Department of Mechanical Engineering Indian Institute of echnology Kanpur PIN 208016 India debjsundar udaya iitka cin httpwww iitkacinkangal KanGAL Report Number 2005012 Abstract Ev olutionary multiobjecti optimization EMO methodologies ha been ampl ID: 33330
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ReferencePointBasedMulti-ObjectiveOptimizationUsingEvolutionaryAlgorithmsKalyanmoyDeb,J.SundarandUdayaBhaskaraRaoN.DepartmentofMechanicalEngineering,IndianInstituteofTechnologyKanpur,PIN208016,Indiafdeb,jsundar,udayag@iitk.ac.inhttp://www.iitk.ac.in/kangal/KanGALReportNumber2005012Abstract:Evolutionarymulti-objectiveoptimization(EMO)methodologieshavebeenamplyappliedtondarepresentativesetofPareto-optimalsolutionsinthepastdecadeandbeyond.Althoughthereareadvan-tagesofknowingtherangeofeachobjectiveforPareto-optimalityandtheshapeofthePareto-optimalfrontieritselfinaproblemforanadequatedecision-making,thetaskofchoosingasinglepreferredPareto-optimalsolu-tionisalsoanimportanttaskwhichhasreceivedaluke-warmattentionsofar.Inthispaper,wecombineonesuchpreference-basedstrategywithanEMOmethod-ologyanddemonstratehow,insteadofonesolution,apreferredsetsolutionsnearthereferencepointscanbefoundparallely.Weproposetwoapproachesforthistask:(i)amodiedEMOprocedurebasedontheelitistnon-dominatedsortingGAorNSGA-II[1]and(ii)apredator-preyapproachbasedonoriginalgridbasedprocedure[2].Ontwo-objectiveto10-objectiveoptimizationtestproblems,themodiedNSGA-IIapproachshowsitsef-cacyinndinganadequatesetofPareto-optimalpoints.Ontwoandthree-objectiveproblems,thepredator-preyapproachalsodemonstrateitsusefulness.Suchproce-dureswillprovidethedecision-makerwithasetofsolu-tionsnearher/hispreferencesothatabetterandamorereliabledecisioncanbemade.Keywords:Referencepointapproach,interactivemulti-objectivemethod,decision-making,predator-preyap-proach,multi-objectiveoptimization.I.IntroductionForthepast15yearsorso,evolutionarymulti-objectiveop-timization(EMO)methodologieshaveadequatelydemon-stratedtheirusefulnessinndingawell-convergedandwell-distributedsetofnearPareto-optimalsolutions[3,4].Duetotheseextensivestudiesandavailablesourcecodesbothcom-merciallyandfreely,theEMOprocedureshavebeenpopu-larlyappliedinvariousproblem-solvingtasksandhavere-ceivedagreatdealofattentionevenbytheclassicalmulti-criterionoptimizationanddecision-makingcommunities.However,recentstudies[5]havediscoveredthatatleastoneoftheEMOmethodologiesNSGA-II[1]facesdifcultyinsolvingproblemswithalargenumberofobjectives.Thedifcultiesareasfollows:(i)thevisualizationoffourormoreobjectivespaceisadifcultywhichmaylimitEMOmethodologiesforndingtheentirePareto-optimalset,(ii)theemphasisofallnon-dominatedsolutionsinapopulationforalargenumberofobjectivesmaynotproduceenoughselectionpressureforasmall-sizedpopulationtomoveto-wardsthePareto-optimalregionfastenoughand(iii)thereisaneedofanexponentiallymorenumberofpointstorepre-sentahigher-dimensionalPareto-optimalfront.Althoughtheuseofalargepopulationandabettervisualizationtechniquemayextendtheirapplicationsinsolvingveorsoobjectives,butif10ormoreobjectivesaretosolved,thereexistsacon-siderableamountofdoubttotheuseanEMOprocedureinndingawell-representativesetofPareto-optimalsolutions.Inlarge-objectiveproblem-solving,EMOmethodologiescanbeputtobenetinndingapreferredandsmallersetofPareto-optimalsolutions,insteadoftheentirefrontier.Thisapproachhasapracticalviewpointandallowsadecision-makertoconcentrateonlytothoseregionsonthePareto-optimalfrontierwhichareofinteresttoher/him.EMOmethodologiesmayprovideanadvantageovertheirclassicalcounterpartsforanotherpragmaticreason,whichwediscussnext.Theclassicalinteractivemulti-criterionoptimizationmeth-odsdemandthedecision-makerstosuggestareferencedi-rectionorreferencepointsorotherclues[6]whichresultinapreferredsetofsolutionsonthePareto-optimalfront.Intheseclassicalapproaches,basedonsuchclues,asingle-objectiveoptimizationproblemisusuallyformedandasin-glesolutionisfound.Asinglesolution(althoughoptimalcorrespondingtothegivenclue)doesnotprovideagoodideaofthepropertiesofsolutionsnearthedesiredregionofthefront.Byprovidingaclue,thedecision-makerisnotusuallylookingforasinglesolution,rathershe/heisinter-estedinknowingthepropertiesofsolutionswhichcorre-spondtotheoptimumandnear-optimumsolutionsrespectingtheclue.Thisisbecausewhileprovidingtheclueintermsofweightvectorsorreferencedirectionsorreferencepoints,1 thedecision-makerhassimplyprovidedahigher-levelinfor-mationabouther/hischoice.Ideally,byprovidinganumberofsuchclues,thedecision-makerinthebeginningisinter-estedinchoosingaregionofher/hisinterest.Weherearguethatinsteadofndingasinglesolutionneartheregionofin-terest,ifanumberofsolutionsintheregionofinterestarefound,thedecision-makerwillbeabletomakeabetterandmorereliabledecision.Moreover,ifmultiplesuchregionsofinterestcanbefoundsimultaneously,decision-makerscanmakeamoreeffectiveandparallelsearchtowardsndinganultimatepreferredsolution.Inthispaper,weusetheconceptofreferencepointmethodol-ogyinanEMOandattempttondasetofpreferredPareto-optimalsolutionsneartheregionsofinteresttoadecision-maker.Wesuggesttwoapproachesforthispurpose.ThemodiedNSGA-IIapproachisabletosolveasmanyas10objectiveseffectivelyandthepredator-preyapproachwithitscurrentimplementationexhibitsitspotentialontwoandthree-objectiveoptimizationproblems.Allsimulationrunsontestproblemsandonsomeengineeringdesignproblemsamplydemonstratetheirusefulnessinpracticeandshowan-otheruseofahybrid-EMOmethodologyinallowingthedecision-makertosolvemulti-objectiveoptimizationprob-lemsbetterandwithmorecondence.II.Preference-BasedEMOAp-proachesInthecontextofndingapreferredsetofsolutions,insteadoftheentirePareto-optimalsolutions,quiteafewstudieshavebeenmadeinthepast.TheapproachbyDeb[7]wasmotivatedbythegoalprogrammingidea[8]andrequiredtheDMtospecifyagoaloranaspirationlevelforeachob-jective.Basedonthatinformation,DebmodiedhisNSGAapproachtondasetofsolutionswhichareclosesttothesuppliedgoalpoint,ifthegoalpointisaninfeasiblesolu-tionandndthesolutionwhichcorrespondtothesuppliedgoalobjectivevector,ifitisafeasibleone.ThemethoddidnotcarendingthePareto-optimalsolutionscorrespondingtothemulti-objectiveoptimizationproblem,ratherattemptedtondsolutionssatisfyingthesuppliedgoals.Theweighted-sumapproachformulti-objectiveoptimizationwasutilizedbyanumberofresearchersinndingafewpreferredsolutions.ThemethodbyCvetkovicandParmee[9]assignedeachcriterionaweightwi,andadditionallyre-quiredaminimumlevelfordominance .Then,thedeni-tionofdominancewasredenedasfollows:x y,Xi:fi(x)fi(y)wi ;withastrictinequalityforatleastoneobjective.Tofacil-itatespecicationoftherequiredweights,theysuggestedamethodtoturnfuzzypreferencesintospecicquantitativeweights.However,sinceforeverycriterionthedominanceschemeonlyconsiderswhetheronesolutionisbetterthananothersolution,andnotbyhowmuchitisbetter,thisap-proachallowsonlyaverycoarseguidanceandisdifculttocontrol.JinandSendhoffalsoproposedawaytocon-vertfuzzypreferencesintoweightintervals,andthenusedtheirdynamicweightedaggregationEA[10]toobtainthecorrespondingsolutions.Thisapproachconvertedthemulti-objectiveoptimizationproblemintoasingleobjectiveopti-mizationproblembyweightedaggregation,butvariedtheweightsdynamicallyduringtheoptimizationrunwithintherelevantboundaries.IntheGuidedMulti-ObjectiveEvolutionaryAlgorithm(G-MOEA)proposedbyBrankeetal.[11],userpreferencesweretakenintoaccountbymodifyingthedenitionofdom-inance.TheapproachallowedtheDMtospecify,foreachpairofobjectives,maximallyacceptabletrade-offs.Forex-ample,inthecaseoftwoobjectives,theDMcoulddenethatanimprovementbyoneunitinobjectivef2isworthadegradationofobjectivef1byatmosta12units.Similarly,againinobjectivef1byoneunitisworthatmosta21unitsofobjectivef2.Thisinformationisthenusedtomodifythedominanceschemeasfollowsfortwoobjectives:x y,(f1(x)+a12f2(x)f1(y)+a12f2(y))^(a21f1(x)+f2(x)a21f1(y)+f2(y));withinequalityinatleastonecase.Althoughtheideaworksquitewellfortwoobjectivesandwaswellutilizedfordis-tributedcomputingpurposeselsewhere[12],providingallpair-wiseinformationinaproblemhavingalargenumberofobjectivesbecomesarealdifculty.InordertondabiaseddistributionanywhereonthePareto-optimalfront,apreviousstudy[13]usedabiasedtnesssharingapproachandimplementedonNSGA.Basedonaweightvectorspecifyingtheimportanceofoneobjectivefunctionovertheother,abiaseddistributionwasobtainedontwo-objectiveproblems.However,theapproachcouldnotbeusedtoobtainabiaseddistributionanywhereonthePareto-optimalfrontandinancontrolledmanner.Recently,BrankeandDeb[14]suggestedamodiedandcontrollablebiasessharingapproachinwhichbyspecify-ingareferencedirection(oralinearutilityfunction),asetofPareto-optimalsolutionsnearthebestsolutionoftheutil-ityfunctionwerefound.Toimplement,allsolutionswereprojectedontothelinearhyper-planeandcrowdingdistancevalueswerecomputedbytheratioofthedistancesofneigh-boringsolutionsintheoriginalobjectivespaceandontheprojectedhyper-plane.Thus,solutionswhichlieonaplaneparalleltothechosenhyper-planewouldhaveacompara-tivelylargecrowdingdistanceandwouldbepreferred.Thecompleteprocesswasshowntoconvergeneartotheoptimalsolutiontotheutilityfunctioninanumberoftwoandthree-objectiveoptimizationproblems.Theproceduredemandedtwouser-denedparameters:areferencedirectionandapa-rameterwhichcontrolstheextentofdiversityneededinthenalsetofsolutions.Theabovepreference-basedproceduresareusefulintheirownmeritsandaresomewaystondapreferredsetofPareto-optimalsolutions.However,eachoftheabovemethodologies,includingthemodiedbiasedsharingap-proach,cannotbeusedforndingpointscorrespondingtomultiplepreferenceconditionssimultaneously.Inthispaper,wemakeusesomeoftheaboveprinciplesandsuggestacou-pleofnewprocedureswhichhavethefollowingcapabilities:1.Multiplepreferenceconditionscanbespeciedsimul-taneously. 2.Foreachpreferencecondition,asetofPareto-optimalsolutionsisthetargetsetofsolutions,insteadofonesolution.3.ThemethodisindifferenttotheshapeofthePareto-optimalfrontier(suchasconvexornon-convex,contin-uousordiscrete,connectedordisconnectedandothers).4.Themethodisapplicabletoalargenumberofobjectives(say,10ormore),alargenumberofvariables,andlinearornon-linearconstraints.Theproceduresofthispaperaresomeotherwaysofndingapreferredsetofsolutionsinaninteractivemulti-objectiveoptimizationproblem,whicharemotivatedbytheclassicalreferencepointapproach,whichwediscussnext.III.ReferencePointInteractiveAp-proachAsanalternativetothevaluefunctionmethods,Wierzbicki[15]suggestedthereferencepointapproachinwhichthegoalistoachieveaweakly,-properlyorPareto-optimalsolutionclosesttoasuppliedreferencepointofaspirationlevelbasedonsolvinganachievementscalarizingproblem.Givenaref-erencepointzforanM-objectiveoptimizationproblemofminimizing(f1(x);:::;fi(x))withx2S,thefollowingsingle-objectiveoptimizationproblemissolvedforthispur-pose:MinimizemaxM=1[wi(fi(x) zi)];Subjecttox2S:(1)Here,wiisthei-thcomponentofachosenweightvectorusedforscalarizingtheobjectives.Figure1illustratestheconcept.Forachosenreferencepoint,theclosestPareto-zAzBf1f2w2w1zz'Figure.1:Classicalreferencepointapproach.optimalsolution(inthesenseoftheweighted-sumoftheob-jectives)isthetargetsolutiontothereferencepointmethod.Tomaketheprocedureinteractiveandusefulinpractice,Wierzbicki[15]suggestedaprocedureinwhichtheobtainedsolutionz0isusedtocreateMnewreferencepoints,asfol-lows:z(j)=z+(z0 z)e(j);(2)wheree(j)isthej-thcoordinatedirectionvector.Forthetwo-objectiveproblemshowninthegure,twosuchnewreferencepoints(zAandzB)arealsoshown.NewPareto-optimalsolutionsarethenfoundbyformingnewachieve-mentscalarizingproblems.Ifthedecision-makerisnotsat-isedwithanyofthesePareto-optimalsolutions,anewrefer-encepointissuggestedandtheaboveprocedureisrepeated.Itisinterestingtonotethatthereferencepointmaybeafea-sibleone(deduciblefromasolutionvector)oraninfeasiblepointwhichcannotbeobtainedfromanysolutionfromthefeasiblesearchspace.IfareferencepointisfeasibleandisnotaPareto-optimalsolution,thedecision-makermaythenbeinterestedinknowingsolutionswhicharePareto-optimalandclosetothereferencepoint.Ontheotherhand,iftheref-erencepointisaninfeasibleone,thedecision-makerwouldbeinterestedinndingPareto-optimalsolutionswhichareclosetothesuppliedreferencepoint.Toutilizethereferencepointapproachinpractice,thedecision-makerneedstosupplyareferencepointandaweightvectoratatime.ThelocationofthereferencepointcausestheproceduretofocusonacertainregioninthePareto-optimalfrontier,whereasasuppliedweightvectormakesanertrade-offamongthetheobjectivesandfo-cussestheproceduretondasinglePareto-optimalsolution(inmostsituations)trading-offtheobjectives.Thus,theref-erencepointprovidesahigher-levelinformationabouttheregiontofocusandweightvectorprovidesamoredetailedinformationaboutwhatpointonthePareto-optimalfronttoconverge.IV.ProposedReferencePointBasedEMOApproachTheclassicalreferencepointapproachdiscussedabove,willndasolutiondependingonthechosenweightvectorandisthereforesubjective.Moreover,thesinglesolutionisspecictothechosenweightvectoranddoesnotprovideanyinfor-mationabouthowthesolutionwouldchangewithaslightchangeintheweightvector.Tondasolutionforanotherweightvector,anewachievementscalarizingproblemneedstobeformedagainandsolved.Moreover,despitesomemodications[16],thereferencepointapproachworkswithonlyonereferencepointatatime.However,thedecision-makermaybeinterestedinexploringthepreferredregionsofPareto-optimalityformultiplereferencepointssimultane-ously.Withtheaboveprinciplesofreferencepointapproachesanddifcultieswiththeclassicalmethods,weproposeanEMOmethodologybywhichasetofPareto-optimalsolutionsnearasuppliedsetofreferencepointswillbefound,therebyelim-inatingtheneedofanyweightvectorandtheneedofapply-ingthemethodologiesagainandagain.Insteadofndingasinglesolutioncorrespondingtoaparticularweightvec-tor,theproposedprocedurewillattempttoandasetofsolutionsintheneighborhoodofthecorrespondingPareto-optimalsolution,sothatthedecision-makercanhaveabetterideaoftheregionratherthanasinglesolution.Toimplementtheprocedure,weusetheelitistnon-dominatedsortingGAorNSGA-II[1].However,asimilarstrategycanalsobeadoptedwithanyotherEMOmethod- ology.Inthefollowing,wedescribeaniterationoftheproposedreference-point-basedNSGA-IIprocedure(wecallhereasR-NSGA-II)forwhichthedecision-makersuppliesoneormorereferencepoints.Asusual,bothparentandoffspringpopulationsarecombinedtogetherandanon-dominatedsortingisperformedtoclassifythecombinedpop-ulationintodifferentlevelsofnon-domination.Solutionsfromthebestnon-dominationlevelsarechosenfront-wiseasbeforeandamodiedcrowdingdistanceoperatorisusedtochooseasubsetofsolutionsfromthelastfrontwhichcan-notbeentirelychosentomaintainthepopulationsizeofthenextpopulation.Thefollowingupdateisperformed:Step1:Foreachreferencepoint,thenormalizedEuclideandistanceofeachsolutionofthefrontiscalculatedandthesolutionsaresortedinascendingorderofdistance.Thisway,thesolutionclosesttothereferencepointisassignedarankofone.Step2:Aftersuchcomputationsareperformedforallrefer-encepoints,theminimumoftheassignedranksisas-signedasthecrowdingdistancetoasolution.Thisway,solutionsclosesttoallreferencepointsareassignedthesmallestcrowdingdistanceofone.Thesolutionshav-ingnext-to-smallestEuclideandistancetoallreferencepointsareassignedthenext-to-smallestcrowdingdis-tanceoftwo,andsoon.Thereafter,solutionswithasmallercrowdingdistancearepreferred.Step3:Tocontroltheextentofobtainedsolutions,allsolu-tionshavingasumofnormalizeddifferenceinobjec-tivevaluesoforlessbetweenthemaregrouped.Arandomlypickedsolutionfromeachgroupisretainedandrestallgroupmembersareassignedalargecrowd-ingdistanceinordertodiscouragethemtoremainintherace.Theaboveprocedureprovidesanequalemphasisofsolu-tionsclosesttoeachreferencepoint,therebyallowingmulti-pleregionsofinteresttobefoundsimultaneouslyinasinglesimulationrun.Moreover,theuseofthe-basedselectionstrategy(whichisalsosimilartothe-dominancestrategiessuggestedelsewhere[17,18])ensuresaspreadofsolutionsnearthepreferredPareto-optimalregions.Intheparlanceoftheclassicalreferencepointapproach,theaboveprocedureisequivalenttousingaweightvec-toremphasizingeachobjectivefunctionequallyorusingwi=1=M.Ifthedecision-makerisinterestedinbiasingsomeobjectivesmorethanothers,asuitableweightvectorcanbeusedwitheachreferencepointandinsteadofempha-sizingsolutionswiththeshortestEuclideandistancefromareferencepoint,solutionswithashortestweightedEuclideandistancefromthereferencepointcanbeemphasized.WereplacetheEuclideandistancemeasurewiththefollowingweightedEuclideandistancemeasure:dij=vMXi=1wifi(x) zifmaxi fmini2;(3)wherefmaxiandfminiarethepopulationmaximumandmin-imumfunctionvaluesofi-thobjective.V.SimulationResultsWenowshowsimulationresultsontwoto10objectivesus-ingtheproposedmethodology.Inallsimulations,weusetheSBXoperatorwithanindexof10andpolynomialmutationwithanindex20.Wealsouseapopulationofsize100andruntill500generationstoinvestigateifagooddistributionofsolutionsremainforalargenumberofiterations.A.Two-ObjectiveZDTTestProblemsInthissection,weconsiderthreeZDTtestproblems.1)TestProblemZDT1First,weconsiderthe30-variableZDT1problem.Thisprob-lemhasaconvexPareto-optimalfrontspanningcontinu-ouslyinf12[0;1]andfollowsafunctionrelationship:f2=1 pf1.Figure2showstheeffectofdifferentvaluesonthedistribution.Tworeferencepointsarechosenforthise=0.005e=0.001e=0.0001Feasibleregione=0.01Reference points 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f1f2Figure.2:EffectofinobtainingvaryingspreadofpreferredsolutionsonZDT1.problemandareshowninlleddiamonds.Fourdifferentvaluesof0.0001,0.001,0.005and0.01arechosen.Solu-tionswith=0:0001areshownonthetruePareto-optimalfront.ItisinterestingtonotehowsolutionsclosetothetwochosenreferencepointsareobtainedonthePareto-optimalfront.SolutionswithothervaluesareshownwithanoffsettothetruePareto-optimalfront.Itisclearthatwithalargevalueof,therangeofobtainedsolutionsisalsolarge.Thus,ifthedecision-makerwouldliketoobtainalargeneighbor-hoodofsolutionsnearthedesiredregion,alargevalueofcanbechosen.Foraparticularpopulationsizeandachosennumberofreferencepoints,theextentofobtainedsolutionsgetsxedbymaintainingadistancebetweenconsecutiveso-lutionsofanamount.Next,weconsidervereferencepoints,ofwhichtwoarefeasibleandthreeareinfeasible.Figure3showstheobtainedsolutionswith=0:001.Nearallvereferencepoints,agoodextentofsolutionsareobtainedonthePareto-optimalfront. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f2f1Figure.3:Preferredsolutionsforvereferencepointswith=0:001onZDT1.Toinvestigatetheeffectofaweight-vectorinobtainingthepreferreddistribution(similartotheclassicalachievementscalarizationapproach),weusethenormalizedEuclideandistancemeasuregiveninequation3.Figure4showstheobtaineddistributionwithR-NSGA-IIwith=0:001onZDT1problemforthreedifferentweightvectors:(0.5,0.5),(0.2,0.8)and(0.8,0.2).Areferencepointz=(0:3;0:3)is(0.2,0.8)(0.8,0.2)Reference point(0.5,0.5)Pareto front 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6f1f2Figure.4:BiasedpreferredsolutionswithdifferentweightvectorsaroundareferencepointforZDT1.used.Asexpected,fortherstweightvector,theobtainedsolutionsareclosesttothereferencepoint.Forthesecondweightvector,moreemphasisonf2isgiven,therebyndingsolutionswhichareclosertominimumoff2.Anoppositephenomenonisobservedwiththeweightvector(0.8,0.2),inwhichmoreemphasisonf1isprovided.Theseresultsshowthatifthedecision-makerisinterestedinbiasingsomeob-jectivesmorethantheothers,abiaseddistributionclosesttothechosenreferencepointcanbeobtainedbytheproposedR-NSGA-II.Inallsubsequentsimulations,weuseauniformweightvector,howeveranon-uniformweight-vectorcanalsobeused,ifdesired.2)TestProblemZDT2The30-variableZDT2problemisconsiderednext.Thisproblemhasanon-convexPareto-optimalfrontranginginf1:f22[0;1]withf2=1 f21.Threereferencepointsarechosenandtheobtainedsetofpointswith=0:001areshowninFigure5.Itcanbeclearlyseenthatnon-convexityofthePareto-optimalfrontdoesnotcauseanydifcultytotheproposedmethodology. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f1f2Figure.5:Preferredsolutionsforthreereferencepointswith=0:001onZDT2.3)TestProblemZDT3The30-variableZDT3problemhasadisconnectedsetofPareto-optimalfronts.Threereferencepointsarechosenandtheobtainedsetofsolutionsfoundusing=0:001areshowninFigure6.Itisinterestingtonotethatcorrespond-ingtothereferencepointlyingbetweenthetwodisconnectedfronts,solutionsonbothfrontsarediscovered.Thisstudyalsorevealsanimportantmatterwiththepro-posedapproach,whichwediscussnext.SincethecompletePareto-optimalfrontisnotthetargetoftheapproachandsincetheproposedprocedureemphasizesnon-dominatedso-lutions,somenon-Pareto-optimalsolutionscanbefoundbytheproposedprocedureparticularlyinproblemshavingnon-continuousPareto-optimalfronts.SolutionA(referFigure6)isonesuchpointwhichisnotaPareto-optimalsolutionbutisfoundasapartofthenalsubpopulationbytheproposedapproach.Thissolutionisnon-dominatedtotherestoftheobtainedsolutions,butisnotamemberofthetruePareto-optimalset.Tomakethissolutiondominated,thereexistnoneighboringsolutionintheobjectivespace.OnlywhensolutionssuchassolutionBarepresentinthepopulation,suchspurioussolutions(likesolutionA)willnotremaininthenalpopulation.However,thechosenreferencepointscanbesuchthatthesolutionBmaynotbeapartofthepreferredsolutions.Insuchsituations,suchspurioussolu-tions(likesolutionA)mayappearinthenalpopulation.However,toensurethePareto-optimalityofasolution,an AB 1 0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1f1f2Figure.6:Preferredsolutionsforthreereferencepointswith=0:001onZDT3.-constraintapproachcanbeappliedwithf1fA1con-straint.IfasolutiondominatingsolutionAisfoundbythe-constraintapproach,thensolutionAcannotbeamemberofthePareto-optimalset.However,inthispaperwerealizetheneedofsuchasecond-leveloptimizationstrategyforen-suringPareto-optimality,butwedonotperformsuchastudyhere.Three-ObjectiveDTLZ2ProblemThe11-variableDTLZ2problemhasathree-dimensional,non-convex,Pareto-optimalfront.WeusetworeferencepointsasshowninFigure7.Weuse=0:01here.Agooddistributionofsolutionsnearthetworeferencepointsareob-tained.Thisindicatestheabilityoftheproposedprocedureinsolvingthree-objectiveoptimizationproblemsaswell.ReferenceReferenceParetoR-NSGA-II 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f1f2f3Figure.7:Preferredsolutionsfortworeferencepointswith=0:01onDTLZ2.C.Five-ObjectiveDTLZ2ProblemNext,weapplytheproposedprocedurewith=0:01tothe14-variableDTLZ2problem.Tworeferencepointsarechosenasfollows:(i)(0.5,0.5,0.5,0.5,0.5)and(ii)(0.2,0.2,0.2,0.2,0.8).Figure8showsthevalue-pathplotoftheve-objectivesolutions.Itisclearthattwodistinctsetsofsolutionsneartheabovereferencepointsareobtainedbytheproposedprocedure.SincethePareto-optimalsolutionsintheDTLZ2problemsatisfyPM=1f2iequaltoone,wecom-putetheleftsideofthisexpressionforallobtainedsolutionsandthevaluesarefoundtoliewithin[1.000,1.044](atmost4.4%fromone),therebymeaningthatallsolutionsareveryclosetothetruePareto-optimalfront. 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5Objective NumberObjective ValueFigure.8:Preferredsolutionsfortworeferencepointswith=0:01onve-objectiveDTLZ2.D.10-ObjectiveDTLZ2ProblemWethenattempttosolve19-variableDTLZ2problemwithonereferencepoint:fi=0:25foralli=1;2;:::;10.Weuse=0:01andtheobtaineddistributionisshowninFig-ure9.Althoughtheobjectivevaluescanvaryin[0,1],the 0.305 0.31 0.315 0.32 0.325 1 2 3 4 5 6 7 8 9 10Objective NumberObjective ValueFigure.9:Preferredsolutionsforonereferencepointwith=0:01on10-objectiveDTLZ2.pointsconcentratesnearfi=1=p10or0.316,whichwouldbetheregionclosesttothechosenreferencepoint.WhenwecomputeP10=1f2iofallobtainedsolutions,theyarefoundtobeexactlyequaltoone,therebymeaningthatallR-NSGA-IIsolutionsareonthetruePareto-optimalfront.Thisstudyshowsthattheproposedprocedureisalsoabletosolvea 10-objectiveproblem,althoughithasbeenshownelsewhere[5]thattheoriginalNSGA-IIfacesdifcultyinndingaconvergedandwell-distributedsetofsolutionsonthetruePareto-optimalfrontforthesame10-objectiveDTLZ2prob-lem.Thus,itcanbeconcludedthatifasmallregiononalarge-dimensionalPareto-optimalfrontisthetarget,thepro-posedprocedureisawaytonditinareasonableamountofcomputations.TwoEngineeringDesignProblemsNext,weapplytheproposedmethodologytotwoengineer-ingdesignproblems,eachhavingtwoobjectives.A.WeldedBeamDesignProblemTheweldedbeamdesignproblemhasfourreal-parametervariablesx=(h;`;t;b)andfournon-linearconstraints.Oneofthetwoobjectivesistominimizethecostoffabricationandotheristominimizetheenddeectionoftheweldedbeam[19]:Minimizef1(~x)=1:10471h2`+0:04811tb(14:0+`);Minimizef2(~x)=2:1952t3b;Subjecttog1(~x)13;600 (~x)0;g2(~x)30;000 (~x)0;g3(~x)b h0;g4(~x)Pc(~x) 6;0000;0:125h;b5:0;0:1`;t10:0:(4)Therearefourconstraints.Therstconstraintmakessurethattheshearstressdevelopedatthesupportlocationofthebeamissmallerthantheallowableshearstrengthofthema-terial(13,600psi).Thesecondconstraintmakessurethatnormalstressdevelopedatthesupportlocationofthebeamissmallerthantheallowableyieldstrengthofthematerial(30,000psi).Thethirdconstraintmakessurethatthicknessofthebeamisnotsmallerthantheweldthicknessfromapracticalstandpoint.Thefourthconstraintmakessurethattheallowablebucklingload(alongtdirection)ofthebeamismorethantheappliedloadF=6;000lbs.Aviolationofanyoftheabovefourconstraintswillmakethedesignun-acceptable.Thestressandbucklingtermsarenon-lineartodesignvariablesandaregivenasfollows[20]: (~x)=q( 0)2+( 00)2+(` 0 00)=p0:25(`2+(h+t)2); 0=6;000p2h`; 00=6;000(14+0:5`)p0:25(`2+(h+t)2)2f0:707h`(`2=12+0:25(h+t)2)g;(~x)=504;000t2b;Pc(~x)=64;746:022(1 0:0282346t)tb3:TheobjectivesareconictinginnatureandNSGA-IIisap-pliedelsewheretondtheoptimizednon-dominatedfronttothisproblem[4].Here,insteadofndingthecompletePareto-optimalfront,weareinterestedinndingtheop-timizedtrade-offregionsclosesttothreechosenreferencepoints:1.(4,0.0030),2.(20,0.0020),and3.(40,0.0002).Figure10showstheobtainedsolutions.Toinvestigatewheretheseregionsarewithrespecttothecompletetrade-offfront,wealsoshowtheoriginalNSGA-IIsolutionswitha`+'.First,theobtainedpreferredsolutionsarefoundtobefallingComplete frontReferenceObtainedpoint 0 0.001 0.002 0.003 0.004 0.005 0.006 0 5 10 15 20 25 30 35 40CostDeflectionFigure.10:Preferredsolutionsforthreereferencepointswith=0:001ontheweldedbeamdesignproblem.onthetrade-offfrontierobtainedusingtheoriginalNSGA-II.Second,solutionsclosetothegivenreferencepointsarefound.Itisinterestingtonotethatalthoughthesecondref-erencepointisfeasible.meaningthattheremayexistaso-lutionvectorx,whichwillproducethegivenreferencepoint(thatis,correspondingtoacostof20unitsandadeec-tionof0.002units),thetaskistond,ifpossible,asetofsolutionswhicharebetterthanthegivenreferencepointinallobjectives.Thegureshowsthatthesuppliedreferencepointisnotanoptimalsolutionandthereexistanumberofsolutionswhichdominatethissolutionx.Althoughshort-estdistancesfromthereferencepointsarepreferred,theem-phasisofnon-dominatedsolutionsoverdominatedsolutionsenablesPareto-optimalsolutionstobefound.Thus,ifthedecision-makerisinterestedinknowingtrade-offoptimalsolutionsinthreemajorareas(minimumcost,inter-mediatetocostanddeectionandminimumdeection)theproposedprocedureisabletondsolutionsnearthesuppliedreferencepoints,insteadofndingsolutionontheentirePareto-optimalfront,therebyallowingthedecision-makertoconsideronlyafewsolutionsandthattoosolutionswhichlieintheregionsofher/hisinterest.B.SpringDesignProblemFinally,weconsideranotherengineeringdesignprobleminwhichtwoofthethreedesignvariablesarediscreteinnature,therebycausingthePareto-optimalfronttohaveadiscretesetofsolutions.Diameterofthewire(d),diameterofthespring(D)andthenumberofturns(N)aretobefoundforminimiz-ingvolumeofspringandminimizingthestressdevelopedduetotheapplicationofaload.Denotingthevariablevector x=(x1;x2;x3)=(N;d;D),wewritethetwo-objective,eight-constraintoptimizationproblemasfollows[21]:Minimizef1(~x)=0:252x22x3(x1+2);Minimizef2(~x)=8KPmaxx3x23;Subjecttog1(~x)=lmax Pmaxk 1:05(x1+2)x20;g2(~x)=x2 dmin0;g3(~x)=Dmax (x2+x3)0;g4(~x)=C 30;g5(~x)=pm p0;g6(~x)=Pmax Pk w0;g7(~x)=S 8KPmaxx3x230;g8(~x)=Vmax 0:252x22x3(x1+2)0;x1isinteger,x2isdiscrete,x3iscontinuous.(5)Theparametersusedareasfollows:K=4C 14C 4+0:615x2x3;P=300lb;Dmax=3in;Pmax=1;000lb;w=1:25in;p=Pk;pm=6in;S=189ksi;dmin=0:2in;G=11;500;000lb=in2;Vmax=30in3;k=Gx248x1x33;lmax=14in;C=x3=x2:The42discretevaluesofdaregivenbelow:00:009;0:0095;0:0104;0:0118;0:0128;0:0132;0:014;0:015;0:0162;0:0173;0:018;0:020;0:023;0:025;0:028;0:032;0:035;0:041;0:047;0:054;0:063;0:072;0:080;0:092;0:105;0:120;0:135;0:148;0:162;0:177;0:192;0:207;0:225;0:244;0:263;0:283;0:307;0:331;0:362;0:394;0:4375;0:5:1ThedesignvariablesdandDaretreatedasreal-valuedpa-rametersintheNSGA-IIwithdtakingdiscretevaluesfromtheabovesetandNistreatedwithave-bitbinarystring,therebycodingintegersintherange[1,32].WhileSBXandpolynomialmutationoperatorsareusedtohandledandD,asingle-pointcrossoverandbit-wisemutationareusedtohandleN.WeapplytheR-NSGA-IIwithtworeferencepoints:(4,180,000)(feasible)and(25,20,000)(infeasible)withauni-formweightvectorandwith=0:001.Figure11showstheR-NSGA-IIsolutionswhicharefoundtobeclosertothetworeferencepoints.Thetrade-offoptimizedsolutionsfoundbytheoriginalNSGA-IIarealsoshown.Itisinterestingtonotehowtheproposedpreferredtechniquecanbeusedtondasetofsolutionsnearsomechosenaspirationpoints,suppliedbythedecision-maker.VII.APredator-PreyApproachBesidestheabovedirectapproachinmodifyingthenichingoperatorofNSGA-IItondapreferredsetofsolutions,thetaskofndingthesolutionscorrespondingtoasetofrefer-encepointsappealamoredirectnaturalapproachtobeap-plied.Byconsideringthereferencepointsaspredatorsandtargetsolutionsclosesttothemaspreys,wemaysimulateapredator-preyhuntingproceduretosolvetheproblem.Laumannsetal.[2]suggestedapredator-preyalgorithminwhichapredatorsandpreysarerandomlyplacedonatoroidalgrid.Eachpredatorworkswithaparticularobjec-tiveanddeletestheworstpreyititsneighborhoodaccordingReference pointR NSGA IIR NSGA IIReference pointPareto optimalpoints 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000 0 5 10 15 20 25 30Volume (in^3)Stress (psi)Figure.11:Preferredsolutionsaroundtworeferencepointsforthespringdesignproblem.toitsobjectivefunction.Sinceeverypredatorworkswithadifferentobjective,attheend,multipleoptimalsolutionsareexpectedtobepresentinthegrid,therebyndingmulti-plePareto-optimalsolutionssimultaneously.Later,Deb[4]extendedtheideatoincludeaweightedsumofobjectivesassignedtoeachpredator.Li[22]extendedtheideatoin-troducedifferingspeedsofpredatorsandpreyswithpreda-torsmakingmovesmoreoftenthanpreysandshowedanim-provementinresults,comparedtotheoriginalmethod.Here,wesuggestasystematicsetofmodicationstotheoriginalmodelofLaumannsetal.[2]byintroducingcrossover,elitepreservation,andanexplicitdiversitypreser-vationmechanismwhichperformedmuchbetterthantheex-istingmethodologies.Step1:Initializesetofpreysrandomlybetweenthevariablelimits.Step2:Placethesepreysontheverticesofundirectedcon-nectedgraph.Step3:Placepredatorsrandomlyontheverticesofthegraph.Step4:Assigneachpredatorwithadistinctweightedsumofobjectivesuniformlycreatedwithin[0;1][0;1]:::[0;1],sothatthesumofweightsisone.Step5:Evaluatepreysaroundeachpredatorandselecttheworstprey.Step6:Createtwooffspringbyapplyingacrossoveroper-ationbetweentherstandthesecondbestpreysintheneighborhoodoftheworstprey.Randomlychooseoneofthetwooffspringandmutateittocreatethechildso-lution.Step7:Childacceptancecriteria:Step7a:Ifthechildsolutionweaklydominatesallex-istingpreysorchildsolutionisnon-dominatedwithallexistingpreys,childbecomesacandidatetoreplacetheworstprey.Ifthechildisnotwithin theinuencingregionofanyexistingprey,itre-placestheworstprey.Predatoralsomovestothepositionoftheworstprey.Step7b:Elseifthechildsolutionisdominatedbyanyexistingpreyorthechildiswithintheinuenc-ingregionofanyexistingprey,thechildisnotac-ceptedandanewchildiscreatedbyStep6.Thecreationofnewchildanditsacceptancetestarecontinuedamaximumof10iterations,afterwhichtheworstpreyisretained.Inthiscase,predatortakesarandomwalktoanypositioninthegrid.Step8:Thiscompletesonegenerationofthepredator-preyalgorithm.RepeatSteps5to7forthenextgeneration.Thesalientfeaturesoftheproposedalgorithmareasfollows:1.Aweighted-sumofobjectivesperpredatorisusedasacriterionfordeletingtheworstprey.2.Acrossoverbetweentwogoodsolutionsandasubse-quentmutationareusedtocreateachildsolution.3.Theelitepreservationanddiversitymaintenanceareen-suredbyacceptinganewlycreatedchildonlywhenitweaklydominatesallexistingpreysanditisnotwithinapredenedregionfromexistingpreys.Forachievingthetaskofndingpreferredsolutions,wefur-thermodifytheaboveprocedureinthefollowingmanner:1.Eachpredatorisassignedtooneofthepreferredpoints.Multiplepredatorassignmenttoasinglereferencepointisalsoallowedandisrecommended.2.Allneighboringpreysaredividedintotwoclasses:(i)onewhichdominatesthepredatorand(ii)theremainingpreysolutions.3.Ifthesecondsetisempty,wedeclarethepreyhavingthesmallestEuclideandistanceastheworstprey.Oth-erwise,wendthepreyinthesecondsethavinglargestEuclideandistanceanddeclareitastheworstprey.4.Thecreationofoffspringisidenticaltotheproposedmethodology.However,ifonlythecreatedoffspringiswithinacriticaldistancefromanyreferencepoint,thisoffspringcanbeconsideredasacandidateforinclusioninthegrid.Iftheoffspringisnotwithinthecriticaldistanceofanyreferencepoint,itissimplydiscarded.Withthesemodications,weapplytheproceduretoanum-berofscenariosonthetwo-objective,ve-variableZDT1testproblem.Alltheseresultsaretakenfor300generations.Thenumberofpreysischoseninproportiontothenumberofreferencepoints(25timesthenumberofreferencepoints).Inallcases,10predatorsareconsideredforeachreferencepoint.12to15showthenalpopulationofpreysfordiffer-entscenarios.Ineachcase,thepredators(orreferencepoints)areshownusingalleddiamond.ItisinterestingtoobservehowtheproposedmethodologyisabletondaconcentratedsetofPareto-optimalsolutionsneareachofthereferencepoints.Itf1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f2Figure.12:Preferredsolutionswithaninfeasiblereferencepoint. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f1f2Figure.13:Preferredsolutionswithafeasiblereferencepoint.alsointerestingtonotethattheprocedureworksequallywellforthereferencepointtolieinsideoroutsidethefeasi-bleobjectivespace.Next,weapplytheproposedpredator-preyproceduretoamodiedthree-objectiveDTLZ2problem,asshowninFig-ure16.TomakethePareto-optimalfrontaconvexfront(sothattheweighted-sumofobjectivescanbeassignedtoeachpredator),wehavemodiedtheoriginalDTLZ2problem[23],bysubtractingeachfunctionvaluefrom(1+g(x)).Onereferencepointz=(0:1;0:4;0:7)isconsidered.10predatorsareused.Apopulationofsize50isusedfor300generations.Thegureshowsaniceconcentrationof50so-lutionsonthetruePareto-optimalfrontnearthesuppliedref-erencepoint.Theprocedureisquitefastcomputationallyandthesimulationresultsdemonstrateitsusefulnessinndingapreferredsetofsolutions.Wearecurrentlyinvestigatingthepotentialofsuchapredator-preyprocedureforhandlingproblemswithalargernumberofobjectives. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f1f2Figure.14:Preferredsolutionswithtworeferencepoints. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1f1f2Figure.15:Preferredsolutionswiththreereferencepoints.VIII.ConclusionsInthispaper,wehaveaddressedanimportanttaskofcom-biningEMOmethodologieswithaclassicalmulti-criteriondecision-makingapproachtonotndasingleoptimalso-lution,buttondasetofsolutionsnearthedesiredregionofdecision-maker'sinterest.Withanumberoftrade-offsolutionsintheregionofinterestswehavearguedthatthedecision-makerwouldbeabletomakeabetterandmorere-liabledecisionthanwithasinglesolution.Thereferencepointapproachisacommonmethodologyinmulti-criteriondecision-making,inwhichoneormoreref-erence(goal)pointsarespeciedbythedecision-makerbe-forehand.ThetargetinsuchanoptimizationtaskisthentoidentifythePareto-optimalregionclosesttothereferencepoints.Wehavesuggestedtwodifferentapproachesforthispurpose.Intherstapproach,thenichingoperatoroftheoriginalNSGA-IIhasbeenupdatedtoemphasizesuchsolu-tions.Theproposedtechniquehasbeenappliedtoanumberoftwoto10-objectiveoptimizationproblemswithtwotoveObtainedPareto 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1f3f2 0 0.2 0.4 0.6 0.8 1f1 1Figure.16:Obtainedsolutionswiththepredator-preyap-proachforthethree-objectivemodiedDTLZ2problem.referencepointsandinallcasesthedesiredsetofsolutionshavebeenobtained.Theapproachinvolvesanewparameter()whichcontrolstheextentofthedistributionofsolutionsneartheclosestPareto-optimalsolution.Themaincruxofthispaperisexploitationofthepopula-tionapproachofanEMOprocedureinndingmorethanonesolutionsnotontheentirePareto-optimalfrontier,butintheregionsofPareto-optimalitywhichareofinteresttothedecision-maker.Thepopulationslotsarewellutilizedinnotonlymakinganimplicitparallelsearch[24],butalsotond(i)multipleregionsofinterestsimultaneouslyand(ii)mul-tipletrade-offsolutionsintheclosevicinityofeachdesiredregionofinterest.Thesecondproposedapproachinvolvesanothernaturaleventofpredatorshuntingpreysoftheirlikings.Bykeepingalivesolutionsclosesttothepredators(modeledforthesup-pliedreferencepoints)andbyemphasizingnon-dominatedsolutionsforconvergencetothePareto-optimalfront,theap-proachhasbeenabletoachievethetaskquitesuccessfullyfortwoandthree-objectiveoptimizationproblems.Asanim-mediateextensiontothiswork,amoredetailedstudymustbemadetofullyexploitthepredator-preyapproachforhigher-objectiveproblems.Havingbeenwelldemonstratingthetaskofndingmulti-plePareto-optimalsolutionsinmulti-objectiveoptimizationproblems,theEMOresearchersandapplicationistsshouldnowconcentrateindevisingmethodologiesofsolvingthecompletetaskofndingpreferredandPareto-optimalsolu-tionsinaninteractivemannerwithadecision-maker.Al-thoughtheultimatetargetinsuchanactivityistocomeupwithasinglesolution,theuseofanEMOprocedurecanbewellappliedwithadecision-makingstrategyinndingasetofpreferredsolutionsinregionsofinteresttothedecision-maker,sothatthesolutionsinaregioncollectivelybringoutpropertiesofthesolutionsthere.Suchanactivitywillthenallowthedecision-makertorstmakeahigher-levelsearchofchoosingaregionofinterestonthePareto-optimalfront,ratherthanusingasinglesolutiontofocusonaparticularsolution. 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