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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,∗aDepartmentofComputerScience,NationalPingtungUniversityofEducation,4-18Min-ShengRoad,Pingtung900,Taiwan,ROC
bInstituteofEngineeringScienceandTechnology,NationalKaohsiungFirstUniversityofScienceandTechnology,1UniversityRoad,Yenchao,Kaohsiung
824,Taiwan,ROC
cDepartmentofElectricalEngineering,NationalKaohsiungUniversityofAppliedSciences,415Chien-KungRoad,Kaohsiung807,Taiwan,ROC
dDepartmentofHealthcareAdministrationandMedicalInformatics,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,thepersonalbest(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+1id =v
t id+c1∗r1∗(p t id−x t id)+c2∗r2∗(p t gd−x t id) (4.1) wherev
tidisvelocityofparticleiatiterationtwithrespecttothedth
dimension.
v
t+1id isnewvelocityofparticleiatiterationt+1with respecttothedthdimension.c1andc2areaccelerationcoefficients.r1andr2areuniformrandomnumbersbetween0and1.tiscurrent
iteration.pt
id ispositionvalueoftheithpbestatiterationtwith
respecttothedthdimension.pt
gdispositionvalueofthegbestat
iterationtwithrespecttothedthdimension. xt+1id =xt
id+
v
t+1id (4.2)wherext
idispositionofparticleiatiterationtwithrespecttothe
dthdimension.xtid+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
tid+1=ωv
tid+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
ωand iterations for
α1,
α20 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
c1and iterations for
β1,
β20 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
c2and iterations for
γ1,
γ2Fig.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 fob2(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
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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