IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION VOL - PDF document

IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002AFastandElitistMultiobjectiveGeneticAlgorithm:KalyanmoyDeb,AssociateMember,IEEE,AmritPratap,SameerAgarwal,andT.MeyarivanMultiobjectiveevolutionaryalgorithms(EAs)thatusenondominatedsortingandsharinghavebeencriti-cizedmainlyfortheir:1) ( computationalcomplexity(where isthenumberofobjectivesand isthepopulationsize);2)nonelitismapproach;and3)theneedforspecifyingasharingparameter.Inthispaper,wesuggestanondominatedsorting-basedmultiobjectiveEA(MOEA),callednondominatedsortinggeneticalgorithmII(NSGA-II),whichalleviatesalltheabovethreedifficulties.Specifically,afastnondominatedsortingapproachwith ( computationalcomplexityispresented.Also,aselectionoperatorispresentedthatcreatesamatingpoolbycombiningtheparentandoffspringpopulationsandselectingthebest(withrespecttofitnessandspread) solutions.Simulationresultsondifficulttestproblemsshowthat (where isthenumberofobjectivesand isthepopulationsize).ThismakesNSGAcomputationallyexpensiveforlargepopu-lationsizes.Thislargecomplexityarisesbecauseofthecomplexityinvolvedinthenondominatedsortingproce-dureineverygeneration.Lackofelitism:Recentresults[25],[18]showthatelitismcanspeeduptheperformanceoftheGAsignificantly,whichalsocanhelppreventingthelossofgoodsolutionsoncetheyarefound.Needforspecifyingthesharingparameter Tradi-tionalmechanismsofensuringdiversityinapopulationsoastogetawidevarietyofequivalentsolutionshavereliedmostlyontheconceptofsharing.Themainproblemwithsharingisthatitrequiresthespecificationofasharingparameter( ).Thoughtherehasbeensomeworkondynamicsizingofthesharingparameter[10],aparam-eter-lessdiversity-preservationmechanismisdesirable.Inthispaper,weaddressalloftheseissuesandproposeanimprovedversionofNSGA,whichwecallNSGA-II.Fromthesimulationresultsonanumberofdifficulttestproblems,wefindthatNSGA-IIoutperformstwoothercontemporaryMOEAs: etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-IIwedescribetheproposedNSGA-IIalgorithmindetails.Sec-tionIVpresentssimulationresultsofNSGA-IIandcomparesthemwithtwootherelitistMOEAs(PAESandSPEA).InSec-tionV,wehighlighttheissueofparameterinteractions,amatterthatisimportantinevolutionarycomputationresearch.ThenextsectionextendsNSGA-IIforhandlingconstraintsandcomparestheresultswithanotherrecentlyproposedconstraint-handlingmethod.Finally,weoutlinetheconclusionsofthispaper.II.EULTIOBJECTIVEVOLUTIONARYDuring1993–1995,anumberofdifferentEAsweresug-gestedtosolvemultiobjectiveoptimizationproblems.Ofthem,FonsecaandFleming’sMOGA[7],SrinivasandDeb’sNSGA[20],andHornetal.’sNPGA[13]enjoyedmoreattention.Thesealgorithmsdemonstratedthenecessaryadditionaloper-atorsforconvertingasimpleEAtoaMOEA.Twocommonfeaturesonallthreeoperatorswerethefollowing:i)assigningfitnesstopopulationmembersbasedonnondominatedsortingandii)preservingdiversityamongsolutionsofthesamenondominatedfront.Althoughtheyhavebeenshowntofindmultiplenondominatedsolutionsonmanytestproblemsandanumberofengineeringdesignproblems,researchersrealizedtheneedofintroducingmoreusefuloperators(whichhavebeenfoundusefulinsingle-objectiveEA’s)soastosolvemultiobjectiveoptimizationproblemsbetter.Particularly,theinteresthasbeentointroduceelitismtoenhancetheconvergencepropertiesofaMOEA.Reference[25]showedthatelitismhelpsinachievingbetterconvergenceinMOEAs.AmongtheexistingelitistMOEAs,ZitzlerandThiele’sSPEA[26],KnowlesandCorne’sPareto-archivedPAES[14],andRudolph’selitistGA[18]arewellstudied.Wedescribetheseapproachesinbrief.Fordetails,readersareencouragedtorefertotheoriginalstudies.ZitzlerandThiele[26]suggestedanelitistmulticriterionEAwiththeconceptofnondominationintheirSPEA.Theysug-gestedmaintaininganexternalpopulationateverygenerationstoringallnondominatedsolutionsdiscoveredsofarbeginningfromtheinitialpopulation.Thisexternalpopulationpartici-patesinallgeneticoperations.Ateachgeneration,acombinedpopulationwiththeexternalandthecurrentpopulationisfirstconstructed.Allnondominatedsolutionsinthecombinedpop-ulationareassignedafitnessbasedonthenumberofsolutionstheydominateanddominatedsolutionsareassignedfitnessworsethantheworstfitnessofanynondominatedsolution.Thisassignmentoffitnessmakessurethatthesearchisdirectedtowardthenondominatedsolutions.Adeterministicclusteringtechniqueisusedtoensurediversityamongnondominatedsolutions.Althoughtheimplementationsuggestedin[26]is ,withproperbookkeepingthecomplexityofSPEAcanbereducedto KnowlesandCorne[14]suggestedasimpleMOEAusingasingle-parentsingle-offspringEAsimilarto(1 1)-evolutionstrategy.Insteadofusingrealparameters,binarystringswereusedandbitwisemutationswereemployedtocreateoffsprings.IntheirPAES,withoneparentandoneoffspring,theoffspringiscomparedwithrespecttotheparent.Iftheoffspringdomi-natestheparent,theoffspringisacceptedasthenextparentandtheiterationcontinues.Ontheotherhand,iftheparentdom-inatestheoffspring,theoffspringisdiscardedandanewmu-tatedsolution(anewoffspring)isfound.However,iftheoff-springandtheparentdonotdominateeachother,thechoicebe-tweentheoffspringandtheparentismadebycomparingthemwithanarchiveofbestsolutionsfoundsofar.Theoffspringiscomparedwiththearchivetocheckifitdominatesanymemberofthearchive.Ifitdoes,theoffspringisacceptedasthenewparentandallthedominatedsolutionsareeliminatedfromthearchive.Iftheoffspringdoesnotdominateanymemberofthearchive,bothparentandoffspringarecheckedfortheirwiththesolutionsofthearchive.Iftheoffspringresidesinaleastcrowdedregionintheobjectivespaceamongthemem-bersofthearchive,itisacceptedasaparentandacopyofaddedtothearchive.Crowdingismaintainedbydividingtheentiresearchspacedeterministicallyin subspaces,where isthedepthparameterand isthenumberofdecisionvariables,andbyupdatingthesubspacesdynamically.Investigatorshavecal-culatedtheworstcasecomplexityofPAESfor evaluations ,where isthearchivelength.Sincethearchivesizeisusuallychosenproportionaltothepopulationsize ,theoverallcomplexityofthealgorithmis Rudolph[18]suggested,butdidnotsimulate,asimpleelitistMOEAbasedonasystematiccomparisonofindividualsfromparentandoffspringpopulations.Thenondominatedsolutionsoftheoffspringpopulationarecomparedwiththatofparentso-lutionstoformanoverallnondominatedsetofsolutions,whichbecomestheparentpopulationofthenextiteration.Ifthesizeofthissetisnotgreaterthanthedesiredpopulationsize,otherindividualsfromtheoffspringpopulationareincluded.Withthisstrategy,heprovedtheconvergenceofthisalgorithmtothePareto-optimalfront.Althoughthisisanimportantachievementinitsownright,thealgorithmlacksmotivationforthesecondtaskofmaintainingdiversityofPareto-optimalsolutions.Anex-plicitdiversity-preservingmechanismmustbeaddedtomakeitmorepractical.Sincethedeterminismofthefirstnondominatedfrontis ,theoverallcomplexityofRudolph’salgo-rithmisalso Inthefollowing,wepresenttheproposednondominatedsortingGAapproach,whichusesafastnondominatedsortingprocedure,anelitist-preservingapproach,andaparameterlessnichingoperator.III.EONDOMINATEDORTINGA.FastNondominatedSortingApproachForthesakeofclarity,wefirstdescribeanaiveandslowprocedureofsortingapopulationintodifferentnondominationlevels.Thereafter,wedescribeafastapproach.Inanaiveapproach,inordertoidentifysolutionsofthefirstnondominatedfrontinapopulationofsize ,eachsolutioncanbecomparedwitheveryothersolutioninthepopulationtofindifitisdominated.Thisrequires comparisonsforeachsolution,where isthenumberofobjectives.Whenthisprocessiscontinuedtofindallmembersofthefirstnondomi-natedlevelinthepopulation,thetotalcomplexityis Atthisstage,allindividualsinthefirstnondominatedfrontarefound.Inordertofindtheindividualsinthenextnondominated IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002front,thesolutionsofthefirstfrontarediscountedtemporarilyandtheaboveprocedureisrepeated.Intheworstcase,thetaskoffindingthesecondfrontalsorequires tions,particularlywhen numberofsolutionsbelongtothesecondandhighernondominatedlevels.Thisargumentistrueforfindingthirdandhigherlevelsofnondomination.Thus,theworstcaseiswhenthereare frontsandthereexistsonlyonesolutionineachfront.Thisrequiresanoverall computations.Notethat storageisrequiredforthispro-cedure.Inthefollowingparagraphandequationshownatthebottomofthepage,wedescribeafastnondominatedsortingapproachwhichwillrequire First,foreachsolutionwecalculatetwoentities:1)domi-nationcount ,thenumberofsolutionswhichdominatethe ,and2) ,asetofsolutionsthatthesolution inates.Thisrequires Allsolutionsinthefirstnondominatedfrontwillhavetheirdominationcountaszero.Now,foreachsolution with wevisiteachmember( )ofitsset andreduceitsdomina-tioncountbyone.Indoingso,ifforanymember thedomi-nationcountbecomeszero,weputitinaseparatelist .Thesemembersbelongtothesecondnondominatedfront.Now,theaboveprocedureiscontinuedwitheachmemberof andthethirdfrontisidentified.Thisprocesscontinuesuntilallfrontsareidentified.Foreachsolution inthesecondorhigherlevelofnondom-ination,thedominationcount canbeatmost .Thus,eachsolution willbevisitedatmost timesbeforeitsdominationcountbecomeszero.Atthispoint,thesolutionisassignedanondominationlevelandwillneverbevisitedagain.Sincethereareatmost suchsolutions,thetotalcom-plexityis .Thus,theoverallcomplexityoftheprocedure .Anotherwaytocalculatethiscomplexityistore-alizethatthebodyofthefirstinnerloop(foreach )isexecutedexactly timesaseachindividualcanbethememberofatmostonefrontandthesecondinnerloop(foreach canbeexecutedatmaximum timesforeachindividual[eachindividualdominates individualsatmaximumandeachdominationcheckrequiresatmost comparisons]resultsintheoverall computations.Itisimportanttonotethatalthoughthetimecomplexityhasreducedto ,thestoragerequirementhasincreasedto B.DiversityPreservationWementionedearlierthat,alongwithconvergencetothePareto-optimalset,itisalsodesiredthatanEAmaintainsagoodspreadofsolutionsintheobtainedsetofsolutions.TheoriginalNSGAusedthewell-knownsharingfunctionapproach,whichhasbeenfoundtomaintainsustainablediversityinapopula-tionwithappropriatesettingofitsassociatedparameters.Thesharingfunctionmethodinvolvesasharingparameter whichsetstheextentofsharingdesiredinaproblem.Thispa-rameterisrelatedtothedistancemetricchosentocalculatetheproximitymeasurebetweentwopopulationmembers.Thepa- denotesthelargestvalueofthatdistancemetricwithinwhichanytwosolutionsshareeachother’sfitness.Thisparameterisusuallysetbytheuser,althoughthereexistsomeguidelines[4].Therearetwodifficultieswiththissharingfunc-tionapproach.1)Theperformanceofthesharingfunctionmethodinmaintainingaspreadofsolutionsdependslargelyonthe value. - - - foreach foreach if thenIf dominates Add tothesetofsolutionsdominatedby elseif then Incrementthedominationcounterof if then belongstothefirstfront Initializethefrontcounter Usedtostorethemembersofthenextfrontforeach foreach if then belongstothenextfront etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-II Fig.1.Crowding-distancecalculation.Pointsmarkedinfilledcirclesaresolutionsofthesamenondominatedfront.2)Sinceeachsolutionmustbecomparedwithallotherso-lutionsinthepopulation,theoverallcomplexityofthesharingfunctionapproachis IntheproposedNSGA-II,wereplacethesharingfunctionapproachwithacrowded-comparisonapproachthateliminatesboththeabovedifficultiestosomeextent.Thenewapproachdoesnotrequireuser-definedparameterformaintainingdiversityamongpopulationmembers.Also,thesuggestedap-proachhasabettercomputationalcomplexity.Todescribethisapproach,wefirstdefineadensity-estimationmetricandthenpresentthecrowded-comparisonoperator.1)DensityEstimation:Togetanestimateofthedensityofsolutionssurroundingaparticularsolutioninthepopulation,wecalculatetheaveragedistanceoftwopointsoneithersideofthispointalongeachoftheobjectives.Thisquantity servesasanestimateoftheperimeterofthecuboidformedbyusingthenearestneighborsasthevertices(callthisthecrowding).InFig.1,thecrowdingdistanceofthe thsolutioninitsfront(markedwithsolidcircles)istheaveragesidelengthofthecuboid(shownwithadashedbox).Thecrowding-distancecomputationrequiressortingthepop-ulationaccordingtoeachobjectivefunctionvalueinascendingorderofmagnitude.Thereafter,foreachobjectivefunction,theboundarysolutions(solutionswithsmallestandlargestfunctionvalues)areassignedaninfinitedistancevalue.Allotherinter-mediatesolutionsareassignedadistancevalueequaltotheab-solutenormalizeddifferenceinthefunctionvaluesoftwoadja-centsolutions.Thiscalculationiscontinuedwithotherobjectivefunctions.Theoverallcrowding-distancevalueiscalculatedasthesumofindividualdistancevaluescorrespondingtoeachob-jective.Eachobjectivefunctionisnormalizedbeforecalculatingthecrowdingdistance.Thealgorithmasshownatthebottomofthepageoutlinesthecrowding-distancecomputationprocedureofallsolutionsinannondominatedset .Here, referstothe thobjectivefunctionvalueofthe thindividualintheset andtheparameters and themaximumandminimumvaluesofthe thobjectivefunc-tion.Thecomplexityofthisprocedureisgovernedbythesortingalgorithm.Since independentsortingsofatmost tions(whenallpopulationmembersareinonefront )arein-volved,theabovealgorithmhas complexity.Afterallpopulationmembersintheset areassignedadistancemetric,wecancomparetwosolutionsfortheirextentofproximitywithothersolutions.Asolutionwithasmallervalueofthisdistancemeasureis,insomesense,morecrowdedbyothersolutions.Thisisexactlywhatwecompareintheproposedcrowded-comparisonoperator,describedbelow.AlthoughFig.1illustratesthecrowding-distancecomputationfortwoobjectives,theprocedureisapplicabletomorethantwoobjectivesaswell.2)Crowded-ComparisonOperator:Thecrowded-compar-isonoperator( )guidestheselectionprocessatthevariousstagesofthealgorithmtowardauniformlyspread-outPareto-optimalfront.Assumethateveryindividual inthepopulationhastwoattributes:1)nondominationrank( 2)crowdingdistance( Wenowdefineapartialorder as if or and Thatis,betweentwosolutionswithdifferingnondominationranks,wepreferthesolutionwiththelower(better)rank.Other-wise,ifbothsolutionsbelongtothesamefront,thenwepreferthesolutionthatislocatedinalessercrowdedregion.Withthesethreenewinnovations—afastnondominatedsortingprocedure,afastcrowdeddistanceestimationproce-dure,andasimplecrowdedcomparisonoperator,wearenowreadytodescribetheNSGA-IIalgorithm.C.MainLoopInitially,arandomparentpopulation iscreated.Thepop-ulationissortedbasedonthenondomination.Eachsolutionisassignedafitness(orrank)equaltoitsnondominationlevel(1isthebestlevel,2isthenext-bestlevel,andsoon).Thus,mini-mizationoffitnessisassumed.Atfirst,theusualbinarytourna-mentselection,recombination,andmutationoperatorsareusedtocreateaoffspringpopulation ofsize .Sinceelitism - - numberofsolutionsin foreach set initializedistanceforeachobjective sort sortusingeachobjectivevalue sothatboundarypointsarealwaysselected to forallotherpoints IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002isintroducedbycomparingcurrentpopulationwithpreviouslyfoundbestnondominatedsolutions,theprocedureisdifferentaftertheinitialgeneration.Wefirstdescribethe thgenerationoftheproposedalgorithmasshownatthebottomofthepage.Thestep-by-stepprocedureshowsthatNSGA-IIalgorithmissimpleandstraightforward.First,acombinedpopulation isformed.Thepopulation isofsize .Then,the issortedaccordingtonondomination.Sinceallpreviousandcurrentpopulationmembersareincludedin elitismisensured.Now,solutionsbelongingtothebestnon-dominatedset areofbestsolutionsinthecombinedpopu-lationandmustbeemphasizedmorethananyothersolutioninthecombinedpopulation.Ifthesizeof issmallerthen wedefinitelychooseallmembersoftheset forthenewpop- .Theremainingmembersofthepopulation arechosenfromsubsequentnondominatedfrontsintheorderoftheirranking.Thus,solutionsfromtheset arechosennext,followedbysolutionsfromtheset ,andsoon.Thisprocedureiscontinueduntilnomoresetscanbeaccommodated.Saythattheset isthelastnondominatedsetbeyondwhichnoothersetcanbeaccommodated.Ingeneral,thecountofsolutionsinallsetsfrom to wouldbelargerthanthepopulationsize.Tochooseexactly populationmembers,wesortthesolutionsofthe usingthecrowded-comparisonoperator indescendingorderandchoosethebestsolutionsneededtofillallpopulationslots.TheNSGA-IIprocedureisalsoshowninFig.2.Thenewpopulation ofsize isnowusedforse-lection,crossover,andmutationtocreateanewpopulation ofsize .Itisimportanttonotethatweuseabinarytournamentselectionoperatorbuttheselectioncriterionisnowbasedonthecrowded-comparisonoperator .Sincethisoperatorrequiresboththerankandcrowdeddistanceofeachsolutioninthepop-ulation,wecalculatethesequantitieswhileformingthepopula- ,asshownintheabovealgorithm.Considerthecomplexityofoneiterationoftheentirealgo-rithm.Thebasicoperationsandtheirworst-casecomplexitiesareasfollows:1)nondominatedsortingis 2)crowding-distanceassignmentis 3)sortingon is Theoverallcomplexityofthealgorithmis ,whichisgovernedbythenondominatedsortingpartofthealgorithm.If Fig.2.NSGA-IIprocedure.performedcarefully,thecompletepopulationofsize notbesortedaccordingtonondomination.Assoonasthesortingprocedurehasfoundenoughnumberoffrontstohave bersin ,thereisnoreasontocontinuewiththesortingpro-Thediversityamongnondominatedsolutionsisintroducedbyusingthecrowdingcomparisonprocedure,whichisusedinthetournamentselectionandduringthepopulationreductionphase.Sincesolutionscompetewiththeircrowding-distance(ameasureofdensityofsolutionsintheneighborhood),noextranichingparameter(suchas neededintheNSGA)isre-quired.Althoughthecrowdingdistanceiscalculatedintheob-jectivefunctionspace,itcanalsobeimplementedintheparam-eterspace,ifsodesired[3].However,inallsimulationsper-formedinthisstudy,wehaveusedtheobjective-functionspaceIV.SIMULATIONESULTSInthissection,wefirstdescribethetestproblemsusedtocomparetheperformanceofNSGA-IIwithPAESandSPEA.ForPAESandSPEA,wehaveidenticalparametersettingsassuggestedintheoriginalstudies.ForNSGA-II,wehavechosenareasonablesetofvaluesandhavenotmadeanyeffortinfindingthebestparametersetting.Weleavethistaskforafuturestudy. combineparentandoffspringpopulation - - - allnondominatedfrontsof and until untiltheparentpopulationisfilled - - calculatecrowding-distancein include thnondominatedfrontintheparentpop checkthenextfrontforinclusion sortindescendingorderusing choosethefirst elementsof - - useselection,crossoverandmutationtocreateanewpopulation incrementthegenerationcounter etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-IITABLEIROBLEMSSEDINTUDY Allobjectivefunctionsaretobeminimized.A.TestProblemsWefirstdescribethetestproblemsusedtocomparedifferentMOEAs.Testproblemsarechosenfromanumberofsignifi-cantpaststudiesinthisarea.Veldhuizen[22]citedanumberoftestproblemsthathavebeenusedinthepast.Ofthem,wechoosefourproblems:Schaffer’sstudy(SCH)[19],FonsecaandFleming’sstudy(FON)[10],Poloni’sstudy(POL)[16],andKursawe’sstudy(KUR)[15].In1999,thefirstauthorsuggestedasystematicwayofdevelopingtestproblemsformultiobjec-tiveoptimization[3].Zitzleretal.[25]followedthoseguide-linesandsuggestedsixtestproblems.WechoosefiveofthosesixproblemshereandcallthemZDT1,ZDT2,ZDT3,ZDT4,andZDT6.Allproblemshavetwoobjectivefunctions.Noneoftheseproblemshaveanyconstraint.Wedescribetheseprob-lemsinTableI.Thetablealsoshowsthenumberofvariables,theirbounds,thePareto-optimalsolutions,andthenatureofthePareto-optimalfrontforeachproblem.Allapproachesarerunforamaximumof25000functionevaluations.Weusethesingle-pointcrossoverandbitwisemutationforbinary-codedGAsandthesimulatedbinarycrossover(SBX)operatorandpolynomialmutation[6]forreal-codedGAs.Thecrossoverprobabilityof amutationprobabilityof or (where isthenumberofdecisionvariablesforreal-codedGAsand isthestringlengthforbinary-codedGAs)areused.Forreal-codedNSGA-II,weusedistributionindexes[6]forcrossoverandmutationoperatorsas and ,respectively.Thepopulationobtainedattheendof250generations(thepopulationafterelite-preservingoperatorisapplied)isusedtocalculateacoupleofperformancemetrics,whichwediscussinthenextsection.ForPAES,weuseadepthvalue tofourandanarchivesize of100.Weuseallpopulationmembersofthearchiveobtainedattheendof25000iterationstocalculatetheperformancemetrics.ForSPEA,weuseapopulationofsize80andanexternalpopulationofsize20(this4:1ratioissuggestedbythedevelopersofSPEAtomaintainanadequateselectionpressurefortheelitesolutions),sothatoverallpopulationsizebecomes100.SPEAisalsorununtil25000functionevaluationsaredone.ForSPEA,weusethe IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002 Fig.3.DistancemetricnondominatedsolutionsofthecombinedGAandexternalpopulationsatthefinalgenerationtocalculatetheperformancemetricsusedinthisstudy.ForPAES,SPEA,andbinary-codedNSGA-II,wehaveused30bitstocodeeachdecisionvariable.B.PerformanceMeasuresUnlikeinsingle-objectiveoptimization,therearetwogoalsinamultiobjectiveoptimization:1)convergencetothePareto-op-timalsetand2)maintenanceofdiversityinsolutionsofthePareto-optimalset.Thesetwotaskscannotbemeasuredade-quatelywithoneperformancemetric.Manyperformancemet-ricshavebeensuggested[1],[8],[24].Here,wedefinetwoper-formancemetricsthataremoredirectinevaluatingeachoftheabovetwogoalsinasolutionsetobtainedbyamultiobjectiveoptimizationalgorithm.Thefirstmetric measurestheextentofconvergencetoaknownsetofPareto-optimalsolutions.Sincemultiobjectiveal-gorithmswouldbetestedonproblemshavingaknownsetofPareto-optimalsolutions,thecalculationofthismetricispos-sible.Werealize,however,thatsuchametriccannotbeusedforanyarbitraryproblem.First,wefindasetof formlyspacedsolutionsfromthetruePareto-optimalfrontintheobjectivespace.Foreachsolutionobtainedwithanalgo-rithm,wecomputetheminimumEuclideandistanceofitfrom chosensolutionsonthePareto-optimalfront.Theaverageofthesedistancesisusedasthefirstmetric (theconver-gencemetric).Fig.3showsthecalculationprocedureofthismetric.TheshadedregionisthefeasiblesearchregionandthesolidcurvedlinesspecifythePareto-optimalsolutions.Solu-tionswithopencirclesare chosensolutionsonthePareto-op-timalfrontforthecalculationoftheconvergencemetricandso-lutionsmarkedwithdarkcirclesaresolutionsobtainedbyanalgorithm.Thesmallerthevalueofthismetric,thebettertheconvergencetowardthePareto-optimalfront.Whenallobtainedsolutionslieexactlyon chosensolutions,thismetrictakesavalueofzero.Inallsimulationsperformedhere,wepresenttheaverage andvariance ofthismetriccalculatedforsolutionsetsobtainedinmultipleruns.EvenwhenallsolutionsconvergetothePareto-optimalfront,theaboveconvergencemetricdoesnothaveavalueofzero.Themetricwillyieldzeroonlywheneachobtainedsolutionliesex-actlyoneachofthechosensolutions.Althoughthismetricalone Fig.4.Diversitymetriccanprovidesomeinformationaboutthespreadinobtainedso-lutions,wedefineandifferentmetrictomeasurethespreadinsolutionsobtainedbyanalgorithmdirectly.Thesecondmetric measurestheextentofspreadachievedamongtheobtainedsolutions.Here,weareinterestedingettingasetofsolutionsthatspanstheentirePareto-optimalregion.WecalculatetheEuclideandistance betweenconsecutivesolutionsintheob-tainednondominatedsetofsolutions.Wecalculatetheaverage ofthesedistances.Thereafter,fromtheobtainedsetofnon-dominatedsolutions,wefirstcalculatetheextremesolutions(intheobjectivespace)byfittingacurveparalleltothatofthetruePareto-optimalfront.Then,weusethefollowingmetrictocal-culatethenonuniformityinthedistribution: Here,theparameters and aretheEuclideandistancesbe-tweentheextremesolutionsandtheboundarysolutionsoftheobtainednondominatedset,asdepictedinFig.4.Thefigureil-lustratesalldistancesmentionedintheaboveequation.Thepa- istheaverageofalldistances , ,assumingthatthereare solutionsonthebestnondomi-natedfront.With solutions,thereare consecutivedistances.Thedenominatoristhevalueofthenumeratorforthecasewhenall solutionslieononesolution.Itisinterestingtonotethatthisisnottheworstcasespreadofsolutionspossible.Wecanhaveascenarioinwhichthereisalargevariancein Insuchscenarios,themetricmaybegreaterthanone.Thus,themaximumvalueoftheabovemetriccanbegreaterthanone.However,agooddistributionwouldmakealldistances equalto andwouldmake (withexistenceofextremesolutionsinthenondominatedset).Thus,forthemostwidelyanduniformlyspreadoutsetofnondominatedsolutions,thenu-meratorof wouldbezero,makingthemetrictotakeavaluezero.Foranyotherdistribution,thevalueofthemetricwouldbegreaterthanzero.Fortwodistributionshavingidenticalvalues and ,themetric takesahighervaluewithworsedistri-butionsofsolutionswithintheextremesolutions.Notethattheabovediversitymetriccanbeusedonanynondominatedsetofsolutions,includingonethatisnotthePareto-optimalset.Using etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-IITABLEIIOWSOWSOFTHE TABLEIIIOWSOWSOFTHE atriangularizationtechniqueoraVoronoidiagramapproach[1]tocalculate ,theaboveprocedurecanbeextendedtoestimatethespreadofsolutionsinhigherdimensions.C.DiscussionoftheResultsTableIIshowsthemeanandvarianceoftheconvergence obtainedusingfouralgorithmsNSGA-II(real-coded),NSGA-II(binary-coded),SPEA,andPAES.NSGA-II(realcodedorbinarycoded)isabletoconvergebetterinallproblemsexceptinZDT3andZDT6,wherePAESfoundbetterconvergence.InallcaseswithNSGA-II,thevari-anceintenrunsisalsosmall,exceptinZDT4withNSGA-II(binarycoded).ThefixedarchivestrategyofPAESallowsbetterconvergencetobeachievedintwooutofnineproblems.TableIIIshowsthemeanandvarianceofthediversitymetric obtainedusingallthreealgorithms.NSGA-II(realorbinarycoded)performsthebestinallninetestproblems.TheworstperformanceisobservedwithPAES.Forillustration,weshowoneofthetenrunsofPAESwithanar-bitraryrunofNSGA-II(real-coded)onproblemSCHinFig.5.Onmostproblems,real-codedNSGA-IIisabletofindabetterspreadofsolutionsthananyotheralgorithm,includingbinary-codedNSGA-II.Inordertodemonstratetheworkingofthesealgorithms,wealsoshowtypicalsimulationresultsofPAES,SPEA,andNSGA-IIonthetestproblemsKUR,ZDT2,ZDT4,andZDT6.TheproblemKURhasthreediscontinuousregionsinthePareto-optimalfront.Fig.6showsallnondominatedsolutionsobtainedafter250generationswithNSGA-II(real-coded).ThePareto-optimalregionisalsoshowninthefigure.ThisfiguredemonstratestheabilitiesofNSGA-IIinconvergingtothetruefrontandinfindingdiversesolutionsinthefront.Fig.7showstheobtainednondominatedsolutionswithSPEA,whichisthenext-bestalgorithmforthisproblem(refertoTablesIIandIII). Fig.5.NSGA-IIfindsbetterspreadofsolutionsthanPAESonSCH. Fig.6.NondominatedsolutionswithNSGA-II(real-coded)onKUR. IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002 Fig.7.NondominatedsolutionswithSPEAonKUR. Fig.8.NondominatedsolutionswithNSGA-II(binary-coded)onZDT2.Inbothaspectsofconvergenceanddistributionofsolutions,NSGA-IIperformedbetterthanSPEAinthisproblem.SinceSPEAcouldnotmaintainenoughnondominatedsolutionsinthefinalGApopulation,theoverallnumberofnondominatedsolutionsismuchlesscomparedtothatobtainedinthefinalpopulationofNSGA-II.Next,weshowthenondominatedsolutionsontheproblemZDT2inFigs.8and9.ThisproblemhasanonconvexPareto-op-timalfront.Weshowtheperformanceofbinary-codedNSGA-IIandSPEAonthisfunction.Althoughtheconvergenceisnotadifficultyherewithbothofthesealgorithms,bothreal-andbinary-codedNSGA-IIhavefoundabetterspreadandmoresolutionsintheentirePareto-optimalregionthanSPEA(thenext-bestalgorithmobservedforthisproblem).TheproblemZDT4has21 or7.94(10 )differentlocalPareto-optimalfrontsinthesearchspace,ofwhichonlyonecorrespondstotheglobalPareto-optimalfront.TheEuclideandistanceinthedecisionspacebetweensolutionsoftwocon-secutivelocalPareto-optimalsetsis0.25.Fig.10showsthatbothreal-codedNSGA-IIandPAESgetstuckatdifferentlocalPareto-optimalsets,buttheconvergenceandabilitytofindadiversesetofsolutionsaredefinitelybetterwithNSGA-II.Binary-codedGAshavedifficultiesinconverging Fig.9.NondominatedsolutionswithSPEAonZDT2. Fig.10.NSGA-IIfindsbetterconvergenceandspreadofsolutionsthanPAESonZDT4.neartheglobalPareto-optimalfront,amatterthatisalsobeenobservedinprevioussingle-objectivestudies[5].Onasimilarten-variableRastrigin’sfunction[thefunction thatstudyclearlyshowedthatapopulationofsizeofaboutatleast500isneededforsingle-objectivebinary-codedGAs(withtournamentselection,single-pointcrossoverandbitwisemutation)tofindtheglobaloptimumsolutioninmorethan50%ofthesimulationruns.Sincewehaveusedapopulationofsize100,itisnotexpectedthatamultiobjectiveGAwouldfindtheglobalPareto-optimalsolution,butNSGA-IIisabletofindagoodspreadofsolutionsevenatalocalPareto-optimalfront.SinceSPEAconvergespoorlyonthisproblem(seeTableII),wedonotshowSPEAresultsonthisfigure.Finally,Fig.11showsthatSPEAfindsabetterconvergedsetofnondominatedsolutionsinZDT6comparedtoanyotheralgorithm.However,thedistributioninsolutionsisbetterwithreal-codedNSGA-II.D.DifferentParameterSettingsInthisstudy,wedonotmakeanyseriousattempttofindthebestparametersettingforNSGA-II.Butinthissection,weper- etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-II Fig.11.Real-codedNSGA-IIfindsbetterspreadofsolutionsthanSPEAonZDT6,butSPEAhasabetterconvergence.TABLEIVEANANDARIANCEOFTHEONVERGENCEANDUPTO500GENERATIONS formadditionalexperimentstoshowtheeffectofacoupleofdifferentparametersettingsontheperformanceofNSGA-II.First,wekeepallotherparametersasbefore,butincreasethenumberofmaximumgenerationsto500(insteadof250usedbefore).TableIVshowstheconvergenceanddiversitymetricsforproblemsPOL,KUR,ZDT3,ZDT4,andZDT6.Now,weachieveaconvergenceveryclosetothetruePareto-optimalfrontandwithamuchbetterdistribution.Thetableshowsthatinallthesedifficultproblems,thereal-codedNSGA-IIhasconvergedveryclosetothetrueoptimalfront,exceptinZDT6,whichprob-ablyrequiresadifferentparametersettingwithNSGA-II.Par-ticularly,theresultsonZDT3andZDT4improvewithgenera-tionnumber.TheproblemZDT4hasanumberoflocalPareto-optimalfronts,eachcorrespondingtoparticularvalueof .Alargechangeinthedecisionvectorisneededtogetoutofalocaloptimum.Unlessmutationorcrossoveroperatorsarecapableofcreatingsolutionsinthebasinofanotherbetterattractor,theimprovementintheconvergencetowardthetruePareto-op-timalfrontisnotpossible.WeuseNSGA-II(real-coded)withasmallerdistributionindex formutation,whichhasaneffectofcreatingsolutionswithmorespreadthanbefore.Restoftheparametersettingsareidenticalasbefore.Theconver-gencemetric anddiversitymeasure onproblemZDT4attheendof250generationsareasfollows: Fig.12.ObtainednondominatedsolutionswithNSGA-IIonproblemZDT4.TheseresultsaremuchbetterthanPAESandSPEA,asshowninTableII.Todemonstratetheconvergenceandspreadofso-lutions,weplotthenondominatedsolutionsofoneoftherunsafter250generationsinFig.12.ThefigureshowsthatNSGA-IIisabletofindsolutionsonthetruePareto-optimalfrontwith V.ROTATEDROBLEMSIthasbeendiscussedinanearlierstudy[3]thatinteractionsamongdecisionvariablescanintroduceanotherlevelofdif-ficultytoanymultiobjectiveoptimizationalgorithmincludingEAs.Inthissection,wecreateonesuchproblemandinvesti-gatetheworkingofpreviouslythreeMOEAsonthefollowingepistaticproblem: minimize where and for AnEAworkswiththedecisionvariablevector ,buttheaboveobjectivefunctionsaredefinedintermsofthevariablevector whichiscalculatedbytransformingthedecisionvariablevector byafixedrotationmatrix .Thisway,theobjectivefunctionsarefunctionsofalinearcombinationofdecisionvariables.InordertomaintainaspreadofsolutionsoverthePareto-optimalregionorevenconvergetoanyparticularsolutionrequiresanEAtoupdatealldecisionvariablesinaparticularfashion.Withagenericsearchoperator,suchasthevariablewiseSBXoperatorusedhere,thisbecomesadifficulttaskforanEA.However,here,weareinterestedinevaluatingtheoverallbehaviorofthreeelitistMOEAs.Weuseapopulationsizeof100andruneachalgorithmuntil500generations.ForSBX,weuse andweuse formutation.TorestrictthePareto-optimalsolutionstolie IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002 Fig.13.ObtainednondominatedsolutionswithNSGA-II,PAES,andSPEAontherotatedproblem.withintheprescribedvariablebounds,wediscouragesolutions byaddingafixedlargepenaltytobothobjec-tives.Fig.13showstheobtainedsolutionsattheendof500generationsusingNSGA-II,PAES,andSPEA.ItisobservedthatNSGA-IIsolutionsareclosertothetruefrontcomparedtosolutionsobtainedbyPAESandSPEA.Thecorrelatedpa-rameterupdatesneededtoprogresstowardthePareto-optimalfrontmakesthiskindofproblemsdifficulttosolve.NSGA-II’selite-preservingoperatoralongwiththereal-codedcrossoverandmutationoperatorsisabletofindsomesolutionsclosetothePareto-optimalfront[with resulting Thisexampleproblemdemonstratesthatoneoftheknowndif-ficulties(thelinkageproblem[11],[12])ofsingle-objectiveop-timizationalgorithmcanalsocausedifficultiesinamultiobjec-tiveproblem.However,moresystematicstudiesareneededtoamplyaddressthelinkageissueinmultiobjectiveoptimization.VI.CInthepast,thefirstauthorandhisstudentsimplementedapenalty-parameterlessconstraint-handlingapproachforsingle-objectiveoptimization.Thosestudies[2],[6]haveshownhowatournamentselectionbasedalgorithmcanbeusedtohandleconstraintsinapopulationapproachmuchbetterthananumberofotherexistingconstraint-handlingapproaches.Asimilarap-proachcanbeintroducedwiththeaboveNSGA-IIforsolvingconstrainedmultiobjectiveoptimizationproblems.A.ProposedConstraint-HandlingApproach(ConstrainedThisconstraint-handlingmethodusesthebinarytournamentselection,wheretwosolutionsarepickedfromthepopulationandthebettersolutionischosen.Inthepresenceofconstraints,eachsolutioncanbeeitherfeasibleorinfeasible.Thus,theremaybeatmostthreesituations:1)bothsolutionsarefeasible;2)oneisfeasibleandotherisnot;and3)bothareinfeasible.Forsingleobjectiveoptimization,weusedasimpleruleforeachCase1)Choosethesolutionwithbetterobjectivefunctionvalue.Case2)Choosethefeasiblesolution.Case3)ChoosethesolutionwithsmalleroverallconstraintSinceinnocaseconstraintsandobjectivefunctionvaluesarecomparedwitheachother,thereisnoneedofhavinganypenaltyparameter,amatterthatmakestheproposedconstraint-handlingapproachusefulandattractive.Inthecontextofmultiobjectiveoptimization,thelattertwocasescanbeusedastheyareandthefirstcasecanberesolvedbyusingthecrowded-comparisonoperatorasbefore.TomaintainthemodularityintheproceduresofNSGA-II,wesimplymodifythedefinitionofbetweentwosolutions and Definition1:Asolution issaidtoconstrained-dominatea ,ifanyofthefollowingconditionsistrue.1)Solution isfeasibleandsolution isnot.2)Solutions and arebothinfeasible,butsolution hasasmalleroverallconstraintviolation.3)Solutions and arefeasibleandsolution dominatessolution Theeffectofusingthisconstrained-dominationprincipleisthatanyfeasiblesolutionhasabetternondominationrankthananyinfeasiblesolution.Allfeasiblesolutionsarerankedaccordingtotheirnondominationlevelbasedontheobjectivefunctionvalues.However,amongtwoinfeasiblesolutions,thesolutionwithasmallerconstraintviolationhasabetterrank.Moreover,thismodificationinthenondominationprincipledoesnotchangethecomputationalcomplexityofNSGA-II.TherestoftheNSGA-IIprocedureasdescribedearliercanbeusedasusual.Theaboveconstrained-dominationdefinitionissimilartothatsuggestedbyFonsecaandFleming[9].Theonlydifferenceisinthewaydominationisdefinedfortheinfeasiblesolutions.Intheabovedefinition,aninfeasiblesolutionhavingalargeroverallconstraint-violationareclassifiedasmembersofalargernondominationlevel.Ontheotherhand,in[9],infeasiblesolu-tionsviolatingdifferentconstraintsareclassifiedasmembersofthesamenondominatedfront.Thus,oneinfeasiblesolutionviolatingaconstraintmarginallywillbeplacedinthesamenondominatedlevelwithanothersolutionviolatingadifferentconstrainttoalargeextent.Thismaycauseanalgorithmtowanderintheinfeasiblesearchregionformoregenerationsbe-forereachingthefeasibleregionthroughconstraintboundaries.Moreover,sinceFonseca–Fleming’sapproachrequiresdomina-tioncheckswiththeconstraint-violationvalues,theproposedapproachofthispaperiscomputationallylessexpensiveandissimpler.B.Ray–Tai–Seow’sConstraint-HandlingApproachetal.[17]suggestedamoreelaborateconstraint-han-dlingtechnique,whereconstraintviolationsofallconstraintsarenotsimplysummedtogether.Instead,anondominationcheckofconstraintviolationsisalsomade.Wegiveanoutlineofthisprocedurehere. etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-IITABLEVROBLEMSSEDINTUDY Allobjectivefunctionsaretobeminimized.Threedifferentnondominatedrankingsofthepopulationarefirstperformed.Thefirstrankingisperformedusing tivefunctionvaluesandtheresultingrankingisstoredina mensionalvector .Thesecondranking isperformedusingonlytheconstraintviolationvaluesofall( ofthem)con-straintsandnoobjectivefunctioninformationisused.Thus,constraintviolationofeachconstraintisusedacriterionandanondominationclassificationofthepopulationisperformedwiththeconstraintviolationvalues.Noticethatforafeasiblesolutionallconstraintviolationsarezero.Thus,allfeasibleso-lutionshavearank1in .Thethirdrankingisperformedonacombinationofobjectivefunctionsandconstraint-violationvalues[atotalof values].Thisproducestheranking .Althoughobjectivefunctionvaluesandconstraintviola-tionsareusedtogether,oneniceaspectofthisalgorithmisthatthereisnoneedforanypenaltyparameter.Inthedominationcheck,criteriaarecomparedindividually,therebyeliminatingtheneedofanypenaltyparameter.Oncetheserankingsareover,solutionshavingthebestrankin arechosenforthenewpopulation.Ifmorepopulationslotsareavailable,theyarecreatedfromtheremainingsolutionssystematically.Bygivingimportancetotherankingin intheselectionop-eratorandbygivingimportancetotherankingin inthecrossoveroperator,theinvestigatorslaidoutasystematicmulti-objectiveGA,whichalsoincludesaniche-preservingoperator.Fordetails,readersmayreferto[17].Althoughtheinvestiga-torsdidnotcomparetheiralgorithmwithanyothermethod,theyshowedtheworkingofthisconstraint-handlingmethodonanumberofengineeringdesignproblems.However,sincenondominatedsortingofthreedifferentsetsofcriteriaarere-quiredandthealgorithmintroducesmanydifferentoperators,itremainstobeinvestigatedhowitperformsonmorecomplexproblems,particularlyfromthepointofviewofcomputationalburdenassociatedwiththemethod.Inthefollowingsection,wechooseasetoffourprob-lemsandcomparethesimpleconstrainedNSGA-IIwiththeRay–Tai–Seow’smethod.C.SimulationResultsWechoosefourconstrainedtestproblems(seeTableV)thathavebeenusedinearlierstudies.Inthefirstproblem,apartoftheunconstrainedPareto-optimalregionisnotfeasible.Thus,theresultingconstrainedPareto-optimalregionisaconcatena-tionofthefirstconstraintboundaryandsomepartoftheuncon-strainedPareto-optimalregion.ThesecondproblemSRNwasusedintheoriginalstudyofNSGA[20].Here,theconstrainedPareto-optimalsetisasubsetoftheunconstrainedPareto-op-timalset.ThethirdproblemTNKwassuggestedbyTanaka[21]andhasadiscontinuousPareto-optimalregion,fallingentirelyonthefirstconstraintboundary.Inthenextsection,weshowtheconstrainedPareto-optimalregionforeachoftheaboveproblems.ThefourthproblemWATERisafive-objec-tiveandseven-constraintproblem,attemptedtosolvein[17].Withfiveobjectives,itisdifficulttodiscusstheeffectoftheconstraintsontheunconstrainedPareto-optimalregion.Inthenextsection,weshowall ortenpairwiseplotsofobtainednondominatedsolutions.Weapplyreal-codedNSGA-IIhere.Inallproblems,weuseapopulationsizeof100,distribu-tionindexesforreal-codedcrossoverandmutationoperatorsof20and100,respectively,andrunNSGA-II(realcoded)withtheproposedconstraint-handlingtechniqueandwithRay–Tai–Seow’sconstraint-handlingalgorithm[17]foramaximumof500generations.Wechoosethisratherlargenumberofgenerationstoinvestigateifthespreadinsolutions IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002 Fig.14.ObtainednondominatedsolutionswithNSGA-IIontheconstrainedproblemCONSTR. Fig.15.ObtainednondominatedsolutionswithRay-Tai-Seow’salgorithmontheconstrainedproblemCONSTR.canbemaintainedforalargenumberofgenerations.However,ineachcase,weobtainareasonablygoodspreadofsolutionsasearlyas200generations.Crossoverandmutationprobabilitiesarethesameasbefore.Fig.14showstheobtainedsetof100nondominatedsolu-tionsafter500generationsusingNSGA-II.ThefigureshowsthatNSGA-IIisabletouniformlymaintainsolutionsinbothPareto-optimalregion.Itisimportanttonotethatinordertomaintainaspreadofsolutionsontheconstraintboundary,thesolutionsmusthavetobemodifiedinaparticularmannerdic-tatedbytheconstraintfunction.Thisbecomesadifficulttaskofanysearchoperator.Fig.15showstheobtainedsolutionsusingRay-Tai-Seow’salgorithmafter500generations.ItisclearthatNSGA-IIperformsbetterthanRay–Tai–Seow’salgorithmintermsofconvergingtothetruePareto-optimalfrontandalsointermsofmaintainingadiversepopulationofnondominatedNext,weconsiderthetestproblemSRN.Fig.16showsthenondominatedsolutionsafter500generationsusingNSGA-II. Fig.16.ObtainednondominatedsolutionswithNSGA-IIontheconstrainedproblemSRN. Fig.17.ObtainednondominatedsolutionswithRay–Tai–Seow’salgorithmontheconstrainedproblemSRN.ThefigureshowshowNSGA-IIcanbringarandompopulationonthePareto-optimalfront.Ray–Tai–Seow’salgorithmisalsoabletocomeclosetothefrontonthistestproblem(Fig.17).Figs.18and19showthefeasibleobjectivespaceandtheobtainednondominatedsolutionswithNSGA-IIandRay–Tai–Seow’salgorithm.Here,thePareto-optimalregionisdiscontinuousandNSGA-IIdoesnothaveanydifficultyinfindingawidespreadofsolutionsoverthetruePareto-optimalregion.AlthoughRay–Tai–Seow’salgorithmfoundanumberofsolutionsonthePareto-optimalfront,thereexistmanyinfeasiblesolutionsevenafter500generations.InordertodemonstratetheworkingofFonseca–Fleming’sconstraint-han-dlingstrategy,weimplementitwithNSGA-IIandapplyonTNK.Fig.20shows100populationmembersattheendof500generationsandwithidenticalparametersettingasusedinFig.18.BoththesefiguresdemonstratethattheproposedandFonseca–Fleming’sconstraint-handlingstrategiesworkwellwithNSGA-II. etal.:AFASTANDELITISTMULTIOBJECTIVEGA:NSGA-II Fig.18.ObtainednondominatedsolutionswithNSGA-IIontheconstrainedproblemTNK. Fig.19.ObtainednondominatedsolutionswithRay–Tai–Seow’salgorithmontheconstrainedproblemTNK.etal.[17]haveusedtheproblemWATERintheirstudy.Theynormalizedtheobjectivefunctionsinthefollowing SincetherearefiveobjectivefunctionsintheproblemWATER,weobservetherangeofthenormalizedobjectivefunctionvaluesoftheobtainednondominatedsolutions.TableVIshowsthecomparisonwithRay–Tai–Seow’salgorithm.Inmostobjectivefunctions,NSGA-IIhasfoundabetterspreadofsolutionsthanRay–Tai–Seow’sapproach.Inordertoshowthepairwiseinteractionsamongthesefivenormalizedobjectivefunctions,weplotall orteninteractionsinFig.21forbothalgorithms.NSGA-IIresultsareshownintheupperdiagonalportionofthefigureandtheRay–Tai–Seow’sresultsareshowninthelowerdiagonalportion.Theaxesofanyplotcanbeobtainedbylookingatthecorrespondingdiagonalboxesandtheirranges.Forexample,theplotatthefirstrowandthirdcolumnhasitsverticalaxisas andhorizontalaxisas Sincethisplotbelongsintheuppersideofthediagonal,this Fig.20.ObtainednondominatedsolutionswithFonseca–Fleming’sconstraint-handlingstrategywithNSGA-IIontheconstrainedproblemTNK.plotisobtainedusingNSGA-II.InordertocomparethisplotwithasimilarplotusingRay–Tai–Seow’sapproach,welookfortheplotinthethirdrowandfirstcolumn.Forthisfigure,theverticalaxisisplottedas andthehorizontalaxisisplotted .Togetabettercomparisonbetweenthesetwoplots,weobserveRay–Tai–Seow’splotasitis,butturnthepage90 theclockwisedirectionforNSGA-IIresults.Thiswouldmakethelabelingandrangesoftheaxessameinbothcases.WeobservethatNSGA-IIplotshavebetterformedpatternsthaninRay–Tai–Seow’splots.Forexample,figures - , - ,and - interactionsareveryclearfromNSGA-IIresults.AlthoughsimilarpatternsexistintheresultsobtainedusingRay–Tai–Seow’salgorithm,theconvergencetothetruefrontsisnotadequate.VII.CWehaveproposedacomputationallyfastandelitistMOEAbasedonanondominatedsortingapproach.Onninedifferentdifficulttestproblemsborrowedfromtheliterature,thepro-posedNSGA-IIwasabletomaintainabetterspreadofsolu-tionsandconvergebetterintheobtainednondominatedfrontcomparedtotwootherelitistMOEAs—PAESandSPEA.How-ever,oneproblem,PAES,wasabletoconvergeclosertothetruePareto-optimalfront.PAESmaintainsdiversityamongsolutionsbycontrollingcrowdingofsolutionsinadeterministicandpre-specifiednumberofequal-sizedcellsinthesearchspace.Inthatproblem,itissuspectedthatsuchadeterministiccrowdingcoupledwiththeeffectofmutation-basedapproachhasbeenbeneficialinconvergingnearthetruefrontcomparedtothedy-namicandparameterlesscrowdingapproachusedinNSGA-IIandSPEA.However,thediversitypreservingmechanismusedinNSGA-IIisfoundtobethebestamongthethreeapproachesstudiedhere.Onaproblemhavingstrongparameterinteractions,NSGA-IIhasbeenabletocomeclosertothetruefrontthantheothertwoapproaches,buttheimportantmatteristhatallthreeapproachesfaceddifficultiesinsolvingthisso-calledhighlyepistaticproblem.Althoughthishasbeenamatterofongoing IEEETRANSACTIONSONEVOLUTIONARYCOMPUTATION,VOL.6,NO.2,APRIL2002TABLEVIOWERANDOUNDSOFTHEBSERVEDINTHEBTAINEDONDOMINATED Fig.21.UpperdiagonalplotsareforNSGA-IIandlowerdiagonalplotsareforRay–Tai–Seow’salgorithm.Comparei;jplot(Ray–Tai–Seow’salgorithm�ij)withj;iplot(NSGA-II).Labelandrangesusedforeachaxisareshowninthediagonalboxes.researchinsingle-objectiveEAstudies,thispapershowsthathighlyepistaticproblemsmayalsocausedifficultiestoMOEAs.Moreimportantly,researchersinthefieldshouldconsidersuchepistaticproblemsfortestinganewlydevelopedalgorithmformultiobjectiveoptimization.Wehavealsoproposedasimpleextensiontothedefinitionofdominanceforconstrainedmultiobjectiveoptimization.Al-thoughthisnewdefinitioncanbeusedwithanyotherMOEAs,thereal-codedNSGA-IIwiththisdefinitionhasbeenshowntosolvefourdifferentproblemsmuchbetterthananotherre-cently-proposedconstraint-handlingapproach.Withthepropertiesofafastnondominatedsortingprocedure,anelitiststrategy,aparameterlessapproachandasimpleyetefficientconstraint-handlingmethod,NSGA-II,shouldfindin-creasingattentionandapplicationsinthenearfuture.[1]K.Deb,MultiobjectiveOptimizationUsingEvolutionaryAlgo-.Chichester,U.K.:Wiley,2001. ,“Anefficientconstraint-handlingmethodforgeneticalgorithms,”Comput.MethodsAppl.Mech.Eng.,vol.186,no.2–4,pp.311–338, ,“Multiobjectivegeneticalgorithms:Problemdifficultiesandcon-structionoftestfunctions,”inEvol.Comput.,1999,vol.7,pp.205–230.[4]K.DebandD.E.Goldberg,“Aninvestigationofnicheandspeciesfor-mationingeneticfunctionoptimization,”inProceedingsoftheThirdIn-ternationalConferenceonGeneticAlgorithms,J.D.Schaffer,Ed.SanMateo,CA:MorganKauffman,1989,pp.42–50.[5]K.DebandS.Agrawal,“Understandinginteractionsamonggeneticalgorithmparameters,”inFoundationsofGeneticAlgorithmsV,W.BanzhafandC.Reeves,Eds.SanMateo,CA:MorganKauffman,1998,pp.265–286.[6]K.DebandR.B.Agrawal,“Simulatedbinarycrossoverforcontinuoussearchspace,”inComplexSyst.,Apr.1995,vol.9,pp.115–148. 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,“Multiobjectiveoptimizationandmultipleconstrainthandlingwithevolutionaryalgorithms—PartII:Applicationexample,”Trans.Syst.,Man,Cybern.A,vol.28,pp.38–47,Jan.1998.[11]D.E.Goldberg,B.Korb,andK.Deb,“Messygeneticalgorithms:Mo-tivation,analysis,andfirstresults,”inComplexSyst.,Sept.1989,vol.3,pp.93–530.[12]G.Harik,“Learninggenelinkagetoefficientlysolveproblemsofboundeddifficultyusinggeneticalgorithms,”llinoisGeneticAlgo-rithmsLab.,Univ.IllinoisatUrbana-Champaign,Urbana,IL,IlliGALRep.97005,1997.[13]J.Horn,N.Nafploitis,andD.E.Goldberg,“AnichedParetogeneticalgorithmformultiobjectiveoptimization,”inProceedingsoftheFirstIEEEConferenceonEvolutionaryComputation,Z.Michalewicz,Ed.Piscataway,NJ:IEEEPress,1994,pp.82–87.[14]J.KnowlesandD.Corne,“TheParetoarchivedevolutionstrategy:Anewbaselinealgorithmformultiobjectiveoptimization,”inProceedingsofthe1999CongressonEvolutionaryComputation.Piscataway,NJ:IEEEPress,1999,pp.98–105.[15]F.Kursawe,“Avariantofevolutionstrategiesforvectoroptimization,”ParallelProblemSolvingfromNature,H.-P.SchwefelandR.Männer,Eds.Berlin,Germany:Springer-Verlag,1990,pp.193–197.[16]C.Poloni,“HybridGAformultiobjectiveaerodynamicshapeoptimiza-tion,”inGeneticAlgorithmsinEngineeringandComputerScience,G.Winter,J.Periaux,M.Galan,andP.Cuesta,Eds.NewYork:Wiley,1997,pp.397–414.[17]T.Ray,K.Tai,andC.Seow,“Anevolutionaryalgorithmformultiobjec-tiveoptimization,”Eng.Optim.,vol.33,no.3,pp.399–424,2001.[18]G.Rudolph,“Evolutionarysearchunderpartiallyorderedsets,”Dept.Comput.Sci./LS11,Univ.Dortmund,Dortmund,Germany,Tech.Rep.CI-67/99,1999.[19]J.D.Schaffer,“Multipleobjectiveoptimizationwithvectorevaluatedgeneticalgorithms,”inProceedingsoftheFirstInternationalConfer-enceonGeneticAlgorithms,J.J.Grefensttete,Ed.Hillsdale,NJ:LawrenceErlbaum,1987,pp.93–100.[20]N.SrinivasandK.Deb,“Multiobjectivefunctionoptimizationusingnondominatedsortinggeneticalgorithms,”Evol.Comput.,vol.2,no.3,pp.221–248,Fall1995.[21]M.Tanaka,“GA-baseddecisionsupportsystemformulticriteriaopti-mization,”inProc.IEEEInt.Conf.Systems,ManandCybernetics-21995,pp.1556–1561.[22]D.VanVeldhuizen,“Multiobjectiveevolutionaryalgorithms:Classifica-tions,analyzes,andnewinnovations,”AirForceInst.Technol.,Dayton,OH,Tech.Rep.AFIT/DS/ENG/99-01,1999.[23]D.VanVeldhuizenandG.Lamont,“Multiobjectiveevolutionaryalgorithmresearch:Ahistoryandanalysis,”AirForceInst.Technol.,Dayton,OH,Tech.Rep.TR-98-03,1998.[24]E.Zitzler,“Evolutionaryalgorithmsformultiobjectiveoptimization:Methodsandapplications,”DoctoraldissertationETH13398,SwissFederalInstituteofTechnology(ETH),Zurich,Switzerland,1999.[25]E.Zitzler,K.Deb,andL.Thiele,“Comparisonofmultiobjectiveevolu-tionaryalgorithms:Empiricalresults,”Evol.Comput.,vol.8,no.2,pp.173–195,Summer2000.[26]E.ZitzlerandL.Thiele,“Multiobjectiveoptimizationusingevolu-tionaryalgorithms—Acomparativecasestudy,”inParallelProblemSolvingFromNature,V,A.E.Eiben,T.Bäck,M.Schoenauer,andH.-P.Schwefel,Eds.Berlin,Germany:Springer-Verlag,1998,pp. KalyanmoyDeb(A’02)receivedtheB.TechdegreeinmechanicalengineeringfromtheIndianInstituteofTechnology,Kharagpur,India,1985andtheM.S.andPh.D.degreesinengineeringmechanicsfromtheUniversityofAlabama,Tuscaloosa,in1989and1991,respectively.HeiscurrentlyaProfessorofMechanicalEn-gineeringwiththeIndianInstituteofTechnology,Kanpur,India.Hehasauthoredorcoauthoredover100researchpapersinjournalsandconfer-ences,anumberofbookchapters,andtwobooks:MultiobjectiveOptimizationUsingEvolutionaryAlgorithms(Chichester,U.K.:Wiley,2001)andOptimizationforEngineeringDesign(NewDelhi,India:Prentice-Hall,1995).Hiscurrentresearchinterestsareinthefieldofevolutionarycomputation,particularlyintheareasofmulticriterionandreal-parameterevolutionaryalgorithms.Dr.DebisanAssociateEditorofIEEETRANSACTIONSONVOLUTIONARYOMPUTATIONandanExecutiveCouncilMemberoftheInternationalSocietyonGeneticandEvolutionaryComputation. AmritPratapwasborninHyderabad,India,onAu-gust27,1979.HereceivedtheM.S.degreeinmath-ematicsandscientificcomputingfromtheIndianIn-stituteofTechnology,Kanpur,India,in2001.HeisworkingtowardthePh.D.degreeincomputerscienceattheCaliforniaInstituteofTechnology,Pasadena,HewasamemberoftheKanpurGeneticAl-gorithmsLaboratory.HeiscurrentlyaMemberoftheCaltechLearningSystemsGroup.Hiscurrentresearchinterestsincludeevolutionarycomputation,machinelearning,andneuralnetworks. SameerAgarwalwasborninBulandshahar,India,onFebruary19,1977.HereceivedtheM.S.degreeinmathematicsandscientificcomputingfromtheIn-dianInstituteofTechnology,Kanpur,India,in2000.HeisworkingtowardthePh.D.degreeincomputerscienceatUniversityofCalifornia,SanDiego.HewasaMemberoftheKanpurGeneticAlgo-rithmsLaboratory.Hisresearchinterestsincludeevo-lutionarycomputationandlearningbothinhumansaswellasmachines.Heiscurrentlydevelopinglearningmethodsforlearningbyimitation. T.MeyarivanwasborninHaldia,India,onNovember23,1977.HeisworkingtowardtheM.S.degreeinchemistryfromIndianInstituteofTechnology,Kanpur,India.HeisaMemberoftheKanpurGeneticAlgorithmsLaboratory.Hiscurrentresearchinterestsincludeevolutionarycomputationanditsapplicationstobiologyandvariousfieldsinchemistry.