Hybrid Taguchi-based particle swarm optimization for flowshop scheduling problem

ContentslistsavailableatScienceDirect

Applied

Soft

Computing

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Hybrid

sliding

level

Taguchi-based

particle

swarm

optimization

for

flowshop

scheduling

problems

Jinn-Tsong

Tsai

a

_,

_Ching-I.

_Yang

b

_,

_Jyh-Horng

_Chou

b,c,d,∗

a_{DepartmentofComputerScience,NationalPingtungUniversityofEducation,4-18Min-ShengRoad,Pingtung900,Taiwan,ROC}

b_{InstituteofEngineeringScienceandTechnology,NationalKaohsiungFirstUniversityofScienceandTechnology,1UniversityRoad,Yenchao,Kaohsiung}

824,Taiwan,ROC

c_{DepartmentofElectricalEngineering,NationalKaohsiungUniversityofAppliedSciences,415Chien-KungRoad,Kaohsiung807,Taiwan,ROC}

d_{DepartmentofHealthcareAdministrationandMedicalInformatics,KaohsiungMedicalUniversity,100Shi-Chuan1stRoad,Kaohsiung807,Taiwan,ROC}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received23May2013

Receivedinrevisedform

11September2013

Accepted5November2013

Availableonline20November2013

Keywords:

Flowshopschedulingproblem

SlidinglevelTaguchi-basedparticleswarm

optimization

Taguchi-basedcrossover

a

b

s

t

r

a

c

t

AhybridslidinglevelTaguchi-basedparticleswarmoptimization(HSLTPSO)algorithmisproposedfor solvingmulti-objectiveflowshopschedulingproblems(FSPs).TheproposedHSLTPSOintegratesparticle swarmoptimization,slidinglevelTaguchi-basedcrossover,andelitistpreservationstrategy.Thenovel contributionoftheproposedHSLTPSOistheuseofaPSOtoexploretheoptimalfeasibleregionin macro-space,theuseofasystematicreasoningmechanismoftheslidinglevelTaguchi-basedcrossovertoexploit thebettersolutioninmicro-space,andtheuseoftheelitistpreservationstrategytoretainthebest particlesofmulti-objectivepopulationfornextiteration.TheslidinglevelTaguchi-basedcrossoveris embeddedinthePSOtofindthebestsolutionsandconsequentlyenhancethePSO.Usingthesystematic reasoningwayoftheTaguchi-basedcrossoverwithconsideringtheinfluenceoftuningfactors˛,ˇand ispresentedinthisstudytosolvetheconflictingproblemofnon-feasiblesolutionsandtofindthe betterparticles.Asaresult,itexhibitsasignificantimprovementinParetobestsolutionsoftheFSP.By combiningtheadvantagesofexplorationandexploitation,fromthecomputationalexperimentsofthe sixtestproblems,theHSLTPSOprovidesbetterresultscomparedtotheexistingmethodsreportedinthe literaturewhensolvingmulti-objectiveFSPs.Therefore,theHSLTPSOisaneffectiveapproachinsolving multi-objectiveFSPs.

©2013ElsevierB.V.Allrightsreserved.

1. Introduction

Inthecurrenteraofglobalindustrialization,resourcescarcity is becoming a critical problem. Therefore, efficient production schedulingisessentialforoptimizingtheuseofavailableresources andalsoforsatisfyingperformancemeasurementcriteria. Multi-objective flowshop scheduling is among the most common flowshopschedulingproblems(FSPs).Generally,theselectionsof performancemeasurementcriteriaarecompletiontimeand tar-dinessproblem.Anefficientproductionschedulingcanincrease machineavailabilitytopromotetheprofitandcompetitivenessof thecompany.Thewidespreadadoptionofjust-in-time manufac-turing,inwhichjobsareprocessedonlyasneeded,hasexpanded therole of tardy production inprocess planning.The tardiness problemsaffectthedelayindeliveryofthenextscheduling.Itmust

∗ Correspondingauthorat:InstituteofEngineeringScienceandTechnology,

NationalKaohsiungFirstUniversityofScienceandTechnology,1UniversityRoad,

Yenchao,Kaohsiung824,Taiwan,ROC.Tel.:+88676011000;fax:+88676011066.

E-mailaddresses:[email protected],[email protected],

[email protected](J.-H.Chou).

notonlycompensateforthecustomerduetothedelayindelivery butalsocausethecompanyreputationandimagetosufferlosses, therefore,reducesmarketcompetitiveness.Toimprovecompletion timeandtominimizethetardinessproblem,effectivesolutionsfor theFSPareneededtominimizebothmakespanandmaximum tar-diness.TheFSPreferstotheproblemofdealingwithnjobsonm machinesorworkcentersinafacilityinwhichalljobsareprocessed onallmachinesinthesamesequence.Theschedulingprocedure knownastheJohnsonruleisusedtosolvethetwo-machine prob-lem[1].Problemsinvolvingmorethantwomachinesorjobsare calledNP-completeorNP-hardproblems[2].

Althoughgeneticalgorithms (GAs)have proven effectivefor solving single-objective optimization problems [3–5], obtain-ing effective solutions for real world problems often requires simultaneousconsiderationofmulti-objectivefunctions.Another frequentlyencounteredpracticalproblemisthataperfect multi-objective solutionthatsimultaneously optimizeseach objective function is virtually precluded by conflicts in the considered objectives.Therefore,whensolvingmulti-objectiveproblems,the ultimate goal is finding the best solution set, i.e., the Pareto bestsolutions.Afterconsideringtradeoffs,thedecisionmakercan thenchoosethepreferredsolution.Multi-objectiveFSPsinvolving

1568-4946/$–seefrontmatter©2013ElsevierB.V.Allrightsreserved.

applicationofmeta-heuristicsgenerallyrequireGAsandparticle swarmoptimization(PSO)tosolveNP-hardproblemseffectively. Many improved GA methods [6–22] have been proposed to solvemulti-objectiveFSPs.ThePSOapproach[34,35]hasproven effectivefor solving both continuous and discrete optimization problems.However,thePSOliteraturereviews[23–33]showvery few studies on solving the multi-objective FSPs. The detailed reviewsabouttheGAandPSOtosolvemulti-objectiveFSPsare describedinSection2.

Further improvement in solution performance is needed, althoughthealgorithmsreportedin[8,11,18,21,31,33]haveshown goodpotential.Therefore,themotivationofthisstudyistoimprove theabovemethods by constructinga novelalgorithm for find-ingParetobestsolutions.TheTaguchimethodisarobust-design approachinnature.Itborrowsfromstatisticalexperimentaldesign conceptsinevaluatingandimplementingimprovementsfor prod-ucts,processesorsystemsofequipment.TheTaguchimethodhas successfullyusedinGAaboutoptimizationdesignand jobshop schedulingproblems[45,50–52].IndevelopinganimprovedPSO algorithm,oneisnaturallycompelledtoaskiftheTaguchimethod can be incorporated to efficiently generate optimal offspring. Motivatedbysuchcuriosity,ahybridslidinglevelTaguchi-based particleswarmoptimization(HSLTPSO)algorithmisproposedto attempttoimprovesearchperformancewithtwofeatures.Because theperformance ofPSO withnon-linear timevaryingevolution (PSO-NTVE)approachdependsonthechoiceoftuningfactors˛, ˇand , thefirstfeature isto usea systematicreasoning way withconsidering theinfluence of tuning factors˛, ˇ and  in theTaguchi-based crossover operationto avoid the scheduling conflictingproblemandtofindanoptimalsolutioninsteadofa crossoveroperationbasedonarandomprocess[8,11].Twomajor toolsusedinthesystematicreasoningwayare(1)the signal-to-noiseratio(SNR)whichmeasuresqualityand(2)theorthogonal array(OA)whichareusedtostudyingmultipleparameters simul-taneously.Thesecondfeatureistouseaneasywaytogenerate theParetobestsolutionsfoundsofarsothatthebestoffspring (solutions)canberetained.For simplicity,twolevelsaresetas ˛,ˇ,∈{0.5,1.5}.Therefore,thetwo-levelOAisadoptedto per-formtheexperiments.ThefirstthreecolumnsofOAareusedfor thesliding level factorsof ˛,ˇ, and . Aparticlecan generate anewparticlethroughPSOprocessbasedonOA.Thefollowing ncolumnsareusedtoallocatethenjobsandperform Taguchi-basedcrossoveroperation.Therefore,slidinglevelTaguchi-based PSO(SLTPSO)combinestheslidinglevelPSOwithTaguchi-based crossovermethod.The particleand thegenerated newparticle areselectedforthelevels1and2ontheTaguchi-basedcrossover operation.ThetuningfactorsintheslidinglevelPSOandjob fac-torsforTaguchi-basedcrossoveroperationarerelated.Factorsare calledrelated when thedesirable experimentalregionof some factorsdependsonthelevelsettingsofotherfactors.Becauseof theuseofTaguchi-basedcrossoveroperationandPSO,the algo-rithmis robustand achievesquick convergence.Therefore, this studyproposesanovelHSLTPSOapproachforfindingtheglobal optimalsolution(GOS)formulti-objectiveFSPsinwhichthe objec-tivesare to minimizeboth makespan and maximum tardiness. TheproposedHSLTPSOalgorithmwascomparedwiththeMOGLS algorithmreportedbyIshibuchiandMurada[8],withthe modi-fiedMOGLSalgorithmreportedbyIshibuchietal.[11],withthe MOGLSalgorithm reported byArroyo and Armentano [21] and withthehybridTaguchi-basedgeneticalgorithm(HTGA)reported byYangetal.[18].Thealgorithmwasthencompared withthe MOPSOalgorithmdevelopedbyLietal.[31]andwiththehybrid Taguchi-based Particle swarm optimization (HTPSO) algorithm reportedbyYangetal.[33].Thecomparisonresultsconsistently showedthattheHSLTPSOalgorithmoutperformsallthesix algo-rithms.

Therestof this paperisorganizedas follows.Theliterature reviewsareshowninSection2.Section3introducestheFSPwith twoobjectivefunctions.Section4introducesthePSO.Section5 explainstheproposedHSLTPSOfor solvingtheFSPs. Numerical examplesaregiventoillustratetheproposedmethodinSection6. Finally,discussionsandconclusionsaregiveninSections7and8, respectively.

2. Literaturereviews

AreviewoftheGAliterature[6–22]showsthatthealgorithms proposedbyIshibuchiandMurata[8]andbyIshibuchietal.[11] forsolvingtheFSParestructurallycomplete.Inthemulti-objective geneticlocalsearch(MOGLS)algorithmreportedin[8]andits mod-ification(modifiedMOGLS)reportedin[11],arandomlyselected weightvaluewasusedtoevaluatethefitnessfunction.Although its modificationreported in [11] hasshown good potential for achievingtheidealoutcomebyselectingonlygoodoffspringas initialsolutionsforlocalsearch,furtherimprovementinsolution performanceisneeded.In[6],amulti-phaseapproachfor minimiz-ingmakespanandtotalweightedtardinessinthehybridflexible flowshopproblemconsideringsequence-dependentsetuptimesis presented.Threephasesareusedinthisalgorithm.Firstphaseuses asimple GAtominimizethecombinationofobjectivefunction. Theothertwophasesareusedtoimprovethesolutionsof previ-ousphase.Paretoarchiveconceptshadbeenimplementedandthe parametersoftheproposedalgorithmwerecalibratedbydesignof experimentmethod.In[7],ageneticalgorithmwashybridizedwith anovelschemeforcombiningtwolocalsearchmethods: inser-tionsearchandinsertionsearchwithcutandrepair.Ishibuchiand Murata[9]proposedanotheralgorithmthatappliedalocalsearch proceduretoeachsolutiongeneratedbygeneticoperation.In[10],a numberofindividualsarerandomlyselectfromcurrentpopulation foreachtime.Twoindividualsareselectedfromthenumberof indi-vidualsforcomparison.Theindividualwithbetterfitnessfunction isselected.Itisagoodindividual.Repeattheprocedureuntilthe goodindividualsisequaltocurrentpopulation.Theselectedgood individualsareusedforlocalsearch.Inordertoallocatethe avail-ablecomputingtimebetweengeneticsearchandlocalsearch,this studystruckabalancebetweengeneticandlocalsearches.In[12], analgorithmwasreportedforselectingindividualsforacrossover operationbasedonaweightedsumofmulti-objectivefunctions withvariableweights.Theproposedpreservationstrategy consid-eredmultipleelitesolutionsinsteadofasingleelitesolution.In [13],theprocedureforselectingindividualsforacrossover oper-ationwasbasedonaweightedsumofmulti-objectivefunctions. For a two-stagebi-criterion FSP,Neppalli etal. [14]considered theobjectiveofminimizingtotal flow timesubject tothe opti-mal makespan. Two GA-based approaches, a vector evaluation approach and a weighted criteria approach, were proposed. In [15],aquantum-inspired GAbasedonQ-bitrepresentation was appliedforexploration.Thepermutation-basedGAwasusednot onlyforexplorationinpermutation-basedschedulingspace,but alsoforstressingexploitationtoachievegoodschedulingsolutions. In[16],asolutionwaspresentedforare-entranthybridFSPwith twoobjectives,maximizingtheutilizationrateinthebottleneck andminimizingthemaximumcompletiontime.Thesolutionwas achievedwithanovelmulti-objectivegeneticalgorithmbasedon theLorenzdominancerelationship.Thealgorithmreportedin[17] introducedanewhybridcontrollerusingartificialintelligenceto improvethedynamic performanceoftheself-excitedinduction generator(SEIG)drivenbywindenergyconversionscheme.The hybridartificialintelligencecompromisesaGAandfuzzylogic con-troller.GAisusedtooptimizetheparametersofthefuzzysetto ensureabetterdynamicperformanceoftheoverallsystem.Inthe HTGA[18],theuseofdynamicweightsselectedrandomlybyfuzzy

inferencesystemandtheapplicationofsystematicreasoningway byTaguchi-basedcrossoveroperationachievedgoodsearch capa-bility[47–49].TheHTGAwascomparednotonlywiththeMOGLS algorithmreportedbyIshibuchiandMurata[8],butalsowiththe modifiedMOGLSalgorithmdevelopedbyIshibuchietal.[11].The comparisonresultsshowthatHTGAisbetterthanboththeoriginal MOGLSalgorithmandthemodifiedMOGLSalgorithm.In[19],the decentralizedmulti-objectivecongestionmanagementproblemin thederegulatedforwardpowermarketwasmodeledunderthe conflictingobjectivesofmaximizingsocialwelfareand minimiz-ingemissionimpacts.Amodifiednon-dominatedsortinggenetic algorithmIIwithcontrolledelitismandadynamiccrowding dis-tancewasapplied.In[20],astaticmixedintegernon-linearmodel fordistributedgenerationwasdefinedandsolvedusingamodified NSGA.Themulti-objectivefunctionsforminimizationweredefined asthetotalactiveloss,investmentandoperationalcost,and envi-ronmentalpollution.ArroyoandArmentano[21]proposedanother MOGLSforamulti-objectiveFSPandcomparedtheperformanceof thealgorithmwithtwomulti-objectivegeneticlocalsearch algo-rithms,Thefirstalgorithmis MOGLSproposedbyIshibuchiand Muratain[8](denotedbyIM-MOGLS)andanotheralgorithmwas proposedbyJaszkiewiczin[22](denotedbyJ-MOGLS)forsolving a50-job,20-machineFSPwithobjectivesofmakespanand maxi-mumtardiness.Fig.8reportedin[21]showstherunningresults achievedbythesameinitialsetforallalgorithms.Their compre-hensiveperformancecomparisonshowedthattheMOGLS[21]was superiortotheIM-MOGLSandJ-MOGLS.

LikeGA,thePSO[35]isinitializedwithapopulationofrandom solutions.However,onedifferenceisthatPSOassignsarandomized velocitytoeachsolution.Eachsolutionisrepresentedbya parti-cleflyingthroughthesolutionspace.ComparedtoGA,itrequires less computation and fewer parameter adjustments. However, althoughthePSOiseasilyimplementedandachievesquick con-vergence,ittendstogetstuckinnear-optimalsolutions,whichare difficulttoimprovebyfurtherfinetuning.AreviewofthePSO liter-ature[23–33]showsveryfewstudiesofthemulti-objectiveFSP.In [23],adiscreteparticleswarmoptimization(DPSO)algorithmwas proposedforsolvingtheno-waitFSPwithbothmakespanandtotal flowtimecriteria.Solutionqualitywasimprovedbyhybridizingthe DPSOalgorithmwiththevariableneighborhooddescentalgorithm. WangandTang[24]introducedanewvelocityandparticleupdate modeltogeneratenewpopulationandproposedaself-adaptively diversitycontrolstrategytoavoidprematureconvergenceforthe discreteparticleswarmoptimizationwithblockingtominimizethe makespan.Alocalsearchmethodnamedstochasticvariable neigh-borhoodsearchwasusedtoimprovethesearchability.In[25],an improvedparticleswarmoptimizationalgorithmbasedonthe“all different”constraintisproposedtosolvetheFSPwiththeobjective ofminimizingmakespan.Itisbecausethattheparticle’scurrent positionandvelocityarebothdenotedaspermutationsofalljobs whichmustsatisfythe“alldifferent”constraint.Thisconstraint forceseverydecisionvariableinagivengrouptoassumeavalue differentfromthevalueofeveryothervariableinthatgroup.This algorithmalsocombinesPSOwithgeneticoperatorstogether effec-tively.Tasgetirenetal.[27,28]firstusedthesmallestpositionvalue (SPV)toconvertthepositionvectortoajobpermutation.Reported two-objectivealgorithmsincludeanalgorithmforminimizingboth makespanandtotalflowtimein[26,27],analgorithmfor minimiz-ingmakespanandmaximumlatenessin[28],andanalgorithmfor minimizingmeancompletiontimeandmeantardinessin[30].Li etal.[31]presentedamulti-objective particleswarm optimiza-tion(MOPSO) for a multi-objective FSP.Based onranked-order values,thecontinuouspositionsofparticlesweretransformedinto jobpermutations.Toenhanceexploitation,alocalsearchbasedon theNawaz-Enscore-Hamheuristicwasappliedtogoodsolutions withaspecifiedprobability.Searchperformancewasalsoenhanced

bydesigningasimulatedannealingwithmultipledifferent neigh-borhoodsandbyapplyinganadaptiveMeta-Lamarchianlearning strategytodecidewhichneighborhoodisused.TheMOPSOapplied a random weighted linear sum function to aggregate a multi-objectivesolutionintoasinglesolutionforevaluationpurposes. TheyalsocomparedMOPSOwithMOGLS(denotedbyIM-MOGLS in[31])intermsofperformanceinsolvinga20-job,10-machine FSPwithobjectivesofminimizingmakespanandmaximum tardi-ness.Fig.5in[31]comparestherunningresultsachievedbythe twoalgorithmsinoneexperiment.Theircomprehensive perfor-mancecomparisonconfirmedthattheMOPSO[31]wassuperiorto theIM-MOGLS.Yangetal.[33]proposedtheHTPSO,whichhasalso showngoodsearchcapability.TheHTPSOoutperformedthe opti-mizationmethodspresentedbyIshibuchiandMurata[8],Ishibuchi etal.[11],Yangetal.[18],andLietal.[31].Proposedtriple-objective algorithmsincludeanalgorithmforsolvingmakespan,totalflow time,andtotalmachineidletimeproposedin[29],analgorithmfor solvingmakespan,averagecompletiontime,andmaximum tardi-nessproposedin[31],andanalgorithmforsolvingmakespan,mean flowtime,andmachineidletimeproposedin[32].

3. Flowshopschedulingproblemwithtwoobjective functions

The following discussion uses the symbols n/m/P/Obj to describetheFSP.Givennjobstobeprocessedonmmachinesin thesameorder,Pindicatesthatonlythepermutationschedulesare considered,andObjdenotestheobjectivefunctionsinwhichthe scheduleistobeevaluated.Theconsideredproblemisfindingthe jobschedulegiventheobjectivesofminimizingbothmakespanand maximumtardiness.Considertheexampleofaten-joband five-machineFSP.Theinputsoftheproblemaretenjobs,fivemachines, theprocessingtimeforeachjoboneachmachine,andtheduedate foreachjob.Theoutputistofindthejobsscheduleforthe multi-objectiveFSP.Theobjectivesaretominimizeboththemakespan andthemaximumtardiness.Themainconstraintoftheproblem mustbeapermutationFSP.Otherconstraintsareasfollows: (1)Alljobsareavailableattimezero.

(2)Physicalbufferspacebetweentwosuccessivemachinesis suf-ficient.

(3)Setuptimesfortheoperationsaresequence-independentand areincludedintheprocessingtimes.

(4)Allmachinesarecontinuouslyavailable. (5)Individualoperationsarenotpreemptive.

Thisstudyappliestheweightedsumapproach[8,11].Because thefeasiblesolutionsarewidelydispersedinthesolutionspace, Ishibuchietal.[8,11]arguedthatthefixedweightmethodmay overlook somebettersolutions becauseitlimits thenumberof searchdirections.Therefore,theyproposedthatdynamicweights findbetterfeasiblesolutionsbyincreasingthenumberofsearching directions.

Thesequence of njobsis denotedbyn dimensionalvectors (J1,J2,...,Jn).Thenjobsareprocessedona seriesofmachines

(M1,M2,...,Mm)inthesamesequencewhereJidenotestheith

processingjobandMjdenotesthejthmachine.Theprocessingtime

ofjobionmachinejispi,j.Thecompletiontimeofjobiisdefinedas

Ci,m,i.e.,thecompletiontimeofjobionthelastmachinem,where

⎧

⎪

⎪

⎨

⎪

⎪

⎩

C1,1=p1,1, C1,j=C1,j−1+p1,j, forj=2,....,m,

Ci,1=Ci−1,1+pi,1, fori=2,....,n,

Ci,j=max{Ci,j−1,Ci−1,j}+pi,j, fori=2,....,n, forj=2,....,m.

ThemakespanofthesequenceofnjobsisdefinedasCmax=Cn,m,

i.e.,themaximumcompletion timeofthelastjobnonthelast machinem.

ThetardinessTiofjobiisdefinedas



Ti=max{(Ci,m−Di),0},

fori=1,2,....,n,andDiistheduedateofjobi.

(3.2) ThemaximumtardinessTmaxofthescheduleisdefinedas

Tmax=max



T1,T2,....,Tn



(3.3) In this study, the proposed HSLTPSO searches for all non-dominatedsolutionsforthemulti-objectiveoptimizationproblem. Considerthefollowingmulti-objectiveoptimizationproblemwith nobjectives:

Minimizef1(x),f2(x),...,fn(x) (3.4)

where f1(x),f2(x),...,fn(x) are n objectives to be minimized.

Whenthefollowinginequalitiesholdtruebetweentwosolutions xandy,solutionyissaidtodominatesolutionx.

∀

i:fi(y)≤fi(x) and

∃

j:fj(y)<fj(x) (3.5)

Asolutionthatisnotdominatedbyanyothersolutionsforthe multi-objectiveoptimizationproblemisaParetooptimalsolution [31].

4. Particleswarmoptimization

Particleswarmoptimization,which wasintroducedby Eber-hartandKennedyin1995[34,35],isanevolutionaryoptimization techniquebasedonmetaphorsforsocialinteractionand commu-nicationsuchasflocksofbirdsandschoolsoffish.Thisstochastic, population-basedapproachhasproveneffectiveforsolvingboth continuousanddiscreteoptimizationproblems.Eachparticleina swarm,whichisanalogoustoabirdinaflockorafishinaschool, movesaroundin ddimensionalsearch space.Based onitsown experienceandthatoftheswarm,itmovestowardthebestposition inthesearchspace.

Thepositionandvelocityofparticleiatiterationtare repre-sentedbyXt

iandV

t

i,whichcanbedefinedasX t i=(x t i1,x t i2,...,x t id) andVt

i =(

v

ti1,

v

ti2,...,

v

tid),respectively.Atiterationt,thepersonal

best(pbest)ofparticleiisrepresentedbyPt

i,whichdenotesthe

positionofparticleiwiththebestfitnessvaluefoundsofarand isdefinedasPt

i =(pti1,pti2,...,ptid).Theglobalbest(gbest)ofall

particlesatiterationtisrepresentedbyPt

g,whichdenotesthebest

positionoftheparticlewiththebestfitnessvalueintheswarm foundsofarand isdefinedasPt

g=(ptg1,pg2t ,...,ptgd).Thenew

velocityandpositionofparticleicanbeobtainedbyEqs.(4.1)and (4.2),respectively:

v

t+1_id =

v

t id+c1∗r1∗(p t id−x t id)+c2∗r2∗(p t gd−x t id) (4.1) where

v

t

idisvelocityofparticleiatiterationtwithrespecttothedth

dimension.

v

t+1_id isnewvelocityofparticleiatiterationt+1with respecttothedthdimension.c1andc2areaccelerationcoefficients.

r1andr2areuniformrandomnumbersbetween0and1.tiscurrent

iteration.pt

id ispositionvalueoftheithpbestatiterationtwith

respecttothedthdimension.pt

gdispositionvalueofthegbestat

iterationtwithrespecttothedthdimension. xt+1_id =xt

id+

v

t+1id (4.2)

wherext

idispositionofparticleiatiterationtwithrespecttothe

dthdimension.xt_id+1ispositionofparticleiatiterationt+1with respecttothedthdimension.

ThefirstpartofEq.(4.1)representsthecurrentvelocity,which providesthemomentumneededfortheparticletoroaminthe

searchspace.Thesecondpartisthecognitioncomponent.The par-ticleinthesearchspacealwaysmovestowarditsownbestposition foundsofar.Thethirdpartisthesocialcomponent.Becauseoftheir cooperativerelationship,theparticlescontinuouslymovetoward thecurrentgbest.

Theeasy implementationof PSOand itsfastconvergenceto areasonablesolutionmakeitaneffectiveheuristicoptimization technique.AlthoughtheoriginalPSOperformedwellinearly iter-ations,ittendedtobecometrappedatthelocalbestsolution,and solutionscouldnotbeimprovedbyfinetuning.Tobalancelocal andglobalsearchduringtheoptimizationprocess,Shiand Eber-hart[36]modifiedEq.(4.1)byintroducingtheconceptofinertia weightω.

ThenewvelocityisexpressedbyEq.(4.3):

v

t_id+1=ω

v

t

id+c1∗r1∗(ptid−xtid)+c2∗r2∗(ptgd−xtid) (4.3)

Theinertiaweightcanbeapositiveconstantorevenapositive linearornonlinearfunctionoftime.Whenω>1.2,thevelocity itembecomesthemainiteminthesearchdirectionofthe parti-cle.Itextendsthesearchareaandfindstheglobaloptimum.When ωisbetween0.8and1.2,threefactors,velocity,pbestandgbest, affectthevelocitycalculationforbothlocalandglobalsearch.When ω<0.8,onlythepbestandgbestaffectthenewvelocitycalculation, whichconvergestothelocaloptimum.Therefore,thevalueof iner-tiaweightωisatradeoffbetweentheglobalsearchandthelocal search.

Aliteraturereview[34–42]showedthattheNTVEmethod, pro-posedbyKoetal.[42],iscurrentlythebesttuningmethod.The inertiaweightisthesameasthatin[41].Theinertiaweightstarts withahighvalueωmaxandnonlinearlydecreasestoωminatthe

maximalnumberofiterations.However,c1startswithahighvalue

c1maxandnonlinearlydecreasestoc1minwhereasc2startswitha

lowvaluec2minandnonlinearlyincreasestoc2max:

ω=ωmin+

_iter max−iter itermax

˛ × (ωmax−ωmin) (4.4) c1=c1min+

_iter max−iter itermax

ˇ × (c1max−c1min) (4.5) c2=c2max+

_iter max−iter itermax

 × (c2min−c2max) (4.6)

whereIter denotesthecurrentnumber ofiterationsand Itermax

denotesthemaximumnumberofiterations.Thevalues˛,ˇ,and areconstantcoefficients.InearlyiterationsofNTVEoptimization, particlesroamthroughoutthesearchspace,andconvergence accel-eratestowardtheglobaloptimumduringlatteriterations.Because ofitsproveneffectivenessinobtainingthebestsolution,NTVEis appliedinthisstudy.

InPSO-NTVE,fivelevelsaresetas˛,ˇ,∈{0,0.5,1,1.5,2}[42]. Thereare53 _{combinations. AnOAL}

25(56)isapplied,as shown

in Table1.For simplicity,two levels for each factor are setas ˛,ˇ,∈{0.5,1.5}.Atfirst,theparticleinthecurrentpopulation pgenerateseightdifferentvaluesofω,c1,andc2.Therelationship

amongω,c1,andc2withiterations(forexample,itermax=3000)are

showninFig.1.Hence,˛,ˇ,and areusedasthefactorsof slid-inglevels.Atwo-levelOAisusedtodealwiththeproblem.First threecolumnsofOAareusedforfactors˛,ˇ,and.Thefollowing ncolumnsareusedfortheallocationofnjobs.Foreachparticleof thecurrentpopulation,theSLTPSOexecutesthefollowingsteps. (1)ForeachexperimentofOA,generatethenewparticlebyusing

thefactorsofslidinglevel˛,ˇ,and.

Firstly,executeslidingleverPSOtogeneratenewvelocityand newposition.Thenewpositionistransformedtoapermutation bySPVrule.AnewparticlePnewisgenerated.

Table1

OrthogonalarrayL25(56).

Experimentno. Factors

A B C D E F Columnnumbers 1 2 3 4 5 6 1 1 1 1 1 1 1 2 1 2 2 2 2 2 3 1 3 3 3 3 3 4 1 4 4 4 4 4 5 1 5 5 5 5 5 6 2 1 2 3 4 5 7 2 2 3 4 5 1 8 2 3 4 5 1 2 9 2 4 5 1 2 3 10 2 5 1 2 3 4 11 3 1 3 5 2 4 12 3 2 4 1 3 5 13 3 3 5 2 4 1 14 3 4 1 3 5 2 15 3 5 2 4 1 3 16 4 1 4 2 5 3 17 4 2 4 3 1 4 18 4 3 1 4 2 5 19 4 4 2 5 3 1 20 4 5 3 1 4 2 21 5 1 5 4 3 2 22 5 2 1 5 4 3 23 5 3 2 1 5 4 24 5 4 3 2 1 5 25 5 5 4 3 2 1

(2)ExecuteTaguchi-basedcrossover

Secondly,selecttheparticlepaslevel1(P1)andnew

parti-clePnewaslevel2(P2)toexecuteTaguchi-basedcrossover.A

particleafterTaguchi-basedcrossoverisgenerated.

(3)Findthebestparticle

Finally,afterallexperimentsarefinished,calculatethe

fit-nessoftheparticlesafterTaguchi-basedcrossover,theSNRs

andeffectsofvariousfactors(Efl)whicharedefinedasthe

fol-lowingsteps.Thebestlevelistheonewithbestlevelvaluefor

eachfactor.Anbestparticleisgenerated.Thedetailsregarding

theTaguchimethodcanbefoundin[43,44].

5. HybridslidinglevelTaguchi-basedparticleswarm optimization

ThissectionintroducestheuseoftheproposedHSLTPSOfor solvingFSPs.Itsobjectiveistominimizebothmakespanand max-imumtardiness.TheHSLTPSOcombinesSLTPSOwithlocalsearch and an elitist preservation strategy. Fig. 2 depicts thesteps of theHSLTPSOapproach,whicharedescribedindetailbelow.The parametersusedinthealgorithmaresetasshowninTable2. 5.1. Step1:solutionrepresentation

InFSP,asequenceofjobsS=(x1,x2,...,xn)representsthejob

processingsequence.Thatis,processingofjobx1isfollowedby

processingofjobx2,andsoon.

InPSO,eachparticlemovesinthendimensionalsearchspace atanassignedvelocity.VectorXt

i =(x t i1,x t i2,...,x t in),which

repre-sentstheithparticleofiterationt,correspondstonjobsoftheFSP. Eachdimensionrepresentsajob.However,sincetheparticleisnot apermutation,theSPVrulemustbeappliedtotransformitintoa permutation[27,28],asshowninTable3.Thejobpermutationsare foundbysortingthedimensionsinascendingorderoftheparticle valuesineachdimension.

0 500 1000 1500 2000 2500 3000 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 iteration ω α1=0.5 α2=1.5

(a) Relationships between

ω

and iterations for

α1

,

α2

0 500 1000 1500 2000 2500 3000 0.5 1 1.5 2 2.5 iteration c1 β1=0.5 β2=1.5

(b) Relationships between

c1

and iterations for

β1

,

β2

0 500 1000 1500 2000 2500 3000 0.5 1 1.5 2 2.5 iteration c2 γ1=0.5 γ2=1.5

(c) Relationships between

c2

and iterations for

γ1

,

γ2

Fig.1. Relationshipsamongω,c1,andc2atvaryingiterations.

5.2. Step2:initialization

Thefirststepisrandomlygeneratingtheinitialpopulationof particleswithpositiveintegers(1,2,...,n),wherenisthe num-ber of jobs.The initialposition and velocity of theith particle in n dimensions of the search space are represented by X0

i = (x0 i1,x 0 i2,...,x 0 id)andV 0 i =(

v

0 i1,

v

0 i2,...,

v

0 id),respectively.

5.3. Step3:generateasetofParetobestsolutions,andevaluate thefitnessfunctionfortheinitialpopulation

(1)GenerateasetofParetobestsolutions.

ThestepsofgeneratingthesetofParetobestsolutionsareas follows:

Start

Initialize

Generate the Pareto optimal solutions and evaluate particle fitness

Initialize the personal best and global best for the initial population

Number of particles finished? Sliding level Taguchi implementation

Local search

Meet stopping condition? Elitist preservation strategy: create the

population for next iteration Find the velocity of the particles of the

population for next iteration

Update the pbest particles and the gbest of the population for next iteration Swarm of particles via sliding level Taguchi implementation is generated Select a suitable two level orthogonal

array for matrix experiments For each particle (p) of current population

Update the Pareto optimal solutions

Display the final Pareto optimal solutions

B A

End Randomly select

three solutions from the Pareto optimal

solutions

A

B For each experiment

Use alpha, beta and gamma as the factors of sliding level and generate a new velocity and a new position. The new position is transformed to a permutation

p_new by SPV rule.

Select parent P1=p and parent P2 =p_new and execute Taguchi-based

crossover

Randomly generate a n job set U

Separate U into U1 and U2. The job numbers in U1 and U2 correspond to factor levels 1 and 2 in executed

experiment, respectively. According to the job numbers in U1 and

U2, we select new P1 and new P2 from P1 and P2 in sequence respectively

According to the executed experiment, a new particle is generated

Number of experiments finished?

Calculate the fitness values and signal-to-noise ratios of the experiments

Calculate the effects of the various factors and find the optimal level of

each factor

One optimal particle is generated based on the optimal level of each factor No Yes No Yes No Yes

Fig.2. ThestepsoftheHSLTPSOforFSP.

1.1Calculatetheobjectivevaluesofthemakespanfob1(x)and

themaximumtardinessfob2(x)ofeachparticleoftheinitial

population.

1.2Rearrange thepopulationby sortingthe valueof fob1(x)

in ascending order and thensorting the valueof f_ob2(x) indescendingorder.Ifonevalueoffob1(x)hasmorethan

two values of fob2(x),select only theminimumvalue of

fob2(x).

1.3Apply Eq. (3.5) to find thePareto best solutions. If the inequalitiesholdtruebetweentwosolutionsxandy,the solutionyisaParetobestsolutionoftheinitialpopulation wherefi(x)=fob1(x)andfj(x)=fob2(x).

(2)Evaluatethefitnessfunctionf(x)bytheweightedsumapproach Thefitnessfunctionisanindexoftheadaptabilityofthe indi-vidualinthepopulation.Ahighfitnessfunctionvalueindicatesa

Fig.11. ParetofrontoftheHSLTPSO,theMOPSO[31],andtheIM-MOGLS[31]for

20-joband10-machineFSP.

Fig.12.ParetofrontoftheHSLTPSOandtheHTPSO[33]for40-joband20-machine

FSP.

of1aresuperiortotheratiosC(MOPSO,HSLTPSO)of0and C(IM-MOGLS,HSLTPSO)of0respectively.Table11showstheresultsof performancecomparisonsoftheHSLTPSO,MOPSO,andIM-MOGLS intermsofSP,maxD,andHA.ThesmallertheSPisandthelarger themaxDandtheHAare,thebetter.TheSPandHAobtainedby theHSLTPSOaresuperiortothoseobtainedbytheMOPSOand IM-MOGLS.ThemaxDobtainedbytheHSLTPSOislargerthanthat obtainedbytheMOPSOandsmallerthanthatobtainedbythe IM-MOGLS.Fromthesimulationresults,theHSLTPSOisbetterthan theMOPSO[31]andIM-MOGLS[31]infindingthebetter Pareto-optimalsolutions.

6.6. Testproblem6

Thisproblemusestheexamplein[33]tocomparetheHSLTPSO withtheHTPSO[33].A40-joband20-machineFSPisrandomly generatedasdescribedinSectionIIIAin[8].

AsystematicreasoningwayofTaguchi-basedcrossoveris per-formedbyOAL64(263).Firstthreecolumnsareusedforfactors˛,ˇ,

and.Thefollowing40columnsareusedfor40jobs.Thefull neigh-borhoodsolutionsand500iterationsareused.Thepopulationsize is20.TheHSLTPSOareconductedin12independentruns.

Table11

ResultsofperformancecomparisonsoftheHSLTPSO,MOPSO,andIM-MOGLS,in

termsofSP,maxD,andHAinthetestproblem5.

Algorithm SP maxD HA

HSLTPSO 18.09 191.82 0.9704

MOPSO 38.08 104.81 0.4790

IM-MOGLS 44.99 415.77 0.4025

Table12

ResultsofperformancecomparisonsofHSLTPSOandHTPSOintermsofSP,maxD, andHAinthetestproblem6.

Algorithm SP maxD HA

HSLTPSO 11.84 360.91 0.8046

HTPSO 72.99 800.81 0.6589

Fig.12showstheParetofrontoftheHSLTPSOandtheHTPSO [33]for40-joband20-machineFSP.ThecoverageratioC(HSLTPSO, HTPSO)of0.733issuperiortotheratioC(HTPSO,HSLTPSO)of0. Table12showstheresultsofperformancecomparisonsofHSLTPSO andHTPSOintermsofSP,maxD,andHA.ThesmallertheSPis andthelargerthemaxDandtheHAare,thebetter.TheSPand HA obtainedby theHSLTPSOare superiortothoseobtainedby theHTPSO,whilethemaxDobtainedbytheHSLTPSOissmaller thanthatobtainedbytheHTPSO.Fromthesimulationresults,the HSLTPSOisbetterthantheHTPSO[33]infindingthebetter Pareto-optimalsolutions.

Thesolutionsfortheabovesixtestproblemsconfirmthatthe proposedHSLTPSOapproacheffectivelysolvestheFSPsand out-performstheotheroptimizationmethods.

7. Discussions

TheHSLTPSOforthesixtestproblemsisimplementedusing MATLAB7intheWindowsXPenvironmentonaPCwithanIntel 1.5GHz CPU and 1GB RAM. The runningtime of the HSLTPSO, accordingtotheFSPcomplexityandmaximumiterations,isshown asfollows.Inthetestproblem1,therequiredtimeoftheHSLTPSOis about5,9,18,and27minutesformaximumiterationsof500,1000, 2000,and3000,respectively.Inthetestproblems2,4,and6,the runningtimeoftheHSLTPSOisabout120minateachrun,because thesameFSPisusedforthetestproblems2,4,and6.Therequired timeoftheHSLTPSOinthetestproblem3isabout180min,because thelargedimensionalFSPhas50jobsand20machines.Therunning timeoftheHSLTPSOinthetestproblem5isabout35min.

Inthisstudy,theweightedsumapproachisusedasthefitness functiontofindtheParetosetsolutions,whicharenotabsolutely betterthananyother.Ishibuchietal. [8,11]confirmedthatthe weightedsumapproach canincrease searchdirectionsandfind betterfeasiblesolutions.Fromtheabovesixtestproblems,itcan beseenthattheapproachcanfindthebetterandmore concen-tratedsolutionsforthedecisionmakerstoselecttheacceptable solutionbasedontheirrequirementsandengineeringpractices,but thelimitationoftheweightedsumapproachistoobtainsmaller maximumspreadcomparedtotheotherPareto-based optimiza-tionapproaches.

8. Conclusions

TheHSLTPSOisproposedtosolvethemulti-objectiveFSPsand variousperformanceindicesareusedformeasuringthequalityof Pareto-optimalsets.TheHSLTPSOcombinesPSO, whichhasthe powerfulglobalexplorationcapabilityfor exploringtheoptimal feasibleregion,theslidinglevelTaguchi-basedcrossovertoexploit thebetteroffspring,andtheelitistpreservationstrategytoretain thebestparticles.Inthisstudy,thedetailedstepsoftheHSLTPSO arepresentedtosolvethemulti-objectiveFSPs.Themajor contri-butionoftheHSLTPSO istheuseofasystematicreasoningway withconsideringtheinfluenceoftuningfactors˛,ˇ,and,and performsTaguchi-basedcrossoverwithouttheconflicting prob-lemofnon-feasiblesolutionstofindtheglobaloptimalsolutions for the FSPs by minimizing both the makespan and maximum tardiness.Ineachcomputationalexperimentofthesixtest prob-lems,thecoverageratio,SP,andHAobtainedbytheHSLTPSOare

superiortothoseobtainedbytheoptimizationmethodsreportedin [8,11,18,21,31,33].Thatis,theproposedHSLTPSOeffectivelysolves theFSPsandoutperformsthoseapproachespresentedbyIshibuchi andMurata[8],Ishibuchietal.[11],ArroyoandArmentano[21], Yangetal.[18],Leetal.,[31],andYangetal.[33].Therefore,the proposedHSLTPSOapproachcanbeusedasamulti-objective opti-mizationmethodforsolvingFSPs.

Acknowledgements

ThisworkwasinpartsupportedbytheNationalScienceCouncil, Taiwan,undergrantnumbersNSC101-2221-E153-003, NSC102-2221-E153-002,andNSC102-2221-E-151-021-MY3.

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