AlsoFinlandDistinguishedProfessorHelsinkiSchoolofEconomicsPOBox1210FIN00101HelsinkiFinlandKalyanmoyDebhse 2simplymaximizingMobjectivefunctionsoneatatimeThisisbecausethesearchoftheworst ID: 383311
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AHybridIntegratedMulti-ObjectiveOptimizationProcedureforEstimatingNadirPointKalyanmoyDeb1??,KaisaMiettinen2,andDeepakSharma11DepartmentofMechanicalEngineeringIndianInstituteofTechnologyKanpur,PIN208016,Indiafdeb,dsharmag@iitk.ac.in2DepartmentofMathematicalInformationTechnologyP.O.Box35(Agora),FI-40014UniversityofJyvaskyla,Finlandkaisa.miettinen@jyu.fiAbstract.AnadirpointisconstructedbytheworstobjectivevaluesofthesolutionsoftheentirePareto-optimalset.Alongwiththeidealpoint,thenadirpointprovidestherangeofobjectivevalueswithinwhichallPareto-optimalsolutionsmustlie.Thus,anadirpointisanimportantpointtoresearchersandpractitionersinterestedinmulti-objectiveopti-mization.Besides,ifthenadirpointcanbecomputedrelativelyquickly,itcanbeusedtonormalizeobjectivesinmanymulti-criteriondecisionmakingtasks.Importantly,estimatingthenadirpointisachallengingandunsolvedcomputingproblemincaseofmorethantwoobjectives.Inthispaper,wereviseapreviouslyproposedserialapplicationofanEMOandalocalsearchmethodandsuggestanintegratedapproachfor ndingthenadirpoint.Alocalsearchprocedurebasedonthesolutionofabi-levelachievementscalarizingfunctionisemployedtoextremesolu-tionsinstabilizedpopulationsinanEMOprocedure.Simulationresultsonanumberofproblemsdemonstratetheviabilityandworkingoftheproposedprocedure.1IntroductionAnadirpointsigni es,inprinciple,oppositetothatmeantbyanidealpoint,inthecontextofmulti-objectiveoptimization.AnidealpointisanM-dimensionalobjectivevector(whereMisthenumberofobjectives)constructedwithbestfeasibleobjectivevaluesandisacomparativelyeasytocompute.Forminimiza-tionproblems,inprinciple,thiscallsforsolvingMsingle-objectiveminimizationproblemsandcollectingeachoptimalobjectivevaluestoformtheidealpoint.Ontheotherhand,anadirpointisconstructedwiththeworstobjectivevaluesofPareto-optimalsolutions.Inminimizationproblems,thistaskisdi erentfrom ??AlsoFinlandDistinguishedProfessor,HelsinkiSchoolofEconomics,POBox1210,FIN-00101,Helsinki,Finland,(Kalyanmoy.Deb@hse. ) 2simplymaximizingMobjectivefunctionsoneatatime.ThisisbecausethesearchoftheworstvalueofanobjectivemustberestrictedwithinthePareto-optimalsolutions.Thisisthereasonwhytheestimationofnadirpointhasbeenfoundtobeacomplextask[13,11]andtheredoesnotexistanyprovablealgo-rithmforthetask,evenforlinearmulti-objectiveoptimizationproblemshavingthreeormoreobjectives.Withtheadventofecientevolutionaryoptimizationproceduresformulti-objectiveoptimization,someattentionhasbeenmadeintherecentpastindevel-opingproceduresforestimatingthenadirpoint.Simplisticideas,suchas ndingasetofPareto-optimalsolutionsbyanEMOprocedureandthenchoosingtheextremesolutionsforestimatingthenadirpoint,tomoresophisticatedideas,suchasreplacingthefocusofEMOto ndawide-spreadedsetofsolutionsontheentirePareto-optimalfrontto ndonlythecriticalextremePareto-optimalpoints[4,16],aresuggested.MostoftheseEMOmethodologieshaveshownto ndanapproximationofthenadirpoint,ratherthantoestimatetheexactnadirpoint.Recentstudies[6,5]suggestedatwo-stepserialprocedureofemployingamodi edNSGA-IIproceduretoidentifyextremenearPareto-optimalsolutionsandthenalocalsearchproceduretoconvergetothetrueextremePareto-optimalpoints.Inthisstudy,wesuggestandsimulateahybridintegratedapproachinwhichalocalsearchprocedureisusedwithinthemodi edNSGA-IIalgorithmspar-inglytoachievethenadirpointestimationtask.Thesuggestedlocalsearchprocedureisbasedonutilizingareferencepointbasedapproach,aso-calledachievementscalarizingfunction[17]whichiswidelyusedintheMCDM eld.Usingthisscalarizedfunction,anypointintheobjectivespacecanbeprojectedontheParetooptimalfrontandthescalarizingfunctiondoesnotneedanyarti- cialinformationlikeweights[14].Intheprocedureproposed,theachievementscalarizingfunctionisusedinabi-levelmannertoguaranteegettingreliableenoughinformationaboutextremevaluesintheParetooptimalfrontforesti-matingthenadirpoint.BasedonastatisticalanalysisoftheperformanceoftheNSGA-IIprocedure,theexecutionofthelocalsearcheventisdecideddy-namicallyateverygeneration.BothNSGA-IIandlocalsearchproceduresareterminatedusingstatisticalperformancecriteria.Simulationresultsonanumberoftestproblemsandthreeengineeringproblemsarepresentedtodemonstratetheecacyoftheproposedprocedure.2NadirObjectiveVectorWeconsidermulti-objectiveoptimizationproblemsinvolvingMcon\rictingob-jectives(fi:S!R)asfunctionsofdecisionvariablesx:minimizeff1(x);f2(x);:::;fM(x)g;subjecttox2S;(1)whereSRndenotesthesetoffeasiblesolutions.Problem(1)givesrisetoasetofPareto-optimalsolutionsoraPareto-optimalfront(P),providingatrade-o 3amongtheobjectives.Inthesenseofminimizationofobjectives,Pareto-optimalsolutionscanbede nedasfollows[14]:De nition1Adecisionvectorx2Sandthecorrespondingobjectivevectorf(x)arePareto-optimaliftheredoesnotexistanotherdecisionvectorx2Ssuchthatfi(x)fi(x)foralli=1;2;:::;Mandfj(x)fj(x)foratleastoneindexj.Inwhatfollows,weassumethatthePareto-optimalfrontisbounded.Wenowde neanadirobjectivevector,thatis,anadirpoint,asfollows.De nition2Anobjectivevectorznad=(znad1;:::;znadM)TconstructedusingtheworstvaluesofobjectivefunctionsinthecompletePareto-optimalfrontPiscalledanadirobjectivevector.Hence,forminimizationproblemswehaveznadj=maxx2Pfj(x).Estimationofthenadirobjectivevectoris,ingeneral,adiculttask.Unliketheidealobjectivevectorz=(z1;:::;zM)T,whichcanbefoundbyminimizingeachobjectiveindividuallyoverthefeasiblesetS(or,zj=minx2Sfj(x)),thenadirpointcannotbeformedbymaximizingobjectivesindividuallyoverS.To ndthenadirpoint,Pareto-optimalityofsolutionsusedforconstructingthenadirpointmustbe rstestablished.Thismakesthetaskof ndingthenadirpointadicultone.Toillustratethisaspect,letusconsiderabi-objectiveminimizationproblemshowninFigure1.Ifwemaximizef1andf2individually,weobtainpointsAandB,respectively.Thesetwopointscanbeusedtoconstructtheso-calledworstobjectivevector,zw.Inmanyproblems(eveninbi-objectiveoptimizationproblems),thenadirobjectivevectorandtheworstobjectivevectorarenotthesamepoint,whichcanalsobeseeninFigure1. Pareto- optimal frontABWorst objective vectorIdealpointobjective spaceFeasiblef1f2 Nadir objective vector Fig.1.Thenadirandworstobjectivevectors. ZnadZ'BIdealpointZ*AB'CA'C'G'F'E'D' 1 0.8 0.6 0.4 0.2 0.4 0.6 0.8 1 0 0.2 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 0f1f2 1.2 1.4f3 Fig.2.Payo tablemaynotproducethetruenadirpoint. 43ExistingMethods3.1Payo TableMethodBenayounetal.[1]introducedthe rstinteractivemulti-objectiveoptimizationmethodforestimatingthenadirpointbyusingapayo table.Tobemorespe-ci c,eachobjectivefunctionis rstminimizedindividuallyandthenatableisconstructedwherethei-throwofthetablerepresentsvaluesofallobjectivefunctionscalculatedatthepointwherethei-thobjectiveobtaineditsminimumvalue.Thereafter,themaximumvalueofthej-thcolumncanbeconsideredasanestimateoftheupperboundofthej-thobjectiveinthePareto-optimalfrontandthesemaximumvaluestogethermaybeusedtoconstructanapproximationofthenadirobjectivevector.Themaindicultyofsuchanapproachisthatsolutionsarenotnecessarilyuniqueandthuscorrespondingtotheminimumsolutionofanobjectivetheremayexistmorethanonesolutionshavingdi er-entvaluesofotherobjectives,inproblemshavingmorethantwoobjectives.Intheseproblems,thepayo tablemethodmaynotresultinanaccurateestimationofthenadirobjectivevector.Toillustrate,considerathree-objectiveproblemshowninFigure2.Minimizationofthe rstobjectivewillresultinanysolutiononthetrapeziumCBB0F0C0C.IfthepointmarkedinasmallcircleonlineCBisobtainedbyanoptimizationalgorithmandsimilarlyothertwocirclesonlinesCAandABareobtainedforminimizationsoff2andf3,respectively,awrongestimate(z0)ofthenadirpoint(znad)willbemade.3.2EvolutionaryApproachesThenadirpointisassociatedwithPareto-optimalsolutionsand,thus,deter-miningasetofPareto-optimalsolutionswillfacilitatetheestimationofthenadirpoint.SinceanEMOalgorithmisaimedat ndingasetofPareto-optimalsolutions,itmaybeanidealwayto ndthenadirobjectivevector.Severalapproachesareproposedrecently.Inthenaiveapproach, rstawell-distributedsetofPareto-optimalsolutionscanbeattemptedto ndbyanEMO[4].Thereafter,anestimateofthenadirobjectivevectorcanbemadebypickingtheworstvaluesofeachobjective[16].InthecontextoftheproblemdepictedinFigure2,thismeans rst ndingawell-representedsetofsolutionsontheplaneABCandthenestimatingthenadirpointfromthem.SinceEMOalgorithmsarenotfoundtoconvergewellandmaintainawell-diversesetofsolutionsformorethanthreeobjectives[7],theaccuracyoftheestimatednadirpointusingthenaiveapproachisquestionable.SzczepanskiandWierzbicki[16]havesimulatedtheideaofsolvingmultiplebi-objectiveoptimizationproblemssuggestedin[8]usinganEMOapproachandconstructthenadirpointbyaccumulatingallbi-objectivePareto-optimalfrontstogether.Asdiscussedinourearlierstudy[5],suchatechniqueisnotgenericandrequiresadditionalobjectiveandvariable-spacenichingtechniquestocorrectlyestimatethenadirpoint.Moreover,theprocedurerequires M2 5bi-objectiveoptimizations,makingitadauntingtaskparticularlyforproblemshavingmorethanthreeobjectives.However,theideaofconcentratingonapreferredregiononthePareto-optimalfront,insteadof ndingtheentirePareto-optimalfront,canbepushedfurther.AnemphasiscanbeplacedinanEMOapproachto ndonlythecriticalextremepointsofthePareto-optimalfront.Ourearlierstudy[4]suggestedtwoapproachesinthecrowdingdistanceoperatoroftheNSGA-IIprocedureandcon-cludedinfavoroftheextremizedcrowdingdistanceapproach.Intheextremized-crowdedNSGA-IIapproach[4],weemphasizedinconcentratingonthebestandworstsolutionsofeachobjective.Inthisapproach,solutionsonaparticularnondominatedfrontare rstsortedfromminimum(withrankR(m)i=1)tomaximum(withrank=Nf)basedoneachobjective.Therankofsolutioniforthem-thobjectiveR(m)iisassignedasmaxfR(m)i;Nf R(m)i+1g.TwoextremesolutionsforeveryobjectivegetarankequaltoNf(numberofsolutionsinthenondominatedfront),thesolutionsnexttotheseextremesolutionsgetarank(Nf 1),andsoon.Afterarankisassignedtoasolutionbyeachobjective,themaximumvalueoftheassignedranksisdeclaredasthecrowdingdistance.Likeotherevolutionaryoptimizationstudies,theproposedextremizedcrowdedNSGA-IIapproachdidnotensureconvergingtothetrueextremesolutionsex-actly,asevolutionaryalgorithmsareexpectedto ndanear-optimalsolution,ratherthanatrueoptimalsolutionina nitenumberofsolutionevaluations.However,inthepursuitofestimatingthenadirpointforthepurposeofnormal-izingobjectivesforexecutingdi erentmulti-objectiveoptimizationalgorithmsorforknowingthetruerangeofPareto-optimalsolutionsfordecision-making,itisimportantto ndthetrueextremePareto-optimalpoints,sothatthenadirpointcanbeestimatedaccurately.Inarecentstudy[6],theextremizedcrowdedNSGA-IIapproachisendedwithabi-levellocalsearchoperationonallextremesolutionstotakethemarbitraryclosertothetrueextremesolutions,sothatthenadirpointcanbeestimatedmoreaccurately.Inthispaper,were-addresstheissueoftheserialapplicationofNSGA-IIandthelocalsearchprocedureandsuggestahybridintegratedapproachforanaccurateestimationofthenadirpoint.4ProposedIntegratedApproachInsteadofapplyingthelocalsearchontheextremesolutionsobtainedbytheex-tremizedcrowdedhybridNSGA-IIprocedure,weproposeanintegratedNSGA-IIapproachinwhichateverygenerationtheextremesolutionsofthebestnon-dominatedfrontaremodi edbythelocalsearchproceduretopushthemtowardstheirtruevalues.Althoughthetaskincreasesthenumberofsolutionevaluationsofeachgeneration,webelievethattheattainedaccuracyoftheintegratedproce-dureisbetterandhasasmallerchanceofgettingstucktointermediatesolutions,whichmaynotleadtoanaccurateestimationofthenadirpoint.Inthefollowing,weoutlineaniterationoftheproposedintegratedNSGA-IIprocedureinwhichthepopulationPtisthecurrentparentpopulationofsizeN.Everymember(i) 6ofPtisalreadyrankedbasedonitsnon-dominationlevel(NDi)anditscrowd-ingdistance(CDi)withinthepopulationmembersofitsownnon-dominationlevel.Step1:PopulationPtisusedtocreateano springpopulationQtbyusingbinarytournamentselection,recombinationandmutationoperators.TwosolutionsarechosenatrandomfromPtandahierarchicalselectionbasedonNDfollowedbyCDisusedtocompletethetournamentselectionoperation.Thereafter,twosuchselectedsolutionsarerecombinedusingthesimulatedbinarycrossoveroperator[3,2]tocreatetwoo springsolutions,eachofwhichisthenmutatedbyusingthepolynomialmutationoperator[2].Theseoperatorsinvolvethefollowingparameters:recombinationprobabilitypc,SBXindexc,mutationprobabilitypm,andmutationindexm.Step2:PopulationsPtandQtarecombinedtogetherandrankedintodi erentlevelsofnon-domination:Pt[Qt=fF1;F2;:::g.ThesetF1containsnon-dominatedsolutionsoflevelone,andsoon.Step3:Dependingonacheckonwhethertoperformthelocalsearchornot(whichwedescribealittlelater),inthesetF1,weidentifytheworstsolution(x(j))withrespecttoeachobjectivej,andmodifyitbyusingalocalsearchprocedure.Themodi edsolution(y(j))replacestheworstpopulationmem-ber.ForMobjectives,thereareMsuchlocalsearchoperationsperformedineachiterationoftheproposedprocedure.Theestimatednadirpoint(zest)atgenerationtisthenformedfromtheextremesolutionsobtainedbythelocalsearches.AllmembersofthesetF1arethenassignedanon-dominationrank(ND)valueequaltoone(beingthe rst-rankedsolutions)andacrowdingdistance(CD)valuebasedontheextremizedcrowdedrankingproceduredescribedearlier[6].Formembersofothernon-dominationlevels(l2),wedonotperformthelocalsearch,butassignanon-dominationrank(ND)equaltolandcrowdingdistance(CD)valuecomputedasabove.Step4:AnewpopulationPt+1isthencreatedbycopyingsolutionsfromthebestnon-dominationlevelF1onwardsoneatatimetillwehaveNpop-ulationmembers.Whenwereachanon-dominationlevelwhichcannotbeentirelyaccepted(tonotincreasethesizeofPt+1overN),weusethecrowd-ingdistance(CD)valuesofthesettodeterminewhichsolutionsshouldbeaccepted.WesimplysortthemembersofthesetaccordingtotheirCDvaluefromhighesttolowestandchooseasmanyweneedto llupthepopulationfromthetopofthesortedlist.ThisprocedureissimilartotheoriginalNSGA-IIprocedure,exceptthatthecrowdingdistancecomputationisdi erentsuitingtheneedforemphasizingex-tremesolutionsforthetaskofestimatingthenadirpointandthatalocalsearchprocedureisusedtoupdatetheextremeobjective-wisesolutionstomakesurethatthenadirpointcanbeestimatedwithadesiredaccuracy.Wenowdescribethelocalsearchprocedurehere.Thebest(fminj)andworst(fmaxj)valuesofeachobjectivejofthesetF1are rstnoted.Weapplyabi-levellocalsearchprocedurefromeachworstsolution(solutionx(j)forwhich 7thej-thobjectivehastheworstvalueinF1)to ndthecorrespondingoptimalsolutiony(j)usingthefollowingbi-leveloptimizationprocedure.Theupper-leveloptimization(describedin(2))usesanobjectivevector(z,referredhereasareferencepoint)asavariablevectorandmaximizesthej-thobjectivevalueoftheoptimalsolutionobtainedbysolvingthecorrespondingaugmentedachievementscalarizingproblem[14](werefertothistaskasthelower-leveloptimizationtask,describedin(3)):maximize(z)fj(z);subjecttozif(j)iEA 0:5(fmaxi fmini);i=1;2;:::;M;zif(j)iEA+1:5(fmaxi fmini);i=1;2;:::;M:(2)Thetermfj(z)istheoptimalvalueofthej-thobjectivefunctionoftheoptimalsolutiontothefollowinglower-leveloptimizationproblemforwhichziskept xed[17]:minimize(y)maxMi=1fi(y) zi fmaxi fmini+PMk=1fk(y) zk fmaxk fmink;subjecttoy2S;(3)Figure3illustratesthislocalsearchprocedure. f*2(A')for B2(C) for ASearch space for reference pointSearch space for reference pointf*f*1(B') QBADC A' Pf2 f1 B' Fig.3.Eacharrowcorrespondstoalower-levelsearchforaspeci edreferencepoint(C,A'orB').Theupper-levelsearch ndsareferencepointhavingoptimalworstobjective(suchasA'orB').Inthelower-leveloptimiza-tionproblem,thesearchisperformedontheoriginalde-cisionvariablespace.Theso-lutiony(j)(z)tothislower-leveloptimizationproblemdeterminestheoptimalobjec-tivevectorffromwhichweextractthej-thcomponentanduseitintheupper-leveloptimizationproblem.Thus,foreveryreferencepointz(asolutionfortheupper-levelproblem),thecorrespondingoptimalaugmentedachieve-mentscalarizingfunctionisfoundinthelower-levelloop.Theupper-leveloptimizationisinitializedwiththeNSGA-IIsolutionz(0)=f(x(j))andthelower-leveloptimizationisinitializedwiththeNSGA-IIsolutiony(0)=x(j).Wenowdiscussthetermi-nationcriterionofeachopti-mizationprocedure.ForterminatingtheoverallNSGA-IIprocedure,wecompute 8anormalizeddistance(ND)metricasfollows:D=vuut 1 MMXi=1zesti zi zwi zi2:(4)Here,thevectorszandzwaretheidealandworstobjectivevectorsoftheop-timizationproblem,respectively.ThesequantitiescanbecomputedoncebeforetheNSGA-IIprocedurebysolving2Mdi erentsingle-objectiveoptimizationsofminimizingandmaximizingeachobjectiveatatime.Sincetheexact nalvalueoftheDmetricisnotknownapriorionanar-bitraryproblem,werecordthechangeinDforthepast (=50usedhere)generations.Say,Dmax,Dmin,andDavg,arethemaximum,minimum,andav-erageDvaluesforthepastconsecutive generations.IfthechangeD=(Dmax Dmin)=Davgissmallerthanathreshold(=1(10 4)isusedhere),theNSGA-IIprocedureisterminated.WeusethesamenormalizeddistancemetrictodecidewhetherthelocalsearchneedstobeperformedinaparticulargenerationofNSGA-II.Atagen-eration,thechangelDinnormalizeddistanceoverthepast l(=20usedhere)generationsisrecorded.IflD(=0:005usedhere),thelocalsearchisperformed.Thisreducesthenumberoflocalsearchesperformedfromnotsogoodsolutions.Whenthebestnon-dominatedfronthasstabilizedsomewhat,theextremesolutionsofthesetaremodi edusingthelocalsearchprocedure.Bothupperandlower-leveloptimizationtasksinthelocalsearchoperationusesapoint-by-pointsearchapproachwhichisterminatedbasedonthechosenoptimizationalgorithmandcodeusedforthepurpose.Inalloursimulations,wehaveusedKNITROforthelower-leveloptimizationtaskinwhichwehavesetaterminationconditionontheKKTerrorvalue(10 6)oramaximumof100iterationswhicheverhappens rst.Fortheupper-leveloptimizationtask,wehaveusedKNITRO'sSQPsolver.Theupper-leveltaskisterminatedifthenormoftheNewton'sdirectionislessthanofequalto0.001oramaximumiterationof100iselapsed.AftertheNSGA-IIrunisterminated,weconstructthenadirpointfromtheworstobjectivevaluesofthe nalnon-dominatedsetF1.5SimulationResultsInthissection,wepresentsimulationresultsoneightproblemshavingthreeormoreobjectives.Inmostoftheseproblems,thenadirpointwasdiculttoobtainusingthepay-o table.Inallproblems,weuseapopulationofsizemax(60;20n)(nisthenumberofvariables),crossoverandmutationprobabilitiesof0.9and1=n,crossoverandmutationindicesof10and50,respectively,and=10 4.Ineachcase,wemake10di erentrunsfromdi erentinitialpopulations,buteverytimetheprocedureisfoundtoconvergenearaparticularsetofextremepoints,therebyleadingto ndingasimilarnadirpointeverytime. 95.1ProblemKMThe rstproblemKM,adaptedfrom[12],isthefollowing:minimize8: x1 x2+51 5(x21 10x1+x22 4x2+11)(5 x1)(x2 11)9=;;subjectto3x1+x2 120;2x1+x2 90;x1+2x2 120;0x14;0x26:(5)Thetruenadirpointofthisproblemisreportedtobeznad=(5;4:6; 14:25)T[9].Table1showsthethreeextremesolutions(x)foundbyourproposedap-proach.Itisclearthatwhentheworstobjectivevaluesarecollectedtogether,weobtainanidenticalpoint(uptotwodecimalpoints)asthatinthetruenadirpoint.Figure4showsthatthenormalizeddistancevaluegetsstabilizedatTable1.Extremepointsfoundbythepro-posedapproachonproblemKM. x Estimatedznad 0.000 0.000 5.000 2.200 -55.000 0.000 6.000 -1.000 4.600 -25.001 3.500 1.501 0.000 -3.100 -14.251 Terminated at gen. 87D stabilized for 50 gen. 0 10 20 30 40 50 60 70 80 90 Fig.4.VariationofDwithgenerationonKM.around40generationandsinceD=50isused,ittookanother50generationstoterminatethehybridprocedure.Interestingly,theDvaluereachesthe nalstabilizedvalueveryquickly,therebyindicatingtheeciencyoftheproposedprocedure.5.2ProblemSW1ThesecondproblemSW1isasfollows[16]:minimize8:f1(x)= (100 7x1 20x2 9x3)f2(x)= (4x1+5x2+3x3)f3(x)= x39=;;subjectto11 2x1+x2+13 5x39;x1+2x2+x310;xi0;i=1;2;3:(6) 10Thepreviousstudy[16]reportedthetruenadirpointtobeznad=( 3:6364;0;0)T.Table2showstwoextremesolutions(x)(hence,thetruenadirpoint)foundbyourproposedapproach.Figure5showstheprogressoftheproposedapproach.Table2.ExtremepointsfoundbytheproposedapproachonproblemSW1. x Estimatedznad 0.0000 3.1818 3.6364 -3.6364 -26.8182 -3.6364 0.0000 0.0000 0.0000 -100.0000 0.0000 0.0000 Stabilized for 50 gen.Terminated at gen. 87 0 10 20 30 40 50 60 70 80 90 Fig.5.VariationofDwithgener-ationonSW1.5.3ProblemSW2ThethirdproblemSW2originatesfrom[16]:minimize8��&#x-2.4;䌡&#x-2.4;䌡:9x1+19:5x2+7:5x37x1+20x2+9x3 (4x1+5x2+3x3) (x3)9&#x-2.4;䌡&#x-2.4;䌡=&#x-2.4;䌡&#x-2.4;䌡;;subjectto1:5x1 x2+1:6x39;x1+2x2+x310;xi0;i=1;2;3:(7)Thetruenadirpointforthisproblemisreportedtobeznad=(94:5;96:3636;0;0)T[16].Theoriginalstudy[16]foundaclosepoint(94:4998;95:8747;0;0)Tusingmultiple,bi-objectiveoptimizationsimulationusinganEMOprocedure.Theoutcomeisnotidenticaltothetruenadirpoint.Table3showsthethreeex-tremesolutionsfoundbyourproposedapproach.Weobtainthetruenadirpoint.Table3.ExtremepointsfoundbytheproposedapproachonproblemSW2. x Estimatedznad 4.0000 3.0000 0.0000 94.5000 88.0000 -31.0000 0.0000 0.0000 3.1818 3.6363 89.3182 96.3636 -26.8182 -3.6363 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 11DuetoanidenticalbehaviorofDvariationwithgenerationnumberonthisandsubsequentproblems,wedonotshowthe gureshere.5.4ProblemKSS1ThelinearKSS1problem[13]wasfoundtobedicultforestimatingthenadirpoint:maximize8:11x2+11x3+12x4+9x5+9x6 9x711x1+11x3+9x4+12x5+9x6 9x711x1+11x2+9x4+9x5+12x6+12x79=;;subjecttoP7i=1xi=1;xi0;i=1;2;:::;7:(8)Thetruenadirpointisreportedtobeznadir=(0;0;0)T[13].Table4showsthethreeextremesolutionsfoundbyourproposedapproach.Ourapproach ndsaTable4.ExtremepointsfoundbytheproposedapproachonproblemKSS1. x Estimatedznad 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 11.000 11.000 0.000 0.994 0.000 0.000 0.000 0.001 0.004 10.910 -0.026 11.006 0.000 0.000 1.000 0.000 0.000 0.000 0.000 11.000 11.000 0.000 nearnadirpointwithaslighterrorinthesecondobjectivevalue(asshowninFigure6theerrorisnotvisuallydetectable).Thisproblemisadicultonetosolveforestimatingtheexactnadirpoint,becauseoftheslowslopeleadingtoeachofthethreeextremepoints,asshownbyasetofrepresentativesolutionsobtainedthroughaclusteredNSGA-II,inwhichNSGA-II'scrowdingdistancemethodisreplacedbythek-meanclusteringmethod[2].Inthisproblem,itiseasytogetstucktoanon-dominatedpointclosetooneormoreextremepoints.Ourapproachseemstohavefoundtheexactextremevaluesfor rstandthirdobjectivesandmanagedtogettoanear-bypointaroundtheextremeofthesecondobjective.5.5ProblemKSS2Next,weconsideranotherlinearproblemKSS2[13]:maximize(x1;x2;x3);subjecttox1+2x2+2x38;2x1+2x2+x38;3x1 2x2+4x312;xi0;i=1;2;3:(9) 12 Nadir point Extreme points (3)by proposed approachClusteredpointsNSGA-II 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 10f3f1f2 8 6 4 2 12 Fig.6.Pareto-optimalfrontshowslongnarrowregionsnearextremepointsinproblemKSS1. Nadir point frontPareto-opt 15 20 25 30 35 40 5000 10000 15000 20000 25000 30000 10Normal StressDeflection 0.012 0.008 0.004CostProposed approachNSGA-II 5 0.0158 0Normal Stress Fig.7.Pareto-optimalfrontandtwoob-tainedpointsforproblemWELD.Table5.ExtremepointsfoundbytheproposedapproachonproblemKSS2. x Estimatedznad 0.000 3.818 0.166 0.000 3.818 0.166 3.344 0.000 0.432 3.344 0.000 0.433 3.253 0.628 0.000 3.253 0.628 0.000 Thenadirpointisreportedtobeznad=(0;0;0)T.Table5presentstheex-tremesolutionsobtainedbyourapproach.Thetruenadirpointisfoundbyourapproachinthisproblem.Nowweconsiderthreemoreproblems,borrowedfromengineering elds.Oneachoftheseproblems,theexactnadirpointisnotknown,butwhereverpossibleweexplaintheaccuracyofthenadirpointobtainedbyourapproach.5.6ProblemWELDTheWELDproblemhasfourvariablesandthreeobjectives,andisformulatedin[6].Ourpreviousstudy[6]introducedtheWELDproblemwhichhasfourvariablesandthreeobjectives.Thenadirpointwasestimatedtobeznad=(36:4209;0:0158;30000)T.Table6presentstwoextremepointsfoundbyourpro-posedapproachofthispaper.Theextremepointsforthesecondandthirdobjec-tivesarefoundtobeidenticalinthisproblem,indicatingthatalthoughtheprob-lemhasthreeobjectivefunctions,thePareto-optimalfrontistwo-dimensional,asisalsocon rmedbytheoriginalNSGA-IIpointsinFigure7.Thenadirpointestimatedbyourapproachis(36:4221;0:0158;30000:1284)T,whichisclosetothatobtainedbytheearlierstudy[6]. 13Table6.ExtremepointsfoundbytheproposedapproachonproblemWELD. x Estimatedznad 1.7356 0.4788 10.0000 5.0000 36.4221 0.000439 1008.0000 0.2444 6.2175 8.2915 0.2444 2.3810 0.015759 30000.1284 5.7ProblemCARTheseven-variable,three-objectiveCARproblemisdescribedin[10]. pointsNSGA-IIapproach (2 pts.)Proposed pointNadir 44 3.7 3.8 3.9 4 10.8 11.2 11.6 12 12.4f1f2f3 24 28 32 36 40 3.6 Fig.8.ExtremeobjectivevectorscoverstheentirePareto-optimalfrontforprob-lemCAR.Nopreviousstudyexistsonthisprob-lemfor ndingthenadirpoint.InTable7,wepresenttwoextremepointsobtainedbyourprocedure.Thus,thenadirpointestimatedbyourapproachforthisproblemisznad=(42:767;4:000;12:521)T.Fig-ure8showsthecompletePareto-optimalfrontwithasetofrepresen-tativeclusteredNSGA-IIsolutions.ItisclearfromtheplotthattheabovetwoextremepointsareadequatetocovertheextremeobjectivevaluesofthePareto-optimalfrontandisabletolocatethenadirpointoftheprob-lem.Table7.ExtremepointsfoundbytheproposedapproachonproblemCAR. x Estimatedznad 1.500 1.350 1.500 1.500 2.625 1.200 1.200 42.767 3.585 10.611 0.500 1.226 0.500 1.208 0.875 0.884 0.400 23.589 4.000 12.521 5.8ProblemWATERFinally,weconsidertheWATERproblem[15],whichisalsodescribedin[2].Forthisproblem,theexactnadirpointisnotknown.However,sincetherearethreevariablesand veobjectives,someredundancyintheobjectivesisexpectedforthePareto-optimalsolutions.AnapplicationofNSGA-IItothisproblem[2](page388)wasfoundtoindicatesomecorrelationsamongtheobtainedrepresen-tativesolutions.Table8presentstheextremepointsobtainedforthisproblem 14Table8.ExtremepointsfoundbytheproposedapproachonproblemWATER. x Estimatedznad 0.010 0.100 0.100 1.038 0.020 0.949 0.075 5.649 0.450 0.098 0.010 0.916 0.900 0.936 0.033 0.002 0.114 0.100 0.010 0.918 0.228 0.951 0.031 0.285 0.098 0.010 0.100 0.918 0.197 0.095 2.671 5.713 byourapproach.Weobservethattheextremepointsforobjectivesf4andf5comefromanidenticalsolution.Theestimatednadirpointusingourprocedureisznad=(1:038;0:900;0:951;2:671;5:713)T.6ConclusionsInthispaper,wehaveextendedourpreviousstudyonaserialimplementationofanEMOprocedurefollowedbyanMCDMbasedlocalsearchapproachto ndextremepointsaccuratelyforestimatingthenadirpointofamulti-objectiveoptimizationproblem.Thenadirpointinmulti-objectiveoptimizationisusedinnormalizingobjectiveswhichisnecessaryfordi erentmulti-criterionopti-mizationalgorithms.Besides,thetaskofestimatingthenadirpointforthreeormoreobjectivesisaopenresearchtaskinmulti-criterionoptimizationliterature.Nadirpointscanonlybeestimatedaccuratelyif(i)objective-wiseextremesand(ii)Pareto-optimalsolutionsarefound.Duetothistwo-prongedrequirements,wehavesuggestedabi-levellocalsearchtask.Thelocalsearchisemployedwithextremenon-dominatedsolutionsonlywhenthebestnon-dominatedfronthasstabilizedsomewhat,therebymakingtheoverallmethodcomputationallytractable.Onasetof vetestproblemsandthreeengineeringdesignproblems,theproposedintegratedprocedurehasableto ndtheexactnadirpointquiteaccurately.Thisworkisalsoimportantfromanotherpointofview.Thisworkdemon-strateshowalocalsearchapproachcanbeintegratedwithanevolutionarypopulation-basedapproachandusedsparinglyforacomplexoptimizationtoensureaccurateconvergence.7AcknowledgmentsAuthorsacknowledgethesupportfromtheAcademyofFinland,FoundationofHelsinkiSchoolofEconomics,andtheJennyandAnttiWihuriFoundationduringthecourseofthisstudy. 15References1.R.Benayoun,J.deMontgol er,J.Tergny,andP.Laritchev.Linearprogrammingwithmultipleobjectivefunctions:Stepmethod(STEM).MathematicalProgram-ming,1(3):366{375,1971.2.K.Deb.Multi-objectiveoptimizationusingevolutionaryalgorithms.Chichester,UK:Wiley,2001.3.K.DebandR.B.Agrawal.Simulatedbinarycrossoverforcontinuoussearchspace.ComplexSystems,9(2):115{148,1995.4.K.Deb,S.Chaudhuri,andK.Miettinen.Towardsestimatingnadirobjectivevectorusingevolutionaryapproaches.InProceedingsoftheGeneticandEvolu-tionaryComputationConference(GECCO-2006),pages643{650.NewYork:TheAssociationofComputingMachinery(ACM),2006.5.K.DebandK.Miettinen.Areviewofnadirpointestimationproceduresusingevolutionaryapproaches:Ataleofdimensionalityreduction.TechnicalReportKanGALReportNumber2008004,KanpurGeneticAlgorithmsLaboratory(Kan-GAL),IndianInstituteofTechnologyKanpur,India,2008.6.K.Deb,K.Miettinen,andS.Chaudhuri.Estimatingnadirobjectivevector:Hybridofevolutionaryandlocalsearch.TechnicalReportElectronicWorkingPaperW-440,HelsinkiSchoolofEconomics,Finland,2008.7.K.DebandD.Saxena.SearchingforPareto-optimalsolutionsthroughdimen-sionalityreductionforcertainlarge-dimensionalmulti-objectiveoptimizationprob-lems.InProceedingsoftheWorldCongressonComputationalIntelligence(WCCI-2006),pages3352{3360,2006.8.M.EhrgottandD.Tenfelde-Podehl.ComputationofidealandnadirvaluesandimplicationsfortheiruseinMCDMmethods.EuropeanJournalofOperationalResearch,151:119{139,2003.9.P.Eskelinen,K.Miettinen,K.Klamroth,andJ.Hakanen.Paretonavigatorforinteractivenonlinearmultiobjectiveoptimization.ORSpectrum,inpress.10.L.Gu,R.J.Yang,C.H.Tho,L.Makowski,O.Faruque,andY.Li.Optimizationandrobustnessforcrashworthinessofsideimpact.InternationalJournalofVehicleDesign,26(4),2001.11.H.IsermanandR.E.Steuer.Computationalexperienceconcerningpayo tablesandminimumcriterionvaluesovertheecientset.EuropeanJournalofOpera-tionalResearch,33(1):91{97,1988.12.K.KlamrothandK.Miettinen.Integratingapproximationandinteractibedecisionmakinginmulticrietriaoptimization.OperationsResearch,56:222{234,2008.13.P.Korhonen,S.Salo,andR.Steuer.Aheuristicforestimatingnadircriterionvaluesinmultipleobjectivelinearprogramming.OperationsResearch,45(5):751{757,1997.14.K.Miettinen.NonlinearMultiobjectiveOptimization.Kluwer,Boston,1999.15.T.Ray,K.Tai,andK.C.Seow.Anevolutionaryalgorithmformultiobjectiveoptimization.EngineeringOptimization,33(3):399{424,2001.16.M.SzczepanskiandA.P.Wierzbicki.Applicationofmultiplecrieterionevolu-tionaryalgorithmtovectoroptimization,decisionsupportandreferencepointapproaches.JournalofTelecommunicationsandInformationTechnology,3:16{33,2003.17.A.P.Wierzbicki.Theuseofreferenceobjectivesinmultiobjectiveoptimization.InG.FandelandT.Gal,editors,MultipleCriteriaDecisionMakingTheoryandApplications,pages468{486.Berlin:Springer-Verlag,1980.