2.Thesis 2

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Journal of Software Engineeringand Applications,2014,7,571-580

PublishedOnlineJune2014 inSciRes. http://www.scirp.org/journal/jseahttp://dx.doi.org/10.4236/jsea.2014.77053

 

 

Comparative Study of Different Representations in Genetic Algorithms for JobShop Scheduling Problem

 

 

Vedavyasrao Jorapur*,V.S. Puranik,A.S.Deshpande,M.R. Sharma

 

Visvesvaraya Technological University, Belgaum, India

Email:*jorapur@fragnel.ac.in

 

Received 13February 2014;revised 10March2014;accepted 18March2014

 

Copyright ©2014by authorsand ScientificResearch PublishingInc.

Thiswork islicensed underthe Creative Commons Attribution International License(CCBY).

http://creativecommons.org/licenses/by/4.0/

 

 

 Abstract

 

Dueto NP-Hardnature of the JobShop Scheduling Problems(JSP),exactmethods failtoprovide the optimalsolutions in quiter easonable computational time.Dueto this natureof theproblem, so many heuristicsand meta-heuristics havebeen proposed in the pasttogetop timalornear- op- timalsolutions foreasy to toughJSP instances in lesser computational time comparedto exact methods.One of suchheuristicsis geneticalgorithm(GA). Representationsin GA will havea direct impact on computationa timeittakesin providing optimalornearoptimalsolutions.Different representationschemesarepossibleincaseofJobScheduling Problems.Theseschemesinturn willhavea higher impacton the performanceof GA.It isintendedto show throughthis paper, how these representationswill perform,bya comparativeanalysisbased on average deviation,evolu- tionof solution over entiregenerationsetc.

 

Keywords

 

JobShopScheduling,GeneticAlgorithm, GeneticRepresentation, Conceptual Model

 

 

 

1.Introduction

 

Scheduling isadecision-making processwhich dealswithallocation of resourcestotasksovergiventime-pe- riodsanditsgoalistooptimizeoneormoreobjectivefunctions.Aschedulingproblem isrepresentedbytriplet α/β/γfielddescribesmachineenvironment;βfieldprovides details of processing characteristicsandcon- straintsandγfielddescribes theobjective functiontobeminimized.Being essentially acombinatorial optimiza- tionproblem,jobshopschedulinghascaughttheattentionof researchersinthelastsomany yearsforoptimized

 

*Correspondingauthor.

 

How tocite this paper:Jorapur,V.,Puranik,V.S.,Deshpande,A.S.andSharma,M.R.(2014)ComparativeStudyofDifferent

RepresentationsinGeneticAlgorithmsforJobShopSchedulingProblem.JournalofSoftwareEngineeringandApplications,

7,571-580. http://dx.doi.org/10.4236/jsea.2014.77053

 

performance.Combinatorialoptimization problemscanbeclassifiedas easyandhard. Problemswhicharepoly- nomialysolvablewithlimitednumberofvariablesaretreatedeasy andarecalledP.Thenotionpolynomial solvabledependsonthetypeofencoding.Itisassumedthatproblemsdescribingnumericaldataarebinary en- codedandthenumberof stepsinvolvedin solvingthese increasesexponentiallywithincreasein length of string andhencecomputationaltimewill beenormously large andtreatedto behardproblems.Jobscheduling prob- lemsbelongtothiscategory andaretermedNP-Hard[1].Inthepracticalmanufacturingenvironment,thescale ofjobshopsisgenerallymuch largerthan thatof JSSPbenchmarkinstancesconsideredintheoreticalresearch. Optimizationalgorithmsforjobshopschedulingusually proceed by BranchandBoundandamongthemostre- centandsuccessful,onesarethoseofCarlierandPinson(1989) andApplegateandCook(1991)[2].Approxi- mationproceduresorheuristicswereinitiallydevelopedonthebasisofpriority rulesordispatching rules.The quality ofsolutionsgeneratedbytheseprocedures leaveplenty of room forimprovement(1998)[3].Therefore, traditionalormeta-heuristicalgorithmscanhardly beabletosolvesuchproblems satisfactorily.Inmanufactur- ingworkshops,availabilityof computationalresourcesismuch lessthanthelaboratorieswhichleadtodifficulty inexploring all possiblefeasiblesolutions.Undersuchcircumstances,itis reasonabletoreducethesearchspace andrangetoonlypromisingareas.Theveryideaof usingconstructiveheuristicsandheuristicsearchalgorithms forlargerproblem sizesof JSSPisthecomputationalexpensivenatureofenumerativetechniques andLagrann- gianalgorithms.AccordingtoOsman(1996),aheuristicsearch“isaniterativegenerationprocesswhichguides asubordinateheuristicby combining intelligently differentconceptsforexploring andexploiting thesearch spaces”.

Extensiveuseofgeneticalgorithmstosolvejobshopschedulingproblemscan beseen throughliteraturesur- vey[4].However,how effectivelygeneticalgorithmscanbeusedinJSSPcaseisnotcompletelyexplored.In this context, some directionis provided byTamer F.Abdelmaguid[5]in hispaper. Gasare based onanabstract modelof naturalevolution,suchthatquality of individualsbuildstothehighestlevelcompatiblewith theenvi- ronment (constraintsofthe problem). (Holland, 1975;Goldberg, 1989)

Representations inGA environmentappliedsofarin jobshopschedulingcanbeclassifiedintoninecatego- ries asgivenbyChengetal. (1996):

1) Operationbased               2)Jobbased                 3)Jobpairrelationbased               4) Completiontime based

5)Randomkeys                 6)Preferencelistbased            7)Priority rulebased          8)Disjunctivegraphbased.

9) Machinebased.

Ninecategoriesmentionedabovecan begroupedintotwobasicencodingapproachesdirectandindirect encoding.Indirectapproach,aΠj    scheduleisencodedasachromosomeandgeneticoperatorsareused  to evolvebetterindividualones.Categories1to5areexamplesofthiscategory.Incaseofindirectapproach,a sequenceofdecisionpreferenceswillbeencodedintoachromosome.Inthis,encoding,geneticoperatorsareappliedtoimprovetheorderingofvariouspreferencesandaΠj  scheduleisthengeneratedfromthesequenceofpreferences.Categories6to9areexamplesof thiscategory[6].Theserepresentationsneedtobestudiedin case ofjobshopschedulingproblemstocomparetheirperformancecriteriatogenerateoptimalornearoptimalsolu- tions,eventhoughcomputationalcomparisonofdifferentrepresentationsisreportedina  tutorialpaperby Cheng,GenandTsujimura[6].AreportbyAnderson,GlassandPotts[7],conductedwithdifferentmetaheu- risticsapproachesincludingfourdifferentGA implementations,lacksinconsistencyaswellascoherenceasre- gards numberoftestproblemsbeingtestedwithrequisitenumber ofruns.

Therestof this paperisorganizedas follows: Wewillstartwithmathematicalmodels withcertain assump- tions thathavebeenusedinnextsectionfollowedbythe literaturereview onthedifferentGA representations usedin thecaseof JSP.Followedby reviewofGArepresentations,wewilldiscussregardingdifferentGAop- eratorsfrequentlyusedby researchersandourownviewsonaddingotheroperatorsnotdiscussedsofar.Now, wewillanalyzetheexperimentalresultsconductedfollowedbytheconclusionprovidedin thefinalpartofthis paper.

 

2.Problem Formulation

 

Sinceitisan importantpracticalproblem,someauthorshaveformulatedvarious JSPmodels basedon different production situationsand problemassumptions. Themostcommonassumptionsincaseof JSPare:

1) A machinemayprocessmorethanone jobata time;

2) Nojobmaybe processedbymorethanonemachineatatime;

3)The sequence of machines whicha jobvisitsiscompletelyspecifiedandhasa linear precedence structure;

4) Processingtimesare known. Allthe processingtimesareassumedto be integers;

5) Eachjobmustbe processed oneachmachineonlyonce. There is no recirculation;

6) Set-uptimesare assumedzero;

7) Pre-emptionis notallowed.

LetJrepresentasetofjobsandeachjobwillbe processedona setof machines ina particular order. LetI=


(1..v)representtheoperationindexes.Theoperationindexesareassignedsuchthatfora job


k  J,the subset


ofconsecutiveindexes  I =


βk,β + 1,β + 2,ωk


I,isasubsetcontainingindexesforthatjob.Nowfrom


thesubsetI dependingonthepriorityoperationwithhigherorlowervalueisprocessedfirst.Letp bethe

processingtimeofithoperation,thejobwhichitbelongstoisj(i)andthemachineonwhichithoperationcar-

riedism(i).


Now theobjectiveofschedulingprocessistodeterminethestarttimesti  ofanoperation


iI.Whileassign-


ingajobtoamachinebasedonabovecalculationsfollowingconstraintsshouldbetakenintoconsiderationviz.

Thetechnological constraintswilltakecareof order of operations tobecarriedout on a jobandasecondsetof constraintwilltakecare of conflictof twojobstobeprocessedonthesamemachinesimultaneously.Accor- dingly:


 

 

and


st +p sti+1..Is the equation to satisfytechnologicalconstraints


(1)


st st +p Or


stj    st +pi


(2)


 

Isthe equationto satisfythe conflictoftwojobsonthe samemachineatthe same time.

i,  j Iwherem(i ) =m( j ) and  j(i )  j( j )

Differenttotalcostfunctionsthatcanbe studiedare

Fmax.(C) :=max.{ fi  (Ci) i=1,,n} IscalledBottleneckobjective and


 

i= 

 fi  (C) = fi  (Ci) IscalledSumObjective.

i=1

The  most  common  objective  functions  are  the  make  span  max


 

 

 

{(Ci) i= 1,,n}


 

 

 

 

and  total  flotime


n

(Ci ) ,andweighted(total)flowtime

i =1


n

 w  C .Wehaveconsideredtheminimizationofmakespanas

i=1


ourobjectivefunction.Mannes[8]proposed anintegerlinearprogrammingmodel(ILP)whichuses different formsof binary variables.Thismodelhasgainedlargerinterestintheresearch community duetosmallnumber ofvariablesconsideredin themodel.Thetechnological constraintsof Equation (1)areanalogoustoaseriesof consecutiveactivities thatarecarried outin projectscheduling.Thisanalogyhasmotivatedimportingproject networksintoJSPenvironment.Torepresentdisjunctiveconstraints as in Equation (2),additionalsets ofarcs required.Thisisachieved ina disjunctive graphmodel[9]andPIANmodel[10].

In thedisjunctivegraphmodel,adisjunctivearc is definedbetweenapairof operationsthatsharethema- chine.Eachdisjunctivearcisassignedabinary decisionvariablesuchthatselectiononthevaluethatvariable definesthelength anddirectionofeach disjunctivearc. ThisistotheMannesmodel.Very efficientalgorithms like immediateselectionsandshifting bottleneckheuristicswere proposedby Carlier[11]andAdams [12]and Lars Monch[13], whichare derivedfromdisjunctive graphmodel.

A variable notationofthe type

m


xi,t


= 1…ifoperationi’isprocessed onmachineminunittime t’

=0...otherwise.


In ILPmodelwas proposed byBowman[14]. Wagner[15]proposedamodelwhereavariablenotation of the


 

x

 
type:

m

i,l


= 1…ifoperationi’takesithpositioninthe processingsequence onmachinem’

=0...otherwise.


 

x

 
And.Mannes[8]proposedamodelwhereavariablenotationofthetype:

processedpriorto operationj’onmachinem.


m

i,j


=1…ifoperationiis

 

= 0...otherwise.


 

3.Representationofthe Problem inGA andGA Operators

 

Darwinsprinciplesurvivalofthefittest”can beusedasastartingpointinintroducingevolutionary computa- tion.Theproblemsofchaos,chance,nonlinearinteractivitiesandtemporality beingsolvedby biologicalspe- cies are provedto beinequivalencewithclassicmethod ofoptimization[15].

Evolutionarycomputationstechniquesthatcontainalgorithmsbasedonevolutionaryprinciplesareusedto

searchforan optimalorbestpossiblesolutionforagivenproblem.Inasearchalgorithm,numberof possible solutionsisavailableandthetask istofindthebestpossiblesolutionin afixedamountoftime. Traditional searchalgorithmsrandomlysearch(e.g.randomwalk)orheuristicallysearch(e.g.gradientdescent),explore onesolution at atime in thesearch spacetofindbest possibleoroptimalsolution,whichiscomputationally in- efficient asthesearchspacegrowsin size. Whereasevolutionaryalgorithmsfromsuchtraditionalalgorithmsare populationbased.Evolutionary algorithmperformsadirectedefficientsearchby adaptationofsuccessivegen- erationsofalargernumberofindividuals.GeneticAlgorithmsisonesuchevolutionary algorithminfindingan optimalornear optimalsolutiontoa problem.Inatraditionalgeneticalgorithm,therepresentationis bitlength string.Itsapproachistogenerateasetof random solutionsfrom theexistingsolutions, sothat there isan im- provement inthequality ofsolutionsthroughoutthegenerations.Thisimplementationis achieved throughmain GAoperatorsviz.random selectionoftwosolutionsfrom individualsintheparentgeneration;performing crossoveroperation onthesetwosolutionstogeneratetwonew childsolutions.Crossoveroperation isper- formedby exchangingspecificelementsof thetwosolutionsselected;andmutationoperationisconductedon childsolutionstofurtherexplorethesearchspaceforbettersolutions.Differentvariations insimpleGA ap- proachcanbefoundinliterature surveyto improveitssearchcapabilities [16]. Representation of solutions of an optimizationproblem istobedoneinasuitableformatin GAtodealwithreproductionandmutation operators. Thisformatorstructurereferredasgenotype,needstobeeasilyinterpretabletoasolutionoftheproblem under study.Inacombinatorialoptimization problem,representationofasolution inGA isdifficultaswellasachal- lengingtask.Theseareproblemscontainingdiscretedecision variablesandare interrelatedby logical relation- ships.Asaresult,differentmathematicalmodelsmayexistforthesamecombinatorialoptimizationproblem andthismayleadto differentrepresentationsfor the same problem.

AsexplainedaboveCheng, GenandTsujimura[6]intheirpaperrepresentationofJSPinGA intodirectand indirecttype.Furthertothat,T.F.Abdelmaguid[17]inhispaperclassifiedGArepresentationsintoModel basedandAlgorithm based.In ouropinion,all representationsarealgorithm basedthoughtheyappeartobe model based.

InPriorityRuleBased(PR)representation, achromosomeisrepresentedasastringof(n1)entries(p1, p2pn)wheren−1isthenumberofoperationsintheproblem instance.Anentryp1  representsapriority rule selectedbeforehand.Accordingly,aconflictintheith   iterationofGifflerandThompsonalgorithm[18]should be resolved usingpriorityrulerepresentedbypi. Itmeans an operation fromtheconflictset hasto be selectedby thepi  tiesarebrokenrandomly.InGA domain,abestsetofpriority rulesshouldbeselected.Heresimplecros- soveryieldsfeasible schedules.

InRandom KeysRepresentation (RK) wasfirstproposedbyBean[19]. Inthisrepresentation,eachgeneis representedwithrandom numbersgeneratedbetween0 and1.These random numbersinagivenchromosome aresortedoutandarereplacedby integers andnow theresultingorderistheorderofoperationsinachromo- some.Thisstringis then interpretedintoafeasibleschedule.Anyviolationofprecedenceconstraintscan be correctedbya correctionalgorithmincorporated.

InOperationbasedrepresentation,each generepresentsanoperation.Achromosomecontainsasmany genesasthenumber of operations.For example, annxm JSPtherewillbenxmgenes in the chromosome. Beirwirth proposedatechniquepermutationwith repetition[20]which issimilartooperation basedrepresen- tation.Fang[21]alsoproposed akindoperation basedrepresentationwherestringcontainsnxmchunkswhich arelargeenoughtoholdthelargestjobnumberforthenxm JSP.Whereas BeirwirthusedaspecialGOXcros- sovertechnique togeneratefeasibleschedule,Fang usedaspecialdecoding approach todecodeachromosome intoa validschedule always.

ThePreferenceListbasedrepresentation(PL)usesastring of operationsforeach machineinsteadof a singlestringforall operationswhichisadirectrepresentation of processingsequencedecisionvariables. Quite oftenviolationofconstraintsisencounteredwhichcanbe overcome byrepair algorithm.

In theMachinebasedrepresentation,[21]thechromosomecontainsastring of lengthequaltothenumber ofmachines.Thesequenceofmachinesinthestringistheorderbywhichamachineistreatedasabottleneck


 

machineinthe shiftingbottleneckalgorithm[12].

IntheJobbasedrepresentation[22] achromosomeisastringoflengthequaltothenumber ofjobsinthe problem understudy.Usingthisrepresentation,asimple algorithmcangenerateafeasibleschedulegivense- quence ofthe jobsonto differentmachines.

 

4.Methodology

 

Thereproductionandmutation operatorsappliedtoJSPmodelaregenerallyadoptedfromTravelling Salesman Problembecause of thesimilarity inrepresentations.Reproductionoperatorsaregenerally requiredinGAto conducttheneighborhoodsearchandamutation operatorgenerallyensuresthatthesolution isnottrappedin localminima.Thedesign of both operatorsiscrucialforthesuccess of GA.Amongthereproduction operators reportedin theliterature, PMX(partiallymatchedcrossover)[23],OX (orderedcrossover)[24]anduniform crossover[25]areextensivelyusedin JSSP.PMXandOX crossovertechniquesuseeithersinglepointortwo pointcrossover.Differentmutation operatorsusedareswapmutation, inversionmutation andinsertion orshift mutationreportedinthe literature[17].

Ingeneral, theflowchartfor GAcanbe representedasshown.

 

5.Results andAnalysis

 

Inourexperiment,fourrepresentationsareusedviz.Operationbased(OB),Jobbased(JB),Machinebased (MB),Priorityrule based (PR). Allexperimentsare conductedwith50generationsanda populationsize of1000. Mutationprobabilityvarieswith 0.1to0.9valuesdynamicallyandelite populationsizeis20%.Reproduction probabilityusedinourexperimentis0.1Parentsinourexperimentareselectedfromtwogroupssortedout basedon fitnessvalue (i.e.minimum makespan). Eachparentisselectedfromthesegroupsprobabilistically.

Inour experimentation,GA isprogrammedwith differentreproductionandmutation operators.Insteadof selectingoperators randomlyas in [17],wehavebuilt-in reproduction operatorsandare beingusedacross the representationsandthebenchmark instances.Thebenchmarkproblemsusedinthispaperaretakenfrom ORli- brary [26]availablein World Wide Web.AlltheexperimentsareconductedwithaPentium-4dualcoreproces- sorwithclockspeed of2.06GHzandRAM of512 Mbs.68benchmarkinstancesare takenandinthe single run, thebestand averagevalues areobtainedandcomparedwith lowerboundoroptimumvalue ofthebenchmark instance.Resultsare showninTable 1. Differentgraphsgeneratedare alsoshownbelow.

 

Table 1.Results ofbenchmarkinstances underdifferentrepresentations.


 

Problem       Size         No.of

Operations


 

BestKnown

Solution


OB         OB            JB              JB            MB          MB          PR        PR Best   Avg.         BestAvg.                 Best         Avg.         Best               Avg.


mt06       6×6          36               55              55        64.889        55         65.712        55         61.822      55    66.648 mt10              10 × 10       100             971       989            1116.02 971               1100.992                         1145.06       958             1100.42 mt20         5× 20        100        1206            1220     1394.47     1206 1383.081245                                1427.24     1242     1426.54 abz05                             10 × 10 100           1259              1275                                1394.79     1259    1386.12                    1287             1409.87                  12671390.94 abz06                               10 × 10      100       971                          958              1072.19971          1075.9996                                1096.13       978      1080.3 abz07                               15 × 20 300                     742         734                                821.16      742     804.892                    751              817.937                  73807.128 abz08                               15 × 20      300       758                          751              833.362758          825.98763                                838.59        755     826.954 abz09                             15 × 20 300                     752         784                                877.468     752     849.838                    773              873.541                    764859.258 car01                               5× 11       55       7038                          7038            8747.847038         8694.017038                                8707.83     7038    8782.28 car02                              4× 13   52             7376              7378            8788.38      7376          8738.23                   7221                           8817    7166          8881.94 car03                               5× 12       60       7725                          7590            9219.197725         9195.367725                                9293.86     7725    9272.51 car04                              4× 14   56             8072              8003            9620.16      8072          9452.62                   8276                         9697.218132            9643.3 car05                                 6× 10       60       7835                          7873            9207.147835         9130.267862                                9251.68     7862      9407 car06                                    9×8    72             8505              8505            10017.7      8505          9886.82                   8505                         10229.58485          9830.33 car07                                 7×7        49       6558                          6576            7673.646558         7782.766627                                7751.89     6632    7738.75 car08                                8×8    64             8407              8407            9436.29      8407          9500.61                   8458                         9470.578366          9470.64 la01                                 5× 10        50        666                            666              783.616666              796.901       674             746.506     666      782.789


 

 

Continued

 

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