sis o fa n int l l ig ntpro t ctio n s st ,a s l l a s a n l ntin a r o t s p r isinga nd co ntro ls st . r s o nsid ring th o p ra ting r gi s o fco o l ing s st s a nd th a ct a l a s r nts a a il a l, t o i po rta ntt p ra t r as r nts, o tto -o ilt p rat r ( ) a nd to p-o ilt p ra t r ( ), a r cho s n fo r th p rpo s o f tra nsfo r r co nditio n o nito ring. In th fo l l o ing,t o th r a l o d l s fo r t p ra t r pr dictio n a nd co nditio n o nito ringo ftra nsfo r rs a r d scri d. . -s s o r np p d (o r ‘na t ra l ’ ) o ilco o l ing o f po r tra nsfo r rs ( ), th o il t p ra t r a tth to p o f indings is a ppro i a t l q a lto th insid th ta nk. H o r, fo r fo rc d o ilcirc l a tio n ( ), th is th s o fth o ilt p ra t r a tth o tto o fth inding, ,a nd th diff r nc t n o ilt p ra t r s a tth to p a nd o tto o fth inding[4] [7 ]. h is d scri d a s fo l l o s: = a nd th is r pr s nt d = + + + () + : + + + (t− ) (2) hr = a intt p ra t r = st a d sta t o ilt p ra t r a t th o tto o fth inding ith th o p ra tingl o ad = o ilt p ra t r ris a o a inta tth o tto o fth inding nd r th ra t d l o ad = th ra tio o fl o ad l o ss (a tra t d l o a d) to no -l o ad l o ss = th ra tio o fth o p ra tingl o a d c rr ntto ra t d l o a d c rr nt, = = st a d sta t o il t p ra t r a tth to p o fth inding ithth o p ra ting l o ad lt p ra t r ris a o a inta tth to p o fth inding nd r th t = oi ra t d l o ad = po n ntr l a t d to o ilt p ra t r ris d to to ta ll o ss s = po n ntr l a t d to indings t p ra t r ris d to th l o a d c rr nt. l lth pa ra t rs in o l d in this o d l ,s cha s , a nd ,a r in d thro gh p ri nto r p rinc . s al ld t r- D e v e lo p m e n t o f P o w e r T ra n s fo rm e r T h e rm a l M o d e ls . r si -s s pp ic 1 9 7 p r i In co ntra stto th indingt p ra t r , th tra nsint a nd ( a nd r sp ct i l ) ca nno ti di a t l r a ch t h co rr spo ndi ng st a d st at al t s nd r cha nging l o a ds, sinc th ir th r a lti co nsta nts a r in th o rd r o fho rs [ ]. cho o s a r c rsi fo r o fth o d l sing th pr io s sa pls to r pr s ntth a nd r sp cti l, hich ca n r fl ctcha ng s o f th th r a lti co nsta nts d to diff r nto p ra ting co nditio ns o fth po r tra nsfo r r. a n a ssist d o ilco o l ingo ftra nsfo r rs is a ctia t d a to a tica l l a cco rding to th o ilt p ra t r . h n th o ilt p ra t r incr a s sa nd c dsa c rta in a l , ,fa ns il l s itch d o n. o disting ishth diff r ntth r a ld na ics a ppro pria t to th p rio ds h n th fa ns a r o n a nd h n th a r o ff, a t o pic o d lis intro d c d. h fo l l o ing o d lis pl o d to pr dictth a nd tha tca n pr ss d th co ina tio n o fa r c rsi o d la nd a fa n f nctio n. h a tti insta nt is: ()= ( − 2) + + A + ( − 2) + ( − )+ ( − 2) + ( − 2) + ( − 2) + () (3) hr ()> o th r is ()= h is d scri d a s: t( ) = t ( − 2)( ( − 2) − + At t ( − ) + t t( )+ t − 2) + + t t + ( − 2) t () t (4) hr () t = t+ ( t − n− n) t t( ) > o th r is In th o d l(3) (4),th co fficints iths script‘ ’d no t tho s a sso cia t d ith a nd ‘ ’ ith . a cht p ra t r is d scri d a t o -pic od l . h co fficints in th o d l s a r sho n in a l , in hich o n s to f co fficints a r a sso cia t d ith th no r a lco nditio n itho tfa n o p ra tio n a nd th o th r a r r l a t d to th ti p rio d d ring hich fa ns a r s itch d o n. ig . ic I t ig t ri r ig s is a po rf ln rica lopti ia tio n t chniq , hichis ro o t d in th cha nis o f o ltio n a nd na t ra l g n tics. sd ri th irstr ngths si l a ting 1 9 8 W .H . T a n g e t a l. th na t ra ls a rcha nd s lctio n pro c ss a sso cia t d ith na t ra lg n tics. s t o fg n s in its ‘chro o so s’d t r in s r o rga nis s id ntit, hich is r frr d to a s a ‘string’in . a t ra ls lctio n ta k s pl a c in s ch a a tha t th s cha ra ct ristics a r i pl icitl s lct d ia th s r ia lo fth tt stcrit rio n. his a l go rith , th o stpo p l a r fo r a to f hich is th ina r g n tic al go rith ,sta rts iths ttingo jcti f nctio ns a s d o n th ph sica l o d lo f pro l s to ca l c l a t tn ss a l s, a nd th r a ft r a s r a ch ina r co d d string’ s str ngth ith its tn ss a l . h stro ng r strings a d a nc a nd a t ith o th r stro ng r strings to pro d c o ffspring. ina l l, th sts r i s. n o fth i po rta nt a d a nta g s is tha t co l d a l to nd o t th gl o al ini o f tn ss inst a d o fa l o ca lso ltio n. h ha s ga in d po p l a rit in r c nt a rs a s a ro sto pti ia tio n to o lfo r a a rit o fpro l s in ngin ring,scinc , co no ics, na nc , tc.[9 ] [ ]. h il lt pica l l pl o thr o p ra to rs: r pro d ctio n, cro sso r a nd ta tio n [ 2]. pica l l, a ch o f th s o p ra to rs is a ppl id to th po p l a tio n o nc p r g n ra tio n, a nd s a l l s ra lg n ra tio ns a r r q ir d to a chi sa tisfa cto r r s l ts. pro d ctio n isa pro c ssin hicha n o l d stringisca rrid thro ghinto a n po p l a tio n d p ndingo n its p rfo r a nc ind (i. . tn ss) a l . ho s strings ith high r tn ss a l s t nd to ha a high r pro a il it o fco ntri tingo n o r o r o ffsprings to th n tg n ra tio n. h cro sso r o p ra to r pro id s ra ndo info r a tio n cha ng . Itis a i d to a rds o ling tt r il ding l o cks (sch a ta ith sho rtd ning lngths a nd high a ra g tn ss a l s). ro sso r po ints a r ra ndo l cho s n. h fr q nc o fcro sso ris go rn d a s r s lct d cro sso rra t o rpro a il it o fcro sso r.Incr a singcro sso rra t incr a s sr co ina tio n o f il ding l o cks, t ith a n incr a singpro a il it o fl o o singgo o d strings. h ta tio n o p ra to r is si pla n o cca sio na lra ndo a l t ra tio n o fa string po sitio n ( a s d o n a pro a il it o f ta tio n). In a ina r co d , this in o l s cha nging a to a a nd ic rsa . h ta tio n o p ra to r h l ps in a o iding th po ssi il it o f ista kinga l o ca l ini fo r a gl o a l ini . ta tio n is s al l s d spa ringl. h n ta tio n is s d in co nj nctio n ithr pro d ctio n a nd cro sso r,iti pro s th gl o a lna t r o fth s a rch. h is pl o d to s a rch fo r th o pti a lpa ra t rs o fth th r a l od l . It is a ina r co d d i pl nta tio n ritt n in is a l + + 5. a nd d l o p d a t th ni rsit o f i rpo o l . Its co p ta tio n pro c ss is l ist d a s fo l l o s: ( ) Initia l i th po p l a tio n. (2) a la t th tn ss o f r stringin th po p l a tio n. (3) K p th ststringin th po p l a tio n. (4) a k a s lctio n o n th po p l a tio n a tra ndo . (5) a k cro sso r o n s lct d strings ith pro a il it . (6 ) a k ta tio n o n s lct d strings ith pro a il it . (7 ) a la t th tn ss o f r stringin th n po p l a tio n. ( ) ak l itis . (9 ) p a t(4) to ( ) ntilth t r ina tio n co nditio n is t. D e v e lo p m e n t o f P o w e r T ra n s fo rm e r T h e rm a l M o d e ls 1 9 9 In th co p ta tio n, th cro sso r pro a il it, , a nd th ta tio n pro a il it, , th si o fth po p l a tio n a nd th a i n r o fg n ra tio ns a r s lct d ‘a prio ri’ . . i ss ci h tn ss f nctio n sho l d r fl ctth d sir d cha ra ct risticso fth s st ing o pti i d. Itis r i po rta ntto s a n a ppro pria t tn ss f nctio n sinc this dri s th o ltio n pro c ss. o a pplth to th po r tra nsfo r r th r a l od l s fo r l a t d in ctio n 2,a tn ss f nctio n a nd o th r r l a ntpa ra t rs o fth ina r g n tic a l go rith sho l d d t r in d. h rro r t n th a s r d a ria ls a nd th o d lo tp ts is d n d a s tn ss. h s,th tn ss f nctio n sho l d a tla stco nta in t o t r s co rr spo ndingto th tn ss o f a nd r sp cti l. o r a ch indiid a l(string) o f a g n ra tio n, its to ta l tn ss a l is ca l c l a t d a s fo l l o s: = + (5) t n n = (k) + t (k) n = n ( (k) − (k)) + ( t (k) − t (k)) hr a nd t a r th tn ss fo r a nd co po n nts o fth o d l r sp cti l; (k) a nd (k ) a r t h rro rs t n th r a l a s r t nta nd th o d lo tp t nd r th a ppro pria t s r ic co nditio n (a int t p ra t r , l o a d ra tio ) r sp cti l; (k) a nd t (k) r pr s nta gro p o f as r nts a nd is th to ta ln r o f as r ntgro ps. (k) a nd (k ) ca n a si l ca l c l a t d t h a o o d l s. t 4 4 . s ts sis r rs h ca l c l a tio n a s a s d o n th da ta pro id d th a tio na l rid o pa n ( ). h da ta co r o th na t ra lo il , na t ra la ir ( ) a nd fo rc d o il ,fo rc d a ir ( ) o p ra tingr gi s hicha r s itch d o r a to a tica l l. o t p s o f o d lha n in stiga t d a s d scri d a o :( ) o n ntio na l st a d -sta t o d l ; (2) ra nsint-sta t o d l ith r c rsio n o fca l c l a t d o tp ts a nd ff ct o ffa ns, hich a ct a l l fa l linto t o gro ps. h rst r fl cts th st a d -sta t r l a tio nship t n th , ,a intt p ra t r a nd tra nsfo r r l o a d ra tio ; th s co nd co nc rns th t p ra t r a ria tio n o r a p rio d ti a s a n in rtia lr spo ns . s sta t d a o ,th a nd a r s lct d a s th o d lo tp ts. h diff r nc a o ng th o d l sl is in th co nstr ctio n o fth fo r l a fo r o th 2 0 0 W .H . T a n g e t a l. th o tto -o ila nd to p-o ilt p ra t r s. h a s r d pa ra t rs, pro id d , a r a intt p ra t r , ra tio o fl o ad , , t a nd co prising 4 gro ps o f a s r nts ith a n sa plint r a lo f in t . h o d lpa ra t rs ha n o pti i d singth ina r g n tic a l go rith a s d o n 4 gro ps o fr a l a s r nts,a nd th n ri d o n th r a ining 56 gro ps. h ca l c l atd r s l tsd ri d fro od l ( ) a nd tho s fro od l (2) a r l ist d in a l . 4 . i -s h st a d -sta t o d lha s n t st d rst. Itis a ss d tha tth a nd ca n i dia t l r a chth ir st a d -sta t a l s a nd r l a t to o nl o n s r ic co nditio n, pr ss d q a tio ns ( ) a nd (2) in s ctio n 2. . h o d lpa ra t rs to d t r in d in o l , , , ra t d a nd ra t d l, th pa ra t rs a r o ta in d pirica l l. h r fo r , t. o r a l to g th r ith l o a d ra tio a nd a intt p ra t r , th r spo ns s o fth pirica l o d lca n ca l c l atd q a tio ns ( ) a nd (2). h co pa riso ns t n th o d lo tp tsa nd r a l o tp tsa r sho n in ig r s a nd 2 r sp cti l. It a no t d tha tth st a d -sta t o d lpro d c s a n nsa tisfa cto r tn ss (in a l ) to th p ri nta lda ta . r si -s pp ic p r i s disc ss d in s ctio n 2.2, th a nd ca nno t i dia t l r a ch th co rr spo nding st a d -sta t a l s nd r cha ngingl o a ds,a s th ir th r a lti co nsta nts a r o f th o rd r o fho rs. h r fo r , th ir tra nsint ha io r sho l d r pr s nt d in a r c rsi fo r singpr io s sa pls. h rs l tso ta in d singth o d l sr l a tingto fa n o p ra tio n ithr c rsio n o fca l c l a t d o tp ts a r sho n in ig r s 3 a nd 4. s p ct d, th s r s l ts pr s ntth sta gr nt ithth as r nts,co pa r d itho th r o d l s. Ifa o r pr cis fa n o p ra tio n ti co l d pro id d, itis p ct d tha ta n n o r a cc ra t r s l t o l d a chi d. co pa riso n ith p ri nta l a s r nts indica t s th tra nsint-sta t o d l ith r c rsio n o fth o tp ts ha s a tt r p rfo r a nc tha n tha to fth st a d -sta t o d l . inc itdo s no tr q ir th a nd as r nts, this il lc rta inl o fgr a tpra ctica la l a s a pr dicti to o l nd r a id r a rit tra nsfo r r l o a dinga nd a intco nditio ns. rti ci r t r s ha s n id l s d to p rfo r co pl f nctio ns in a rio s l ds o f a ppl ica tio n incldingpa tt rn r co gnitio n,cl a ssi ca tio n a nd co ntro l s st s. h t chniq is a s d o n th th o r o f io l o gica ln r o ss st s a nd in o l s s lctio n o finp t, o tp t, n t o rk to po l o g a nd ight d co nn ctio ns o fth no d s [ 3]. In o r ca s ,th n t o rkinp ts a r cho s n a s l o a d ra tio a nd a int D e v e lo p m e n t o f P o w e r T ra n s fo rm e r T h e rm a l M o d e ls 2 0 1 n iro n ntt p ra t r a nd th n t o rk o tp tis th or o fth tra nsfo r r. h tra ining po chs a r 5 . l lth n ro n f nctio ns a r cho s n a s th sig o id f nctio n c ptth o tp tn ro n hich pl o sa p r l in a rf nctio n. 7 % o f to ta lda ta a r s d to tra in th n t o rk a nd th r a ining 3 % a r r s r d fo r a la tio n o fth p rfo r a nc o fth a pping. h dir ct a pping pl o s th inp ts: [ ( − ) ( − 2) ( − 3) ( − ) ( − 2) ( − 3)] a nd th o tp t t ( ) o r ( ). 6 9 is cho s n fo r th dir ct a pping. h rs l ts a r sho n in ig r s 5a nd 6 . h o d lpro id s a go o d a pping t n th inp ts a nd o tp ts. Ho r itdo s no tpo ss ss a n ph sica l a ning a nd itha s n no t d tha t, fo r dir ct a pping,th o d lo nlr pr s nts th inp ta nd o tp tr l a tio nship a cc ra t l ithin th ra ng co r d th tra iningda ta . tsid this ra ng ,th o d lr spo ns a nd a s r d da ta do no ta gr ith a ch o th r sa tisfa cto ril. It o nl d o nstra t s th po ssi il it o fo n st p pr dictio n o fth tra nsfo r r t p ra t r s. l so a sth o d l do sno tha a n ph sica l a ning,itpo ss ss s l ittlpo t ntia lfo r co nditio n o nito ring. 6 c si s h a ppl ica tio n o fth o d l sto th la rning tho d,pr s nt d in thispa p r sho tha t high pr cisio n r s l ts r a tta in d h n co pa r d to th r s l ts d ri d fro th o d l s itho tr c rsio n a nd th o d l so f .Itca n s n tha tth tra nsint-sta t o d l ith r c rsio n o fca l c l a t d o tp ts, inclding th ff cto ffa n o p ra tio n, ha s a po t ntia lfo r r pr s nting a cc ra t l th r a l tra nsfo r r th r a ld na ics. o thth o d lpa ra t rs a nd r spo ns s in th o d lappl ica lto fa n o p ra tio n ca n s d fo rpr dictingth r a l distri tio n a nd o nito ringth s r ic co nditio n o ftra nsfo r rs. r c s . . H . r a n a na , . . a tt ns, ra nsf rm r c nditin m nit ring r a l iing a n int rgra t d a da pti a na lsis s st m, I 2 s ssin,3 g st-5 pt mb r. 2. . ibfrid, . n rr, . ir ck, n-l in m nt ring f p r tra nsf rm rstr nds, n d lpm nta nd first p rinc s, I s ssin. 3. I. . mp, a rtia ldischa rg pl a nt-m nit ringt chn lg : r s nta nd f t r d lpm nts, rc di gs ci c , as r ta d ch g, l . 42, n . , an ar 5, pp.4- . 4. a m s, . ta l , d l -ba s d m nit ring ftra nsf rm rs, a ssa ch s tts Instit t f chn lg , a b ra t r f r lctr ma gn tic a nd lctr nic st ms, 5. 5. ra nsf rm rs mmitt fth I r ngin ring cit, g id t a di g i ra i i rs d tra sf r r, td 57. 5 , h Instit t f lctrica la nd lctr nics ngin rs,Inc., ,345 a st47 th tr t,N rk, N 7, . 2 0 2 W .H . T a n g e t a l. 6 . Int rna tina llctr t chnica lc mmissin, 5 a di gg id f r i i rs d r tra sf r rs, q i a t t / 5 6 ( hi s ta da rd), , n` , iss . 7 . hin s sta nda rds b a rd, 5, r tra sf r rs a rt : ra t r ris , 5. . . . , p ra tin fp rtra nsf rm rs, hi s ctric r b ish r, n 3. . . H. , . . a, r s st m ptima lr a cti dispa tch sing ltina r pr gra mming, ra s r st s, l . , n .3, 5, pp. 243- 24. . .H . , . . a , . . n, ptima l r a cti p rdispa tch singa n a da pti g n tic a l g rithm, t r a ti a r a f ctrica ra d rg st s, l .2 , n . , , pp.56 3-56 . . . . a, . H . , n ra t r pa ra m t r id ntifica tin sing ltina r pr gra mming, t r a ti a r a f ctrica ra d rg st s, l . 7, n .6 , 5, pp.47 -423. 2. . il l a , .N l a n, ppl ica tin fg n tic a l g rithms t m t r pa ra m t r d t rmina tin f r tra nsintt rq ca l c l a tins, ra sa cti s d str i ca ti s, l .33, n .5, pt mb r/ ct b r 7 , pp. 27 3- 2 2. 3. . .Z a ma n, p rim nta l t sting fth a rtificia l n ra l n t rkba s d pr t ctin fp r tra nsf rm rs, ra sa cti s r i r, l . 3, n .2, pirl , pp.5 -5 7 . 7RSRLO WHPSHUDWXUH GHJUHH   PHDVXUHG WRSRLO WHPS ZLWKRXW UHFXUVLRQ         i. .      1XPEHU RI VDPSOHV 6DPSOH LQWHUYDO  PLQXWH p- ilt mp ra t r   fth m d l ith tr c rsin D e v e lo p m e n t o f P o w e r T ra n s fo rm e r T h e rm a l M o d e ls 2 0 3 %RWWRPRLO WHPSHUDWXUH GHJUHH  PHDVXUHG ERWWRPRLO WHPS ZLWKRXW UHFXUVLRQ             1XPEHU RI VDPSOHV 6DPSOH LQWHUYDO  PLQXWH tt m- ilt mp ra t r i. .   fth m d l ith tr c rsin  PHDVXUHG WRSRLO WHPS ZLWK UHFXUVLRQ RI FDOFXODWHG YDOXHV DQG HIIHFW RI IDQV 7RSRLO WHPSHUDWXUH GHJUHH               1XPEHU RI VDPSOHV 6DPSOH LQWHUYDO  PLQXWH i. . p- ilt mp ra t r ff ct ffa ns   f th m d l ith r c rsin f ca l c l atd tp ts a nd  %RWWRPRLOWHPSHUDWXUH GHJUHH PHDVXUHGERWWRPRLOWHPS ZLWKUHFXUVLRQRIFDOFXODWHGYDOXHVDQGHIIHFWRIIDQV                1XPEHURIVDPSOHV 6DPSOHLQWHUYDOPLQXWH i. . tt m- ilt mp ra t r ff ct ffa ns fth m d l ith r c rsin fca l c l atd tp ts a nd W .H . 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PHDVXUHG WRSRLO WHPS $11 PDSSLQJ 7RSRLO WHPSHUDWXUH GHJUHH               1XPEHU RI VDPSOHV 6DPSOH LQWHUYDO  PLQXWH p- ilt mp ra t r i. . fth   N N ma pping  %RWWRPRLO WHPSHUDWXUH GHJUHH 2 0 4 PHDVXUHG ERWWRPRLO WHPS $11 PDSSLQJ               1XPEHU RI VDPSOHV 6DPSOH LQWHUYDO  PLQXWH tt m- ilt mp ra t r i.6 . . fth N N ma pping s l ts fth m d l s c nstr ct d b d l () itn ss t 4. . 3 . 3 5.7 4 .6 5 32 d l (2) b tt m 2. 3 2 . t tp mpirica la l s . .23 t t .5 .2 .2 t t 25.2 .2 - . 3 . . t . . t t .2 t . 2 -3.6 4- . 4 . 2 5 .6 = 5, = . h si fp p l a tin is 5 a nd th ma im m n mb r f g n ra tins is 2 . h t ta lgr ps fda ta f r ca l c l a tin is 4 . h d finitin fpa ra m t rs a r sh n in q a tins (3) (4). Automatic Validation of Protocol Interfaces Described in VHDL Fulvio Corno, Matteo Sonza Reorda, Giovanni Squillero Politecnico di Torino Dipartimento di Automatica e Informatica Corso Duca degli Abruzzi 24 I-10129, Torino, Italy {corno, sonza, squillero}@polito.it Abstract. In present days, most of the design activity is performed at a high level of abstraction, thus designers need to be sure that their designs are syntactically and semantically correct before starting the automatic synthesis process. The goal of this paper is to propose an automatic input pattern generation tool able to assist designers in the generation of a test bench for difficult parts of small- or medium- sized digital protocol interfaces. The proposed approach exploit a Genetic Algorithm connected to a commercial simulator for cultivating a set of input sequence able to execute given statements in the interface description. The proposed approach has been evaluated on the new ITC 99 benchmark set, a collection of circuits offering a wide spectrum of complexity. Experimental results show that some portions of the circuits remained uncovered, and the subsequent manual analysis allowed identifying design redundancies. 1 Introduction In the past years, the design flow of protocol interfaces, and Application Specific Integrated Circuits (ASICs) in general, experienced radical changes. Due to the maturity of automatic logic synthesis tools most of the design activity is now performed at high level of abstraction, such as register transfer level (RT), instead of low level such as gate. The new methodology dramatically increases designer productivity since high-level descriptions are more readable and considerably smaller. One important step of the new design flow consists of design validation, i.e., the verification that the design is syntactically and semantically correct before starting automatic logic synthesis. Although many techniques have already been proposed in the CAD literature (e.g., static checks, formal verification [HuCh98], mutation testing [AHRo98]), none has gained enough popularity to compete with the current industrial practice of validation by simulation. Verification engineers resort to extensive simulation of each design, and of the complete system, in order to gain confidence over its correctness. This situation is far from ideal, and designers need to face many difficulties. At the present days, simulation technology is effective enough for synthesized circuits. But when it comes to mixed-signal circuits, or to circuits containing embedded cores, or to S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 0 5 − 2 1 4 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 0 6 F . C o rn o , M . S o n z a R e o rd a , a n d G . S q u ille ro a complete system composed of a network of several interconnected interfaces, simulation is unable to provide the needed versatility. Even restricting our attention to medium-sized, digital interfaces, the fundamental issue of measuring the test bench quality can be considered still unanswered. Many metrics have been proposed to evaluate the thoroughness of a given set of input stimuli, often adopted from the software testing domain [Beiz90], ranging from statement or branch coverage, state coverage (for finite state machine controllers), condition coverage (for complex conditionals), to the more complex path coverage. Many variants have been developed, mainly to cater for observability [DKGe96] and for the inherent parallelism of hardware descriptions [TAZa99], that are not taken into account by standard metrics. Since no well established metric is yet widely accepted for validation, some authors also propose to measure the quality of validation patterns with the stuck-at fault coverage. Several products (normally integrated into existing simulation environments) are now available that provide the user with the possibility of evaluating the coverage of given input stimuli with respect to a selected metric. Designers can therefore pinpoint the parts of their design that are poorly tested, and develop new patterns specifically addressing them. Currently, this is a very time consuming and difficult task, since all the details of the design must be understood for generating suitable input sequences. The right trade-off between designer s time and validation accuracy is often difficult to find, and this often results in under-verified circuits. Moreover, in the generation of test vectors the designer may be biased by his knowledge of the desired system or module behavior, so that he often fails in identifying input sequences really able to activate possible critical points in the description. When faced with this problem, the CAD research community traditionally invested in formal verification [GDNe91] [HuCh98], in the hope that circuits can be proven correct by mathematical means. Although formal verification tools give good results on some domains, they still have too many limitations or they require too much expertise to be used as a mainstream validation tool. Designers are left waiting for the perfect formal verification system, while few or no innovative tools help them with simulation-based validation. The goal of this paper is to propose GIP-PI (Genetic Input Pattern generator for a Protocol Interface). GIP-PI is an automatic input pattern generation tool able to assist designers in the generation of a test bench for difficult parts of small- or mediumsized digital protocol interfaces. The proposed approach belongs to a brand new framework that can be called approximate validation, which explicitly relinquishes exactness in order to gain the ability of dealing with realistic designs. This philosophy has already been successfully applied in different areas: validation of the implementation of protocol interfaces [CSSq99c]; automatic test pattern generation [CSSq99a]; low-level [CSSq98] and mixed-level [CSSq99b] equivalence validation. Although the goal of this paper is completely different from the previous applications, all these methodologies share a common quality: being able to deal with real circuits exploiting an evolutionary algorithm. GIP-PI employs a Genetic Algorithm, interacting with a VHDL simulator, for deriving an input sequence able to execute a given statement, or branch, in the highlevel description. Whenever the test bench quality, as measured by one of the proposed metrics, is too low, our tool can be used to generate test patterns that are A u to m a tic V a lid a tio n o f P ro to c o l In te rfa c e s D e s c rib e d in V H D L 2 0 7 able to stimulate the parts of the design that are responsible for the low metric. The designer must manually analyze only those parts of the description that the tool failed to cover. Experimental results show that only a small fraction of difficult statements remain uncovered, and that many of them, upon closer inspection, indeed contain design errors or redundancies. While no metric is yet widely accepted by validation teams, we aimed at evaluating the effectiveness of our approach using some pre-defined metric. The algorithm is quite easily adapted to different metrics, but for the sake of the experiments we adopted branch coverage as a reference. We developed a prototypical system for generating test patterns based on branch coverage, applicable to synthesizable VHDL descriptions. We aim at addressing moderately sized circuits, that usually can not be handled by formal approaches, and at working directly on the VHDL description, without requiring any transformation nor imposing syntax limitations. The approach has been evaluated on the new ITC 99 benchmark set [ITC99], a collection of circuits described in high-level (RT) VHDL that offers a wide spectrum of complexity. Manually derived validation suites did not adequately cover all parts of the designs, and new sequences have been generated by the tool to increase the overall coverage. Experimental results show that some portions of the circuits remained uncovered, and the subsequent manual analysis allowed identifying design redundancies. Section 2 gives an overview over the proposed approach for test bench generation, experimental are presented in Section 3 and Section 4 concludes the paper. 2 RT-level Test Bench Generation The goal of test bench generation is to develop a set of input sequences that attain the maximum value of a predefined validation metric. Despite this implementation of GIP-PI is tuned for simulating high-level VHDL network interfaces only, the proposed method could be easily extended to deal with any simulable descriptions. For instance, given a protocol specification in ESTELLE, and with a reduced effort, GIP-PI could eventually generate a set of stimuli (events) to validate the protocol description itself. 2.1 Adopted Metric Most available tools grade input patterns according to metrics derived from software testing [Beiz90]: statement coverage and branch coverage are the most widely known, but state/transition coverage (reaching all the states/transitions of a controller) and condition coverage (controlling all clauses of complex conditionals) are also used in hardware validation. Path coverage, although often advocated as the most precise one, is seldom used due to its complexity, and because it loses meaningfulness when multiple execution threads run concurrently in parallel processes. Some recent work extends those metrics to take also into account observability [DGKe96] and the structure of arithmetic units [TAZa99]. Those 2 0 8 F . C o rn o , M . S o n z a R e o rd a , a n d G . S q u ille ro extensions are essential when the sequences have to be used as test patterns to cover stuck-at faults, but for validation they have lower importance since internal values are available. The metric we adopt in this paper is branch coverage, although the tool can be easily adapted to more sophisticated measures. Also, since synthesizable VHDL is a structured language, complete statement coverage implies complete branch coverage, and the tool takes advantage of this simplification. 2.2 Overall Approach The adopted approach is an evolution of the one presented in [CPSo97], where a Genetic Algorithm uses a simulator to measure the effectiveness of the sequences it generates. Instead of trying to justify values across behavioral statements, that would require solving Boolean and arithmetic constraints [FADe99], thanks to the nature of Genetic Algorithms we just need to simulate some sequences and analyze the propagation of values. Each sequence is therefore associated with the value returned by a fitness function, that measures how much it is able to enhance the value of the validation metric, and the Genetic Algorithm evolves and recombines sequences to increase their fitness. The fitness function needs to be carefully defined, and accurately computed. In particular, the fitness function can not be just the value of the validation metric: it must also contain some terms that indicate how to increase the covered branches, not just to count the already covered ones. In a sense, the fitness function includes a dominant term, that measures the accomplished tasks (covered branches), and secondary terms, that describe sub-objectives to be met in order to cover new branches. The computation of such function is accomplished by analyzing the simulation trace of the sequence, and by properly weighting the executed assignments, statements, and branches according to the target statements. In the implementation, to avoid arbitrary limitations in the VHDL syntax, simulation is delegated to a commercial simulator that runs an instrumented version of the VHDL code and records the simulation trace in the transcript file. Such trace is then interpreted according to control- and data-dependencies, that are extracted from a static analysis of the design description. Figure 1 shows a simplified view of the overall system architecture. 2.3 VHDL Analysis The goal of the algorithm is to achieve complete coverage, but for efficiency reasons we do not consider each statement separately, and we group them into basic blocks [ASUl86]: a basic block is a set of VHDL statements that are guaranteed to be executed sequentially, i.e., they reside inside a process and do not contain any intermediate entry point nor any control statement (i f , c a s e , ). All the operations required for code instrumentation, dependency analysis, branch coverage evaluation, and fitness function computation are performed at the level of basic blocks. A u to m a tic V a lid a tio n o f P ro to c o l In te rfa c e s D e s c rib e d in V H D L 2 0 9 Since the Genetic Algorithm exploits the knowledge about data and control dependencies, we need to extract that information from the VHDL code: for this reason, we build a database containing a simplified structure and semantics of the design. The database is structured as follows: • The hierarchy of component instantiations inside different entities is flattened (C1 and C2 in the figure 2). A dictionary of signal equivalencies is also built, that allow us to uniquely identify signals that span multiple hierarchical levels. 9+'/ $1$/<=(5 ,167580(17(5 seqs *(1(7,& $/*25,7+0 &200(5&,$/ 6,08/$725 trace Figure 1: System architecture • All VHDL processes occurring in the flattened circuit are given a unique identifier (Pi in the figure 2). This operation also converts standalone concurrent statements into their equivalent process. The design is thus represented as a network of processes interconnected by signals. E1 E2 P1 P3 P4 C1 P5 P6 C2 P2 ')* A C BB1 BB2 BB3 BB4 BB5 BB6 BB7 P7 + B Figure 2: Abstract representation of RT-level designs • • Each process is analyzed to define its interface, in terms of signals that it reads and writes. The sequential part of each process is analyzed, its control flow graph (CFG) is extracted, and statements are grouped in basic blocks (BBs). The control structure of the process is described as a control flow of basic blocks (figure 2 reports the CFG for process P2). 2 1 0 • F . C o rn o , M . S o n z a R e o rd a , a n d G . S q u ille ro A dependency matrix between basic blocks is computed, by assigning a probability that a basic block will be executed, given that another block has just been executed. These correlation probabilities take into account the branching and looping nature of the control flow. Each basic block is entered, and the data flow graph (DFG) of the operations that occur inside each basic block is extracted. Since a basic block consists of multiple statements and/or conditions, multiple dependencies are associated to a single block. Fig. 2 shows the DFG for the basic block BB4 of process P2. • 2.4 Genetic Algorithm The Genetic Algorithm (GA) is based on encoding potential test sequences as variable length bit matrices. A number of such sequences are randomly generated and constitute an initial population: the goal of the GA is to evolve this population to increase its fitness value. The fitness function measures the closeness of a sequence to the goal. Currently, in GIP-PI the genetic algorithm is run several times, each time for a different target. Moreover, the fitness function assumes two different forms: • In the initial phase, all basic blocks are considered simultaneously. The goal is to generate a set of sequences S that activate most of the blocks, for identifying easy-to-execute blocks. The fitness function is simply the number of activated blocks over the total number of blocks: activated _blocks(S ) fitness(S ) = tot _ blocks • Subsequently, GIP-PI is targeting a specific block T and the goal is to generate a sequence S able to cover it. In this phase sequences are targeted and the executed blocks are weighted by their correlation probability with respect to the target, measured as the weighted average of the execution counts of the basic blocks in the input cone (taking into account both control and data dependencies, thus potentially spanning several processes) of the target. The adopted weights take into account the probabilities of conditional execution that were statically computed in the database. fitness(S , T ) = ∑ correlation(b, T ) b∈ covered_ bb ( S ) In both phases, a saturation mechanism prevents easy-to-execute but not-sorelevant statements from diverting the attention of the GA. Moreover, during each run of the GA, an heuristic mechanism detects individuals that may be useful in a subsequent run. When such an individual is found, it is saved and later it is inserted in the initial population of the correct run of the GA. These predefined individuals may never exceed 5% of the initial population. The GA in GIP-PI evolves a population of µ individuals and in each generation λ new sequences are first generated, then selection is performed on the whole set of µ +λ individuals. Individuals are selected for reproduction using a roulette wheel mechanism based on their linearized fitness. In p of the cases, the new individual is built mutating a single parent: the original sequence can be shortened, or enlarged, or some bits may be flipped. In 1-p of the cases, the new individual is built mating two A u to m a tic V a lid a tio n o f P ro to c o l In te rfa c e s D e s c rib e d in V H D L 2 1 1 different parents: the offspring sequence can inherit the beginning from one parent and the end from the other, or some entire bit column from each parent. 3 Experimental results To test the effectiveness of the tool in generating test benches, we selected a set of VHDL benchmarks from the ITC 99 benchmark set [ITC99]. The first columns report some data about the RT-level descriptions, in terms of VHDL lines, VHDL processes (with hierarchy unflattened), and overall number of extracted basic blocks (BB). To have a better idea about circuit size, some characteristics of the synthesized netlists are reported in the last columns: number of Primary Inputs, Primary Outputs, FlipFlops, and combinational gates. These benchmarks have been publicly released in September 1999 at the IEEE International Test Conference, and there are no published results, yet, to compare with. We compared our results against a pure random approach, to evaluate the effectiveness of the GA, but it was so easily overcome that results are not reported. The implementation consists of about 4,700 lines of C code for VHDL code analysis and instrumentation, linked to the LEDA LPI interface [LEDA95], and of 2,700 lines of C code for the Genetic Algorithm and the interface to the simulator. All experiments were run on a Sun Ultra 5 running at 333 MHz with 256MB of memory. We adopt a population of µ = 30 individuals, with λ = 20 new individuals in each generation. The mutation probability was set to p = 0.3. 2 1 2 F . C o rn o , M . S o n z a R e o rd a , a n d G . S q u ille ro Circ b01 b02 b03 b04 b05 b06 b07 b08 b09 b10 b11 b12 b13 b14 b15 b20 b21 VHDL Lines Proc 111 1 71 1 142 1 103 1 333 3 129 1 93 1 90 1 104 1 168 1 119 1 570 4 297 5 510 1 672 3 1,085 3 1,089 3 BB 28 17 27 23 94 25 21 14 16 38 37 118 74 244 171 491 491 GATE PI PO FF 2 2 5 1 1 4 4 4 30 11 8 66 1 36 34 2 6 9 1 8 49 9 4 21 1 1 28 11 6 17 7 6 31 5 6 121 10 10 53 32 54 245 36 70 449 32 22 490 32 22 490 Gate 46 28 149 597 963 60 420 167 159 189 481 1,036 339 4,775 8,893 9,419 9,803 Table 1: Benchmark characteristics In Table 2, we report the experiments we obtained with our prototypical tool in terms of percent number of covered branches, number of generated vectors, and required CPU time. These data demonstrate that for most descriptions, our method is able to reach a complete or very high branch coverage. There are a few circuits (e.g., b05 and b12) where the obtained coverage is low: this is due to the specific characteristics of these circuits, which include highly nested conditional statements that the current version of our algorithm can hardly go through. It is worth noting that a manual analysis of the branches left uncovered proved that many of them were effectively unreachable, in most cases due to e l s e or d e f a u l t statements that are required by the synthesis tool not to infer sequential logic, but are redundant since all cases have already been considered in previous tests. A u to m a tic V a lid a tio n o f P ro to c o l In te rfa c e s D e s c rib e d in V H D L CIRCUIT b01 b02 b03 b04 b05 b06 b07 b08 b09 b10 b11 b12 b13 b14 b15 b20 b21 2 1 3 Cov % #VECT CPU [s] 100,00 259 439.0 100,00 114 41.3 100,00 174 55.5 100,00 83 425.1 52,13 68 2014.9 100,00 125 52.4 95,24 351 920.6 100,00 1,005 971.1 100,00 958 511.5 100,00 364 122.1 94,59 1,222 1,410.2 36,44 155 1,022.3 100,00 3,303 4,203.9 93,03 4,597 7,875.5 91,81 2,838 9,369.2 93,48 7,784 27,286.5 93,69 6,376 28,878.9 Table 2: Experimental results 4 Conclusions This paper presented an automatic input pattern generation tool able to assist designers in the generation of a test bench for difficult parts of small- or mediumsized digital protocol interfaces. The approach resorts to a Genetic Algorithm that interacts with a simulator to generate new sequences able to increase the coverage of the test bench with respect to a predefined validation coverage metric. The methodology has been tested on the new ITC 99 benchmark set [ITC99]. Experimental results prove that the method is able to increase the quality of the validation process both over manual simulation and pseudo-random sequence generation. However, the proposed method could be easily extended to deal with any simulable descriptions, like the ESTELLE specification of a network protocol. The tool results have also been useful as a feedback for better understanding the most difficult parts of the design from the validation point of view. 5 References [AHRo98] G. Al-Hayek, C. Robach: From Design Validation to Hardware Testing: A Unified Approach, JETTA: The Journal of Electronic Testing, Kluwer, No. 14, 1999, pp. 133-140 2 1 4 F . C o rn o , M . S o n z a R e o rd a , a n d G . S q u ille ro [ASUl86] [Beiz90] [CPSo97] [CSSq98] [CSSq99a] [CSSq99b] [CSSq99c] [DGKe96] [FADe99] [FDKe98] [GDNe91] [HuCh98] [ITC99] [LEDA95] [TAZa99] A.V. Aho, R. Sethi, J.D. Ullman, Compilers, Principles, Techniques, and Tools, Addison-Wesley Publishing Company, 1986 B. Beizer, Software Testing Techniques (2nd ed.), Van Nostrand Rheinold, New York, 1990 F. Corno, P. Prinetto, M. Sonza Reorda: Testability analysis and ATPG on behavioral RT-level VHDL, Proc. IEEE International Test Conference, 1997, pp. 753-759 F. Corno, M. Sonza Reorda, G. Squillero, VEGA: A Verification Tool Based on Genetic Algorithms, Intl. Conf. on Circuit Design, 1998, pp. 321-326 F. Corno, M. Sonza Reorda, G. Squillero, Improved Test Pattern Generation on RT-level VHDL descriptions, ITSW 99: International Test Synthesis Workshop, 1999 F. Corno, M. Sonza Reorda, G. Squillero, Simulation-Based Sequential Equivalence Checking of RTL VHDL, ICECS’99: 6th IEEE Intl. Conf. on Electronics, Circuits and Systems, 1999 F. Corno, M. Sonza Reorda, G. Squillero, Approximate Equivalence Verification for Protocol Interface Implementation via Genetic Algorithms, Evolutionary Image Analysis, Signal Processing and Telecommunications First European Workshops, EvoIASP’99 and EuroEcTel’99, 1999, pp. 182-192 S. Devadas, A. Ghosh, K. Keutzer: An Observability-Based Code Coverage Metric for Functional Simulation, Proc. ICCAD 96 F. Fallah, P. Ashar, S. Devadas: Simulation Vector Generation from HDL Descriptions for Observability-Enhanced Statement Coverage, Proc. 36th DAC, New Orleans, 1999, pp. 666-671 F. Fallah, S. Devadas, K. Keutzer: OCCOM: Efficient Computation of Observability-Based Code Coverage Metrics for Functional Verification, Proc. 35th DAC, 1998 A. Ghosh, S. Devadas, A.R. Newton, Sequential Logic Testing and Verification, Kluwer, 1991 S.-Y. Huang, K.-T. Cheng, Formal Equivalence Checking and Design Debugging, Kluwer, 1998 h t t p : / / w w w . i t c t e s t w e e k . o r g / b e n c h m a r k s . h t m l LVS System User s Manual, LEDA Languages for Design Automation, Meylan (F), April 1995 P.A. Thaker, V.D. Agrawal, M.E. Zaghloul: Validation Vector Grade (VVG): A New Coverage Metric for Validation and Test, VTS 99: IEEE VLSI Test Symposium, 1999, pp. 182-188 ti M d i r I tr si il ippo / N t r Tr D t cti c ri - ni rsit o f imo nt rinta l, o rso o rsa l ino 54, 5 lssa ndria ( L), Ita l neri@di.unito.it, neri@al.unipmn.it c . h d t ctio n o fintrusio ns o r co mput r n t o rks ca n b ca stto th ta sk o fd t cting a no ma l o us pa tt rns o fn t o rk tra ffic. 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( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 1 4 − 2 2 3 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 E v o lu tiv e M o d e lin g o f T C P /IP N e tw o rk T ra ffic fo r In tru s io n D e te c tio n 2 1 5 al go rithms), [ io rda a a d ri, 9 9 5, ria d a itta , 9 9 6]. h rst s st m ha s b s lct d b ca s o fits pr io s s [ ta l ., 9 9 9 ]; it il l th s a ct a s b chma r . h s co d s st m ha s b s lct d b ca s its i tri sica l l sto cha stic b ha io r a d sho l d al l o th a cq isitio o fmo r ro b st mo d l s [ ria d a itta , 9 9 6]. I th fo l l o i g, a d scriptio o fth s st m ( ctio 2) a d o fth p rim ts p rfo rm d i th I a d co t ts ( ctio 3 a d ctio 4) a r r po rt d. i a l l, th co clsio s a r dra . The Systems Regal and Ripper o r spa c r a so il lpro id h r a a bstra ct d scriptio o f bo th la r i g s st ms a d I a s th ir f l ld scriptio s ha a l r ad b p bl ish d. h s st m is a a il a blfo r fr fro m th a tho r. [ io rda a a d ri, 9 9 5, ria d a itta , 9 9 6] is a la r i g s st m, ba s d o a distrib t d g tic a l go rithm ( ). Itta s a s i p ta s to f da ta (tra i i gi sta c s) a d o tp ts a s to fs mbo l ic cl a ssi ca tio r ls cha ra ct rii gth i p tda ta . s s a l , la r i gis a chi d b s a rchi ga spa c o f ca dida t cl a ssi ca tio r ls. h l a g ag s d to r pr s tcl a ssi ca tio r lsisa H o r cl a s l a g ag i hich t rms ca b a ria bls o r disj ctio s o fco sta ts, a d ga tio o cc rs i a r strict d fo rm [ icha l s i, 9 3]. a mplo fa a to mic pr ssio co ta i i g a disj cti t rm is r( r ), hich is s ma tica l l q ia l t to r( ) r r( r ). ch fo rm l a s a r r pr s t d a s bitstri gs tha ta r a ct a l l th po p l a tio i diid a l s pro c ss d b th . l a ssica lg tic o p ra to rs, o p ra ti g o bi a r stri gs, ith th a dditio o f ta s o ri t d s a a d ra cro sso rs a r pl o it d, i a a da pti a , i sid th s st m (fo r d ta il s s [ io rda a a d ri, 9 9 5]. is a distrib t d g tic a l go rithm tha t ff cti lco mbi s th h o r o f ich s a d p cis o f io l o gica l o ltio to g th r ithpa ra l llpro c ssi g. h s st m a rchit ct r is ma d b a s to f t d d impl tic l go rithms ( ) [ ol db rg, 9 9 ], hich co o p ra t s to si a d scriptio spa c , a d b a p r iso r pro c ss tha tco o rdi a t s th s ffo rts b a ssig i gto a ch o fth m a diff r tr gio o fth ca dida t r l spa c to b s a rch d. I pra ctic this is a chi d b d a mica l l d isi g s bs ts o fth da ta s tto b cha ra ct ri d b a ch . h s st m I [ o h , 9 9 5] is ba s d o th it ra t d a ppl ica tio o f a gr d h ristic, simil a r to th I fo rma tio ai m as r [ il a , 9 9 3], to b il d co j cti cl a ssi ca tio r ls. t a chit ra tio ,tho s tra i i gi sta c s co rr ctlcl a ssi d b th fo d r lsa r r mo d a d th a l go rithm co c tra t o la r i g a cl a ssi ca tio r lfo r th r ma i i g o . h s st m o tp ts a o rd r d l ist o f cl a ssi ca tio r ls (po ssibl a sso cia t d to ma cl a ss s) to b a ppl id i tha tsa m o rd r to cl a ssif a i sta c . it r sti gfa t r s o f th m tho d is tha tit pl o its o -l i r lpr i g hili cr m ta l l b il di g a cl a ssi ca tio r lto a o id o r tti g. 2 1 6 3 F . N e ri tr si d t cti t tc t st i t r ti p r ti a la tio o f o r a itr sio d t ctio ta s b pl o iti g da ta fro m th I fo rma tio pl o ra tio ho o to t ro jct (I ) is r po rt d i this s ctio . h I ma d a a il a bl t o r l o gs pro d c d b ’ tcpd mp’fo r a l a ti gda ta mi i gto o lo r l a rg s to fda ta . h s l o gs r co l lct d a tth ga t a b t a t rpris a d th o tsid - t o r (I t r t). I th I co t t, d t cti gitr sio s m a s to r co g i th po ssiblo cc rr c o f a tho ri d (’ ba d’ ) da ta pa c ts it rla d ith th a tho ri d (’ go o d’ )o s o r th t or d r mo ito ri g. h I ’ s pro jctma s a a il a blfo r tor l o gs: o is g a ra t o tto co ta i a itr sio a tt mpts, h r a s th o th r o s do i cld bo th o rma ltra ffic a d itr sio s a tt mpts. I th I co t t, o cl a ssi ca tio fo r a ch da ta pa c tis r q st d, i st a d a o ra l l cl a ssi ca tio o fa b ch o fth t o r tra ffic, a s co ta i i g a tta c s o r o t, is d sir d. a ppro a ch to itr sio d t ctio , ba s d o a o ma l d t ctio , ha s b s lct d. pro c d a s fo l l o s. I da ta ca b pa rtitio d, o th ba sis o fth ir I a ddr ss s,ito pa c ts iti gth r fr c i sta l l a tio ( tgo i g), t ri gth i sta l l a tio (I co mi g) a d bro a dca st d fro m ho stto ho sti sid th i sta l l a tio (I t rl a ). hr mo d l s o fth pa c ttra ffic, o fo r a ch dir ctio , ha b b il tfro m th itr sio -fr da ta s t. h , th s mo d l s ha b a ppl id to th thr da ta s ts co ta i i gitr sio s. p ctto o bs r a sig i ca t a ria tio i th cl a ssi ca tio ra t b t itr sio -fr l o gs a d l o gs co ta i i g itr sio s b ca s o fth a r a cha ra ct ristics o fth tra ffic pro d c d b th itr si b ha io r. Ifthis o l d a ct a l lo cc r, co l d a ss rttha tth la r d tra ffic mo d l s co rr ctlca pt r th ss tia lcha ra ct ristics o fth itr sio -fr tra ffic. o t tha t h th r this a ppro a ch sho l d or , co l d co cld tha t a itr si a tt mptis ha pp i g b ca s o fa diff r t a tra a tt r . p rim ts ha b p rfo rm d bo th ith I a d . h I is a ppl id to th I da ta , th cl a ssi ca tio ra t a pp a ri g i a bl b co m s id t[ ta l ., 9 9 9 ]. a chta bl tr r pr s ts a cl a ssi ca tio rro r a s m a s r d o r th fo r t o r l o gs, o fo r a ch l i ,a d ith r sp ctto th thr cl a ss l ab l s: tgo i g, I t rl a , a d I co mi g. h co rr ctcl a ssi ca tio ra t s ca b o bta i d b s btra cti ga tr ’ s a l fro m . h s th rstro sho th miscl a ssi d o rma ltra ffic pa c ts, h r a s th o th r o s sho s th miscl a ssi d pa c ts d ri ga itr sio a tt mpt. hs rs l ts ha b o bta i d b a ppli g I to th da ta a s a a il a blfro m th tcpd mp d ls(s pp di ). o pr pro c ssi go rth da ta , s cha s fa t r co str ctio ,ha s b a ppl id. h p rim ta l di gs sho s tha tth a cq ir d mo d l s do o t hibit r diff r tcl a ssi ca tio ra t h a ppl id to l o gs co ta i i g itr sio s ith r sp ctto itr sio -fr l o gs. h s di gs ma s gg sttha tth pl o it d da ta r pr s ta tio is to o d ta ild ith r sp ctto th ca pa bil it o fth la r i gs st m. I t r , this ca s s th la r d mo d l s to miss th i fo rma tio cha ra ct rii gitr sio -fr tra ffic. E v o lu tiv e M o d e lin g o f T C P /IP N e tw o rk T ra ffic fo r In tru s io n D e te c tio n . p rim nta l r sul ts o fa ppling I to I da ta s ts usingth ra r pr s nta tio n. a ch ta bl ntr sta t s a cl a ssifica tio n rro r. 2 1 7 da ta a ta s t int rl a n inco mingo utgo ing no rma ltra ffic .4 .4 .4 intrusio n .23 .7 .4 intrusio n2 .9 .7 .5 intrusio n3 . .4 .4 . p rim nta l r sul tso fa ppling I da ta r pr s nta tio n. to I da ta s tsusinga co mpr ss d a ta s t int rl a n inco mingo utgo ing no rma ltra ffic .2 .5 .4 intrusio n . . .2 intrusio n2 .3 .3 .2 intrusio n3 . .2 .2 ol l o i g this o bs r a tio , d l o p a mo r co mpa ct r pr s ta tio fo r th pa c ts tha tco sists i ma ppi g a s bs to ffa t r ’ s a l s ito a si gl a l , th s r d ci g th ca rdi a l it o fpo ssibl fa t r s a l s (s pp di ). pl o iti g this r pr s ta tio , I ’ s p rfo rma c s b co m th o s r po rt d i a bl 2 a d ’ s p rfo rma c s pl o iti g th sa m co mpa ct da ta r pr s ta tio a pp a r i a bl3. h o bs r d g r s sho a mo r sta bl cl a ssi ca tio b ha io r o fth mo d l s a cro ss diff r ttra ffic co ditio s. l so a mo r disti ctcl a ssi ca tio p rfo rma c b t th itr sio -fr l o g a d th l o gsi cldi gitr sio sis id t. co mpr ssio -ba s d r pr s ta tio isth a a la bl a o fi cr a si gcl a ssi ca tio p rfo rma c s itho titro d ci gco mpl fa t r tha tma i o l s a dditio a lpro c ssi go rh a d. a la tio o f th ff ctca s d b th a dditio o fco mpl fa t r s to th ra t o r da ta r pr s ta tio ha s b p rfo rm d i [ ta l ., 9 9 9 ]. o r th sa o fcl a rit, a a mpl o fr lcha ra ct rii g itr sio -fr I co mi gpa c ts,la r d b ,a pp a rs i ig r . h I co mi gpa c ts a r cha ra ct ri d i t rm o fth a l so fth fa t r sfro m th ir / I h a d r. his r ls cc ssf l lco rs 7 349 I co mi gpa c ts itho tb i gfo o ld b a . p rim nta l r sul ts o fa ppling da ta r pr s nta tio n. Lto I da ta s ts usinga co mpr ss d a ta s t int rl a n inco mingo utgo ing no rma ltra ffic .2 .4 .4 intrusio n .2 .5 . intrusio n2 .6 . .2 intrusio n3 .2 .5 . 2 1 8 F . N e ri I srcprt( ,[[ ,2 ],[4, ],[ 5 ,2 ],[ 5 ]]) a nd dstprt( ,[ 24]) a nd fla g( ,[ ,pt]) a nd s q ( ,[[ , 5 ],[2 ,3 ],[5 ,5 ],[ ]]) a nd s q2( ,[[5 , ],[2 ,3 ],[5 ,2 ]]) a nd a ck( ,[[ ,3 ],[5 , ]]) a nd in( ,[[ ,2 ],[ 3 ]]) a nd buf( ,[ =5 2]) H Inco ming a ck t( ) o ra g : (Int rl a n, Inco ming, utgo ing) = ( , 7 349 , ) i. . a mplo fa rulcha ra ct riingpa rto fth inco mingtra ffic. h ruld scrib s 7 349 inco mingpa ck ts itho utco nfusing th m ith a n o utgo ing o r int rl a n pa ck t. I t rl a o r tgo i go s. d scriptio o fth pr dica t s a pp a ri gi th r l is pro id d i pp di . tr si d t cti i t t cti ti r r tr si al so p rfo rm d a a dditio a l a la tio o f o r a ppro a ch o r tor l o gs fro m 9 9 I tr sio t ctio a la tio ro gra mm [ ippma ta l ., 9 9 9 ] ho s o b jcti a s to s r a d a la t r s a rch i itr sio d t ctio . sta da rd s to fda ta to b a dit d, hichi cld s a id a rit o fitr sio s sim l a t d i a mil ita r t or iro m t, a s pro id d. pl o it d da ta a a il a blfro m th K ’ 9 9 I tr sio t ctio o t st. h ra tra i i gda ta a s a bo tfo r giga b t s o fco mpr ss d bi a r d mp da ta fro m s s o f t o r tra ffic. his a s pro c ss d ito a bo t mil l io co ctio r co rds. imil a rl, th t o s o f t st da ta il d d a ro d t o mil l io co ctio r co rds. co ctio is a s q c of pa c ts sta rti g a d di g a tso m l ld d tim s, b t hich da ta flo s fro m a so rc I a ddr ssto a ta rg tI a ddr ss d rso m l ld d pro to co l . a ch co ctio is l a b ld a s ith r o rma l , o r a s a a tta c , ith a ctl o sp ci c a tta c t p . a chco ctio r co rd co sists o fa bo t b t s. tta c s fa l lito fo r ma i ca t go ris: : d ia l -o f-s r ic , .g. s flo o d; 2 : a tho ri d a cc ss fro m a r mo t ma chi , .g. g ssi gpa ss o rd; 2 : a tho ri d a cc ss to l o ca ls p r s r (ro o t) priilg s, .g., a rio s “ b ff r o rflo ” a tta c s; ro b : s r il l a c a d o th r pro bi g, .g., po rtsca i g. I pra ctic t o da ta ls co ta i i gcl a ssi d co ctio s a r a a il a bl: o ha s to b s d fo r a cq iri ga mo d lo fth tra ffic a d th o th r o fo r t sti g its Info rma tio n a bo ut K ’ 9 9 Intrusio n t ctio n http:/ / . psil o n.co m/ kdd9 / ta sk.html . o nt st is a a il a bl o n-l in at E v o lu tiv e M o d e lin g o f T C P /IP N e tw o rk T ra ffic fo r In tru s io n D e te c tio n 2 1 9 p rfo rma c s. h disti ctio is impo rta tb ca s th t st lco ta i s a tta c t p s o to cc rri gi th la r i g l. his is it d d to ma th ta s mo r r al istic. I g r 2 a d g r 3,p rfo rma c s o f I pls ta - a r i g 1 Probe Dos U2r R2l 0.8 0.6 0.4 0.2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 t ctio n p rfo rma nc s hibit d b I pl us ta -L a rning o n th t st da ta . n t nd d r pr s nta tio n o f th da ta a nd a co mpl la rning a ppro a ch (m ta -l lla rning) ha b n pl o it d. i. . (a s s d i [ ta l ., 9 9 9 ]) a d o r ’ s da ta a r r sp cti l sho . I this ca s p rfo rma c s a r sho b pl o iti g c i r p ra ti g r s a s do h a la ti gda ta mi i gto o l s.I th g r s,th a is r pr s ts th fa l s al a rm ra t , i. . th p rc ta g o f’ o rma l ’co ctio s l a b ld a sitr sio s, h r a sth a isr pr s tsth d t ctio ra t ,i. .th p rc ta g o fitr sio s tha tha b co rr ctlr co g i d. a chl i is a sso cia t d to a diffr ta tta c t p . his i d o fgra phis s d to sho ho a cl a ssi rs b ha io r d gra d s h ”r l a i g” its ma tchi gco ditio s: g ra l l mo r itr sio a r d t ct d hila l so co ri gmo r fa l s al a rms. H o r i th ca s o fa s to f s mbo l ic cl a ssi ca tio r ls (itis a ct a l l b tt r to sa i th ca s o f I a d ) th r a r o co ditio s to b r l a d tha tis th r a so o fth fla tl i a tth to p. h r po rt d p rfo rma c s ha b o bta i d o th co ctio s o cc rri g i th t st l. h r po rt d gra phs sho simil a r d t ctio p rfo rma c s, b t th mo d l s a cq ir d b th s st ms, fo r ro b a d mo t - o - o ca l ( 2l ) a tta c s t p s. I st a d, ’ s mo d lp rfo rms sl ightl b tt r o t p a tta c s b t o rsto s r- o - o o t( 2r) a tta c s. tco sid r, o ,th mo d l i ga ppro a ch s pl o it d b th t o s st ms. a d to l fo [ ta l ., 9 9 9 ] r I o ra t d d da ta r pr s ta tio o fth tcp co ctio i cldi g, i a dditio to th ba sic tcp fa t r s, d ri d i fo rma tio s ch a s: th mb r o fco ctio s to th sa m ho st i th pa st 2 2 0 F . N e ri 1 Probe Dos U2r R2l 0.8 0.6 0.4 0.2 0 0 0.02 0.04 i. . t ctio n p rfo rma nc s ta -L a rningha s b n us d). 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 hibitb Lo n t stda ta (no a dditio na l co mpr ss d da ta r pr s nta tio n ha s b n pl o it d. t o s co ds (’ co t’ ), th mb r o fco ctio s to th sa m s r ic , a s th c rr tco ctio , i th pa stt o s co ds (’ sr -co t’ ). h s fa t r s ha b cho s o th ba sis o fth a tho rs p rtis . pr pro c ssi g o fth ra t or l o gs is r q ir d i o rd r to pl o its this fa t r s. ra lcl a ssi rs (r ls ts) fo r a ch a tta c t p ha b o bta i d. t al l m ta -la r i g, i. . la r i g a t th cl a ssi r l l , ha s b a ppl id to pro d c th r po rt d p rfo rma c s. ,o th co tra r ,ha sb r a ft ra ppli ga co mpr ssio ma ppi g to th fa t r a l s, a s d scrib d i pp di . l th ba sic fa t r s o fa co ctio ha b co sid r d s ch a s: ’ d ra tio ’ , sta ti g th l gth ( mb r o fs co ds) o fth co ctio , ’ pro to co l -t p ’ , sta ti g th t p o fth pro to co l( .g. tcp, dp, tc.), o r ’ src-b t s’ , sta ti g th mb r o fda ta b t s fro m so rc to d sti a tio . o a dditio a lm ta -la r i gpha s is c ssa r . c si s r po rt p rim tstha tsho th po t tia l it o fa distrib t d g tic la r r to th mo d l i g o f t o r da ta . o diff r ts t- ps to d a l ith d t cti g itr sio s ha b pl o r d. a a l d a da ta pa c tr pr s ta tio pl o iti gco mpr ssio o fth fa tr’ s a l s i th ffo rtto r d c th co mpl it o fa cq iri g mo d lo fth tra ffic. b l i this b i ga impo rta tr q isit fo r th a to ma tic mo d l ig a d th o -l i d pl o m to fitr sio d t ctio s st m. h p rim ta lr s l ts s ppo rt s o fth co mpr ssio o fth fa t r a l s a s a a la blm tho d to i cr a s d t ctio p rfo rma c s hila o idi gth s o fd ri d a d co mpl fa t r s tha ti o l a dditio a lco mp ta tio a lo rh a d. E v o lu tiv e M o d e lin g o f T C P /IP N e tw o rk T ra ffic fo r In tru s io n D e te c tio n c d 2 2 1 ts a tha s to th a o impro i gth pa p r. mo s r i pp di . r d t r pr s t ti rs ho s ti s f lco mm p r ti ts h l p dm i t tr h I da ta (a a il a bl o l i a thttp:/ / iris.cs. ml . d ) ha b co l lct d b m a s o fth til it. a i g ito a cco tpria c co c r s, th da ta po rtio o f a ch pa c tha s b dro pp d. o r a ch pa c ti th da ta s ts th fo l l o i ga ttrib t s a r a a il a bl: tim - co rt d to flo a ti gpts co ds .. hr* 36 + mi * 6 + s cs. a ddr a d po rt- (j stg trid o f . .256 .256 .po rt) h rstt o l ds o fth src a d d sta ddr ss ma p th fa a ddr ss, so th co rt d a ddr ss a s ma d a s: + * 256 . fla g - a dd d a ” ” fo r dp da ta (o l ha s l ) - m a s pa c t a s a a m s r r r q sto r r spo s . h I # a d r sto fda ta is i th ”o p” l d. (s tcpd mp d scriptio ) -m a s th r r o po rts...fro m ”fra gm t d pa c ts”. s q - th da ta s q c mb r o fth pa c t. s q2 - th da ta s q c mb r o fth da ta p ct d i r t r . b f- th mb r o fb t s o fth r c i r b ff r spa c a a il a bl. a c - th s q c mb r o fth tda ta p ct d fro m th o th r dir ctio o this co ctio . i -th mb r o fb t s o fr c i b ff r spa c a a il a blfro m th o th r dir ctio o this co ctio . l - ifa dp pa c t, th l gth. o p - o ptio a li fo s ch a s (df) ... do o tfra gm t. a rtic l a r a tt tio ha s to b ta h d al i g ith l ds l i ’ o p’tha tco ta i s a l a rg a mo to f a l s. pp di . d t c pr ss d t r r pr s t ti o m fa t r s o fth I da ta ma a ss m a l a rg s to f a l s ith r co ti o s o r discr t . h s l a rg s ts do impa cto r cl a ssi ca tio p rfo rma c s o f th la r d mo d l s b ca s o fth itri sic diffic l t o fa cq iri g r l ha i g a g ra lsco p . h , a r d ctio o fth ra g o fpo t tia la l s is d sira bl to i cr a s bo th th g ra l it o fth la r d mo d la d to r d c th la r i g co mp ta tio a lco mpl it. al t r a ti a ppro a ch to this pro blm co sists i a ddi g/ b il di g mo r co mpl fa t r s, co mbi i gth ba sic o s,to th o rigi a lda ta r pr s ta tio . do o tfo l l o thisa ppro a chi this o r ,b ca s b l i tha tth pr io s a ppro a chis simplr a d sho l d b th rstto b a a l d. 2 2 2 F . N e ri rigina l a l u srcpo rt 5 5 srcpo rt ... skipp d t st... srcpo rt 2 ... skipp d t t... o p co nta ins ” ” o p co nta ins ” o ma in” o p co nta ins Y H L . o mpr ssio n ma ppinga ppl id al u srcpo rt= srcpo rt= ... skipp d t t... srcpo rt= ... skipp d t t... o p= o p=2 o p=3 h n d al ing ith I n t o rk da ta . s a i sta c o fr d ci gth ra g o fth fa t r a l s,co sid rs tha tth fa t r ’ srcpo rt’(s pp di fo r a d scriptio ) ma irt a l l a ss m a it g r mb r fro m to 6 5536 . l so , th fa t r ’ o p’ma a ss m h dr ds o fdiscr t a l s. a i g ito a cco tba sic o ldg a bo tth do ma i , ma a l ld l o p d th r d ctio ma ppi g sho i a bl4. his ma ppi g is o tto b co sid r d a s th b sto b ta s a pro o ftha ta simplr d ctio o f th fa t r a l s ma po siti l impa cto r th r co g itio ca pa bil itis. r c s [ o h n, 9 9 5] o h n, . ( 9 9 5). a st ff cti rulinductio n. In r di gs fI t rti hi ri g fr , La k a ho , . o rga n Ka ufma nn. [ nning, 9 7 ] nning, . ( 9 7 ). n intrusio n d t ctio n mo d l .I r s ti ft r gi ri g, - 3(2):222–232. [ ho sh ta l ., 9 9 9 ] ho sh, ., ch a rt ba rd, ., a nd cha t , . ( 9 9 9 ). L a rning pro gra m b ha io rpro fils fo rintrusio n d t ctio n. In I rksh I tr si t ti d t rk it ri g. I sso cia tio n. [ io rda na a nd ri, 9 9 5] io rda na , . a nd ri, . ( 9 9 5). a rch-int nsi co nc pt inductio n. ti r m t ti , 3 (4):37 5–46 . [ ol db rg, 9 9 ] o l db rg, . ( 9 9 ). ti grithms i rh timi ti d hi r i g. ddiso n- sl , a ding, a . [Kuma r a nd pa ffo rd, 9 9 4] Kuma r, . a nd pa ffo rd, . ( 9 9 4). pa tt rn ma tching mo d lfo r misus d t ctio n. In ti m tr rit fr ,pa g s –2 , al timo r . [La n a nd ro dl , 9 9 7 ] La n , . a nd ro dl , . ( 9 9 7 ). n a ppl ica tio n o fma chin la rningto a no ma ld t ctio n. In ti I f rm ti st ms rit fr , al timo r . [La n a nd ro dl , 9 9 ] La n , .a nd ro dl , .( 9 9 ). ppro a ch sto o nl in la rninga nd co nc ptua l driftfo r us rid ntifica tio n in co mput rs curit. chnica l r po rt, a nd th La bo ra to r , urdu ni rsit, o a st 9 - 2. [L ta l ., 9 9 9 ] L , ., to l fo , ., a nd o k, K. ( 9 9 9 ). ining in a da ta -flo niro nm nt: p rinc in n t o rk intrusio n d t ctio n. In dg is r d t iig ’ , pa g s 4– 24. r ss. E v o lu tiv e M o d e lin g o f T C P /IP N e tw o rk T ra ffic fo r In tru s io n D e te c tio n 2 2 3 [Lippma nn ta l ., 9 9 9 ] Lippma nn, ., unningha m, ., rid, ., ra f, I., K nda l l , K., bst r, .,a nd Z issma nn, . ( 9 9 9 ). sul ts o fth 9 9 o ffl in intrusio n d t ctio n a l ua tio n. In t d s i I tr si t ti I ’ , . La fa tt , I . urdu ni rsit. [ icha l ski, 9 3] icha l ski, . ( 9 3). th o r a nd m tho do l o g o finducti la rning. In icha l ski, ., a rbo n l l , ., a nd itch l l , ., dito rs, hi ri g rtifi i I t ig r h, o l um I, pa g s 3– 34. o rga n Ka ufma nn,Lo s l to s, . [ ria nd a itta , 9 9 6 ] ri, . a nd a itta , L. ( 9 9 6 ). pl o ringth po r o fg n tic s a rchin la rnings mbo l ic cl a ssifi rs. I r s. tt r sis d hi I t ig , I- : 35– 42. [ uinl a n, 9 9 3] uinl a n, . . ( 9 9 3). 4. rgrms f r hi r i g. o rga n Ka ufma nn, a l ifo rnia . M u ltim o d a l P e r fo r m a n c e P r o file s o n th e A d a p tiv e D is tr ib u te d D a ta b a s e M a n a g e m e n t P r o b le m M . O a te s 1, D . C o rn e 2, a n d R . L o a d e r 1 2 2 B ritis h T e le c o m A d a s tra l P a rk , M a rtle s h a m H e a th , S u ffo lk , E n g la n d , IP 5 3 R E D e p a rtm e n t o f C o m p u te r S c ie n c e , U n iv e rs ity o f R e a d in g , R e a d in g , R G 6 6 A Y A b s tr a c t. P re v io u s p u b lic a tio n s b y th e a u th o rs h a v e d e m o n s tra te d a b im o d a l p e rfo rm a n c e p ro file fo r s im p le e v o lu tio n a ry s e a rc h o n v a ria n ts o f th e A d a p tiv e D is trib u te d D a ta b a s e M a n a g e m e n t P ro b le m (A D D M P ) a n d o th e r p ro b le m s o v e r a ra n g e o f e v a lu a tio n lim its . T h is p a p e r e x a m in e s a n a n o m a ly s e e n in o n e o f th e s e p ro file s a n d to g e th e r w ith re s u lts fro m a ra n g e o f o th e r p ro b le m s , s h o w s th a t w ith s u ffic ie n tly h ig h e v a lu a tio n lim its , a m u ltim o d a l p e rfo rm a n c e p ro file is a p p a re n t in s e a rc h s p a c e s w ith s ig n ific a n t n u m b e rs o f d e c e p tiv e lo c a l o p tim a . T h is is p a rtic u la rly a p p a re n t in th e p e rfo rm a n c e p ro file o f th e H ie ra rc h ia l If a n d o n ly If p ro b le m (H -IF F ) w h e re th e re g u la r s tru c tu re o f th e s e a rc h s p a c e p ro d u c e s s e v e ra l d is tin c t p e a k s a n d tro u g h s in th e p e rfo rm a n c e p ro file , p o s s ib ly in d ic a tiv e o f a ra n g e o f s p e c ific ‘fitn e s s b a rrie rs ’ w h ic h a re s u rm o u n ta b le b y s p e c ific ra te s o f m u ta tio n . T h is o b s e rv a tio n c o u ld p ro v e im p o rta n t in g e n e ra l E A p a ra m e te r tu n in g o v e r a ra n g e o f p ro b le m s w ith s im ila r c h a ra c te ris tic s . F u rth e r, th e e x is te n c e o f o p tim a l m u ta tio n ra te s in d u c in g a m in im u m in s ta n d a rd d e v ia tio n o f ru n -tim e , is o f c ritic a l im p o rta n c e in th e a p p lic a tio n o f E A s to re a ltim e , re a l-w o rld p ro b le m s . 1 I n tr o d u c tio n M a n y r e a l w o r ld , r e a l tim e a p p lic a tio n s o f E v o lu tio n a r y A lg o r ith m s ( E A s ) [ 1 ,3 ,4 ,6 ] re q u ire th e s e a rc h p ro c e s s to re lia b ly p ro d u c e q u a lity s o lu tio n s in a fix e d n u m b e r o f e v a lu a tio n s . T o in c re a s e th e lik e lih o o d o f th is , it is o fte n e s s e n tia l to tu n e th e a lg o rith m b y s e le c tio n o f s u ita b le p a ra m e te r v a lu e s s u c h a s p o p u la tio n s iz e a n d m u ta tio n ra te . O n e s u c h a p p lic a tio n w h ic h h a s b e e n s tu d ie d e x te n s iv e ly b y th e a u th o rs [ 1 1 ,1 2 ,1 3 ,1 4 ,1 5 ,1 6 ,1 7 ] is th e A d a p tiv e D is tr ib u te d D a ta b a s e M a n a g e m e n t P r o b le m (A D D M P ) w h ic h a tte m p ts to b a la n c e a n u m b e r o f u s e r lo a d s o n to a ra n g e o f a v a ila b le s e rv e rs o v e r a c o m m u n ic a tio n s n e tw o rk to m a x im is e a g iv e n q u a lity o f s e rv ic e m e tric . T h e p ro b le m is re d u c e d to o n e o f c o m b in a to ria l o p tim is a tio n , w h e re a ‘s o lu tio n v e c to r’ g e n e ra te d b y th e E A d e fin e s fo r e a c h c lie n t n o d e (d e te rm in e d b y lo c u s ), w h ic h s e rv e r to u s e (d e te rm in e d b y a lle le v a lu e ) u s in g a n a tu ra l k -a ry re p re s e n ta tio n (w h e re a lle le s a re in te g e rs in th e ra n g e 1 th ro u g h k ). T h is e ffe c tiv e ly d e te rm in e s a ro u te th ro u g h th e c o m m u n ic a tio n s n e tw o rk , a n d th e c o m b in e d d e la y im p o s e d b y th e s e rv e r p ro c e s s in g tim e a n d th e c o m m u n ic a tio n s la te n c y is c a lc u la te d b y a p e rfo rm a n c e m o d e l u tilis in g th e p rin c ip le s o f M M 1 q u e u in g a n d L ittle ’s L a w . T h e p ro b le m h a s b e e n lik e n e d to a fo r m o f c o m p le x ‘b in -p a c k in g ’ a n d fu r th e r d e ta ils a r e a v a ila b le in [ 1 2 ,1 3 ] w ith e x a m p le c o d e a n d d a ta s e ts a v a ila b le a t h ttp ://w w w .d c s .n a p ie r /a c /u k /e v o n e t p a r t S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 2 4 − 2 3 4 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 M u ltim o d a l P e rfo rm a n c e P ro file s 2 2 5 o f th e E C T E L N E T w e b s ite (E C T E L N E T is th e T e le c o m m u n ic a tio n s s u b g ro u p o f th e E u ro p e a n N e tw o rk o f E x c e lle n c e in E v o lu tio n a ry C o m p u ta tio n ) C lie n t : W h ic h S e rv e r to u s e : 1 2 1 3 4 4 3 5 4 6 3 7 2 8 2 9 3 4 1 0 1 F ig . 1 . E x a m p le s o lu tio n v e c to r. M a n y in s ta n c e s o f th e A D D M P e x is t, w ith d iffe re n t u s e r lo a d p a tte rn s , d iffe re n t s e rv e r p ro file s a n d d iffe re n t c o m m u n ic a tio n s n e tw o rk to p o lo g ie s , a n d s o m e o f th e s e a re d is c u s s e d in [1 3 ]. S o m e o f th e s e in s ta n c e s h a v e b e e n s h o w n to p re s e n t c o n s id e ra b le d iffic u lty to a r a n g e o f a lg o r ith m s [ 1 1 ,1 4 ] a n d in g e n e r a l, a n E v o lu tio n a r y a lg o r ith m , c o rre c tly tu n e d , h a s b e e n s h o w n to g iv e th e m o s t c o n s is te n t p e rfo rm a n c e . P re v io u s ly p u b lis h e d s tu d ie s o f o n e o f th e s e in s ta n c e s [ 1 6 ,1 7 ] s h o w a th r e e d im e n s io n a l p e rfo rm a n c e p ro file fo r a s im p le g e n e ra tio n a l B re e d e r E A [7 ] u s in g 5 0 % e litis m , u n ifo rm c ro s s o v e r [1 9 ], a n d u n ifo rm ly d is trib u te d a lle le re p la c e m e n t m u ta tio n a t a fix e d ra te p e r g e n e . E x p e rim e n ts w e re ru n o v e r a w id e ra n g e o f m u ta tio n ra te s , fro m 1 E -7 to 0 .8 3 , th e la tte r e ffe c tiv e ly d e g e n e r a tin g th e E A in to r a n d o m s e a r c h , a n d a w id e ra n g e o f p o p u la tio n s iz e s ty p ic a lly 1 0 to 5 0 0 m e m b e rs in s te p s o f 1 0 . E a c h ru n w a s a llo w e d 2 0 ,0 0 0 e v a lu a tio n s a n d e a c h e x p e r im e n t w a s r e p e a te d 5 0 tim e s . D u r in g e a c h ru n , a n o te w a s m a d e o f th e fitn e s s o f th e b e s t s o lu tio n fo u n d , a n d th e e v a lu a tio n n u m b e r a t w h ic h it w a s firs t fo u n d . F o r th e 5 0 ru n s a ll w ith th e s a m e e x p e rim e n ta l p a ra m e te rs , th e m e a n o f th e ‘tim e to b e s t s o lu tio n ’ w a s p lo tte d a g a in s t p o p u la tio n s iz e a n d m u ta tio n ra te , a n d th e re s u ltin g g ra p h is s h o w n in F ig u re 2 . A s w a s c o m m e n te d o n in [1 6 ], th is p ro d u c e s , fo r a g iv e n , lo w p o p u la tio n s iz e , a b im o d a l p e rfo rm a n c e p ro file , w h e re th e n u m b e r o f e v a lu a tio n s ta k e n to firs t fin d th e b e s t s o lu tio n to b e fo u n d in th e ru n firs t ris e s , th e n fa lls , th e n ris e s a g a in b e fo re fa llin g in to ra n d o m s e a rc h . T h e s e fe a tu re s a re in te rp re te d a s fo llo w s : th e lo w v a lu e s o f e v a lu a tio n s u s e d a t lo w m u ta tio n ra te s re p re s e n t p re m a tu re c o n v e rg e n c e o n p o o r s o lu tio n s , ra p id ly e x h a u s tin g th e lim ite d d iv e rs ity a v a ila b le in th e in itia l p o p u la tio n w h ic h c a n n o t b e s u p p le m e n te d b y m u ta tio n d u e to its lo w ra te ; a s m u ta tio n ra te s in c re a s e , th e E A is a b le to p ro g re s s fu rth e r w ith its s e a rc h , a s d e p le te d a lle le v a lu e s a re re -in je c te d in to th e p o p u la tio n w ith in c re a s in g fre q u e n c y b y th e in c re a s in g m u ta tio n ra te ; a p o in t is re a c h e d w h e re th e re is a s u ffic ie n tly h ig h ra te o f m u ta tio n to a llo w th e s e a rc h to ty p ic a lly u tilis e in e x c e s s o f 8 0 % o f th e a v a ila b le e v a lu a tio n s in te rm s o f fitn e s s im p ro v e m e n t – th is is th e le ft h a n d p e a k o n F ig u re 2 ; fu rth e r in c re a s e s in m u ta tio n ra te a llo w g o o d s o lu tio n s to th e p ro b le m to b e fo u n d in fe w e r e v a lu a tio n s , le a d in g to a p o in t w h e re th e m u ta tio n ra te a llo w s g o o d s o lu tio n s to b e fo u n d in a m in im u m o f e v a lu a tio n s – th e tro u g h in F ig u re 2 ; a s m u ta tio n ra te is in c re a s e d fu rth e r, p e rfo rm a n c e d e te rio ra te s a s th e m u ta tio n ra te b e g in s to d e s tro y p ro g re s s a lm o s t a s fa s t a s it is m a d e – th e rig h t h a n d p e a k o n F ig u re 2 ; fin a lly a t v e ry h ig h ra te s o f m u ta tio n , in e x c e s s o f 4 0 % , p e rfo rm a n c e d e te rio ra te s to ra n d o m s e a rc h w ith th e m e a n n u m b e r o f e v a lu a tio n s u s e d te n d in g to 5 0 % o f th o s e a llo w e d . T h is in te rp re ta tio n is s u p p o rte d b y e x a m in a tio n o f th e m e a n fitn e s s e s o f s o lu tio n s fo u n d a s m u ta tio n ra te s a re in c re a s e d w h ic h s h o w : p o o r fitn e s s a t lo w m u ta tio n ra te s ; a s te a d y im p ro v e m e n t in fitn e s s fo r m u ta tio n ra te s 2 2 6 M . O a te s , D . C o rn e , a n d R . L o a d e r a b o v e th e firs t p e a k in F ig u re 2 ; a p la te a u o f g o o d fitn e s s c o in c id in g m in im a l e v a lu a tio n s in F ig u re 2 ; a n d a d e te rio ra tio n in fitn e s s a p p ro a c h th e rig h t h a n d s id e o f F ig u re 2 . T h is c o -in c id e n c e o f tro a n d p e a k in fitn e s s o c c u rs a t m u ta tio n ra te s a ro u n d 1 / L (w c h r o m o s o m e le n g th , h e r e e q u iv a le n t to a r o u n d 2 .5 % ) , w h ic h th e o re tic a l s tu d ie s b y , a m o n g s t o th e rs , B a e c k [1 ], M ü h le n b e in [8 ] a s tu d ie s in c lu d in g [ 2 ,8 ,9 ,1 8 ] b y D e b e t a l, v a n N im w e g e n a n d C r u tc a n d th is a n d o th e r fe a tu r e s a r e d is c u s s e d in m o r e d e ta il in [ 1 6 ,1 7 ] . w ith th e tro u g h o f a s m u ta tio n ra te s u g h in e v a lu a tio n s h e re L = b in a ry is in s u p p o rt o f n d m a n y e m p iric a l h fie ld , O a te s e t a l, O f p a rtic u la r in te re s t h o w e v e r in th is p a p e r is th e a n o m a ly w h ic h c a n b e s e e le ft h a n d e d g e o f th e rig h t h a n d p e a k in F ig u re 2 a t lo w p o p u la tio n s iz e s (to w b a c k o f th e fig u re ). H e re , th e ris e in m e a n e v a lu a tio n n u m b e r w ith in c re a s in g c a n b e s e e n to b e n o t s m o o th , w ith a rid g e fe a tu re a p p a re n t o v e r a ra n g e p o p u la tio n s iz e s . S u c h a n a n o m a ly w a s n o t s e e n a t lo w e r e v a lu a tio n lim its e x te n s iv e in v e s tig a tio n s o f s im p le ‘u n im o d a l’ s e a rc h s p a c e s s u c h a s p re s e n te ‘M a x -O n e s ’ p ro b le m . T h e re m a in d e r o f th is p a p e r e x p lo re s th is p h e n o m e n o n d e ta il, w ith s e c tio n 2 d e s c rib in g th re e o th e r te s t p ro b le m s a n d th e ir e x p e rim u p ; s e c tio n 3 d e s c r ib in g th e r e s u lts a t b o th 2 0 ,0 0 0 a n d 1 m illio n e v a lu a tio n s ; d ra w in g s o m e p re lim in a ry c o n c lu s io n s fro m th e s e re s u lts a n d s e c tio n s a c k n o w le d g in g s u p p o rt a n d re fe re n c e d m a te ria l. n o n th e a rd s th e m u ta tio n o f lo w e r , n o r o n d b y th e in m o re e n ta l s e t s e c tio n 4 5 a n d 6 2 M e th o d T o e x a m in e th is ra n g e o f ‘s ta n d v a lu e s , a n d a s e lim it a n d m o re r a n o m a ly in a rd ’ te s t p c o n d s e rie s e fin e d s c a le m o r ro b le o f e o f m e d e ta il, m s o v e x p e rim e u ta tio n a s e rie s o f e x p e rim e n ts w a s c o n d u c te d w ith a r s im ila r ra n g e s o f e x p e rim e n ta l p a ra m e te r n ts c o n d u c te d w ith a m u c h h ig h e r e v a lu a tio n ra te s . T h e firs t p ro b le m to b e lo o k e d a t w a s th e ‘M a x -O n e s ’ p ro b le m w h e re in a b in a ry s trin g , fitn e s s is c a lc u la te d to b e th e n u m b e r o f ‘1 ’s p re s e n t in th e c h ro m o s o m e . W ith s ta n d a rd re p re s e n ta tio n , s te a d y s ta te , s in g le 3 w a y to u rn a m e n t s e le c tio n [3 ], u n ifo rm c ro s s o v e r a n d s im p le p e r g e n e m u ta tio n , th is p re s e n ts a s im p le u n im o d a l s e a rc h la n d s c a p e w h ic h c a n b e a s c e n d e d b y e v e n th e s im p le s t ‘h illc lim b in g ’ a lg o rith m . 5 0 tria ls w e re ru n fo r e a c h p a ra m e te r s e ttin g , w h ic h ra n g e d fro m p o p u la tio n s iz e s o f 2 to 1 0 0 a n d p e r g e n e m u ta tio n r a te s fr o m 1 E -7 to .8 3 . M e a n e v a lu a tio n s to fir s t fin d th e b e s t s o lu tio n fo u n d in 2 0 ,0 0 0 e v a lu a tio n s w e r e n o te d . C h r o m o s o m e le n g th fo r r e s u lts s h o w n h e re w a s 5 0 b its , h o w e v e r s im ila r e x p e rim e n ts h a v e a ls o b e e n d o n e a t 3 3 , 3 0 0 a n d 1 6 3 0 b its , g iv in g c o rre s p o n d in g re s u lts . T h e s e c o n d p ro b le m to b e in v e s tig a te d w a s a 6 4 b it im p le m e n ta tio n o f W a ts o n ’s H I F F ( H ie r a r c h ic a l I f a n d o n ly I f) p r o b le m [ 2 0 ,2 1 ] , w h ic h in c r e a s in g ly r e w a r d s e v e r la rg e r a lig n e d b lo c k s o f c o n tig u o u s ‘1 ’s o r ‘0 ’s a n d c a n b e re p re s e n te d a s : M u ltim o d a l P e rfo rm a n c e P ro file s f(B ) = 1 , |B | + f ( B L ) + f ( B f(B L ) + f(B R ), R 2 2 7 i f |B | = 1 i f ( | B | > 1 ) a n d ( "i { b i = 0 } o r "i { b i = 1 } ) , o th e rw is e ), w h e r e B i s a b l o c k o f b i t s , { b 1 , b 2 , … b n } , |B | i s t h e s i z e o f t h e b l o c k = n , b i i s t h e i t h e l e m e n t o f B , a n d B L a n d B R a r e t h e l e f t a n d r i g h t h a l v e s o f B ( i . e . B L = { b 1 , … b n /2 } , B R = { b n /2 + 1 , … b n } . N m u s t b e a n i n t e g e r p o w e r o f 2 . H -IF F th e re fo re h a s tw o g lo b a l o p tim a , o n e a t a ll ‘1 ’s a n d o n e a t a ll ‘0 ’s . T h e re a re s e c o n d a ry o p tim a a t s trin g s o f 3 2 ‘1 ’s fo llo w e d b y 3 2 ‘0 ’s a n d v ic e v e rs a , a n d a ra n g e o f s u b -o p tim a a t c o m b in a tio n s o f a lig n e d b lo c k s o f ‘1 ’s a n d ‘0 ’s e a c h o f le n g th 1 6 , 8 , 4 a n d 2 . T h e ‘s e a rc h la n d s c a p e ’ c a n b e c o n s id e re d to b e ru g g e d b u t in a h ig h ly s tru c tu re d fa s h io n . S im p le h illc lim b e rs p e rfo rm in a d e q u a te ly o n th is s u rfa c e a s th e b a s in o f a ttra c tio n o f th e g lo b a l o p tim a is v e ry s m a ll, a n d th e re a re m a n y lo c a l o p tim a w h e n s e e n b y s in g le p o in t m u ta tio n a lo n e . E v e n s ta n d a rd o n e -p o in t a n d tw o -p o in t c ro s s o v e r o p e ra to rs h a v e b e e n s e e n to re q u ire v e ry h ig h p o p u la tio n s iz e s to a c h ie v e re a s o n a b le a n d c o n s is te n t p e rfo rm a n c e o n th is p ro b le m u n le s s a n a p p ro p ria te d iv e rs ity m a in te n a n c e te c h n iq u e is e m p lo y e d . T h e th ird p ro b le m re p o rte d o n is th is p a p e r is a 5 0 -8 K a u fm a n N K la n d s c a p e [5 ] g e n e ra te d b y d e fin in g a ta b le o f ra n d o m n u m b e rs o f d im e n s io n 5 0 b y 5 1 2 . F o r a b in a ry c h ro m o s o m e o f le n g th N = 5 0 , s ta rtin g a t e a c h o f th e 5 0 g e n e p o s itio n s , K + 1 (h e re 9 ) w e ig h te d c o n s e c u tiv e g e n e s a re u s e d to g e n e ra te a n in d e x (in th e ra n g e 0 5 1 1 ), a n d th e 5 0 v a lu e s s o in d e x e d fro m th e ta b le a re s u m m e d to g iv e a fitn e s s v a lu e . F o r N K la n d s c a p e s w ith a K v a lu e o f 0 , e a c h g e n e p o s itio n c o n trib u te s in d iv id u a lly to fitn e s s , h e n c e p ro d u c in g a u n im o d a l la n d s c a p e to s in g le p o in t m u ta tio n h illc lim b e rs . H o w e v e r a s K is in c re a s e d , a n y s in g le p o in t m u ta tio n w ill a ffe c t K + 1 in d e x e s , ra p id ly in tro d u c in g p o s itio n a l lin k a g e in to th e p ro b le m a n d p ro d u c in g a ru g g e d a n d in c re a s in g ly u n s tru c tu re d la n d s c a p e . T h e A D D M P , M a x -O n e s , H -IF F a n d N K 5 0 -8 p ro b le m s w e re e v a lu a tio n s in th e firs t s e rie s o f e x p e rim e n ts , w ith re s u lts s h o w n re s p e c tiv e ly . A s e c o n d s e rie s o f e x p e rim e n ts w a s th e n p e rfo 1 ,0 0 0 ,0 0 0 e v a lu a tio n s , fo r A D D M P ( F ig u r e 6 ) o v e r a r a n g e o f 3 0 0 in s te p s o f 1 0 ) a n d fo r M a x -O n e s a n d H -IF F a t a fix e d (F ig u re s 7 a n d 8 ). F o r e a s ie r c o m p a ris o n o f re s u lts , th e A D D M P ru n s a t a p o p u la tio n s iz e o f 2 0 a re s h o w n m o re c le a rly in F ig u re a n d m in u s o n e s ta n d a rd d e v ia tio n s e e n o v e r th e 5 0 tria ls a n d s h o w n in e a c h o f th e fix e d p o p u la tio n s iz e g ra p h s . ll tr ia lle d a t 2 0 ,0 0 0 F ig u r e s 2 ,3 ,4 a n d 5 e d w ith a lim it o f o p u la tio n s iz e s (1 0 p o p u la tio n s iz e o f 2 0 1 ,0 0 0 ,0 0 0 e v a lu a tio n 9 . P lo ts s h o w in g p lu s m e a n fitn e s s a re a ls o U n w e c ro th e D D t G o in b le le s s re c sso v o b v o th o n d e r. io u e rw is e u c te d T h e H s in a p p s ta te d , a ll e x w ith a s te a d -IF F e x p e rim ro p ria te n e s s p e rim y s ta e n ts o f th e e n ts te , 3 w e re u n if (w ith w a y c o n d o rm o th e e x c s in g le u c te d u p e ra to r e p tio n o f to u rn a m s in g o n e o n th is p A e n -p ro a in rm p M P a t 2 0 K e v a ls ) A u s in g u n ifo rm t c ro s s o v e r d u e to m . M . O a te s , D . C o rn e , a n d R . L o a d e r Mean Evaluations 12000 20000 18000 16000 10000 10 50 90 130 170 210 250 8000 6000 20 80 140 200 12000 10000 8000 260 6000 320 4000 380 330 2000 440 370 0 500 410 290 4000 14000 Pop Size 14000 18000-20000 16000-18000 14000-16000 12000-14000 10000-12000 8000-10000 6000-8000 4000-6000 2000-4000 0-2000 2E 0 -0 8E 7 3. 07 2E 1. -0 28 6 E 5. -0 12 5 E 0. -0 00 5 0 0. 20 00 5 0 0. 81 00 9 3 0. 27 01 7 3 0. 10 05 7 2 0. 42 20 9 9 0. 71 83 5 88 61 16000 16000-18000 14000-16000 12000-14000 10000-12000 8000-10000 6000-8000 4000-6000 2000-4000 0-2000 Mean Evaluations 18000 Pop Size 2 2 8 2000 450 490 Mutation 2E 0 -0 8E 7 3. -0 7 2E 1. -0 28 6 E 5. -0 12 5 0. E-0 00 5 0 0. 20 00 5 0 0. 81 00 9 3 0. 27 01 7 3 0. 10 05 7 2 0. 42 20 9 9 0. 71 83 5 88 61 0 Mutation Rate F ig . 2 . A D D M P a t 2 0 K e v a lu a tio n s F ig . 4 . H -IF F 6 4 a t 2 0 K e v a lu a tio n s 18000 16000 12000 14000 2 16 30 8000 6000 44 58 4000 Pop Size 10000 12000 10000 8000 2 14 26 38 50 6000 4000 62 72 Mutation Rate F ig . 3 . O n e M a x a t 2 0 K e v a lu a tio n s 86 98 0 3. 2E 2. 06 56 E05 0. 00 02 05 0. 00 16 38 0. 01 31 07 0. 10 48 5 0. 8 83 88 61 100 0 0 74 -0 7 86 2000 4E 2000 0 4E -0 3. 7 2E 2. -06 56 E 0. -05 00 02 0 0. 00 5 16 3 0. 01 8 31 0 0. 10 7 48 5 0. 83 8 88 61 Mean Evaluations 14000 16000-18000 14000-16000 12000-14000 10000-12000 8000-10000 6000-8000 4000-6000 2000-4000 0-2000 16000 Mean Evaluations 18000 18000-20000 16000-18000 14000-16000 12000-14000 10000-12000 8000-10000 6000-8000 4000-6000 2000-4000 0-2000 Mutation F ig . 5 . N K 5 0 -8 a t 2 0 K e v a lu a tio n Pop Size 20000 M e a n E v a ls +sd -s d F itn e s s 1000000 Mean Evaluations Pop 150 Size 10 8 7 5 0 0 0 -1 0 0 0 0 0 0 7 5 0 0 0 0 -8 7 5 0 0 0 6 2 5 0 0 0 -7 5 0 0 0 0 5 0 0 0 0 0 -6 2 5 0 0 0 3 7 5 0 0 0 -5 0 0 0 0 0 2 5 0 0 0 0 -3 7 5 0 0 0 1 2 5 0 0 0 -2 5 0 0 0 0 0 -1 2 5 0 0 0 290 800000 350 400000 300 200000 250 0 200 -2 0 0 0 0 0 150 M u ta tio n R a te e v a lu a tio n s F ig . 8 . H -IF F 6 4 a t 1 M e v a ls , P o p s iz e = 2 0 3. 2E -0 6 2. 56 E05 0. 00 02 05 0. 00 16 38 0. 01 31 07 0. 10 48 58 0. 83 88 61 4E 0 40 -0 7 0 M u ta t io n R a t e e v a ls , p o p s iz e = 2 0 0 -2 0 0 0 0 0 M u ta t io n R a t e F ig . 9 . A D D M P a t 1 M e v a ls , p o p s iz e = 2 0 M e a n E v a ls +sd -s d F itn e s s 2 2 9 F ig . 7 . O n e M a x a t 1 M 200000 1 42 400000 0. 1 44 600000 0. 01 200000 46 0. 00 1 400000 48 0. 00 01 R a w E v a ls +sd -sd F itn e s s 600000 M u ltim o d a l P e rfo rm a n c e P ro file s 800000 3500 3000 2500 2000 1500 1000 500 0 800000 1E -0 7 0. 00 00 01 0. 00 00 1 50 Mean Evaluations 1000000 1000000 Mean Evaluations 400 600000 M u t a t io n R a te F ig . 6 . A D D M P a t 1 M 450 1. 00 E 3. -0 36 7 E1. 0 13 7 E 3. -0 81 6 E 1. -0 28 6 E4. 0 31 5 E 1. -0 45 5 E 4. -0 87 4 E1. 0 64 4 E 5. -0 51 3 E 1. -0 85 3 E 6. -0 23 2 E 2. -0 10 2 E 7. -0 05 1 E01 1. 0 6E 5. -06 12 E 0. -05 00 16 3 0. 05 8 24 29 Mean Evaluations 1000000 875000 750000 625000 500000 375000 250000 125000 0 2 3 0 M . O a te s , D . C o rn e , a n d R . L o a d e r 3 R e s u lts A n in te rp re ta tio n o f F ig u re 2 h a s a lre a d y b e e n g iv e n in S e c tio n 1 o f th is p a p e r, a n d a s im ila r p e r fo r m a n c e p r o file a t 2 0 ,0 0 0 e v a lu a tio n s c a n a ls o b e s e e n fo r th e M a x -O n e s p ro b le m in F ig u re 3 . A g a in , a t lo w p o p u la tio n s iz e s a n d lo w m u ta tio n ra te s , th e s e a rc h is s e e n to s ta ll a fte r v e ry fe w e v a lu a tio n s , a s d iv e rs ity in th e in itia l p o p u la tio n is ra p id ly d e p le te d . V e ry lo w ra te s o f m u ta tio n p re v e n t re -o c c u rre n c e o f p o te n tia lly u s e fu l a lle le s a n d th u s th e m e a n n u m b e r o f e v a lu a tio n s u s e d is v e ry lo w . A s m u ta tio n ra te s in c re a s e , th e s e a rc h is a b le to p ro g re s s to h ig h e r m e a n n u m b e rs o f e v a lu a tio n s , u n til a p o in t is re a c h e d w h e re m u ta tio n ra te s a llo w , o n a v e ra g e , m o s t o f th e e v a lu a tio n s to b e u tilis e d in th e s e a rc h fo r g o o d s o lu tio n s . A s m u ta tio n ra te s in c re a s e fu rth e r, th e s e g o o d s o lu tio n s a re fo u n d in le s s a n d le s s e v a lu a tio n s , u n til th e tro u g h p o in t in F ig u re 3 is re a c h e d w h e re g lo b a l o p tim u m s o lu tio n s a re fo u n d in th e m in im u m n u m b e r o f e v a lu a tio n s . B e y o n d th is ra te o f m u ta tio n , p e rfo rm a n c e d e te rio ra te s e v e n tu a lly d e g e n e ra tin g in to ra n d o m s e a rc h . H o w e v e r in c o n tra s t to A D D M P (F ig 2 ), th is c a n b e s e e n to d e te rio ra te s m o o th ly , w ith n o a n o m a ly o n th e le ft h a n d e d g e o f th e rig h t h a n d p e a k . T h e m u ta tio n ra te in d u c in g th e firs t p e a k c a n b e s e e n to b e e ffe c tiv e ly in d e p e n d e n t o f p o p u la tio n s iz e (o v e r its v is ib le ra n g e ) a n d o c c u r s a t a r a te a r o u n d .0 0 0 1 . T h e tr o u g h r a te is c e n tr e d a r o u n d a m u ta tio n r a te o f .0 5 (= 5 % c h a n c e o f a p p ly in g a N e w R a n d o m A lle le (la te r re fe rre d to a s N R A ), th is c o u ld b e w r itte n a s 2 .5 % c h a n c e o f a p p ly in g ‘g u a r a n te e d flip ’ m u ta tio n 1 / 5 0 ). F ig u r e 4 s h o w s th e p e r fo r m a n c e p r o file fo r th e 6 4 b it H -I F F p r o b le m a g a in a t 2 0 ,0 0 0 e v a lu a tio n s . H e re , in s ta rk c o n tra s t to F ig u re 3 , a th ird p e a k c a n b e s e e n in th e p e rfo rm a n c e p ro file , p a rtic u la rly a t lo w p o p u la tio n s iz e s . It is b e lie v e d th a t th e le ft h a n d tro u g h c o rre s p o n d s to a lo w m u ta tio n ra te w h ic h re p re s e n ts a lo w e r b o u n d th a t is u s e fu l fo r fin d in g a c e rta in s e t o f s u b -o p tim a in th e p ro b le m s p a c e – fo r e x a m p le b lo c k s o f s iz e 4 . H o w e v e r, a s th e m u ta tio n ra te is in c re a s e d , th is ra te b e c o m e s s u b o p tim a l fo r th is b lo c k s iz e , a n d a n o th e r m u ta tio n ra te is fo u n d w h ic h p ro v e s u s e fu l in fin d in g la rg e r b lo c k s iz e s – p o s s ib ly o f s iz e 8 o r 1 6 . T h is s u p p o s itio n is s u p p o rte d b y th e fa c t th a t a p lo t o f m e a n fitn e s s e s fo u n d a g a in s t m u ta tio n ra te s h o w s m a rk e d s te p im p ro v e m e n ts b e tw e e n e a c h o f th e s e tro u g h s in e v a lu a tio n s u s e d , im p ly in g th a t th e E A w a s s u d d e n ly a b le to fin d b e tte r o p tim a a t e a c h o f th e s e s p e c ific m u ta tio n ra te b o u n d a rie s . T h is o b s e rv a tio n c a n b e s e e n in F ig u re 8 a n d is n o w th e s u b je c t o f fu rth e r in v e s tig a tio n w h ic h w ill b e re p o rte d o n in d u e c o u rs e . It is lik e ly th a t th e re e x is t o th e r ‘o p tim a l’ m u ta tio n ra te s fo r o th e r b lo c k s iz e s in th is p ro b le m , b u t th e s e a re n o t d is tin g u is h a b le a t th is s c a le o f re s o lu tio n o n th e m u ta tio n ra te a x is a n d th is is a g a in fu rth e r e x p lo re d in th e d is c u s s io n o f F ig u re 8 . A s m u ta tio n is in c re a s e d b e y o n d a n u p p e r th re s h o ld , p e rfo rm a n c e is s e e n to d e te rio ra te , a s in th e p re v io u s tw o p ro b le m c a s e s , in to ra n d o m s e a rc h . T h e m u ta tio n ra te in d u c in g th e firs t a n d s e c o n d p e a k s a re s e e n to b e .0 0 0 1 a n d .0 1 3 r e s p e c tiv e ly . F ig u r e 5 s h o w s th e 2 0 ,0 0 0 e v a lu a tio n p e r fo r m a n c e p r o file fo r th e N K 5 0 -8 p r o b le m . H e re it c a n b e s e e n , a s w a s a ls o v is ib le in th e A D D M P p ro file (F ig u re 2 ), th a t th e d e te rio ra tio n o f p e rfo rm a n c e fro m th e o p tim u m tro u g h is n o t s m o o th , w ith a ‘rid g e - M u ltim o d a l P e rfo rm a n c e P ro file s 2 3 1 lik e ’ a n o m a ly a g a in p re s e n t o n th e le ft h a n d e d g e o f th e rig h tm o s t p e a k o v e r a ra n g e o f lo w e r p o p u la tio n s iz e s . F ig u re 6 s h o w s a p e rfo rm a n c e p ro file fo r th e s a m e A D D M P in s ta n c e a s F ig u re 2 , h o w e v e r th is tim e th e E A is a llo w e d 1 ,0 0 0 ,0 0 0 e v a lu a tio n s o v e r a p o p u la tio n r a n g e o f 1 0 th ro u g h 3 0 0 in s te p s o f 1 0 . In fa c t re s u lts w e re o b ta in e d a t e a c h o f fifty th o u s a n d , o n e h u n d re d th o u s a n d , tw o h u n d re d th o u s a n d a n d fiv e h u n d re d th o u s a n d e v a lu a tio n s , w ith th e o b s e rv e d a n o m a ly s im p ly b e c o m in g m o re d is tin c t a s e v a lu a tio n lim it in c re a s e d . In d e e d , th e m u ta tio n ra te in d u c in g th e firs t p e a k , firs t tro u g h a n d a n o m a ly fe a tu re s w e re s e e n to d e c re a s e w ith in c re a s e d e v a lu a tio n lim it, im p ly in g a re la tio n s h ip to to ta l n u m b e r o f m u ta tio n s u tilis e d a s w e ll a s a c tu a l m u ta tio n ra te . T h is o b s e rv a tio n is a ls o c u rre n tly th e s u b je c t o f fu rth e r in v e s tig a tio n a n d w ill b e re p o rte d o n in d u e c o u rse . A s c a n c le a rly b e s e e n in F ig u re 6 , th e m u ta tio d r o p p e d fr o m 4 E -4 a t 2 0 ,0 0 0 e v a lu a tio n s ( F ig u r e 2 T h e s e c o n d a ry a n d te rtia ry p e a k s ‘e m e rg e d ’ fro m p e a k w ith h ig h e r e v a lu a tio n s lim its u n til h e re a t 1 b e s e e n to b e p e rfo rm a n c e la n d s c a p e fe a tu re s in th ra n g e o f p o p u la tio n s iz e s . In d e e d , th is p lo t s h o w s IF F p ro b le m (s e e F ig u re s 4 a n d 8 ). n ra te in d u c in g th e fir ) to 1 .3 E -5 a t 1 m illio n th e le ft h a n d e d g e o f th m illio n e v a lu a tio n s th e y e ir o w n rig h t, p e rs is te n t s im ila ritie s w ith re s u lts st p e a k h a s e v a lu a tio n s . e rig h t h a n d c a n c le a rly o v e r a w id e fro m th e H - F ig u re 7 s h o w s th e 2 d im e n s io n a l p lo t a t a fix e d p o p u la tio n s iz e o f 2 0 o n th e ‘M a x O n e s ’ p r o b le m a t 1 ,0 0 0 ,0 0 0 e v a lu a tio n s . H e r e , th e p o s itio n o f th e fir s t p e a k c a n b e s e e n to b e a t a c o n s id e ra b ly lo w e r m u ta tio n ra te th a n in F ig u re 3 . T h e tro u g h o f lo w e v a lu a tio n s is s e e n to b e m u c h w id e r, in d ic a tin g a w id e r ra n g e o f m u ta tio n ra te s c a p a b le o f fin d in g th e g lo b a l o p tim u m in a lo w n u m b e r o f e v a lu a tio n s . F ig u re 7 a ls o s h o w s p lo ts o f th e m e a n n u m b e r o f e v a lu a tio n s p lu s a n d m in u s 1 s ta n d a rd d e v ia tio n o f re s u lts o v e r th e 5 0 ru n s . A s c a n b e s e e n , w h e re m u ta tio n ra te s a re to o lo w , p ro c e s s v a ria tio n is h ig h , b u t a s m u ta tio n ra te s a p p ro a c h o p tim u m (th e rig h t h a n d e d g e o f th e le ft h a n d p e a k ), th is v a ria tio n is re d u c e d . O n c e m u ta tio n ra te s b e c o m e c o u n te r p ro d u c tiv e (a b o v e 2 0 % N R A ), d iv e rs ity is s e e n to in c re a s e a g a in a s th e E A d e g e n e ra te s in to ra n d o m s e a rc h . A ls o s h o w n is th e m e a n fitn e s s o f th e b e s t s o lu tio n o f e a c h o f th e 5 0 ru n s a t e a c h m u ta tio n ra te . T h e p la te a u re p re s e n tin g ru n s fin d in g th e g lo b a l o p tim u m 5 0 tim e s o u t o f 5 0 is s e e n to c o in c id e w ith th e tro u g h in e v a lu a tio n s in d ic a tin g a n id e a lly tu n e d E A . F ig u re 8 s h o w s a 2 d im e n s io n a l p e rfo rm a n c e p ro file a t 1 m illio n e v a lu a tio n s o n th e H IF F 6 4 p ro b le m w ith a p o p u la tio n s iz e o f 2 0 . H e re 3 d is tin c t p e a k s a n d tro u g h s c a n b e s e e n b e fo re d e te rio ra tio n in to ra n d o m s e a rc h , w ith s p e c ific p e a k s a t a m u ta tio n ra te o f 1 E -6 , a n o th e r a t 1 .6 E -3 a n d a th ir d a t 4 .4 E -2 . T h is is a n in c r e a s e in th e n u m b e r o f fe a tu r e s s e e n a t 2 0 ,0 0 0 e v a lu a tio n s ( F ig u r e 4 ) , a n d a g a in it is n o te d th a t th e fe a tu r e s o c c u r a t lo w e r m u ta tio n ra te s g iv e n th e h ig h e r e v a lu a tio n lim it. It c a n a ls o b e c le a rly s e e n o n th e le ft m o s t tw o tro u g h s th a t th e s ta n d a rd d e v ia tio n in th e tro u g h s is a t a d is tin c t m in im u m o n th e le ft h a n d s id e o f e a c h tro u g h in e v a lu a tio n s . A ls o , th e a v e ra g e 2 3 2 M . O a te s , D . C o rn e , a n d R . L o a d e r fitn e s s p lo t c le a rly s h o w s m a rk e d in c re a s e s b e tw e e n th e tro u g h s s h o w in g th a t th e E A is a b le to fin d s te p -lik e im p ro v e m e n t a s m u ta tio n ra te s a re in c re a s e d . F in a lly , F ig u re 9 s h o w s th e 2 d im e n s io n a l p e rfo rm a n c e p ro file fo r o u r A D D M P s c e n a rio a t 1 m illio n e v a lu a tio n s w ith a p o p u la tio n s iz e o f 2 0 . W h ils t th e firs t p e a k a n d tr o u g h a r e c le a r ( a t m u ta tio n r a te s o f 1 .3 E - 5 a n d 8 .2 E - 4 r e s p e c tiv e ly , th e s e c o n d a n d th ird p e a k a re le s s d is tin c t, a lth o u g h c le a rly e x is t a s fe a tu re s in F ig u re 6 . T h e re a re c le a r ly r e g io n s o f lo w s ta n d a r d d e v ia tio n c o r r e s p o n d in g to tr o u g h s a t 8 .2 E - 4 , 2 .5 E - 3 a n d 0 .2 , a n d a r e g io n o f h ig h e r m e a n a n d m u c h h ig h e r d e v ia tio n b e tw e e n m u ta tio n r a te s o f 2 .5 E -3 a n d 0 .1 6 . T h is r e la tiv e d is o r d e r is to b e e x p e c te d a s a n y s tr u c tu r e in th e s e a rc h s p a c e is lik e ly to b e fa r le s s re g u la r in th e A D D M P s c e n a rio th a n in th e c a s e o f H -IF F . N o n e th e le s s , F ig u re s 6 a n d 9 c le a rly s h o w th e re to b e s o m e fe a tu re s fo r w h ic h k e y ra te s o f m u ta tio n a re e ith e r h ig h ly o p tim a l o r h ig h ly s u b -o p tim a l. T h e p lo t o f m e a n fitn e s s d o e s n o t s h o w s u c h c le a rly d e fin e d ‘s te p -lik e ’ im p ro v e m e n t a s in F ig u re 8 , h o w e v e r o n c e a g a in it c a n b e s e e n th a t th e re is g e n e ra l im p ro v e m e n t to w a rd s h ig h e r m u ta tio n ra te s . In te re s tin g ly , a n d in c o n tra s t to o th e r re s u lts s h o w n h e re , fitn e s s d e te rio ra tio n a t e x c e e d in g ly h ig h ra te s o f m u ta tio n is n o t a s m a rk e d , a g a in s u g g e s tin g a la c k o f s tru c tu re in th e s e a rc h s p a c e , fa v o u rin g a ‘ra n d o m s e a rc h ’ lik e p ro c e s s . O f c o u rs e th e m a rk e d in c re a s e in b o th m e a n a n d s ta n d a rd d e v ia tio n o f th e n u m b e r o f e v a lu a tio n s ta k e n to fin d th e s e g o o d s o lu tio n s s h o w s th e re la tiv e in e ffic ie n c y o f ra n d o m s e a rc h w ith re s p e c t to a w e ll tu n e d E A o n th is p ro b le m . T h e id e a l m u ta tio n r a te h e r e b e in g 0 .3 1 ( N R A ) in d u c in g h ig h f itn e s s s o lu tio n s , in a m in im u m o f e v a lu a tio n s w ith a m in im u m o f p ro c e s s v a ria tio n . 4 C o n c lu s io n s F o r th e s im p le u m o d e ra te ra te o f b o th in te rm s o e v a lu a tio n s ta k e n ra n g e o f s u ita b le n im o d a l s e a rc h s p a c e m u ta tio n (u s u a lly c lo s f c o n s is te n t q u a lity o to fin d th e m . W h e re a m u ta tio n ra te s e x te n d s e x p lo re d h e to 1 / L ) f s o lu tio n h ig h e r n u m to w a rd s e v e e re , c a n fo u n b e r r lo w it p r d o f e c a n c le a rly o d u c e o p tim a n d m in im e v a lu a tio n s r v a lu e s . F o r th e s tru c tu re d m u lti-m o d a l s e a rc h s p a c e , th e re a p p e a r to b e a m u ta tio n ra te s , c a p a b le o f e x p lo itin g c e rta in fe a tu re s o f th e s e a rc h o f H -IF F th is is lik e ly to b e a lig n e d b lo c k s o f c o n s ta n t le n g th e a c h H a m m in g d is ta n c e . V a lu e s a ro u n d th e s e ‘o p tim a l’ ra te s a re s e e n o p tim a l in te rm s o f p ro c e s s re p e a ta b ility (in c re a s e d m e a n a n d s ta n ‘g lo b a lly o p tim u m ’ m u ta tio n ra te a g a in s e e m s to e x is t c lo s e to o p tim u m fitn e s s s o lu tio n s , b u t in th is c a s e , n o t in a n o p tim a lly m e v a lu a tio n s (h o w e v e r it is p o s s ib le th a t fu rth e r e x p e rim e n ta tio n w m u ta tio n ra te ra n g e m a y s h o w th is n o t to b e th e c a s e ). b e se e n a l p e rfo u m n u m is a llo w th a t a rm a n c e b e r o f e d , th is s e rie s o f ‘o p tim a l’ s p a c e . In th e c a s e s e p a ra te d b y e q u a l to b e c le a rly s u b d a rd d e v ia tio n ). A 1 / L w h ic h g iv e s in im u m n u m b e r o f ith a fin e r g ra in e d T h e p e rfo rm a n c e p ro file o f th e A D D M P h a s b e e n s h o w n to e x h ib it s o m e fe a tu re s in c o m m o n w ith b o th s tru c tu re d a n d u n s tru c tu re d m u lti-m o d a l s e a rc h s p a c e s . It is c le a rly m u ltim o d a l o v e r a ra n g e o f p o p u la tio n s iz e s w h e re s u ffic ie n t e v a lu a tio n s a re M u ltim o d a l P e rfo rm a n c e P ro file s p e rm itte d . T h e e x is te n c e o f k e n u m b e r o f e v a lu a tio n s is h ig h ly th is p ro b le m d o m a in a llo w in g T h e fa c t th a t o n c e a g a in a m u s o lu tio n s in a lo w n u m b e r o f e a llo w e d ra n g e o f e v a lu a tio n s w re a l tim e a p p lic a tio n o f th e s e te 2 3 3 y m u ta tio n ra te s in d u c in g a lo w s ta n d a rd d e v ia tio n in s ig n ific a n t w h e n c o n s id e rin g th e a p p lic a tio n o f E A s to a h ig h d e g re e o f c o n fid e n c e in p ro c e s s re p e a ta b ility . ta tio n ra te c lo s e to 1 / L is s e e n to g iv e h ig h fitn e s s v a lu a tio n s w ith lo w s ta n d a rd d e v ia tio n , e v e n w h e n th e a s c o n s id e ra b ly h ig h e r, is o f c ritic a l im p o rta n c e to th e c h n iq u e s to a re a l w o rld p ro b le m d o m a in . 5 A c k n o w le d g e m e n ts T h e a u th o rs a re g ra te fu l to B ritis h T e le c o m m u n ic a tio n s P lc fo r o n g o in g s u p p o rt fo r th is re s e a rc h . R e fe r e n c e s 1 . T B ä c k , E v o lu tio n a r y A lg o r ith m s in T h e o r y a n d P r a c tic e , O x fo rd U n iv e rs ity P re s s , 1 9 9 6 2 . K D e b a n d S A g ra w a l : U n d e r s ta n d in g In te r a c tio n s a m o n g G e n e tic A lg o r ith m in F o u n d a tio n s o f G e n e tic A lg o rith m s 1 9 9 8 , M o rg a n K a u fm a n n . P a r a m e te r s . 3 . D G o ld b e rg (1 9 8 9 ), G e n e tic A lg o r ith m s in S e a r c h O p tim is a tio n a n d M a c h in e L e a r n in g , A d d is o n W e s le y . 4 . J H o lla n d , A d a p ta tio n in N a tu r a l a n d A r tific ia l S y s te m s , M IT p re s s , C a m b rid g e , M A , 1 9 9 3 5 . K a u ffm a n , S .A ., T h e O r ig in g s o f O r d e r : S e lf- O r g a n iz a tio n O x fo rd U n iv e rs ity P re s s , 1 9 9 3 a n d S e le c tio n in E v o lu tio n , 6 . Z M ic h a le w ic z , G e n e tic A lg o r ith m s + 1 9 9 6 . 7 . H M ü h le n b e in a n d D S c h lie rk a m p -V o o s e n (1 9 9 4 ), T h e S c ie n c e o f B r e e d in g a n d its a p p lic a tio n to th e B r e e d e r G e n e tic A lg o r ith m , E v o lu tio n a ry C o m p u ta tio n 1 , p p . 3 3 5 -3 6 0 . 8 . H M ü h le n b e in , H o w g e n e tic a lg o r ith m s r e a lly w o r k : I. M u ta tio n a n d h illc lim b in g , in R . M a n n e r , B . M a n d e r i c k ( e d s ) , P r o c . o f 2 n d I n t ’l C o n f e r e n c e o n P a r a l l e l P r o b l e m S o l v i n g fro m N a tu re , E ls e v ie r, p p 1 5 -2 5 . 9 . E v a n N im w e g e n a n d J S iz e In d e p e n d e n t T h e o C o m p u te r M e th o d s in a n d G e n e tic A lg o rith m D e b , e d ito rs , 1 9 9 8 . D a ta S tr u c tu r e s = C ru tc h fie ld : O p tm iz in g r y , S a n ta F e In s titu te W A p p lie d M e c h a n ic s a n d s in C o m p u ta tio n a l M e c E p o c h a l E o rk in g P a p E n g in e e rin h a n ic s a n d E v o lu tio n P r o g r a m s , S p rin g e r, v o lu tio n a r y e r 9 8 -0 6 -0 4 g , s p e c ia l is E n g in e e rin g S e a rc h 6 , a ls o su e o n , D G o : P su E v ld b o p u la tio b m itte d o lu tio n a e rg a n d n to ry K 1 0 . E v a n N im w e g e n a n d J C ru tc h fie ld : O p tm iz in g E p o c h a l E v o lu tio n a r y S e a r c h : P o p u la tio n S iz e D e p e n d e n t T h e o r y , S a n ta F e In s titu te W o rk in g P a p e r 9 8 -1 0 -0 9 0 , a ls o s u b m itte d to M a c h in e L e a rn in g , 1 9 9 8 . 2 3 4 1 1 . M M . O a te s , D . C o rn e , a n d R . L o a d e r O a te s , D C o rn e a n d R L o a d e r, In v e s tig a tin g E v o lu tio n a r y A p p r o a c h e s fo r S e lf-A d a p tio n in L a r g e D is tr ib u te d D a ta b a s e s , in P ro c e e d in g s o f th e 1 9 9 8 IE E E IC E C , p p . 4 5 2 -4 5 7 . 1 2 . M O a te s a n d D C o rn e , Q o S b a s e d G A P a r a m e te r S e le c tio n fo r A u to n o m o u s ly M a n a g e d D is tr ib u te d In fo r m a tio n S y s te m s , in P ro c s o f E C A I 9 8 , th e 1 9 9 8 E u ro p e a n C o n fe re n c e o n A rtific ia l In te llig e n c e , p p . 6 7 0 -6 7 4 . 1 3 . M O a te s a n d D C o rn e , In v e s tig a tin g E v o lu tio n a r y A p p r o a c h e s to A d a p tiv e D a ta b a s e M a n a g e m e n t a g a i n s t v a r i o u s Q u a l i t y o f S e r v i c e M e t r i c s , L N C S , P r o c s o f 5 th I n t l C o n f o n P a ra lle l P ro b le m S o lv in g fro m N a tu re , P P S N -V (1 9 9 8 ), p p . 7 7 5 -7 8 4 . 1 4 . M O a te s , A u to n o m o u s M a n a g e m e n t o f D is tr ib u te d In fo r m a tio n S y s te m s u s in g E v o lu tio n a r y C o m p u tin g T e c h n iq u e s , C o m p u tin g A n tic ip a to ry S y s te m s , A IP C o n f P ro c s 4 6 5 , 1 9 9 8 , p p . 2 6 9 -2 8 1 . 1 5 . M O a te s , D C o rn e a n d R L o a d e r, S k e w e d C r o s s o v e r a n d th e D y n a m ic D is tr ib u te d D a ta b a s e P r o b le m , A rtific ia l N e u ra l N e tw o rk s a n d G e n e tic A lg o rith m s 1 9 9 9 , D o b n ik a r e t a l (e d s ), S p rin g e r p p 2 8 0 -2 8 7 . 1 6 . M O a te s , D C o rn e a n d R L o a d e r , In v e s tig a tio n o f a C h a r a c te r is tic B im o d a l C o n v e r g e n c e tim e /M u ta tio n -r a te F e a tu r e in E v o lu tio n a r y S e a r c h , in P ro c s o f C o n g re s s o n E v o lu tio n a ry C o m p u ta tio n 9 9 V o l 3 , IE E E , p p . 2 1 7 5 -2 1 8 2 1 7 . O a te s M , C o rn e D a n d L o a d e r R , V a r ia tio n in E v o lu tio n a r y A lg o r ith m P e r fo r m a n c e C h a r a c te r is tic s o n th e A d a p tiv e D is tr ib u te d D a ta b a s e M a n a g e m e n t P r o b le m , in P ro c s o f G e n e tic a n d E v o lu tio n a r y C o m p u ta tio n C o n fe r e n c e 9 9 , M o r g a n K a u fm a n n , p p .4 8 0 - 4 8 7 1 8 . M . O a te s , J . S m e d le y , D . C o rn e , R . L o a d e r, B im o d a l P e r fo r m a n c e P r o file o f E v o lu tio n a r y S e a r c h a n d th e E ffe c ts o f C r o s s o v e r , in P ro c s o f 1 9 9 9 E v o n e t S u m m e r S c h o o l o n T h e o re tic a l a s p e c ts o f E v o lu tio n a ry C o m p u ta tio n . 1 9 . G S y s w e rd a (1 9 8 9 ), U n ifo r m C r o s s o v e r in G e n e tic A lg o r ith m s , in S c h a ffe r J . (e d ), P ro c s o f th e T h ird In t. C o n f. o n G e n e tic A lg o rith m s . M o rg a n K a u fm a n n , p p . 2 – 9 2 0 . W a ts o n R A , H o rn b y G S , a n d P o lla c k J B , M o d e llin g B u ild in g -B lo c k In te r d e p e n d e n c y , L N C S , P r o c s o f 5 th I n t l C o n f o n P a r a l l e l P r o b l e m S o l v i n g f r o m N a t u r e , P P S N - V ( 1 9 9 8 ) , p p . 9 7 -1 0 6 . 2 1 . W a ts o n R A , P o lla c k J B , H ie r a r c h ic a lly C o n s is te n t T e s t P r o b le m s fo r G e n e tic A lg o r ith m s, in P ro c s o f C o n g re s s o n E v o lu tio n a ry C o m p u ta tio n 9 9 V o l 2 , IE E E , p p . 1 4 0 6 -1 4 1 3 r c l ic ic o l as s r c i Usi rc c iq s a rpls & Ia ak a ni rsit f uss a l r, right n, H nichs@cogs.susx.ac.uk & ianw@cogs.susx.ac.uk c h c nstructin ftra nsp rtpr t c l s hich ff r r l ia bl c unica tin is a c pl ica t d ta sk. h c unica ting a g nt ust quickl a da ptt cha ng s in th n t rk in rd r t a inta in pti a l p rf r a nc . his a da pti l ntis a n tr ldifficul tc p n nt t c nstructa s th highl d na ic n ir n ntin hich th pr t c l p ra t s isdifficul tt pr dict. ur rka tt ptst a ut a t th d sign pr c ss b c n rtingitfr a d sign pr bl t n f pti isa tin,in hichg n tic a l g rith s a r us d t s a rchth spa c fp ssiblpr t c l d signs in a n a tt ptt find th pti a ls l utin. pr s ntr sul ts fr p ri nts in hich ha l d al t rna tingbitpr t c l s a nd al s ind d fl c ntr l pr t c l s, hichha highcha nn l util isa tin. r ci is o rk pl o r s t po ssibil it o f a ppli g o ltio a r s a rc stra t gis to t s t sis o f t o rk co ica tio s pro to co l s. pri cipls a a o pt r o rigi a l lpio r it lo f o ltio a r o bo tics[3]a [2] ic s s g tic a l go rit s to s a rc t sig spa c o fro bo tco tro la rc it ct r s fo r s st s ic ibita sir b a io r. l ti a t go a lofo r r s a rc is to l o p a r sa bl t o o l o g o r co c pt a lfra o rk fo r t co str ctio o ft a a pti a l go rit s a pro to co l s pl o b tra spo rt l a r pro to co l s fo r pa ck ts itc t o rks. p ci ca l l a r fo c si go t l op to fco g stio co tro la a o ia c al go rit s s c a s t o s pl o b rl ia blpro to co l ss c a st ra spo rt o tro l ro to co l ( ). i c is so t si l s itis cla r t a ta s al li pro ti its a bil it to til is r so rc s a a a ptto c a g il lgr a tli pro gl o ba l t o rk p rfo r a c . H o r a - sig o ft s co c rr tpro c ss s ic st co -o p ra t o r a rl ia bla co ti a l lc a gi gco ica tio i is a p rpl i ga fo r i a blta sk. sig fla s a r iffic l tto pr ict a fo r t o stpa rt t ct a s pro bl s i t sig il lo lb co a ppa r t t al go rit is i pl t at ac o o ft t o rk. ta p r lso ft a r l lt I t r tisco po s o f a s a l lit ra cti g pro c ss s a c o f ic a s a go a lt a tis cla rl a tt l lo ft a t pro c ss: t a t o fco ica ti g a ta to a r c ii g p r pro c ss. t a ig r l lt co l lcti it ra ctio o ft s pro c ss s is a ir ctr s l to ft r ls S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 3 5 − 2 4 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 3 6 N . S h a rp le s a n d I. W a k e m a n t a tgo r a si glco ica tio . H o r lss a r ca r f la tt l o r l lt rg t a ic a tt a t ig r l l il lb pr icta bl la i g to t o rk co g stio a al l co l l a ps . o b ca r f l l op ta tt is l o r l l ca co tro lt ig r-l l rg t a ic. l o pi g a a pti tra spo rt co tro lpro to co l s fo r co ica tio s to rks a s l a rg a ig l a ic a s t I t r t a s i r ta co flicti g pro bl s. pri a r f ctio o fr l ia blco ica ti g a ta i o l s t co o r i a tio o fr o t pro c ss s. is is a iffic l tpro bl ic is co pl ica t f rt r t co ica tio i s ibits pr icta bll o ss c a ra ct ristics. i g to t is iffic l t co ica tio stb p rfo r i a ffici t a r. pro to co l st s t t o rk r so rc s i a fri l a r it r sp ctto p r pro to co l s ic is i co flict it t a i o f ffici c . It stb a blto t ctc a g s i t t o rk iro ta co s it r o r o r lss o fits r so rc s p i go t circ sta c s a ro it. pro to co l stsa tisf a l lo ft s a s it o tgl o ba li fo r a tio a bo tt sta t o ft t o rk a f rt r stil l it o tgl o ba lco tro lo r t pa t a pa ck t a ta k . f a ta lgo a loft is r s a rc is to l op a to ol og ic il l al l o t s t sis o fa a pti pro to co l sa pro i i sig ts ito t sig spa c o fs c a l go rit s. a t rt a a pro to co l sig r a -cra fti gt it sta t a c i a tt a rto ft pro to co l / s il lsp cif t so ltio spa c a a t ss f ctio ic il la ss ss so ltio s a ga i st t pro to co l r q ir ts. sig r il lt s tic a rc t c iq s to l o ca t so ltio s ic t s t t ss f ctio a so t t r q ir ts. I ctio 2 so o t a rio s co po ts o fo r to ol o g o rk to g t r to o l o rki gpro to co l s. o ft o sti po rta tf ctio s o fa co ica ti gs st is t a bil it to co ica t a ta r l ia bl si g a rl ia bl i . Ifa s st ca b l op i ic r l ia blpro to co l s ca b s t sis t it il lgo so a to pro i gt a t ig r-l lb a io rs a r a l so a tta i a bl si g t sa t c iq . I o r rst p ri t scrib i s ctio 3. s o t a tr l ia bl pro to co l s si g iff r tt c iq s s c a s a ck o l g ts a r a t tra s issio s a r o l r iff r tsi l a tio co itio s. fa cto r c a lca pa cit ito o r pro bl spa c c a l til isa tio b co s a i po rta t a s r o ft t ss o ft so ltio . I t s co p ri t scrib i ctio 3.2 s o t a t i o i g pro to co l s ar ol ic a t a bil it to r t c a la t % til isa tio . co cl it a isc ssio a gi po it rs a s to o pro to co l s ic a a ptto co g stio ca b ol . l t o rkca b i a s a irt a l iro t. pro to co lis a co ica ti g tit o ft a t iro t o s a ctio s a r icta t b its rli g it sta t a c i . i c i t is r sp cta pro to co lis sta co tro la rc it ct r itis cla r t a ti o r r to o l a a pti co ica tio pro to co l s ca s t P ro to c o l C o n s tru c tio n U s in g G e n e tic S e a rc h T e c h n iq u e s sa o ltio a r t c iq s a s t o s s a rc it ct r s. I o r r to ol t o rk pro to co l s fo l l o i gco po ts: i t pro r q ir t 2 3 7 ctio o fro bo tco tro l i pl ta tio o ft pr s ta tio o fpro to co l a rc stra t g co i gsc p ra to rs to a ip l a t co i g t o rksi l a tio it ss f ctio . pr s t t fpr t c r pr s ta tio a c o s to s is a t sio o f t it sta t ac i ( ) ca l l t co ica ti g it sta t a c i ( ). a c o s to s t a so rr pr s ta tio fo rt o i po rta tr a so s. irstl pro to co l s a r a sil r pr s t si gt a t is a s b a o pt b t pro to co lr s a rc co it a s a sta a r r pr s ta tio [6 ] [4]. co t as b s i co p t r sci c a a rti cia lit l l ig c fo r a a rs a its c a ra ct ristics a r l lo c t a rsto o . s a rc rs a s t a s a r pr s ta tio fo r o pti isa tio pro bl s fo r so ti si gbo t tra itio a la o ltio a r s a rc stra t gis [ ]. iff rs fro t i t a t it ca a ff ct a s l la s r a ct to t iro ti ic ito p ra t s. pa ss s ssa g s o r sig a l s ia bo I q s. a c ac i as a ip ta a o tp t q a c t s a t o -st p a l go rit . ri gt rstst p i p tsig a l s a r a la t a a tra sitio r ls lct . I t s co st p t o tp tsig a lis p a t a t a c i c a g s sta t . If o o ft r ls fo r a sta t ca b c t t a c i r a i s i its c rr tsta t . . rc tr t g a ta lto t is o rk is t a bil it to i t pro to co la s a co tro la rc it ct r ic ca b r pr s t i so s bo l ic fo r . is r pr s ta tio stb o t a tca b ta t o r s cc ssi it ra tio s t ro g t s a rc spa c o fpo ssiblso ltio s. is tra sfo r s t pro bl fro o o fco tro l lr l op t ito o o fco tro l lr t s t . ta sk o fg ra ti g co pl co tro la rc it ct r s si g s a rc a s b a cco pl is i o t r ls si g tic a l go rit s( s). il la tt pt to co str ctco ica tio pro to co l s si gsi il a r pri cipls. . c g c co i g sc s to r pr s t t cl o s lti to t s a rc stra t g pl o . so ltio i t pro bl spa c is o b io sc o ic o fr pr s ta tio 2 3 8 N . S h a rp le s a n d I. W a k e m a n si g g tic r pr s t b a s q b t sta a r g to a ria bll gt g o a r pr s g o is tra sfo r sig spa c . .4 al go rit s is a bi a r stri g i ic a c tra sitio is c o fbit a l s.I a itio to a l l o i g a s a ip l a tio tic o p ra to rs t is r pr s ta tio il la l so l its l f a sil co i g. bi a r co i g o ft is k o a s t ts a so ltio i t s a rc spa c . rio r to a la tio t ito t p o a i sta tia tio o f t so ltio i p r t rs t pu t c g g tic a l go rit pl o ri gt is r s a rc s s t sta a r g tic o p ra to rs i. . ta tio cro sso r a itio a s btra ctio . o r t sa k o f cl a rit t fo l l o i gs ctio s ta ilt a ct a li pl ta tio s . ut t . g r issp ci fo r a c r r l a ti gto t b ro f ta tio s p r g o . ri g a ti g a c biti t stri gis gi a i ii a lc a c o f ta tio a c c a c b i ga q a lpro po rtio o ft ta tio ra t fo r t is g o . a c biti t stri g is t c ck a ga i stt r ca l c l at ta tio ra t a a ta tio flips t bit a l . r ss r. a fo r bitba sis. o r a l lt cro sso r a s pro a a p rg ba sis r s o fcro sso r ista ca o p ra t o a p r g o r p r p ri ts ca rri o t ri g t is r s a rc si glpo it q at. o rt o stpa rtt is a s b i pl t o t cro sso r po itis r strict to g bo a ris. t u tr ct . a itio a s btra ctio o p ra to rs a l l o fo r a t o ro g pl o ra tio o ft s a rc spa c a i g o r r o i gg s fro t g o ra o l. itio a s a ra o l co str ct g to t g o . l t o g o s lctio pr ss r fo r si is itro c b t t ss f ctio t r a c il o g r g o s cr a t s ro b stso ltio s it r ga r ta tio . is is b ca s t o rki g g s o ft so ltio (t pa rts o ft it sta t a c i t a tco trib t to t ss) ca i t s l s a o gstt r a tg s o ft g o . . t r u t orl i a r o pti isa tio pro bl s a c ca i a t so ltio is a la t a ga i st a sta tic t ss f ctio ic pro i s a a s r o f its a bil it to so l t pro bl . o r o -l i a r pro bl s s c a s t o s fa c r st a la t a pro to co l ’ s f ctio a l it i its o rki g iro t. o r a a ppl ica tio s t is is i pra ctica l to t a o to fti r q ir fo r a a la tio a a co p t r o l stb s to si l a t t a g t’ s iro ta its it ra ctio it it. is is o tt ca s fo r t a ppl ica tio pro po s ; a la tio o f a pro to co l sp rfo r a c co lta k pl a c i a r a l t o rk it o t r a so a bl P ro to c o l C o n s tru c tio n U s in g G e n e tic S e a rc h T e c h n iq u e s 2 3 9 ti o r a a l t o g a tt is sta g r a l o rl a la tio o la cssa r co pl it. a tis r q ir t is a t stb i ic to a la t t p rfo r a c o fa pro to co l it i its o rki g iro t o r a s cl o s to t a t iro ta s is po ssibl. fo l l o i g p ri ts s a si pls r-r c i r t o rk si l a tio a a plo f ic ca b s i g r . o i itia t a si l a tio t o p o s a r co str ct : a s ra a r c i r. co po to f a c p o isg ra t fro t g o to b a la t . o t s ra r c i r a a a sso cia t co ica tio c a lt c a ra ct ristics o f ic a r p to t p ri tb i g p rfo r . pica l l ac p o al so a s a tra s issio b ff r a a r c i b ff r. t s i g a c i t tra sissio b ff r co ta i s t ssa g to tra s it. I a itio ac ac i al so a s so fo r o fc rr t or i ic a ta it s ca b l t po ra ril b fo r b i gpl a c ito b ff rs o r ssa g s. si l a tio c t s fo r a sp ci c b r o f ti st ps. ri g a c st p t s ra t t r c i rp o s g ta c a c to c t . ri g c tio ac a c i it ra t st ro g t tra sitio so fitsc rr tsta t fro rstto l a st. a ctio a sso cia t it t tra sitio is a la t ift a ctio ca b p rfo r t tra sitio rs. I ff ct a c tra sitio fo r s a bo o la pr ssio . Ift pr ssio o ls t t ac i a k s t tra sitio to t tsta t . t a c ti st p i t si l a tio r po ssibltra sitio o ft c rr tsta t is a tt pt tila c tio is ca pa blo f ri g. tt is po it t a c i c a g s sta t to t sta t i ica t b t t-sta t l o ft tra sitio . t r o rk il li o l l a rg rsi l a tio s i ic a si l at t o rk o s il lco ica t a cro ss a si gl s a r c a l it a c o pl o i gt sa pro to co la s i t r a l o rl. a la tio il lb a as r o f t gl o ba ltra ffic t ro g -p ta cro ss t c a l . o ti o rk a s t c rr tr a l - o rl I t r ti pl a c o ft si l a tio . .6 t ss u ct p rpo s o ft t ssf ctio isto a la t t p rfo r a c o ft pro to co l ri g its c tio i t t o rk si l a tio a pro i so q a ti a bl a s r o fitsf ctio a l it. a s r its l fs o li cl a sf co po ts a s po ssibl si c t gr a t r t b r o fco po ts t lss pr icta blt po p l a tio ’ s co rg c o t sir po iti so ltio spa c . is as r ca b ca l c l a t i t o a s:t o st iffic l tis to ga t r i fo r a tio ri g t si l a tio a ca l c l at t t ss fro t a ta co l lct ;a a sir o ptio is to as r t t ss o ft sta t o ft t o rk o s a ft r t si l a tio a s is . st b a r i i t a t a ss ssi g t a l o fa so ltio si g o l a si gl si l a tio o s o tpro c a a cc ra t a s r o fits a bil it si c it is po ssiblt a tt pro to co lis pl o iti gso a sp cto ft t st a ta s . I o r r to ga i a fa ir a s r o ft t ss o fa so ltio ac stb t st a b ro fti s si g iff r tt st a ta . stb ca r f l s lcti gt 2 4 0 N . S h a rp le s a n d I. W a k e m a n b r o fr -t sts to p rfo r gra a if s to o f la s s so c o ic a s to so fo r o f a . ro g a l g ra t al i ac i t is is t a l a s 3 p ri if s to o a r -t sts t p rfo r a c il l t so ltio il l o tb a q a t l t st . is ic a l s o l b c o s :t ig st l o sto r p ri ta tio fo t a t si g t a s i t la st b r o fg ra tio s. r fo r i t fo l l o i g p ri ts. s fo l l o i g s ctio s ta ilt o o f t p ri ts rta k to a t . I a c ca s t p ri t a s p rfo r si ga istrib t po p l a tio o ffo r r i ii a l s o l fo r o t o sa g ra tio s. ta sk a s to co ica t t a ta it s fro t s r to t r c i r. . t I o r i itia l p ri t s to tto s t sis a f l lr l ia bl co ica tio pro to co la i o i g so pro t a tt co tro la rc it ct r rli g a co ica tio pro to co lca b g ra t si g o ltio a r s a rc . o pro i a a s r o ft q a l it o ft s t sis pro to co l a c ca i a t so ltio a s a la t si g a t o rk si l a tio ic l ats t iro tfo r ic t a l t r a ti gbitpro to co lis a o pti a lso ltio . It a s o p t a tt s st o l pro c a a c i it si il a r sta t tra sitio s a t r a lb a io r. al t r a ti gbitpro to co lis a si plpro to co lt a tpro i s r l ia bl co ica tio si g a a l t r a ti g a l i a tio bit iti ica t s t ror ot a ssa g a s l o st ri g tra s issio . Itis l lo c t i t l it ra t r a a s rstpro po s i [ ]. t r u t . o g ra t a s r si il a r to t a to fo r b c a rk a c i t si l a tio stcl o s l o lt o p ra ti g iro to ft o rigi a lpro to co l . o r t is t c a l st a t ca pa cit to o l o l a si gl ssa g . I a cco r a c it t r fr c pro to co l ;o l ssa g s ca b l o sta ifa ssa g is l o st t c a l il lflip t biti t ssa g a r. o r a sc a tic o r i o ft si l a tio s gr . i ct s. fo l l o i ga ctio s a b tra po l a t fro i t o rigi a la l t r a ti gbitpro to co l . t a c i sp c- : a ssa g o t co ica tio s c a lta ki g a ta fro o r ifr q ir . ci : ci a ssa g fro t co ica tio s c a la pl ac t a ta ito or. q : l a c t it i o r ito t r c i b ff r. q : l ac t rst l to ft tra s itb ff r ito or. P ro to c o l C o n s tru c tio n U s in g G e n e tic S e a rc h T e c h n iq u e s Transmition Buffer Transmition Buffer 0:4 0:3 0:2 0:1 2 4 1 Channel 0:0 Data FSM FSM Channel Receive Buffer Receive Buffer Data bit : seq Memory t rksi ul a tin f r l l: Memory l utin fr l ia blc unica tingfinit sta t a chin . l la ctio a la s o rks a c o ft s a ctio s r t r s a tr o r fa l s bo o la a l p i go its a bil it to co plt t sp ci ta sk. l fo r t s a ctio is t is o tt ca s it il lr t r tr r al lco itio s to pr t it tio a lf ba ck fro t r c ii g a c i . I t ca s r t c a la l ra co ta i s a ssa g t ssa g is isca r . t ss u ct . t ss fo r a si glpro to co lis ba s o t pro po rtio o fa ssa g r c i a ft r a b r o fti st ps. o s r a rl ia bl pro to co lis pro c o lt i -o r r po rtio o ft r c i ssa g is co si r . g o il lsco r ro ifa l lb tt rst ssa g a s s cc ssf l l r c i . Ift is a s o tt ca s a t tir pro po rtio o ft ssa g a s co si r t t po p l a tio o l pr a t r l co rg o a rl ia bl so ltio a b co i ca pa bl o f ba ck-tra cki g t ro g t s a rc spa c to rl ia blso ltio s. su ts. so i g r 3 asb g ra t ri ga t pica lr o ft s st . s it a t pica lsta t tra sitio ia gra ac o r pr s ts a sta t . a c tra sitio is r pr s t a s a a rc fro t sta t i ic itis c t to t t tsta t . tra sitio is a rk it t a ctio to b ta k a t pa ra t rsfo rt a ta ctio . o a ctio sr q ir o pa ra t rs i ic ca s t is r a t co i g i t tra sitio a s tra l it to t g o : i cr a si g its ro b st ss it r ga r to ta tio . tra sitio s a r a rk it t o r r i ic t a r to b a tt pt s o i sq a r bra ck ts. s ca b s i t ia gra t s r co ti a l ls s a ssa g it i ti r “ ”. la ft r r c ii ga a ck o l g tfro t r c i r o s it q t t a ta it . is is a si plsto p-go stra t g a is t cl o s st a co to g ra ti g t a l t r a ti g bitpro to co l . o t : a c tra sitio a s t sa b r o fpa ra t rs a l t o g fo r q q a l l t s a r s p rfl o s a isca r ri g c tio . 2 4 2 N . S h a rp le s a n d I. W a k e m a n 1 Fitness Value 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 Generations itn ss a l u bta in d b b stindiidua l during 0 N u ll a c k 0 [2 ] D e q u e u e a c k 1 [0 ] 1 R e c e iv e m e s g 1 [1 ] R e c e iv e a c k 0 [0 ] S e n d m e sg 1 [0 ] 6 pt S e n d m e sg 1 [1 ] u 7 E n q u e u e m e sg 0 [0 ] S e n d a c k 0 [0 ] h t pica lr sul t fth r l ia blc . l utin fa r l ia blpr t c l . 2 unica tin p ri nt,a st p g stra t g . c t I t s co s to f p ri ts s to tto l o p pro to co l s ic a tt pt to a k o pti a l s o f t ba i t a a il a bl. r fo r a go o so ltio o lb o ic tra s itt a ssa g ri g a c ti st p. I a l l o ll ik to g ra t a a c i ic ibits a i o i gb a io r. t r u t . fcritica li po rta c fo rt is p ri tist l at c co t r ri gco ica tio . o si l a t t is t co ica tio c a l s o ft p o a b co rt fro t sta a r q str ct r s i t sta a r ito a rr o fsi q a lto t l a t c r q ir . ri g a c si l a tio ti st p ssa g s o t c a la r o o pl ac al o gt a rra . s it t r l ia bil it p ri ts l o ss is ca l c l at ri gt r c i p a s o fco ica tio . Ifa ssa g is l o st ri gco ica tio itis o tpa ss to t r c ii gp o . I t pr io s p ri t a c ac i a s p r itt o a ctio p r ti st p. H o r i itia l p ri ts r r a l t a tso ltio s g ra t si gt is ti i gsc o rco rl ia bil it b P ro to c o l C o n s tru c tio n U s in g G e n e tic S e a rc h T e c h n iq u e s 2 4 3 s ig l tiplpa ck ts. ti sp tsto ri ga pa ck ti t r tra s issio b ff r a s s to s a s co pa ck t s cc ssi pa ck ts a i gfa rlssc a c o fl o ss t a t pr io s. o fo rc t l op to fa r tra s issio sc a c tra sitio asb a ssig a pa ra t r ic sp ci s t l gt o fti it a ta k to c t t a ctio . Transmission Buffer 0:4 0:3 0:2 0:1 Transmission Buffer 0:0 Receive Buffer FSM Re-transmission Buffer Memory FSM Latency Variable B Variable B t rksi ul a tin f r l utin fr l ia blc hich a l s c unica t s da ta in pti a lti . ct Re-transmission Buffer Latency Variable A Receive Buffer Variable A Memory unica tingfinit sta t a chin , s. : a ssa g o t co ica tio s c a lta ki g a ta fro o r ifr q ir . ci : ci a ssa g fro t co ica tio s c a la pl ac t a ta ito or. q : l a c t it i o r ito t r c i b ff r. q : l ac t rst l to ft tra s itb ff r ito or. q tra s it: l a c t it i o r ito t r tra s itb ff r. q tra s it: l ac t rst l t o ft r tra s itb ff r ito or. o ro tra s it: o t c rr tit o f a ta fro t r tra sissio b ff r. t a ria bl : t a ria bl it t c rr tit i or. t a ria bl : tt o r to t a l o f a ria bl . t a ria bl : t a ria bl it t c rr tit i or. t a ria bl : tt o r to t a l o f a ria bl . l l: l la ctio a la s o rks .I t p o s al so q ipp to a ip l at so ltio s to o p a bls a blt fl l . a itio s pr it a a ta . ra t o po ra r to t sta a r tra s it r c i a o r b ff rs o f io sl t p o s fo r t is s to f p ri ts is it r -tra s issio b ff r a t o a ria bls it ic r -tra s issio b ff r a s b a to co ra g s ts o ft o ra l l ssa g . tra o r a risto ra g o f a ta t r -tra s issio b ff r b co s 2 4 4 N . S h a rp le s a n d I. W a k e m a n t ss u ct o r t is p ri t t t ss f ctio st a ss ss t so ltio ’ s a bil it to r l ia bl co ica t t a ta . I a itio it st a l so pro i so a s r o ft ti ta k fo r t co ica tio to ta k pl ac ; so ltio s ic co ica t o stq ickl sco r a ig r t ss a l . o t r i t is a l a co bi t o rigi a lf ctio s to l op t rl ia blpro to co l it a a itio a la l ic i ica t s t pro po rtio o ft si l a tio -ti ta k to co plt t a ta c a g . m t 2 − m m t + tm omm t t h su ts. gra p s pr s t i g r s o t a ra g a o to fsi l a tio ti ta k to s t co plt ssa g . Ift tir ssa g a s o t r c i b t irt ti st ps t si l a tio as tr iat . l so s o is t a ra g b r o f a ta it s s cc ssf l lr c i ri g a a la tio . gra p i g r 7 s o s t co rr spo i g t ss a l o ft ig stsco ri g i ii a la t a c g ra tio . expB 100 80 Time 60 40 20 0 0 100 200 300 400 500 600 700 800 900 1000 Generations i 4 isc ssi ak n t r unica t plt ssa g r is pa p r a s s o t a tt s a rc spa c o f it sta t a c i s fo r co ica tio pro to co l s ca b pl or si gg tic s a rc t c iq s. I itia l l a tt pt to co str ct l lk o pro to co l s s c as t al t ra ti gbitpro to co l . tt pti gto fo rc t s a rc pro c ss to g ra t sp ci c pro to co l s pro iffic l t it t sir pro to co l r b ig o l . H o r al i pro to co l s ic p rfo r t q ia l tta sk r pro c t s P ro to c o l C o n s tru c tio n U s in g G e n e tic S e a rc h T e c h n iq u e s 2 4 5 expB 10 Data Items 8 6 4 2 0 0 100 200 300 400 500 600 700 800 900 1000 Generations u b r f a ta It s 6 unica t d 1 Fitness Value 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 Generations 7 itn ss a l u bta in d b b stindiidua l during l utin fa pti a l pr t c l . 0 [2 ] D e q u e u e R e tra n s m it 6 [3 ] R e c e iv e 1 4 [1 ] D e q u e u e [1 ] G e t V a ria b le B [0 ] E n q u e u e R e tra n s m it 3 [0 ] D e q u e u e R e tra n s m it [0 ] D e q u e u e [1 ] E n q u e u e R e tra n s m it [0 ] S e n d [0 ] E n q u e u e [2 ] N u ll [2 ] E n q u e u e [1 ] G e t V a ria b le B [2 ] E n q u e u e [3 ] R e c e iv e 8 1 0 [2 ] R e c e iv e [1 ] R e m o v e V a ria b le fro m 1 1 R e tra n s m it b u ffe r 1 5 h t pica lr sul t fth pro i g t a t pro to co l s t is t c iq . ibiti g a pti a lc sir b unica tin a io r ca p ri nt. b g ra t si g 2 4 6 N . S h a rp le s a n d I. W a k e m a n o rk rta k r as s o t a tt t pro c ss o fco str cti g al i t o rksi l a tio s is a iffic l tpro bl . H o r fo r co pl pro to co l s ic o p ra t i ig l a ic iro ts t iffic l t i co str cti gt si l a tio il lb ga t b t iffic l tis o ftr i g to pr ictt rg t a ic. a a ta g o f si g t is a ppro a c is its a bil it to gi s ir ct f ba cko t pro to co l ’ sp rfo r a c i t si l at iro ti o r rto ri t sig pro c ss si c t t ss f ctio s s a si l a tio o ft to rk iro tto t r i t p rfo r a c o ft so ltio s. t ig t si l a tio spa c fro a si glpa ir o fpro to co l titis to l tiplco ica ti g titis ca a ss ss t i pa cto fa si glco ica tio stra t g o t gl o ba l t o rk a ic. f i g-ba ck si l a tio p rfo r a c ito t sig pro c ss ca s r t a t t pro to co lpro c is bo t ro b sta o pti a l it r sp ctto t gl o ba l t o rk a ic. r t p ri ts il la l tipl titis co ica ti go ra si gl c a l si g t sa pro to co l . t ss f ctio il l s so o tio o f fa ir ss a s l la s r l ia bil it a til isa tio to s r t a tt o l pro to co l r a cts to co g stio . fr c s . ca ntlbur . . a rtlttK. . a nd . . il kins n. n t n r l ia blful l -dupl tra ns issin r ha l f-dupl l in s. t t , (5): 6 – 6 5, 6 . . . l iff, I. H a r , a nd .H usba nds. Incr nta l l utin fn ura ln t rk a rchit ctur s f r a da pti b ha iur. chnica l p rt gniti cinc s a rch ap r 56 , ch l f gniti a nd puting cinc s, ni rsit f uss , right n H , ngl a nd, K, . 3. . l iff, . H usba nds, a nd I. H a r . ling isua l l guid d r b ts. chnica l p rt gniti cinc s a rch a p r , ch l f gniti a nd puting cinc s, ni rsit f uss , right n H , ngl a nd, K, . 4. ra nd . a nd Z a fir pul . n c unica ting finit sta t a chin s. r t , 3 ( ):3 3–34, 3. 5. a r nc . g l . rt t g tr g s t t . hn il a nd ns Inc, 6 6 . 6 . . . ch a nn. init sta t d scriptins fc unica tin pr t c l s. tr t rs, (4/ 5):36 –37 , 7 . P r e d ic tio n o f P o w e r R e q u ir e m e n ts fo r H ig h -S p e e d C ir c u its F . C O R N O , M . R E B A U D E N G O , M . S O N Z A R E O R D A , M . V IO L A N T E D ip . A u to m a tic a e In fo rm a tic a P o lite c n ic o d i T o rin o h t t p : / / w w w . c a d . p o l i t o . i t A b s tr a c t. M o d e rn V L S I d e s ig n m e th o d o lo g ie s a n d m a n u fa c tu rin g te c h n o lo g ie s a re m a k in g c irc u its in c re a s in g ly fa s t. T h e q u e s t fo r h ig h e r c irc u it p e rfo rm a n c e a n d in te g ra tio n d e n s ity s te m s fro m fie ld s s u c h a s th e te le c o m m u n ic a tio n o n e w h e re h ig h s p e e d a n d c a p a b ility o f d e a lin g w ith la rg e d a ta s e ts is m a n d a to ry . T h e d e s ig n o f h ig h -s p e e d c irc u its is a c h a lle n g in g ta s k , a n d c a n b e c a rrie d o u t o n ly if d e s ig n e rs c a n e x p lo it s u ita b le C A D to o ls . A m o n g th e s e v e ra l a s p e c ts o f h ig h -s p e e d c irc u it d e s ig n , c o n tro llin g p o w e r c o n s u m p tio n is to d a y a m a jo r is s u e fo r e n s u rin g th a t c irc u its c a n o p e ra te a t fu ll s p e e d w ith o u t d a m a g e s . In p a rtic u la r, to o ls fo r fa s t a n d a c c u ra te e s tim a tio n o f p o w e r c o n s u m p tio n o f h ig h s p e e d c irc u its a re re q u ire d . In th is p a p e r w e fo c u s o n th e p ro b le m o f p re d ic tin g th e m a x im u m p o w e r c o n s u m p tio n o f s e q u e n tia l c irc u its . W e fo rm u la te th e p ro b le m a s a c o n s tra in e d o p tim iz a tio n p ro b le m , a n d s o lv e it re s o rtin g to a n e v o lu tio n a ry a lg o rith m . M o re o v e r, w e e m p iric a lly a s s e s s th e e ffe c tiv e n e s s o f o u r p ro b le m fo rm u la tio n w ith re s p e c t to th e c la s s ic a l u n c o n s tra in e d fo rm u la tio n . F in a lly , w e re p o rt e x p e rim e n ta l re s u lts a s s e s s in g th e e ffe c tiv e n e s s o f th e p ro to ty p ic a l to o l w e im p le m e n te d . 1 . I n t r o d u c t i o n M o d e rn te le c o m m u n ic a tio n s y s te m s m u s t e n s u re h ig h b a n d w id th a n d h ig h p e rfo rm a n c e to c o p e w ith th e in c re a s in g a m o u n t o f d e liv e re d d a ta . D iffe re n t s o lu tio n s a t d iffe re n t le v e ls o f a b s tra c tio n c a n b e e x p lo ite d : im p ro v e d c o m m u n ic a tio n c h a n n e ls , b e tte r c o m m u n ic a tio n p ro to c o l a n d h ig h -s p e e d c o m m u n ic a tio n e q u ip m e n t. M o d e rn V L S I d e s ig n m e th o d o lo g ie s a n d m a n u fa c tu rin g te c h n o lo g ie s a re m a k in g c irc u its e v e ry d a y fa s te r. T o c o p e w ith th e c o m p le x ity o f d e s ig n in g h ig h -s p e e d c irc u its d e s ig n e rs m u s t e x p lo it s u ita b le C A D to o ls . A m o n g th e s e v e ra l a s p e c ts o f h ig h -s p e e d c irc u it d e s ig n , p o w e r c o n s u m p tio n is to d a y a m a jo r is s u e . In th e la s t y e a rs , d e s ig n fo r lo w -p o w e r c o n s u m p tio n h a s b e c a m e a w id e s p re a d d e s ig n p a ra d ig m . D e s ig n te c h n iq u e s to c o n tro l p o w e r c o n s u m p tio n a re m a n d a to ry b e c a u s e e x c e s s iv e p o w e r d is s ip a tio n c a n c a u s e p e rfo rm a n c e d e g ra d a tio n , ru n -tim e e rro rs , o r d e v ic e d e s tru c tio n d u e to o v e rh e a tin g . L a rg e in s ta n ta n e o u s p o w e r d is s ip a tio n m a y a ls o c a u s e lo c a l h o t s p o ts th a t h a v e n e g a tiv e im p a c t o n c irc u it re lia b ility . W ith in c re a s in g d e m a n d s fo r re lia b le h ig h -s p e e d c irc u its a c c u ra te e s tim a tio n o f p e a k -p o w e r d is s ip a tio n d u rin g d e s ig n p ro c e s s is b e c o m in g e s s e n tia l. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 4 7 − 2 5 4 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 4 8 F . C o rn o e t a l. In th is p a p e r w e a d d re s s th e p e a k -p o w e r e s tim a tio n p ro b le m fo r h ig h -s p e e d s e q u e n tia l c irc u its . P o w e r c o n s u m p tio n in s e q u e n tia l c irc u its is a n o n -lin e a r fu n c tio n , 3 , o f th r e e v a r ia b le s : th e c ir c u it in itia l s ta te , S , a n d tw o v e c to r s ( 9− 1 , 9 ) th a t a r e I W a p p lie d to th e c irc u it in p u ts a t tim e t-1 a n d t. W e re p re s e n t th e m W a s a 3 - tu p le ( S , 9− 1 , W 9 ). T h e p ro b le m o f c o m p u tin g th e p e a k -p o w e r c o n s u m p tio n o f s e q u e n tia l c irc u its c a n b e f o r m u la te d a s th e p r o b le m o f f in d in g th e 3 - tu p le ( S , 9− 1 , 9 ) th a t m a x im iz e s W W W th e fu n c tio n 3 . I S e v e ra l te c h n iq u e s h a v e b e e n p ro p o s e d to s o lv e th is p ro b le m re s o rtin g to e ith e r e x a c t a p p ro a c h e s [1 ] o r te c h n iq u e s b a s e d o n A u to m a tic T e s t P a tte rn G e n e ra to rs [2 ]. B u t o n ly in [3 ] a n d [4 ] a n a lg o rith m is p ro p o s e d a b le to c o p e w ith la rg e s e q u e n tia l c irc u its . In p a rtic u la r th e p e a k -p o w e r p ro b le m is fo rm u la te d a s a n u n c o n s tra in e d o p tim iz a tio n p ro b le m a n d a te c h n iq u e b a s e d o n G e n e tic A lg o rith m s is p ro p o s e d . T h is te c h n iq u e is s tro n g ly a p p ro x im a te d s in c e th e re a c h a b ility p ro p e rty o f th e s ta te S is n e g le c te d d u rin g th e o p tim iz a tio n p ro c e s s . A s a c o n s e q u e n c e , th is a p p ro a c h c o u ld le a d to s o lu tio n s w h e r e th e 3 - tu p le ( S , 9− 1 , 9 ) e m b e d s a s ta te S th a t th e s e q u e n tia l c irc u it n e v e r re a c h e s . T o m a k e th e 3 -tu p le a fe a s ib le s o lu tio n , th e a u th o rs o f [3 ] a n d [4 ] p ro p o s e to re p la c e a t th e e n d o f th e o p tim iz a tio n a lg o rith m th e s ta te S w ith a re a c h a b le o n e a s c lo s e a s p o s s ib le to S . In th e fo llo w in g , w e w ill a n a ly z e th e b e h a v io r o f th is a lg o rith m , a n d e x p e rim e n ta lly s h o w th a t it c a n u n d e re s tim a te p e a k -p o w e r w h e n th e n u m b e r o f re a c h a b le s ta te s is a s m a ll fra c tio n o f a ll th e p o s s ib le s ta te s . W e w ill th e n p ro p o s e a n a p p ro a c h th a t o v e rc o m e s th is d ra w b a c k b y e x p lo itin g th e k n o w le d g e a b o u t th e re a c h a b le s ta te s e t d u rin g th e o p tim iz a tio n p ro c e s s . In th is p a p e r w e p ro p o s e a n d e v a lu a te a n im p le m e n ta tio n o f o u r p e a k -p o w e r e s tim a tio n a lg o rith m th a t is b a s e d o n th e S e lfis h G e n e a lg o rith m [5 ], a n e v o lu tio n a ry o p tim iz a tio n a lg o rith m w e d e v e lo p e d . W e s u p p o rt o u r a p p ro a c h b y re p o rtin g e x p e rim e n ta l re s u lts a s s e s s in g th e e ffe c tiv e n e s s o f o u r a p p ro a c h . T h e a lg o rith m p re s e n te d h e re is b a s e d o n a p re v io u s w o rk [8 ] w h e re G e n e tic A lg o rith m s w e re u s e d to s o lv e th e p e a k -p o w e r p ro b le m . In [8 ] a G e n e tic A lg o rith m e v o lv e s a p o p u la tio n w h e re e a c h in d iv id u a l is a 3 -tu p le c o m p o s e d o f a re a c h a b le s ta te a n d tw o v e c to rs . T o g u a ra n te e th a t th e p o p u la tio n c o n ta in s o n ly v a lid in d iv id u a ls , e a c h tim e c ro s s o v e r a n d m u ta tio n o p e ra to rs a re a p p lie d th e s ta te in th e n e w in d iv id u a l is a n a ly z e d , a n d re p la c e d w ith a re a c h a b le o n e w h e n re q u ire d . T h is a p p ro a c h c a n lim it th e s e a rc h c a p a b ility o f G e n e tic A lg o rith m s a n d th u s th e e ffe c tiv e n e s s o f o u r p e a k p o w e r e s tim a tio n a lg o rith m . B y re s o rtin g to th e p e c u lia ritie s o f S e lfis h G e n e , s ta te m o d ific a tio n is n o lo n g e r re q u ire d , a n d th u s th e e ffe c tiv e n e s s o f o u r a lg o rith m c a n b e im p ro v e d . T h e re m a in d e r o f th e p a p e r is o rg a n iz e d a s fo llo w s . S e c tio n 2 p re s e n ts a fo rm a l d e s c rip tio n o f th e p ro b le m . S e c tio n 3 d e s c rib e s th e e v o lu tio n a ry a lg o rith m w e e x p lo it in o u r to o l, a n d S e c tio n 4 p re s e n ts th e p e a k -p o w e r e s tim a tio n a lg o rith m w e d e v e lo p e d . S e c tio n 5 re p o rts e x p e rim e n ta l re s u lts o n s o m e b e n c h m a rk c irc u its . F in a lly , S e c tio n 6 d ra w s s o m e c o n c lu s io n s . W W Prediction of Power Requirements for High-Speed Circuits 249 2 .  Pro b l e m f o rmul at io n We assume that the sequential circuits we are analyzing are manufactured with CMOS technology. Power consumption in CMOS circuits is mainly due to the number of times the output of circuit gates switches from 0 to 1 and vice-versa. Peakpower consumption is defined as the maximum power consumption that takes place when a couple of input vectors are applied to the circuit inputs, starting from a given initial state. The procedure to measure peak-power can be outlined as follows: 1. the circuit memory elements are initialized to a given state S; 2. an input vector 9 1 is applied to the circuit, and a clock pulse is applied; W 3. an input vector 9 is applied to the circuit, a clock pulse is applied and the power consumption is measured during the transient period. The first two steps lead the circuit to a configuration where the circuit nodes hold known values (either 0 or 1). We refer to the power measured during the last step as 3 , and use to notation 3 ( 6 , 9 1 , 9 ) to indicate that power is function of the 3-tuple W I I W W (S, 9 1 , 9 ). The classical peak-power problem formulation that can be found in [3] and [4] is the following: W W 33 0$; ( 3I ( 6 , 9W 1 , 9W )), 6  % Q , 9  % P (1) where PP is the peak-power consumption, % {0,1} , n is the number of memory elements in the circuit and m is the number of circuit inputs. This formulation neglects the fact that in sequential circuits the number of reachable states is often less than 2 Q . Therefore, algorithms that solve equation (1) could compute 3-tuples that correspond to PP values that cannot be obtained, since the memory elements can never reach the configuration S during circuit operations. Power figures attained by solving equation (1) are overestimation of actual peak-power consumption, and if adopted during design could lead to unnecessary expensive design solutions. The authors of [3] and [4] suggest replacing an unreachable state in the solution of (1) with the closest reachable one. This approach weakens the correlation between S and (S, 9 1 , 9 ) and usually lead to peak-power figures that underestimate the actual power consumption. As a consequence, the obtained peak-power prediction could lead to wrong design solutions. The problem formulation we propose is the following: W 33 0$; ( 3I ( 6 ,9W 1 ,9W )), 6  6,9  % P W (2) where 6 Ž % Q is the set of states that the sequential circuit under analysis can reach starting from the reset state, i.e., the state where all the memory elements are set to 0. By solving equation (2) and therefore by considering reachability of the state S as a dimension of the search space, we compute peak-power figures more accurate than what equation (1) provides. 2 5 0 F . C o rn o e t a l. 3 . T h e S e l f i s h G e n e a l g o r i t h m T h e S e lfis h G e n e a lg o r ith m (S G ) is a n e v o lu tio n a ry o p tim iz a tio n a lg o rith m b a s e d o n a re c e n t in te rp re ta tio n o f th e D a rw in ia n th e o ry . It e v o lv e s a p o p u la tio n o f in d iv id u a ls s e e k in g fo r th e fitte s t o n e . In th e s e lfis h g e n e b io lo g ic a l th e o ry , p o p u la tio n its e lf c a n b e s im p ly s e e n a s a p o o l o f g e n e s w h e re th e n u m b e r o f in d iv id u a ls , a n d th e ir s p e c ific id e n tity , a re n o t o f in te re s t. T h e re fo re , d iffe re n tly fro m o th e r e v o lu tio n a ry a lg o rith m s , th e S G re s o rts to a s ta tis tic a l c h a ra c te riz a tio n o f th e p o p u la tio n , b y re p re s e n tin g a n d e v o lv in g s o m e s ta tis tic a l p a ra m e te rs o n ly . E v o lu tio n p ro c e e d s in d is c re te s te p s : in d iv id u a ls a re e x tra c te d fro m th e p o p u la tio n , c o lla te d in to u rn a m e n ts a n d w in n e r o ffs p rin g is a llo w e d to s p re a d b a c k in to th e p o p u la tio n . A n in d iv id u a l is id e n tifie d b y th e lis t o f its g e n e s . T h e w h o le lis t o f g e n e s is c a lle d g e n o m e a n d a p o s itio n in th e g e n o m e is te rm e d lo c u s . E a c h lo c u s c a n b e o c c u p ie d b y d iffe re n t g e n e s . A ll th e s e c a n d id a te s a re c a lle d th e g e n e a lle le s . In th e c o n te x t o f a n o p tim iz a tio n p ro b le m , lo o k in g fo r th e fitte s t in d iv id u a l c o rre s p o n d s to d e te rm in e th e b e s t s e t o f g e n e s a c c o rd in g to th e fu n c tio n to b e o p tim iz e d . S in c e th e S G “ v irtu a l” p o p u la tio n is u n lim ite d , in d iv id u a ls c a n b e c o n s id e re d to b e u n iq u e , b u t s o m e g e n e s w o u ld c e rta in ly b e m o re fre q u e n t th a n o th e rs m ig h t. A t th e e n d o f th e e v o lu tio n p ro c e s s , th e fr e q u e n c y o f a g e n e m e a s u re s its s u c c e s s a g a in s t its a lle le s . H o w e v e r, a t th e b e g in n in g o f th e e v o lu tio n p ro c e s s , th e fre q u e n c y c a n b e re g a rd e d a s th e g e n e d e s ir a b ility . W h e n th e m a jo rity o f a p o p u la tio n is c h a ra c te riz e d b y th e p re s e n c e o f a c e rta in c h a ra c te ris tic , n e w tra its m u s t h a rm o n iz e w ith it in o rd e r to s p re a d . JHQRPH S G ( 9LUWXDO3RSXODWLRQ3) { JHQRPH%(67, *, *, ZLQQHU, ORVHU; it e r = 0 ; %(67 = se l e c t _ individual ( 3) ; EHVWVRIDU do { ++it e r; * = se l e c t _ individual ( 3) ; * = se l e c t _ individual ( 3) ; t o urname nt ( *, *) ;LGHQWLI\ZLQQHUDQGORVHU inc re ase _ al l e l e _ f re que nc ie s( 3, ZLQQHU) ; de c re ase _ al l e l e _ f re que nc ie s( 3, ORVHU) ; if ( ZLQQHU is pre f e rabl e t o %(67) %(67 = ZLQQHU; } wh il e ( st e ady_ st at e ( ) ==F AL SE && it e r<max_ it e r) ; re t urn %(67; } F ig . 1 : S e lfis h G e n e a lg o rith m T h im p le m o tiv d e a l w e p m e n a tio ith s e u d o -c o d e o f th e S ta tio n d e ta ils a b o u t th n s a re b e tte r a n a ly z e d m o re c o m p le x fitn e s s G e S in la n a lg o rith a lg o r [1 0 ]. A d sc a p e s G m ith n is p se u d o -c o d e . c o re is a re a v e x te n s io n d e s c rib e d m re p o rte d in F ig . 1 . F u rth e r a ila b le in [9 ], w h ile b io lo g ic a l to th e b a s ic S G a lg o rith m to in [1 1 ]. Prediction of Power Requirements for High-Speed Circuits 251 4.  Po w e r e st imat io n al go rit h m The peak-power estimation algorithm we developed is composed of the following steps: 1. the set of the reachable states 6 Ž % Q is computed, which cardinality is 6 . It can be computed either resorting to exact symbolic calculation techniques [7] or through logic simulation; 2. peak-power estimation is performed. The SG is run to solve equation (2) where the genome represents the 3-tuple (S, 9 1 , 9 ) and is composed of 1  2 ˜P loci. The W W first locus is an index ranging from 0 to 6 1 representing a reachable state 6  6 . The remaining loci are binary values, coding the couple of vectors ( 9 1 , 9 ). W W During step 2, the power consumption 3 ( 6 , 9 1 , 9 ) is computed resorting to a I W W unit-delay logic simulator. The adoption of a logic simulator is a well-known effective approach to measure power consumption in CMOS circuits [6], since it conjugates simulation speed with accuracy. 5 .  Expe rime nt al re sul t s A prototypical version of our algorithm named SG-ALPS, Selfish Gene-based AnaLyzer of Power in Sequential circuits, has been written, which implements the above-introduced procedures. The tool consists of about 500 lines of ANSI C code and exploits the SG and logic simulation packages developed at our institution. Reachability analysis has been performed resorting to exact calculation techniques exploiting the BDD [7] package developed at our institution. The subset of ISCAS’89 sequential circuits tractable with symbolic calculation techniques have been used to evaluate the performance of SG-ALPS: all the experiments have been performed on a Sun UltraSparc 5/333 with 256 MB RAM. To compare SG-ALPS with a state-of-the-art tool, we have re-implemented the algorithm proposed in [4]. Experimental results of our re-implementation of [4] are the same of [4]; we can therefore perform a fair comparison between the two tools. Two sets of experiments have been performed. The first one aims at empirically showing that solving equation (1) as done by [4] could greatly underestimate peakpower consumption. Conversely, the second set of experiments aims at assessing the effectiveness of the approach we propose. Table 1 reports results we gathered with our implementation of [4]. The first column reports the benchmark name, PPU and PPR report respectively the peakpower consumption obtained by solving equation (1) (which takes into account also unreachable states) and the power obtained by replacing the state in the solution with the closest reachable one. Column ' reports the difference between these power figures, while column CPU reports the time requirements. By observing the column ', one can observe that several circuits exist where peak- 252 F. Corno et al. power strongly depends on the reachability of the initial state S and on the correlation between S and the vectors ( 9 1 , 9 ). Where a significant loss in peak-power is found, a large portion of the state S has been modified to make it reachable, thus a high difference exists between the ideal initial state and the selected reachable one. W Circ s208 s298 s344 s349 s382 s386 s400 s420 s444 s499 s510 s526 s641 s713 s820 s832 s1196 s1238 s1488 s1494 Avg. PPU 0.900 1.007 1.573 1.020 0.961 0.855 1.000 0.893 1.077 0.595 0.860 0.915 2.753 2.793 0.951 0.927 1.035 1.015 1.235 1.234 W PPR 0.900 0.833 1.552 0.869 0.790 0.855 0.435 0.893 0.551 0.230 0.860 0.631 2.739 0.837 0.951 0.927 1.020 1.000 1.156 1.155 ' [%] 0.00 -17.28 -1.34 -14.80 -17.79 0.00 -56.50 0.00 -48.84 -61.34 0.00 -31.04 -0.50 -70.02 0.00 0.00 -1.45 -1.47 -6.40 -6.39 -15.96 CPU [s] 3.2 3.5 4.7 4.6 4.9 4.6 5.2 5.3 6.3 4.9 6.7 7.0 17.0 16.5 9.9 9.9 14.7 15.3 22.7 22.6 10.4 T ab l e 1 . Analysis of algorithm [4] Table 2 reports results obtained by running SG-ALPS when max_iter (Fig 1) is set to 1000. Column SG-ALPS reports the peak-power figures predicted by our algorithm, while the third column reports the best results attained by the algorithm proposed in [4]. We compare the two algorithms in column '. Finally, the CPU time requirements are reported. As far as peak-power estimation accuracy is concerned, we can conclude that our approach is superior to [4]. Even if on the average SG-ALPS attains PP figures 21% higher than [4], several circuits exist where SG-ALPS computes power figures 50% higher than [4], thus showing the importance of considering reachability during the optimization phase. As far as CPU time is concerned, the algorithm proposed in [4] is far more effective than SG-ALPS. This is mainly due to two factors: SG requires more time to converge than Genetic Algorithms, and guaranteeing state reachability during the optimization process is a time consuming operation. P re d ic tio n o f P o w e r R e q u ire m e n ts fo r H ig h -S p e e d C irc u its 2 5 3 W e d o p o in t o u t th a t th e C P U re q u ire m e n ts o f S G -A L P S , a lth o u g h h ig h e r th a n th o s e o f p re v io u s ly p ro p o s e d a p p ro a c h e s , s till re m a in in th e o rd e r o f m in u te s , w h ic h is n e g lig ib le w ith re s p e c t to th e tim e re q u ire d b y m o s t s te p s in th e c u rre n t c irc u it d e s ig n flo w . T h e in c re a s e in th e C P U tim e is w o rth p a id b y th e im p ro v e m e n t in th e a tta in e d e s tim a tio n q u a lity . W h e n c o m p a re d w ith th e m e th o d p ro p o s e d in [8 ], im p le m e n tin g a s im ila r a p p ro a c h u s in g G e n e tic A lg o rith m , w e o b s e rv e d th a t th e a lg o rith m p ro p o s e d h e re p ro v id e s c o m p a ra b le re s u lts . H o w e v e r, th e a d o p tio n o f th e S G a lg o rith m a llo w s to m o re e a s ily d e a l w ith o p tim iz a tio n p ro b le m s fo r w h ic h a h e te ro g e n e o u s s o lu tio n e n c o d in g is m o re s u ita b le . C irc s2 0 s2 9 s3 4 s3 4 s3 8 s3 8 s4 0 s4 2 s4 4 s4 9 s5 1 s5 2 s6 4 s7 1 s8 2 s8 3 s1 1 s1 2 s1 4 s1 4 A v 8 8 4 9 2 6 0 0 4 9 0 6 1 3 0 2 9 6 3 8 8 8 9 4 g . S G -A L P S [4 ] P P P P 0 .9 0 .8 1 .5 0 .8 0 .7 0 .8 0 .4 0 .8 0 .5 0 .2 0 .8 0 .6 2 .7 0 .8 0 .9 0 .9 1 .0 1 .0 1 .1 1 .1 0 .9 0 .8 1 .5 1 .5 0 .8 0 .8 0 .8 0 .8 0 .9 0 .2 0 .8 0 .7 2 .7 2 .1 0 .9 0 .9 1 .0 1 .0 1 .1 1 .1 0 0 3 3 5 3 3 5 5 9 7 2 5 0 9 3 4 2 3 0 6 9 3 7 3 9 5 5 6 2 6 7 4 4 5 0 5 7 5 6 ∆ [% ] 0 0 3 3 5 2 6 9 7 9 0 5 5 3 5 9 9 3 5 1 7 3 0 6 0 3 1 1 3 9 3 7 5 1 2 7 2 0 0 0 5 6 5 5 1 5 C P U [s] 0 .0 0 0 .0 0 0 .0 3 6 .6 5 8 .7 2 2 .0 1 5 .4 3 0 .0 0 0 .9 6 0 .0 0 1 .0 5 6 .8 0 0 .0 0 7 .3 6 1 .1 6 4 .3 5 2 .3 7 5 .0 0 0 .0 4 0 .0 8 2 1 .0 2 2 1 1 1 3 1 5 1 1 1 2 8 5 6 1 8 3 1 9 8 3 7 1 0 2 7 0 3 1 5 4 5 9 9 5 8 9 8 5 0 7 1 6 8 9 5 1 7 1 6 3 0 .1 .6 .0 .9 .5 .5 .6 .1 .7 .5 .0 .7 .3 .2 .5 .4 .8 .6 .8 .4 .9 T a b le 2 . S G -A L P S re s u lts 6 . C o n c l u s i o n s H ig h re q u e st c o m m u c o n su m -s p e e d te le c o m m u n ic a tio n e q u ip m e n fo r h ig h -s p e e d d a ta d e liv e rin g . W n ic a tio n e q u ip m e n t, p a rtic u la r c a re m p tio n . In p a rtic u la r, p e a k -p o w e r e s tim t is h e n u st a tio re q u ire d d e s ig n in b e p o se n is m a n to c o p e h ig h -s d to th e d a to ry fo g w ith th e p e e d c irc u c o n tro l o f r d e s ig n in g m a its p o to rk e t fo r w e r d a y 2 5 4 F . C o rn o e t a l. h ig h -s p e e d c irc u its . In th is p a p e r th e p e a k -p o w e r p re d ic tio n p ro b le m h a s b e e n fo rm u la te d a s a c o n s tra in e d o p tim iz a tio n p ro b le m , a n d a n a lg o rith m b a s e d o n a n e w e v o lu tio n a ry p a ra d ig m h a s b e e n p ro p o s e d . T h e a lg o rith m is d e s ig n e d fo r a d d re s s in g s e q u e n tia l c irc u its a n d th a n k s to its a b ility o f g u a ra n te e in g fe a s ib ility o f s o lu tio n s d u rin g th e o p tim iz a tio n p ro c e s s it is m o re e ffe c tiv e th a n o th e r a p p ro a c h e s . M o re o v e r, in th is w o rk w e e x p e rim e n te d th a t S G is b e tte r s u ite d to a d d re s s o p tim iz a tio n p r o b le m s h a v in g h e te r o g e n e o u s s o lu tio n s , i.e ., s o lu tio n s m ix in g c o m p o n e n ts d e fin e d o v e r d iffe re n t d o m a in s . A s a n e x a m p le , in p e a k -p o w e r e s tim a tio n th e s o lu tio n e m b e d s a s ta te th a t is n a tu ra lly e x p re s s e d a s a n in d e x in a s ta te ta b le , a n d a c o u p le o f v e c to rs , re p re s e n te d a s a c o lle c tio n o f b in a ry v a lu e s . W e a re c u rre n tly w o rk in g to w a rd a n im p le m e n ta tio n o f S G -A L P S w h e re a p p ro x im a te d re a c h a b ility a n a ly s is is p e rfo rm e d th ro u g h lo g ic s im u la tio n . T h a n k s to th is im p ro v e m e n t, w e w ill b e a b le to a d d re s s la rg e s e q u e n tia l c irc u it c u rre n tly n o t tra c ta b le b y s y m b o lic c a lc u la tio n te c h n iq u e s . 7 . R e f e r e n c e s [1 ] [2 ] [3 ] [4 ] [5 ] [6 ] [7 ] [8 ] [9 ] [1 0 ] [1 1 ] S . M a n n e , A . P a rd o , R . I. B a h a r, G . D . H a c h te l, F . S o m e n z i, E . M a c ii, M . P o n c in o , “ C o m p u tin g th e m a x im u m p o w e r c y c le s o f a s e q u e n tia l c irc u it” , P ro c . o f IE E E /A C M D A C , 1 9 9 5 , p p . 2 3 -2 8 7 C .- Y . W a n g , K . R o y , “ M a x im u m C u r r e n t E s tim a tio n in C M O S C ir c u its U s in g D e te rm in is tic a n d S ta tis tic a l T e c h n iq u e s ” , IE E E T ra n s . o n V L S I S y s te m s , M a rc h 1 9 9 8 , p p . 1 3 4 -1 4 0 M .S . H s ia o , E .M . R u d n ic k , J . P a te l, “ K 2 : A n E s tim a to r f o r P e a k S u s ta in a b le P o w e r o f V L S I C irc u its ” , P ro c . o f In t. S y m p . o n L o w P o w e r E le c tro n ic s a n d D e s ig n , 1 9 9 7 , p p . 1 7 8 -1 8 3 M .S . H s ia o , E .M . R u d n ic k , J . P a te l, “ E f f e c ts o f D e la y M o d e ls o n P e a k P o w e r E s tim a tio n o f V L S I S e q u e n tia l C irc u its ” , P ro c . o f IE E E /A C M IC C A D , 1 9 9 7 , p p . 4 5 -5 1 F . C o rn o , M . S o n z a R e o rd a , G . S q u ille ro , “ O p tim iz in g D e c e p tiv e F u n c tio n s w ith th e S G C l a n s A l g o r i t h m ” , C E C '9 9 : 1 9 9 9 C o n g r e s s o n E v o l u t i o n a r y C o m p u t a t i o n , W a s h i n g t o n D C (U S A ), J u ly 1 9 9 9 , p p . 2 1 9 0 -2 1 9 5 A . G h o s h , S . D e v a d a s , K . K u e tz e r, J . W h ite , “ E s tim a tio n o f a v e ra g e s w itc h in g a c tiv ity in c o m b in a tio n a l a n d s e q u e n tia l c irc u its ” , P ro c . o f IE E E /A C M D A C , 1 9 9 2 , p p . 2 5 3 -2 5 9 R . E . B ry a n t, “ S y m b o lic B o o le a n M a n ip u la tio n w ith O rd e re d B in a ry D e c is io n D ia g r a m s ,” A C M C o m p u tin g S u r v e y s , V o l. 2 4 , N o . 3 , 1 9 9 2 , p p . 2 9 3 - 3 1 8 F . C o rn o , M . R e b a u d e n g o , M . S o n z a R e o rd a , M . V io la n te , “ A L P S : A P e a k -P o w e r E s tim a tio n A lg o rith m f o r S e q u e n t i a l C i r c u i t s ” , G L S - V L S I '9 9 : 8 t h G r e a t L a k e s S y m p o s iu m o n V L S I, 1 9 9 9 , p p . 3 5 0 -3 5 3 F . C o rn o , M . S o n z a R e o rd a , G . S q u ille ro , “ T h e S e lfis h G e n e A lg o rith m : a N e w E v o l u t i o n a r y O p t i m i z a t i o n S t r a t e g y ” , S A C '9 8 : 1 3 t h A n n u a l A C M S y m p o s iu m o n A p p lie d C o m p u tin g , 1 9 9 8 , p p . 3 4 9 -3 5 5 F . C o rn o , M . S o n z a R e o rd a , G . S q u ille ro , “ A N e w E v o lu tio n a ry A lg o rith m In s p ire d b y t h e S e l f i s h G e n e T h e o r y ” , I C E C '9 8 : I E E E I n t e r n a t i o n a l C o n f e r e n c e o n E v o l u t i o n a r y C o m p u ta tio n , 1 9 9 8 , p p . 5 7 5 -5 8 0 F . C o rn o , M . S o n z a R e o rd a , G . S q u ille ro , “ O p tim iz in g D e c e p tiv e F u n c tio n s w ith th e S G C l a n s A l g o r i t h m ” , C E C '9 9 : 1 9 9 9 C o n g r e s s o n E v o l u t i o n a r y C o m p u t a t i o n , 1 9 9 9 , p p . 2 1 9 0 -2 1 9 5 A Communication Architecture for Multi-Agent Learning Systems N. Ireson, Y. J. Cao, L. Bull and R. Miles Intelligent Computer Systems Centre Faculty of Computer Studies and Mathematics University of the West of England, Bristol, BS16 1QY, UK Bristol, BS16 1QY, UK Abstract. This paper presents a simple communication architecture for Multi-Agent Learning Systems. The service provided by the communication architecture allows each agent to connect to the user interface, the application and the other agents. The communication architecture is implemented using TCPIP. An application example in a simplied trafc environment shows that the communication architecture can provide reliable and e cient communication services for Multi-Agent Learning Systems. 1 Introduction Many researchers in the eld of Distributed Articial Intelligence are beginning to build agents that can work in a complex, dynamic multi-agent domains 1. Such domains include virtual theater 2, realistic virtual training environments 1, RoboCup robotic and virtual soccer 3 and robotic collaboration by observation 4. This is because that there is a realisation of the benets of using problem-solving models based upon an interacting group of agents rather than a single agent and multi-agent systems can benet from the inherent properties of distributed systems, i.e. parallelism, robustness, scalability. Learning in multi-agent systems has been seen as important both in removing the need to hard code" the agent behaviour, as for certain problems the appropriate behaviour is unknown 5, 6. It is motivated by the insight that it is impossible to determine a-priori the complete knowledge that must exist within each component of a distributed, heterogeneous system in order to allow satisfactory performance of that system. Especially if we want to exploit the potential of modularity, such that it is possible for individual agents to join and leave the multi-agent system, there is a constant need for the acquisition of new and the adaptation of already existing knowledge, i.e., for learning. Within this setting, dierent kinds of learning tasks must be investigated, such as `traditional' single agent learning tasks, learning in teams, learning to act within a team, and learning to cooperate with other agents. To solve any of these tasks, communication, i.e., the existence of appropriate information that can be communicated to the learning agents is of primary importance. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 5 5 − 2 6 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 5 6 N . Ire s o n e t a l. In this work, we consider the communication issues in the cooperative multi-agent learning systems. In a cooperative multi-agent system, the agents can cooperate either implicitly or explicitly. With implicit cooperation agents act selshly to satisfy their individual goals but their actions can have benecial eects upon other agents. With explicit cooperation, agents share information and rewards, thus perform actions which provide mutual benet, involving some form of direct communication. Sen et al showed that agents attempting to optimise the use of limited resources converge to optimal states faster with a limited view 7 . They also investigated the formation of agent coalitions, where agents within a coalition share information about their resource utilisation intentions. The use of such coalitions further improves the system's convergence to the optima. Prasad and Lesser examined an explicitly cooperative directly communicating system in which agents learning to communicate only relevant information dependant on situation specics 8 . The learning algorithm uses the Instance-Based learning paradigm. At the end of a particular problem solving run, agents assess their coordination strategy according to four performance measures. Agents then derive new strategies and broadcast these to the other agents. Seredynski et al developed a coevolutionary multi-agent system showing that global behaviour evolves via only local cooperation between agents acting without global information about the system. Seredynski introduces an exchange process into the game, which redistributes the payo 9, 10 . The players are placed in a ring and play a set number of games with a selected number of neighbours, i.e. local interactions. Three exchange schemes were tested: 1. no payo exchange, no cooperation 2. payo is exchanged amongst interacting players, local cooperation sharing 3. payo is exchanged amongst all players, global cooperative sharing. They considered two simple coevolutionary schemes: loosely coupled GA and loosely coupled classier systems. The term loosely coupled" refers to the individuals in a population being evaluated only on their local rather than a global tness function. The experiments showed that with no payo exchange the players evolve to the Nash equilibrium both defecting, whilst with both the local and global payo exchange players evolved to cooperate. In fact the payo exchange alters the game's payo matrix and encourage cooperation. Seredynski applied the technique of payo exchange to a dynamic mapping and scheduling problem 9 . Bull et al applied the learning classier system design to a multiagent environment in which each of a number of classier systems represents a co-operating communicating agent 11 . It is demonstrated that the evolution of multiple co-operating agents can give improved performance over an equivalent single agent model. The performances of the varying components, such as reinforcements, discovery system, in a multi-agent environment, were examined in detail in 12 . In this work, we extend the framework of the multi-agent systems in 10, 12 to a general version and aim at developing a communication architecture for these multi-agent learning systems. In these environments, each agent is dened by a classier system which evolves a set of control rules and is concerned with agents as software rather than theoretical constructs. In this communication architecture, each agent is allowed to connect to user interface, the application A C o m m u n ic a tio n A rc h ite c tu re fo r M u lti-a g e n t L e a rn in g S y s te m s 2 5 7 A g e n t C la s s ifie r S y s te m C o m m u n ic a tio n S e rv e r Figure 1: Structure of the agent and the other agents. An application example is given to show that the communication architecture can provide reliable and ecient communication services. 2 The Framework of the Multi-Agent Learning System Multi-agent learning systems usually consist of a number of collaborative agents and consider how these agents can interact to eectively cooperate in problem solving tasks. They have many applications in process control, network management, scheduling, etc. 9, 13, 17 . As with a number of contemporary elds on computer science, such as AI and Articial Life ALife the denition of an agent ranges from a strong to a weak notion. On the stronger end of the scale an agent is deemed to possess properties akin to those found in humans, such as knowledge, belief, intention, etc. The weaker notion of an agent tends to be more pragmatic and associated with agent software engineering. Such agents can be dened as possessing less anthropomorphic properties, for example 18 : autonomy: operate without the direct intervention of others, and have some control over actions and internal states communication: interactions with other agents reactivity: perception of their environment and timely response to changes that occur in it. Pro-activeness: not merely reacting to events but exhibit pre-active goaldirected behaviours. The multi-agent learning system considered in this paper contains a number of distributed, communicating agents, where each agent, as shown in Figure 1, has a learning classier system providing the rule base and control actions and a communication server which is used to connect the agent to the user interface, the application and to other agents. These two elements of the agent are separate since as messages are passed around the agent network, the communication server acts independently of the classier system to route the message to its neighbours. Another reason for keeping the communication server distinct 2 5 8 N . Ire s o n e t a l. S e n s o rs / A c tu a to rs U s e r I n te r f a c e M o n ito r In te rf a c e s C o n tro l T C P /I P A g e n t S ta te D e te c to r R M S h a r e d I n fo r m a t io n I A c t io n E ffe c to r S h a r e d R e w a r d P e r fo r m a n c e E v a lu a t io n A g e n t A p p lic a tio n Figure 2: Interaction structure of the agent from the classier system is that communication is likely to be implementation specic even in the test applications. Thus it is necessary when specifying the communication server to consider the general requirements of setting up and maintaining the communication in a multi-agent learning system rather than those in a specic software and hardware implementation. The communication server must provide the following services: Interaction between the application and the agents: The interface between the agents and the application is dened by the agents' detectors and e ectors. The application must package the system state into a message which is in the representation expected by the agents. The application receives an action from the detectors which is interpreted to e ect the system. Interaction between the User Interface and the agents: The amount of performance information reported by the agents is parameterised. During the development of an application it is likely that the user will wish to monitor performance more closely to ensure the system has been correctly congured. Another consideration is whether information performance statistics, warnings, error, etc. is logged to a le or sent to the user interface directly. Inter-agent communication: The communication strategy between the agents is e ected by the application and the learning strategy to be employed. Each agent can communicate with a specic other agent or group of agents, known as an agents neighbourhood, the user species the agents contained in each agents neighbourhood. The Figure 2 shows a representation of the interactions between an agent and A C o m m u n ic a tio n A rc h ite c tu re fo r M u lti-a g e n t L e a rn in g S y s te m s 2 5 9 A g e n t N e tw o rk 2 1 3 6 5 7 1 0 9 1 3 4 8 1 1 1 4 1 2 1 5 1 6 Figure 3: Structure of the agent network its environment which contains three types of actors: the user, the application, and other agents. While it is important to examine the research into agent languages and implementations such as KQML, such work provides functionality beyond the needs of these systems. The communication in such learning systems will have a limited syntax and have a well dened content, thus the language required will be fairly simplistic. We have specied the nature of the communication, which includes: utility measure enabling tness" sharing environmental state actions performed or to perform intentions shared classiers. 3 The Proposed Communication Architecture 3.1 The Whole Agent Network An agent network is dened by the characteristics of the application, which determines the appropriate distribution and connections between agents. Each agent is allocated a specic address, and a port for each channel of communication, this includes communication with the user interface, application and other agents. There are two basic protocols of communication. The simplest is to broadcast the message which is received by all the agents, the message can include a tag to identify the sender and nature of the message. It is the responsibility of the receiving agent to determine whether to utilise or ignore the information in the message. The second means of communication is to send messages to the agents in the local neighbourhood. In this approach each agent is the centre of a neighbourhood, those agents contained in the neighbourhood from none, i.e. no communication, to all other agents receive any messages sent by the central agent. As neighbourhoods might overlap an agent can be a member of a number of neighbourhoods, for example, for the agent network, shown in Figure 3, neighbourhoods are constructed from an agents nearest neighbours, thus sixteen 2 6 0 N . Ire s o n e t a l. groups, each centred on one agent, are formed: Group 1: 1, 2, 5 Group 2: 2, 1, 3, 6 Group 3: 3, 2, 4, 7 ... Group 6: 6, 2, 5, 7, 10 Group 7: 7, 3, 6, 8, 11 ... Group 16: 16, 12, 15. This approach is general known as multicast or one-to-many communication. There are a number of possible approaches to this form of communication in distributed systems from ooding where each node sends a copy of the incoming message to all the connected nodes except for the message source node to routing where messages follow a pre-specied path to their destination node. The principal problem with ooding is that it causes a great deal of redundant communication, also as each node must check if it has already received each incoming message, with frequent communication this can lead to a communication overload. Routing requires a more complex initialisation process but minimises the communication trac. Each node when it receives a message refers to a lookup table giving the nodes to which copies of the message are sent. Unlike ood broadcast there is no redundant communication, thus the severing of a communication link will cause at least one node to fail to receive messages. In practice the choice of method should reect the constraints of the system, i.e. the trade-o between fault tolerance and communication load. 3.2 Initialisation of the Agent Network The initialisation process rstly involves the setup and opening of communication channels from agent to user interface, agent to application and agent to other agents. Once this has been successfully completed, the classier system can be initialised, the agent then waits for the rst message from the application to begin its control process. The initialisation of communication involve each agent connecting to the user interface, application and other agents. All these channels might involve two-way communication. During the initialisation the agents open a communication channel and await a connection message. The channel is tested to ensure the communication is setup correctly as although the conguration parameters have been previously checked for consistency, the parameters may be inconsistent with the physical communication process, also this process might be faulty. Although the term socket" is used in the specication as the medium to connect communication channels in implementation other methods can be used, such as calls to remote objects, when using RMI or DCOM. The basis of the communication initialisation and run-time processes are not aected. The creation of the communication object and binding in a remote registry on a given hostname and port replaces the creation of a server socket and calls to the remote object replace read and write calls to the sockets. Note that it is possible for the communication server to create separate processes to listen on the communication channel for messages, this allows the A C o m m u n ic a tio n A rc h ite c tu re fo r M u lti-a g e n t L e a rn in g S y s te m s C o m m u n ic a tio n C o n fig u ra tio n 2 6 1 P a ra m e te rs C o m m u n ic a tio n S e rv e r 2 . C re a te S e rv e r 3 . C o n n e c t (C o m m u n ic a tio n S e rv e r) C o n n e c to r 4 . R e q u e st C o n n e c tio n 5 . R e q u e st C o n n e c tio n U s e r In te rfa c e , A p p lic a tio n o r A g e n t N e ig h b o u r 1 . C re a te M o n ito r 6 . A d d C o n n e c tio n C o n n e c to r M o n ito r 7 . A c c e p t C o n n e c tio n C o m m u n ic a tio n S e rv ic e 8 . In itia lis e S tre a m s C o n n e c tio n Figure 4: Structure of communication server agent to be reactive to external messages. The communication with neighbours requires a single channel for incoming messages, and separate channels from sending to each neighbour except if the messages are broadcast on sent via a proxy. The initialisation of the communication server, as shown in Figure 4, involves the following steps: 1. The Communication Server object creates the speci c Communication Services Application, User Interface or Neighbourhood as speci ed by the con guration. 2. The Communication Server object create a monitor which maintains the list of current connections. 3. The Communication Server object passes the Communication Service object and connection con guration information to the Connector object which, for connection with the User Interface and Application and incoming channel from the neighbouring agents, opens a Server Socket on the speci ed port and waits for a request to connect. For the outgoing channel to the neighbouring agents the Connector object intermittently requests a connection to the neighbours speci ed port. 4. The User Interface, Application or Neighbouring Agent sends a request to connect. 5. The request to connect is accepted by the neighbour's server socket. 6. The Connector sends the Communication Service object and open socket to the Communication Monitor. 7. The Communication Monitor object tests the communication channel, if the test succeeds the Communication Service is passed to the Connection object, otherwise the socket is closed and the failure reported. 2 6 2 N . Ire s o n e t a l. Figure 5: Class diagram of the communication 8. The Connection object starts the thread to handle the connection and passes the input and output streams to the Communication Service object. 3.3 Logical View of the Communication A class diagram of the communication is given in Figure 5. The function of each class object in the communication is described briey as follows: Communication Server: The Communication server supports a number of multi-threaded channels. It creates or opens each channel on a speci ed port allowing the Agent to communicate with the User Interface, Application and other Agents. It provides the ability to send messages whilst listening on the port for incoming messages which are passed to the appropriate objects. Connection Monitor: The Connection Monitor object maintains a list of the current connections. The thread waits to be noti ed if a connection terminates and updates the list. Connector: The Connector class either listens for a connection on a speci ed port using a server socket or connects to another agent's server socket. Once accepted the socket is sent to the Connection Monitor. Connection: Connection objects are created by the Connector thread using the Communication Monitor method addConnection. It simply creates a thread to handle the connection. A C o m m u n ic a tio n A rc h ite c tu re fo r M u lti-a g e n t L e a rn in g S y s te m s I II III IV 2 6 3 Figure 6: The simulated trac environment Communication Service: A general class for each of the types of communication required. Listener: An object that listens on a communication channel for an incoming message. User Interface: The communication service require by the User Interface. This object handles messages received from or sent to the User Interface. Incoming messages are interpreted and, if necessary, call the appropriate function. Outgoing messages are packaged and sent to the User Interface. Application: The communication service require by the Application. This object handles messages received from or sent to the Application. Incoming messages generally system state or rewards are interpreted and, if necessary, sent to the appropriate objects such as classi ers in the classi er system. Outgoing messages general actions are packaged and sent to the Application. Neighbourhood: The communication service require by the Agent Neighbourhood. This object handles messages received from or sent to the Agent Neighbourhood. Incoming messages are interpreted and, if necessary, sent to the appropriate functions in the classi er system. Outgoing messages are packaged and sent to the neighbouring agents. 4 Application Example Optimization of a group of trac signals over an area is typical multi-agent type real-time planning problem without precise reference model given. To do this planning, each signal should learn not only to acquire its control plans individually through reinforcement learning but also to cooperate with each other. This requires communication between the agents. In this example, we developed a multi-agent learning system, which is aimed at learning the ecient control rules for the dynamic trac environment and with the communication provided by 2 6 4 N . Ire s o n e t a l. 5.5 5 4.5 Traffic Speed 4 3.5 3 2.5 2 1.5 0 1000 2000 3000 4000 5000 6000 Time Steps 7000 8000 9000 10000 Figure 7: Performance comparison of dierent control strategies the developed communication architecture. Each agent has a classier system providing the control strategy and a communication server which is used to connect the agent to the user interface, the application and to other agents. To control a tra c network, we associate an agent to each junction of the tra c network. The agents are initialised according to the tra c network conguration and user-specied parameters. For the simulated 2 2 tra c network, as shown in Figure 6, four agents, i.e., agents I, II, III, and IV, associating with junctions I, II, III, and IV, are need to provide comprehensive control of the network. Agent I has the neighbouring agents II and III, and agent II has the neighbouring agents I and IV, etc. The communication server in each agent provides the control actions of its neighbouring agents, and these information is used to construct control rules for its junction. The classier system employed is a version of Wilson's zeroth-level" system ZCS 19, with some changes on the classier representation 20. The condition part of each classier consists of six bits, which reects the scalar level of queue length from each direction and the previous actions of the neighbouring agents. In this application, the scalar level of the queue length is set to 4, which ranges from 0 to 3, corresponding to the four linguistic variables, fzero small medium large g. The action part indicates the required state of the signal. For instance, for junction I, the rule 130201:1 says that if the queue from directions east and west are small 1 and zero 0, but the queue from directions south and north are large 3 and medium 2, and the previous neighbourhood junction controllers' actions are vertically red 0 junction II and green 1 junction III, then the tra c light stays green vertically 1 for a xed period of time. The performance evaluation, reinforcement learning strategy, genetic algorithm and the simulated tra c environment are all similar to those used in 20. For comparison purpose, two types of control strategies are employed: random control strategy and the developed multi-agent learning system MALS strategy. The random control strategy determines the tra c light's state  0 or 1 randomly at 50 of probability whilst MALS strategy determines the tra c light's state according to the action of the winning classier of the agent. A C o m m u n ic a tio n A rc h ite c tu re fo r M u lti-a g e n t L e a rn in g S y s te m s 2 6 5 Experiments were carried out for three dierent types of trac conditions. In these simulations, the mean arrival rates for the cars are the same but the number of cars in the area is limited to 30, 60, and 90, corresponding to a sparse, medium, and crowded trac condition. In all cases, the MALS strategy is found to learn how to reduce the average queue length and improve the trac speed in the network. For example, Figure 7 shows the average performances of the random control strategy and MALS strategy respectively over 10 runs in the crowded case, where the solid line represents MALS strategy and the dotted line represents random control strategy. It can be seen that the MALS strategy consistently learns and improves the trac speed over 10,000 iterations. 5 Conclusion and Future Work We have extended the framework of the multi-agent learning systems in 10, 12 to a general case and developed a simple communication architecture for these systems. The service provided by the communication architecture allows each agent to connect to the user interface, the application and the other agents. An application example shows that the communication architecture is reliable and ecient. Although the communication architecture is implemented using TCPIP, it can also be implemented using RMI or DCOM, via binding the objects in the remote registry and making calls to the remote objects. 6 Acknowledgment This work was carried out as part of the ESPRIT Framework V Vintage project ESPRIT 25.569. References 1 Tamble, M., Rosenbloom, P. S.: RESC: An approach for real-time, dynamic agent tracking. In Proc. of the International Joint Conference on Articial Intelligence, Montreal, Canada, 1995 2 Hayes-Roth, B., Brownston, L., Gen, R. V.: Multiagent collaboration in directed improvisation. In Proc. of International Conference on Multi-Agent Systems. USA 1995 3 Kitano, H., Asada, M. Kuniyoshi, Y., Noda, I., Osawa, E.: The robot world cup initiative. In Proc. IJCAI-95 Workshop on Entertainment and AIAlife, Montreal, Canada 1995 4 Kuniyoshi, Y., Rougeaux, S., Ishii, M., Kita, N., Sakane, S., Kakikura, M.: Cooperation by observation: the framework and the basic task pattern. In Proc. IEEE International Conference on Robotics and Automation. 1994 5 Weiss, G. and Sen, S. eds : Adaptation and Learning in Multi-Agent Systems. Springer-Verlag, Berlin, Heidelberg, New York, 1995 6 Sen, S. ed : AAAI Spring Symposium on Adaptation, Coevolution and Learning in Multiagent Systems. AAAI Press, 1996 7 Sen, S. Sekaran, M. and Hale: Learning To Coordinate without Sharing Information. In Proceedings of the Twelfth National Conference on Articial Intelligence, 1994 426-431. 2 6 6 N . Ire s o n e t a l. 8 Prasad, M.V.N. and Lesser, V.R.: Learning Problem Solving Control in Cooperative Multi-Agent Systems. Workshop on Multi-Agent Learning AAAI-97, 1997 9 Seredynski, F.: Coevolutionary Game-Theoretic Multi-Agent Systems: the Application to Mapping and Scheduling Problems Technical Report TR-96-045 Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland. 1996 10 Seredynski, F., Cichosz, P. and Klebus, G. P: Learning classier systems in MultiAgent Environments, In Proc. First IEEIEEE International Conference on Genetic Algorithms in Engineering: Innovations and Applications, 1995 287292 11 Bull, L., Fogarty, T. C., and Snaith, M.: Evolution in Multi-Agent Systems: Evolving Communicating Classier Systems for Gait in a Quadrupedal Robot. In Eshelman, L. J. ed : Proceedings of the Sixth International Conference on Genetic Algorithms, Morgan Kaufmann, 1995 382388 12 Bull, L: On ZCS in Multi-Agent Environments. Parallel Problem Solving From Nature - PPSN V, Springer Verlag 1998 471480 13 Fleury, G., Goujon, J., Gourgand, M. and Lacomme, P., Multi-agent approach and stochastic optimization: random events in manufacturing systems. Journal of Intelligent Manufacturing, 10, 1, 1999 81102 14 Cao, Y. J. and Wu, Q. H.: A mixed-variable evolutionary programming for optimisation of mechanical design. International Journal of Engineering Intelligent Systems, 7, 2, 1999 7782 15 Cao, Y. J. and Wu, Q. H.: An improved evolutionary programming approach to economic dispatch. International Journal of Engineering Intelligent Systems, 6, 2, 1998 187194 16 Cao, Y. J. and Wu, Q. H.: Optimisation of control parameters in genetic algorithms: a stochastic approach. International Journal of Systems Science, 20, 2, 1999 551559 17 Kouiss, K., Pierreval, H. and Mebarki, N., Using multi-agent architecture in FMS for dynamic scheduling. Journal of Intelligent Manufacturing, 8, 1, 1997 4148 18 Wooldridge, M. and Jennings, N.R.: Intelligent agents: theory and practice. In The Knowledge Engineering Review, 10 2, 1995 115-152. 19 Wilson, S. W.: ZCS: A zeroth level classier system. Evolutionary Computation, 2, 1994 118 20 Cao, Y. J., Ireson, N. I., Bull, L. and Miles, R.: Design of Trac Junction Controller Using a Classier System and Fuzzy Logic. In Computational Intelligence: Theory and Applications, Reusch, B. ed, Lecture Notes in Computer Sciences, 1625, Springer Verlag, 1999 342353 A n A m b u la n c e C r e w R o s te r in g S y s te m P . V . G . B r a d b e e r † , C . F in d la y ‡ a n d T .C F o g a r t y .¶ † F if e C o lle g e o f F u r th e r a n d H ig h e r E d u c a tio n ., p v g b @ c it.f if e .a c .u k F ife A m b u la n c e S e rv ic e . ¶ N a p ie r U n iv e r s ity ., t.f o g a r ty @ d c s .n a p ie r .a c .u k ‡ A b str a c t T h e p r o d u c tio n o f a r o s te r fo r p r a c tic a l, m a n a g e r ia l a n d s o c o f a n in v e s tig a tio n in to th e c h a r e p o r ts o n th e s u c c e s s to d a te o c e p ta b le s o lu tio n to th e p r o b le a q u ic k e r w a y o f te s tin g th e a fo r m m u ltip le tim e c o n s u m in g th e d u tie s o f a m b u la n c e c r ia l c o n s tr a in ts . T h is d o c u m r a c te r is tic s o f th e s e a r c h s p f a n e v o lu tio n a r y a lg o r ith m m . T h e v is u a lis a tio n m e th o p p r o p r ia te n e s s o f r e p r e s e n e x p e r im e n ts . e w is s u b je c t to a v a r ie ty o f e n t d e s c r ib e s th e fir s t s te p s a c e fo r s u c h a p r o b le m a n d a p p r o a c h in fin d in g a n a c d d e s c r ib e d is s u g g e s te d a s ta tio n s th a n h a v in g to p e r - S e c tio n 1 - I n tr o d u c tio n R e c e n tly , th e d e c is io n w a s m a d e to re -e s ta b lis h th e a m b u la n c e s e rv ic e s u b -s ta tio n a tta c h e d to th e V ic to ria H o s p ita l in K irk c a ld y . T h e p re v io u s a rra n g e m e n t w a s to p ro v id e a ll a m b u la n c e s e rv ic e s fro m a c e n tra l b a s e in G le n ro th e s . ‘F ro n t L in e ’ p e rs o n n e l fo r th e n e w s e rv ic e c o m p ris e s a g ro u p o f e ig h t s ta ff, fo u r o f w h o m a re d e s ig n a te d p a ra m e d ic s a n d h a v e re c e iv e d s p e c ia lis e d tra in in g fo r th e ir d u tie s , a n d fo u r s ta ff w h o a re d e s ig n a te d m e d ic a l te c h n ic ia n s , a ls o a fte r a p p ro p ria te tra in in g . T h is g ro u p o f s ta ff a re re q u ire d to p ro v id e tw e n ty fo u r h o u r a d a y , s e v e n d a y a w e e k c o v e ra g e . E a c h a m b u la n c e is c re w e d a t a ll tim e s b y a te a m c o n s is tin g o f o n e p a ra m e d ic a n d o n e te c h n ic ia n , w o rk in g a tw e lv e h o u r s h ift. C le a rly s u c h a c o v e ra g e s c h e m e re q u ire s c a re fu l a rra n g e m e n t to e n s u re th a t s e n s ib le ro s te rs a re p ro d u c e d fo r a ll o f th e in d iv id u a ls c o n c e rn e d . T h e re a re a n u m b e r o f c o n s tra in ts o n w h e n in d iv id u a ls a re a v a ila b le to c re w a v e h ic le . G iv e n th e n a tu re o f th e d u tie s th a t c re w m e m b e rs u n d e rta k e it is n o t a p ra c tic a b le o p tio n to a llo w d o u b le s h ift w o rk in g , fo r b o th th e s a fe ty o f u s e rs o f th e s e rv ic e a n d th e c re w s th e m s e lv e s . T h is c o m p le m e n t o f s ta ff im p lie s th a t a c re w m e m b e r is re q u ire d to p e rfo rm a n a v e ra g e o f th re e a n d a h a lf s h ifts p e r w e e k . In p ra c tic e th is tra n s la te s to e ith e r th re e o r fo u r s h ifts in o n e s e v e n d a y p e rio d . E x p e rie n c e h a s s h o w n th a t re g u la rly w o rk in g e x c e s s s h ifts in a w e e k to b e b o th p h y s ic a lly a n d e m o tio n a lly d ra in in g , a n d th is is to b e a v o id e d a s fa r a s p o s s ib le . In a d d itio n to th e p ra c tic a l c o n s tra in ts d e s c rib e d , m a n a g e m e n t h a v e d e c id e d a g a in s t fo rm in g p e rm a n e n t p a irin g s . T h is m e a n s th a t e a c h p o s s ib le c o m b in a tio n o f p a ra m e d ic a n d te c h n ic ia n m u s t w o rk to g e th e r a t s o m e s ta g e in th e s c h e d u le . T h is is s e e n to b rin g w ith it S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 6 7 − 2 7 9 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 6 8 P .V .G . B r a d b e e r , C . F in d la y , a n d T .C . F o g a r ty th e b e n e fits o f p ro m o tin g ‘s k ill p u ll-th ro u g h ’ a n d e n s u rin g th a t th e re la tio n s h ip s b e tw e e n p a ra m e d ic s a n d te c h n ic ia n s re m a in p ro d u c tiv e . In th e in te re s ts o f ‘fa irn e s s ’ it is d e s ira b le th a t e a c h o f th e s ix te e n (fo u r p a ra m e d ic s a n d fo u r te c h n ic ia n s ) p o s s ib le te a m s a re a c tiv e a n e q u a l n u m b e r o f tim e s . A fu rth e r c o m p lic a tio n a ris e s fro m th e a m b u la n c e s ta ff th e m s e lv e s . M a n y o f th e m h a v e fa m ilie s , a n d a re d e s iro u s o f s p e n d in g ‘q u a lity tim e ’ w ith th e m . T h is n o t o n ly p ro m o te s d e v e lo p m e n t o f th e fa m ily u n it, b u t a ls o a llo w s th e s ta ff to u n w in d a fte r s o m e o f th e m o re h a rro w in g in c id e n ts th a t th e y a re re q u ire d to a tte n d . P o s s ib ly th e m o s t im p o rta n t o p p o rtu n ity is to s p e n d tim e w ith a fa m ily is d u rin g th e w e e k e n d , w h e n c h ild re n a re n o t a t s c h o o l. It is c le a rly n o t p o s s ib le to a v o id w e e k e n d w o rk , b u t th e s ta ff h a v e a s ta te d p re fe re n c e to e ith e r w o rk a c o m p le te w e e k e n d , o r to h a v e n o s h ifts in a w e e k e n d a t a ll. T h e in te rm e d ia te s itu a tio n w h e re o n ly o n e s h ift is w o rk e d in a w e e k e n d is re fe rre d to in h o u s e a s a ‘ru in e d w e e k e n d ’. T h e re is th u s a re q u e s t th a t s in g le s h ift w e e k e n d s a re a v o id e d b y a n y s y s te m u s e d to g e n e ra te a m b u la n c e s ta ffin g ro s te rs . T h e re is a ls o th e n o tio n o f p e rc e iv e d ‘fa irn e s s ’ w ith in a ro s te r. T h is is b e s t v ie w e d a s a d e s ire to h a v e a ll m e m b e rs o f s ta ff w o rk in g a n e q u a l n u m b e r o f d a y /n ig h t s h ifts a n d th e s a m e n u m b e r o f w e e k e n d /w e e k d a y s h ifts . T h is p re v e n ts a n y a c c u s a tio n s o f b ia s w h e n s lig h t im b a la n c e s a re n o tic e d in ro s te rs . S ta ff le a v e re q u ire s n o s p e c ia l tre a tm e n t, a s s ta ff a re s e c o n d e d fro m th e c e n tra l a m b u la n c e b a s e in G le n ro th e s to fill in fo r s c h e d u le d h o lid a y s . T h is m a y c a u s e s lig h t im b a la n c e s in th e fre q u e n c y w ith w h ic h in d iv id u a ls a re p a rtn e re d , b u t th is is n o t re g a rd e d a s im p o rta n t. S e c tio n 2 - A p p r o a c h 1 A tte m p ts to p ro d u c e a ro s te r b y h a n d p ro v e d th a t th e p ro b le m w a s q u ite d iffic u lt, a n d a c o m p u te ris e d s o lu tio n w a s s o u g h t. A t firs t g la n c e th e d e v e lo p m e n t o f a s o ftw a re s y s te m to a u to m a tic a lly g e n e ra te u s e a b le a m b u la n c e ro s te rs w ith in th e c o n s tra in ts d e s c rib e d in s e c tio n 1 lo o k s a s th o u g h a s im p le d e p th firs t re c u rs iv e tre e s e a rc h w o u ld b e a fe a s ib le a p p ro a c h , a s th e c h o ic e o f th e ‘n e x t s h ift’ is s o h ig h ly c o n s tra in e d th a t c o n s id e ra b le p ru n in g o f th e s e a rc h tre e w o u ld b e p o s s ib le . U n fo rtu n a te ly th e e ffe c t o f s o m e o f th e c o n s tra in ts c a n o n ly b e a s s e s s e d w h e n th e tre e h a s b e e n g ro w n to s o m e d e p th . T h is h a s th e e ffe c t o f m a k in g e x h a u s tiv e s e a rc h a n n o n -v ia b le o p tio n e v e n if v e ry fa s t c o m p u ta tio n a l m a c h in e ry w e re a v a ila b le . T h is le a d s u s in to th e a re a w h e re a s e a rc h b a s e d o n e v o lu tio n a ry te c h n iq u e s s u g g e s ts its e lf a s a p o s s ib ility . E v e n fro m th e o u ts e t, th e c o n s tra in ts p la c e d o n th e s y s te m s u g g e s t th a t fin d in g a s o lu tio n w ill p re s e n t a c o n s id e ra b le c h a lle n g e , b u t it w a s fe lt th a t g iv e n th e p a s t s u c c e s s e s in s o lv in g s c h e d u le b a s e d p ro b le m s w ith g e n e tic a lg o rith m s (G A s ), in c lu d in g th o s e d e s c rib e d in L a n g d o n [L a n g d o n 1 9 9 5 ] W re n a n d W re n [W re n 1 9 9 5 ] a n d F a n g e t. a l. [F a n g 1 9 9 3 ], th e re w a s a re a s o n a b le c h a n c e o f s u c c e s s . A n A m b u la n c e C re w R o s te rin g S y s te m 2 6 9 T h e a lp h a b e t in itia lly c h o s e n to re p re s e n t th e g e n e tic m a te ria l in th e c a n d id a te s o lu tio n s h a s s ix te e n c h a ra c te rs , o n e fo r e a c h o f th e d iffe re n t te c h n ic ia n p a ra m e d ic p a irin g s . T h is w a s p a rtly to a llo w e a s y c h e c k in g to s e e if th e p a rtn e rs h ip c o n s tra in ts w e re b e in g m e t. T h e s e le c tio n o f th e le n g th o f th e g e n e tic m a te ria l (c h ro m o s o m e le n g th ) is a ls o a n is s u e , a s th e le n g th o f th e ro s te r is n o t s p e c ifie d . U n d e r s u c h c irc u m s ta n c e s a n a p p lic a tio n o f m e s s y G A s [G o ld b e rg 1 9 9 0 ] m a y p ro v e p a rtic u la rly a p p ro p ria te , b u t fo r a n in itia l in v e s tig a tio n a fix e d le n g th re p re s e n ta tio n w a s c h o s e n . A s th e re a re s ix te e n te a m s to a c c o m m o d a te th e fix e d le n g th w a s s e t a t s ix te e n w e e k s , g iv in g a p a tte rn th a t re p e a ts th re e tim e s a y e a r. E a c h o f th e s e s ix te e n w e e k c y c le s w o u ld id e a lly h a v e e a c h p a rtn e rs h ip a c tiv e fo u rte e n tim e s (s e v e n o n d a y s h ift a n d s e v e n o n n ig h ts ). A y e a r c o u ld th u s c o m p ris e th re e o f th e s e c y c le s , le a v in g a g a p o f a b o u t fo u r w e e k s . T h is is c o n v e n ie n t fo r th e d is ru p tio n in p a tte rn re q u ire d fo r th e C h ris tm a s /N e w Y e a r p e rio d w h e n th e w o rk in g re q u ire m e n ts a re s u b je c t to c h a n g e . T h e fe s tiv e p e rio d is a b u s y tim e fo r th e e m e rg e n c y s e rv ic e s . N o te th a t if w o rk in g to th e s e s ix te e n w e e k ro s te rs th a t th e ‘w ra p a ro u n d ’ e ffe c t fro m th e e n d o f o n e c y c le to th e b e g in n in g o f th e n e x t m u s t b e ta k e n in to a c c o u n t. F a ilu re to d o s o m a y re s u lt in d o u b le s h ift w o rk in g b e tw e e n th e e n d o f a c y c le , a n d th e b e g in n in g o f th e n e x t. T h e firs t in c a rn a tio n o f g e n e tic s e a rc h w a s la rg e ly to g a in in s ig h t in to th e c h a ra c te ris tic s o f th e s p a c e , a n d u s e d a fa irly tra d itio n a l a p p ro a c h , in th a t e a c h lo c u s o n th e c h ro m o s o m e w a s a llo w e d to ta k e a n y o f th e a llo w a b le a lle le s , a lo w m u ta tio n ra te , u n ifo rm c ro s s o v e r [S y s w e rd a 1 9 8 9 ] a n d b in a ry to u rn a m e n t s e le c tio n [B rin d le 1 9 8 1 ] (w ith 1 0 0 % c h a n c e o f th e b e tte r o f th e tw o c a n d id a te s p ro g re s s in g ). F o llo w in g th e s u c c e s s re p o rte d b y M ille r e t. a l. [M ille r1 9 9 5 ], J o n e s a n d B ra d b e e r [J o n e s 1 9 9 4 ] a n d C h is h o lm a n d B ra d b e e r [C h is h o lm 1 9 9 7 ] u s in g s m a ll b re e d in g p o o l s iz e s , a re la tiv e ly s m a ll b re e d in g p o o l o f th irty w a s in itia lly a d o p te d . A ra n d o m s a m p le o f fiv e th o u s a n d in d iv id u a ls g e n e ra te d th e d is trib u tio n s h o w n a s fig u re 1 . T h is s u g g e s ts th a t a la rg e p o rtio n o f th e p o p u la tio n h a v e p o o r fitn e s s , a n d le a v e s u s h o p in g th a t th e re a re v e ry lo n g (if th in ) ta ils to th e d is trib u tio n . T h is ty p e o f d is trib u tio n , a llie d w ith th e fa c t th a t th e re a re m a n y c o n s tra in ts o n th e s e a rc h s p a c e re in fo rc e s th e s u s p ic io n th a t th is w ill b e a d iffic u lt p ro b le m fo r th is b ra n d o f G A . It is w o rth c o m m e n tin g th a t th e re w a s little c o d e p ro d u c e d fo r th is in itia l s y s te m th a t w a s n o t to b e o f u s e in s u b s e q u e n t im p le m e n ta tio n s . T h e fitn e s s fu n c tio n u s e d fo r e v a lu a tio n p u rp o s e s is b a s e d o n th e a c c ru a l o f p e n a ltie s , w ith d iffe re n t w e ig h ts b e in g a s s ig n e d to b re a c h e s o f d iffe re n t c o n s tra in ts . E a c h tim e a c o n s tra in t is v io la te d th e in te g e r re p re s e n tin g th e fitn e s s o f th e c a n d id a te s o lu tio n is in c re a s e d . T h is m e a n s th a t in th e re p o rtin g o f re s u lts a lo w fitn e s s n u m b e r in d ic a te s a a g o o d s o lu tio n . 2 7 0 P .V .G . B r a d b e e r , C . F in d la y , a n d T .C . F o g a r ty Initially the following penalties were allocated C o n s tra in t b re a c h P e n a lty le v e l D a y /n ig h t im b a la n c e 4 8 h o u r+ p e r se v e n d a y s 1 D o u b le S h ift 1 1 R u in e d W e e k e n d s 1 Table 1: Initial Penalty Scheme T h is a p p o rtio n m e n t o f p e n a ltie s im p lie s th a t a ll b re a c h e s o f c o n s tra in ts a re e q u a l, w h ic h is c le a rly a n o v e rs im p lific a tio n o f th e p ro b le m . In re a lity , a s ix te e n w e e k ro s te r th a t p ro d u c e d o n e o r tw o b re a c h e s o f th e 4 8 h o u r ru le w o u ld p ro b a b ly b e a c c e p ta b le , w h e re a s a s c h e m e w ith th e s a m e n u m b e r o f d o u b le s h ifts w o u ld n o t b e a c c e p ta b le . T h o s e c o n s tra in ts th a t m u s t n o t b e b re a c h e d a re u s u a lly re fe rre d to a s h a r d c o n s tr a in ts , w h ile th o s e fo r w h ic h m in o r o r lim ite d b re a c h e s c o u ld b e to le ra te d a re re fe rre d to a s s o ft c o n s tr a in ts . U s u a lly a h a rd c o n s tra in t v io la tio n w o u ld a ttra c t a h ig h e r p e n a lty v a lu e th a n a s o ft p e n a lty . M ic h c h a le w ic z [M ic h c h a le w ic z 1 9 9 4 ] p ro v id e s a d is c u s s io n o f s o m e o f th e s o ft c o n s tra in ts th a t c a n a p p e a r in tim e ta b lin g p ro b le m s , a n d th e c o n tra s tin g h a rd c o n s tra in ts . T h e a p p o rtio n m e n t o f re la tiv e p e n a lty le v e ls is a n a re a o f in te re s t in its o w n rig h t, a n d c a n e ith e r b e s ta tic , o r d y n a m ic a s c a n b e s e e n fo r e x a m p le in E ib e n e t. a l. [E ib e n 1 9 9 8 ] w h e re d iffe re n t a p p ro a c h e s a re c o m p a re d . A s th is is o f th e s y s te m , a n p e n a lty re s u lt. F o r m o re m o re a tte n tio n , [P a e c h te r1 9 9 8 ] w a n in itia l s tu d y h o w e v e r w e a re m o re in te re s te d in th e g ro s s p e rfo rm a n c e d m e re ly n o te th a t a n id e a l s o lu tio n to th is p ro b le m w o u ld p ro d u c e a z e ro d e ta ile d s tu d y , th e e x a c t w e ig h t g iv e n to e a c h c la s s o f b re a c h w o u ld n e e d p o s s ib ly u s in g a n a p p ro a c h s u c h a s th a t d e ta ile d b y P a e c h te r e t. a l. h e re a ‘fro n t p a n e l’ is a tta c h e d to th e s y s te m a llo w in g d y n a m ic a lte ra tio n A n A m b u la n c e C re w R o s te rin g S y s te m 2 7 1 o f th e w e ig h ts . It is c e rta in ly p o s s ib le th a t d iffe re n t p e n a lty le v e ls c o u ld c h a n g e th e tra je c to ry o f th e s e a rc h . Fitness Distribution (30000 samples) 4000 3500 Frequency 3000 2500 2000 1500 1000 500 580 560 540 520 500 480 460 440 420 400 380 360 340 320 300 280 0 Fitness F ig u re 1 : D is trib u tio n o f fitn e s s fo r ra n d o m T h is s a a b o u t fo u r h u n th is v ic in ity . It tw e n ty c o n s tra m p le in d re d a n is d iffic in ts , s o d ic a te s th a t th e d fo rty c o n s tra in u lt to b e lie v e th a th a t le a v e s a c o n s a m p le u s in g ‘u n c o n s tra in e d ’ re p re s e n ta tio n . a v e ra g e t re q u ire m t a n a c c e p s id e ra b le ra n d o m ly e n ts , a n d ta b le s o lu c h a lle n g e g th tio fo e n e e b n w r th ra te u lk ill b e se d in d iv id o f th e p o re a c h m o a rc h m e c u a p u re h a l b re a c h e s la tio n is in th a n a b o u t n is m . In o rd e r to p ro g re s s th e s e a rc h , s o m e s o rt o f m o v e o p e ra to r m u s t b e e m p lo y e d . It is u s e fu l to h a v e a n in d ic a tio n o f th e lik e ly e ffe c tiv e n e s s o f o p e ra to rs . R a n d o m m u ta tio n is o n e o f th e o p e ra to rs tra d itio n a lly u s e d , a n d th e e ffe c t o f th is o p e ra to r in th is re p re s e n ta tio n in th is p ro b le m is s u m m a ris e d in fig u re 2 . M u ta tio n in th is c a s e in v o lv e s th e s e le c tio n o f a s in g le lo c u s o n th e c h ro m o s o m e , a n d re p la c in g it w ith a s in g le ra n d o m ly g e n e ra te d le g a l v a lu e . T h e fig u re is g e n e ra te d fro m a s a m p le o f 3 0 0 0 0 in d iv id u a ls , e a c h o f w h ic h re c e iv e s o n e m u ta tio n . T h e re s u ltin g c h a n g e in fitn e s s d u e to th e m u ta tio n is n o te d a g a in s t th e fitn e s s o f th e o rig in a l. R a th e r th a n k e e p re c o rd s o f in d iv id u a l fitn e s s n u m b e rs , th e o u tc o m e s a re g ro u p e d in to ra n g e s , a n d a v e ra g e d . T h e g ra p h s h o w s th e p h e n o m e n o n o f a b o v e a v e ra g e fitn e s s in d iv id u a ls g e n e ra lly b e in g a d v e rs e ly a ffe c te d b y ra n d o m m u ta tio n s , w h e re a s b e lo w a v e ra g e fitn e s s in d iv id u a ls a re o n a v e ra g e im p ro v e d . T h e k in k s a t th e e x tre m itie s c a n b e e x p la in e d a s d u e to s m a ll s a m p le s iz e s a t th e ta ils o f th e d is trib u tio n . 2 7 2 P .V .G . B r a d b e e r , C . F in d la y , a n d T .C . F o g a r ty N o s te p s w e re ta k e n to p re v e n t th e m u ta tio n re p lic a tin g th e o rig in a l v a lu e . A s a n a lp h a b e t o f s ix te e n w a s u s e d , a p p ro x im a te ly o n e in s ix te e n m u ta tio n s re s u lt in n o c h a n g e . T h is m a y b e v ie w e d a s re d u c in g th e s lo p e o f th e g ra p h s lig h tly . A g r in d u c e s s e v A se ly p o o r re s u a p h e re rie s lts w ith a g ra e p is ta tic e o f ru n s (w to c o n firm d ie ffe ith th n ts lo w a c ts , o r th a ra n g e a t a b e tte s th is a t th e o f p a r r re p r c a n p ro a m e s e b e ta b le m e te r s n ta tio k e n sp a e ts ) n sh a s in d ic c e h a s n u s in g th o u ld b e a tin g th a t th e re p re s e n ta tio n o s tru c tu re . is a p p ro a c h g a v e s u ffic ie n ts o u g h t. Average Change due to Mutation 7 6 5 Change 4 3 2 1 560 540 520 500 480 460 440 420 400 380 360 340 -1 320 0 -2 Original fitness F ig u r e 2 : A v e r a g e c h a n g e in fitn e s s u n d e r m u ta tio n .( 3 0 0 0 0 s a m p le s ) S e c tio n 3 - A p p r o a c h 2 In s tru m e n ta tio n o f th e s o u rc e o f p e n a lty p o in ts w a s d n u m b e r o f tim e s e a c h te a m w a s e n ta tio n . A s a w a y o f re d u c in e m p lo y e d , w ith e a c h te a m a p p T h is c h a n g e la rg e ly fitte d in to c o d e c h a n g e s. A s b e fo re , a sa m c e rta in if a n y b e n e fit a c c ru e d . ru u s g n s o f th e firs t a p p ro a c h s e e m e d to in d ic a te th a t th e e to d iffic u lty in m a in ta in in g th e e v e n s p re a d b e tw e ro s te re d . C le a rly , th e n e x t s te p w a s to re c o n s id e r th e th e im b a la n c e p e n a ltie s a p e rm u ta tio n re p re s e n ta tio e a rin g a g iv e n n u m b e r o f tim e s w ith in a s ix te e n w e e k th e fra m e w o rk o f th e p re v io u s s y s te m , w ith re la tiv e ly p le o f ra n d o m ly g e n e ra te d in d iv id u a ls w a s e v a lu a te d T h e re s u lts a re n o te d in fig u re 3 . m a jo r e n th e re p re n w a s ro s te r. s m a ll to a s - A n A m b u la n c e C re w R o s te rin g S y s te m Permutation vs Unconstrained representation s 2 7 3 Permutation Unconstrained 4000 3500 Frequency 3000 2500 2000 1500 1000 500 570 550 530 510 490 470 450 430 410 390 370 350 330 310 290 270 0 Fitness F ig u re 3 : D is trib u tio n o f fitn e s s fo r ra n d o m s a m p le u s in g s in g le p e rm u ta tio n c h o m o s o m e c o m p a re d to u n c o n s tra in e d c h ro m o s o m e (3 0 0 0 0 s a m p le s ). C o m p a rin g fig u re 3 w ith fig u re 1 w e s e e th a t th e d is trib u tio n h a s m o v e d fa r e n o u g h to p ro v id e e n c o u ra g e m e n t to c o d e th e re s t o f th e s y s te m to p e rfo rm fu rth e r te s t ru n s . Effect of mutation on permutation representation 6 4 2 -4 -6 -8 -10 -12 -14 Fitness 570 550 530 510 490 470 450 430 410 390 370 350 330 310 Change 0 -2 2 7 4 P .V .G . B r a d b e e r , C . F in d la y , a n d T .C . F o g a r ty F ig u re 4 : A v e ra g e c h a n g e in fitn e s s to p e rm u ta tio n re p re s e n ta tio n u n d e r m u ta tio n .(3 0 0 0 0 s a m p le s ) O n c e a g a in p e rfo rm in g th e c h a n g e u n d e r m th e re s p o n s e is v e ry lo w , a g a in le a d in g u s to s u s p e c tio n le ft in th e re p re s e n ta tio n . U s in g P M X [G o ld b e rg 1 9 8 5 ] a s th e c ro s s o v p e rfo rm e d . V a rio u s p a ra m e te r s e ts fa ile d to p ro d u c e re p re s e n ta tio n o f th e c a n d id a te s o lu tio n s p e rm itte d It is b e lie v e d th a t th is in tu rn le d to a h ig h ly m o d a l th e s e a rc h a lg o rith m . u ta tio n te s t, w e s e e th a t th e g ra d ie n t o f t th a t th e re is a la rg e a m o u n t o f in te ra c e r e v fa r se a m e th o e n n e a to o m rc h sp d , a r a c a n y a c e n u m b c e p ta b c o n s tr th a t c a e r o f le re s a in ts u se d te s t ru u lts . A to b e v d iffic u n s w e r g a in th io la te d ltie s fo e e . r S e c tio n 4 - A p p r o a c h 3 T h e re p re s e n ta tio n d e s c rib e d in th e p re v io u s s e c tio n h a d th e e ffe c t o f fo rc in g a ll c a n d id a te s o lu tio n s to o b e y th e c o n s tra in t re q u irin g a ll te a m p a irin g s to b e e q u a lly re p re s e n te d , b u t th e ‘e q u ita b ility ’ re q u e s t is le ft to th e e v o lu tio n a ry m e c h a n is m . In o rd e r to re d u c e th e n u m b e r o f u n d e s ira b le c o m b in a tio n s fu rth e r, a th ird e n c o d in g w a s d e v is e d . T h is w a s m a d e u p fro m fo u r s e p a ra te c o m p o n e n ts (c h ro m o s o m e s ), e a c h e n c o d in g a p e rm u ta tio n o f te a m s . T h e firs t tw o , e a c h o f le n g th s ix te e n , d e te rm in e w h ic h w e e k e n d th e te a m w ill w o rk a d a y s h ift a n d w h ic h w e e k e n d th e y w o rk a n ig h t s h ift. If a te a m is s c h e d u le d to w o rk b o th d a y a n d n ig h t s h ift th e n th is w ill b e d e te c te d a n d a c c ru e th e a s s o c ia te d p e n a lty . T h is a p p ro a c h g u a ra n te e s th a t n o t o n ly d o e s e a c h te a m w o rk a n e q u a l n u m b e r o f w e e k e n d d a y a n d n ig h t s h ifts , b u t a ls o th a t th e re q u ire m e n t to a v o id ru in e d w e e k e n d s is a v o id e d . T h is re n d e rs h a lf o f th e fitn e s s fu n c tio n re d u n d a n t, th u s s p e e d in g th e e v a lu a tio n p o rtio n o f th e s y s te m b y a b o u t 2 5 % . S im ila rly th e th ird a n d fo u rth c h ro m o s o m e s , e a c h o f le n g th e ig h ty e n c o d e fiv e w e e k d a y s h ift a n d fiv e w e e k n ig h t s h ift a p p e a ra n c e s fo r e a c h te a m . F ig u re 5 c o m p o f th is a d m itte d ly s m th e m u ta tio n o p e ra tio s u lts fro m th e p re v io u a re s a ll s n . E s tw th e fitn e s a m p le s e e x a m in a tio o re p re se n s d is m s o n o f ta tio trib u tio n o f th is re p re s e n ta tio n n ly a little b e tte r. F ig u re 6 e x th e s lo p e o f th is g ra p h in c o m n s re v e a ls th a t it is m o re p ro n o , w a m p a u n h ic in e ris o c e d h o n th e b s th e e ffe c n w ith th e . T h is is ta a s is t o f re k e n A n A m b u la n c e C re w R o s te rin g S y s te m 2 7 5 4000 3500 3000 2500 2000 1500 1000 500 0 Permutation Unconstrained 570 540 510 480 450 420 390 360 330 300 4 Chromosome 270 Frequency a s in d ic a tio n th a t th e re is le s s e p is ta s is e v id e n t in th e re p re s e n ta tio n . E v e n s o th e s lo p e is s till re la tiv e ly s h a llo w , in d ic a tin g th a t th e p ro b le m is s till ‘h a rd ’. Fitness F ig u re 5 : D is trib u tio n o f fitn e s s fo r ra n d o m s a m p le u s in g 4 c h ro m o s o m e a rra n g e m e n t, c o m p a re d to s in g le p e rm u ta tio n c h o m o s o m e a n d u n c o n s tra in e d c h ro m o s o m e (3 0 0 0 0 s a m p le s ). Effect of mutation on 4 Chomosome permutation representation 10 530 510 490 470 450 430 410 390 370 350 330 310 -10 290 0 270 Average change 20 -20 -30 Fitness F ig u re 6 : A v e ra g e c h a n g e in fitn e s s to 4 C h ro m o s o m e p e rm u ta tio n re p re s e n ta tio n u n d e r m u ta tio n .( 3 0 0 0 0 s a m p le s ) P re lim in a ry ru n s w ith th o rd e r o f te n c o n s tra in t b re a c h e v e a le d th a t a b o u t th re e o f th e s e th u s n o t a c c e p ta b le . R a th e r th a n o c c u rre n c e o f d o u b le s h ifts , it w th e v a rio u s b re a c h e s . T h e v a lu e th a t th is w o u ld d riv e th e s e a rc h n e s s d is trib u tio n v a lu e s a lre a d y is re p re s e n ta tio n g a v e a s e rie s o f re s u lts w ith fitn e s s in th e s . C lo s e r e x a m in a tio n o f th e re s u ltin g ro s te r h o w e v e r re w e re b re a c h e s o f th e ‘n o d o u b le s h ift’ c o n s tra in t, a n d w e re try to re c o d e th e re p re s e n ta tio n fu rth e r to to ta lly a v o id th e a s d e c id e d to re b a la n c e th e w e ig h t o f p e n a lty a c c ru e d b y fo r d o u b le s h ift v io la tio n s w a s in c re a s e d to 2 5 , in th e h o p e in to e lim in a tin g d o u b le s h ifts . T h is o f c o u rs e a lte rs th e fitp ro d u c e d , b u t a s w e w e re n o w n e a rin g th e id e a l s o lu tio n s 2 7 6 P .V .G . B r a d b e e r , C . F in d la y , a n d T .C . F o g a r ty th is w a s fe lt to b e a s m a ll p ric e to p a y . It is a ls o p o s s ib le o r e v e n lik e ly th a t th e s e c h a n g e s w ill c h a n g e th e s e a rc h s p a c e s lig h tly , w ith a re s u ltin g c h a n g e in p ro g re s s S e c tio n 5 - R e s u lts A fte r m a k in g th tio n , th e s y s te m w a s ru 2 0 , P M X , b in a ry to u rn o n ly 5 m in o r b re a c h e s tio n to th e p ro b le m , a n w ith p a ra m e te rs , s o fa r fitn e s s v a lu e . e c h a n g e n w ith v a a m e n t se o f th e 4 8 d is s h o w n o o th e r d e s c rib e d to p e n a lty w rio u s p a ra m e te rs . O n e le c tio n a n d a 1 0 % m u h o u rs p e r w e e k ru le . T n in fig u re 7 . D e s p ite s o lu tio n s h a v e b e e n g e e ig h tin g s g iv e n in th o f th e firs t ru n s , u s in ta tio n ra te p ro d u c e d h is w a s fe lt to b e a p a c e rta in a m o u n t o f e n e ra te d th a t h a v e s u c e p re v io u s s e c g a p o o l s iz e o f a s o lu tio n w ith ra c tic a b le s o lu x p e rim e n ta tio n h a n a c c e p ta b le F ig u re 7 : S c re e n d u m p o f b e s t re s u lt s o fa r. S e c t io n 6 - C o n c lu s io n s T h e s y s te m , d e v e lo p e d a g a in s t a p ra c tic a l n e e d a n d re la tiv e ly u n d e v e lo p e d a s it is h a s p ro d u c e d a w o rk a b le ro s te r fo r a h ig h ly c o n s tra in e d p ro b le m . A n A m b u la n c e C re w R o s te rin g S y s te m 2 7 7 A s th e re a re a n u m b e r o f d iffe re n t lo c a l o p tim a b e in g fo u n d , it is p o s s ib le to g iv e th e ‘c lie n t’ a n u m b e r o f d iffe re n t c a n d id a te s o lu tio n s . T h is w ill a llo w c o n s id e ra tio n o f o th e r ‘s o c ia l’ fa c to rs n o t e x p lic itly m e n tio n e d in th e in itia l o u tlin e o f th e p ro b le m . T h is a p p ro a c h o f lo o k in g a t th e b e h a v io u r o f th e s e a rc h s p a c e h a s s a v e d a c o n s id e ra b le a m o u n t o f tim e in ru n n in g e x p e rim e n ts a n d a n a ly s in g th e re s u lts fro m th e m . W h ile th e re la tiv e s h o rtc o m in g s o f th e firs t tw o re p re s e n ta tio n s a re to a n e x te n t p re d ic ta b le , g iv e n th e n a tu re o f th e c o n s tra in ts , it is u s e fu l to b e a b le to c o n firm th is w ith o u t e x p e n d in g la rg e a m o u n ts o f c o m p u te c y c le s . S e c tio n 7 - F u r th e r W o r k T h is d o c u m e n t d e s c rib e s th e firs t s te p s to w a rd s s o lv in g th is p ro b le m . T h e a re a w a re th a t m a n y v a ria n ts a re p o s s ib le , a n d e v e n d e s ira b le . A s m e n tio n e d in th e th e te x t, th e le n g th o f th e c y c le is n o t c le a r, s o a n a p p ro a c h a llo w in g a v a ria b le le n g is o n e p o s s ib le d ire c tio n fo r fu rth e r s tu d y . It w o u ld a ls o b e o f in te re s t to d is c o v e r m o re a b o u t th e m o d a lity o f th e s e a rc a n d it is p la n n e d to u s e a re v e rs e h ill-c lim b in g te c h n iq u e , s u c h a s d e s c rib e d b [Jo n e s1 9 9 5 ]. P ro d u c in g m o re in fo rm a tio n o n th e s e n s itiv ity o f th e p ro b le m to d iffe re n t te rs , s u c h a s p o o l s iz e a n d c ro s s o v e r m e c h a n is m re m a in s a n ite m o n th e a g e n d a . It h a s b e e n n o te d th a t th e a d d itio n o f h e u ris tic s c o u ld b e u s e d to im p ro v e p a n c e , p o s s ib ly in a s im ila r fa s h io n to th a t re p o rte d b y H a rt e t. a l. [H a rt1 9 9 8 ]. a u th o rs b o d y o f th c y c le h sp a c e , y J o n e s p a ra m e e rfo rm - A c k n o w le d g e m e n ts T h e c o n s tru c tiv e c o m m e n ts o f th e a n o n y m o u s re fe re e s a re n o te d a n d a p p re c ia te d . T h a n k s a ls o to C o lin W ils o n fo r h is h e lp . B ib lio g r a p h y [B rin d le 1 9 8 1 ]. B rin d le , A , “ G e n e tic a lg o rith m s fo r fu n c tio n o p tim iz a tio n ” , D o c to ra l D is s e rta tio n a n d T e c h n ic a l R e p o rt T R 8 1 -2 , D e p a rtm e n t o f C o m p u te r S c ie n c e , U n iv e rs ity o f A lb e rta , E d m o n to n , 1 9 8 1 . [ C h is h o lm 1 9 9 7 ] . C h is h o lm K .J . a n d B r a d b e e r P .V .G . “ U s in g a G e n e tic A lg o r ith m to O p tim is e a D ra u g h ts P ro g ra m B o a rd E v a lu a tio n F u n c tio n ” , P ro c e e d in g s o f IE E E IC E C ’9 7 , In d ia n a p o lis , 1 9 9 7 . [E ib e n 1 9 9 8 a ] E ib e n A .E ., B a c k T ,. S c h o e n a u e r M , a n d S c h w e fe l (e d s .) “ P ro c e e d in g s o f P a ra lle l P ro b le m S o lv in g F ro m N a tu re - P P S N V ” , L N C S 1 4 9 8 , S p rin g e r V e rla g 2 7 8 P .V .G . B r a d b e e r , C . F in d la y , a n d T .C . F o g a r ty [ E ib e n 1 9 9 8 b ] E ib e n A .E ., v a n H e m e r t J .I ., M a r c h io r i E . a n d S te e n b e e k A .G , “ S o lv in g B in a ry C o n s tra in t S a tis fa c tio n P ro b le m s U s in g E v o lu tio n a ry A lg o rith m s w ith a n A d a p tiv e F itn e s s F u n c tio n ” , in [E ib e n 1 9 9 8 a ] [F a n g 1 9 9 3 ]. F a n g H -L , R o s s P . a n d C o rn e D . “ A p ro m is in g G e n e tic A lg o rith m a p p ro a c h to jo b -s h o p S c h e d u lin g , re s c h e d u lin g a n d o p e n -s h o p s c h e d u lin g p ro b le m s ” , in [F o rre s t1 9 9 3 ]. [ F o r r e s t1 9 9 3 ] . F o r r e s t S ., ( e d .) “ P ro c e e d in g s o f th e F if th I n te rn a tio n a l C o n f e r e n c e o n G e n e tic A lg o rith m s ” , M o rg a n K a u fm a n n , S a n M a te o , 1 9 9 3 . [ F o g a r ty 1 9 9 5 ] . F o g a r ty T .C ( e d .) “ P r o c e e d in g s o f th e A I S B W o rk s h o p o n E v o lu tio n a r y C o m p u tin g ” , S h e ffie ld , L N C S 9 9 3 , S p rin g e r V e rla g , B e rlin 1 9 9 5 . [ G o ld b e r g 1 9 9 0 ] . G o ld b e r g D .E . “ M e s s y G e n e tic A lg o r ith m s : M o tiv a tio n , A n a ly s is a n d F irs t R e s u lts ” , C o m p le x S y s te m s , V o l. 3 . [G o ld b e rg 1 9 8 5 ]. G o ld b e rg D .E . a n d L in g le , R ., “ A lle le s , lo c i, a n d th e tra v e llin g s a le s m a n p ro b le m ” , in [G re fe n s te tte 1 9 8 5 ]. [ G r e f e n s te tte 1 9 8 5 ] . G r e f e n s te tte J .J . ( e d .) “ P r o c e e d in g s o f th e F ir s t I n te r n a tio n a l C o n f e r e n c e o n G e n e tic A lg o rith m s ” , L a w re n c e E a rlb a u m , H ills d a le , 1 9 8 5 [ H a r t1 9 9 8 ] H a r t E ., N e ls o n J . a n d R o s s P ., “ S o lv in g a R e a l- W o r ld P r o b le m U s in g a n E v o lv in g H e u ris tic a lly D riv e n S c h e d u le B u ild e r” , E v o lu tio n a ry C o m p u ta tio n V 6 (1 ):6 1 -8 0 . [J o n e s 1 9 9 5 ]. J o n e s T ., “ E v o lu tio n a ry A lg o rith m s , F itn e s s L a n d s c a p e s a n d S e a rc h ” , P h D D is s e rta tio n , U n iv e rs ity o f N e w M e x ic o , 1 9 9 5 . [ J o n e s 1 9 9 4 ] . J o n e s P .A . a n d B r a d b e e r P .V .G ., “ D is c o v e r y o f o p tim a l w e ig h ts in a c o n c e p t s e le c tio n s y s te m ” , in [L e o n 1 9 9 4 ]. [ L a n g d o n 1 9 9 5 ] . L a n g d o n W .B . “ S c h e d u lin g P la n n e d M a in te n a n c e o f th e N a tio n a l G r id ” , in [F o g a rty 1 9 9 5 ]. [ L e o n 1 9 9 4 ] . L e o n R ., ( e d .) “ P r o c e e d in g s o f th e 1 6 th R e s e a rc h C o llo q u iu m fo rm a tio n R e trie v a l S p e c ia lis t G ro u p ” , T a y lo r G ra h a m , 1 9 9 4 o f th e B C S In - [M ic h c h a le w ic z 1 9 9 4 ] M ic h c h a le w ic z Z ., “ G e n e tic A lg o rith m s + D a ta S tru c tu re s = E v o lu tio n P ro g ra m s ” , (2 n d E d itio n ) p 2 5 6 , S p rin g e r V e rla g , 1 9 9 4 . A n A m b u la n c e C re w R o s te rin g S y s te m 2 7 9 [ M ille r 1 9 9 5 ] . M ille r J .F ., T h o m s o n P . a n d B r a d b e e r P .V .G . “ T e rn a r y D e c is io n D ia g r a m O p tim iz a tio n o f R e e d -M u lle r L o g ic F u n c tio n s u s in g a G e n e tic A lg o rith m fo r V a ria b le a n d S im p lific a tio n R u le O rd e rin g ” , in [F o g a rty 1 9 9 5 ]. [ P a e c h te r 1 9 9 8 ]. P a e c h te r B ., R a n k in R .C ., C u m m in g A . a n d F o g a r ty T .C ., “ T im e ta b lin g th e C la s s e s o f a n E n tire U n iv e rs ity w ith a n E v o lu tio n a ry A lg o rith m .” in [E ib e n 1 9 9 8 a ] [ S c h a f f e r 1 9 9 3 ] . S c h a f f e r J .D . “ P r o c e e d in g s o f th e T h ir d I n te r n a tio n a l C o n f e r e n c e o n G e n e tic A lg o rith m s ” , M o rg a n K a u fm a n n , S a n M a te o , 1 9 9 3 . [S y sw e rd a 1 9 8 9 ]. [S c h a ffe r1 9 9 3 ] S y sw e rd a G . “ U n ifo rm C ro s so v e r in [ W r e n 1 9 9 5 ] . W r e n A . a n d W r e n D . O ., “ A G e n e tic a lg o r ith m s c h e d u lin g ” , C o m p u te rs in O p e ra tio n s R e s e a rc h , 2 2 (1 ), 1 9 9 5 . G e n e tic A lg o rith m s ” , in fo r p u b lic tra n s p o rt d riv e r A Systematic Investigation of GA Performance on Jobshop Scheduling Problems Emma Hart, Peter Ross Division of Informatics, University of Edinburgh, Edinburgh EH1 2QL, Scotland femmah,peterg@dai.ed.ac.uk Abstract. Although there has been a wealth of work reported in the literature on the application of genetic algorithms GAs to jobshop scheduling problems, much of it contains some gross over-generalisations, i.e that the observed performance of a GA on a small set of problems can be extrapolated to whole classes of other problems. In this work we present part of an ongoing investigation that aims to explore in depth the performance of one GA across a whole range of classes of jobshop scheduling problems, in order to try and characterise the strengths and weaknesses of the GA approach. To do this, we have designed a con gurable problem generator which can generate problems of tunable di culty, with a number of di erent features. We conclude that the GA tested is relatively robust over wide range of problems, in that it nds a reasonable solution to most of the problems most of the time, and is capable of nding the optimum solutions when run 3 or 4 times. This is promising for many real world scheduling applications, in which a reasonable solution that can be quickly produced is all that is required. The investigation also throws up some interesting trends in problem di culty, worthy of further investigation. 1 Introduction Since the rst applications of GAs to scheduling problems, 4, there have been many reported applications of GAs to scheduling problems in general, with the jobshop problem receiving a great deal of attention, for example 2, 8, 9, and more recently 3, 13. A large variety of representations and operators have been reported, each showing impressive performance of a GA on some small subset of benchmark problems. Often this performance is then extrapolated to claim that the GA in question is a good algorithm for solving job-shop scheduling problems. However, in order to properly evaluate the quality of a method, it is important to show that it works over a wide range of problems. This requires testing the method over an extremely large number of problems  the problems should be chosen such that they exhibit a variety of features, and should vary in diculty in some congurable way. Rather than use benchmarks problems, it is more useful to use a parameterised problem generator which can generate problem instances at random, and in some tunable manner, so that many dierent instances of problem classes can be generated. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 7 7 − 2 8 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 7 8 E . H a rt a n d P . R o ss Therefore, in this article we describe a problem generator which we have used to generate instances of jobshop scheduling problems falling into 280 dierent problem classes. We present the preliminary results from an ongoing investigation, which allows us to come to some general conclusions about the robustness of a GA as a technique for solving scheduling problems. We evaluate our results not only in terms of the capability of the GA to nd the optimum solution to the problems we generate which all have a known optimum value , but also in terms of its accuracy at producing 'reasonable' quality solutions on a reliable basis. For real-world problems, it is seldom necessary to produce optimum solutions to problems | good enough fast enough" will generally suce. Therefore, it is important to take this into account when making statements about the performance of the GA. We rst describe the features of the problem generator, then present a summary of the results from 28000 experiments on a range of scheduling problems. Some general conclusions about the performance of the GA are then drawn. 2 The Problem Generator We designed a parameterised, tunable problem generator which generates a solution to a jobshop scheduling problem, i.e. a gantt chart, characterised by P O M D I S , where O is the number of operations to be scheduled, M is the number of machines, D de nes the distribution that the operation sizes are drawn from, either Gaussian or uniform , I is the total amount of idle time per machine, and S is the amount of slack in the arrival and due dates of each job. The parameters are described in more detail below: OperationsJobsMachines The operations, O, are divided equally between the number of machines, m. A schedule is generated at random, by placing operations into the gantt chart. The operations are then assigned to jobs, j , such that each job is processed only once on each machine, and an operation of a job on a machine cannot begin until its operation on the previous machine has nished. Therefore, although the minimum number of resulting jobs is O=M , the actual number depends on the exact manner in which the tasks are allocated to jobs. Sizes of Operations The size of each operation in O is drawn from either a uniform distribution between min and max, U min max , or a Gaussian distribution of speci ed mean and deviation, G m d . Idle Time The amount of idle time on each machine is speci ed as a percentage of the total processing time of operations on that machine. The idle time is randomly and uniformally distributed between the actual operations. Slack The maximum amount of slack in the arrival A and due-dates D of each job is also speci ed as a percentage of the total processing time pt of the job. If there is no slack, then the arrival date of the job is set to be equal to the A S y s te m a tic In v e s tig a tio n o f G A P e rfo rm a n c e o n J o b s h o p S c h e d u lin g P ro b le m s 2 7 9 time slot at which the job is rst processed in the generated schedule, and the due-date is set to the time-slot in which the job nishes. If slack is specied, then the arrival and due dates of each job are altered randomly such that A ! A , random0 S   pt D ! D + random0 S   pt Therefore, no job is ever tardy, and the maximum tardiness objective, Tmax for each generated problem has an optimum value of 0. For each solution, there is also a known upper bound on the makespan of each problem | if the problem has no idle time, then the optimum makespan is known exactly. 3 Experimental Parameters 3.1 The Genetic Algorithm The GA used is HGA which was described by the authors in 6 . This GA outperformed other heuristic combination methods, and compared well to the most recently published results on a number of benchmark problems. It uses an indirect representation, in which each gene on the chromosome encodes a pair Method Heuristic. The method gene denotes the methodology that should be used to calculate a conicting set of schedulable operations at each iteration of the algorithm. An operation is then chosen using the heuristic denoted by the heuristic gene. The method applied is either the Gier and Thompson algorithm, 5 , which produces an active schedule, or a modied G&T algorithm which produces a non-delay schedule. The GA used a population size of 100, uniform crossover and a swap mutation operator. Each experiment is run for a maximum of 500 generations, or until it nds the optimum solution. The maximum tardiness of the best solution is noted, with the number of generations required to nd it if the optimum was discovered. The best solutions for each of the 100 trials in each problem class are averaged, and we also record the minimum and maximum solution quality in each class. 3.2 The Problem Classes In the experiments presented in this article we x the number of operations, and hence the chromosome length, at 60. This facilitates a thorough and fair investigation of the eects of the other four parameters on problem diculty. Furthermore, as each GA experiment uses a chromosome population of identical size and length, we can x the GA" parameters such as the number of generations to be the same in all experiments. The other four parameters are varied as shown in table 1, resulting in a total of 280 problem classes, each dened by a tuple P O M D I S . For all experiments in which S  0, then I is xed at 0, and vice versa. For each of the 280 problem classes, 10 problems are generated from dierent random number 2 8 0 E . H a rt a n d P . R o ss seeds. We then run a GA 10 times on each problem instance, and using max as the objective function which we know to have an optimum solution of 0, average the results over the 100 experiments in that class. In the remainder of this article, the term problem class refers to the general class of problems dened by a tuple  . The term problem instance refers to one of the 10 problems generated for a problem class. T P O M D I  S Parameter Value Total Number of Operations 60 Number of Machines 2,3,4,5,6 Task Size Distribution U 0,10 , U 0,30 , U 0,50 , U 0,100 G 50,1 , G 50,10 , G 50,20 , G 50,40 Idle Time 5 , 10 , 25 Slack 10 , 25 , 50 Table 1. Experimental Parameters 4 Results As space limitations do not allow us to present the results of all 280 experiments here, we attempt to present summaries of the ndings and general trends observed, and discuss some individual problem classes in more detail. The complete set of results can be found at 1 . 4.1 Overall Performance Firstly, we note that in every problem class, the GA is able to nd the optimum solution for at least 1 instance of the problems in that class. On the whole, better performance is observed in the 140 experiments in which the operation sizes were distributed uniformally, regardless of distribution size. For example, in 81 of problem classes that had a uniform distribution of task size, then running the GA on problem instances of that class resulted in the optimum solution being found in at least half of all experiments. This compares to a gure of 59 for problem classes with a Gaussian distribution of task size. These statements should be treated with some caution however as they may simply be an artifact of the parameters chosen to dene the uniform and Gaussian distributions, and the two series of experiments cannot be directly compared. Turning our attention to those problem classes which are solved to reasonable" accuracy, we note that 55 of the problem classes with Gaussian task size distribution are solved to within 10 time units of the optimum solution. This increases to 96 for problem classes with uniformally distributed operation sizes. Tables 2 and 3 show the problem classes in which all generated instances where solved with 100 accuracy. There is no obvious pattern, except that it A S y s te m a tic In v e s tig a tio n o f G A P e rfo rm a n c e o n J o b s h o p S c h e d u lin g P ro b le m s Idle Time Slack Std.Dev Machines 10 0 10 2 25 0 1 3 25 0 1 2 25 0 10 2 25 0 20 2 25 0 40 2 25 0 40 3 0 50 40 6 Table 2. Problems with Gaussian Operation Size Distribution which are solved with 100 accuracy 2 8 1 Idle Time Slack Max Size Machines 25 0 10 2 25 0 30 2 25 0 50 2 25 0 100 2 25 0 30 4 0 25 10 5 0 25 10 6 0 50 30 6 0 50 50 6 Table 3. Problems with Uniform Operation Size Distribution which are solved with 100 accuracy appears that for perfect performance, either a large value of idle time or large slack is required in all cases, and that for Gaussian distributions of operation size, a large value of standard deviation helps. For problems with a Gaussian distribution of operation size, the worst performance is observed in 2 problem classes which both result in the optimum solution only being in 1 run of 1 of the 10 problem instances. These problems classes are P 60 5 G50 10  0 0 and P 60 6 G50 10  0 0 . In the uniform case, the worst performance, again with only 1 optimum solutions is for a problem class P 60 6 U 50  0 0 . 4.2 Number of Machines Intuitively, it would be expected that as the number of machines increases for a xed number of operations, O, then the problems would become easier to solve. This is because the number of schedulable operations in the con ict set produced as a result of applying the G&T or non-delay algorithm must reduce as the number of machines increases. Table 4 shows the percentage of problem classes with m 2 2 3 4 5 6 in which the optimum solution was found. For a uniform distribution of jobs, the results are as expected. However, for the Gaussian distributions, we see exactly the opposite | i.e the percentage of optimum solutions decreases as the number of machines is increased. Number of Machines 2 3 4 5 6 Gaussian 63 56 55 55 58 Uniform 62 65 74 75 73 Table 4.  of problem classes resulting in an optimum solution vs no. machines Slack Parameter 0.0 0.1 0.25 0.5 Gaussian 33 32 60 82 Uniform 46 50 76 80 Table 5.  of problem classes resulting in an optimum solution vs slack 2 8 2 E . H a rt a n d P . R o ss 4.3 Slack Table 5 shows the percentage of problem classes with slack 2 0:0 0:1:0:25:0:5 in which the optimum solution was found. For problems with Gaussian distributions of task size, we observe that increasing the slack in due and arrival dates to 10 does not increase performance, as would be expected, and in fact results in a slight decrease. Increasing the slack to larger values however increases performance. This trend is not observed with uniform distributions, which tend to become easier to solve as the slack parameter is increased. Examining the e ect of the slack parameter in more detail in the Gaussian experiments, we notice a correlation between the number of machines parameter m and the slack S in determining solution quality. Table 6 shows the minimum value of the slack parameter S that was required for each set of problem classes with machine m before an improvement was observed in solution quality compared to the equivalent experiment with S = 0:0. Where no improvement was observed, adding slack had a detrimental e ect on solution quality. Number of Machines Minimum Percentage Standard Deviation Machines Slack Required 2 0.5 20,40 0.25 all values of std. dev. 3 4 0.1 all values of std. ded. 5 0.05 all values of std. dev. 0.05 all values of std. dev. 6 Table 6. Minimum Value of Slack Parameter Required to Improve Solution Quality In problem classes where operation size was uniformally distributed, then adding slack generally increases performance. The only exceptions where a signicant di erence occurs are for the problem classes P 60 2 U 0max 0:0 0:1, where max 2 10 30 50 100. 4.4 Idle Time For problems with both uniform and Gaussian task distributions, adding idle time generally improves performance, as is seen in table 7, which shows the percentage of problem classes in which the optimum solution was found for various values of I . This is as expected | inserting idle time into the schedule allows some exibility in the exact placement of operations in the schedule, without necessarily decreasing schedule quality. 5 A Phase Transition in Problem Classes with Gaussian Distributions In the majority of problems with a Gaussian distribution of operation sizes, we notice an interesting transition in the diculty of the problems as the standard A S y s te m a tic In v e s tig a tio n o f G A P e rfo rm a n c e o n J o b s h o p S c h e d u lin g P ro b le m s Table 7. time 2 8 3 Idle Time 0.0 0.05 0.1 0.25 Gaussian 33 48 55 85 Uniform 46 65 75 89 of problem classes in which the optimum solution was achieved vs idle deviation of the distribution is varied.The GA performs best on those problems with large standard deviations. For very small standard deviations, i.e when the all tasks have very similar sizes, then the GA also performs reasonably well. However, for a range of values of standard deviation in the middle, performance decreases considerably. For example, gure 1 shows an expanded graph for the problem class 60 5 50 0 0 0 0 , in which some extra experimental points have been added. A clear peak in di culty is seen, centered around a standard deviation of 7. P  G  sd  :  : A v e ra g e T a rd in e s s o f S o lu tio n s 5 0 4 5 4 0 3 5 3 0 2 5 2 0 1 5 1 0 0 5 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 S ta n d a rd D e v ia tio n o f D is trib u tio n Fig. 1. Solution Quality vs Standard Deviation of Task Size Distribution This appearance of this phenomenum shows remarkably similar properties to earlier work performed by the authors in the timetabling domain. Work described in 11 showed that there was a clear phase transition in the performance of a GA on a sequence of solvable timetabling problems designed to be of increasing di culty the GA tested could solve very lightly constrained, and also very highly constrained problems, however, for moderately constrained problems, the GA would often fail to nd a solution. The appearance of similar phase-transition regions has also been reported in other classes of constraint satisfaction problems, for example see 7, 10, 12. In the timetabling case, it was noted that other non-evolutionary algorithms also failed on the same subset of problems, suggesting that it was not the GAs fault", but that the problems were intrinsically di cult. We have not yet investigated the performance of other non-evolutionary methods on these problems, but expect to see a similar pattern in performance. 2 8 4 E . H a rt a n d P . R o ss 1. Calculate the set C of all operations that can be scheduled next 2. Calculate the completion time of all operations in C , and let m equal the machine on which the minimum completion time t is achieved. 3. Let G denote the conict set of operations on machine m - this is the set of operations in C which take place on m , and whose start time is less than t. 4. Select an operation from G to schedule 5. Delete the chosen operation from C and return to step 1. Fig. 2. Gi er and Thompson Algorithm A possible reason, still to be looked at in more detail, is the use of the Gier and Thompson G&T algorithm or the modied non-delay version in constructing the con ict sets of operations at each iteration. The G&T algorithm is shown in gure 2. For problems in which there is a large deviation in operation sizes, then it is possible that the con ict set is generally smaller at each iteration, and therefore it is more straightforward to choose the 'correct' operation. When all the operations are of similar size, then it seems likely that the size of the con ict set is non-trivial, and hence it is more di cult to choose the 'correct' operation. Figure 3 shows the size of the con ict set at each iteration, averaged over 100 runs of experiments in which the standard deviation of operations sizes was set to 5, and then to 40, using a constant mean of 50. Early in the scheduling process, there is a small region, highlighted on the gure, in which there a fewer items in the con ict set for the case when sd = 40. As we know that the placement of operations early in the schedule is crucial to the success of the algorithm, this may provide a clue, however the matter needs further attention. 6 Problem Classes with Uniform Distributions As noted earlier, better results appear to be obtained when the operation sizes are uniformally distributed, compared to those problems with gaussian distribution of operation sizes. Solution quality tends to decrease as the range of the distribution increases. This information is summarised in table 8 which reports the percentage of optimum solutions obtained for instances of problems tested with each dierent range value. A S y s te m a tic In v e s tig a tio n o f G A P e rfo rm a n c e o n J o b s h o p S c h e d u lin g P ro b le m s 2 8 5 4 S iz e o f c o n flc it s e t 3 .5 S ta n d a rd d e v ia tio n 4 0 3 S ta n d a rd d e v ia tio n 5 c r u c ia l r e g io n ? 2 .5 2 1 .5 1 0 1 0 2 0 3 0 4 0 5 0 6 0 Ite ra tio n s o f a lg o rith m Fig. 3. Size of Conict Set For Dierent Distributions of Operation Sizes Maximum Task Size 10 30 50 100  optimum solutions 80 76 71 68 Table 8. Percentage of problem instances resulting in an optimum solution vs maximum task size 7 Conclusion This article has presented some initial observations made whilst attempting to perform a systematic investigation on the performance of a genetic algorithm on a range of job-shop scheduling problems. The investigation involved a total of 280 dierent problem classes, each containing 10 randomly generated problems. Initial ndings suggest that although the GA does not nd the optimum solution for all problem classes, for most instances of all problem classes it is capable of nding the optimum in at least 1 in 10 trials, and that the quality of solution is generally satisfactory. This suggests the GA appears to be a relatively robust method of tackling such problems, and that since runs are reasonably fast, running the GA several times to nd a solution is a viable strategy. Some interesting trends have been observed | we now intend to try and understand and explain these trends, and to determine whether they are a feature of the problems themselves or due to the genetic algorithm itself. This study concentrated on problem classes which all contained a xed number of operations. Therefore, in order to complete the study, we will also use the generator to test GA performance on a sequence of much larger problems. Although we expect performance to degrade as the problems get bigger, it is fruitful to determine at what point this happens. Finally, further work is planned to compare the performance of other nonevolutionary algorithms on the same set of problem classes, to see if similar trends are observed, and to compare overall performance. These other methods will include constraint satisfaction techniques, and simple scheduling rules. 2 8 6 E . H a rt a n d P . R o ss Acknowledgements Emma Hart is supported by EPSRC grant GRL22232. References 1. http:www.dai.ed.ac.uk emmahjobshop-expts.html. 2. Sugato Bagchi, Serdar Uckun, Yutaka Miyabe, and Kazuhiko Kawamura. Exploring problem-specic recombination operators for job shop scheduling. In R.K. Belew and L.B. Booker, editors, Proceedings of the Fourth International Conference on Genetic Algorithms, pages 10 17. San Mateo: Morgan Kaufmann, 1991. 3. Brizuela. C.A. and N. Sannomiya. A diversity study in genetic algorithms for jobshop. In Proceedings of the Genetic and Evolutionary Computation Conference, pages 75 83, 1999. 4. L. Davis. Job shop scheduling with genetic algorithms. In J. J. Grefenstette, editor, Proceedings of the International Conference on Genetic Algorithms and their Applications, pages 136 140. San Mateo: Morgan Kaufmann, 1985. 5. B. Gier and G.L. Thompson. Algorithm for solving production scheduling problems. Operations Research, 84:487 503, 1960. 6. E. Hart and P. Ross. A heuristic combination method for jobshop scheduling problems. In Parallel Problem Solving from Nature, PPSN-V, pages 845 854, 1998. 7. T. Hogg, A. Huberman, and C.P. Williams. Phase transitions and the search problem. Articial Intelligence, 811-2:1 15, 1996. 8. S-C. Lin, E.D. Goodman, and W.F. Punch. A genetic algorithm approach to dynamic job-shop scheduling problems. In Thomas Back, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, pages 481 489. MorganKaufmann, 1997. 9. R. Nakano and T. Yamada. Conventional genetic algorithms for job shop problems. In R.K. Belew and L.B. Booker, editors, Proceedings of the Fourth International Conference on Genetic Algorithms, pages 474 479. San Mateo: Morgan Kaufmann, 1991. 10. P. Prosser. An empirical study of phase transitions in binary constraint satisfaction problems. Articial Intelligence, 811-2:81 109, 1996. 11. P Ross, E Hart, and D Corne. Some observations about ga-based exam timetabling. In Practice and Theory of Automated Timetabling, pages 115 130, 1997. 12. B.M. Smith and M.E. Dyer. Locating the phase transitions in binary constraint satisfaction problems. Articial Intelligence, 811-2:155 181, 1996. 13. P. Van Bael, D. Devogelaere, and M. Rijckaert. The job shop problem solved with simple, basis evolutionary search elements. In Proceedings of the Genetic and Evolutionary Computation Conference, pages 665 670, 1999. An Ant Algorithm with a New Pheromone Evaluation Rule for Total Tardiness Problems Daniel Merkle1 and Martin Middendorf2 Institute for Applied Computer Science and Formal Description Methods, University of Karlsruhe, Germany f1 merkle,2middendorfg@aifb.uni-karlsruhe.de Abstract. Ant Colony Optimization is an evolutionary method that has recently been applied to scheduling problems. We propose an ACO algorithm for the Single Machine Total Weighted Tardiness Problem. Compared to an existing ACO algorithm for the unweighted Total Tardiness Problem our algorithm has several improvements. The main novelty is that in our algorithm the ants are guided on their way to good solutions by sums of pheromone values. This allows the ants to take into account pheromone values that have already been used for making earlier decisions. 1 Introduction Ant Colony Optimization ACO is an evolutionary metaheuristic to solve combinatorial optimization problems by using principles of communicative behaviour found in real ant colonies for an introduction and overview see 5. Recently the ACO approach has been applied to scheduling problems, like Job-Shop 2, 7, Flow-Shop 13, and the Single Machine Total Tardiness problem 1. Bullnheimer et al. 1 have compared an ACO algorithm with several other heuristics to solve the Single Machine Total Tardiness problem e.g. decomposition heuristics, interchange heuristics and simulated annealing. They have shown that the ACO algorithm found the optimal solution of 125 benchmark problems more often than the other heuristics these benchmark problems where generated with the same method from 12 as the benchmarks problems used in this paper . In this paper we propose alternative and improved ways to solve the Single Machine Total Tardiness problem by ACO. Moreover, we also study the weighted version of the total tardiness problem. In ACO algorithms several generations of articial ants search for good solutions. Every ant of a generation builds up a solution step by step going through several probabilistic decisions until a solution is found. In general, ants that found a good solution mark their paths through the decision space by putting some amount of pheromone on the edges of the path. The following ants of the next generation are attracted by the pheromone so that they will search in the solution space near good solutions. In addition to the pheromone values the ants will usually be guided by some problem specic heuristic for evaluating the possible decisions. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 2 8 7 − 2 9 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 2 8 8 D . M e rk le a n d M . M id d e n d o rf The approach used in 1 and 13 to solve scheduling problems with ACO algorithms is to use a pheromone matrix = f g where pheromone is added to an element of the pheromone matrix when a good solution was found where job is the th job on the machine. The following ants of the next generation then directly use the value of to estimate the desirability of placing job as the th job on the machine when computing a new solution. Here we propose P a dierent approach. Instead of using only the value of the ants use =1 to compute the probability of placing job as the th on the machine. A problem with using only can occur when the ant does not chose job as the th job in the schedule. Because, if the +1 , +2 values are small then job might be scheduled much later than at the th place and possibly long after its due date . It is likely that this will not happen P when using =1 . Note, that this approach diers from nearly all other ant algorithms proposed so far, in that we base one possible decision of an ant on several pheromone values. The only other work that uses several pheromone values to estimate the quality of one possible decision is 11. Moreover, we let the ants make optimal decisions when this is possible and use a heuristic that is a modi cation of the heuristic used in 1. This paper is organized as follows. The Single Machine Total Weighted Tardiness Problem is de ned in Section 2. In Section 3 we describe an ACO algorithm for the unweighted problem. The pheromone summation rule is introduced in Section 4. Section 5 contains further variants and improvements. The choice of the parameter values of our algorithms used in the test runs and the test instances and are described in Section 6. The results are reported in Section 7. A conclusion is given in Section 8. T Tij Tij j i Tij j i Tij i Tkj k j i Tij j i Ti j Ti j i k j  : : : i Tkj 2 The Single Machine Total Weighted Tardiness Problem The Single Machine Total Weighted Tardiness Problem SMTWTP is to nd for jobs, where job , 1   has a processing time , a due date , , a non-preemptive one machine schedule that minimizes = Pand=1a weight  maxf0 , g where is the completion time of job . is called the total weighted tardiness of the schedule. The unweighted case, i.e. = 1 for all 2 f1 g, is the Single Machine Total Tardiness Problem SMTTP . It is known that SMTTP is NP-hard in the weak sense 8 and SMTWTP is NP-hard in the strong sense 10. A pseudopolynomial time algorithm for SMTWTP in case that the weights agree with the processing times i.e. implies  was given in 10. Observe, that the last result implies that SMTTP is pseudopolynomial time solvable. For an overview over dierent heuristics for SMTWTP see 4. n n j j j n pj dj wj wj T  Cj dj Cj j T wj j  : : : n pj  ph wj wh 3 ACO Algorithm for SMTTP The ACO algorithm of Bullnheimer et al. 1 is described in this section. The general idea was to adapt an ACO algorithm called ACS-TSP for the traveling A n A n t A lg o rith m w ith a N e w P h e ro m o n e E v a lu a tio n R u le 2 8 9 salesperson problem of Dorigo et al. 6 for the SMTTP. In every generation each of m ants constructs one solution. An ant selects the jobs in the order in which they will appear in the schedule. For the selection of a job the ant uses heuristic information as well as pheromone information. The heuristic information, denoted by ij , and the pheromone information, denoted by ij , are an indicator of how good it seems to have job j at place i of the schedule. The heuristic value is generated by some problem dependent heuristic whereas the pheromone information stems from former ants that have found good solutions. With probability q0, where 0 q0  1 is a parameter of the algorithm, the ant chooses a job j from the set S of jobs that have not been scheduled so far which maximizes ij  ij  where  and  are constants that determine the relative in uence of the pheromone values and the heuristic values on the decision of the ant. With probability 1 , q0 the next job is chosen according to the probability distribution over S determined by   pij = P ij  ij   h2S ih  ih  The heuristic values ij are computed according the Modi ed Due Date rule MDD, i.e., 1 ij = maxfT 1+ p  d g j j where T is the total processing time of all jobs already scheduled. After an ant has selected the next job j, a local pheromone update is performed at element i j of the pheromone matrix according to ij = 1 ,   ij +   0 for some constant , 0   1 and where 0 = m  T1 EDD and TEDD is the total tardiness of the schedule that is obtained when the jobs are ordered according to the Earliest Due Date heuristic EDD, i.e., with falling values of 1=dj . The value 0 is also used to initialize the elements of the pheromone matrix. After all m ants have constructed a solution the best of these solutions is further improved with a 2-opt strategy. The 2-opt strategy considers swaps between all pairs of jobs in the sequence. Then it is checked whether the so derived schedule is the new best solution found so far. The best solution found so far is then used to update the pheromone matrix. But before that some of the old pheromone is evaporated according to 2 9 0 D . M e rk le a n d M . M id d e n d o rf = 1 ,   ij The reason for this is that old pheromone should not have a too strong inuence on the future. Then, for every job j in the schedule of the best solution found so far some amount of pheromone is added to element ij  of the pheromone matrix where i is the place of job j in the schedule. The amount of pheromone added is =T where T is the total tardiness of the best found schedule, i.e., ij ij = 1 ij +   T The algorithm stops when some stopping criterion is met, e.g. a certain number of generations has been done or the best found solution has not changed for several generations. 4 The Pheromone Summation Rule In this section we describe a new approach of using the pheromone values which is used in our ACO algorithm for SMTTP. In general, a high pheromone value ij means that it is advantageous to put job j at place i in the schedule. Assume now that by chance an ant chooses to put some job h at place i of the schedule that has a low pheromone value ih instead of a job j that has a high pheromone value ij . Then in order to have a high chance to still end up with a good solution it will likely be necessary for the ant to place job j not too late in the schedule when j has a small due date. To some extend the heuristic values lj for l  i will then force the ant to choose j soon. But a problem occurs when the values lj are small because no good solutions have been found before that have job j at some place l  i. Then the product lj    lj  is small and it is likely that the ant will not choose j soon. In this case the ant will end up with a useless solution having a high total tardiness value. To handle this problem we propose to let a pheromone value ij also inuence later decisions when choosing a job for some place l  i. A simple way to guaranty this inuence is to use the sum of all pheromone values for every job from the rst row of the matrix up to row i when deciding about the job for place i. When using this pheromone summation rule we have the following modi ed decision formulas. An ant chooses as next job for place i in the schedule with probability q0 the job j 2 S that maximizes X  i k=1 kj   kj  2 and with probability 1 , q0 job j 2 S is chosen according to the probability distribution over S determined by A n A n t A lg o rith m pij w ith a N e w P h e ro m o n e E v a lu a tio n R u le 2 9 1 Pi kj  ij  kP =1  i kh  ih =P h2S 3 k=1 5 Further Variations and Improvements In this section we describe further variations and improvements that we used in our ACO algorithm. 5.1 Modied Heuristic A problem when using the heuristic values according to formula 1 is that the values of max + pj  dj become much larger | due to | when deciding about jobs to place at the end of the schedule than they are when placing jobs at the start of the schedule. As a consequence the heuristic di erences between the jobs are, in general, small at the end of the schedule. To avoid this e ect we used the following modi ed  values 1 4 ij = max + pj  dj For the weighted problem SMTWTP we multiplied every value on the right side of equation 4 with the weight wj of job j . Note that jobs with a small weighted processing time pj =wj have a high heuristic value when + pj dj . fT g T fT g, T T  5.2 Deterministic Scheduling Between Due Dates Consider the construction of a schedule for the unweighted problem SMTTP. Assume that some jobs have already been scheduled. Assume further that the sum of the processing times of all jobs scheduled so far lies between some due date dj and a due date dh  dj and every other due date is smaller than dj or larger than dh . For this case it is easy to show that it is optimal to schedule all jobs with a due date dj before scheduling a job with a due date dh as long as the sum of the processing times of the scheduled jobs is at most dh . Moreover when there are several jobs with due date dj it is optimal to schedule these jobs ordered by increasing processing times. If the ants apply this deterministic rule whenever possible we say that the ants work locally deterministic. Then the ants will switch between probabilistic and deterministic behaviour. T    6 Test Instances and Parameters We tested the di erent variants of ACO algorithms on 125 benchmark instances for SMTWTP of size 100 jobs that are included in the OR-Library 14. These benchmark instances were generated as follows: for each job j 1 : 125 an integer processing time pj is taken randomly from the interval 1 : 100, an 2 2 9 2 D . M e rk le a n d M . M id d e n d o rf integer weight wj is taken randomly from the interval 1 : 10 and an integer due date dj is taken randomly from the interval 3 2 125 125 X X 4 pj 1 , TF , RDD  pj 1 , TF + RDD 5 j =1 2 j =1 2 The value RDD relative range of due dates determines the length of the interval from which the due dates were taken. TF tardiness factor P determines the relative position of the centre of this interval between 0 and 125 j =1 pj . The values for TF and RDD are chosen from the set 0:2 0:4 0:6 0:81:0 . The benchmark set contains ve instances for each combination of TF and RDD values. For the unweighted problem SMTTP we used the same benchmark instances but ignored the dierent weights. Our results for SMTWTP were compared to the best known results for the benchmark instances that are from 3 and can be found in 14. The parameters used for the test runs are:  = 1,  = 1,  = 0:1, q0 0 0:9 . The number of ants in every generation was m = 20. Every test was performed with 4 runs on every instance. Every run was stopped after 500 generations. We used a 2-opt strategy to improve the best solution that was found in every generation which diers slightly from the 2-opt strategy used in 1. For every pair of jobs it was checked exactly once whether a swap of these jobs improves the schedule. A swap that improves the schedule was xed immediately. Thus we tried exactly 4950 swaps per generation. In the following ACS-SMTTP or short ACS denotes the algorithm of 1 as described in Section 3 but with the new 2-opt strategy described in the last paragraph. Our algorithm ACS-SMTWTP- is similar to ACS but uses the pheromone summation rule as described in Section 4. Algorithm ACS-SMTWTPH is similar to ACS but uses the new heuristic from Section 5.1. Algorithm ACSSMTWTP-D is similar to ACS but additionally uses the deterministic strategy from Section 5.2 for scheduling between due dates. Algorithms that use combinations of new features are denoted by ACS-SMTWTP-XYZ where X,Y,Z , H, D e.g. ACS-SMTWTP-H uses the new heuristic and the pheromone summation rule . For shortness we write ACS-XYZ for ACS-SMTWTP-XYZ. f g 2 f g 2 f g 7 Experimental Results The inuence of the pheromone summation rule called -rule in the following and the modied heuristic was tested on weighted and unweighted problem instances. Since the parameter q0 has some inuence on the results we performed tests with q0 = 0 and q0 = 0:9. Table 1 shows the results for SMTWTP. The average total tardiness values found by the ACO algorithms for SMTWTP were compared to the average total tardiness of the best known solutions that are from 3. The average total tardiness per instance of the best solutions from 3 is 217851:34. Table 1 shows that ACS-H performed better than ACS-H and also that ACS- performed A n A n t A lg o rith m w ith a N e w P h e ro m o n e E v a lu a tio n R u le 2 9 3 better than ACS this holds for both cases 0 = 0 and 0 = 0 9. In all cases the dierence of the total tardiness values compared to the best known solutions are at least 61 1 lower for the ACO algorithm with -rule 79 5 for ACS- H compared to 204 5 for ACS-H with 0 = 0 9. Moreover, the ACO algorithms with -rule found for more instances a better total tardiness than their counterparts without -rule at least 5 3 times as often. The dierences of the total tardiness values compared to the best known values over the rst 200 generations are shown in Figure 1. The best solution of ACS- H was found after an average of 80 generations, which was after less than 3.5 seconds on a 450 MHz Pentium-II processor. Table 1 also shows that the ACO algorithms with modied heuristic performed in all cases better than their counterparts using the heuristic from 1. For 0 = 0 9 the advantage of the modied heuristic is smaller than for 0 = 0 e.g. for 0 = 0 9 ACS- H has a 60 2 smaller dierence to optimal total tardiness than ACS- compared to a 92 9 smaller dierence for 0 = 0. q q :  : q : :  :   :  q : q q :  :  : q Table 1. Inuence of pheromone summation rule and new heuristic on solution quality for SMTWTP. Total Tardiness: average dierence to total tardiness of best found solutions from 3 average over 500 test runs, 125 instances and 4 runs for each instance Better: comparisons between ACS- H and ACS-H respectively ACS- and ACS, number of instances with smaller average total tardiness average over 125 instances and 4 runs for each instance. weighted ACS- H ACS-H ACS- ACS Total q0 = 0 191.8 3024.7 946.1 9914.7 Tardiness q0 = 0:9 79.5 204.5 200.0 1198.6 q 97 2 106 0 0 =0 Better q = 0:9 86 16 97 3 0 Table 2 shows the results for the unweighted problem SMTTP. The results are compared with the average of the best total tardiness values we found for the unweighted instances, i.e. 54309 5. Similarly as for the weighted problem in all cases the ACO algorithms with -rule are better than their counterparts without -rule. Also the modied heuristic performed better in all cases than the heuristic from 1. Since the 2-opt strategy signicantly inuences of the quality of the solutions we also compared the ACS- H with ACS-H when using no 2-opt strategy. The results can be found in Table 3 for SMTWTP and in Table 4 for SMTTP. The only case where ACS- H performed not signicantly better than ACS-H is the unweighted case with 0 = 0 9. In this case ACS-H found a slightly better average total tardiness ACS- H dierence is 331 5 for ACS-H and 332 3 for ACS- H. On the other hand ACS- H found for more instances better solutions than ACS-H For 65 instances ACS- H found better solutions than ACS-H whereas ACS-H performed better than ACS- H for 33 instances. :     q :  :    :  2 9 4 D . M e rk le a n d M . M id d e n d o rf Fig. 1. SMTWTP: Average dierence to total tardiness of best found solutions from 3 over the rst 200 generations. ACS- H ACS-H ACSACS 2000 1500 1000 500 0 0 50 100 150 200 Inuence of pheromone summation rule and new heuristic on solution quality for SMTTP. Total Tardiness: average dierence to total tardiness of best found solutions average over 500 test runs, 125 instances and 4 runs for each instance Better" as in Table 1. unweighted ACS- H ACS-H ACS- ACS Total q0 = 0 47.9 48.5 112.9 256.4 Tardiness q0 = 0:9 7.0 19.0 8.7 26.3 q 53 32 82 17 0 =0 Better q = 0:9 53 22 67 14 Table 2. 0 Inuence of pheromone summation rule and new heuristic on solution quality for SMTWTP when using no 2-opt. Total Tardiness" as in Table 1 Better" as in Table 1 but comparison between ACS- H and ACS-H. Table 3. no 2-0pt, weighted ACS- H ACS-H Total q0 = 0 11894.4 22046.8 Tardiness q0 = 0:9 1733.2 1793.5 0 76 48 Better qq0 = 67 42 0 = 0:9 A n A n t A lg o rith m w ith a N e w P h e ro m o n e E v a lu a tio n R u le 2 9 5 Table 4. Inuence of pheromone summation rule and new heuristic on solution quality for SMTTP when using no 2-opt. Total Tardiness" as in Table 2 Better" as in Table 1 but comparison between ACS- H and ACS-H. no 2-0pt, unweighted ACS- H ACS-H Total q0 = 0 3943.7 4515.8 Tardiness q0 = 0:9 332.3 331.5 q 59 50 0 =0 Better q = 0:9 65 33 0 The inuence of the deterministic strategy for scheduling between due dates for SMTTP has only a minor inuence on the results for the unweighted benchmark instances from the OR-Library. The reason is that these instances have small gaps between the due dates. Thereby, the deterministic strategy does come into play only rarely. Hence, we created new test instances which have two neighboured due dates that have a large gap in between. We changed each of the problem instances from the OR-Library as follows. The jobs were ordered by their due dates and the due dates of jobs 41 to 59 were set to the same due date that job 40 has. The average of the best total tardiness values we found for these modi ed instances was 56416 3. Table 5 shows for 0 = 0, that ACS- HD performed much better than ACS- H and also that ACS-HD performed much better than ACS-H. For 0 = 0 9 the ACS- HD algorithm could not pro t from the deterministic scheduling between due dates. : q   q :  Table 5. Inuence of deterministic strategy between due dates on solution quality for SMTTP and problem instances with modied due dates. Total Tardiness as in Table 2 Better" as in Table 1 but comparison between ACS- H and ACS- HD. unweighted ACS- H ACSTotal q0 = 0 101.4 Tardiness q0 = 0:9 2.9 q 8 0 =0 Better q = 0:9 36 0 HD ACS-H ACS-HD 45.7 120.1 3.8 8.7 11.1 9.2 78 1 92 14 29 36 8 Conclusion We have introduced a new method to use the pheromone values in an Ant Colony Optimization ACO algorithm for the Single Machine Total Weighted Tardiness problem. An ACO algorithm using this pheromone summation rule gives better solutions for 125 benchmark than its counterpart that does not use the summation rule. This holds also for the unweighted total tardiness problem. Moreover, 2 9 6 D . M e rk le a n d M . M id d e n d o rf we proposed a new heuristic that can be used by the ants when searching for a solution. For the unweighted problem we have shown that the ACO algorithm can prot from ants that switch between a deterministic behaviour in case that optimal decisions can be made and the standard" probabilistic behaviour. References 1. A. Bauer, B. Bullnheimer, R.F. Hartl, C. Strauss: An Ant Colony Optimization Approach for the Single Machine Total Tardiness Problem in: Proceedings of the 1999 Congress on Evolutionary Computation CEC99, 6-9 July Washington D.C., USA, 1445-1450, 1999. 2. A. Colorni, M. Dorigo, V. Maniezzo, M. Trubian: Ant System for Job-Shop Scheduling JORBEL - Belgian Journal of Operations Research, Statistics and Computer Science, 34: 3953 1994. 3. R.K. Congram, C.N. Potts, S. L. van de Velde: An iterated dynasearch algorithm for the single-machine total weighted tardiness scheduling problem submitted to INFORMS Journal on Computing. 4. H.A.J. Crauwels, C.N. Potts, L.N. Van Wassenhove: Local Search Heuristics for the Single Machine Total Weighted Tardiness Scheduling Problem INFORMS Journal on Computing, 10: 341359 1998. 5. M. Dorigo, G. Di Caro: The ant colony optimization meta-heuristic in: D. Corne, M. Dorigo, F. Glover Eds., New Ideas in Optimization, McGraw-Hill, 1999, 11-32. 6. M. Dorigo, L. M. Gambardella: Ant colony system: A cooperative learning approach to the travelling salesman problem IEEE Trans. on Evolutionary Comp., 1: 53-66 1997. 7. M. Dorigo, V. Maniezzo, A. Colorni: The Ant System: Optimization by a Colony of Cooperating Agents IEEE Trans. Systems, Man, and Cybernetics  Part B, 26: 29-41 1996. 8. J. Du, J.Y.-T. Leung: Minimizing the Total Tardiness on One Machine is NP-hard Mathematics of Operations Research, 15: 483496 1990. 9. P. Forsyth, A. Wren: An Ant System for Bus Driver Scheduling Report 97.25, University of Leeds - School of Computer Studies, 1997. 10. E.L. Lawler: A `pseudopolynomial' algorithm for sequencing jobs to minimize total tardiness Annals of Discrete Mathematics, i: 331342 1977. 11. R. Michels, M. Middendorf: An Ant System for the Shortest Common Supersequence Problem in: D. Corne, M. Dorigo, F. Glover Eds., New Ideas in Optimization, McGraw-Hill, 1999 692-701. 12. C.N. Potts, L.N. Van Wessenhove: Single machine tardiness sequencing heuristics IEE Transactions, 23: 346-354 1991 13. T. Stutzle: An ant approach for the ow shop problem in Proc. of the 6th European Congress on Intelligent Techniques & Soft Computing EUFIT '98 , Vol. 3, Verlag Mainz, Aachen, 1560-1564, 1998. 14. http:mscmga.ms.ic.ac.ukjeborlibwtinfo.html. $1HZ*HQHWLF5HSUHVHQWDWLRQDQG &RPPRQ&OXVWHU&URVVRYHU IRU-RE6KRS6FKHGXOLQJ3UREOHPV 7(=8.$ 0DVDUX +,-, 0DVDKLUR 0,<$%$<$6+, .D]XQRUL DQG 2.8085$ .HLJR +LWDFKL 7RKRNX 6RIWZDUH /WG  +RQFKR $REDNX 6HQGDL  0L\DJL -DSDQ ^WH]XND KLML PL\DED\ RNXPXUD`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T h is p a p e r e x a m in e s tw o te c h n iq u e s fo r s e ttin g th e p a ra m e te rs o f a n e v o lu tio n a r y A lg o r ith m ( E A ) . T h e e x a m p le E A u s e d f o r te s t p u r p o s e s u n d e r ta k e s a s im p le s c h e d u lin g p r o b le m . A n in itia l v e rs io n o f th e E A w a s te s te d u tilis in g a s e t o f p a ra m e te rs th a t w e r e d e c id e d b y b a s ic e x p e rim e n ta tio n . T w o s u b s e q u e n t v e r s io n s w e re c o m p a r e d w ith th e in itia l v e rs io n , th e firs t o f th e s e a d ju s te d th e p a ra m e te rs a t ru n tim e , th e s e c o n d u s e d a s e t o f p a r a m e te r s d e c id e d o n b y r u n n in g a m e ta - E A . T h e a u th o r s h a v e b e e n a b le to c o n c lu d e th a t th e u s a g e o f a m e ta - E A a llo w s a n e f fic ie n t s e t o f p a r a m e te r s to b e d e r iv e d fo r th e p ro b le m E A . 1 . A D e s c r ip tio n o f th e P r o b le m T h e u s e o f E v o lu tio n a ry A lg o rith m s (E A s ) fo r s o lv in g tim e ta b lin g a n d s c h e d u lin g p ro b le m s h a s b e c o m e c o m m o n p la c e in re c e n t y e a rs [1 ], [9 ]. T h is p a p e r e x a m in e s tw o m e th o d s fo r o p tim is in g th e p a ra m e te rs u s e d b y th e a lg o rith m . A lth o u g h th e c o n c e p ts a n d m e th o d s u s e d fo r d e v e lo p in g a n d o p tim is in g th e e v o lu tio n a ry a lg o rith m a re in te n d e d to b e g e n e ra l, fo r th e p u rp o s e s o f p re s e n ta tio n w ith in th is p a p e r th e y w ill b e a p p lie d to a s im p le s c h e d u lin g p ro b le m . T h e s c h e d u lin g p ro b le m u n d e r c o n s id e ra tio n re q u ire s a n u m b e r o f jo b s to b e p ro c e s s e d th ro u g h a fa c to ry . E a c h jo b h a s s ta rt a n d e n d tim e s th a t fo rm its tim e w in d o w . W ith in th is w in d o w th e jo b m u s t “ v is it” a ll th e re s o u rc e s re q u ire d fo r th is jo b , w ith in a s p e c ifie d o rd e r. E a c h re s o u rc e is m u tu a lly e x c lu s iv e (ie o n ly o n e jo b m a y m a k e u s e o f a re s o u rc e a t a n y o n e tim e ). T h e v a rio u s c o n flic tin g c o n s tra in ts fo r th e s c h e d u lin g ta s k c a n b e d iv id e d in to tw o c a te g o rie s h a rd c o n s tra in ts a n d s o ft c o n s tra in ts , a s s h o w n in ta b le 1 . A h a rd c o n s tra in t is o n e th a t m u s t b e s a tis fie d in o rd e r to p ro d u c e a fe a s ib le s c h e d u le , s o ft c o n s tra in ts a lte r th e q u a lity o f th e s c h e d u le . 2 . A D e s c r ip tio n o f th e E A T h e g e n e ra l c o n c e p ts o f e v o lu tio n a ry a lg o rith m s a re w e ll u n d e rs to o d . E a c h in d iv id u a l re q u ire s a re p re s e n ta tio n (a g e n o ty p e ) th a t m a y b e d e c o d e d in to a s p e c ific s c h e d u le (a S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 0 7 − 3 1 8 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 3 0 8 N . U rq u h a rt, K . C h is h o lm , a n d B . P a e c h te r p h e n o ty p e ). E a c h (s to re d a s a re a l c h ro m o s o m e , th e re s o u rc e a re s to re c o n s id e re d a d ire c d ire c t re p re s e n ta tio a c t d ire c tly o n th e w o u ld re p re s e n t a p ro c e s s in g w o u ld b g e n e w ith in th e g e n o ty p e re p re s e n ts th e s ta rt tim e o f th e e v e n t n u m b e r). W ith in e a c h g e n o ty p e th e e v e n t g e n e s m a k e u p a e v e n t g e n e s a re g ro u p e d s o th a t a ll th e g e n e s fo r a p a rtic u la r d to g e th e r in o rd e r o f s ta rt tim e . T h is re p re s e n ta tio n m a y b e t re p re s e n ta tio n , in th a t th e g e n e s w ith in th e c h ro m o s o m e a re a n o f th e fin is h e d s c h e d u le . T h e m u ta tio n a n d c ro s s o v e r o p e ra to rs ite m s in th e s c h e d u le . W ith a n in d ire c t re p re s e n ta tio n th e g e n e s s e t o f in s tru c tio n s fo r b u ild in g th e s c h e d u le [1 0 ], th u s fu rth e r e re q u ire d to c o n v e rt th e g e n o ty p e in to a s c h e d u le . T a b le 1 . H a rd a n d s o f t c o n s tra in ts C o E v E v T h T h T h T h n s tr a in t e n t m u s t s ta r t a f te r th e e n t m u s t s ta r t a f te r th e e f ir s t e v e n t m u s t s ta r t e la s t e v e n t m u s t fin ish e e v e n t s h o u ld ta k e p la e e v e n t s h o u ld ta k e p la In th e c o u rs e s u m m a rise d a s : o f th e se p re v io u s p re v io u s a f te r th e a fte r th e c e in n o r c e o n w e e v e v jo b jo m a e k e n t o n e n t in s “ a v a b s “ d u l w o rk d a y s e x p e rim e n ts 3 th is r e s th is jo b ila b le ” e ” tim e in g h o u d a ta T y H a H a H a S o S o S o o u r c e h a s fin is h e d . h a s fin ish e d . tim e rs se ts w e re u tilis e d , th e y p e rd rd rd f t1 ft ft m a y b e D a ta s e t 1 : 3 5 E v e n ts , w ith in 1 6 J o b s u s in g 4 R e s o u rc e s , o v e r a 9 d a y p e rio d D a ta s e t 2 : 4 1 E v e n ts , w ith in 1 5 J o b s u s in g 8 R e s o u rc e s , o v e r a 4 d a y p e rio d D a ta s e t 3 : 5 0 E v e n ts , w ith in 1 8 J o b s u s in g 8 R e s o u rc e s , o v e r a 4 d a y p e rio d E a c h re so u rc e h a s a ‘q m in u te . T h u s fo r a n e v e p e rio d th e re a re 1 4 4 0 * 4 p d a y ). T h u s if th e re w e re 1 n u m b e r o f p o te n tia l q u e d iffe re n t q u e u e s . T h e to ta th e p ro d u c t o f th e s iz e o f th e s e s c h e d u le s a re in fe a p ro d u c tio n . 2 .1 u e u e ’ o f e v e n ts , e a c h e v e n t h a s a s ta rt tim e s e t to th e n e a re s t n t b e in g p la c e d w ith in a s c h e d u le b e in g b u ilt fo r a 4 d a y o s s ib le v a lu e s fo r th e s ta rt tim e (th e re a re 1 4 4 0 m in u te s in a 0 e v e n ts s c h e d u le d to u s e a re s o u rc e o v e r a 2 d a y p e rio d th e u e s fo r th a t re s o u rc e w o u ld b e (4 8 * 1 4 4 0 )* 1 0 = 6 1 9 2 0 0 l n u m b e r o f p o te n tia l s o lu tio n s to th e s c h e d u lin g p ro b le m is th e p o te n tia l q u e u e s fo r e a c h re s o u rc e . T h e v a s t m a jo rity o f s ib le , a n d th e re s tric te d m u ta tio n o p e ra to r d is c o u ra g e s th e ir B a s ic O p e r a tio n s W ith in th e s y s te m tw o -p o in t c ro s s o v e r, b a s e d o n th e D a v is s y s te m is u s e d to c re a te c h ild re n , b a s e d o n e x c h a n g in g s ta rt tim e g e n e s . T h e c h ild re p la c e s o n e o f th e ra n d o m ly s e le c te d p a re n ts . T h e E A u s e s s te a d y s ta te p o p u la tio n to re d u c e c o m p le x ity . 1 A jo b w h ic h is n o t c o m p le te d u n til a fte r th e e n d o f its tim e w in d o w d e s ira b le , b u t w ill s till re s u lt in a s c h e d u le th a t is fe a s ib le . m a y n o t b e O p tim is in g a n E v o lu tio n a ry A lg o rith m U s e is m a d e o f e litis m , th is e n s u re s th a t th e b e s t in d iv id u a ls o f p re s e rv e d fo r in c lu sio n in th e n e x t g e n e ra tio n . T h e m u ta tio n o p e ra to r c o n s id e rs e a c h m e m b e r o f th e p o p m u ta tio n , th e c h a n c e o f a s p e c ific m e m b e r b e in g s e le c te d is b a ra te w h ic h is e x p re s s e d a s a p e rc e n ta g e . F o r in s ta n c e w ith a m th e re is a 3 5 % c h a n c e o f a n y in d iv id u a l b e in g s e le c te d . W h s e le c te d , o n e g e n e is s e le c te d fo r m u ta tio n w ith a s e le c tio n p re s s u th e le n g th o f th e c h ro m o s o m e ). T h e fitn e s s fu n c tio n is b a s e d o n th e c o n s tra in ts o u tlin e d in s a tis fa c tio n o f h a rd c o n s tra in ts b e in g g iv e n g re a te r re w a rd s th a s o ft c o n s tra in ts . T h e fitn e s s fo r e a c h in d iv id u a l e v e n t is c a lc u la s h o w n in ta b le 2 . fo r S c h e d u lin g 3 0 9 e a c h g e n e ra tio n a re u la tio n fo r p o s s ib le s e d o n th e m u ta tio n u ta tio n ra te o f 3 5 % e n a n in d iv id u a l is re o f 1 /L (w h e re L is s e c tio n 1 , w ith th e n th e s a tis fa c tio n o f te d u s in g th e v a lu e s T a b le 2 . I n itia l w e ig h t v a lu e s u s e d in th e f itn e s s fu n c tio n C o n s tra in t C u rre n t e v A ll e v e n ts C u rre n t e v C u rre n t e v E v e n t ta k e E v e n t ta k e E v e n t ta k e e n t is th a t m e n t ta e n t ta s p la c s p la c s p la c “ p la c e d ” (d o e s n o t c o n flic t w ith a n y o th e r e v e n t) a k e u p th e c u rre n t jo b p la c e d k e s p la c e b e fo re th e jo b is d u e fo r c o m p le tio n k e s p la c e a fte r th e jo b is “ a v a ila b le ” e b e tw e e n M o n d a y a n d F rid a y e b e fo re “ n ig h t tim e ” e a fte r “ d a y tim e ” R e w a rd 1 2 0 1 0 0 5 0 5 0 2 5 4 4 T h e E A w a s a llo w e d to ru n u n til a ll th e e v e n ts w e re p la c e d (ie a ll th e h a rd c o n s tra in ts a re s a tis fie d ). T h e re m a y a ris e s itu a tio n s w h e re th e re is n o fe a s ib le s c h e d u le fo r a g iv e n d a ta s e t (ie if 2 5 h o u rs o f w o rk a re s c h e d u le d to ta k e p la c e in o n e d a y ) th e re fo re a lim it is s e t o n th e to ta l n u m b e r o f g e n e ra tio n s th a t m a y b e c a lc u la te d . T h is lim it is c a lc u la te d a s n o o f e v e n ts * 1 5 0 . T h ro u g h in itia l e x p e rim e n ta tio n it w a s fo u n d th a t a llo w in g th e E A to c o n tin u e ru n n in g fo r a s h o rt p e rio d a fte r a fe a s ib le s o lu tio n h a d e v o lv e d a llo w e d fu rth e r im p ro v e m e n t w ith re g a rd s to th e s o ft c o n s tra in ts . T h e n u m b e r o f g e n e ra tio n s u s e d fo r th is ru n -o n p e rio d w a s c a lc u la te d a s to ta l n u m b e r o f e v e n ts * 5 . 2 .2 I m p r o v in g th e Q u a lity o f S c h e d u le s u s in g L a m a r c k is m T o im p ro v e th e q u a lity o f s c h e d u le s c o n s tru c te d b y th e s y s te m , th e a u th o rs im p le m e n te d L a m a rc k ia n w rite b a c k [1 1 ],[5 ]. L a m a rc k ia n th e o ry s u g g e s ts th a t k n o w le d g e a c q u ire d b y in d iv id u a ls d u rin g th e ir life tim e m a y b e p a s s e d g e n e tic a lly o n to th e ir d e s c e n d a n ts . T h is th e o ry h a s s in c e b e e n d is c re d ite d b y b io lo g is ts w h o m a in ta in th a t c ro s s o v e r a n d ra n d o m m u ta tio n a re th e m a in in flu e n c e s o n e v o lu tio n a ry d e v e lo p m e n t. T h e ty p e o f im p ro v e m e n t m a d e to a s c h e d u le is illu s tra te d in fig u re 1 , it m a y b e a c c e p te d th a t a n y s o lu tio n w h e re e v e n ts w ith in a jo b “ o v e rla p ” is n o t fe a s ib le a n d th u s o f n o p ra c tic a l v a lu e . T h e w o rk o f [1 0 ] o u tlin e s th e u s e o f L a m a rc k ia n is m w ith in a n E A u s e d fo r s o lv in g th e N a p ie r U n iv e rs ity tim e ta b lin g p ro b le m . 3 1 0 2 .3 N . U rq u h a rt, K . C h is h o lm , a n d B . P a e c h te r T a r g e te d M u ta tio n In [1 0 ] u s e is m a d e o f w h a t is k n o w n a s “ T a rg e te d m u ta tio n ” . T h is m o d ifie s th e m u ta tio n o p e ra to r to in c re a s e th e c h a n c e s o f g e n e s w ith s p e c ific c h a ra c te ris tic s b e in g s e le c te d fo r m u ta tio n . T h e a c tu a l c h a ra c te ris tic s th a t a re u s e d to s e le c t a g e n e fo r a n in c re a s e d c h a n c e o f m u ta tio n w ill v a ry d e p e n d in g o n th e a p p lic a tio n . In m o s t c a s e s th e g e n e s th a t w ill h a v e th e e x tra b ia s in fa v o r o f m u ta tio n w ill b e th o s e th a t e x h ib it s o m e d e fe c t re q u irin g im p ro v e m e n t. A n e x tra w e ig h t v a lu e w ill b e a d d e d to th e c h a n c e s o f g e n e s e x h ib itin g th a t d e fe c t. In th e c a s e o f th e p ro b le m u n d e r c o n s id e ra tio n th e a u th o rs d e c id e d to im p le m e n t th e b ia s s o th a t g e n e s re la tin g to e v e n ts b re a k in g a h a rd c o n s tra in t h a v e a h ig h e r c h a n c e o f b e in g s e le c te d fo r m u ta tio n . T h e m u ta tio n fu n c tio n w a s b e e n m o d ifie d s o th a t if a g e n e s e le c te d fo r m u ta tio n re p re s e n ts a n e v e n t th a t d o e s n o t b re a k a h a rd c o n s tra in t th e re is a c h a n c e (e x p re s s e d a s a p e rc e n ta g e ) th a t a n o th e r g e n e m a y b e s e le c te d . T a b le 3 s h o w s th e re s u lts o f u s in g a 4 0 % , 5 0 % , a n d 6 0 % b ia s a g a in s t th e m u ta tio n o f p la c e d g e n e s . F ig . 1 . A n e x a m p le o f p a r t o f a s c h e d u le b e f o r e w rite b a c k ( le ft) a n d a fte r (r ig h t) w r ite b a c k 2 .4 P r e m a tu r e C o n v e r g e n c e a n d R e g e n e r a tio n W ith in th e E A it is d e s ir a b le to m a in ta in a d iv e r s e p o p u la tio n . W h e n a g o o d s o lu tio n is fo u n d , th is in d iv id u a l w ill b e g in to d o m in a te a n d th u s r e d u c e th e b io -d iv e r s ity w ith in th e p o p u la tio n . T h e c o n v e r g e n c e o f m a n y p o p u la tio n m e m b e r s to a s im ila r p o in t w ith in th e s e a r c h s p a c e is fe a tu r e o f a ll E A s . It is d e s ir a b le th o u g h , to tr y to p r e v e n t c o n v e r g e n c e u n til th e E A b e in g s to c o n v e r g e o n th e o p tim u m s o lu tio n . O n e m e th o d o f e lim in a tin g th e p r e m a tu r e c o n v e r g e n c e e ffe c t is to im p le m e n t a r e g e n e ra tio n o p e ra to r . W ith in th e r e g e n e ra tio n o p e r a to r a n u m b e r o f in d iv id u a ls h a v e th e ir g e n e s ra n d o m ly r e s e t. T h u s in th e c a s e o f th e s c h e d u lin g a lg o r ith m u n d e r c o n s id e r a tio n a n u m b e r o f m e m b e r s o f th e p o p u la tio n w ill h a v e th e tim e s o f th e ir a ll e v e n ts ra n d o m ly m u ta te d . C o n s id e r a tio n m u s t b e g iv e n to th e fr e q u e n c y o f th e u s a g e o f th e o p e r a to r a n d th e p e r c e n ta g e o f th e p o p u la tio n in v o lv e d . If th e r e g e n e r a tio n O p tim is in g a n E v o lu tio n a ry A lg o rith m o p e r a to r is u tilis e d to o fr e q u e n tly th e n th it a ffe c ts to o m a n y in d iv id u a ls w ith a g e n e tic m a te ria l. T h e o p e ra to r a s im p le m e n te d in itia lly , ra n d o m ly e v e ry 5 0 g e n e ra tio n s . T h e re m u ta tio n o p e ra to r to a ll o f th e g e n e s in in d iv id u a l. T h e tim e s ta k e n to p ro d u c e s h o w n in ta b le 3 . It m a y b e s e e n th e re fo r (a p p ro x im a te ly 4 0 % ) s a v in g in tim e . 3 . fo r S c h e d u lin g 3 1 1 e p o p u la tio n w ill n o t h a v e tim e to e v o lv e , if h ig h fitn e s s th e n th e E A m a y lo o s e u s e fu l re in itia lis e s g e n e ra tio n th e s c h e d u a fe a s ib le e th a t u s in g th e o p e le , sc h re g lo w e r 5 0 % o f th ra to r in p ra c tic e th u s c re a tin g a n e d u le u s in g re g e e n e ra tio n o ffe rs e p o p u la tio n a p p lie s th e e w , ra n d o m n e ra tio n a re a s u b s ta n tia l S e ttin g th e E A P a r a m e te r s T h e p a ra m e te rs m u ta tio n ra te a n d E A . C o n v e n tio n a w ith re fe re n c e to a lte rn a tiv e m e th o in itia l p a ra m e te rs u s re lly p d s w e d w g e n e th e re v io o f a a s c o ith in th e E A to s p e c if ra tio n h a v e th e b ig g e s t b e E A d e s ig n e r u s u a lly s e ts u s e x p e rim e n ta tio n . T h e rriv in g a t a se t o f p a ra m e n s tru c te d a n d w ill b e re fe y p o p u la tio n s iz e , a rin g o n th e s u c c e s s th e s e p a ra m e te rs to a u th o rs d e c id e d to te rs . A v e rs io n o f th rre d to a s v e rs io n C P c ro o r v a in v e E . sso v fa ilu lu e s e s tig A u e r ra te , re o f a n d e c id e d a te tw o s in g th e T a b le 3 . T im e in s e c o n d s to p r o d u c e a fe a s ib le s c h e d u le u s in g d a ta s e t 2 5 4 6 R e g e a n d 5 3 .1 N o 0 % 0 % 0 % n e r 0 % R u B ia B ia B ia B ia a tio b ia n 1 s s s s n 5 7 1 1 2 3 5 0 0 8 1 5 7 2 1 1 7 8 3 5 1 6 1 1 8 8 0 1 6 3 5 9 1 1 4 3 1 9 1 7 3 4 3 7 7 2 4 5 0 2 8 1 6 1 9 3 2 2 2 0 4 4 8 1 4 1 5 1 1 1 6 5 5 7 9 9 6 3 0 8 3 1 A v e 7 5 7 6 3 ra 6 8 6 3 1 6 4 6 4 3 g e .5 .3 .3 .9 .6 s M o d ify in g th e P a r a m e te r s a t R u n T im e T h e firs t m e th o d in v e s tig a te d is b a s e d o n a s im p lifie d v e rs io n o f th e s y s te m d e s c rib e d in [1 3 ] a n d [5 ]. E a c h o f th e s e p a ra m e te rs s ta rts o ff s e t to a lo w v a lu e , w h e n th e in d iv id u a l a t th e h e a d o f th e p o p u la tio n (ie th e m o s t fit in d iv id u a l) c h a n g e s th e p a ra m e te r th a t in v o k e s th e o p e ra to r re s p o n s ib le fo r c re a tin g th a t in d iv id u a l is in c re a s e d s lig h tly . T h is is s im p lifie d fro m th e o rig in a l s y s te m p ro p o s e d b y D a v is [1 3 ] w h ic h n o t o n ly to o k in to a c c o u n t th e m o s t fit in d iv id u a l, b u t a ls o th e o p e ra to rs u s e d to c re a te th e a n c e s to rs o f th a t in d iv id u a l fo r s e v e ra l g e n e ra tio n s p re v io u s to th e c u rre n t g e n e ra tio n . A n e w v e rs io n o f th e E A in c o rp o ra tin g th e a b ility to m o d ify its p a ra m e te rs a t ru n tim e w a s c o n s tru c te d . W h e n e v e r th e m o s t fit in d iv id u a l c h a n g e s th e p a ra m e te r c o n tro llin g th e c re a tin g o p e ra tio n is m o d ifie d to in c re a s e th e u s a g e o f th e o p e ra to r th a t c a u s e d th e im p ro v e m e n t. F o r in s ta n c e if a n e w m o s t-fit in d iv id u a l is c re a te d b y th e m u ta tio n o p e ra to r th e n th e m u ta tio n ra te is in c re a s e d b y 0 .1 % , in th e e v e n t o f th e c ro s s o v e r o p e ra tio n c re a tin g a n e w m o s t-fit in d iv id u a l th e n th e c ro s s -o v e r ra te is 3 1 2 N . U rq u h a rt, K . C h is h o lm , a n d B . P a e c h te r in c re a s e d b y 0 .1 % . In th e e v e n t o f th e re g e n e ra tio n o p e ra to r b e in g re s p o n s ib le th e n th e fre q u e n c y o f th e re g e n e ra tio n o p e ra to r is in c re a s e d b y 1 g e n e ra tio n . A n e w v e rs io n o f th e E A im p le m e n tin g th is o p e ra to r w a s c o n s tru c te d , it w ill b e re fe rre d to a s V P 1 . In itia l v a lu e s fo r th e m u ta tio n ra te a n d c ro s s o v e r ra te a re 7 5 % a n d 5 0 % w ith th e re g e n e ra tio n in te rv a l b e in g s e t a t 5 0 g e n e ra tio n s , a s u s e d in C P . T h e re s u lts o b ta in e d w h e n u tilis in g d a ta s e t2 m a y b e s e e in ta b le 4 . In o rd e r to a tte m p t to d e c re a s e th e tim e ta k e n to b u ild a s c h e d u le , it w a s d e c id e d to m o d ify th e s ta rtin g v a lu e s , th u s a llo w in g th e E A its e lf to m a k e o n e o r th e o th e r b e c o m e th e d o m in a n t o p e ra to r. A s e c o n d v e rs io n (V P 2 ) w ith s ta rtin g m u ta tio n a n d c ro s s o v e r ra te s w e re s e t to 1 0 % a n d th e s ta rtin g re g e n e ra tio n in te rv a l w a s s e t to 1 0 0 0 g e n e ra tio n s , th u s n o o p e ra to r is a llo w e d to d o m in a te a t th e s ta rt. T a b le 4 . F itn e s s o f s c h e d u le s g e n e r a te d u s in g d a ta s e t 2 . ( C P – C o n s ta n t p a r a m e te r s ; V P 1 – V a r ia b le p a r a m e te r s , s ta rts w ith 7 5 % c r o s s o v e r a n d 5 0 % m u ta tio n r a te s ; V P 2 – V a ria b le p a r a m te r s s ta r ts w ith 1 0 % c r o s s o v e r a n d m u ta tio n ra te s ) R u C P V P V P n . 2 1 1 3 4 1 6 2 .9 3 4 5 9 8 .1 3 4 2 1 4 .1 2 3 4 0 1 9 .4 3 4 4 8 7 .4 3 4 3 8 2 .8 3 3 4 1 9 3 .2 3 4 4 9 7 .6 3 4 4 2 4 .7 4 3 4 0 6 2 .4 3 4 3 5 2 .3 3 4 3 7 2 .8 F ro m th e fig u re s in ta b le 4 it m a y b e s e e n th a t v e rs io n V P 1 g iv e s th e h ig h e s t e ra g e fitn e s s w h e n b u ild in g s c h e d u le s b a s e d o n d a ta s e t 2 . T h e re s u lts w o u ld g g e s t th a t in o rd e r to b e e ffe c tiv e th e ru n tim e p a ra m e te rs m u s t h a v e s ta rtin g v a lu e s a t a re p ro v e n to p ro v id e a n e ffic ie n t re s u lt w h e n u s e d w ith o u t m o d ific a tio n . It w a s u n d th a t w h e n s ta rtin g w ith lo w in itia l v a lu e s th e E A is u n a b le to p ro g re s s d u e to e la c k o r c ro s s o v e r, m u ta tio n o r re g e n e ra tio n . B a s e d o n th e re s u lts o b ta in e d fro m th e te s tin g o f v e rs io n s V P 1 a n d V P 2 , p tim is in g th e p a ra m e te rs a t ru n -tim e d o e s n o t a p p e a r to g iv e a n y a d v a n ta g e s o v e r s in g p a ra m e te rs th a t th e u s e r h a s a rriv e d a t u s in g “ tria l a n d e rro r” m e th o d s . T h e e a s o n s fo r fa ilu re m a y b e c o n n e c te d w ith th e in itia l p a ra m e te r v a lu e s . If th e a ra m e te rs a re in itia lly s e t to s m a ll v a lu e s (a s in v e rs io n V P 2 ), th e re is a la c k o f c tiv ity w ith in th e G A , a n d th e p a ra m e te rs c a n n o t in c re a s e , a n d th u s in c re a s e th e c tiv ity w ith in th e a lg o rith m u n til n e w m o s t-fit in d iv id u a ls a re b e in g c re a te d . a v su th fo th o u r p a a 3 .2 U s in g a M e ta -L e v e l E A to O p tim ise th e P a r a m e te r s T h e p a ra m e te rs o f a n E A m a y th e m s e lv e s b e o p tim is e d b y u s in g a n o th e r E A [1 3 ], [4 ], k n o w n a s a m e ta -E A , to a tte m p t to e v o lv e a s e t o f e ffic ie n t p a ra m e te rs . W ith in th is m e ta -le v e l E A th e g e n e s re p re s e n t th e p a ra m e te rs o f th e E A th a t w e w is h to o p tim is e (h e re in a fte r re fe rre d to a s th e p ro b le m E A ). T h e fitn e s s fu n c tio n o f th e m e ta -E A is b a s e d o n th e re s u lts o f ru n n in g th e p ro b le m E A w ith th e p a ra m e te rs s to re d in e a c h in d iv id u a l w ith in th e m e ta -E A p o p u la tio n . T h e p a ra m e te rs o f th e p ro b le m E A to b e o p tim is e d a re lis te d in ta b le 5 . T h e fitn e s s o f e a c h in d iv id u a l w ith in th e m e ta -E A w a s c a lc u la te d b y ru n n in g th e s c h e d u lin g -E A u s in g th e p a ra m e te rs e n c o d e d w ith in th a t in d iv id u a l, th e fitn e s s w a s c a lc u la te d a s th e fitn e s s o f th e fin a l s c h e d u le le s s a tim e v a lu e . O p tim is in g a n E v o lu tio n a ry A lg o rith m fo r S c h e d u lin g 3 1 3 T a b le 5 . P a ra m e te rs u se d fo r th e m e ta -E A P a r a m e te P o p u la tio E lite s iz e M u ta tio n C ro sso v e r V a lu e 1 0 1 4 0 % 5 0 % n s iz e D u rin g in itia lis a tio n th e firs t 2 m p a ra m e te rs u s e d w ith v e rs io n C P ra n d o m ly . In o rd e r to a s c e rta in th e a b ility p a ra m e te rs th e fo llo w in g e x p e rim e w o rk s ta tio n s w e re a llo w e d to ru n th th e E A s w a s ru n n in g c o m p le te ly in d w e re e v a lu a te d , th is w o u ld m e a n c o n fig u ra tio n s , w h ic h w o u ld b e 1 2 .5 R a te r R a te e m b e rs o f th e p o p u la tio n w e re s e t to e q u a l th e a n d th e re m a in in g m e m b e rs w e re in itia lis e d o f th e n t w a s e m e ta -E e p e n d e n e a c h P m in s p e m c a A tly e ta -E A to rrie d o u t fo r a p p ro . In to ta l 3 C e v a lu a te d r e v a lu a tio n . e v o o v e r x im a 9 6 1 o n lv e a n e ffic ie n t s e t o 6 d a y s . S ix id e n tic a te ly 1 3 6 h o u rs , e a c h o d iffe re n t c o n fig u ra tio n a v e ra g e 6 6 0 d iffe re n f l f s T a b le 6 . P a ra m e te rs r e q u irin g o p tim is a tio n b y th e m e ta -E A C h ro m o so m e P o p s iz e E lite s iz e M u ta tio n r a te C ro ss o v e r ra te B fa v o R R ia s u r o e g e e g e m u ta tio n in f f itn e s s n e r a te tim e n e ra te P C M in 1 0 0 0 F a lse 0 0 M a 2 0 1 0 1 0 1 0 x N o te 0 0 0 T ru e 1 0 0 1 0 0 I f 0 th e n e litis m is s w itc h e d o f f . c h a n c e o f a n y o n e in d iv id u a l b e in g m % c h a n c e o f a n y in d iv id u a l b e in g s e le c c ro sso v e r. I f tr u e p r o v id e in c r e a s e c h a n c e s o f b e in m u ta te d to th e m o s t fit e x a m p le s N o o f g e n e r a tio n s b e tw e e n re - g e n e r a tio P e r c e n ta g e o f p o p u la tio n to r e - g e n e ra te % u ta te d . te d f o r g n . In a ll o f th e ru n s th e m e ta E A fitn e s s in itia lly in c re a s e d ra p id ly , b e fo re s lo w in g . It s h o u ld b e re m e m b e re d th a t th e g ra p h c o v e rs a p e rio d o f a p p ro x im a te ly 6 d a y s . If it h a d b e e n p ra c tic a l to h a v e a llo w e d th e m e ta -E A e x p e rim e n t to c o n tin u e th e g ra p h s u g g e s ts th a t s o m e , if n o t a ll o f th e fitn e s s ra te s w o u ld h a v e c o n tin u e d to g ro w . A fte r th e m e ta E A e x p e rim e n t h a d fin is h e d th e p a ra m e te r s e t w a s e x tra c te d fro m th e in d iv id u a l w ith th e h ig h e s t fitn e s s . T h is is s h o w n in ta b le 7 . T h e p e rc e n ta g e o f th e p o p u la tio n to b e re g e n e ra te d h a s b e e n d e c re a s e d a n d th e in te rv a l b e tw e e n re g e n e ra tio n in c re a s e d fro m th e s e ttin g s u s e d in v e rs io n C P o f th e G A . T h e p o p u la tio n a n d th e e lite s iz e s h a v e b o th b e e n in c re a s e d . T h e s e p a ra m e te rs h a v e b e e n in c o rp o ra te d in a n e w v e rs io n o f th e E A k n o w n a s M P . It is n o w p o s s ib le to c o m p a re v e rs io n M P w ith v e rs io n V P 1 a n d v e rs io n C P , th e re s u lts a re s h o w n in ta b le 7 . t 3 1 4 N . U rq u h a rt, K . C h is h o lm , a n d B . P a e c h te r T a b le 7 . :T h e p a r a m e te r s e v o lv e d u s in g th e m e ta - G A P a ra m e te r C ro s s o v e r R a te M u ta tio n R a te R e g e n e ra tio n In te rv a l R e g e n e ra tio n % o f p o p u la tio n P o p u la tio n s iz e E lite s iz e B ia s m u ta tio n in fa v o u r o f fitn e s s V a lu e 6 8 3 0 7 4 2 9 9 5 4 Y e s T a b le 8 . A c o m p a r is o n o f th e fitn e s s v a lu e o f s c h e d u le s R u n C P V P 1 M P 1 3 4 5 0 0 .0 8 3 4 4 8 5 .0 8 3 4 5 4 5 .0 5 2 3 4 7 1 5 .8 3 4 5 9 8 .1 3 4 4 5 4 .8 4 . C o n c lu s io n s a n d F u tu r e W 4 .1 S o m e G e n e r a l C o n c lu s io n s o n th e E A 3 3 4 1 6 2 3 4 4 8 7 .4 3 4 4 3 5 .8 4 3 2 8 9 1 3 4 5 1 8 3 4 6 1 1 o r k T h e re p re s e n ta tio n u s e d is a b a s ic d ire c t re p re s e n ta tio n . A lth o u g h m a n y E A s c h e d u lin g te c h n iq u e s , m a k e e x te n s iv e u s e o f m e m e tic a lg o rith m s , th e b a s ic s c h e d u lin g p ro b le m o u tlin e d in th is re p o rt w a s s o lv e d w ith o u t re c o u rs e to a p u re ly m e m e tic a lg o rith m . A n u m b e r o f c h a n g e s fro m th e b io lo g ic a l m o d e l w e re n e c e s s a ry to a llo w th e s y s te m to p ro d u c e s c h e d u le s w ith a h ig h fitn e s s in a re a s o n a b le tim e . T h e s e c h a n g e s w e re th e u s e o f a s te a d y -s ta te p o p u la tio n , th e u s e o f re s tric te d m u ta tio n a n d th e u s e o f L a m a rk ia n w rite b a c k . T h e m u ta tio n o p e ra to r w a s e x te n s iv e ly m o d ifie d a n d it w a s re s tric te d to o n ly m u ta tin g e v e n t s ta rt tim e s to w ith in th e s ta rt a n d e n d tim e fo r th a t jo b , ra th e r th a n a n y w h e re o n th e s c h e d u le . T h e m u ta tio n o p e ra to r w a s a ls o g iv e n a b ia s to d is c o u ra g e th e m u ta tio n o f g e n e s th a t re p re s e n te d a lre a d y p la c e d e v e n ts . It m a y b e c o n c lu d e d th a t th e m u ta tio n o p e ra to r w o rk s m o s t e ffe c tiv e ly w h e n it is re s tric te d fro m p ro d u c in g s c h e d u le s th a t a re o b v io u s ly in fe a s ib le . T h e m u ta tio n o p e ra to r’s p o w e r s te m s fro m its a b ility to c re a te c h a n g e s th a t a re n o t in flu e n c e d b y e x is tin g g e n e tic m a te ria l w ith in th e p o p u la tio n . T h is a b ility to m a k e “ ra d ic a l” c h a n g e s c a n c a u s e a b ig g e r d e c re a s e th a n in c re a s e in fitn e s s , if th e c h a n g e s a re d e trim e n ta l to th e o v e ra ll p ro b le m th a t th e E A is a tte m p tin g to s o lv e . T h e d ire c te d m u ta tio n o p e ra to r s h o w e d a s ig n ific a n t im p ro v e m e n t, th e a v e ra g e tim e s to b u ild a fe a s ib le s c h e d u le d e c re a s e d fro m 7 6 8 .5 to 5 6 3 .3 s e c o n d s fo r v e rs io n s w ith n o b ia s in fa v o u r o f b a d ly p la c e d e v e n ts , a n d 5 0 % b ia s re s p e c tiv e ly . T h e s ta n d a rd d e v ia tio n in tim e ta k e n w ith o u t b ia s w a s 6 3 2 .3 s e c o n d s , b u t a d d in g a 5 0 % b ia s , th is d e c re a s e s to 5 8 4 s e c o n d s . T o fu rth e r e x a m in e th e e ffe c ts o f ta rg e te d O p tim is in g a n E v o lu tio n a ry A lg o rith m fo r S c h e d u lin g 3 1 5 m u ta tio n tw o m o re v e rs io n s o f th e E A w e re c o n s tru c te d w ith 4 0 % a n d 6 0 % b ia s a g a in s t th e m u ta tio n o f p la c e d e v e n ts . A m o re d e ta ile d c o m p a ris o n o f ty p ic a l ru n s is s h o w n in fig u re 2 . T h e fitn e s s fu n c tio n is o f p a ra m o u n t im p o rta n c e to th e E A , a n d a s u b tle a lte ra tio n to th e fitn e s s fu n c tio n w ill ra d ic a lly a ffe c t th e o u tp u t o b ta in e d fro m th e E A . T h e e x a c t v a lu e s u s e d a s w e ig h ts w ith in th e fitn e s s fu n c tio n w e re e s ta b lis h e d b y “ tria l a n d e rro r” . It w a s n o t p o s s ib le to a llo w th e m e ta -E A to o p tim is e th e fitn e s s fu n c tio n v a lu e s d u rin g th e m e ta -E A e x p e rim e n t. If th is h a d ta k e n p la c e it is lik e ly th a t th e m e ta -E A w o u ld h a v e a lte re d th e fitn e s s fu n c tio n v a lu e s to g iv e a h ig h fitn e s s re g a rd le s s o f th e a c tu a l fitn e s s o f th e s c h e d u le . It is a c k n o w le d g e d th a t a m o re e ffic ie n t s e t o f w e ig h ts m a y e x is t. S o m e re s e a rc h in to e s ta b lis h in g a n o p tim u m s e t o f w e ig h ts fo r a fitn e s s fu n c tio n w ith in a d ra fts p ro g ra m [4 ] h a s b e e n c a rrie d o u t w ith c o n s id e ra b le s u c c e s s T h e u s e o f th e re -g e n e ra tio n o p e ra to r p ro v id e d a m a rk e d im p ro v e m e n t. In itia lly th is o p e ra to r a p p e a re d to b e v e ry d is ru p tiv e a n d c o u ld p o s s ib ly c a u s e th e lo s s o f g e n e tic m a te ria l. T h e re -g e n e ra tio n o p e ra to r re q u ire s s k ill in its u s a g e , if to o m a n y in d iv id u a ls a re re -in itia lis e d o r th e o p e ra to r in v o k e d to o fre q u e n tly th e n u s e fu l g e n e tic m a te ria l w ill b e d e s tro y e d . F u rth e r re s e a rc h to fin d if th e re is a n o p tim u m p e rc e n ta g e o f p o p u la tio n to re -in itia lis e a n d a n o p tim u m fre q u e n c y o f u s e w o u ld b e d e s ira b le . T h e m e ta -E A lo w e re d th e p e rc e n ta g e o f th e p o p u la tio n to b e re -in itia lis e d to 2 9 % . T h e re -g e n e ra tio n in te rv a l w a s in c re a s e d to 7 4 g e n e ra tio n s . 4 .2 F ig M P d e c v e r re p S o m e C o n c lu s io n s o n P a r a m e te r O p tim is a tio n u re 3 s h o w s th e fitn e s s g ro w th ra te s fo r ty p ic a l ru n s u s in g v e rs io n s C P , V P 1 a n d , w h e n ru n n in g d a ta s e t 2 . V e rs io n C P re p re s e n ts th e E A ru n n in g w ith p a ra m e te rs id e d b y th e u s e r u s in g “ tria l a n d e rro r” , v e rs io n V P 1 re p re s e n ts th e b e s t o f th e s io n s th a t a llo w e d th e p a ra m e te rs to b e o p tim is e d a t ru n tim e , fin a lly v e rs io n M P re s e n ts th e re s u lt o f th e m e ta -E A . W h e n try in g to p ic k a “ b e s t” v e rs io n o f th e E A it is im p o rta n t to n o tic e th a t d iffe re n t v e rs io n s d is p la y d iffe re n t q u a litie s . F o r in s ta n c e v e rs io n C P p ro d u c e d s c h e d u le s w ith a s lig h tly h ig h e r a v e ra g e fitn e s s (fo r d a ta s e t 3 , s e e ta b le 1 0 ) th a n M G , b u t th e tim e re q u ire d to p ro d u c e th e s c h e d u le s is s ig n ific a n tly le s s (s e e ta b le 1 1 ). It is th e c o n c lu s io n o f th e a u th o rs th a t v e rs io n M P a p p e a rs to re p re s e n t th e b e s t “ tra d e -o ff” in te rm s o f a c h ie v in g a n a c c e p ta b le le v e l o f fe a s ib le s c h e d u le s , a c h ie v in g a h ig h e r a v e ra g e fitn e s s th a n th e v e rs io n s a n d c o m p le tin g th e ru n s in s ig n ific a n tly le s s tim e . V e rs io n s V P 1 a n d V P 2 w e re u n a b le to p ro d u c e s c h e d u le s th a t w e re e ffic ie n t w h e n c o m p a re d w ith th e re s u lts o b ta in e d u s in g C P a n d M P (s e e ta b le 9 ). T h e m a in re a s o n fo r th is a p p e a rs to b e d u e to th e s u b tle n a tu re o f th e lin k s b e tw e e n th e p a ra m e te rs . A d ju s tin g th e p a ra m e te rs b a s e d o n ly o n th e m o s t fit in d iv id u a l in th e p o p u la tio n is n o t n e c e s s a rily p ro d u c tiv e . T h e e n tire p o p u la tio n c o n trib u te s to e a c h o th e r fitn e s s th ro u g h c ro s s o v e r th e re fo re a lth o u g h o n e o p e ra to r m a y a p p e a r to b e d o m in a n t in te rm s o f c re a tin g th e in d iv id u a l w ith th e h ig h e s t fitn e s s a n d th is o p e ra to r m a y n o t b e d o m in a n t th ro u g h o u t th e p o p u la tio n . 3 1 6 N . U rq u h a rt, K . C h is h o lm , a n d B . P a e c h te r 35000 Fitnes 34500 34000 33500 33000 2704 2545 2386 2227 2068 1909 1750 1591 1432 1273 1114 955 796 637 478 319 1 32000 160 32500 Generation No bias. 50 % bias 40 % bias F ig . 2 . F itn e s s G r o w th w ith 0 ,4 0 ,5 0 a n d 6 0 % 60 % bias m u ta tio n b ia s u s in g d a ta s e t 2 A m a jo r c o n c lu s io n th a t m a y b e d ra w n fro m th e w o rk is th a t th e o p tim is a tio n a n d s e le c tio n o f th e E A p a ra m e te rs m a y b e b e s t c a rrie d o u t b y a m e ta -E A . In th is c a s e th e m e ta -E A p a ra m e te rs re c o rd e d a s m a ll b u t s ig n ific a n t in c re a s e in p e rfo rm a n c e fro m th o s e o b ta in e d u s in g th e m o re tra d itio n a l “ tria l a n d e rro r” m e th o d s . W ith in v e rs io n C P , th e v e rs io n th a t u s e d th e “ tria l a n d e rro r” p a ra m e te rs 8 6 % o f a ll th e E A ru n s re s u lte d in fe a s ib le s c h e d u le s , u s in g v e rs io n M P th a t in c re a s e d to 9 3 % . It m a y b e s e e n th a t u s in g p a ra m e te rs d e riv e d b y th e m e ta -E A it is p o s s ib le to a c h ie v e a fa s te r a n d h ig h e r g ro w th ra te . O p tim is in g w ith th e m e ta -E A g iv e s s u p e rio r re s u lts to th o s e a c h ie v e d u s in g p a ra m e te rs d e c id e d b y th e u s e r (v e rs io n C P ) o r b y m o d ify in g th e p a ra m e te rs a t ru n tim e (V P 1 a n d V P 2 , s e e fig u re 4 ) 35000 Fitnes 34500 CP 34000 VP1 MP 33500 33000 1111 963 889 815 741 667 593 519 445 371 297 223 149 1 75 32000 1037 32500 Generation F ig . 3 . F itn e s s g ro w th c o m p a ris o n fo r ty p ic a l ru n s o f v e rs io n s C P , V P 1 a n d M P O p tim is in g a n E v o lu tio n a ry A lg o rith m fo r S c h e d u lin g 3 1 7 T a b le 9 . T o ta l jo b s p la c e d (a v e ra g e d o v e r 1 0 ru n s ) D a ta se t 1 (1 6 jo b s ) 1 6 1 5 .8 1 5 .9 1 6 C P V P 1 V P 2 M P D a ta 1 4 1 4 1 2 1 5 se t 2 (1 5 Jo b s) .7 .4 .1 D a ta 1 7 1 7 1 3 1 7 se t 3 (1 8 jo b s ) .7 .6 .8 T a b le 1 0 . S c h e d u le fitn e s s (A v e ra g e d o v e r 1 0 ru n s ) C P V P 1 V P 2 M P D a ta se t 1 7 4 6 2 6 .5 7 4 5 0 0 .1 3 7 4 5 9 2 .8 1 7 4 6 0 0 .8 1 D a ta 3 4 4 3 3 4 2 6 3 3 9 5 3 4 4 7 se t 2 3 .3 8 3 0 .0 4 1 0 .8 6 8 .6 3 4 D a ta 5 3 6 9 5 3 6 2 5 3 0 0 5 3 6 8 se t 3 8 .6 0 8 .3 3 9 .9 2 1 .2 6 T a b le 1 1 . T im e ta k e n f o r r u n ( s e c o n d s a v e r a g e d o v e r 1 0 r u n s ) C P V P 1 V P 2 M P D a ta se t 1 2 0 7 .2 2 1 4 .5 4 1 2 .1 1 5 4 .3 D a ta se t 2 8 0 5 6 7 8 .4 1 6 5 7 .6 3 6 7 .5 D a ta se t 3 1 3 8 8 .3 1 2 6 0 .9 2 4 3 1 3 3 8 400 214.5 350 412.1 300 154.3 250 CP 200 VP1 150 VP2 100 VP3 50 MP 0 CP VP1 VP2 VP3 MP F ig . 4 . A v e r a g e tim e ( s e c o n d s ) to b u ild a f e a s ib le s c h e d u le 4 .3 F u tu r e W o r k T h e a u th o rs p la n to a p p ly m e ta -E A p rin s y s te m u s e d to s o lv e a ro u tin g p ro b le m . R ra is e d th e p o s s ib ility o f u s in g d iffe re n tia l u s e d w ith in th e fitn e s s fu n c tio n [2 ]. It e v o lu tio n to o p tim is e th e p a ra m e te rs fo r m e ta -E A is th e p ro c e s s in g tim e re q u ire d , re s u lts to th e m e ta -E A , b u t w ith fe w e r C P U c ip le s to th e o p tim is in g o f a n E A b a s e d e c e n t w o rk in to d iffe re n tia l e v o lu tio n h a s e v o lu tio n to o p tim is e th e w e ig h tin g v a lu e s m a y a ls o b e p o s s ib le to u s e d iffe re n tia l th e w h o le E A . A m a jo r p ro b le m w ith th e d iffe re n tia l e v o lu tio n m a y a c h ie v e s im ila r c y c le s b e in g re q u ire d . 3 1 8 N . U rq u h a rt, K . C h is h o lm , a n d B . P a e c h te r R e fe r e n c e s [ 1 ] P r o d u c tio n S c h e d u lin g a n d R e s c h e d u lin g w ith g e n e tic a lg o rith m s . B ie r w ir th C , M a ttf e ld D . E v o lu tio n a r y C o m p u ta tio n v o lu m e 7 , N o 1 . M I T P re s s 1 9 9 9 . [ 2 ] C o - E v o lv in g D ra u g h ts S tra te g ie s w ith D if fe r e n tia l E v o lu tio n , C h a p 9 p p 1 4 7 - 1 5 8 in N e w I d e a s in O p tim is z a tio n , C o r n e D , D o r ig o M , G lo v e r F E d s . M c G r a w - H ill 1 9 9 9 . [ 3 ] B u ild in g a n d O p tim is in g a S c h e d u lin g G A . B S c H o n o u rs d is s e r ta tio n . U r q u h a r t, N C h is h o lm , K .( s u p e r v is o r ) . N a p ie r U n iv e r s ity , E d in b u r g h 1 9 9 8 . [ 4 ] M a c h in e L e a r n in g U s in g a G e n e tic A lg o rith m to O p tim is e a D ra u g h ts P r o g r a m B o a r d E v a lu a ti o n F u n c t io n . C h i s h o lm K .J , B r a d b e e r P .V .G .. P r o c e e d i n g s o f I E E E I C E C ’9 7 , I n d ia n a p o lis , U S A , 1 9 9 7 . [ 5 ] A n in tr o d u c tio n to G e n e tic A lg o rith m s , M itc h e ll, M . M I T p r e s s 1 9 9 6 . [ 6 ] . E x te n s io n s to a M e m e tic T im e ta b lin g S y s te m . P a e c h te r B , N o r m a n M , L u c h ia n H . P r a c tic e a n d th e o r y o f A u to m a te d T im e ta b lin g , B u r k e a n d R o s s E d s . S p rin g e r V e rla g 1 9 9 6 . [ 7 ] E v o lu tio n a r y C o m p u ta tio n , F o g e l D B . I E E E P r e s s 1 9 9 5 . [ 8 ] S p e c ia lis e d R e c o m b in a tiv e O p e ra to rs f o r T im e ta b lin g P r o b le m s , B u r k e E , E llim a n D , W e a r e R . P r o c e e d in g o f E v o lu tio n a r y C o m p u tin g A I S B W o r k s h o p S h e ffie ld U K A p ril 1 9 9 5 e d F o g a r ty , T . S p r in g e r - V e rla g 1 9 9 5 . [ 9 ] O p tim is in g a P re s e n ta tio n T im e ta b le U s in g E v o lu tio n a r y A lg o rith m s . P a e c h te r B . L e c tu r e N o te s I n C o m p u te r S c ie n c e N o 8 6 4 , S p rin g e r- V e rla g 1 9 9 4 . [ 1 0 ] T w o s o lu tio n s to th e G e n e r a l T im e ta b le P r o b le m U s in g E v o lu tio n a r y M e th o d s , P a e c h te r B , C u m m in g A , L u c h ia n H , P e tr iu c , M . P r o c e e d in g s o f th e I E E E W o r ld C o n g r e s s o n C o m p u ta tio n a l I n te llig e n c e , J u n e 1 9 9 4 . [ 1 1 ] A C a s e f o r L a m a r c k ia n E v o lu tio n , A c k le y D H , L ittm a n M L . A r tific ia l L ife I II, L a n g to n C e d . A d d is o n -W e sle y 1 9 9 4 . [ 1 2 ] G e n e tic A lg o r ith m s in S e a r c h O p tim is a tio n a n d M a c h in e L e a r n in g , G o ld b e r g D . A d d is o n W e sle y 1 9 8 9 . [ 1 3 ] A d a p tin g O p e ra to r P ro b a b ilitie s in G e n e tic A lg o r ith m s , D a v is L . P r o c e e d in g s o f th e th ir d I n te r n a tio n a l C o n fe r e n c e o n G e n e tic A lg o r ith m s . S c h a f f e r J . e d . M o r g a n K a u fm a n n 1 9 8 9 . [ 1 4 ] O p tim is a tio n & C o n tr o l P a r a m e te r s f o r G e n e tic A lg o rith m s , G r e fe n s te tte J . I E E E T r a n s a c tio n s o n S y s te m s , M a n a n d C y b e r n e tic s 1 9 8 6 . [ 1 5 ] F a s t P r a c tic a l E v o lu tio n a r y T im e ta b lin g .. C o r n e D , R o s s P , F a n g H .. L e c tu r e N o te s I n C o m p u te r S c ie n c e N o 8 6 4 , S p r in g e r - V e r la g 1 9 9 4 . [ 1 6 ] A d a p tin g o p e r a to r s e ttin g s in G e n e tic A lg o rith m s . T u s o n A , R o s s P . E v o lu tio n a r y C o m p u ta tio n V o l 6 N o 2 . M a s s a c h u s e tts I n s titu te o f T e c h n o lo g y . O n - l i n e E v o l u t i o n o f C o n t r o l f o r a Fo u r - Le g g e d R o b o t Us i n g G e n e t i c P r o g r a m m i n g B jö rn A n d e rs s o n , P e r S v e n s s o n , M a ts N o rd a h l a n d P e te r N o rd in C o m p le x S y s te m s , C h a lm e rs U n iv e rs ity o f T e c h n o lo g y , S E -4 1 2 9 6 G o th e n b u rg , S w e d e n {t f e m n , n o r d i n }@ f y . c h a l m e r s . s e A b s to w G P R /C tio n 1 tr a c t. W a lk d y n s y s te m . se rv o s. a n d th e e e v o lv e a ro b o tic c o n tro lle r fo r a fo u ra m ic a lly . E v o lu tio n is p e rfo rm e d o n -lin T h e ro b o t h a s e ig h t d e g re e s o f fre e d o m D iffe re n t w a lk in g s tra te g ie s a re s h o w n e v o lv in g s y s te m is ro b u s t a g a in s t m e c h a le g g e d re a l ro b o e b y a lin e a r m a n d is b u ilt fr b y th e ro b o t d n ic a l fa ilu re s . t e n a c h o m u rin a b lin g it in e c o d e s ta n d a rd g e v o lu - I n tr o d u c tio n R o b o ts o n le g s c o n s titu te b o th o n e o f th e la rg e s t p o te n tia ls a n d o n e o f th e la rg e s t c h a lle n g e s fo r in te llig e n t ro b o tic c o n tro l. A p p lic a tio n s a re n u m e ro u s in a ll e n v iro n m e n ts a c c e s s ib le to h u m a n s a n d a n im a ls b u t in a c c e s s ib le to w h e e le d a u to n o m o u s a g e n ts . In g e n e ra l th e fle x ib ility o f th e le g g e d ro b o t in c re a s e s w ith d e c re a s in g n u m b e r o f le g s , u n fo rtu n a te ly s o d o e s a ls o c o n tro l c o m p le x ity . It is le s s c o m p lic a te d to c o n tro l a ro b o t w ith s ix le g s o r m o re , s in c e th e ro b o t c a n h a v e fo u r le g s o n th e g ro u n d a ll th e tim e p r o v i d i n g s t a b l e s t a t i c b a l a n c e [ 2 , 4 , 9 ] . A l r e a d y t h e f o u r -l e g g e d c a s e b e c o m e s m o r e d i f f i c u l t . A f o u r -l e g g e d c r e a t u r e c a n c r a w l l i k e a t u r t l e w i t h p a r t i a l s u p p o r t b y i t s b o d y o r b y a ta il. In th is w a y it is p o s s ib le to w a lk o n fo u r le g s w ith s ta tic b a la n c e . S ta tic b a la n c e m e a n s th a t th e a g e n t is in b a la n c e a t a ll m o m e n ts s c a rify in g s o m e e ffic ie n c y a n d fle x ib ility b y d ra g g in g th e ir b o d y o v e r th e g ro u n d . H o w e v e r, it is m o re a d v a n ta g e o u s in g e n e ra l to u s e a d y n a m ic w a lk w h e re th e a g e n t w ill fa ll if in te rru p te d in th e m id d le o f a m o v e m e n t. W e h a v e e v o l v e d t h e f i r s t c o n t r o l l e r f o r a f o u r -l e g g e d r o b o t , l e a r n u s i n g a g e n e t i c p ro g ra m m in g s y s te m a n d a re a l ro b o t. T h e e v o lu tio n o f b e h a v io r p a s s e s th ro u g h s e v e ra l s ta g e s s ta rtin g w ith s im p le p a d d lin g b e h a v io r, c o n tin u in g th ro u g h c ra w lin g , " c a m e l w a lk " , a n d fin a lly g a llo p in g w ith d y n a m ic e le m e n ts in th e w a lk . 2 E x p e r im e n ts S in c e w e a re u s in g a re a l ro b o t th e e x p e rim e n t c o n s is ts o f o n e h a rd w a re p a rt, th e ro b o t, a s w e ll a s a s o ftw a re p a rt, th e G P s y s te m . T h e re is a ls o a c o m m u n ic a tio n p a rt o f S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 1 9 − 3 2 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 3 2 0 th se re th B . A n d e rs s o n e t a l. e s y s te m n so rs. M a so n w e e ro b o t b , se n d o st e x n e e d e a c k if in g p e r d a it h c im s a o n tro e n ts p e c ia d a d v l c w l a a n o m e re u to c e d m a n d ru n o m a tic to o f to th e a c tu a to rs a n d re c e iv in g fe e d b a c k fro m th e n a lo n g tim e s c a le o f a b o u t 2 0 h o u rs a n d fo r th is s y s te m th a t m o n ito rs th e e x p e rim e n t a n d p u lle d a r. 2 .1 T h e f o u r le g g e d r o b o t T h e 1 . T N c m w ith n e e d fo u r-le g h e se rv o . T h e se a s e ria l e d fo r th g e d ro b o t is b u ilt w ith e ig h t s ta n s a re o f o n e o f th e s m a lle s t a v a rv o s a re c o n tro lle d b y a s ta n d a rd p o rt to c o n tro l u p to e ig h t s e rv o is e x p e rim e n t. d a ila se s. rd R b le rv o N o /C s e rv o b u t s till c o n tro lle in p u t s e n s a s a c tu a to h a v e a m o m r c a rd , w h ic s o rs o n th e rs, se e e n tu m h a llo w d o g its F ig o f s a e lf u re 1 1 P C a re F ig u r e 1 : T h e fo u r-le g g e d ro b o t d o g . F itn e s s is g e n e ra te d th ro u g h a c o m p u te r m o u s e , w h ic h th e ro b o t d ra g s b e h in d its e lf. P o s itiv e re in fo rc e m e n t is g iv e n b y d ra g g in g th e m o u s e in th e fo rw a rd d ire c tio n . D ra g g in g b a c k w a rd s g iv e s a n e g a tiv e fitn e s s s ig n a l. J u s t m o v in g th e ro b o t g iv e s a s m a ll p o s itiv e c o n trib u tio n to fitn e s s . T h e e x p e rim e n t s e t-u p c o n s is ts o f a re c ta n g u la r b o x ( 0 .3 m X 1 .2 m ) w h e r e th e a g e n t c a n le a r n f o r w a r d w a lk in g . I f it r e a c h e s th e e n d o f th e b o x th e n it is a u to m a tic a lly p u lle d b a c k w a rd s to th e o th e r e n d b y a n e le c tric m o to r. S in g le e x p e rim e n ts ra n a u to m a tic a lly w ith th is s e t-u p fo r m o re th a n 1 8 h o u rs . 2 .2 C o n t r o l s y s t e m T h b in e ra P C e c o a ry tin g a n d n tro l m a c h o n a n o rm s y s te m is b u ilt in e c o d e [ 1 ,5 ] . T s e t o f re g is te rs o a l m o u se c o m m a r h e f a u n o u n d a e v o lv e re g is te ic a tio n lin e a r g e n d s tru c tu re r m a c h in e . is u s e d fo r e tic is a T h e fe e p ro g ra m lin e a r s c o n tro l d b a c k w m in g trin g o s y s te m h ile th s y s te m e v f in s tru c tio is e x e c u te e se rv o s a r o lv n s d o e c in o p n o n g a - O n -l i n e E v o l u t i o n o f C o n t r o l f o r a F o u r - L e g g e d R o b o t tro lle d th ro u g h s e ria l c o m m u e v o lv e d p ro g ra m s a re p la c e d w h ile th e o u tp u t v e c to r is w h p ro g ra m . E a c h in d iv id u a l th a t e ig h t o u tp u ts a re s e n t a s a n g le in p u t to th e n e x t ite ra tio n e n a b th e e ig h th ite ra tio n s (a n d b a s ic m o v e d fo rw a rd ). S o m e in d iv w h ic h d o n o t m o v e th e s e rv o s 3 2 1 n ic a tio n w ith a s e rv o c o n tro l c a rd . T h e in p u ts to th e in to th e re g is te rs o f th e re g is te r m a c h in e (p ro c e s s o r) a te v e r re m a in s in th e re g is te rs u p o n te rm in a tio n o f th e is e v a lu a te d is ite ra te d e ig h t tim e s . In e a c h ite ra tio n th e s to th e s e rv o s a n d th e p re v io u s o u tp u t v e c to r is s e n t a s lin g a s e q u e n c e to b e e v o lv e d . F itn e s s is m e a s u re d a fte r a lly c o n s is ts o f th e n u m b e r o f " tic k s " th a t th e m o u s e h a s id u a ls a re re m o v e d b e fo re e v a lu a tio n , s u c h a s th o s e , a t a ll. F ig u r e 2 : F e e d b a c k th ro u g h c o m p u te r m o u s e F ig u r e 3 : C a b le s to P C a n d p o w e r s u p p ly T h e p o p u la tio n c o n s is te d o f 1 0 0 in d iv id u a ls a n d th e g e n e tic o p e ra to rs w h e re c ro s s o v e r a n d m u ta tio n . C ro s s o v e r w a s p e rfo rm e d b o th a s tw o -p o in t s trin g c ro s s o v e r a n d a s h o m o lo g o u s lin e a r c r o s s o v e r [ 3 ,7 ] . T h e s y s te m u s e d a s m a ll to u r n a m e n t f o r s e le c tio n 3 2 2 B . A n d e rs s o n e t a l. a n d o p e ra te s u n d e r s te a d y s ta te , s e e F ig u re 4 . T h e fu n c tio n s e t c o n s is te d o f a rith m e tic o p e r a to r s a n d s q u a r e r o o t. A s in g le c o n s ta n t o f 0 .5 w a s u s e d in th e te r m in a l s e t. F ig u r e 4 : O n -lin e G P s y s te m O u r m e th o d fo r u s in g G P w ith a re a l-tim e a p p lic a tio n is b a s e d o n a p ro b a b ilis tic s a m p lin g o f th e e n v iro n m e n t [6 ]. D iffe re n t s o lu tio n c a n d id a te s (p ro g ra m s ) a re e v a lu a te d in d iffe re n t s itu a tio n s . T h is is u n fa ir b e c a u s e a g o o d in d iv id u a l d e a lin g w ith a h a rd s itu a tio n c a n b e re je c te d in fa v o r o f a b a d in d iv id u a l d e a lin g w ith a v e ry e a s y s itu a tio n . F o r in s ta n c e , a n in d iv id u a l th a t g e ts s tu c k n e a r th e w a ll o f th e b o x a n d d o e s a g o o d jo b o f m o v in g a w a y fro m th e w a ll b u t a d v a n c e s little in th e fo rw a rd d ire c tio n m ig h t g e t a lo w s c o re w h ile a p o o r in d iv id u a l in th e m id d le o f th e b o x m ig h t p e rfo rm b e tte r. O u r e x p e rie n c e is , h o w e v e r, th a t a g o o d o v e ra ll in d iv id u a l te n d s to s u rv iv e a n d re p ro d u c e in th e lo n g te rm . T h e s o m e w h a t p a ra d o x ic a l fa c t is th a t s p a rs e tra in in g d a ta s e ts o r p ro b a b ilis tic s a m p lin g in e v o lu tio n a ry a lg o rith m s o fte n b o th in c re a s e s p e e d to w a rd th e g o a l a n d k e e p th e d iv e rs ity h ig h e n o u g h to e s c a p e lo c a l o p tim a d u rin g s e a rc h . 3 E v o d o m m o u k in d b a re w h e R e s u lts lv in g w a in k n o se o n a o f c h a ly v is ib re th e r a lk in w le d ro d , o tic le . A o b o t g b e h a v io r, th ro u g h c o o rd in a tio n o f e ig h t s e rv o s a n d g e o r g u id a n c e a n d w h e re th e o n ly fe e d b a c k is fro m is n o t a triv ia l p ro b le m . T h e firs t e m e rg in g b e h a v io r is p a d d lin g , w h ic h s lo w ly m o v e s th e ro b o t fo rw a rd e v e n c o m m o n lo c a l o p tim u m a fte r th e p a d d lin g is a s tra s ta n d s u p rig h t o n a ll le g s c a re fu lly b a la n c in g a n d m o w ith o u t a n y a c o m p u te r u s u a lly s o m e th o u g h it is n g e s tra te g y , v in g q u ic k ly O n -l i n e E v o l u t i o n o f C o n t r o l f o r a F o u r - L e g g e d R o b o t 3 2 3 b a c k a n d fo rth w ith o u t re a lly a d v a n c in g th e ro b o t. It is u n c le a r h o w th is s tra te g y e m e rg e s b u t it c o u ld b e th e re s u lt o f s o m e la c k o f s y m m e try in th e h a rd w a re a n d m e a s u re m e n t s y s te m . T h e n e x t s tra te g y is o fte n s o m e k in d o f c ra w lin g s im ila r to th a t o f tu rtle . O th e r o b s e rv a b le b e h a v io rs a re a " c a m e l w a lk " w h e re th e le g s m o v e in p a ra lle l a n d p a ir w is e o n e a c h s id e . T h e m o s t e ffic ie n t s tra te g y fo r w a lk in g th a t h a s e v o lv e d is a ls o th e m o s t d iffic u lt to le a rn a n d g a llo p in g o n ly a p p e a rs a fte r m a n y h o u rs o f tra in in g . H e re b o th th e fro n t le g s a n d b a c k le g s a re p a ra lle l a n d th e ro b o t o fte n s ta n d s u p o n its b a c k le g s b e fo re p u s h in g fo rw a rd . It u s e s th e d y n a m ic in te ra c tio n w ith th e h e a v y m o u s e lo d e to a c h ie v e m a x im u m s p e e d fo rw a rd . E m e rg e n c e o f b e h a v io r d o e s n o t a lw a y s p a s s th ro u g h a ll o f th e s e b e h a v io rs b u t u s u a lly m o s t o f th e m a p p e a r in o rd e r. S u m m a r 1 2 3 4 y o . . . . f c o m c h a o c ra w " c a m g a llo m o n tic p a lin g e l w p in g e v o lu tio n o f b e h a v io r o v e r tim e c o u ld lo o k lik e : d d lin g b e h a v io r a lk " w ith d y n a m ic e le m e n ts in th e w a lk L ik e m a n y e v o lu tio n a ry c o n tro l s c h a n ic a l fa ilu re . F ig u re 5 s h o w s th h o u rs . D u rin g e v o lu tio n a tre e s e rv m a n a g e d to in c re a s e th e s p e e d a fte c u rre d in g e n e ra tio n 2 0 , 7 5 a n d 1 0 d e c re a s e s ra p id ly b u t th e s y s te m m a g e d ) h a rd w a re c o n fig u ra tio n . y s te m s e e v o lu o s b ro k r th e fa 0 . T h e a n a g e s w e o b s e rv e s o m e ro b u s tn e s s a g a in s t m e tio n o f s p e e d o v e r 1 1 1 g e n e ra tio n s a n d 1 5 e d o w n b u t th e c o n tro l s y s te m a d a p te d a n d ilu re s , s e e F ig u re 5 . S e rv o b re a k d o w n s o c fig u re c le a rly in d ic a te s h o w fitn e s s a t firs t to re le a rn a n d a d a p t to th e c h a n g e d (d a m - F ig u r e 5 : T h e v e lo c ity (in m /h ) o f th e d o g w h e n c o n tro lle d b y th e (c u rre n tly ) b e s t in d iv id u a l d u rin g 1 1 7 g e n e ra tio n s . S e rv o fa ilu re o c c u rre d in g e n e ra tio n 2 0 , 7 5 a n d 1 0 0 . 3 2 4 B . A n d e rs s o n e t a l. 4 F u tu r e W O u m a fo u o n r in te n n o id p r le g g th e E L tio n ro je e d w V IS is c t, a lk h u o r k to se in m u s e th e F ig u g a n d a n o id e re s u lts re 6 . H e r d e m a n d s c o n firm s o b ta in e d e th e a im m u c h m o th e fe a s ib w ith th is b i-p re e m p ility o f e fo e d a h a s th e u r-le g l w a lk is o n b a p p ro g e d in g a la a c h ro b o t fo r a la rg e r h u , w h ic h is h a rd e r th a n n c e . P re lim in a ry w o rk [8 ]. F ig u r e 6 : E L V IS h u m a n o id a b ip e d a l w a lk in g ro b o t M a k in g th e ro b o t fu lly a u to n o m o u s is a n o th e r a m b itio n fo r th e fu tu re . T h a t m e a n th a t th e e x p e rim e n ts w ill n o t b e h in d e re d b y a ll th e c a b le s a n d to ta l a u to w o u ld a ls o b e in te re s tin g fro m a m o re p h ilo s o p h ic a l s ta n d p o in t. F ig u re 7 s h c h a o tic ro b o t m a d e o f " g a rb a g e " w h ic h h a s le a rn e d to m o v e b y th e s a m e m e c h a s th e fo u r-le g g e d ro b o t b u t w h ic h is fu lly a u to n o m o u s . T h e o n -b o a rd G P s y s te m o n a s m a ll, e m b e d d e d P IC -c h ip a n d e v o lv e s b in a ry c o d e fo r th is tin y p ro c e s s o p la n is to m o v e th is s y s te m to th e w a lk in g e x p e rim e n ts . w o n o o w a n u ld m y s a is m ru n s r. O u r O n -l i n e E v o l u t i o n o f C o n t r o l f o r a F o u r - L e g g e d R o b o t F ig u r e 7 : A n a u to n o m o u s c h a o tic ro b o t m a d e fro m te m in a n e m b e d d e d c h ip . 5 3 2 5 " tra s h " w ith a n o n -b o a rd G P s y s - S u m m a r y a n d C o n c lu s io n s W e h a v e e v o lv e d th e firs t c o n tro lle r fo r a fo u r-le g g e d ro b o t, w h ic h le a rn o n -lin e u s in g a g e n e tic p ro g ra m m in g s y s te m a n d a re a l ro b o t. T h e e v o lu tio n is o f b e h a v io r p a s s e s th ro u g h s e v e ra l s ta g e s s ta rtin g w ith s im p le p a d d lin g b e h a v io r, c o n tin u in g th ro u g h c ra w lin g , " c a m e l w a lk " , a n d fin a lly g a llo p in g w ith d y n a m ic e le m e n ts in th e w a lk . A h ig h d e g re e o f ro b u s tn e s s is s e e n fo r m e c h a n ic a l fa ilu re - th e s y s te m is o b s e rv e d to a d a p t to c h a n g e s in th e m e c h a n ic a l c o n fig u ra tio n d u e to c o m p o n e n t fa ilu re . A c k n o w le d g e m e n t P e te r N o rd in g ra te fu lly a c k n o w le d g e s s u p p o rt fro m T F R a n d N U T E K . R e fe r e n c e s 1 . B a n z h a f , W ., N o r d in , P . K e lle r , R . E ., a n d F r a n c o n e , F . D .( 1 9 9 8 ) G e n e tic P r o g r a m m in g : A n In tr o d u c tio n o n th e A u to m a tic E v o lu tio n o f C o m p u te r P r o g r a m s a n d Its A p p lic a tio n s . M o rg a n K a u fm a n n , G e rm a n y . 2 . T . d im g ra c h a B r e n m m p te o u s io in r 1 g h n a g . 4 . to n l d In A c , P . e s ig n P e te r a d e m S . C o a te s , a n d H w o rld s u s in g L in B e n tle y , e d ito r, E v ic p re s s , L o n d o n , U . J a c k s o n . (1 9 9 9 ) E x p lo rin g th re e d e n m e y e r s y s te m s a n d g e n e tic p ro o lu tio n a ry D e s ig n U s in g C o m p u te rs , K ,. 3 2 6 B . A n d e rs s o n e t a l. 3 . F r a n c o n e F .D ., C o n r a d s M ., N o r d in J .P .a n d B a n z h a f W .( 1 9 9 9 ) H o m o lo g o u s C ro s s o v e r in G e n e tic P ro g ra m m in g , In P ro c e e d in g s o f: G e n e tic a n d E v o lu tio n a ry C o m p u ta tio n C o n fe re n c e (G E C C O 9 9 ) M o rg a n -K a u fm a n n 4 . M . A n th o n y g ra m m in g a p in g ro b o t. In b o tic s a n d A tro n ic a B k s . L e w is , A n d re p ro a c h to th e P ro c e e d in g s u to m a tio n , w H . F c o n s tr o f th e p a g e s a g g , a n d A la n S o lid u m . (1 u c tio n o f a n e u ra l n e tw o rk 1 9 9 2 IE E E In te rn a tio n a lC 2 6 1 8 -2 6 2 3 , N ic e , F ra n c e , 9 9 c o o n M 2 ) G e n e tic n tro l o f a w fe re n c e o n a y 1 9 9 2 . E p ro a lk R o le c - 5 . N o r d in , J .P . ( 1 9 9 7 ) E v o lu tio n a r y P r o g r a m In d u c tio n o f B in a ry M a c h in e C o d e a n d its A p p lic a tio n s . K re h l V e rla g , M u e n s te r, G e rm a n y 6 . N to In U 7 . N o r c h in T o O ’R 8 . N o r d in J . P ., N o r d a h l M . ( 1 9 9 9 ) : A n E v o lu tio n a r y A r c h ite c tu r e F o r A H u m a n o id R o b o t, In P ro c e e d in g o f: T h e F o u rth In te rn a tio n a l S y m p o s iu m o n A rtific ia l L ife a n d R o b o tic s (A R O B 4 th 9 9 ) O ita J a p a n 9 . G ra h a m F . S p e n c e r. A u to m a tic g e n e ra tio n o f p ro g ra m s fo r c ra w lin g a n d w a lk in g ( 1 9 9 4 ) . I n K e n n e th E . K in n e a r , J r ., e d ito r , A d v a n c e s in G e n e tic P r o g ra m m in g , c h a p te r 1 5 , p a g e s 3 3 5 -3 5 3 . M IT P re s s , 1 9 9 4 . o r d in , J .P ., B a n z h a f W .( 1 9 9 7 ) A n O n - lin e M e th o d to E v o lv e B e h a v io r a n d c o n tro l a M in ia tu re R o b o t in R e a l T im e w ith G e n e tic P ro g ra m m in g : T h e te rn a tio n a l J o u rn a l o f A d a p tiv e B e h a v io r, (5 ) p p 1 0 7 - 1 4 0 M IT P re s s , S A . d in J . e C o d a p p e a e illy , P ., e f r A n B a n z h a f o r C IS C in A d v a g e lin e , S W ., a n d A rc h ite c n c e s in p e c to r, M F ra n tu re s G e n IT -P c o n e u s in e tic re ss F . (1 9 9 9 ) E ffic ie n t E v o lu tio n o f M a g B lo c k s a n d H o m o lo g o u s C ro s s o v e r. P ro g ra m m in g III, (E d s ) L a n g d o n , , U S A O p tim iz e d C o llis io n F r e e R o b o t M o v e S ta te m e n t G e n e r a tio n b y th e E v o lu tio n a r y S o ftw a r e G L E A M C h ris tia n B lu m e F a c h h o c h s c h u le K ö ln A m S a n d b e rg 1 , D -5 1 6 T e l. + 4 9 /2 2 6 1 /8 1 9 6 -2 9 6 , -3 3 e m a i l : b l u m e A b e v o g e n tim ro b c o m ro b , A 4 3 0 o @ g str a c t. T h e G L E A M a lg o rith m a lu tio n a ry m e th o d a p p lic a tio n in th e fie e ra te s c o n tro l c o d e fo r re a l in d u s tria l e re la te d d e s c rip tio n o f th e ro b o t m o v e o t a rm c o n fig u ra tio n s ). T h is in te rn a m a n d s is m a p p e d to a re p re s e n ta tio n o t la n g u a g e , w h ic h c a n b e lo a d e d a t th e b te ilu n g G u m m e r r -3 3 2 , F m . f h - k G u m m e rsb a c h sb a c h , G e rm a n y a x : + 4 9 /2 2 6 1 /8 1 9 6 1 5 o e l n . d e n d its im p le m e n ta tio n a re a n e w ld o f ro b o tic s . T h e G L E A M s o ftw a re ro b o ts . T h e re fo re G L E A M a llo w s a m e n t (n o t o n ly a s ta tic d e s c rip tio n o f l re p re s e n ta tio n o f p rim itiv e m o v e o f m o v e s ta te m e n ts o f a n in d u s tria l ro b o t c o n tro l a n d e x e c u te d . I n tr o d u c tio n T h e re a re m a n y d iffe re n t id e a s a n d p ro c e d u re s a b o u t G e n e tic A lg o rith m s a n d E v o lu tio n a ry P ro g ra m m in g , b u t o n ly s o m e o f th e s e le a d to im p le m e n ta tio n s w h ic h s o lv e „ re a l w o rld p ro b le m s “ fo r in d u s tria l a p p lic a tio n s . G L E A M (G e n e tic L e a r n in g A lg o r ith m a n d M e th o d s ) is a n a lg o rith m b a s e d o n e v o lu tio n a ry s tra te g y , w h ic h w a s u s e d to im p le m e n t a s o ftw a re to o l fo r s o lv in g u s e fu l a n d n o t o n ly a c a d e m ic p ro b le m s , se e [1 ] a n d [2 ]. T h e a im o f th e G L E A M a p p lic a tio n to in d u s tria l ro b o ts w a s a n im p le m e n ta tio n w ith re s p e c t to p ra c tic a l re q u ire m e n ts a n d re s tric tio n s to p ro o f th e G L E A M m e th o d . T h e re fo re a firs t p ro to ty p e im p le m e n ta tio n w a s d o n e a t A B B in V ä s te ra s (S w e d e n ) a n d a s e c o n d o n e w ith im p ro v e d fa c ilitie s a t D a im le r C h ry s le r in B e rlin (G e rm a n y ). B o th im p le m e n ta tio n s in c lu d e th e o u tp u t o f ro b o t la n g u a g e c o d e fo r th e s p e c ific in d u s tria l ro b o t (A B B IR B 2 4 0 0 a n d K U K A K R 6 ). T h e E v o lu tio n a r y A lg o r ith m T h e a re d e sc o n e p rin c ip le s o f G L E A M in fo rm u la te d w ith c lo s e rip tio n in [3 ]). T h e g e n o s te p o f a p la n to b e e x e c c lu d in c o n n e ty p e o u te d . T G L E A M g its g e n e tic o c tio n to th e f G L E A M is h e p la n e x e c u p e ra to rs a b io lo g ic a c a lle d a n tio n is d o n d l „ a n e g e n o ty p e e v o lu tio n c tio n “ , w h b y s im u la S . C a g n o n i e t a l. (E d s .): E v o W o rk s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 2 7 -3 3 8 , 2 0 0 0 . © S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 re p re se (se e d ic h re p tio n a s n ta tio n e ta ile d re s e n ts p a rt o f 3 2 8 C . B lu m e th e e v o lu tio n a lg o rith m to g e t a v a lu e fo r th e „ fitn e s s “ o f th e G L E A M g e n e ra te s a s e q u e n c e o f b a s ic a c tio n s o r s ta te m e n ts , w h ic o f th e p la n , w h ic h is a m e m b e r o f th e e v o lu tio n p o p u la tio n . A p la n th e g e n e tic in fo rm a tio n , s e e [1 ]. T h e p u rp o s e o f th e p la n is n o t e v o lu tio n its e lf, th e re fo re th e k e rn e l o f G L E A M in c lu d in g th e e v o lu a p p lie d to d iffe re n t p ro b le m s w ith m in o r c h a n g e s . F o r e x a m p le , th th e b a s ic c o m m a n d s o f a s im p le ro b o t c o n tro lle r o r a llo c a tio n m a c h in e in a p ro d u c tio n p la n . p la n . In h a re th e re p re se n o f in te re tio n a lg o e s e a c tio s te p s to p a rtic u la r e le m e n ts ts d ire c tly s t fo r th e rith m w a s n s c a n b e re se rv e a T h e G L E A M m e th o d w a s im p le m e n te d fo r s e v e ra l a p p lic a tio n s : œ p la n n in g c o llis io n fre e m o v e s fo r in d u s tria l ro b o ts , s e e [4 ] a n d [1 1 ] œ g e n e ra tin g p ro d u c tio n p la n s s o lv in g th e jo b -s h o p -p ro b le m , s e e [5 ] œ p ro c e s s s c h e d u lin g in c h e m ic a l in d u s try , s e e [6 ] G L E A M is im p le m e n te d a s w e ll in P a s c a l a s in C o n d iffe re n t h a rd w a re p la tfo rm s , e .g . o n P C , w o r k s ta tio n , a n d a p a r a lle l c o m p u te r s y s te m . I t is p o r ta b le , b e c a u s e th e im p le m e n ta tio n is s tru c tu re d m o d u la r a n d c o n s is ts o f th e fo llo w in g m a in p a rts : B a s ic m a c h in e : E .g . in itia liz a tio n , d a ta s tr u c tu r e c o n s tr u c tio n a n d m a n a g e m e n t, e rro r h a n d lin g O p tim iz a tio n k e rn e l: E v o lu tio n fu n c tio n s fo r m u ta tio n , re c o m b in a tio n , p o p u la tio n m a n a g e m e n t S im u la tio n a n d e v a lu a tio n : E x e c u tio n o f p la n s , c rite rio n ’s c h e c k , re s tric tio n s , fitn e s s c a lc u la tio n O v e ra ll c o n tro l: M a n a g e m e n t o f p a ra m e te r in p u t, d is p la y o f re s u lts , in te rru p tin g s im u la tio n o r o p tim iz a tio n a n d o th e rs O v e r a ll c o n tr o l U s e r S im u la tio n & e v a lu a tio n I n te r f a c e O p tim iz a tio n k e r n e l B a s ic m a c h in e G L E A M im p le m e n ta tio n s tr u c tu r e Im p le m e n ta tio n e ffo r t to a p p ly G L E A M to a n o th e r a p p lic a tio n R o b o t s p e c ific m o d u le : G e n e o f r c o n s ta te r a tio n o b o t tr o l m e n ts O p tim iz e d C o llis io n F re e R o b o t M o v e S ta te m e n t G e n e ra tio n 3 2 9 U s e r in te rfa c e : C rite rio n ’s p rio rity a n d fitn e s s d e fin itio n , ta rg e t d e fin itio n , s im u la tio n v is u a liz a tio n , p la n d e s c rip tio n a n d o th e rs R o b o t c o d e g e n e ra tio n m o d u le , g e n e ra te s a ro b o t p ro g ra m p a rt in c lu d in g th e m o v e s ta te m e n ts S im u la tio n a n d e v a lu a tio n n e e d th e m a in a m o u n t o f im p le m e n ta tio n e ffo rt to a p p ly th e G L E A M im p le m e n ta tio n to o th e r a p p lic a tio n s . It m e a n s o n th e o th e r s id e , th a t d u e to th e g e n e ra lity o f th e m e th o d a n e w a p p lic a tio n re q u ire s le s s im p le m e n ta tio n c o s ts b e c a u s e th e b a s ic m a c h in e , o p tim iz a tio n k e rn e l a n d o v e ra ll c o n tro l a re n o t m u c h e ffe c te d . A p p lic a tio n o f G L E A M to R o b o tic s T h e firs t a p p lic a tio n o f G L E A M to ro b o tic s w a s p e rfo rm e d b e c a u s e o f tw o re a s o n s : 1 . T h e a u th o r is a n e x p e rt in in d u s tria l ro b o t p ro g ra m m in g a n d c o n tro l (C o n v e n o r o f a n IS O w o rk in g g ro u p ), th e re fo re a p ra x is o rie n te d w o rk w a s p e rfo rm e d , s e e [7 ] 2 . T h e c a lc u la tio n fo r c o n tro llin g a n in d u s tria l ro b o t to m o v e o n a p re d ic te d tra je c to ry w ith o u t c o llis io n is v e ry c o m p le x b u t th e re s u lt is e a s y to e v a lu a te : e v e ry b o d y c a n s e e , if th e m o v e m e n t is c o llis io n fre e a n d ta k e s a s h o rte r tim e . T h e a p p lic a tio n o f G L E A M w a s s o m e th in g lik e a b e n c h m a rk e s p e c ia lly to s h o w , th a t G L E A M is a p o w e rfu l p la n n in g a n d o p tim iz a tio n to o l fo r th e c o n tro l o f d y n a m ic p ro c e s s e s lik e th e m o v e m e n t o f in d u s tria l ro b o ts . T h e G L E A M m e t h o d a p p l i e d t o r o b o t c o l l i s i o n f r e e m o v e s d o e s n 't p e r f o r m a n e x p l i c i t s e a r c h o f t h e c o n f i g u r a t i o n s p a c e , b e c a u s e t h e s e a r c h p a r a m e t e r s d o n 't i n c l u d e c o n fig u ra tio n s , lik e in [8 ] o r [9 ]. W ith G L E A M th e m o v e s ta te m e n ts fo r p e rfo rm in g th e c o llis io n fre e m o v e a re g e n e ra te d d ire c tly . T h e G L E A M m e th o d a v o id s th e p ro b le m o f c a lc u la tin g a s m o o th p a th b e tw e e n th e c o n fig u ra tio n s a n d o f b u ild in g u p 3 3 0 C . B lu m e a n d s to rin g th e c o n fig u ra tio n s p a c e , s e e [1 0 ]. T h e c o llis io n fre e p a th is o p tim iz e d b y c r ite r io n ’ s s e le c te d b y th e u s e r , e .g . th e c r ite r io n c o u ld b e a s h o r t C a r te s ia n p a th . O th e r c rite rio n ’s lik e e n e rg y o r m o v e e x e c u tio n tim e a re in te g ra te d in to th e o p tim iz a tio n p ro c e s s . T h e c rite rio n ’s c o u ld b e c o n tra d ic tio n a ry . T h e re fo re e v e ry c rite rio n h a s a p rio rity a n d th e re s u lts o f th e o p tim iz a tio n re fle c t th e d iffe re n t p rio ritie s . T h e e v o lu tio n p e rfo rm e d b y G L E A M s ta rts b y g e n e ra tin g a c tio n re s . p rim itiv e s ta te m e n t s e q u e n c e s fo r c o n tro llin g th e ro b o t m o v e m e n t. T h e s e b a s ic a c tio n s fo r th e ro b o t m o v e s lo o k lik e fo llo w s : - m o v e r o b o t a x is k w ith a v e lo c ity o f r d e g r e e s p e r s e c o n d w ith a n a c c e le r a tio n o f b d e g r e e s p e r s e c o n d 2 (E v o lu tio n p a ra m e te rs a re k , r, a n d b ) - r o b o t a x is m w ith a s lo w d o w n o f n d e g r e e s p e r s e c o n d 2 (E v o lu tio n m e te rs a re m a n d n ) e s s o f a s e q u e n c e c a n b e c a lc u la te d a s a fu n c tio n o f th e fo llo w in g c rite rio n ’s : o s itio n p r e c is io n , i.e . th e d is ta n c e o f th e e n d p o in t o f th e m o v e m e n t a n d th e g iv e n ta rg e t p o in t o r ie n ta tio n p r e c is io n , i.e . th e d if f e r e n c e b e tw e e n th e p la n n e d a n d r e a c h e d o rie n ta tio n o f th e g rip p e r o r ie n ta tio n c h a n g e s , i.e . th e o r ie n ta tio n m o v e s o f th e g r ip p e r to r e a c h th e ta r g e t o rie n ta tio n q u a lity o f th e m o v e tr a je c to r y , i.e . th e d if f e r e n c e b e tw e e n th e m o v e p a th a n d a s tra ig h t lin e le n g th o f th e m o v e tra je c to ry f a s tn e s s o f m o v e e x e c u tio n , i.e . th e d u r a tio n o f m o v e m e n t p r o g r a m le n g th , i.e . th e n u m b e r o f a c tio n s r e s . s ta te m e n ts r e q u ir e d to p e r f o r m th e ro b o t m o v e I n te r m e d ia te p o in ts , i.e . th e r o b o t m o v e p a s s e s s p e c if ie d in te r m e d ia te p o in ts e c o n o m y , i.e . th e e n e r g y n e e d e d f o r th e m o v e e x e c u tio n s to p p a ra T h e fitn œ p œ œ œ œ œ œ œ œ B e fo re th e e v a lu a tio n , th e s ta te m e n t s e q u e n c e is p ro o fe d fo r its p la u s ib ility , fo r e x a m p le : a s to p -a x is -s ta te m e n t b e fo re a s ta rt-m o v e -a x is -s ta te m e n t is n o t m e a n in g fu l (a n d th e re fo re c a n c e le d ). A fte r th e p la u s ib ility -c h e c k th e e v o lu tio n c o n tin u o u s b y th e m u ta tio n o f th e e v o lu tio n p a ra m e te rs o f th e s ta te m e n ts a n d b y re c o m b in a tio n . A s th e s e q u e n c e o f e le m e n ta ry m o v e s is e s s e n tia l fo r th e re s u ltin g o v e ra ll m o v e m e n t th e re a re s o m e m u ta tio n s a lte rin g o n ly th e s e q u e n c e o f a c tio n s . T h e c o m p le te s e q u e n c e o f a c tio n s c o n tro ls th e ro b o t m o v e to a p o in t n e a r th e ta rg e t p o s itio n w ith o u t c o llis io n s . It is a s s u m e d , th a t th e ro b o t c o n tro l is b a s e d o n a fix e d c o n tr o l c y c le tim e , e .g . o f 5 0 m s . E v e r y c o n tr o l s ta te m e n t is e x e c u te d b y th e s im u la tio n to o l w ith re s p e c t to th is c y c le tim e , w h ic h c a n b e d e fin e d a s a c o n tro l c h a ra c te ris tic . T h e s im u la tio n p e rfo rm s th e ro b o t m o v e a n d s to re s th e a x is v a lu e s fo r e v e ry c o n tro l c y c le . T o m e a s u re th e d is ta n c e fro m th e ta rg e t p o s itio n a n d th e p a th le n g th , th e g e n e ra l fo rw a rd tra n s fo rm a tio n fro m ro b o t to C a rte s ia n c o o rd in a te s is a p p lie d . T h e G L E A M m e th o d is a b le to d e v e lo p a s ta te m e n t s e q u e n c e fo r a n y k in d o f ro b o t. A s o ftw a re to o l R O B M O D E F (R o b o t M o d e l D e fin itio n ) w a s im p le m e n te d O p tim iz e d C o llis io n F re e R o b o t M o v e S ta te m e n t G e n e ra tio n p ro v id in g a g ra p h ic s u p p 1 6 ro ta tio n a l ro b o t a x e s , z e ro p o s itio n ), a n d w h ic h T h e u s e r c a n d e fin e a v o id a n c e . T h e d e fin e d a c tio n d e fin itio n is c a lle d u s e r a n d s to re d fo r fu rth e o rte d ro b o t d e fin itio n . T h e w h ic h c a n b e d ire c te d in to c a n ro ta te a b o u t th e x -, y o b s ta c le s in th e ro b o t m ro b o t m o d e l, th e o b s ta c le th e " a c tio n m o d e l" . It c a n r e x p e rim e n ts . u s e r c a n b u ild th e x - o r z -c o o r o r z -c o o rd in a te o v e m e n t a re a d e fin itio n s a n e a s ily c h a n g e d a ro b o t w d in a te d ir a x is . to s h o w d th e (p r o r e x te n d 3 3 1 ith u p to e c tio n (in c o llis io n e d e fin e d ) e d b y th e G e n o ty p e R e p r e s e n ta tio n a n d D a ta S tr u c tu r e T h e a c tio n o f G L E A M is o n e g e n e a n d c o n s is ts o f th e a c tio n c o d e a n d a n u m b e r o f p a ra m e te rs . T h e p a ra m e te rs c a n b e o f in te g e r, re a l o r c h a ra c te r ty p e . T h e a c tio n c o d e a n d p a ra m e te r d e fin itio n is s to re d in a n (re a d a b le ) file a n d c a n b e c h a n g e v e ry e a s ily . T h e n u m b e r o f a c tio n s , w h ic h b u ild u p a m e m b e r o f th e e v o lu tio n p o p u la tio n , is n o t lim ite d . T h e le n g th o f s u c h a n a c tio n c h a in is fle x ib le a n d a n o p tim iz a tio n c rite ria o f th e e v o lu tio n . G L E A M w a s d e s ig n e d fo r th e o p tim iz a tio n o f d y n a m ic p ro c e s s e s w ith re s p e c t to th e n a tu ra l e v o lu tio n . T h e re fo re th e e v o lu tio n p ro c e s s o p e ra te s a b o u t g e n e s re s . p la n s o f v a ria b le le n g th . D u e to th is a ttrib u te th e ta s k o f re c o m b in a tio n is m o re c o m p lic a te d th a n in e v o lu tio n a lg o rith m s o f o th e r a u th o rs . T h e m o s t im p o rta n t im p ro v e m e n t o f G L E A M is th e in tro d u c tio n o f th e c o n c e p t o f s o c a lle d „ s e c tio n s “ . A s e c tio n is fo rm e d b y a ( v a r ia b le ) n u m b e r o f g e n e s r e s . a c tio n s a s a s u b s tr u c tu r e o f th e p la n , i.e . th e a c tio n c h a in is p a rtitio n e d in to s e g m e n ts . A s e g m e n t c a n b e re g a rd e d a s a c h ro m o s o m e o f th e b io lo g ic a l g e n e tic in fo rm a tio n . S o , o n e m e m b e r, th a t m e a n s th e a c tio n c h a in o f th e p la n , c o n s is ts o f s e g m e n ts , a n d e a c h s e g m e n t o f a n u m b e r o f a c tio n s . T h e re a re n e w d e fin e d e v o lu tio n o p e ra to rs (fo r m u ta tio n a n d re c o m b in a tio n ) to b e a p p lie d to th e s e c tio n s , lik e d e le te a s e g m e n t o r m o v e a s e g m e n t to a n o th e r p la c e in th e a c tio n c h a in . T h is c o n c e p t e n a b le s th e re c o m b in a tio n o f „ g o o d “ s u b -s tru c tu re s o f a 3 3 2 C . B lu m e p ro b le m s o lv in g p ro c e s s to s p e e d u p th e e v o lu tio n . If tw o p la n s a re m e rg e d b y th e re c o m b in a tio n , th e c o m b in a tio n o f a s e g m e n t o f p la n A w ith a s e g m e n t o f p la n B c a n b e tre a te d a s a c o m b in a tio n o f tw o s u b -s o lu tio n s . If th e tw o s u b -s tru c tu re s re s . s e g m e n ts a re g o o d in s in g le , th e y b o th to g e th e r in th e re s u ltin g p la n w ill g iv e a m u c h h ig h e r fitn e s s v a lu e th a n th e fitn e s s o f th e b o th p a re n t p la n s . T o c o m e to s u c h a g o o d s o lu tio n o n ly b y m u ta tio n o f th e g e n e s ta k e s p ro b a b ly m u c h lo n g e r tim e . T h e c o n c e p t o f s e g m e n ts (th e „ c h ro m o s o m e s “ ) a n d a c tio n s (th e „ g e n e s “ ) w a s v e ry s u c c e s s fu l fo r p la n n in g a n d o p tim iz a tio n ta s k s o f d y n a m ic p ro c e s s e s lik e ro b o t m o v e s . S im ila r to th e „ tra d itio n a l“ a rtific ia l in te llig e n c e la n g u a g e L IS P , th e d a ta s tru c tu re o f G L E A M is b a s e d o n d y n a m ic lis ts . T h e a c tio n c h a in s ta rts w ith a h e a d e r n o d e s to r in g a ll n e c e s s a r y in f o r m a tio n a b o u t th e a c tio n c h a in , e .g . c h a in le n g th ( e .i. n u m b e r o f a c tio n s a n d s e g m e n ts ), fitn e s s v a lu e , a n d o th e rs . T h e h e a d e r is fo llo w e d b y a c tio n s , a ll lin k e d b y p o in te rs . A n a c tio n c h a in c a n g ro w o r b e re d u c e d , it d e p e n d s o n th e e v o lu tio n . O f c o u rs e , if a n a c tio n c h a in (th a t m e a n s a p ro b le m s o lu tio n ) g e ts a g o o d fitn e s s v a lu e , a n d a n o th e r c h a in g e ts th e s a m e fitn e s s ra n k b u t c o n s is ts o f a lo w e r n u m b e r o f a c tio n s , th e la s t o n e is b e tte r a n d w ill s u rv iv e . R e p r e s e n ta tio n o f a c tio n : A c t io n c o d e A c t io n p a r a m e t e r s A c tio n c h a in : 1 .A c tio n M o v e A x is 1 H e a d e r o f th e a c tio n c h a in R e fe re n c e to th e n e x t a c tio n 2 .A c tio n M o v e A x is 3 ... E x a m p le s fo r th e m u t a tio n o p e r a to r s fo r a c tio n s : D e le t e : A c tio n K A c tio n L A c tio n M In se rt: A c tio n K A c tio n L A c tio n M A c tio n X C h a n g e : A c tio n M A c tio n L A c tio n K O p tim iz e d C o llis io n F re e R o b o t M o v e S ta te m e n t G e n e ra tio n 3 3 3 G e n o ty p e O r ie n te d C o d e a n d R o b o t C o d e G e n e r a tio n T h e g e n o ty p e o rie n te d c o d e o f G L E A M fo r ro b o ts c o n s is ts o f p rim itiv e m o v e m a n d s . T h e c o d e is s im u la te d d u rin g th e e v o lu tio n (w ith o u t a n y o u tp u t, o n ly th e e s s v a lu e is c a lc u la te d d u rin g th e s im u la tio n ), a n d if th e u s e r w a n ts to s e e th e re s u lt th e p la n n in g a n d o p tim iz a tio n p ro c e s s , th e c o d e o f s u c c e s s fu l a c tio n p la n s is u la te d w ith g ra p h ic a l d is p la y o f th e ro b o t m o v e m e n ts . A s m e n tio n e d a b o v e , th e ro b o t c o n tro l s im u la tio n c a lc u la te s a m o v e s te p b y s te p , s e e [1 1 ]. B e tw e e n e v e ry c a lc u la te d v ia p o s itio n o f th e tra je c to ry a u s e r d e fin e d c y c le tim e h a s p a s s e d . T h e m o v e d e s c rip tio n b y p rim itiv e c o m m a n d s (u s e d in te rn a lly b y G L E A M ) h a s to b e tra n s fe rre d to c o d e fo rm u la te d in a ro b o t p ro g ra m m in g la n g u a g e , b u t p ro g ra m m in g th e s a m e ro b o t m o v e . A n e w m o d u le „ ro b o t c o d e g e n e ra to r“ w a s im p le m e n te d , w h ic h c a lc u la te s th e p a ra m e te r fo r a m o v e p ro g ra m m in g b y ro b o t m o v e s ta te m e n ts . A fte r e v e ry c o n tro l c y c le th e v a lu e s fo r th e ro b o t a x e s a n d th e C a rte s ia n c o o rd in a te s fo r th e to o l c e n te r p o in t T C P a re c a lc u la te d a n d tra n s fe rre d to th e ro b o t c o d e g e n e ra to r (if th e u s e r h a s m a rk e d „ c o d e g e n e ra tio n “ w ith th e h e lp o f m e n u in p u t). T h e firs t s te p o f th e c o d e g e n e ra tio n is th e d e c la ra tio n o f p o s itio n d a ta a n d a n a s s ig n m e n t o f v a lu e s . T h is p o s itio n d a ta c o n s is ts o f th e in te rm e d ia te p o in ts o f th e m o v e a n d th e ta rg e t p o in t. T h e ta rg e t p o in t is g iv e n b y th e u s e r in C a rte s ia n c o o rd in a te s , th e c o d e g e n e ra to r c a lc u la te s th e v a lu e s fo r th e p a ra m e te rs to d e s c rib e th e ro b o t s p e c ific o rie n ta tio n a n d c o n fig u ra tio n d a ta . A fte r th e d a ta d e fin itio n , a s im p le m a in fu n c tio n is g e n e ra te d c o n s is tin g o f m o v e c o m m a n d s a n d re fe rrin g to th e p o s itio n d a ta g e n e ra te d b e fo re . T h e ro b o t c o d e c a n b e d o w n lo a d e d to th e ro b o t c o n tro l a n d e x e c u te d . c o m fitn o f s im E x a m p le to p r o g r a m th e A B B IR B 2 4 0 0 r o b o t: ! ! ! C O M M E N T C O M M E N T C O M M E N T M O D U L E V A R C O N C O N C O N ! . . . . . C O N P o s i t i o n l i s t w i t h t h e m o v e ( r e p r e s e n t i n g p o s i t i o n f o r e v e r y t h e t r a j e c t o r y ) : d e m o 1 r o b t a r g e t t a r g e t : = [ [ [ 0 . 1 2 3 6 7 , 0 . 6 9 6 2 1 , 0 . 1 2 3 [ 9 E S T j o i n t t a r g e t P 1 : = [ [ 0 . 0 , - 4 5 . 3 , S T j o i n t t a r g e t P 2 : = [ [ 0 . 0 , - 4 5 . 3 , S T j o i n t t a r g e t P 3 : = [ [ 0 . 0 , - 4 4 . 6 , . . . S T j o i n t t a r g e t P 7 4 : = [ 1 6 . 9 , 4 . 1 , - 1 - 1 6 4 6 7 , 9 , 9 E 0 0 . 0 ] 0 0 . 0 ] 0 0 . 0 ] 7 . 0 . 9 , . 5 , . 7 , . 7 , 0 , 6 9 9 E , [ 9 , [ 9 , [ 9 - 6 6 2 9 , 4 E 9 4 E 9 4 E 9 0 4 1 ] 9 E 5 . , 5 . , 5 . , . 2 , [ 9 , 3 , . . 3 , . . 3 , . . , 1 3 7 - 2 , 0 9 E 9 , 4 5 . . , 4 5 . . , 4 5 . . , [ - 1 6 0 . 7 , 4 5 . 2 , 1 7 . 0 ] , [ 9 E 9 , . . . . C O M M E N T M o v e s t a t e m e n t s f o r t h e P R O C m a i n ( ) M o v e A b s J P 1 , v 4 0 0 , z 2 0 , t o o l x ; R A P I D c y c l e 4 . , 0 9 E . 3 9 E . 3 9 E . 3 9 E 7 ] , 0 9 ] , 9 ] , 9 ] , 9 ] , ] , ] ; ] ; ] ; ] ; 0 . 6 , , 9 E 9 ] ] ; p r o g r a m : 3 3 4 C . B lu m e M o M o . . . . M o M o E N D P R E N D M O v e A b s J v e A b s J P 2 , v 4 0 0 , z 2 0 , t o o l x ; P 3 , v 4 0 0 , z 2 0 , t o o l x ; v e A b s J P 7 4 , v 4 0 0 , z 2 0 , t o o l x ; v e L t a r g e t , v 1 0 0 , z 2 0 , t o o l x ; O C D U L E E x e c u tin g th e s e ro b o t s ta te m e n ts , th e A B B c o n tro l p e rfo rm s o n ly o n e ro b o t m o v e to th e ta rg e t p o in t u s in g th e m a n y g e n e ra te d p o in ts to p e rfo rm a „ s m o o th p a th “ . T h e re fo re th e re s u ltin g m o v e m e a h u m a n a rm . T h e re a s o n is , th o n ly a fe w (ty p ic a lly 3 to 1 0 ) th e m , a n d th e ro b o t m o v e lo o k s o f th e n e w m e th o d is th e g e n e ra d ire c tio n . T h is le a d s to a c o n tr m o to rs a n d g e a rs . A ls o th e g e n e s to p s . n t lo o k s m u c h m o re „ n a tu ra a t th e p ro g ra m m e r u s u a lly in te rm e d ia te p o in ts w ith a th e n m o re „ m a c h in e -lik e “ o tio n o f m o v e s , w h ic h a v o id o l b e h a v io r m o re a v o id in g ra te d m o v e w a s c h e c k e d to l“ , i.e . lik e a m o v e m e n t o f „ d iv id e s “ a ro b o t m o v e in g re a te r d is ta n c e b e tw e e n r „ c o rn e re d “ . A s id e e ffe c t su d d e n c h a n g e o f sp e e d o r w e a r a n d te a r o f th e a x is a v o id ru n n in g a g a in s t a x is O p tim iz e d C o llis io n F re e R o b o t M o v e S ta te m e n t G e n e ra tio n 3 3 5 R e s u lts o f th e I m p le m e n ta tio n s fo r I n d u s tr ia l R o b o ts T h e im p le m e n ta tio n s o f G L E A M to in d u s tria l ro b o ts h a v e b e e n d o n e to d e m o n s tra te , th a t G L E A M is a b le to p ro d u c e ro b o t c o d e in a n in d u s tria l e n v iro n m e n t. T h e s o ftw a re fo r th e A B B ro b o t p ro g ra m m in g fa c ility g e n e ra te s ro b o t p ro g ra m s w ith c o llis io n fre e ro b o t m o v e s . T h e c o llis io n fre e p a th is o p tim iz e d b y c rite rio n ’s o f d iffe re n t p rio ritie s s e le c te d b y th e u s e r . E .g . th e c r ite r io n c o u ld b e a s h o r t C a r te s ia n p a th o f th e T C P o r a s h o rt d is ta n c e o f o n e o re m o re ro b o t a x e s . T h e p ro c e s s o f p ro g ra m m in g o p tim iz e d ro b o t m o v e s c a n b e d o n e b y th e s o ftw a re to o l, w h ic h re d u c e s th e n e e d e d m a n p o w e r fo r th e u s e r. A n o th e r re s u lt is th e d e m o n s tra tio n , th a t th e m e th o d G L E A M is a p p lic a b le to a n in d u s tria l ro b o t c o n tro l o f th e u s e r. T h e a d v a n ta g e s a re c o n tro llin g ro b o ts w ith o u t c o m p lic a te d m a th e m a tic s c a lc u la tio n s a n d e a s y a d a p ta tio n o f d iffe re n t e n v iro n m e n ts a n d ro b o t m o d e ls . T h e re s u lt c a n b e u s e d fo r o ff-lin e p a th p la n n in g in c lu d in g a m u ltic rite ria o p tim iz a tio n . T h e g e n e ra te d m o v e tra je c to ry is a n o p tim iz a tio n o f a ll c rite rio n ’s , a v o id in g p ro b le m s o f d e fin in g th e c o n fig u ra tio n s p a c e o r tra n s fe r p o s itio n s . T h e im p le m e n ta tio n f o r D a im le r C h r y s le r in c lu d e s s e v e r a l im p r o v e m e n ts , e .g . th e d e fin itio n o f in te rm e d ia te p o in ts b y th e u s e r. T h e s e in te rm e d ia te p o in ts h a v e to b e p a s s e d b y th e ro b o t tra je c to ry (m o re o r le s s ), th e u s e r c a n in flu e n c e th e ro b o t p a th a n d h e lp th e s y s te m to fin d a c o llis io n fre e p a th in a s h o rte r tim e . T h e im p le m e n ta tio n s h a v e b e e n p e rfo rm e d o n a P C , th e y a llo w a lo w c o s t s o lu tio n a n d w ill b e a c c e p te d a ls o fo r s m a lle r c o m p a n ie s . T h e s o ftw a re fo r th e G L E A M m o d u le s a re w ritte n in C . T h e c o m p a n ie s A B B a n d D a im le rC h ry s le r c a n te s t th e n e w m e th o d a n d d e c id e , if it w ill b e in te g ra te d in to th e ir p ro g ra m m in g s o ftw a re to o ls , th e im p le m e n ta tio n s a re th e b a s e o f fe a s ib ility s tu d ie s to a n a ly z e th e n e w m e th o d . 3 3 6 C . B lu m e O u tlo o k T h e G L E A M c o n c e p t w a s re a liz e d b y im p le m e n ta tio n s fo r d iffe re n t ro b o ts : th e M its u b is h i R V -M 2 ro b o t, th e A B B IR B 2 4 0 0 in d u s tria l ro b o t , a n d th e K U K A K R 6 in d u s tria l ro b o t. T h e y g e n e ra te s ta te m e n ts fo r th e ro b o t c o n tro ls to m o v e th e ro b o t o n a n o p tim iz e d c o llis io n fre e tra je c to ry to a g iv e n m o v e ta rg e t. In fu tu re a n im p le m e n ta tio n fo r a p ro fe s s io n a l s im u la tio n to o l fo r ro b o ts w ill b e o f in te r e s t, e .g . f o r th e I G R I P s y s te m o f th e D e n e b c o m p a n y . S u c h a n im p le m e n ta tio n w ill d e m o n s tra te th e a p p lic a tio n o f th e G L E A M m e th o d to a h ig h le v e l la n g u a g e s im u la tio n c o m m a n d la n g u a g e lik e G S L , a n d a c o m p le x m o d e le d e n v iro n m e n t o f o b s ta c le s a n d ro b o ts . T h e fo llo w in g p ic tu re s h o w s th e in fo rm a tio n flo w b e tw e e n th e s y s te m c o m p o n e n ts . T h e u s e r w o rk s w ith th e s im u la tio n to o l a s b e fo re a n d m o d e ls its m a c h in e s , w o rk c e ll, o b s ta c le s , a n d o th e rs . H e c a n p ro g ra m th e ro b o t m o v e s in a s im u la tio n la n g u a g e a n d s im u la te th e m o v e s . U s e r In p u t to th e s im u la tio n to o l C o m m a n d s in c l. c o llis io n fre e m o v e fu n c tio n S im u la tio n w ith g ra p h . re p re s e n ta t. S im u la tio n to o l w ith m o d e llin g , ro b o t m o v e s, s im u la tio n la n g u a g e s ta te m e n t g e n e ra tio n S im u l.s ta fitn e s s v c o llis io n tio n to G tm e n ts , a lu e s , d e te c L E A M S im u la tio n s ta tm e n ts to th e s im u l. to o l G L E A M o p tim iz a tio n a n d g e n e r a tio n o f c o llis io n fr e e ro b o t m o v e s S im u la tio n la n g u a g e s ta tm e n ts P o s tp ro c e s s o r to g e n e ra te s ta te m e n ts fo r th e ro b o t c o n tro l M o v e s ta tm e n ts fo r d iffe re n t ro b o t c o n tro ls R o b o t C o n tr o l U n it A n e w s im u la tio n p ro g ra m m e ro b o t m o v e fu n c tio n „ o p s y s te m . T h e d b y th e u s e r s (s h o rte r e x e tim iz e m s im u la tio to th e G c u tio n tim o v e s n to L E A e ) w w ith o o l se n M c o m ith re s u t c o llis io n “ d s th e s im u p o n e n t. G L E p e c t to c o llis w ill b e in tro d u c la tio n s ta te m e n ts A M trie s to o p tim io n a v o id a n c e . D u e d a lre iz e rin g in to a d y th e th e O p tim iz e d C o llis io n F re e R o b o t M o v e S ta te m e n t G e n e ra tio n 3 3 7 o p tim iz a tio n p ro c e s s G L E A M s e n d s c h a n g e d s ta te m e n ts in s im u la tio n la n g u a g e c o d e to th e s im u la tio n s y s te m , w h ic h w ill e x e c u te th e s ta te m e n ts in a „ s ile n t“ b a c k g ro u n d m o d e w ith o u t g ra p h ic a l o u tp u t. T h e s im u la tio n to o l s e n d s b a c k to G L E A M o n ly th e in fo rm a tio n a b o u t th e „ fitn e s s “ o f th e s im u la te d m o v e s ta te m e n ts , e s p e c ia lly if th e re w a s a c o llis io n o r n o t. A fte r th e o p tim iz a tio n p ro c e s s G L E A M s e n d s th e re s u lt to th e s im u la tio n to o l: th e o p tim iz e d c o llis io n fre e m o v e s ta te m e n ts in th e s im u la tio n la n g u a g e . T h e u s e r c a n s im u la te th is re s u lt a n d te s t, if it is b e tte r th a n h is o w n s ta te m e n ts . A t th e e n d , th e s im u la tio n to o l g e n e ra te s p ro g ra m c o d e fo r d iffe re n t ro b o t c o n tro l u n its . It is im p o rta n t, th a t G L E A M c a n u s e s o lu tio n s a lre a d y p ro g ra m m e d b y th e u s e r a s a n in p u t. G L E A M trie s to fin d a b e tte r s o lu tio n (if th e re e x is ts o n e ), a n d it g e n e ra te s a tra je c to ry w ith o u t c o llis io n s . T h e u s e r c a n v is u a liz e th e re s u lt b y s im u la te it, a n d h e c a n u s e th e g e n e ra te d s ta te m e n ts fo r o th e r p ro g ra m s o r to p ro g ra m a n y ro b o t. R e fe r e n c e s 1 . B lu m e , C ., J a k o b , W .: C lo s in g th e O p tim iz a tio n G a p in P r o d u c tio n b y G e n e tic A lg o rith m s . P ro c . o f th e E u r o p e a n C o n g r e s s o n In te llig e n t T e c h n iq u e s a n d S o ft C o m p u tin g (E U F IT 9 3 ), 1 9 9 3 , A a c h e n 2 . B lu m e , C .: I n d u s tr ie lle A n w e n d u n g e n E v o lu tio n ä r e r A lg o r ith m e n ( I n d u s tr ia l A p p lic a tio n s o f e v o lu tio n a ry a lg o rith m s ). C o n tr ib u tio n : P la n u n g k o llis io n s fre ie r B e w e g u n g e n fü r In d u s trie ro b o te r (P la n n in g o f c o llis io n fre e m o v e m e n ts o f in d u s tr ia l r o b o ts ) . E d .: S . H a f n e r , R . O ld e n b o u r g V e r la g , M ü n c h e n W ie n 1 9 9 8 3 . B lu m e , C .: G L E A M - A S y s te m f o r S im u la te d „ I n tu itiv e L e a r n in g “ . P r o c e e d in g s o f th e 1 s t In te r n a tio n a l W o r k s h o p o n P r o b le m S o lv in g fr o m N a tu r e , D o rtm u n d , G e rm a n y , O c to b e r 1 -3 , 1 9 9 0 4 . B lu m e , C ., J a k o b , W ., K r is c h , S .: R o b o t T r a je c to r y P la n n in g w ith C o llis io n A v o id a n c e u s in g G e n e tic A lg o rith m s a n d S im u la tio n . P r o c . o f th e 2 5 th I n te r n a tio n a l S y m p o s iu m o n I n d u s tr ia l R o b o ts , 2 5 .- 2 7 . A p r il 1 9 9 4 , H a n n o v e r , p p . 1 6 9 -1 7 5 5 . P la n n in g a n d O p tim iz a tio n o f S c h e d u lin g in In d u s tria l P ro d u c tio n b y G e n e tic A lg o rith m s a n d E v o lu tio n a ry S tra te g y . P r o c . o f th e S e c o n d B ie n n ia l E u r o p e a n J o in t C o n fe r e n c e o n E n g in e e r in g S y s te m s D e s ig n a n d A n a ly s is (E S D A ), J u ly 4 -7 , 1 9 9 4 , L o n d o n , E n g la n d 6 . B lu m e , C ., G e r b e , M .: D e u tlic h e S e n k u n g d e r P r o d u k tio n s k o s te n d u r c h O p tim ie ru n g d e s R e s s o u rc e n e in s a tz e s . (R e d u c tio n o f p ro d u c tio n c o s ts b y o p tim iz in g th e re s o u rc e p la n n in g ). a tp - A u to m a tis ie r u n g s te c h n is c h e P r a x is , 3 6 (1 9 9 4 ) p p . 5 9 7 . B lu m e , C ., F r ü a u f , P .: S ta n d a r d iz a tio n o f P r o g r a m m in g M e th o d s a n d la n g u a g e s f o r M a n ip u la tin g In d u s tria l R o b o ts . 2 7 th In te r n a tio n a l S y m p o s iu m o n In d u s tr ia l R o b o ts , O k to b e r 1 9 9 6 , M a ila n d 8 . D a i, F .: C o llis io n - F r e e M o tio n o f a n A r tic u la te d K in e m a tic C h a in in a D y n a m ic E n v iro n m e n t, IE E E C o m p u te r G r a p h ic s & A p p lic a tio n , J a n u a ry 1 9 8 9 , p p . 7 0 -7 4 3 3 8 C . B lu m e 9 . G o ld b e r g , D ., P a r k e r , J ., K h o o g a r , A .: I n v e r s e K u s in g G e n e tic A lg o rith m s . IE E E In te r n a tio n a l A u to m a tio n 7 , (1 9 8 9 ). p p . 2 7 1 -2 7 6 1 0 . H e in e , R ., S c h n a r e , T .: K o llis io n s f r e ie B a h n p la n p a th p la n n in g fo r ro b o ts ). R o b o te r s y s te m e 7 , (1 9 9 1 1 1 . G e n e ra tio n o f O p tim iz e d C o llis io n F re e R o b o t M o A lg o rith m s . P r o c e e d in g s o f th e W o r ld A u to m a tio n R o b o tic a n d M a n u fa c tu r in g S y s te m s , M a y 2 8 -3 0 , 8 9 - 9 4 in e m a tic s o f R e d u n d a n t R o b o ts C o n fe r e n c e o n R o b o tic s a n d u n g ). p p v e S C o n 1 9 9 fü r R o b o te r (C o . 1 7 -2 2 ta te m e n ts B a s e d g re ss (W A C ‘9 6 ) 6 , M o n tp e llie r , F llis io n fre e o n G e n e tic V o lu m e 3 : ra n c e . p p . S e lf-A d a p tiv e M u ta tio n L a rry In te llig F a c u lty o f U n iv e B { L a r r y .B e n t C C o m rs ity ris to u ll,J B u ll & o m p u o f l B a c o p u te te r S th e S 1 6 b 3 .H in Z C S Ja c o b H u rst r S y s te m tu d ie s & W e st o f 1 Q Y , U u rs t} @ u s C e n M a th E n g la .K . w e .a c .u C o n tr o lle r s tre e m a tic s n d k A b s tr a c t. T h e u s e a n d b e n e fits o f s e lf-a d a p tiv e m u ta tio n o p e ra to rs a re w e ll-k n o w n w ith in e v o lu tio n a ry c o m p u tin g . In th is p a p e r w e e x a m in e th e u s e o f s e lf-a d a p tiv e m u ta tio n in M ic h ig a n -s ty le C la s s ifie r S y s te m s w ith th e a im o f im p ro v in g th e ir p e rfo rm a n c e a s c o n tro lle rs fo r a u to n o m o u s m o b ile ro b o ts . In itia lly , w e im p le m e n t th e o p e ra to r in th e Z C S c la s s ifie r a n d e x a m in e its p e rfo rm a n c e in tw o a n im a t e n v iro n m e n ts . It is s h o w n th a t, a lth o u g h n o s ig n ific a n t in c re a s e in p e rfo rm a n c e is s e e n o v e r re s u lts p re s e n te d in th e lite ra tu re u s in g a fix e d ra te o f m u ta tio n , th e o p e ra to r a d a p ts t o a p p r o x i m a t e l y t h i s r a t er e g a r d l e s s o f t h e i n i t i a l r a n g e . 1 I n tr o d u c tio n W ith in G e n e tic A lg o rith m s (G A s ) [H o lla n d 1 9 7 5 ] a n d G 1 9 9 1 ] th e m u ta tio n ra te is tra d itio n a lly a g lo b a l p a ra m e te r H o w e v e r, in E v o lu tio n a ry S tra te g ie s [R e c h e n b e rg 1 9 E v o lu tio n a ry P ro g ra m m in g (M e ta -E P ) [F o g e l 1 9 9 2 ], th e e v o lv in g e n tity in its e lf , i.e . it a d a p ts d u r in g th e s e a r c h fo rm o f m u ta tio n n o t o n ly re d u c e s th e n u m b e r o f h a n e v o lu tio n a ry a lg o rith m , it h a s a ls o b e e n s h o w n to im p [B c k 1 9 9 2 ] fo r re s u lts w ith a s e lf-a d a p tiv e G A ). In th is a s e lf-a d a p tiv e m u ta tio n o p e ra to r w ith in M ic h ig a n -s ty le [H o lla n d e t a l. 1 9 8 6 ], m o re s p e c ific a lly in W ils o n ’s Z C S T h e p e rfo rm a n c e o f th e n e w o p e ra to r w ith in Z C S a u to n o m o u s e n tity /ro b o t - a n im a t [W ils o n 1 9 8 5 ] e a c h o f w h ic h w e re o rig in a lly u s e d b y W ils o n to b o th c a s e s it is fo u n d th a t n o b e n e fits in p e rfo p re s e n te d b y W ils o n u s in g a fix e d m u ta tio n ra te . ra n g e fo r th e a d a p tin g m u ta tio n ra te s , th e fin a l C S s a m e m u ta tio n ra te a s th a t u s e d b y W ils o n . T h a t a d a p ta tio n w o rk s w ith in th e C la s s ifie r S y s te m fra m e n e tic P ro g ra m m in g [K o z a w h ic h is c o n s ta n t o v e r tim e . 7 3 ] a n d la te r fo rm s o f m u ta tio n ra te is a lo c a lly p ro c e s s . T h is s e lf-a d a p tiv e d -tu n a b le p a ra m e te rs o f th e r o v e p e r f o r m a n c e ( e .g . s e e p a p e r w e e x a m in e th e u s e o f C la s s ifie r S y s te m s (C S s ) [W ils o n 1 9 9 4 ] s y s te m . is e x a m in - ta s k s : W in tro d u c e rm a n c e a re H o w e v e r, a n im a t c o n is , w e s h e w o rk . e d u s in o o d s 1 a n d in v fo u n d re g a rd le tro lle rs o w th e g tw a n d e s tig o v e ss o h a v e p rin o a r f c s im W o o te Z th e th e ro u g ip le u la te d d s 7 , C S . In re s u lts in itia l h ly th e o f s e lf- T h e p a p e r is a rra n g e d a s fo llo w s : th e n e x t s e c tio n in tro d u c e s Z C S . S e c tio n 3 d e s c rib e s h o w s e lf-a d a p tiv e m u ta tio n is im p le m e n te d a n d S e c tio n 4 d e s c rib e s th e ta s k s a n d e x a m in e s th e e ffe c ts o f th e o p e ra to r. F in a lly , a ll re s u lts a re d is c u s s e d . S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 3 9 − 3 4 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 340 L. Bull and J. Hurst 2 Z C S ZCS is a "Zeroth-level" Michigan-style Classifier System without internal memory, where the rule-base consists of a number (N) of condition/action rules in which the condition is a string of characters from the usual ternary alphabet {0,1,#} and the action is represented by a binary string. Associated with each rule is a strength scalar which acts as an indication of the perceived utility of that rule within the system. This strength of each rule is initialised to a predetermined value termed S0. Reinforcement in ZCS consists of redistributing strength between subsequent "action sets", or the matched rules from the previous time step which asserted the chosen output or "action". A fixed fraction (E) of the strength of each member of the action set ([A]) at each time-step is placed in a "common bucket". A record is kept of the previous action set [A]-1 and if this is not empty then the members of this action set each receive an equal share of the contents of the current bucket, once this has been reduced by a pre-determined discount factor (J). If a reward is received from the environment then a fixed fraction E of this value is distributed evenly amongst the members of [A]. Finally, a tax (W) is imposed on all matched rules that do not belong to [A] on each time-step in order to encourage exploitation of the stronger classifiers. Hence this is different from the traditional "Bucket-brigade" algorithm [Holland et al. 1986] and is known [Wilson 1994] to be similar to Watkin’s Q-learning [1989] reinforcement algorithm. ZCS employs two discovery mechanisms, a panmictic GA and a covering operator. On each time-step there is a probability p of GA invocation. When called, the GA uses roulette wheel selection to determine two parent rules based on strength. Two offspring are produced via mutation (probability P) and crossover (single point with probability F). The parents then donate half of their strengths to their offspring who replace existing members of the rule-base. The deleted rules are chosen using roulette wheel selection based on the reciprocal of rule strength. If on some time-step, no rules match or all matched rules have a combined strength of less than I times the rule-base average, then a covering operator is invoked. The default parameters presented for ZCS, and unless otherwise stated for this paper, are: N = 400, S0=20, E = 0.2, J = 0.71,W = 0.1, F = 0.5, P = 0.002, p = 0.25, I = 0.5 Thus ZCS represents a "basic classifier system for reinforcement learning that retains much of Holland’s original framework while simplifying it so as to increase understandability and performance" [Wilson 1994]. For this reason the ZCS architecture has been chosen to examine the basic behaviour of classifier systems with self-adaptive mutation rates. The reader is referred to [Wilson 1994] for full details of ZCS. Self-Adaptive Mutation in ZCS Controllers 341 3 Se l f - A dapt ive C l assif ie r Syst e m C o nt ro l l e rs 3. 1 Se l f - A dapt at io n In this paper we use the same form of self-adaptive mutation as in Meta-EP. That is, each rule has its own mutation rate P, stored as a real number, which is passed to its offspring, either under recombination or directly (depending upon the satisfaction of F). The offspring then applies its mutation rate to itself using a Gaussian distribution, i.e. Pi’ = Pi + N(0,Pi), before mutating the rest of the rule at the resulting rate. It is noted that this form of self-adaptation is simpler than that typically used in Evolutionary Strategies, where a Lognormal is applied to P, however the simpler form is shown to be adequate here and has been suggested to work better in noisy environments [Angeline et al. 1996]. We also note that this is in contrast to the adaptive form of crossover introduced by Wilson (1987) for CS, under which a system entropy measure was used to alter the operator rate; Wilson showed benefits from increasing crossover as entropy dropped using predetermined rules of change. 3. 2 C l assif ie r Syst e ms in Evo l ut io nary Ro b o t ic s A number of investigators have examined the use of Classifier Systems in evolutionary robotics. Dorigo, in conjunction with many others (see [Dorigo & Colombetti 1999] for a comprehensive overview), has used multiple CSs in a hierarchy to control an autonomous robot in a variety of environments. To our knowledge this remains the only hardware implementation to date. A large body of work exists on the use of CSs to control simulated robots however, e.g. [Riolo 1991], [Cliff & Bullock 1993], [Donnart & Meyer 1994], [Stolzmann 1999], etc. The reader is referred to [Lanzi et al. 2000] for a full CS bibliography. The performance of a Michigan-style classifier system - ZCS - with self-adaptive mutation in simulated evolutionary robotics tasks is now examined, with the aim of determining ways to improve their use in real environments. 4 Re sul t s in Wo o ds 1 and Wo o ds 7 4. 1 T h e T ask s Wilson [1994] introduced two multi-step "woods" environments with which to examine the performance of ZCS. Woods 1 is a two dimensional rectilinear 5x5 toroidal grid. Sixteen cells are blank, eight contain rocks and one contains food. ZCS is used to develop the controller of a robot/animat which must traverse the map in search of food. It is positioned randomly in one of the blank cells and can move into any one of the surrounding eight cells on each discrete time step, unless occupied by a rock. If the animat moves into the food cell the system receives a reward from the 3 4 2 L . B u ll a n d J . H u rs t e n v ir o n m e n t ( 1 0 0 0 ) , a n d th e ta s k is r e s e t, i.e . f o o d is r e p la c e d a n d th e a n im a t r a n d o m ly re lo c a te d (F ig u re 1 ). O n e a c h tim e s te p th e a n im a t re c e iv e s a s e n s o ry m e s s a g e w h ic h d e s c rib e s th e e ig h t s u rro u n d in g c e lls . T h e m e s s a g e is e n c o d e d a s a 1 6 -b it b in a ry s trin g w ith tw o b its re p re s e n tin g e a c h c a rd in a l d ire c tio n . A b la n k c e ll is re p re s e n te d b y 0 0 , fo o d (F ) b y 1 1 F ig . e n v ir C S c a d ja c * O O O O T h e W m e n t s tro lle d t to th o o d s 1 h o w in g th e a n im a t * e fo o d g o a l. F O O 1: o n o n e n O O a n d ro c k s (O ) b y 1 0 (0 1 h a s n o m e a n in g ). T h e m e s s a g e is o rd e re d w ith th e c e ll d ire c tly a b o v e th e a n im a t re p re s e n te d b y th e firs t b it-p a ir, a n d th e n p ro c e e d in g c lo c k w is e a ro u n d th e a n im a t. T h e tria l p re v io u s o n e a c h w h ils t th W o o d s 7 F ifty -s e v e ro c k s p o s is b la n k . o p tim u m Z C S h a s c a n n o t b e h o w w e ll W o o d s 7 is re p e a te 5 0 tria ls ) tria l. If it e o p tim u m n 1 0 ,0 0 0 tim e o f h o w m a n y m o v e d ra n d o m is s a id to b e s a n s te p ly W 1 .7 d a s it ils o s te p re c o rd is k e p t o f a m o v in g a v e ra g e (o v e r th e ta k e s fo r th e a n im a t to m o v e in to a fo o d c e ll n c a lc u la te s p e rfo rm a n c e a t 2 7 s te p s p e r tria l, s. is a m o re c o m p le x a n d n o n -M a rk o v v e rs io n o f c e lls e v e n ly s c a tte re d a ro u n d th e m a p c o n ta in itio n e d ra n d o m ly in tw o o f th e e ig h t s u rro u n d in g W ils o n s ta te s th a t ra n d o m s e a rc h w ill ta k e 4 1 is 2 .2 s te p s p e r tr ia l ( n o t s h o w n - s e e [ W ils o n 1 n o te m p o ra ry m e m o ry (s e e [C liff & R o s s 1 9 9 5 ][T e x p e c te d to s o lv e W o o d s 7 o p tim a lly , h o w e v e r [it] c a n d o [W ils o n 1 9 9 4 ]. A ll in p u ts a n d o th e r a s W o o d s 1 . A ll re s u lts in 4 .2 d th is p a p e r a re th e a v e ra g e o f te n W o o d s 1 o n a 5 8 x 1 8 g rid . fo o d . E a c h o f th e s e h a s c e lls . T h e re s t o f th e m a p s te p s to fo o d , w h ils t th e 9 9 4 ]). It is n o te d th a t, s in c e o m lin s o n & B u ll 1 9 9 8 ]), it it is s till o f in te re s t to s e e ta s k d e ta ils a re th e s a m e in ru n s. R e s u lts F ig u re 2 s h o w s th e p e rfo rm a n c e o f th e s e lf-a d a p tiv e m u ta tio n o p e ra to r w ith in Z C S o n W o o d s 1 . H e re th e in itia l p o s s ib le ra n g e o f m u ta tio n ra te s w a s c e n tre d a ro u n d th e f i x e d r a t e o f 0 .0 0 2 u s e d b y W i l s o n , i .eµ. < 00 <.0 0 4 . I t c a n b e s e e n t h a t t h e u s e o f s e l f - S e lf-A d a p tiv e M u ta tio n in Z C S C o n tro lle rs 3 4 3 a d a p ta tio n h a s h a d n o re a l b e n e fic ia l/d e trim e n ta l e ffe c ts o n p e rfo rm a n c e , if a n y th in g le a rn in g is a little q u ic k e r (F ig . 2 a ). E x a m in a tio n o f th e a v e ra g e m u ta tio n ra te in th e F i g . 2: S h o w i n g ru le -b 0 .0 0 2 th e re b y W (a ) th e b e h a v io u r o f s e lf-a d a p tiv e m u ta tio n in W o o d s 1 . a s e ( F ig . 2 b ) s h o w s th a t it r is e s s lig h tly f r o m th e m e a n o f 0 .0 0 2 0 u p to a r o u n d 5 . T h a t is , a lth o u g h n o s ig n ific a n t im p ro v e m e n ts in p e rfo rm a n c e a re s e e n h e re , is o b v io u s ly a s lig h t s e le c tiv e p re s s u re fo r a h ig h e r m u ta tio n ra te th a n th a t u s e d ils o n . T h e r e s u ltin g r u le - b a s e s a t th e e n d o f th e 1 0 ,0 0 0 tr ia ls h a v e b e e n e x a m in e d . A lth o u g h th e a v e ra g e m u ta tio n ra te a p p e a rs ro u g h ly e q u a l to th e fix e d ra te u s e d b y W ils o n , a n a ly s is s h o w s a w id e ra n g e o f in d iv id u a l ra te s . T y p ic a lly , a la rg e p ro p o rtio n o f th e F i g . 3: S h o w i n g (a ) th e e ffe c ts o f a h ig h e r in itia l m u ta tio n r u l e s h a v e n o m u t a t i o n , i . µe .i = 0 , w h i l s t o t h e r s h T h e fo rm e r o f th e s e is a s s o c ia te d w ith ru le s fo e n v iro n m e n t, w h ils t th e la tte r a re a s s o c ia te d w ith H e n c e it a p p e a r s th a t th e d e g r e e o f e v o lu tio n a r s itu a tio n /c e ll, d u r in g th e le a r n in g p r o c e s s , is d ir t h e r u l e s i n t h e i n d u c t i v e / r e i n f o r c e m e n t c h a i nR . u n tria ls s h o w th e a v e ra g e m u ta tio n ra te e v e n tu a lly ra te s e e d in W o o d s 1 . a v e m u ta tio n ra te s u p to a n d o v e r 1 . r c e lls c lo s e s t to th e fo o d g o a l in th e ru le s fo r c e lls fu rth e s t fro m th e g o a l. y s e a r c h o n th e r u le s o f a p a r tic u la r e c tly c o r r e la te d w ith th e p o s itio n o f s o v e r a m u c h la rg e r n u m b e r o f g o in g to z e ro (n o t s h o w n ). 3 4 4 L . B u ll a n d J . H u rs t W e h a v e a ls o e x a m in e d th e ro b u s tn e s s o f th e s e lf-a d a p tiv e a p p ro a c h to s ta rtin g th e s y s te m w ith a n in a p p ro p ria te m e a n m u ta tio n ra te . T h a t is , w e w e re in te re s te d in w h e th e r s e lf-a d a p ta tio n c a n b e u s e d to re m o v e th e m u ta tio n p a ra m e te r fro m th e (a ) F i g . 4: S h o w i n g (b th e e ffe c ts o f s e lf-a d a p tiv e m u ta tio n d e s ig n e r’s c o n tro l s o m p a rtic u la rly e v o lu tio n a 0 < µ < 0 .5 . I t c a n a g a in fix e d m u ta tio n ra te (F to w a r d s 0 .0 0 2 a f te r a e w h a t a ry ro b o b e se e n ig . 3 a ), s lig h t d n d h e n tic s . F th a t n b u t th a e la y (F c e e a s e th ig u re 3 s o re a l b e n t n o w th e ig . 3 b ). e u se h o w s e fit/d e a v e ra in o f re s trim g e W o o d s 7 . C la s u lts e n t m u ta s ifie rs fro m is fo u tio n r in c o m p le x ta s a n in itia l ra n g n d o v e r th e u s e a te c o n tin u a lly f k s, e o f a a lls R e s u lts fro m th e m o re c o m p le x a n d n o n -M a rk o v W o o d s 7 ta s k w e re v e ry s im ila r to th o s e a b o v e . F ig u re 4 s h o w s th a t th e re is n o s ig n ific a n t c h a n g e in p e rfo rm a n c e w h e n F i g . 5: S h o w i n g (a ) th e e ffe c ts o f a h ig h e r in itia l m u ta tio n th e in itia l m u ta tio n ra te s T h e re is a ra p id in c re a s e w h ic h th e n d e v ia te s a ro u n in a d iffe re n t e n v iro n m e n m u ta tio n ra te . T e s ts fo r 0 < µ < 0 .5 ) a ls o s h o w e d n o a re se e d e d in th e a v e d 0 .0 0 3 , a t w e se e a ro b u s tn e s s c h a n g e in a ro u n ra g e m s lig h tly d iffe re w ith p e rfo r d W ils o u ta tio n h ig h e r n t fo rm m u c h h m a n c e a n ’s ra te ra te o f ig h e n d ra te s e e d in W o o d s 7 . f ix e d r a te o f 0 .0 0 2 ( F ig . 4 a ) . , u p to a r o u n d 0 .0 0 4 ( F ig . 4 b ) , th a n s e e n in W o o d s 1 . T h a t is , s e lf-a d a p ta tio n o c c u rrin g in th e r in itia l m u ta tio n r a te s ( e .g . th a t th e a v e ra g e m u ta tio n ra te S e lf-A d a p tiv e M u ta tio n in Z C S C o n tro lle rs fa lls to w a rd 0 .0 0 2 , a lth o u g h m o re q u ic k ly th a n in W o o d s 1 3 4 5 (F ig u re 5 ). D u e to th e c o m p le x ity o f th e e n v iro n m e n t a n a ly s is o f th e re s u ltin g ru le -b a s e s is m o re d iffic u lt h e re , b u t th e s a m e g e n e ra l c o rre la te d e ffe c t in te rm s o f m u ta tio n /s tre n g th c o n v e rg e n c e a p p e a rs to o c c u r a s d e s c rib e d a b o v e in W o o d s 1 . 5 C o n c lu s io n s In th is p a p e r it h a s b e e n s h o w n th a t it is p o p e ra to r w ith in M ic h ig a n -s ty le C la s s ifie r S y s te m th e a im o f im p ro v in g th e ir p e rfo rm a n c e a s c o n W e a re n o w m o v in g th e s e e x p e rim e n ts o n to a th e In te llig e n t A u to n o m o u s S y s te m s L a b o ra to r F u rth e r e n h a n c e m e n ts to th e s e lf-a d a p tiv e m in v e s tig a te d , a s w e ll a s im p le m e n tin g th e m in [W ils o n 1 9 9 5 ]. o s s ib le to u s e a s e lf-a d a p tiv e s - s p e c ific a lly W ils o n ’s Z C S tro lle rs fo r a u to n o m o u s m o b ile re a l ro b o t p la tfo rm in c o n ju n c y , F a c u lty o f E n g in e e rin g a t e c h a n is m a re a ls o c u rre n tly th e m o re s o p h is tic a te d X C S m u ta tio n - w ith ro b o ts . tio n w ith U W E . b e in g s y s te m A c k n o w le d g e m e n ts T h a n k s to A n d y T o m lin s o n fo r a n u m b e r o f u s e fu l d is c u s s io n s d u rin g th is w o rk . R e fe r e n c e s A n g e lin e , P .J ., F o g e l, D .B ., F o g e l, L .J . ( 1 9 9 6 ) A C o m p a r is o n o f S e lf - A d a p ta tio n M e th o d s f o r F in ite S ta te M a c h in e s in a D y n a m ic E n v ir o n m e n t. I n L .J . F o g e l, P .J . A n g e l i n e , & T . B c k ( e d s E . )v o l u t i o n a r y P r o g r a m m i n g V , M I T P r e s s , p p . 4 4 1 - 4 4 9 . B c k , T . ( 1 9 9 2 ) S e lf - A d a p ta tio n in G e n e tic A lg o r ith m s . I n F .J . V a r e la & P . B o u r g in e ( e d s .) T o w a r d a P r a c tic e o f A u to n o m o u s S y s te m s : P r o c e e d in g s o f th e F ir s t E u r o p e a n C o n f e r e n c e o n A r t i f i c i a l L i ,f e M I T P r e s s , p p 2 6 3 - 2 7 1 . C liff, D . & B u llo c k , S . (1 9 9 3 ) A d d in g B e h a v io r 2 (1 ):4 7 -7 0 . C liff, D . & R o s s , S . (1 9 9 5 ) A d d in g 3 (2 ): 1 0 1 -1 5 0 . D o n n a rt, J -Y . & M e y e r, P o s itio n in g w ith M o n a L y s W ils o n ( e d s .) F r o m A n im a C o n fe r e n c e o n S im u la tio n J-A a . ls o f . (1 9 9 4 In P . M to A n im A d a p tiv `F o v e a l V is io n ' to T e m p o ra ry ) S a e s a ts e B M e m o ry p a tia l E x p lo ra , M . M a ta ric , 4 : P r o c e e d in g e h a v i o uM r ,I T P r tio n J-A s o e ss, D o r i g o , M . & C o l o m b e t t i , M . ( 1 9 9 R9 ) o b o t S h a p i n g : A n E n g in e e r in g . M IT P re s s . W i l s o n ' s A nA i d m a a p t t . i v e t o AZ dC a S p .t i v e B e h a v i o r , M . M f th p p a p L e e y e r, e F o u 2 0 4 -2 a rn in g , a n d S e lfJ . P o lla c k & S .W . r th In te r n a tio n a l 1 3 . E x p e r im e n t in B e h a v io r 3 4 6 L . B u ll a n d J . H u rs t F o g e l , D . B . ( 1 9 9 2 )E v o l v i n g C a lifo rn ia . A r t i f i c i a l I n t e l l i g e n c .e P h D H o lla n d , J .H . ( 1 9 7 5 )A d a p ta tio n M ic h ig a n P re s s . in N a tu r a l a n d d is s e rta tio n , U n iv e rs ity A r t i f i c i a l S y s t e . m sU n i v e r s i t y H o lla n d , J .H ., H o ly o a k , K .J ., N is b e tt, R .E . & T h a g a rd , P r o c e s s e s o f I n f e r e n c e , L e a r n i n g a n d D i s c o v e. r M y I T P r e s s . K o z a , J . R . ( 1 9 9 1 )G e n e t i c P r o g r a m m i n g . M I T P .R . o f o f ( 1 9 I n8 d6 u) c t i o n : P re ss. L a n z i , P - L . , S t o l z m a n n , W . & W i l s o n , S . W . ( e d s . ) ( 2 0 P 0 r0 o ) c e e d i n g s o f t h e S e c o n d I n t e r n a t i o n a l W o r k s h o p o n L e a r n i n g C l a s s i f i e r S y s t e ,m Ss p r i n g e r - V e r l a g . R e c h e n b e r g , I . ( 1 9 7 3 )E v o lu tio n s s tr a te g ie ; O p tim ie r u n g te c h n is c h e r P r in z ip e n d e r b io lo g is c h e n E v o lu tio.n F r o m m a n n - H o lz b o o g V e r la g . R io J-A In te 3 2 6 S y s te m e n a c h lo , R . (1 9 9 1 ) L o o k a h e a d P la n n in g a n d L a te n t L e a rn in g in a C la s s ifie r S y s te m . In . M e y e r & S . W . W i l s o n ( e d Fs . r ) o m A n i m a l s t o A n i m a t s : P r o c e e d i n g s o f t h e F i r s t r n a t i o n a l C o n f e r e n c e o n S i m u l a t i o n o f A d a p t i v e B e h a v i ,o u M r I T P r e s s , p p 3 1 6 . S to lz m a n n , W . (1 9 9 9 ) L a te n t L e a rn in g in K h e p ra R o b o ts w ith A n tic ip a to ry C la s s ifie r S y s t e m s . I n A . S . W u ( e d . )P r o c e e d i n g s o f t h e 1 9 9 9 G e n e t i c a n d E v o l u t i o n a r y C o m p u t a t i o n C o n f e r e n c e W o r k s h o p P r o g r a ,m M o r g a n K a u f f m a n , p p 2 9 0 - 2 9 7 . T o m lin s o n , A . & B u ll, L . ( 1 9 9 8 ) A C o r p o r a te C la s s if ie r S y s te m . I n A .E . E ib e n , T . c k , M . S c h o e n a u e r & H - P . S c h w e f e l ( e dP s a . )r a l l e l P r o b l e m S o l v i n g f r o m N a t u r e - P P S N V, S p r i n g e r , p p . 5 5 0 - 5 5 9 . B W a t k i n s , C . ( 1 9 8 9 )L e a r n i n g C a m b rid g e . fr o m D e la y e d R e w a r d. s P h D d is s e rta tio n , U n iv e rs ity o f W ils o n , S .W . ( 1 9 8 5 ) K n o w le d g e G r o w th in a n A r tif ic ia l A n im a l. I n J .J . G r e f e n s te tte ( e d .) P r o c e e d in g s o f th e F ir s t I n te r n a tio n a l C o n fe r e n c e o n G e n e tic A lg o r ith m s a n d t h e i r A p p l i c a t i o n s, L a w r e n c e E r l b a u m A s s o c i a t e s , p p 1 6 - 2 3 . W ils o n , S .W . ( 1 9 8 7 ) C la s s if ie r S y s te m s a n d 2 :1 9 9 -2 2 8 . W ils o n , S .W . ( 1 9 9 4 ) C o m p u ta tio n 2 (1 ):1 -1 8 . Z C S : A W ils o n , S .W . ( 1 9 9 5 ) C la s s if ie r C o m p u ta tio n 3 (2 ):1 4 9 -1 7 7 . t h e A n i m a t P r o b lM e m a c . h i n e L e a r n i n g Z e ro th -le v e l F itn e s s C la s s ifie r B a se d o n S y s t e mE v . o l u t i o n a r y A c c u r a c E y v. o l u t i o n a r y Using a Hybrid Evolutionary-A* Approach for Learning Reactive Behaviours Carlos Cotta and Jose M. Troya Dept. of Lenguajes y CC.CC., University of Malaga, Complejo Tecnologico 3.2.49 , Campus de Teatinos, E-29071, Malaga, Spain fccottap, troyag@lcc.uma.es Abstract. A hybrid approach for learning reactive behaviours is presented in this work. This approach is based on combining evolutionary algorithms EAs with the A* algorithm. Such combination is done within the framework of Dynastically Optimal Forma Recombination, and tries to exploit the positive features of EAs and A* e.g., implicit parallelism, accuracy and use of domain knowledge while avoiding their potential drawbacks e.g., premature convergence and combinatorial explosion . The resulting hybrid algorithm is shown to provide better results, both in terms of quality and in terms of generalisation. 1 Introduction The control of autonomous mobile agents is a complex task to which great eorts are devoted due to its practical applications. In general, such control is achieved by means of both planning and reactive components 13 . Each of these components has it own particularities, and can be examined in combination e.g., 1, 2, 7  or in isolation e.g., 11, 12 . In line with the latter, this work focuses on the acquisition of reactive behaviours in mobile agents. Reactive behaviours are driven by a stimulus-to-response mapping, i.e., the agent receives some information about its local environment and decides the action s to carry on exclusively on the basis of such information. This kind of behaviour has usually the advantage of not requiring any underlying global model of the world in which the agent is located. The obvious drawback of reactive systems is the fact that they can get stuck into dead-ends, situations in which the correct action does not only depend on the locally available information but also the structure of the world at a higher-level hence the necessity of longterm planning capabilities. Nevertheless, reactive systems have been shown to provide a very good performance in a wide variety of scenarios and remain a very suitable option when response-time is critical. There exist several techniques for designing reactive systems. These can be typically classied into reinforcement learning and optimisation techniques. Algorithms such as Holland's bucket brigade, Sutton's temporal dierence learning or Watkins's Q-learning lie in the rst class. Within the second class, evolutionary algorithms deserve special attention because of their power and exibility. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 4 7 − 3 5 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 3 4 8 C . C o tta a n d J .M . T r o y a Regarding the use of these techniques for this purpose e.g., 8, 9, 11 , a critical point is the use of as much domain knowledge as possible. Otherwise, the user would be relying on a fortuitous matching between her algorithm and the problem under consideration 14 . Such specialised algorithms are usually termed hybrid evolutionary algorithms 6 . This work presents a hybrid evolutionary algorithm for acquiring reactive behaviours. In the proposed algorithm, domain knowledge is included by using a specialised technique the A* algorithm as an internal operator. The remainder of the article is organised as follows. First, the agent and the worlds used in the experiments are described Sect. 2 . Next, the classical A* approach for solving the posed problem is shown Sect. 3 . Then, the hybrid algorithm is introduced Sect. 4 . Subsequently, experimental results are presented Sect. 5 . Finally, some conclusions are extracted and future work is outlined Sect. 6 . 2 The Agent and its World The agent used in this work is located in a two-dimensional toroidal grid-world in which several obstacles are distributed. The purpose of the agent is to reach a certain target point from its initial location within an allowed time. To do so, the agent is capable of making some elementary actions such as moving straight ahead a single grid square, turning 90o to its left, or turning 90o to its right. Obviously, the agent must avoid obstacles while navigating through its world. For this purpose, it is equipped with proximate sensors that can inform of the presence or absence of obstacles in front of the agent, 90o to its left, or 90o to its right see Fig. 1, left . In addition, these sensors can also detect whether the target point is in any of these three locations or not. Fig. 1. Left Structure of the agent used in experiments. Right Example world and regions into which it is divided according to the location of the target point. U s i n g a H y b r i d E v o l u t i o n a r -y A* Approach for Learning Reactive Behaviours 3 4 9 The agent is equipped with a direction sensor as well. This sensor allows determining in which of four imaginary regions of the world the target point is located. These regions are illustrated in Fig. 1 right. It must be noted that these regions are not absolute but relative to the agent's actual orientation. For example, the agent is facing North in Fig. 1 and hence the target point is in zone 1. Now, if the agent turned 90o to its right, the target would be in zone 0. Notice also that these regions are determined taking into account the toroidal shape of the world. Thus, if the agent were a few positions South from the location shown in the previous example, the target point might happen to be in zone 2. According to this description, the goal is to design a reactive behaviour allowing the agent reaching its target in as many situations as possible. Such reactive behaviour can be de ned in a variety of ways, e.g., using a neural network 15, a fuzzy rule-base 8, a cellular automata 3, etc. This work is in line with the latter approach. To be precise, a lookup-table is sought relating every possible sensorial input with a primitive action. At each time-step, the agent must look up the action that corresponds to the current inputs and carry it out. Since each proximate sensor can provide three dierent inputs OBSTACLE, NO-OBSTACLE, TARGET, and the direction sensor can return four values, the resulting table has 33 4 = 108 entries. Since three primitive actions  MOVE-AHEAD, TURN-LEFT, TURN-RIGHT  are available, this implies a search space of 3108 3 1051 tables.  3 A Classical Approach: A* A classical approach for nding the lookup-table mentioned above is the utilisation of the A* algorithm. Based on incrementally constructing solutions in an intelligent fashion, this technique constitutes a powerful tool for solving search problems to optimality. Before getting into the application of this technique to the design of reactive behaviours, some notation details must be given. Let be the current world, and let be the con guration of the agent position and orientation. Now, let    be the sensorial input of the agent when con gured according to . Let M^ be a possibly underspeci ed function relating sensorial inputs with actions, and let M^ 1 M^ 2 whenever M^ 2 provides the same outputs that M^ 1 does and M^ 2 is de ned in at least one case in which M^ 1 is not. Finally, let  be the maximum allowed time for reaching the target and let  be a function such that     M^  =  1   k  . This function provides a trace of the agent trajectory across con guration space when behaving according to M^ . The value  is an indication of the nal status of the agent: AT-TARGET, COLLISION, TIMED-OUT or UNKNOWN. This latter value is returned whenever no action is speci ed in M^ for the current input. Now, the application of the A* algorithm requires the availability of an optimistic evaluation function  such that    M^  provides a lower bound on the number of steps necessary for reaching the target when the agent is congured as and behaves according to any M^  M^ M^ . It is easy to see that making  return the Manhattan distance from the agent's current location to W I W  W h i W 0  0 3 5 0 C . C o tta a n d J .M . T r o y a the target point fulls this requirement. Having dened this function, the whole process is as follows: 1. Let P00 = 0  M^ 0  0 t0 , where 0 is the initial conguration of the agent, M^ 0 is a fully underspecied function, and t0 = W  0  M^ 0 . Let P = M^ 0  1 be the current best solution. Insert P00 in the node queue. ^ h t be the 2. If the node queue is empty, go to 3. Otherwise let P =  M rst element in the queue. a Let  W   M^ = h1     i  . b if  6= UNKNOWN then ^ t0 +  , where i. if  = COLLISION or  = TIMED-OUT then P = M 0 ^ t = h + k + W    M . ^ h+k . ii. if  = AT-TARGET then P = M  iii. If P is better than P , update the latter and purge nodes in the queue. iv. Go to 2. c Create three nodes P 3+1+1 , P 3+1+2 , and P 3+1+3 from P . Each node is P 3+1+ =   M^  h0  t , where M^ is obtained by extending M^ to return the rth possible action when the input is I W   , h0 = h + k, and t = W    M^ . Insert these nodes in the queue keeping it ordered according to the sum of the last two components of each node. d Go to 2. 3. Return P i j k k j j i r k r r i j i j i i j r k r k r This algorithm will thus return the lookup-table allowing the agent reach the target in minimal time from the given starting point. Since the problem has been posed with the goal of obtaining a generalisable reactive behaviour, the process must be slightly modied. To be precise, a training set is selected and the A* algorithm tries to nd the table that minimises the sum of the times required to reach the target in each training case or, if such a solution is not possible, a table that rstly maximises the number of training cases solved and secondly minimises the total time. Notice that no global model of the world i.e., highlevel knowledge about the distribution of obstacles is required. All information used for nding the optimal solution is locally obtained through simulation. This algorithm has been evaluated on a set of nine dierent worlds. These worlds are named as Wxy, where x 2 f10 25 50g indicates the dimension of the world each world is a x  x grid , and y 2 fa b cg indicates the density of obstacles 5, 10 and 20 respectively . For each world, a training set of ve cases has been selected. Subsequently, the best solution found has been tested for generality on a test set whose size depends on the dimension of the world 50, 400 and 2000 cases respectively . The results are shown in Table 1. These results are very indicative of the two main drawbacks of the A* algorithm. On the one hand, it is very sensitive to the size of the task to be solved. As it can be seen, the algorithm expended a high computational eort for solving W25b and ran out of memory in three cases W25c, W50b, and W50c . Moreover, it did not nd any fully satisfactory solution for all training cases in W50b Using a Hybrid Evolutionary- A* Approach for Learning Reactive Behaviours 3 5 1 Results of the A* algorithm on nine dierent worlds. The cost values are measured as the number of single simulation steps carried out. Table 1. World Timeout Optimal solution Iterations W10a 13.20 11615 W10b 25 13.80 19664 W10c 14.00 26429 W25a 32.40 75818 W25b 150 31.60 222788 W25c 26.40, 51.00 300000 W50a 54.60 119864 W50b 400 45.60, 479.40 150000 W50c 34.80, 1629.80 110000 Performance Cost on test set 289857 74 300391 88 281091 58 10352732 63 29083614 48 25000000 36 36861614 66 37000000 14 16000000 2 and W50c 1 and 4 training cases were left unsolved1. On the other hand, the solutions found are not very generalisable. This is a direct consequence of the internal functioning of the A* algorithm. Assume that the nal solution is found when evaluating node . This node was obtained as successive extensions of n3 n9 0 0 . Hence, it contains information regarding the best decisions ,1 ,2 to be taken only in the situations found during this optimal path, i.e., the path from the root node of the implicitly de ned search tree to the optimal leaf node. All that may have been learnt in solving other situations is discarded since these situations do not take place in this optimal path. i Pj i P j i P j  P Growth of the computational cost of the A* algorithm when the number of training cases is increased. Fig. 2. 1 These results were not bad a priori since there might exist no better solution. However, further experimentation with the hybrid EA showed that this was not the case. 3 5 2 C . C o tta a n d J .M . T r o y a This generalisation problem could be solved by considering a larger training set whose optimal solution covered all possible situations. However, the subsequent combinatorial explosion makes this approach unrealistic. This is illustrated in Fig. 2. As it can be seen, the computational cost of the algorithm grows very fast when the size of the training set is increased. For this reason, it is clear that alternative approaches must be found. These will be discussed in next section. 4 The Hybrid EA-A* Approach Evolutionary algorithms constitute a very suitable alternative to A* for nding the lookup-table. A nave approach for applying EAs to this problem would rstly consist of dening an encoding function for storing the lookup-table into an individual, e.g., a linear chromosome in which the rows of the table are consecutively arranged. Since this is an orthogonal representation 10 i.e., all combinations of genes are feasible, the next step would simply involve selecting any of the standard genetic operators that can be found in the literature e.g., single-point crossover  SPX , uniform crossover  UX , etc.. However, such a simple approach is likely to provide very poor results. Recall that this is highly epistatic problem in which the value of each gene i.e., a specic action to be carried out when a certain sensorial input is received does not contribute with a xed amount to the tness of an individual. On the contrary, the goodness of the reactive behaviour dened is determined by the interplay between all genes. For this reason, a blind recombination operator that randomly shues the genetic material of recombined solutions will provably produce solutions with a phenotype reactive behaviour completely unrelated to the parents, even when the latter are genotypically similar. In an extreme situation, it may even reduce to macromutation. The algorithm would be largely more eective if it were able to extract positive behavioural patterns from existing solutions and transmit them to the ospring. This can be achieved within the framework of Dynastically Optimal Forma Recombination 4 DOR. This framework comprises a family of recombination operators of the form DOR : S  S  S ! 0 1 1 where S is the search space and DOR  is the P probability of generating  = 1, the when recombining and . Besides the obvious z2S DOR probability distribution induced by these operators verify that x y z z x x y z y DOR  0  f 2 f g ^ 8 2 f g :      g 2 where is the tness function to be minimised without loss of generality and f g is the dynastic potential 10 of and , i.e., the set of solutions that can be built using nothing but the information contained in and . Thus, the solutions created by DOR are the best that can be constructed using the genetic material of the parents. On the one hand, this implies that x y z  z , x y w , x y  w  z  , x y x y x y Using a Hybrid Evolutionary- A* Approach for Learning Reactive Behaviours 3 5 3 DOR is a fully transmitting operator, i.e., no implicit mutation genetic information not present in any of the parents is introduced in the ospring. On the other hand, the tness-oriented functioning of DOR makes valuable portions of solutions be transmitted to ospring only if they contribute to a good resulting behaviour. In other words, DOR is capable of identifying valuable high-order formae, preventing their disruption. This intelligent combination of information has provided very good results on epistatic problems 5 . In order to implement DOR, it is required to use an embedded A*-like mechanism so as to nd the best solution in the dynastic potential of the parents. In this case, the algorithm described in Sect. 3 can be used. It is only necessary to modify step 2c by considering that the possible actions to be taken in a given situation are just those present in any of the parents for . Notice that the search carried out by this subordinate A* algorithm is thus restricted to small portions of the search space and hence its computational cost is largely reduced with respect to the original unrestricted version. Moreover, individuals in the population tend to be more similar as the EA converges and, subsequently, the dynastic potential of selected solutions tends to be smaller and DOR is less computationally expensive. This combination of EAs and A* has an additional advantage. Each individual carries an information that reects its past evolution in fact, the evolution of its ancestors. This way, things that were learnt in the past are retained as long as they do not negatively aect the present behaviour. This accumulated history" eect is also present in a simple EA, but the learning capabilities of the hybrid algorithm are larger. For this reason, solutions obtained with the hybrid EA are expected to be more general than either the EA or the A* algorithm by themselves. This will be studied in next section. I I 5 Experimental Results Experiments have been done with a steady-state EA popsize = 100, c = 9, m =1chromosomeLength using ranking selection  + = 2 0 , = 0 0. This algorithm has been run 40 times for each operator and test world. In order to make a fair comparison between DOR and the other simpler operators, each run is terminated when a xed number of simulation steps  105 in these experiments, where is the timeout value is reached. Thus, the internal calculations performed by DOR are eectively accounted. As in Sect. 3, a training set of ve cases is used in the tness function. First of all, Fig. 3 shows how the hybrid EA is much more successful in solving the training cases. As it can be seen, while standard operators only provide an acceptable performance on the smallest instances and with the lowest obstacle density, DOR consistently yields satisfactory results: above a 70 of the runs provide a fully successful solution for the training set the percentage is 100 for 5 out of 9 test worlds. The exception is world W50c for which none of the operators could nd a full solution it must be noted that such a solution may p p  :     : : 3 5 4 C . C o tta a n d J .M . T r o y a Fig. 3. Number of runs in which each operator provided a fully satisfactory solution for the training set. not exist. Nevertheless, DOR was capable of solving 3 out of the 5 training cases while SPX could only solve one and UX could not solve any of them. Table 2 shows a more detailed summary of the results. Notice that DOR is not only more eective in nding satisfactory solutions, but also provides higher-quality results. By comparing the median values2 provided by DOR with the optimal best-known solutions see Table 1, it can be seen that DOR yields near-optimal solutions. Moreover, the lower variance of DOR results with respect to SPX and UX indicates a more stable algorithm. Table 2. Comparison of dierent genetic operators on nine dierent environments. All results correspond to series of forty runs. SPX UX DOR World Timeout mean median mean median mean median W10a 17.36 12.06 16.00 15.88 7.47 15.20 13.85 0.87 13.80 W10b 25 22.15 25.96 16.20 18.10 9.26 16.20 14.35 0.74 14.20 W10c 51.49 38.39 59.60 53.85 38.81 59.60 18.75 10.75 14.00 W25a 176.70 210.00 45.60 121.63 193.32 40.00 33.17 2.14 32.40 W25b 150 376.08 285.43 399.30 380.96 272.08 468.40 36.65 6.67 34.00 W25c 520.32 179.07 616.80 502.23 218.40 616.40 99.92 107.70 38.80 W50a 442.39 348.27 438.00 423.87 335.45 438.00 61.17 11.72 58.80 W50b 400 1532.52 554.56 1636.60 457.65 457.65 1636.60 337.32 506.03 103.8 W50c 2021.45 60.77 2031.20 2031.20 0.00 2031.20 1638.78 538.75 1650.60 2 The median value seems to be a more representative measure of the quality of the results than the mean value since the former is much less sensitive to outliers. Furthermore, it provides an reasonable alternative to averaging the tness of solutions that solve the whole training set with solutions that do not solve any training case. Using a Hybrid Evolutionary- A* Approach for Learning Reactive Behaviours 3 5 5 Fig. 4. Percentage of the test set solved for each of the techniques considered. Finally, the results obtained with the EA are tested for generality. Fig. 4 shows the results. Firstly, notice the poor results of standard EAs. The solutions provided by UX and SPX do not reach 50 success in 6 out of 9 worlds. The A* algorithm performs better than standard EAs, but its performance quickly drops when the density of obstacles is increased. The hybrid EA provide the overall best results, outperforming both A* and standard EAs on all worlds. Moreover, this improvement is larger on instances with higher obstacle densities. It must be noted that the results on W50c are not satisfactory for any algorithm although the hybrid algorithm remains the best . This is a really hard instance as mentioned before, and may require longer evolution times and or a larger training set to cope with such a tough environment. 6 Conclusions This work has presented a hybrid approach for learning reactive rule-bases. By combining EAs with the A* algorithm, a synergetic system has been achieved. This hybrid algorithm has been shown to provide higher-quality results than standard EAs. These results are also better than those of the A* algorithm in terms of their generalisation to previously unseen test cases. Furthermore, the hybrid EA is capable of tackling instances in which the A* algorithm would suer the eects of the combinatorial explosion. Future work will try to extend these results to more sophisticated agents. In this sense, notice that most details of the agent are encapsulated within the simulation function and hence they do not aect the presented algorithm qualitatively. Nevertheless, it is clear that issues regarding simulations of higher computational cost are worth studying. Work is in progress in this area. Additionally, new environments and tasks to be solved will be tackled as well. 3 5 6 C . C o tta a n d J .M . T r o y a Acknowledgement This work is supported by the Spanish Tecnologa Comision Interministerial de Ciencia y CICYT under grant TIC99-0754-C03-03. References 1. K. Ali and A. Goel. Combining navigational planning and reactive control. In Theories of Action, Planning, and Robot Control. Bridging the Gap: Proceedings of the 1996 AAAI Workshop, pages 19, Menlo Park, CA, 1996. AAAI Press. 2. C.T.C. Arsene and A.M.S. Zalzala. Control of autonomous robots using fuzzy logic controllers tuned by genetic algorithms. In Proceedings of the 1999 Congress on Evolutionary Computation, pages 428435. IEEE NNC - EP Society - IEE, 1999. 3. T.D. Barfoot and D'Eleuterio G.M.T. An evolutionary approach to multiagent heap formation. In Proceedings of the 1999 Congress on Evolutionary Computation, pages 420427. IEEE NNC - EP Society - IEE, 1999. 4. C. Cotta, E. Alba, and J.M. Troya. Utilising dynastically optimal forma recombination in hybrid genetic algorithms. In A.E. Eiben, Th. Back, M. Schoenauer, and H.-P. Schwefel, editors, Parallel Problem Solving From Nature V, volume 1498 of Lecture Notes in Computer Science, pages 305314. Springer-Verlag, Berlin, 1998. 5. C. Cotta and J.M. Troya. Tackling epistatic problems using dynastically optimal recombination. In B. Reusch, editor, Computational Intelligence. Theory and Applications, volume 1625 of Lecture Notes in Computer Science, pages 197205. Springer-Verlag, Berlin Heidelberg, 1999. 6. L. Davis. Handbook of Genetic Algorithms. Van Nostrand Reinhold Computer Library, New York, 1991. 7. J.-Y. Donnart and J.-A. Meyer. Learning reactive and planning rules in a motivationally autonomous animat. IEEE Transactions on Systems, Man, and Cybernetics, 263:381195, 1996. 8. F. Homann and G. Pster. Learning of a fuzzy control rule base using messy genetic algorithms. In F. Herrera and J.L. Verdegay, editors, Genetic Algorithms and Soft Computing, pages 279305. Physica-Verlag, Heidelberg, 1996. 9. J.R. Koza. Genetic Programming. MIT Press, Cambridge MA, 1992. 10. N.J. Radclie. The algebra of genetic algorithms. Annals of Mathematics and Arti cial Intelligence, 10:339384, 1994. 11. A.C. Schultz and J.J. Grefenstette. Using a genetic algorithm to learn behaviours for autonomous vehicles. In Proceedings of the AIAA Guidance, Navigation and Control Conference, pages 739749, Hilton Head SC, 1992. 12. C. Thornton. Learning where to go without knowing where that is: the acquisition of a non-reactive mobot behaviour by explicitation. Technical Report CSRP-361, School of Cognitive and Computing Sciences, University of Sussex, 1994. 13. G. Weiss. Multiagents Systems: a Modern Approach to Distributed Arti cial Intelligence. The MIT Press, Cambridge MA, 1999. 14. D.H. Wolpert and W.G. Macready. No free lunch theorems for search. Technical Report SFI-TR-95-02-010, Santa Fe Institute, 1995. 15. B. Yamauchi and R. Beer. Integrating reactive, sequential and learning behaviour using dynamical neural networks. In D. Cli, P. Husbands, J.-A. Meyer, and S. Wilson, editors, From Animals to Animats 3: Proceedings of the Third International Conference on Simulation of Adaptive Behaviour, pages 382391, Cambridge MA, 1994. MIT PressBradford Books. Supervised Evolutionary Methods in Aerodynamic Design Optimisation D.J. Doorly, S Spooner and J. Peiro Aeronautics Department, Imperial College, London SW7 2BY, UK Abstract. This paper outlines the application of evolutionary search methods to problems in aeronautical design optimisation. The procedures described are based on the genetic algorithm GA and may be applied to other areas. Although easy to implement, a simple genetic algorithm is often found in applications to be of low e ciency and to su er from premature convergence. To improve performance, two alternative strategies are investigated. In the rst, a learning classi er scheme is used to tune the GA for a particular class of problems. The second strategy uses a parallel distributed genetic algorithm supervised by single or competing agents. The implementation of each procedure, and results for typical design problems are outlined. The agent supervised distributed genetic algorithm is found to provide a model with a very high degree of adaptibility, and to lead to considerably improved e ciency. 1 Introduction The principles of evolutionary computation are well established and are described in texts such as 1, 3, 2, 4 . These methods continue to grow in diversity, and are becoming more commonly used for many problems in engineering design. In aerospace vehicle design, the genetic algorithm GA has been applied to a range of problems, as described for example in 5, 6, 7, 8, 9 . The apparent robustness of the GA, the ease with which it can be applied, and its ability to handle discontinuous or even discrete data make it attractive as a search procedure. There are other search procedures however including various gradient search methods which are more ecient than the GA, albeit that they may only work for a more restricted class of problems. Furthermore, although evolutionary methods have been found to be quite adept at locating global optima in highly multimodal problems, there is usually no guarantee that they will, and their rate of convergence to the optimum solution can be very slow. Ideally one would like an optimisation procedure to work well across a broad range of problems. Bearing in mind the no free lunch `rule' however, some degree of matching of technique to problem seems inevitable. Three simple ways of improving an evolutionary search procedure for a particular type of application are: 1. tuning the parameters to suit the class of problem, 2. improving its ability to adapt to the problem, 3. hybridisation with other techniques appropriate to the class of problem. S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 5 7 − 3 6 6 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 3 5 8 D .J . D o o r ly , S . S p o o n e r , a n d J . P e ir ó We discuss only the rst two of these here, and will use the GA as our basic evolutionary procedure. The construction of a suitable hybrid is very problem dependent, and we consider that an improved GA may either outperform a hybrid routine or may at worst lead to more eective hybrids. Hybrid routines will in any case be discussed elsewhere. The rest of the paper is organised as follows. The common design problem of shape or form optimisation is introduced in the context of aerodynamic or aeroelastic design. The application of a real encoded GA to a typical problem is also briey outlined. After this, we examine the use of a learning classier system to tune the GA parameters, for a particular class of such design problems. The distributed genetic algorithm or DGA is then introduced, and nally the use of agent and multi-agent supervision to improve the adaptibility of the DGA is described. 2 Shape optimisation & outline of basic GA A frequent task in aeronautical design is to nd the `best' aerodynamic shape of an airfoil in 2D or a wing in 3D, subject to certain constraints. The basic GA we use for this task is constructed as follows. The encoding species the shape of a trial solution, though it may also specify structural parameters such as type of material, weight, rigidity etc. For simplicity, let us consider the case of a 2D wing section or airfoil. Then an array cj  may be used to specify the ordinates of a B-spline control polygon, which in turn describes an airfoil shape, as shown in g. 1. We use a real array of 20 control point ordinates to encode the shape. For ... ... C h ro m o so m e P o p u la tio n k ... ... C h r o m o s o m e a r r a y : C k = ( Y 1 , ... ,Y n ) j i 0.06 0.04 Y y/c 0.02 i 0.0 Y -0.02 j -0.04 -0.06 0.0 Fig. 1. 0.1 0.2 0.3 0.4 0.5 x/c 0.6 0.7 0.8 0.9 1.0 Representation of airfoil geometry. aeroelastic optimisation of wings instead 13, we encode the design as a sur- S u p e rv is e d E v o lu tio n a ry M e th o d s in A e ro d y n a m ic D e s ig n O p tim is a tio n 3 5 9 face interpolated between a series of spanwise sections, with the encoding specifying a standard basic prole, together with the chord, thickness, twist and structural parameters for each section. An initial population of random airfoils is generated, each initial shape is ensured to be valid, i.e. encodings which produce surface crossings or violate thickness constraints are discarded. The tness is evaluated using a CFD ow solver or a CFD + structural solver. Either an unstructured mesh based solution of the Euler equations, or a viscous inviscid panel ow solver is used, 11 . In a direct optimisation, the search is for the shape which best meets the requirements e.g. high lift drag L D, whereas in an inverse optimisation, the shape which best matches a given pressure coe cient distribution  p  is sought. Fitness values are range-scaled and remapped to the interval 0 1 , with individuals below the mean tness assigned a base value typically 0 4, and those above the mean assigned a value scaled quadratically up to 1. Roulette wheel is used for selection in the basic GA and DGA operations, though the learning GA allows other methods binary tournament etc. Two point crossover is again the basic crossover type, with one point, uniform etc. allowed in the learning GA. Mutation and crossover are applied as separate operators. The probability of mutation of a given gene is low, typically 0.005, but is commonly increased at later stages. Elitist population replacement is applied, where the best of each generation is automatically carried through, here together with a slightly mutated copy of the best. The population replacement routine ensures that no overreplication i.e. excess identical or almost identical duplication occurs. C  O : 3 Classier Learning Directed GA The classier learning system is described in standard texts on machine intelligence much of the research in this area also follows from ideas put forward by Holland 14 . The classier learning system adds a layer on top of the GA. Rules to control operators mutation crossover, type of crossover, selection scheme etc., and parameter values e.g. mutation rate are prescribed, and their effectiveness when implemented in the GA are assessed. In a static mode, the entire set of rules can undergo genetic operations to evolve better rules and rule combinations. In a dynamic mode, a system of reward paybacks can be used to determine rule selection. The objective of the procedure here was to train the GA in a static mode for a particular problem type. The e ciency of the trained and basic GAs were then compared rstly when given problems of a very similar nature to those used in training, and secondly for slightly dierent problems. In the learning GA outlined in g. 2, an initial population of 30 rules was used, which were randomly initialised. The rules were encoded as 15 genes in an IF-THEN-AND conguration, and were designed to work in groups, with the nal 2 genes being 3 6 0 D .J . D o o r ly , S . S p o o n e r , a n d J . P e ir ó L e a r n in g G A G A r u le c o n tr o l S T A R T G A T e s t n e w r u le s e t & F lo w S o lv e r M a tu r e r u le s E n d : D e v e lo p e d s o lu tio n Fig. 2. Classier Learning SystemGA index references to other rules to be used in combination with the current one. The rule gene string comprised: two IF operators, with parameter values determining the generation number, and average gene diversity respectively which triggered operation of the rule, the following 11 gene values type THEN set operator type and parameters, the nal two type AND determined the combination of rules to be used. The procedure was repetitively applied to the inverse design of respectively symmetric and non-symmetric airfoils. To recall, in inverse design, the problem is to nd the shape which matches a specied pressure coe cient Cp  distribution it is often required in real applications, and for comparative tests of search methods purposes it is preferrable to direct design, as the target is given. Each rule was tested twice, and the top 10 peforming rules were then isolated and repetitively tested over 15 separate runs. The GA was trained for inverse optimisation of symmetric and non-symmetric airfoil sections, respectively using NACA airfoil types -0012,-0022,-0024 for symmetric training, and -23015, -4421 for non-symmetric training. Figure 3 shows results for the inverse design of a non-symmetric airfoil 23015, comparing the performance of the GA using: rules developed for this type of airfoil, rules developed for a dierent symmetric type, and the basic untrained GA. It can be seen that the trained GA outperforms the baseline GA for inverse optimisation whether it is trained on a class of similar airfoils, or a dissimilar class. However, the dierence in performance resulting from the class of problem used for training shows how very specic training may be needed to obtain the highest gains. At present, the cost of CFD evaluations is generally so high, that the benets of training do not appear worthwhile. However other possibilities, such as using a simpler approximate evaluation method for training only, or reducing the complexity of the scheme, and hence degree of training, may yet render it more practical in this area. Also, the classier system may still prove useful within the context of an agent supervised DGA. A v e ra g e b e s t fitn e s s (1 5 ru n s ) S u p e rv is e d E v o lu tio n a ry M e th o d s in A e ro d y n a m ic D e s ig n O p tim is a tio n 3 6 1 R u le s fro m n o n s y m m e tric a irfo il tra in in g R u le s fro m s y m m e tric a irfo il tra in in g } 0 2 0 0 B a s e lin e G e n e ra tio n n o . Application of learning GA to inverse design of NACA-23015 airfoil. Comparison of performance with rule sets developed for similar airfoils, for di erent airfoils, and basic GA performance. Fig. 3. 4 Distributed Genetic Algorithm DGA Previous work has already shown that the DGA outperforms the GA on many test problems and in design optimisation, 10, 16. Applications of the DGA to aeronautical design problems are also described in 11, 13. Brie y the DGA di ers from the standard GA in that the population of trial solutions is split into semi-isolated subpopulations or `demes'. The demes are considered analogous to island populations, where geography acts as a barrier to exchange. Restrictions on the recombination and genetic exchange between subpopulations are imposed the exchange is limited to the migration of a few individuals often only the best one or two from one neighbouring deme to another every m generations, with typically m = 5 or m = 10 in our implementations. Thus the parameter set for the basic GA operator probabilities, selection mode, etc. is enlarged to include the number of individuals migrating, barriers to acceptance of immigrants, exchange frequency, geographical exchange radius and topology of the demes. With limited exchanges between demes, the DGA is then ideally suited for coarse grain parallelisation g.4 whether on a parallel supercomputer or network of workstations. Nang 12 surveys the parallel GA, of which the `stepping stone' connected DGA shown in g. 4 is one type. For a workstation network 11, provided the ow solution can be run on a single workstation, the only communication required between processors involves the exchange of a limited number of chromosomes, at intervals of several generations. The communication requirements are extremeley low, given that by far the bulk of the computational e ort is devoted to the ow solution which performs the evaluation. Each processor may be responsible for a number of demes. For a heterogeneous network of processors of di erent speeds or loadings, load balancing can be achieved by varying the mapping of demes to processors, or by altering the number of individuals treated by a given processor. 3 6 2 D .J . D o o r ly , S . S p o o n e r , a n d J . P e ir ó C F D Fig. 4. G A C F D G A ( I ,J ) D E M E ( I + 1 ,J ) D E M E C F D G A ( I ,J - 1 ) D E M E Distributed GA DGA mapped to processor array. 4.1 Application to aeronautical design Inverse design Comparison of the convergence behaviour obtained using a distributed 13 and a conventional GA, with the same total population and number of evaluations, for the viscous inverse design of a NLF1 -0115 airfoil is shown in g. 5. For the DGA, the population of 180 was distributed onto 9 subpopulations of 20, each residing on a dierent processor, as in g.4 above. The migration between the islands occurred in a stepping stone fashion, ie. migration occurred only between immediate neighbours every ve generations. As the results indicate, the DGA greatly outperforms the conventional GA algorithmically. This gain is then further multiplied almost ninefold with the distributed processor implementation. The implementation was done using the MPI standard on a network of workstations it has also been implemented on a multiprocessor machine . Comparison of convergence rate of parallel DGA and single population version. If computing speed up rate nearly 9 were applied, parallel DGA performance gain would appear even more dramatic Fig. 5. S u p e rv is e d E v o lu tio n a ry M e th o d s in A e ro d y n a m ic D e s ig n O p tim is a tio n 3 6 3 Direct Airfoil Optimisation Results Application of the DGA to problems of direct airfoil optimisation for low speed and transonic cases are described in 6, 11. For both inverse and direct optimisation problems however, the use of an agent to supervise the operation of the DGA has been found to improve the performance, as described next. 5 Agent Supervision of DGA The better ability of the DGA at maintaining population diversity appears to account for its notable gain in performance over the single population GA. Eventually however, the population on each island converges. Adding an agent to supervise the operation of the parallel GA provides a capability whereby the DGA can adapt more generally than is possible with a sequential GA. Although adaptation can be built into a GA e.g. in the adaptive operator tness of Davis 4, the use of agents provides a more general framework by decoupling the tasks of higher level supervision from the lower level optimisation. The agent supervised paradigm is very well suited to a distributed computing environment, where agents can direct the operation of the GA on local or global populations, and can additionally direct processing resources. At one extreme, the agent layer may be combined with the DGA software to execute as a single albeit distributed entity, or at another, it may run as an entirely separate distributed program, communicating with the DGA by reading external output messages and writing to action inputs. A simple agent supervision of the DGA is as follows. The agent receives status messages from the DGA, i.e. generation number, measures of population convergence, tness changes etc. , and parameter settings local mutation and crossover rates types etc. . The agent layer then instructs the DGA to take either global or local action. Examples of such actions could be the introduction of a mechanism to improve diversity between island populations, actions to favour speci c local niches within islands, or actions to improve the parallel load balancing by adjusting the deme placing or sizing. 5.1 Infection Agent A simple implementation of agent supervision of the DGA which was used for direct airfoil design employs a single agent supervisor 15 to act as a vector for infections, with low population diversity encouraging epidemics. On infected islands, individuals close to the global best have greatly reduced tness, and undergo increased mutation results 15 shows the pattern of infection changes dynamically. When implemented for the problem of inverse design optimisation, there was a gain in late solution convergence, beginning just beyond the point when the population on all the islands initially converges towards a global `champion'. Applying the procedure to direct design viscous L D optimisation of a low speed airfoil section, at operating points of 3 and 8 degrees incidence, and a 3 6 4 D .J . D o o r ly , S . S p o o n e r , a n d J . P e ir ó Reynolds number of 4 million, with a moment constraint  Cm 0:97 produced the shape and Cp distributions shown in g. 6, though this is not yet fully a converged solution. j j Fig. 6. LD optimization: Distribution of Cp on the surface of the `best' airfoil. This computation was performed using XFOIL. The addition of the infection in these problems was found to show a clear improvement over the solution obtained in 11 for corresponding eort the improvement in the tness after island convergence is shown in g. 8a further below. 5.2 Evolutionary Agent Supervision In the learning classi er method presented earlier, adaptation occurs through repetitively solving a problem or class of problems. The agent supervision can adapt dynamically however, especially if a large number of generations are to be evaluated. Agent or Multi-Agent A COMPETE A A Actions (Local or Global) DGA Messages Fig. 7. Messages Competing agent supervised DGA For example, one may place several islands in a group under the control of one agent, and other groups of islands under the control of other agents, with each agent adopting dierent strategies. If migration between groups is eliminated, the S u p e rv is e d E v o lu tio n a ry M e th o d s in A e ro d y n a m ic D e s ig n O p tim is a tio n 3 6 5 relative improvement over a number of generations may be compared. The worse performing agents then modify either the parameter settings or the rules which they apply to the DGA under their control, and may also replace their population partly or fully with that of the best group. An example of the eectiveness of the approach is shown in g.8b, where the simple agent infection approach described previously is supervised by a pair of competing agents. A population of 200 10 individuals per deme demes connected in a 4 x 5 array was used in this case for the inverse design of a NACA 0012 airfoil. The agents apply dierent genetic operators to their respective groups of subpopulations, and their performance as managers is compared after a certain number of generations. In the example, only the mutation rate was altered by the supervising agents at a lower level, the infection strategy was still implemented. As can be seen from g. 8b, these early results are encouraging further work will consider the eects of controlling dierent parameters, and dierent competition mechanisms. 18 30 16 D G A + A g e n t C o m p e tin g A g e n ts 25 14 20 In fe c tio n 10 F itn e s s F itn e s s 12 8 6 15 10 D G A 4 S in g le A g e n t 5 2 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 G e n e ra tio n G e n e ra tio n a b 250 300 350 400 Agent supervision: a Comparison of DGA and simple agent supervised DGA bComparison of simple agent and competing agent supervised DGA. Note dierence in tness scales results in b are also averaged over more trials Fig. 8. 6 Conclusion The distributed genetic algorithm DGA has been applied to a number of aeronautical design optimisation problems. Earlier results indicated that the method has better convergence behaviour than the single population GA the present work outlines agent supervision strategies to improve the DGA further. A learning classier scheme applied to the single population GA showed some improvement, but appears very costly at present. In contrast, the use of competing agents which evolve appears quite promising for further investigation. Applica- 3 6 6 D .J . D o o r ly , S . S p o o n e r , a n d J . P e ir ó tions of the DGA to low speed airfoil optimisation, demonstrate that the method is straightforward to implement, and can be easily applied to dierent problems. References 1. Goldberg D E, Genetic Algorithms in Search, Optimisation and Machine Learning, Addison-Wesley, 1988. 2. Back T, Evolutionary Algorithms in Theory and practice, Oxford, 1996. 3. Schwefel H P, Evolution and Optimum Seeking, Wilrey New York, 1995. 4. Davis L, Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York 1991. 5. Obayashi, S., Yamaguchi Y., and Nakamura, T. Multiobjective genetic algorithm for multidisciplinary design of transonic wing planfoem, J. Aircraft 34, 5, pp 690 693, 1997 6. Doorly D J, Ch. 13 of Genetic Algorithms in Engineering and Computer Science, ed. G. Winter et al., Wiley, 1995. 7. Quagliarella D and DellaCioppa A, Genetic Algorithms Applied to the Aerodynamic Design of Transonic Airfoils, J. Aircraft 32, 889891, 1995. 8. Poloni C, Ch. 20 of Genetic Algorithms in Eng. and Comp. Sci., ed. G. Winter et al., Wiley, 1995. 9. Yamamoto K, and Inoue O, Applications of Genetic Algorithms to Aerodynamic Shape Optimisation, AIAA-95-1650-CP, 1995 10. Tanese R, Distributed Genetic Algorithms, PhD thesis, U. Michigan, 1989. 11. Doorly D J, Peiro J, Kuan T, and Oesterle J-P, Optimisation of Airfoils Using Parallel Genetic Algorithms, in Proc. 15th Int. Conf. Num. Meth. Fluid Dyn., Monterey, 1996. 12. Nang J and Matsuo K, A Survey of Parallel Genetic Algorithms, J. SICE 33, 6, 500509, 1994. 13. Doorly D J, Peiro J, and Oesterle J-P, Optimisation of Aerodynamic and Coupled Aerodynamic-Structural Design using Parallel Genetic Algorithms, in Proc. Sixth AIAANASAISSMO Symposium on Multidisciplinary Analysis and Optimization, 401409, 1996. 14. Holland J H, Adaptation in Natural and Articial Systems, MIT Press, 1992. 15. Doorly D J and Peiro , Supervised parallel genetic algorithms in Aerodynamic Optimisation, AIAA paper 97-1852, 1997. 16. Oesterle J-P, Aeronautical optimisation using parallel genetic algorithms, MSc thesis, Aeronautics Dept.,Imperial College London, 1996. An Evolutionary Algorithm for Large Scale Set Covering Problems with Application to Airline Crew Scheduling Elena Marchiori1 and Adri Steenbeek2 Free University Faculty of Sciences, Department of Mathematics and Computer Science De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands 1 elena@cs.vu.nl CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands 2 adri@cwi.nl Abstract. The set covering problem is a paradigmatic NP-hard com- binatorial optimization problem which is used as model in relevant applications, in particular crew scheduling in airline and mass-transit companies. This paper is concerned with the approximated solution of large scale set covering problems arising from crew scheduling in airline companies. We propose an adaptive heuristic-based evolutionary algorithm whose main ingredient is a mechanism for selecting a small core subproblem which is dynamically updated during the execution. This mechanism allows the algorithm to nd covers of good quality in rather short time. Experiments conducted on real-world benchmark instances from crew scheduling in airline companies yield results which are competitive with those obtained by other commercial academic systems, indicating the e ectiveness of our approach for dealing with large scale set covering problems. 1 Introduction The set covering problem SCP is one of the oldest and most studied NP-hard problems cf. 14 . Given a m-row, n-column, zero-one matrix a , and an n-dimensional integer vector w , the problem consists of nding a subset of columns covering all the rows and having minimum total weight. A row i is covered by a column j if the entry a is equal to 1. This problem can be formulated as a constrained optimization problem as follows: ij j ij minimize P n j =1 wx j j subject to the constraints 8 x 2 f0 1g  : P =1 a x j n j ij j j = 1 : : :  n  1 i = 1 : : :  m: S . C a g n o n i e t a l. ( E d s .) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 6 7 − 3 8 1 , 2 0 0 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 3 6 8 E . M a rc h io ri a n d A . S te e n b e e k The variable xj indicates whether column j belongs to the solution xj = 1 or not xj = 0. The m constraint inequalities are used to express the requirement that each row be covered by at least one column. The weight wj is a positive integer that speci es the cost of column j . When all wj 's are equal to 1, then the SCP is called unicost SCP. Relevant practical applications of the SCP include crew scheduling 1, 2, 12, 15: nd a set of pairings having minimum-cost which covers a given set of trips, where a pairing is a sequence of trips that can be performed by a single crew. A widely used approach to crew scheduling works as follows. First, a very large number of pairings is generated. Next, a SCP is solved, having as rows the trips to be covered, and as columns the pairings generated. When this approach is used in mass-transit applications, very large scale SCP instances may arise, involving thousands of rows and millions of columns. The most successful heuristic algorithms for large scale SCP's are based on Lagrangian relaxation 13. Lagrangian relaxation is used to compute the score of a column according to its likelihood to be selected in an optimal solution. These scores are employed in simple greedy heuristics for computing a solution. A very eective heuristic algorithm for large scale SCPs based on this approach is 7. We refer the reader to 8 for a recent survey on exact and heuristic algorithms for SCP. All eective heuristics for large scale SCP's act on a subset of the columns, called core, which is selected before the execution of the algorithm. In the static approach the core remains the same during the execution cf. 6, 11, while in the dynamic approach it is updated using an adaptive mechanism e.g. 7, 10, 9. In this paper we propose a novel heuristic algorithm for large scale SCPs arising from crew sheduling problems in airline companies. At each iteration a near optimal cover is constructed using the information provided by the previous iterations to guide the search. The nal solution is the best cover found in all the iterations. Given a problem instance, the algorithm extracts an initial core from the set of columns given in the input. Then the algorithm consits of the iterated application of the following three steps: 1 First, an approximated solution to the actual SCP core is constructed by means of a novel greedy heuristic. 2 Next, a local search optimization algorithm is applied to the resulting solution. 3 Finally, some columns that occur in the best solution found in all iterations up to now are selected for forming the initial partial solution for the next iteration. The size of the core is determined by an adaptive size parameter, while the selection of a column is speci ed by a suitable merit criterion. During the execution, the score of the columns is modi ed as well as the size parameter, and the core is dynamically updated. This algorithm can be viewed as a hybrid 1 + 1 steady-state evolutionary algorithm, where at each iteration a child is generated from the parent using the above described heuristic, and the best between the parent and the child survives. A n E v o lu tio n a ry A lg o rith m fo r L a rg e S c a le S e t C o v e rin g P ro b le m s 3 6 9 In order to assess the performance of the algorithm, we conduct extensive experiments on real-world problem instances arising from crew scheduling in airlines, as well as on other benchmark instances from the literature. The results of the experiments are rather satisfactory: our algorithm is able to nd covers of very good quality in a short amount of time, yielding results which are competitive with those reported by the best industrial as well as academic methods for solving large set covering problems. The rest of the paper is organized as follows. In the next subsections we briey discuss some related work, and set up the notation and terminology used throughout the paper. In Section 2 we introduce the overall method and present in detail the four main modules of the algorithm. In Section 3 we report the results of extensive computational experiments. We conclude with some nal remarks on the present investigation and on future work. 1.1 Related Work An experimental comparison of the most e ective exact and heuristic algorithms for the weighted SCP is given in a recent paper by Caprara et al 8 . A rather e ective heuristic algorithm based on Lagrangian relaxation is the CFT algorithm 7 by Caprara et al. This algorithm has been tested also on large scale problem instances arising from crew scheduling in railway, yielding rather satisfactory results. In 15 a approximation algorithm for solving large 0-1 integer programming problems is proposed. This algorithm is used in the CARMEN system for airline crew scheduling, a industrial system used by several major airlines. Research based on evolutionary computation includes the following two papers. Beasley and Chu in 6 introduce a genetic algorithm for the SCP. The authors employ a representation where a chromosome is a bit string of lenght equal to the number of columns, one bit for each column, representing the set of columns whose bit in the string are equal to 1. The algorithm employs a heuristic repair mechanism for transforming infeasible chromosomes into solutions. Moreover, a core is used for constructing the chromosomes of the initial population. A genetic algorithm based on a non-binary representation has been proposed by Eremeev in 11 . Here a chromosome is a string of lenght equal to the number of rows, where the i-th entry contains the index of a column covering the ith row. As a consequence, all chromosomes are feasible solutions, thus they do not need to be repaired as in 6 . Moreover, heuristics are used for eliminating redundant columns as well as for dening the crossover operator. In Section 3 we will compare experimentally the above mentioned algorithms with the algorithm introduced in this paper. 1.2 Notation and Terminology In order to describe our method, we use the following terminology and notation. 3 7 0 E . M a rc h io ri a n d A . S te e n b e e k In the sequel, the indexes i j denote a generic row and column, respectively. A column will also be denoted by c, and a row by r, possibly subscripted. Moreover, S denotes a set of columns. Let cov S  be the set of rows that are covered by the columns in S : cov S  = fi j a = 1 for some j 2 S g: ij For simplicity, we write cov j  instead of cov fj g. We say that a column j is redundant with respect to S if cov S n fj g = cov S . A partial cover also called partial solution is a set of columns containing no redundant column. Let cov j S  be the set of rows which are covered by column j , but are not covered by any column in S n fj g: cov j S  = fi j a = 1 and a = 0 for all j 2 S n fj g g: Moreover, let min weighti be the minimum weight of the columns that cover i: min weighti = minimum fw j i 2 cov j  g: We can now de ne the function cov val , called cover value, which is used to evaluate a column j with respect to a partial cover S in order to select a column to be added resp. removed to resp. from S : 0 ij 0 ij j cov val j S  = X i2cov jS  min weighti: A convenient property of cov val is that cov val j S  = cov val j S n fj g. This allows one to compute the cover value of a column without taking into account whether it belongs to the partial solution S or not. Moreover, we can characterize the redundancy of a column by means of the condition cov val j S  = 0. The cover value is used to de ne the selection value sel val j S  of a column j with respect to the partial cover S :  if j redundant wrt S , sel val j S  = Lim w =cov val j S  otherwise. The selection value of redundant columns is set to a very big constant Lim. In this way, redundant columns do not have any chance of being selected. j 2 The Overall Method The algorithm we propose consists of an iterated procedure, where each iteration generates an approximated solution using only columns from the actual core. Roughly, at each iteration a greedy heuristic is used to construct incrementally a cover starting from a partial cover: in the rst iteration the partial cover is empty, while in the following iterations the partial cover is a proper subset of the best cover found in all iterations up to now. The cover found after the A n E v o lu tio n a ry A lg o rith m fo r L a rg e S c a le S e t C o v e rin g P ro b le m s 3 7 1 application of the greedy heuristic is given as input to an optimization procedure which tries to improve the partial solution. The core is updated from time to time during the execution. The nal result is the best cover found in all iterations. The corresponding algorithm WSCP Weighted Set Covering Problem is illustrated below in pseudo-code, where Sbest represents the best cover found so far, S denotes the actual partial P cover, and valueS is the sum of the weights of the columns of S, that is, j2S wj . Therefore the optimal cover is the cover S having minimum valueS. FUNCTION WSCP BEGIN RECOMPUTE_CORE Sbest -  1..ncol  S -   FOR 1 .. param.number_of_iterations DO IF  core_selection  RECOMPUTE_CORE ENDIF S - GREEDYS S - OPTIMIZES IF  valueS = valueSbest  THEN Sbest - S ENDIF S - SELECT_PARTIAL_COVERSbest ENDFOR RETURN Sbest END 2.1 Greedy Heuristic Our greedy heuristic GREEDY is described in pseudo-code below. Lines starting with "==" are comments. The algorithm constructs a solution a cover, starting from a possibly empty partial cover S . Columns are added resp. removed to resp. from S until S covers all the rows. extend S until it is a cover: FUNCTION GREEDY var S  BEGIN WHILE  S is not a cover  DO select and add one column to S S - S + select_add remove 0 or more columns from S WHILE  remove_is_okay  DO S - S - select_rmv ENDWHILE ENDWHILE S is a cover, without redundant columns return S END 3 7 2 E . M a rc h io ri a n d A . S te e n b e e k The function select add selects a column j not in S having minimum selection value sel val j S . The test remove is okay determines whether columns should be removed from S . If S is empty it returns false if S contains at least one redundant column then it returns true otherwise, with probability param:p rmv typical value 0:3 it returns true, otherwise false. Finally, the function select rmv selects a column in S having maximum selection value. 2.2 Local Optimization The local optimization procedure OPTIMIZE is based on the following idea. Given a cover S , suppose there is a column j 62 S such that S  fj g contains at least two columns other than j , say j1  : : :  j , with l  2 that are redundant, and of their weights is greater than the weight of j , that is, P such wthatk thew .sum Then S n fj1  : : :  j g  fj g is a better cover than S . In this =1 case we call j a superior column. The gain of j is de ned by l k :::l j j l X gain j  = k =1 w k ,w : j j :::l So a best superior column is the one having highest gain. Note that the optimization procedure operates on a cover containing no redundant columns. The optimization algorithm OPTIMIZE in pseudo-code is given below. S is a cover, without redundant columns FUNCTION OPTIMIZE var S  BEGIN Sup - select_superior WHILE  Sup not empty DO select best column from Sup best - select_best Sup - Sup - best add superior and remove redundant columns from S IF  best superior  S - S + best S - S - select_redundant ENDIF ENDWHILE S is a cover, without redundant columns return S END First, the function select superior  is used, which generates the list Sup consisting of all the superior columns ordered in decreasing order according to their gain. Next, the list Sup is scanned in the WHILE loop. At each iteration, the head of Sup is removed and memorized in the variable best using A n E v o lu tio n a ry A lg o rith m fo r L a rg e S c a le S e t C o v e rin g P ro b le m s 3 7 3 the function select best. If the selected column is still superior that is if the test best superior is satised then it is added to S , and the set of redundant columns are removed from the resulting partial cover S using the function select redundant. 2.3 Restoring Part of the Actual Best Solution In the rst iteration of WSCP the heuristic GREEDY constructs a cover starting from the empty set in the following iterations, GREEDY builds a cover starting from a subset of the best cover found so far. For a column j , we keep track of the number chosen j  of times that j has been part of a best solution. The function SELECT PARTIAL COVER considers the set E of so-called elite columns, consisting of those columns j of the best solution Sbest such that cov val j Sbest  wj . Then SELECT PARTIAL COVER selects from E the set of columnsPhaving low chosen j  in our implementation chosen j  has to be smaller than j chosen j =neli 10, where neli is the number of elements of E  while the remaining columns of E are selected with a probability that is set to a random value between 0:1 and 0:9. 2.4 Selecting the SCP Core This is a fundamental step in the design of an algorithm for dealing with large SCP instances. We introduce the following method for constructing an SCP core, which has been implemented in the function RECOMPUTE CORE. The SCP cover is constructed from the empty set by incrementally adding columns according to the following criterion. Columns are selected in increasing order according to their selection value. Suppose column j has been selected: 1. if j is an elite column then with probability close to 1 it is added to the actual SCP core 2. otherwise, j is added if there exists a row i such that j covers i and wj  min weighti  K0 , with K0 a given constant real value greater or equal than 1 3. otherwise, j is added if there exists a row i such that j covers i and i is covered by less than K1 columns of the actual SCP core, with K1 a given constant integer value greater or equal than 1. Note that K0  K1 are parameters which are chosen depending on the class of problems one considers. Condition 3 implies that the SCP core contains for each row, at least the rst K1 best columns according to the ordering induced by the selection value function that cover that row. The function core selection determines when the actual SCP core has to be recomputed. In our implementation, we recompute the SCP core every 100 iterations of WSCP. During the execution of GREEDY, when ninety per cent of a cover has been constructed, the min weight of those rows that are not yet covered is increased 3 7 4 E . M a rc h io ri a n d A . S te e n b e e k by a small quantity in our implementation min weighti is multiplied by 1:1. This aects the selection value of the columns, hence their order of selection in the construction of the SCP core changes during the execution of the overall algorithm WSCP. 3 Experimental Evaluation The algorithm WSCP has been tested on large set covering problems arising from crew scheduling applications in various airline companies. Moreover, we have considered the weighted SCP instances from the OR library maintained by J.E. Beasley 1 . These instances are considered standard benchmarks for testing the eectiveness of exact and heuristic algorithms for the SCP. In particular, they have been used in 8 for comparing experimentally various exact and heuristic algorithms for SCP. WSCP has been implemented in C++. The algorithm was run on a Sun Ultra 10 UltraSPARC-IIi 300MHz. The results of the experiments are based on 10 runs on each problem instance of the OR Library, and on 5 runs on the other instances. In each table, the entry labeled Id contains the name of the problem instance. The label BK denotes the best known solution for that instance Bst denotes the best result found by the algorithm Fbst indicates the frequency of obtaining the best solution runs Apd denotes the average percentage deviation Pk=110 zink ,thez performed =10  z   100, where zk is the solution found in the k-th run, and z is the optimal or best known solution. Tbst denotes the average cpu time for obtaining the best solution Bst, while Tsol denotes the average cpu time for nding a solution. Finally, Ibst and Isol denote the average number of iterations of obtaining the best solution Bst, and a solution, respectively. 3.1 Experiments on Airline Crew Scheduling Problem Instances We consider three sets of benchmark instances from real-world airline crew scheduling problems. A set of instances from a major airline company, here called AIR instances, the airline scheduling instances from Wedelin 15 , and the instances from Balas and Carrera 3 . The characteristics of these problems are reported in Tables 2, 3, and 7, respectively. Observe that in many instances, like, e.g., the Wedelin instances, the weights of the columns are very large numbers, because the weight represents the cost of a pairing and takes into account several factors. We compare experimentally WSCP with the industrial system used by an airline company on the AIR instances, with the CFT algorithm by Caprara et al 7 , and with the Wedelin algorithm 15 . The results of the experiments are given in Tables 2, 4, 5, and 7. Note that the results for the Wedelin and CFT algorithms are taken from the paper 7 , where the cpu time is estimated in DECstation 1 see http:mscmga.ms.ic.ac.ukjeborlibscpinfo.html A n E v o lu tio n a ry A lg o rith m fo r L a rg e S c a le S e t C o v e rin g P ro b le m s 3 7 5 5000240 CPU seconds. Only the value of the best solution is reported. For the CFT algorithm, the time for nding the best solution is given, while for the Wedelin algorithm, only the overall execution time Texe of the algorithm is reported. The authors do not specify the setting of the various parameters in their algorithms, and the total number of trials performed. Id Rows Columns Density  Weight Range A01 A02 A03 A04 A05 A06 A07 A08 A09 A10 A11 A12 258303 0.167 1319-35302 19441 0.135 1437-37206 40580 0.092 1337-37148 79481 0.123 1460-37142 72377 0.126 1411-37251 23741 0.135 1437-37037 32363 0.15 1319-36370 45286 0.18 1345-36370 50047 0.19 1361-36370 49525 0.18 1344-36370 389388 5.55 1800-18768 642613 1.45 1630-19000 Table 1. Characteristics of AIR instances Id Industry Bst A01 16351667 A02 12879297 A03 15663720 A04 16110608 A05 16315241 A06 13162511 A07 13301520 A08 13510606 A09 13489489 A10 13571530 A11 247775 A12 732587 5265 3878 4965 4916 4656 1971 4203 4320 4287 4369 150 682 WSCP Bst Fbst Apd Tbst Tsol Ibst Isol 16351667 1.0 0.0 550.9 550.9 919.8 919.8 12879297 1.0 0.0 131.0 131.0 822.4 822.4 15663688 1.0 0.0 254.2 254.2 1004.4 1004.4 16110608 1.0 0.0 363.1 363.1 1085.0 1085.0 16315070 0.3 0.0001 923.5 848.1 3501.6 3203.2 13162511 1.0 0.0 156.9 156.9 907.6 907.6 13301520 1.0 0.0 200.9 200.9 945.6 945.6 13510584 1.0 0.0 254.4 254.4 946.2 946.2 13489489 1.0 0.0 235.4 235.4 944.2 944.2 13571530 1.0 0.0 237.3 237.3 933.8 933.8 247775 1.0 0.0 224.2 224.2 1087.8 1087.8 732587 0.3 0.11 1064.9 896.8 1460.6 1167.4 Table 2. Results for AIR instances On three AIR instances WSCP found a solution which is better than the best solution found by the industrial system, while on the other instances WSCP found solutions of equal value as those found by the industrial system. 3 7 6 E . M a rc h io ri a n d A . S te e n b e e k Id B727scratch ALITALIA A320 A320coc SASjump SASD9imp2 Rows Columns Density  Weight Range 29 157 8.2 1600-11850 118 1165 3.1 2200-2110900 199 6931 2.3 1600-2111450 235 18753 1.9 1900-1812000 742 10.370 0.6 4720-55849 1366 25032 0.3 3860-35200 Table 3. Characteristics of Wedelin instances Id CFT Wedelin Bst Tbst Bst Texe B727scratch 94400 94.400 0.3 94400 4.7 ALITALIA 27258300 27258300 6.2 27258300 37.2 A320 1262100 1262100 79.5 1262100 216.9 A320coc 14495500 14495600 577.8 14495500 1023.7 SASjump 7338844 7339537 396.3 7340777 806.8 SASD9imp2 5262190 5263640 2082.1 5262190 1579.7 Table 4. Results of CFT and Wedelin on Wedelin instances Id BK BK WSCP Bst Fbst Apd Tbst Tsol Ibst Isol B727scratch 94400 94400 1.0 0.0 0.018 0.018 38.4 38.4 ALITALIA 27258300 27258300 1.0 0.0 0.63 0.63 106.8 106.8 A320 1262100 1262100 1.0 0.0 17.34 17.34 326.2 326.2 A320coc 14495500 14495500 0.2 0.0006 651.08 446.20 3494.5 2402.0 SASjump 7338844 7339541 0.1 0.02 269.3 200.98 4635.0 3454.6 SASD9imp2 5262190 5263590 0.1 0.04 741.9 608.452 4603.0 3671.4 Table 5. Results of WSCP on Wedelin instances A n E v o lu tio n a ry A lg o rith m fo r L a rg e S c a le S e t C o v e rin g P ro b le m s Id AA03 AA04 AA05 AA06 AA11 AA12 AA13 AA14 AA15 AA16 AA17 AA18 AA19 AA20 BUS1 BUS2 Rows Columns Density  Weight Range 106 8661 4.05 91-3619 106 8002 4.05 91-3619 105 7435 4.05 91-3619 105 6951 4.11 91-3619 271 4413 2.53 35-2966 272 4208 2.52 35-2966 265 4025 2.60 35-2966 266 3868 2.50 35-2966 267 3701 2.58 35-2966 265 3558 2.63 35-2966 264 3425 2.61 35-2966 271 3314 2.55 35-2966 263 3202 2.63 35-2966 269 3095 2.58 35-2966 454 2241 1.89 120-877 681 9524 0.51 120-576 Table 6. Characteristics of Balas and Carrera instances CFT WSCP Id Bst Tbst Bst Fbst Apd Tbst Tsol Ibst Isol AA03 33155 61.0 33155 1.0 0.0 1.26 1.266 40.0 40.0 AA04 34573 3.6 34573 1.0 0.0 1.73 1.73 74.6 74.6 AA05 31623 3.1 31623 1.0 0.0 0.48 0.48 9.6 9.6 AA06 37464 5.2 37464 1.0 0.0 2.67 2.67 128.2 128.2 AA11 35384 193.7 35384 1.0 0.0 19.11 19.11 755.4 755.4 AA12 30809 53.8 30809 1.0 0.0 7.88 7.88 350.8 350.8 AA13 33211 8.3 33211 1.0 0.0 2.32 2.32 103.8 103.8 AA14 33219 30.3 33219 1.0 0.0 11.74 11.74 557.8 557.8 AA15 34409 18.8 34409 1.0 0.0 8.92 8.92 485.6 485.6 AA16 32752 33.6 32752 1.0 0.0 4.63 4.63 257.4 257.4 AA17 31612 10.9 31612 1.0 0.0 4.69 4.69 262.2 262.2 AA18 36782 13.5 36782 0.1 0.01 17.1 6.94 1108.0 433.0 AA19 32317 5.9 32317 1.0 0.0 2.73 2.73 175.4 175.4 AA20 34912 13.6 34912 1.0 0.0 4.76 4.76 318.4 318.4 BUS1 27947 5.0 27947 1.0 0.0 8.19 8.19 382.6 382.6 BUS2 67760 19.2 67760 1.0 0.0 37.24 37.24 616.2 616.2 Table 7. Results of CFT and WSCP on Balas and Carrera instances 3 7 7 3 7 8 E . M a rc h io ri a n d A . S te e n b e e k On the instances from Wedelin the performance of WSCP is comparable to the one of the CFT and Wedelin algorithms. Finally, on the instances from Balas and Carrera, both WSCP and CFT are always able to nd the optimal solution. In the AA instances WSCP is faster that CFT, while in the BUS instances CFT nds the optimum in a shorter time. The results of the experiments indicate that WSCP is a rather powerful tool for solving large real-life airline crew scheduling problems. 3.2 Experiments on the OR Library SCP Instances We consider the families A-D from 4, and the NRE-NRH from 5, consisting of randomly generated SCP instances. Each class contains 5 instances. The values of the characteristic parameters of these problem classes, like number of rows and columns, are given in Table 8. We compare experimentally WSCP with the genetic algorithms by Beasley and Chu 6, and by Eremeev 11, and with the CFT algorithm by Caprara et al 7. The results of the experiments are summarized in Tables 9, 10, and 11. The results for the CFT, Beasley Chu, and Eremeev algorithms are from 11. In particular, the cpu time is estimated in 100MHz Pentium CPU seconds. All the algorithms are able to solve the instances of the classes A-D. On these instances, WSCP seems to have a more robust behaviour that the two genetic algorithms, nding the optimum in each of the 10 trials. The performance of WSCP on the other problem instances of classes E-H is rather satisfactory, both in terms of quality of the solutions as well as running time. On each instance, WSCP is able to nd the optimum or best known solution, while the two genetic algorithms BC and Er do not nd the optimum value on instances H1 and H2. Moreover, WSCP nds the solutions for instances in the harder classes G and H in a much shorter time than all the other algorithms. Id Rows Columns Density  Weight Range A B C D E F G H 300 300 400 400 500 500 1000 1000 3000 3000 4000 4000 5000 5000 10000 10000 2 5 2 5 10 20 2 5 1-100 1-100 1-100 1-100 1-100 1-100 1-100 1-100 Table 8. Characteristics of Classes A, B, C, D 4 Conclusion In this paper we have introduced a novel heuristic method for solving large weighted set covering problems. The results of the experiments indicate that WSCP is able to nd covers of satisfactory quality in short running time. A n E v o lu tio n a ry A lg o rith m Id CFT Tbst A 47.15 B 3.34 C 29.23 D 7.64 fo r L a rg e S c a le S e t C o v e rin g P ro b le m s 3 7 9 Beasley Chu Fbst Apd Tsol 0.86 0.20 65.98 1.00 0.00 68.63 0.68 0.41 87.93 0.96 0.06 101.70 Eremeev WSCP Fbst Apd Tbst Tsol Fbst Apd Tbst Tsol Ibst Isol 0.44 0.35 82.00 71.8 1.00 0.00 0.98 0.98 108.0 108.0 1.00 0.00 20.80 20.80 1.00 0.00 0.30 0.30 7.8 7.8 0.74 0.26 53.50 52.40 1.00 0.00 0.76 0.76 72.6 72.6 0.94 0.08 26.62 23.33 1.00 0.00 0.40 0.40 26.0 26.0 Table 9. Results for Classes A, B, C, D Id BK CFT Beasley Chu Eremeev Bst Tbst Bst Fbst Apd Tsol Bst Fbst Apd Tbst Tsol E1 29 29 11.5 29 1.0 0.0 16.9 29 1.0 0.0 1.0 1.0 E2 30 30 180.5 30 0.4 2.0 266.9 30 1.0 0.0 94.8 94.8 E3 27 27 41.7 27 0.3 2.6 85.1 27 1.0 0.0 23.1 23.1 E4 28 28 11.6 28 1.0 0.0 238.5 28 1.0 0.0 11.0 11.0 E5 28 28 16.2 28 1.0 0.0 15.5 28 1.0 0.0 2.1 2.1 F1 14 14 14.7 14 1.0 0.0 33.8 14 1.0 0.0 7.9 7.9 F2 15 15 13.8 15 1.0 0.0 34.5 15 1.0 0.0 1.3 1.3 F3 14 14 110.0 14 1.0 0.0 117.9 14 1.0 0.0 55.4 55.4 F4 14 14 13.7 14 1.0 0.0 92.6 14 1.0 0.0 20.4 20.4 F5 13 13 89.0 13 0.3 5.4 67.1 13 0.3 5.4 497.4 151.2 G1 176 176 65.0 176 0.2 1.0 451.3 176 0.7 0.3 115.0 96.0 G2 154 154 346.6 155 0.5 1.5 159.3 154 0.5 0.65 318.3 226.6 G3 166 166 432.7 166 0.1 1.1 312.1 166 0.1 0.8 627.6 319.1 G4 168 168 105.0 168 0.2 1.4 665.4 168 0.4 0.7 160.0 172.5 G5 168 168 105.0 168 0.2 0.8 242.6 168 0.7 0.05 161.2 170.4 H1 63 63 642.1 64 1.0 1.6 743.0 64 1.0 1.6 90.5 90.5 H2 63 63 392.5 64 1.0 1.6 234.3 64 1.0 1.6 34.7 34.7 H3 59 59 690.4 59 0.9 0.2 796.6 59 1.0 0.0 493.2 493.2 H4 58 58 105.1 58 0.4 91.6 62.9 58 1.0 0.0 218.2 218.2 H5 55 55 68.8 55 0.9 0.2 198.6 55 1.0 0.0 25.2 25.2 Table 10. Results of CFT, Beasley and Chu, and Eremeev on Classes E, F, G, H 3 8 0 E . M a rc h io ri a n d A . S te e n b e e k Id BK E1 29 E2 30 E3 27 E4 28 E5 28 F1 14 F2 15 F3 14 F4 14 F5 13 G1 176 G2 154 G3 166 G4 168 G5 168 H1 63 H2 63 H3 59 H4 58 H5 55 WSCP Bst Fbst Apd Tbst Tsol 29 1.0 30 1.0 27 1.0 28 1.0 28 1.0 14 1.0 15 1.0 14 1.0 14 1.0 13 1.0 176 1.0 154 0.5 166 0.2 168 0.4 168 0.9 63 0.2 63 1.0 59 0.3 58 0.8 55 1.0 Table 11. Results Ibst Isol 0.0 1.8 1.8 2 2 0.0 2.7 2.7 62.9 62.9 0.0 2.3 2.3 48.3 48.3 0.0 2.1 2.1 31.1 31.1 0.0 1.8 1.8 5.0 5.0 0.0 3.6 3.6 19.6 19.6 0.0 3.6 3.6 9.0 9.0 0.0 6.2 6.2 153.2 153.2 0.0 3.6 3.6 13.1 13.1 0.0 34.1 34.1 2061.1 2061.5 0.0 2.2 2.2 29.7 29.7 1.1 8.9 4.0 315 107.9 0.6 30.2 14.9 1433.5 640.3 0.8 18.4 25.0 812.3 1114.7 0.6 5.9 5.7 207.7 197.4 1.1 9.5 11.7 161.5 269.2 0.0 50.4 50.4 1872 1872 1.1 25.2 21.6 778.3 678.9 0.3 28.1 23.9 1016.3 834.1 0.0 5.5 5.5 64.1 64.1 of WSCP on Classes E, F, G, H In all the experiments we have worked with a core which is a proper subset of the set of all columns. The size of the core depends on the problem instance. However, in general a small fraction which varies from 10 per cent to 50 per cent of the set of columns is used as core. Using small covers helps the e ciency of the algorithm. Moreover, extensive experiments with di erent core sizes have revealed a somehow counter intuitive phenomenon: in many instances, the quality of the results become worse by using a larger core, even if the same number of iterations is used. This seems to indicate that the merit criterion used in WSCP is not the best possible, because it can make the wrong decision when all the columns are present in the core. We are actually investigating the use of alternative merit criteria and their relationship with the selection of the core. Future work concerns the investigation of how to tune automatically the parameters 0 1 for determining the core problem, and how the value of thesecan be adaptively change during the execution. K K Acknowledgements We would like to thank Thomas Baeck and Martin Schuetz for interesting discussions on the subject of this paper. A n E v o lu tio n a ry A lg o rith m fo r L a rg e S c a le S e t C o v e rin g P ro b le m s 3 8 1 References 1. E. Andersson, E. Housos, Kohl, and D. Wedelin. Crew pairing optimization. In Operation Research in the Airline Industry. Kluwer Scientic Publishers, 1997. 2. J.P. Arabeyre, J. Fearnley, F.C. Steiger, and W. Teather. The airline crew scheduling problem: A survey. Transportation Science, 3 :140163, 1969. 3. E. Balas and M.C. Carrera. A dynamic subgradient-based branch-and-bound procedure for set covering problem. Operations Research, 44:875890, 1996. 4. J.E. Beasley. An algorithm for set covering problem. European Journal of Operational Research, 31:8593, 1987. 5. J.E. Beasley. A lagrangian heuristic for set covering problems. Naval Research Logistics, 37:151164, 1990. 6. J.E. Beasley and P.C. Chu. A genetic algorithm for the set covering problem. European Journal of Operational Research, 94:392404, 1996. 7. A. Caprara, M. Fischetti, and P. Toth. A heuristic method for the set covering problem. In W.H. Cunningham, T.S. McCormick, and M. Queyranne, editors, Proc. of the Fifth IPCO Integer Programming and Combinatorial Optimization Conference. Springer-Verlag, 1996. 8. A. Caprara, M. Fischetti, and P. Toth. Algorithms for the set covering problem. Technical report, DEIS Operation Research Technical Report, Italy, 03 1998. 9. S. Ceria, P. Nobili, and A. Sassano. A Lagrangian-based heuristic for large-scale set covering problems. Mathematical Programming, 1995. to appear. 10. H.D. Chu, E. Gelman, and E.L. Johson. Solving large scale crew scheduling problems. European Journal of Operational Research, 97:260268, 1997. 11. A.V. Eremeev. A genetic algorithm with a non-binary represenation for the set covering problem. In Proc. of OR'98, pages 175181. Springer-Verlag, 1998. 12. M.M. Etschmaier and D.F. Mathaisel. Airline scheduling: An overview. Transportation Science, 19 :127138, 1985. 13. M.L. Fisher. An application oriented guide to Lagrangian relaxation. Interfaces, 15 2 :1021, 1985. 14. M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco, 1979. 15. D. Wedelin. An algorithm for large scale 0-1 integer programming with application to airline crew scheduling. Annals of Operational Research, 57:283301, 1995. Design, Implementation, and Application of a Tool for Optimal Aircraft Positioning J. Pfalzgraf 1 , K. Frank 1 , J. Weichenberger 1 , S. Stolzenberg 2 1 Department of Computer Science,University of Salzburg jpfalz@cosy.sbg.ac.at, 2 Deutsche Lufthansa AG, FrankfurtMain Siegfried.Stolzenberg@dlh.de Abstract. Optimal positioning of aircraft at a specic airport is a very dicult problem involving the modeling of many constraints. Lufthansa AG formulated this problem eld for the airport FrankfurtMain. In this contribution we describe the development of a tool for nding solutions to positioning problems automatically. Our approach consists of two parts. A generic airport model is developed where the notion of logical berings plays a basic role. The optimization task is treated by application of modied and extended genetic algorithms. A system has been implemented which is capable of computing concrete positioning plans that can be used by a human operator for further processing. This leads to a considerable speed up in the generation of positioning plans for aircraft in comparison with the former method. The application of the aircraft positioning tool to a real world scenario airport Frankfurt is brie y presented. Keywords: optimal aircraft positioning, logical berings, genetic algorithms, hybrid problem solving 1 Introduction This contribution deals with the general problem of optimal positioning of aircraft at an airport. The problem formulation has been provided by Lufthansa AG for the concrete case of airport FrankfurtMain. The main task in the eld of aircraft positioning is to nd an optimal schedule for all incoming and outgoing aircraft with respect to their position at corresponding gates of an airport. One has to take into account many constraints and requests, such as neighborhood relationships, runway crossings, critical passenger connections, aircraft types, airline requests, special gates, security constraints, and others, depending on particular situations. The basic task is to fulll all these constraints and requests in an optimal way so that the yield converges to a predened maximum. Our work consists of two main parts. The development of a generic airport model" not depending on the choice of a specic airport and thus reusable and the application of extended and modied genetic algorithms to work on the optimization problem. A rather complicated tness function cost function S . C a g n o n i e t a l . ( E d s . ) : E v o W o r k s h o p s 2 0 0 0 , L N C S 1 8 0 3 , p p . 3 8 3 2 , − 2 30 90 0 .  S p rin g e r-V e rla g B e rlin H e id e lb e rg 2 0 0 0 D e s i g n , I m p l e m e n t a t i o n , a n d A p p l i c a t i o nof a Tool for Optimal Aircraft Positioning 3 8 3 has been developed in cooperation with Lufthansa. In establishing the airport model, the concept of logical berings plays a basic role. This is a logical modeling approach which allows to describe what we call the logical state space of a virtual airport distribution of logics. Thus, for example, a gate and a position has its own local" logic, respectively, and they can communicate via non-classical logical operations called transjunctions. In principle, it would be possible to apply methods from multi-agent systems MAS. In our case a gate or an airplane position could be interpreted as an agent, respectively. Application of MAS techniques is intended future work. In terms of logical berings an agent" corresponds to a local ber". For the treatment of the optimization problem we develop and apply modied and extended genetic algorithms. A suitable tness function has been devised which encodes the basic information and constraints underlying the positioning task. A prototype of an aircraft positioning tool has been implemented. A detailed description of the generic airport model, the evolutionary optimization approach, the system design and real world applications can be found in FW99 . As far as we know no such approach to treat the problem eld as described above has been tried before. This has been conrmed by somebody from the EvoNet project. And no such implemented system existed before, as Lufthansa states. There is a considerable increase of performance by our system in comparison with the procedure in use so far. Our PC-based system needs about one hour to calculate a positioning plan which can then be used by a human operator to process it further. In contrast to that, with the former method, the human planner needed about one week to establish a similar plan by hand" supported by a graphical computing system, interactively. 2 On logical berings The purpose of this section is twofold: to give a very brief introduction to the elementary notions of logical berings for later use and to bring them to the attention of the reader. The concept of logical berings originates in an industrial project  a case study on so-called polycontextural logics and their possible applications to complex communication and information systems. Subsequently we present only the elementary notion of a logical bering and point to practical applications in the area of cooperating agents in particular cooperating robots scenarios. A detailed introduction to logical berings with background information and motivating comments can be found in Pfa91 . The notion of a logical bering is inspired by the mathematical modeling language of ber bundles. This very expressive and powerful notion integrates dierent structures, namely geometric, topological, and algebraic structures. A ber bundle consists of a whole bunch of bers which form a so-called total space ditributed over a base manifold. Each ber is mapped via a projection map onto its base point in the base space. Thus, for example, in a vector bundle the typical ber is a vector space of a given dimension. So we can say that in a ber bundle vertically one does algebra and horizontally geometry, topology. 3 8 4 J . P fa lz g ra f e t a l. Concerning a logical bering, now the idea is to use a ber bundle and take logics as bers, thus, vertically one does logic. In Pfa91 we take as typical ber a classical two valued logic. Thus a logical bering is an abstract ber space or bundle with typical ber F = L, a classical rst order logical space or a sub- space generated by a set of formulas. The base space B of the bering will often be denoted by B = I , the indexing set, the total space is denoted by E , and  : E ,! I is the corresponding projection map. For i 2 I we have the ber  ,1 i over i, namely  ,1 i = fx 2 E j x = ig. It has the structure of a logic as mentioned above. We note that E is decomposed into the bers ,1 i for all i 2 I. The simplest form of a bering is the "trivial bering" having total space E = I  L, base space I and  i l = i. Therefore the ber over i 2 I is L :=  ,1 i = fig  L. Such a trivial bering is a parallel system" of logics L over the index set I as base space. We can think of reasoning processes running in parallel within each ber L = ,1 i . A ber L is interpreted as a local logical system a subsystem of the whole bering . Transition communication betweeen bers is described with the help of suitable maps cf. Pfa91 . Such a trivial bering as previously considered will also be called "free parallel system" with total space the disjoint union of the bers L . Each subsystem L has local classical truth values  = fT  F g. The global set of truth values is denoted by  I . In a free parallel system  I is just the disjoint union of the i for i 2 I . Logical connectives can be introduced by taking "berwise" logical operations. For example, one can form logical expressions like the following for a system with three bers using vector notation : x1 ^ y1  x2  y2  x3 _ :y3 , etc.. For more details we refer to Pfa91. A special nonclassical bivariate operation arises naturally: a local pair xi  yi in Li  Li , i 2 I , can be mapped into dierent subsystems L  L  :::: Taking truth values as input we can observe that for the four possible input pairs in i  i for a locally dened bivariate operation there can be maximally four dierent subsystems where that function can be evaluated. We can say that the values will be distributed over the subsystems. Such an operation is called transjunction. Below we give an example of a conjunctional" transjunction  we just display the truth table i i i i i i i i i T0 F0 T0 T F  F0 F F Every transjunction can be described by such a table or T-F-pattern" together with the indices f   g. If f   g 6= fig then we obtain a transjunction  in the previous example we have the type of a conjunction as can be seen by omitting the indices. If f   g = fig then we have a classical conjunction remaining in subsystem Li . In work cited below, we introduced generalized transjunctions having more than 4 input pairs. The example which we present in section 3 deals with such a generalized trunsjunction. Design, Implementation, and Application of a Tool for Optimal Aircraft Positioning 3 8 5 We applied transjunctions for the logical control of cooperating robots scenarios. Cf.Pfa97 for a brief discussion of such an application. For further information on the subject we refer to PSS96a, PSS96b. In the framework of this contribution here we apply methods from logical berings to support logically the modeling of the aircraft positioning system as described subseqently. Generally spoken, we consider the concept of logical berings as a natural logical modeling approach for multi-agent systems MAS . This has been discussed in Mei99. Future work is planned, especially with respect to an extended generic airport model using MAS techniques. 3 Development of an aircraft positioning system 3.1 Design concept of the positioning tool First of all, we want to give a short problem description and explain what we mean by aircraft positioning. The main task of aircraft positioning is to nd schedules for all incoming and outgoing aircraft at an airport. Di cult constraints have to be considered, like runway crossings, optimal passenger connections, aircraft types, special ights, particular gates, airline requests, security problems, and others. The main problem in the eld of aircraft positioning is to nd an optimal schedule for all incoming and outgoing aircraft with respect to their positions and corresponding gates. Three subtasks have to be distinguished: long-time, short-time and the actual day scheduling. To each subtask corresponds an individual knowledge about the aircraft which inuences the positioning. The knowledge changes rapidly during scheduling. The external state of the airport can change rapidly too, caused by construction work, for example. Thus an important design objective is to build a tool which is able to react to rapidly changing situations. As previously mentioned, we decided to choose logical berings as a logical modeling approach. This decision was naturally motivated by our problem analysis. Figure 1 shows the global system design. Our system includes various agencies and data areas, namely the kernel agency, the airport agency and the external data areas, like airport database, temporary ightplan. Another part of our system is the output unit and the output communication unit, which will be used to visualize an airport utilization. The communication between the various parts of our model will be handled by a negotiation protocol. This allows us to handle the basic communication in our system for the short-time planning task. An augmented nal version of the tool will be able to treat the two remaining cases too, namely long time planning and the actual day. 3.2 External data areas The external data areas are specialized data storages for, e.g. airport description, ight characteristics, airport characteristics , airline characteristcs. We use an airport database, which includes basic airport information like the number of 3 8 6 J . P fa lz g ra f e t a l. Fig. 1. Global system design Design, Implementation, and Application of a Tool for Optimal Aircraft Positioning 3 8 7 positions and gates. Besides that there are the infoserver, which is our main data source for ight information, the airline preferences, the ight specialities and the gate and position specialities. All these areas contain positioning relevant data which are necessary to calculate an optimal solution for the positioning problem. 3.3 Airport Agency The airport agency is used to model a virtual airport. The virtual airport model can be applied to an existing airport, in our case airport FrankfurtMain. The design of the virtual airport is based on the concept of logical berings. The airport agency, respectively the virtual airport, consists of serveral units. Two units are interfaces for communication with the enviroment and all others are internal units. First, there is the precalculating ltering hierarchical list of positions and gates p.f.H.L.P.G and second the output communication unit O.C.U which will not be discussed in this contribution. These two units manage the data transfer with the enviroment and prepare input data for the internal use in the airport agency and the kernel agency which will be decribed later in this contribution. Figure 2 gives an overview of the design of the airport agency with their units and the internal communication paths. Fig. 2. Design of the Airport Agency The airport agency, shown in Figure 2, includes a number of virtual clusters, e.g. "AWEST", "AOST" or "BWEST". These clusters include the bers of our 3 8 8 J . P fa lz g ra f e t a l. airport. We use two dierent types of bers one for positions and the other one for gates because they have a lot of dierent characteristics. Some typical characteristics of a position ber are the maximum valid wing code or the buertime, on the other side some typical characteristics of a gate ber are the maximum number of allowed passengers in the gate area or the time to bring the passengers to the aircraft. In terms of logical berings to each agent" corresponds a local" ber. These bers are connected via communication paths with the p.f.H.L.P.G. and they can have further connections with their neighbors. The connections communication with neighbors are modeled by transjunctions. A transjunction can be used to control the state spaces of neighbored connected bers. A special eect in an application of a transjunction can be described as follows: if a local ber A corresponds to, for example, an aircraft or group of passengers GoPax , and there is a transjunction from A to a ber B, then the state space of ber B will be downgraded by the transjunction. This downgrade of the state space of ber B will be cancelled again as soon as ber A is no longer used by attached to" an aircraft or GoPax. Example:Transjunction rule 'A10' to 'A12' if 11 then 5 The eect of this transjunction is, that the state space of 'A12' will be downgraded to SWC 5, if the current SWC of 'A10' is 11. Here SWC is the short notation for wing code". One has to take into account that dierent wing codes have to be distinguished there exist priorities which must be taken into consideration. The complete transjunction corresponding to this example has 3 truth values in the local ber A and 12 values in local ber B and therefore represents a generalized transjunction in the sense of section 2. The complete truth table of the transjunction is displayed in FW99. Figure 3 shows a special cluster with its communication paths and neighborhood relationships. Moreover, the complexity of the state space of an agent depends on the number of allowed aircraft types for that agent. 3.4 Kernel Agency The kernel agency includes currently 4 dierent algorithms, the adaptive longtime scheduling algorithm, the genetic short-time scheduling algorithm, the random scheduling algorithm, the conict solving algorithm. Another part of the kernel is the kernel communication unit. In this contribution we discuss the genetic short-time scheduling algorithm. Treatment of the other scheduling algorithms is planned as future work. Genetic short-time scheduling algorithm This genetic algorithm Mic96, Hof96 is designed to solve the problems occurring in short time scheduling. It uses an already pre-optimized season plan generated by a human operator or by Design, Implementation, and Application of a Tool for Optimal Aircraft Positioning Fig. 3. 3 8 9 Internal structure of cluster 'AWest' Fig. 4. First attempt a long time scheduler together with actual changes and additional information including, among others, actually ying passengers, planes available, passengers that need to get a connection. We decided to use evolutionary computing to solve this problem. The original idea was to treat the positioning problem with a usual genetic algorithm, encoding the positions of a given aircraft in a standard way gure 4. Although this approach works it causes a crucial problem: almost no valid valid for a nal solution individuals are created this way since most solutions will position one of the planes either on an impossible position plane is too large, for example or will put planes at the same time on the same position. Of course, such solutions would get a very low tness, so they are not likely to produce o spring. Still we get the problem that the resulting algorithm spends most of the time searching for a solution where the planes actually t and not for an optimal positioning of the planes thus wasting valuable time. 3 9 0 J . P fa lz g ra f e t a l. Fig. 5. Second attempt Therefore, another approach was chosen. First we changed the coding such that the aircraft are mapped onto positions gure 5. This enables us to use the complete rule set which is provided by the airport agency. So we are now able to allow only such genetic operators which produce a valid individuum. For example, if the aircraft number one would be a B747 and the only positions which were allowed for such a plane would be position one, two, and three, then a mutation on plane number one could only mutate it towards position one, two, or three. This second model works very well with the mutation operator, but new problems arise when using a crossover operator. The loss of data or the duplication of data can happen. These problems are well known since they also occur when trying to nd solutions to a traveling salesman problem with the help of GAs. When doing a crossover not only the parts that are actually selected are exchanged, but also other parts which are necessary to maintain consistency. To this end we have to check for each exchanged plane whether its counterpart" is also moved. If not, we have to set this aircraft also on the exchange list. On rst view this seems to be a very useful trick to solve the problem, but it also produces a problematic side e ect. Every time we wish to do a crossover at a certain point we also exchange planes on quite randomly chosen other positions. This can destroy building blocks, especially in a problem as big as the positioning task. For this reason we decided not to implement a crossover operator in the classical way. The Condense Operator Mutation 1 This is the simplest mutation operator which processes all planes that are not yet at valid positions. It aims at positioning such planes correctly. The Replace Operator Mutation 2 This second mutation operator is almost as simple as the rst one, but works in the opposite way. It is mainly used to maintain the diversity in a population. To do this it moves a plane from one position to another one. Technically this is also done in three steps. First one aircraft is randomly selected. This aircraft is moved to the Temp place. Temp denotes the set of planes not yet positioned. Then the algorithm selects randomly Design, Implementation, and Application of a Tool for Optimal Aircraft Positioning 3 9 1 a position where the plane ts. Then the selected plane is force-positioned on this selected position. It will be positioned there in any case removing disturbing planes if necessary. Later the removed planes will be reordered using the random scheduler. Chromosome Repositioning Mutation 3 This is the most 'advanced' mutation operator that will be used in our implementation of the GA. It does not operate on single planes like the previous ones, but on complete positions. Guided Replace Crossover 1 As already mentioned, many diculties can arise when using crossover operators in a given problem. Therefore we decided not to use them in a classical way, but in form of a new class of genetic operators. This class works like a mixture of crossover and mutation. It behaves like a normal crossover functionally, but it has a probability of occurrence like a mutation. So this kind of crossover operators can be considered as guided mutations. The Guided Replace operator is the simplest case of an operator of this type. Guided Chromosome Repositioning Crossover 2 This crossover-like operator has similarity to the operator Mutation 3". The idea is to make a transfer of a perfect distribution of planes corresponding to a certain position in an individual to another individual. The exact algorithm becomes a rather complicated ruleset. A detailed description can be found in FW99 . Cluster Crossover Crossover 3 This is the most advanced crossover operator. Since the airport contains groups clusters of positions it is plausible to use these in the optimization process. The internal algorithm of the operator has close links with the second crossover operator. In fact, we can use this operator for the cluster crossover if we select more than one position all positions in the given cluster. Fitness Function To nd an appropriate tness function was one of the main problems of the complete optimization process. We designed the function in close cooperation with the experts from Lufthansa. The optimization process is inuenced by many factors which have to be taken into account. Furthermore, the approach should be exible and it should be possible to cope with an optimization task which depends on selected factors only. The complete tness function consists of several constituents.   is based on four dierent main optimization criteria with respect to an individual : F F i i  =  +  +  +   F i P i C i S i Q i :   denotes the part that tries to optimize the number of passengers.   is responsible for reducing the connecting time.   describes the service aspects and   the quality of the solution. The Greek letters in the subsequent formulas denote parameters which can be tuned problem dependent. P i C i S i Q i 3 9 2 J . P fa lz g ra f e t a l. P i =  IP +  OP +  IB +  OB + " IY +  OY The passenger part takes care that a maximum amount of passengers can leave or enter a plane directly from the gate IP and OP . Additionally, also the yield of passengers can be taken into account IY and OY . P i models the request that a maximum amount of aircraft are well positioned right in front of the building IB and OB . C i =  CT + GD: The connexe connecting passengers part optimizes the time needed to get from one plane to the other. To this end two options are possible. Either the buer time maximal possible time to reach the plane - walking time is maximized CT  or special gates are used for planes with many connecting passengers GD. S i = BN + BY + P D: S i represents the service part of the tness function. Three factors are summarized here. BN and BY reduce the amount of bus transfers or the amount of bus transfers for valuable passengers, respectively. The third factor, P D, makes the time between boarding and take o as small as possible. Q i =  P C +  GC +  CC . Finally, Q i is responsible for the fact that the solution is admissible at all. P C and GC model the constraints that all positions and gates are valid. CC is responsible for the request that all connecting passengers are able to catch their planes. 4 Test of the tool in a real world scenario In FW99 a real world example is presented which corresponds to a typical scenario to be handled every day by Lufthansa and FAG the Frankfurt airport operating center. The initial input to our system is a concrete schedule elaborated by the long-term scheduling team of FAG. The syntax of the input data is of the following type example we display three lines only. inbound LH 00201 sta 07:45 in type I in ight type P inpaxcode 3 outbound LH 03720 std 08:40 out type S outpaxcode 3 out ight type P dest VIE air type A321 swc 11 pos V123 out gate B13 The complete input comprises about 1200 such 3-line units. Evaluating the performance of the tool in its application to a real problem situation it turned out that the system found better solutions than the human expert. In order to produce an optimized solution to the position scheduling task the human Design, Implementation, and Application of a Tool for Optimal Aircraft Positioning 3 9 3 operator needs about one week, whereas, working on the same task, our system needs about one hour on a standard modern PC and it produces even better positioning plans. More details and an example of an airport resource utilization plan produced by the tool can be found in FW99. 5 Conclusions In the previous sections we presented work on the hard problem of optimal positioning of aircraft at an airport. Many real world constraints have to be considered. The original problem has been described by Lufthansa AG focusing on airport Frankfurt Main. We developed a general generic airport model using, among others, the concept of logical berings. The optimization problem was treated by modi ed and extended genetic algortihms. On the basis of these approaches an aircraft positioning tool was developed and implemented, especially tailored for computing positioning con gurations of the airport Frankfurt Main. A prototypic rst version of the system is currently being tested with Lufthansa at Frankfurt airport. Our new system achieves much better performance than the methods applied before. In addition to the previously described methods, in future work we intend to use also methods from arti cial neural networks for modeling position constraints and optimization, make systematic applications of multi-agent systems techniques and rule based systems. References FW99 K. Frank and J. Weichenberger. Design and implementation of an aircraft positioning tool using hybrid problem-solving methods. Master's thesis, Institut fur Computerwissenschaften, Universitat Salzburg, Austria, 1999. Hof96 Frank Ho mann. Automatischer Entwurf von Fuzzy-Reglern mit genetischen Algorithmen. PhD thesis, Mathematisch-Naturwissenschaftliche Fakultat, Christian-Albrechts Universitat zu Kiel, Germany, 1996. Mei99 W. Meixl. Logical berings. a general decomposition method for many-valued logics and a modeling approach for multi-agent systems. Master's thesis, Institut fur Computerwissenschaften, Universitat Salzburg, Austria, 1999. Mic96 Zbigniew Michalewicz. Genetic Algorithms + Data Structures = Evolution Programs. Springer Verlag Berlin Heidelberg, 1996. Pfa91 J. Pfalzgraf. Logical berings and polycontextural systems. In Fundamentals of Articial Intelligence Research, Ph.Jorrand, J.Kelemen eds. . Lecture Notes in Computer Science 535, Subseries in AI, Springer Verlag, 1991. Pfa97 J. Pfalzgraf. On geometric and topological reasoning in robotics. Annals of Mathematics and Articial Intelligence, 19:279 318, 1997. PSS96a J. Pfalzgraf, U. Sigmund, and K. Stokkermans. 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