A NEW EVOLUTIONARY ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION BASED ON ENDOCRINE PARADIGM

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1 Proceedngs of he Inernaonal Conference on Theory and Applcaons of Mahemacs and Informacs ICTAMI 003, Alba Iula A NEW EVOLUTIONARY ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION BASED ON ENDOCRINE PARADIGM by Corna Roar Absrac: Mulobjecve opmzaon problems, due o her complexy, are suable for neresng evoluonary approaches. Many evoluonary algorhms have been developed for solvng mulobjecve problems, appealng or no o he Pareo opmaly concep. Alhough, evoluonary echnques for mulobjecve opmzaon confron wh several ssues as: elsm, dversy of he populaon, or effcen sengs for he specfc parameers of he algorhm. In hs paper, we propose a new evoluonary echnque, whch s nspred by he behavor of he endocrne sysem and uses he Pareo non-domnance concep. Therefore, he populaon s members are no more called chromosomes bu hormones and even f hey evolve accordng o he genec prncples (selecon, crossover, and muaon), a supplemenary mechansm, based on he, s conneced wh sandard approach o deal wh mulobjecve opmzaon problems. Moreover, he proposed algorhm, n order o manan populaon s dversy, uses a specfc scheme of fness sharng, elmnang he nconvenen of defnng an approprae value of sharng facor. Keywords: mulobjecve opmzaon,, evoluonary algorhm.. Inroducon Mulobjecve opmzaon (MOO) occupes a large area n echncal leraure. The opmzaon problems became dffcul n more han a sngle objecve s cases, so, he opmum concep s reformulaed accordngly. Vlfredo Pareo offers he mos common defnon of opmum n mulobjecve opmzaon (896). The evoluonary echnques, whch make use of Pareo concep of opmaly, are so called Pareo-based approaches and were frsly suggesed by Goldberg (989). Snce hen, many Pareo-based evoluonary echnques have been developed for mulobjecve opmzaon problems. Our paper proposes a new evoluonary algorhm for mulobjecve opmzaon based on. Evoluonary algorhms for MOO deal wh specfc ssues as fness assgnmen, dversy preservaon or elsm. We also sugges n our paper a new paradgm, whch could be a fruful source of nspraon for evoluonary compuaon:. Ths naural sysem proves self as a complex, sem-ndependen and adapve sysem quales ha could be exploed n evoluonary echnques.. Saemen of he Problem Mulobjecve opmzaon problems, also called mulcreral opmzaon problems MOPs, nvolves he search of he opmum soluons accordng wh more 7

2 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on han one creron. Generally, hose crera are conflcng and hs fac makes he MO problems o be dffcul. Formally, MOP s gven as follows: Consderng m nequaly consrans (), p equaly consrans (3) and k crera (), we are neresed n fndng he decson varable vecor n he doman D, D R n * * * * : x = ( x, x,..., xn ), whch opmzes he vecor funcon (), and sasfes he consrans () and (3): ( x) 0 ( ) ( x) f ( x) = f ( x), f ( x) f ( x) F =,..., () k g,,, K, m = (), ( x) = 0 h, =,, K, p (3) Snce, we hardly ever deec a real-world suaon where all he objecve funcons have he same opmum n he search space D, he concep of opmaly should have been revsed n he mulobjecve opmzaon s conex. 3. Pareo Opmaly Vlfredo Pareo formulaed he mos common concep of opmaly for mulcreral opmzaon problems. In order o esmae he qualy of a parcular elemen from he search space, a relaon of domnance was esablshed. We consder a k-objecve mnmzaon problem and a search doman D, D R n. Defnon : We sae ha he vecor a = ( a,a,...,a k ) domnaes he vecor b = ( b,b,...,b k ) f and only f he followng asseron f verfed: {,,...,k} : a b ) ( {,,...,k} : a b ) ( < (4) Defnon : We sae ha he soluon x D s weakly non-domnaed, f: {,,...,k} y D for whch : b < a (5), where f ( y) ( b b,..., b ) and f ( x) ( a, a,..., ) =, k = a k. Defnon 3: We sae ha he soluon x D s srongly non-domnaed, f: {,,...,k} : b a ) ( {,,...,k} : b a ) y D for whch ( < (6) where f ( y) ( b b,..., b ) and f ( x) ( a, a,..., ) =., k = a k Defnon 4: We sae ha he soluon x D s Pareo opmal f and only f he nex asseron s verfed: y D for whch f ( y) = (b,b,...,b ) domnaes f ( x) = (a,a,...,a ) (7) k k 7

3 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on Pareo fron consss of he non-domnaed vecors f(x) ha corresponds o he Pareo opmal soluons x, x D. For example, an opmzaon problem wh wo objecve funcons: f and f (fgure). Fgure : The Pareo fron s marked wh bold lne 4. Dversy versus Elsm Many researchers emphaszed he mporance of hese wo feaures of he evoluonary approaches n he cases of mulobjecve opmzaon [6], [7], [5], [7]. They also developed varous mechansms for elsm and dversy preservaon, whch leaded o a subsanal ncrease of he algorhms performance. Whle elsm sresses he local search he examnaon of he bes areas of he search space, dversy of he populaon guaranees an effcen exploraon of he enre search space. A good equlbrum beween local search and space s exploraon could be mananed by he equlbrum beween dversy preservaon mechansm and elsm echnque. The echncal leraure specfes how mporan elsm s and sresses he fac ha els approaches offer beer soluons han non-els approaches. Evoluonary mulobjecve opmzaon, due o s specfcy, requres a closer aenon regardng hose wo menoned feaures. Fndng mulple soluons, whch are dversely dsrbued along he Pareo froner, represens a dffcul problem. 73

4 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on 5. Endocrne Paradgm Many evoluonary echnques for solvng dfferen ypes of problems are developed and nspred from naure (as: mmune sysems, an colony, heredary prncples, and so on). Our proposed algorhm for mulobjecve opmzaon apples several prncples nspred of endocrne sysem. Even f endocrne sysem has a parcular dependence of nervous sysem, due o he nrnsc mechansms of conrol and s funcons, represens by self a suable paradgm n evoluonary compuaon s landscape. The endocrne sysem, hrough s funcon and anaomy s relevan as a complex and adapable sysem. Tha s why an arfcal shapng of hs one seems o be suable for solvng some complex problems. A quck vew on endocrne sysem reveals how complex and ngenous he naure can be. Endocrne sysem s represened by a collecon of glands, whch produce chemcal messengers called hormones. These sgnals ge hrough he blood sysem o he arge organs, whch have cells conanng approprae recepors. The recepors of he cells recognze and e one ype of hormone. Hormones provoke profound changes a he level of he arge cells. Every ype of hormone has a specfc shape, recognzed only by he arge cells. The man glands of endocrne sysem are: puary gland (hypophyss), hyrod gland, he pancreas, he gonads, he adrenal glands and pneal gland. Puary gland s consdered maser gland of he body. Ths descrpon s based on he conrol, whch hs one exercses. A he puary gland level are produced hose hormones whch nfluence and conrol he cells and he process of he organsm. Is role of supervsor s offered n realy by he funcon of he hypohalamus, hs one represenng he bound beween he nervous and endocrne sysems. Hypohalamc neurons secree hormones ha regulae he release of hormones from he puary gland. Hypohalamc hormones are wo ypes: releasng and nhbng hormones, reflecng he nfluence, whch hey have over he producng of puary hormones, called ropes. In order o beer undersand he self-conrol process of he endocrne sysem, he followng example s aken n consderaon: releasng hyrodan hormone (TRH) produced by hypohalamus acons on hypophyss deermnng he secreon of he hyrodan smulaon hormone (TSH). Ths one wll ac on he arge organ.e. hyrod gland. The acon of TSH hormone on hyrod s concrezed by he secreon of he specfc hyrodan hormones. Thus, when he concenraon of he specfc hyrodan hormones becomes oo hgh or oo low, hrough a negav feedback process, he hypohalamus s announced abou hs aspec. So, usng nhbng or releasng hormones he hypohalamus acs on hypophyss producng an nhbon/releasng n he producon of hyrodan smulaon hormones (TSH) wh a drec effec on hyrod gland acvy and hus, a comng back o normal of hyrodan hormones concenraon. 74

5 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on Concludng, he concenraon of specfc hormones s conrolled rough a feedback mechansm. So, a specal ype of hormones generaed by hypophyss, called ropes, have he role o supervse and o mprn he releasng or nhbng of specfc hormones. Ths prncple s adoped n our algorhm for dversy preservaon. Smplfyng, endocrne sysem s mechansm for conrollng he hormonal concenraon s shown n he nex fgure. Fgure Conrol scheme of endocrne sysem 6. A New Evoluonary Algorhm for Mulobjecve Opmzaon The prncple of he proposed mehod reles on keepng wo populaons: an acve populaon of hormones, H, and a passve populaon of non-domnaed soluons, A. The members of he passve populaon behave as a populaon of ele and also have a supplemenary funcon: o lead he hormones oward he Pareo fron, keepng hem as much as possble well dsrbued among he search space. These wo populaons correspond o he wo classes of hormones n :. Specfc hormones, whch are released by dfferen glands of he body he acve populaon H.. The hormones of conrol (rop), whch are produced by conrol level of he endocrne sysem (hypohalamus and hypophyss) n order o supervse he densy of each ype of hormone - he passve populaon A. Populaon of conrol, A, s modfed a each generaon. The new populaon A + gahers all non-domnaed soluon from he populaon U, whch has resuled from mergng curren populaon of hormones H and he prevous populaon A. 75

6 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on Ths manner of changng he populaon of conrollers assures us ha he nondomnaed soluons from he prevous populaon A canno be los f hey sll reman non-domnaed afer he acve populaon H has changed. The passve populaon, A, doesn suffer modfcaon a ndvdual s level, under he varaon operaors as crossover and muaon. I behaves as an ele populaon of non-domnaed soluons from he curren generaon, whch s only acualzed a each generaon. Fnally, populaon A conans a predeermned number of non-domnaed vecors and provdes a good approxmaon of Pareo fron. A each generaon, he members of H are classfed no sa classes. A correspondng conroller from A supervses a parcular class. The dea s ha each hormone h from H s supervsed by he neares conroller a from A : C ( a ) = { h H ds( h, a ) = mn( ds( h, a ), j,,...,sa) }, (6) j = The dea s ha each member a from se A has a smlar conrol funcon as a rop from. Anoher specfc ssue of he proposed algorhm s he manner o selec and recombne he hormones n order o generae descendans. A specal knd of sharng among he members of he curren populaon resuls. Due o he fac ha each ndvdual a from populaon hormones: C ( ) a H A conrols a class of, hs ndvdual of conrol, a, mprns o each hormone a probably of selecng as frs paren. The selecon of frs paren s made proporonal o he value of s class, whch s calculaed accordng o formula: val = (7). szeof ( C( a )) Frs paren s seleced from populaon H, proporonally wh value of s class, so, we can affrm ha he hormones from a parcular class locally share he resources. By hs manner of selecng frs paren, he less crowded hormones are preferred and hose zones from he search space, whch are apparenly dsappearng, would be revved. Le h be he seleced frs paren and a, s conroller. k The second paren h, he mae of he paren h, s seleced only from he l k 76

7 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on C a k. Paren h s proporonally seleced wh s performance, where l performance value of a hormone h s gven by he formula: h s class, ( ) nr _ domnaed performance( h) = (8) sh and nr_domnaed - represens he number of soluons from H, whch are domnaed by he hormone h. By selecng he second paren proporonally wh s performance, he mehod assures a faser convergence of he populaon oward Pareo fron. Crossover operaor recombnes wo parens and produces a sngle descendan, whch s acceped no he nex populaon. MENDA echnque. Inalze_he_populaons... = 0 (number of curren generaon);.. sh = sa. (nal sze of he populaon A s he same wh he sze of he populaon H).3. Generae randomly he populaons: H = { h,h,...,h sh }, populaon of hormones, and A = { a,a,...,a sa }, populaon of conrollers, where: sh = sze ( H ) and sa = sze ( A ).. Repea.. Jon he populaons H and A, resulng U. The populaon U conans all ndvduals from H and A. sha = sze ( U ) = sze ( H )+sze ( A )... Generae A +. Populaon A + embodes all non-domnaed soluon from U : A + = U { u U,u - non - domnaed }, sa = card ( A + ).3. Classfy he hormones from H accordng o A + and se he crowdng degrees for hormones. Each ndvdual from A + conrol he hormones from s neghborhood. Furher, a parcular hormone can recombne only wh hormones from s class..4. Evaluae H. The performance of each hormone h s proporonal wh he number of oher ndvduals of H, whch are domnaed by h. 77

8 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on.5. Generae H + : ; H Φ + = For each h from degree do: H, whch s seleced proporonally o s crowdng Selec a mae h from he class of h hormone. Recombne h and h, for creang he descendan d. Include he descendan no he nex generaon: H + = H + U {d}.6. = +. Unl ( sa = sh ). Remarks: The algorhm ends when all ndvduals from H became nondomnaed. Condon (sa = sh) could be replaced wh anoher condon, as aanng a prefxed number of generaons. Numercal expermens proved ha for a populaon of sze 00, he condon (sa = sh) s sasfacory for deecng an approxmae Pareo fron. We also noced ha less crowded ndvduals or hose ndvduals, whch are dsan from he se A, have a sgnfcan role for dversy preservaon. So, he se A could also embody hose ndvduals. 7. Numercal Expermens Example. We consder a mnmzaon problem wh wo objecves of wo varables. Ths es funcon was proposed by Ls and Eben []. 8 ( x x ) = x f +, x ( ) 4 f x, x = ( x 0.5) + ( x 0.5 where, x [ 5,0] ) x., Pareo fron s dsconnuous and concave, and nex fgures show he populaon a dfferen generaon, =,3,4,5. Populaon sze s

9 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on Fgure : = Fgure 3: =3 Fgure 4: =4 Fgure 5: =5 Example. We consder nex a es b-objecve funcon, wh one varable, proposed by Schaffer []. x, x - + x, < x x, 3 < x x, 4 < x ( x) =, f ( x) = ( x ), where, x [ 5,0] f Pareo fron s dsconnuous. 5 x., 79

10 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on Fgure 6: Pareo fron, generaon =3 Fgure 7: Pareo fron, generaon =0 Example 3: Our es funcons also ncluded a popular dffcul es su (ZDT -6 bobjecve problems), proposed by Zzler, Deb and Thele n [7]. F s a b-objecve x = f x f x and he opmzaon problem s: funcon ( ) ( ( ) ( ) For example: ZDT problem: f x = g h ( ) x F, ( x,..., xm ) = + 9 ( f g) =, Mnmze F(x), subjec o f (x) = g(x, K,x )h(f (x ),g(x, Kx ) m m where x = (x, K,x ) m f g x m = m ZTD problem: ( x) f = x m x g( x,..., xm ) = + 9 = m f h( f, g) = g and ZTD3 problem: f x = g ( ) x ( x,..., xm ) = + 9 x m = m 80

11 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on h f ( f, g) = sn( 0πf ) g g In all cases m=30, x [ 0,] f, and Pareo fron s formed wh g(x)=. A specfc characersc of hose funcons made he populaon o converge oward a segmen of he Pareo fron, a segmen ha corresponded wh he lowes values of he frs funcon. I can be noced ha f he frs objecve funcon depends only of one varable (for example: f ( x) = x ), he second objecve represens a funcon wh m- varables, where m s usually 30. We observed ha for m=, he Pareo fron s deeced usng our algorhm. Inuvely, we assume ha he proposed evoluonary algorhm mnmzes he values of he frs funcon more quckly han he values of he second one. Thus, he fnal soluons correspond o a small par of he Pareo fron, where he frs objecve s values are lower. Due o hs fac, a modfcaon mus be made n order o oban he enre Pareo fron. Our proposal s o change he varaon operaor, respecvely he crossover operaor. The sandard crossover operaor s appled on wo seleced parens x = ( x, x,..., x m ) and y = ( y, y,..., y m ), resulng he unque descendan z = ( z, z,..., z m ), where: z = ( q) x + q y and q randomly generaed. The modfed operaor generaes he unque descendan z = ( z, z,..., z m ), oo. Consderng λ from [0,] (usually 0.5), represenng a parameer of he algorhm, he modfed crossover operaor performs as follows: Modfed Crossover Operaor For each from o m z = ( q) x + q y endfor Randomly generae p from [0,]. If p> λ hen z = ( q ) x + q y else endif. randomly generae a value for z from he specfc doman. Ths aleraon of crossover operaor gves good resuls. The algorhm works wh a populaon of sze 00. The algorhm ends when he populaon conans only non-domnaed soluons. 8

12 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on The nex fgures show he fnal Pareo Fron for ZDT, and 3 problems. Fgure 8 ZDT3, Pareo fron afer 5 generaons Fgure 9 ZDT, Pareo fron afer 5 generaons Fgure 0: ZDT3, Pareo fron afer 0 generaons 8. Conclusons Naure offers complex phenomenon, processes and sysems, whch can nspre us n many ways. An example of hs knd s he naural endocrne sysem. I reveals us as a dynamc, adapve and sem-ndependen sysem. Is complex srucural and funconal feaures sugges us a possble maon of s characerscs for developng new echnques for mulobjecve opmzaon. I s clearly ha hormonal sysem was gnored by now. Our work essenally aemps o mmc hormones behavor n order o solve dffcul problems lke mulobjecve opmzaon. Consequenly, we propose a new echnque based on, called MENDA. I provdes sasfacory soluons n our ess. Furher drecons could be mprovng he proposed algorhm and more ess wh dfferen mulobjecve funcon. 8

13 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on References:. Bäck T., Evoluonary Algorhms n Theory and Pracce, Oxford Unversy Press, Coello C. A. C., A Shor Tuoral on Evoluonary Mulobjecve Opmzaon, In Eckar Zzler, Kalyanmoy Deb, Lohar Thele, Carlos A. Coello Coello, and Davd Corne, edors, Frs Inernaonal Conference on Evoluonary Mul- Creron Opmzaon, Sprnger-Verlag. Lecure Noes n Compuer Scence No. 993, Coello C. A.. C., An Updaed Survey of Evoluonary Mulobjecve Opmzaon Technques : Sae of he Ar and Fuure Trends, In 999 Congress on Evoluonary Compuaon, Washngon, D.C., Deb, K., Mul-objecve genec algorhms: Problem dffcules and consrucon of es funcons. Evoluonary Compuaon, 7(3), Dumrescu D., Algorm Genec ş Sraeg Evoluve aplcaţ în Inelgenţa Arfcală ş în domen conexe, Edura Albasră, Cluj Napoca, Romana, Dumrescu D., Lazzern B., Jan L.C., Dumrescu A., Evoluonary Compuaon, CRC Press, Boca Raon London, New York, Washngon D.C., Fonseca, C.M., Flemng, P.J., Mulobjecve Opmzaon and Mulple Consran Handlng wh Evoluonary Algorhms I: A Unfed Formula, Research Repor no. 564, Sheffeld, Uned Kngdom, Fonseca, C.M., Flemng, P.J., Mulobjecve Opmzaon and Mulple Consran Handlng wh Evoluonary Algorhms II: Applcaon Examples, Research Repor no. 565, Sheffeld, Uned Kngdom, Goldberg D. E., Rchardson J., Genec algorhms wh sharng for mulmodal funcon opmzaon., Proc. nd In. Conf. on Genec Algorhms (Cambrdge, MA), ed J. Greffensee, Goldberg D.E., Genec Algorhms n Search, Opmzaon, and Machne Learnng, Addson-Wesley Publshng Company, Inc., Laumanns, M., Thele, L., Zzler, E., Deb, K., On he Convergence and Dversy Preservaon Properes of Mul-Objecve Evoluonary Algorhms, TIK - Repor no. 08, Swzerland, 00.. Ls J., Eben A. E., A Mul-sexual Genec Algorhm for Mulobjecve Opmzaon, n Proceedngs of he 996 Inernaonal Conference on Evoluonary Compuaon, Nagoya, Japan, Roar C., Ileană I., Models of Populaon for Mulmodal Opmzaon. A New Evoluonary Approach, Proc. 8 h In. Conf. on Sof Compung, Mendel00, Czech Republc, Tooanu I., Ghe., Inroducere n fzologa clnca a ssemulu endocrn, Targu- Mures, Romana,

14 Corna Roar - A new evoluonary algorhm for mulobjecve opmzaon based on 5. Van Veldhuzen, D., Lamon, G., On Measurng Mulobjecve Evoluonary Algorhms Performance, Congress on Evoluonary Compuaon, CEC 000, Pscaaway, NJ, Van Veldhuzen, D., Lamon, G., Mulobjecve Evoluonary Algorhm Research: A Hsory and Analyss, Techncal Repor: TR-98-03, Zzler, E., Deb, K., Thele, L, Comparson of Mulobjecve Evoluonary Algorhms: Emprcal Resuls, Evoluonary Compuaon, vol. 8, no., Zzler, E., Thele, L., Mulobjecve Opmzaon Usng Evoluonary Algorhms A Comparave Case Sudy, In Proceedngs 5 h Parallel Problem Solvng from Naure, Neherlands, Zzler, E., Thele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G., Performance Assessmen of Mulobjecve Opmzers: An Analyss and Revew, TIK-Repor No. 39, Swzerland, 00. Auhor: Corna Roar, Decembre 98 Unversy, Alba Iula, Compuer Scence Deparmen, croar@lmm.uab.ro 84