Springer.Stochastic.Modeling.Of.Manufacturing.Systems.Advances.In.Design.Performance.Evaluation.And.Control.Issues.[2006.Isbn3540265791]

《Springer.Stochastic.Modeling.Of.Manufacturing.Systems.Advances.In.Design.Performance.Evaluation.And.Control.Issues.[2006.Isbn3540265791] 》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

StochasticModelingofManufacturingSystemsAdvancesinDesign,PerformanceEvaluation,andControlIssues G.Liberopoulos·C.T.Papadopoulos·B.TanJ.MacGregorSmith·S.B.GershwinEditorsStochasticModelingofManufacturingSystemsAdvancesinDesign,PerformanceEvaluation,andControlIssuesWith121Figuresand91Tables123 GeorgeLiberopoulosJ.M.SmithDepartmentofMechanicalDepartmentofMechanicalandIndustrialEngineeringandIndustrialEngineeringUniversityofThessalyUniversityofMassachusetts38334VolosAmherst,Massachusetts01003GreeceUSAE-mail:glib@mie.uth.grE-mail:jmsmith@ecs.umass.eduChrissoleonT.PapadopoulosStanleyB.GershwinDepartmentofEconomicSciencesDepartmentofMechanicalEngineeringAristotleUniversityofThessalonikiMassachusettsInstituteofTechnology54124ThessalonikiCambridge,Massachusetts02139-4307GreeceUSAE-mail:hpap@econ.auth.grE-mail:gershwin@mit.eduBarıs¸TanGraduateSchoolofBusinessKoçUniversity80910Sariyer,IstanbulTurkeyE-mail:btan@ku.edu.trPartsofthepapersofthisvolumehavebeenpublishedinthejournalORSpectrum.LibraryofCongressControlNumber:2005930501ISBN-103-540-26579-1SpringerBerlinHeidelbergNewYorkISBN-13978-3-540-26579-5SpringerBerlinHeidelbergNewYorkThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialiscon-cerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,repro-ductiononmicrofilmorinanyotherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.ViolationsareliableforprosecutionundertheGermanCopyrightLaw.SpringerisapartofSpringerScience+BusinessMediaspringeronline.com©Springer-VerlagBerlinHeidelberg2006PrintedinGermanyTheuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Coverdesign:ErichKirchnerProduction:HelmutPetriPrinting:StraussOffsetdruckSPIN11506560Printedonacid-freepaper–42/3153–543210 Editorial–StochasticModelingofManufacturingSystems:AdvancesinDesign,PerformanceEvaluation,andControlIssuesManufacturingsystemsrarelyperformexactlyasexpectedandpredicted.Unex-pectedeventsalwayshappen:customersmaychangetheirorders,equipmentmaybreakdown,workersmaybeabsent,rawpartsmaynotarriveontime,processedpartsmaybedefective,etc.Suchrandomnessaffectstheperformanceofthesys-temandcomplicatesdecision-making.Respondingtounexpecteddisturbancesoccupiesasignificantamountoftimeofmanufacturingmanagers.Therearetwopossibleplansofactionforaddressingrandomness:reduceitorrespondtoitinawaythatlimitsitscorruptingeffectonsystemperformance.Thisvolumeisdevot-edtothesecond.Itincludesfifteennovelchaptersonstochasticmodelsforthedesign,coordination,andcontrolofmanufacturingsystems.Theadvantageofmodelingisthatitcanleadtothedeepestunderstandingofthesystemandgivethemostpracticalresults,providedthatthemodelsapplywelltotherealsystemsthattheyareintendedtorepresent.Thechaptersinthisvolumemostlyfocusonthedevelopmentandanalysisofperformanceevaluationmodelsusingdecomposition-basedmethods,Markovianandqueuinganalysis,simulation,andinventorycon-trolapproaches.Theyareorganizedintofourdistinctsectionstoreflecttheirsharedviewpoints.SectionIincludesasinglechapter(Chapter1)onfactorydesign.Inthischapter,Smithraisesseveralconcernsthatmustbeaddressedbeforeevenchoosingamodelingapproachanddevelopingandtestingamodel.Specifically,hediscussesanumberofdilemmasinfactorydesignproblemsandtheparadoxesthattheyleadto.Theseparadoxesgiverisetonewparadigmsthatcanbringonnewapproachesandinsightsforsolvingthem.SectionIIincludesChapters2–7onunreliableproductionlineswithin-processbuffers.Morespecifically,inChapter2,Enginarlar,Li,andMeerkovanalyzeatandemproductionlineanddeterminetheminimumbufferlevelsthatarenecessarytoobtainadesiredline-efficiency.Theworkconsiderstandemlineswithnon-exponentialstationsandextendspriorworkontandemlineswithexponentialservers.Afairlydetailedsimulationstudyisconductedtoanalyzetheperformanceofthetandemlines.Theresultsareusedtoderiveanempiricallawthatprovidesanupperboundonthedesiredbufferlevels.InChapter3,HelberusesdecompositiontoanalyzeflowlineswithCox-2dis-tributedprocessingtimesandlimitedbuffercapacity.First,hederivesanexactsolutionforatwo-stationline.Basedonthissolution,hethenderivesanapproxi-mate,decomposition-basedsolutionforlargerflowlines.Finally,hecomparesthe VIEditorialresultsobtainedbyhisdecompositionmethodagainstthoseobtainedbyBuzacott,Liu,andShanthikumar.InChapter4,Colledani,Matta,andToliopresentadecompositionmethodtoevaluatetheperformanceofaproductionlinewithmultiplefailuremodesandmultipleproducts.Theysolveanalyticallythetwo-part-type,two-machinelineandderivethedecompositionequationsforlongerlines.TheyuseanalgorithmsimilartotheDDXalgorithmtosolvetheseequationstodeterminetheproductionrateandotherperformancemeasuresapproximately.Inthenextchapter(Chapter5),Matta,Runchina,andTolioaddressthequestionofhowtoincreasetheproductionrateofproductionlinesbyusingasharedbufferwithinthesysteminordertoavoidblocking.Simulationisusedtodemonstratethegaininthemeanproductionratewhenacommonbufferisused.Inaddition,anapplicationofthesharedbufferapproachtoarealcaseisreported.InChapter6,KimandGershwinaskwhathappensifmachinesinaproductionlinecaneitherfailcatastrophically(stopproducing),orfailtoproducegoodpartswhilecontinuingtoproduce.First,theydevelopaMarkovprocessmodelformachineswithbothqualityandoperationalfailures.Then,theydevelopmodelsfortwo-machinesystems,forwhichtheycalculatetotalproductionrate,effectiveproductionrate,andyield.Usingthesemodels,theyconductnumericalstudiesontheeffectofthebuffersizesontheeffectiveproductionrate.Finally,inChapter7,LeeandLeeconsideraflowlinewithfinitebuffersthatrepetitivelyproducesmultipleitemsinacyclicorder.Theydevelopanexactmethodforevaluatingtheperformanceofatwo-stationlinewithexponentiallyorphase-typedistributedprocessingtimesbymakinguseofthematrixgeometricstructureoftheassociatedMarkovchain.Theythenpresentadecomposition-basedapprox-imationmethodforevaluatinglargerlines.Theyreportontheaccuracyoftheirproposedmethodandtheydiscusstheeffectsofjobvariationandjobsequenceonperformance.SectionIIIincludesChapters8–13onqueueingnetworkmodelsofmanufac-turingsystems.Morespecifically,inChapter8,VanVuuren,Adan,andResing-Sassenconsidermulti-servertandemqueueswithfinitebuffersandgenerallydistributedservicetimes.Theydevelopaneffectiveapproximationtechniquebasedonaspectralexpan-sionmethod.Numerousexperimentsareutilizedtodemonstratetheeffectivenessoftheirperformancemethodologywhencomparedwithsimulationofthesamesystems.Theirapproximationmethodologyshouldbeveryusefulforproductionlinedesign.InChapter9,KoukoumialosandLiberopoulospresentananalyticalapproxi-mationmethodfortheperformanceevaluationofmulti-stage,serialsystemsoperatingundernestedorechelonkanbancontrol.Fulldecompositionisutilizedalongwithanassociatedsetofalgorithmstoeffectivelyanalyzetheperformanceofthesesystems.Finally,theseapproximationalgorithmsareutilizedtoaccuratelyoptimizethedesignparametersofthesystem.Inthenextchapter(Chapter10),Spanjers,vanOmmeren,andZijmconsiderclosed-loop,two-echelonrepairableitemsystemswithrepairfacilitiesatanumberoflocalservicecentersandatacentrallocation.Theyuseanapproximationmethod EditorialVIIbasedonageneralmulti-classmarginaldistributionanalysisalgorithmtoevaluatetheperformanceofthesystem.Theperformanceevaluationresultsarethenusedtofindthestocklevelsthatmaximizetheavailabilitygivenafixedconfigurationofmachinesandserversandacertainbudgetforstoringitems.InChapter11,VanNyen,Bertrand,vanOoijen,andVandaelepresentaheuris-ticthatminimizestherelevantcostsandsatisfiesthecustomerservicelevelsinmulti-product,multi-machineproduction-inventorysystemscharacterizedbyjob-shoproutingsandstochasticarrival,set-up,andprocessingtimes.Thenumericalresultsderivedfromtheheuristicarecomparedagainstsimulation.InChapter12,VanHoutum,Adan,Wessels,andZijmstudyaproductionsystemconsistingofseveralparallelmachines,whereeachmachinehasitsownqueueandcanproduceaparticularsetofjobtypes.Whenajobarrivestothesystem,itjoinstheshortestqueueamongallqueuescapableofservingthatjob.Undertheassump-tionofPoissonarrivalsandidenticalexponentialprocessingtimestheyderiveupperandlowerboundsforthemeanwaitingtimeandinvestigatehowthemeanwaitingtimeiseffectedbythenumberofcommonjobtypesthatcanbeproducedbydif-ferentmachines.Finally,inChapter13,GeraghtyandHeaveyreviewtwoapproachesthathavebeenfollowedintheliteratureforovercomingthedisadvantagesofkanbancontrolinnon-repetitivemanufacturingenvironments.Thefirstapproachhasbeencon-cernedwithdevelopingnew,orcombiningexisting,pullcontrolstrategiesandthesecondapproachhasfocusedoncombiningJITandMRP.AcomparisonbetweenaProductionControlStrategy(PCS)fromeachapproachispresented.Also,acom-parisonoftheperformanceofseveralpullproductioncontrolstrategiesinanenvi-ronmentwithlowvariabilityandalight-to-mediumdemandloadiscarriedout.Thelastsection(SectionIV)includesChapters14and15onproductionplan-ningandassembly.InChapter14,Axsäterconsidersamulti-stageassemblynetwork,whereanum-berofenditemsmustbedeliveredatcertainduedates.Theoperationtimesatallstagesareindependentstochasticvariables.Theobjectiveistochoosestartingtimesfordifferentoperationsinordertominimizethetotalexpectedholdingandback-ordercosts.Anapproximatedecompositiontechnique,whichisbasedonrepeatedapplicationofthesolutionofasimplersingle-stageproblem,isproposed.Theper-formanceoftheapproximatetechniqueiscomparedtoexactresultsinanumericalstudy.InChapter15,Yıldırım,Tan,andKaraesmenstudyastochastic,multi-periodproductionplanningandsourcingproblemofamanufacturerwithanumberofplantsandsubcontractorswithdifferentcosts,leadtimes,andcapacities.Thedemandforeachproductineachperiodisrandom.Theypresentamethodologyfordecidinghowmuch,when,andwheretoproduce,andhowmuchinventorytocarry,givencertainservicelevelconstraints.Therandomnessindemandandrelatedprobabilisticservicelevelconstraintsareintegratedinadeterministicmathematicalprogrambyaddinganumberofadditionallinearconstraints.Theyevaluatetheperformanceoftheirmethodologyanalyticallyandnumerically.ThisvolumeisareprintofaspecialissueofORSpectrum(Vol.27,Nos.2–3)onstochasticmodelsforthedesign,coordination,andcontrolof VIIIEditorialmanufacturingsystems,withtheadditionofChapters7and12thatappearedasarticlesinotherissuesofORSpectrum.ThatspecialissueofORSpectrumorigi-natedfromthe4thAegeanInternationalConferenceonAnalysisofManufacturingSystems,whichwasheldinSamosIsland,Greece,inJuly1–42003.Thepurposeofthatissuewasnottosimplypublishtheproceedingsoftheconference.Ratheritwastoputtogetheraselectsetofrigorouslyrefereedarticles,eachfocusingonanoveltopic.Collectedintoasingleissuethearticlesaimedtoserveasausefulreferenceformanufacturingsystemsresearchersandpractitioners,andasreadingmaterialsforgraduatecoursesandseminars.WewishtothankProfessorDr.Hans-OttoGuenther,ManagingEditorofORSpectrum,andhisstaffforsupportingthespecialissueofORSpectrumandseeingthatitbecomesapublishedrealityaswellasforsupportingitssubsequentreprintintothisvolumewiththeadditionofChapters7and12.G.Liberopoulos,UniversityofThessaly,GreeceC.T.Papadopoulos,AristotleUniversityofThessaloniki,GreeceB.Tan,Koc¸University,TurkeyJ.M.Smith,UniversityofMassachusetts,USAS.B.Gershwin,MassachusettsInstituteofTechnology,USA ContentsSectionI:FactoryDesignDilemmasinfactorydesign:paradoxandparadigmJ.MacGregorSmith..............................................3SectionII:UnreliableProductionLinesLeanbufferinginserialproductionlineswithnon-exponentialmachinesEmreEnginarlar,JingshanLiandSemyonM.Meerkov..................29AnalysisofflowlineswithCox-2-distributedprocessingtimesandlimitedbuffercapacityStefanHelber...................................................55PerformanceevaluationofproductionlineswithfinitebuffercapacityproducingtwodifferentproductsM.Colledani,A.MattaandT.Tolio.................................77AutomatedflowlineswithsharedbufferA.Matta,M.RunchinaandT.Tolio..................................99IntegratedqualityandquantitymodelingofaproductionlineJongyoonKimandStanleyB.Gershwin..............................121Stochasticcyclicflowlineswithblocking:MarkovianmodelsYoung-DooLeeandTae-EogLee...................................149SectionIII:QueueingNetworkModelsofManufacturingSystemsPerformanceanalysisofmulti-servertandemqueueswithfinitebuffersandblockingMarcelvanVuuren,IvoJ.B.F.AdanandSimoneA.E.Resing-Sassen.......169AnanalyticalmethodfortheperformanceevaluationofechelonkanbancontrolsystemsSteliosKoukoumialosandGeorgeLiberopoulos.......................193 XContentsClosedlooptwo-echelonrepairableitemsystemsL.Spanjers,J.C.W.vanOmmerenandW.H.M.Zijm....................223Aheuristictocontrolintegratedmulti-productmulti-machineproduction-inventorysystemswithjobshoproutingsandstochasticarrival,set-upandprocessingtimesP.L.M.vanNyen,J.W.M.Bertrand,H.P.G.vanOoijenandN.J.Vandaele...253PerformanceanalysisofparallelidenticalmachineswithageneralizedshortestqueuearrivalmechanismG.J.VanHoutum,I.J.B.E.Adan,J.WesselsandW.H.M.Zijm............289Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategiesJohnGeraghtyandCathalHeavey..................................307SectionIV:StochasticProductionPlanningandAssemblyPlanningorderreleasesforanassemblysystemwithrandomoperationtimesSvenAxsäter....................................................333AmultiperiodstochasticproductionplanningandsourcingproblemwithservicelevelconstraintsIs¸ılYıldırım,Barıs¸TanandFikriKaraesmen..........................345 SectionI:FactoryDesign Dilemmasinfactorydesign:paradoxandparadigm
J.MacGregorSmithDepartmentofMechanicalandIndustrialEngineering,UniversityofMassachusetts,Amherst,MA01003,USA(e-mail:jmsmith@ecs.umass.edu)Abstract.Theproblemsoffactorydesignarenotoriousfortheircomplexity.Itisarguedinthispaperthatfactorydesignproblemsrepresentaclassofproblemsforwhichtherearecrucialdilemmasandcorrespondinglydeep-seatedunderlyingparadoxes.Theseparadoxes,however,giverisetonovelparadigmswhichcanbringaboutfreshapproachesaswellasinsightsintotheirsolution.Keywords:Factorydesign–Dilemmas–Paradox–Paradigm1IntroductionThepurposeofthispaperistodevelopanewparadigmforfactorydesignthatintegratesmuchofthetheoreticalunderpinningsoftheproblemsandprocessesencounteredintheauthor’sexperienceswithfactorydesign.Asasidebenefittothispaper,manyoftheideasdiscussedwithinpointtowardsanewdirectionforwhichmanufacturingandindustrialengineeringprofessionalsmightre-alignthemselves,sincetheparadigmswhichhaveguidedthesefieldsareinneedofanewvisionandrepair.1.1MotivationTheoriginsofthispaperstemfromaninvitationtogiveakeynoteaddressataconferenceontheAnalysisofManufacturingSystems1wheretheideaofthe
Iwouldliketothanktherefereesfortheirinsightsandsuggestionsandpointingoutsomeproblemsinearlierdrafts.Myapproachtofactorydesignhasevolvedovertheyears,andisstillevolving,anditislargelyduetotheinfluenceofProfessorHorstRittel,myprofessorattheUniversityofCaliforniaatBerkeleyduringmyformativeundergraduatedays,whoinstilledmuchofthebasisofthisphilosophy.14thAegeanConferenceon:“TheAnalysisofManufacturingSystems”,SamosIslandGreece,July1st-July4th,2003 4J.MacGregorSmithaddresswastorecounttheauthor’sphilosophyaboutmanufacturingsystemsdesignandinparticularanapproachtofactorydesignproblems.Concurrentlywiththeconferencethereappearedarelatedconundrumontheemaillistserv:iefac.list@deming.ces.clemson.eduoftheIndustrialEngineeringfac-ultyaboutan“identity”crisiswithintheindustrialengineeringcommunityandthedirectionoftheprofessionandmorepracticallyspeakingwhatfundamentalcoursesshouldbetaughtstudentsofindustrialengineering.Itisnotthefirsttimethisiden-titycrisishasariseninIE,noristhecrisisoneexclusivetoindustrialengineers,asitcommonlyoccursthroughoutmostprofessionsfromtime-to-time.Paradoxically,allprofessionshaveavestedinterestintheirclients,butcannotbetrustedtoactintheirclientsbestinterests,“aconspiracyagainstthelaity.”[21,17].Since,theFactoryDesignProblem(FDP)isaveryimportantaspectwithinmanufacturingandindustrialengineering,itbecameobviousthatthesubjectmatterofthekeynoteaddressandthecrisisinindustrialengineeringeducationaretwocloselyrelatedmatters.Sowhilenotattemptingtobepresumptuous,theresultingpaperwasaresponsepartlytothiscrisisandalsomoreimportantlytodemonstratetheauthor’sphilosophyaboutfactorydesign.Theviewpointandconclusionsinthepapermayalsoapplytotheproblemsoffactoryplanningandcontrol,butthefocusforthepresentpaperisontheFDPproblem.1.2OutlineofpaperSection2ofthispaperprovidesnecessarybackground,definitions,andnotationontheproblemoffactorydesign.Section3describesacasestudyusedtoillustratemanyoftheideaswithinthepaper,whileSection4providesthetheoreticalback-groundofthemanyconceptsinthepaper.Section5describestheimplicationforthemanufacturingandIEprofessionandSection6concludesthepaper.2BackgroundManymanufacturingandindustrialengineeringprofessionalsviewtheFDPasacomplexqueueingnetwork,whereonehastomanufactureorproduceaseriesofproducts(1,2,...,n)fromdifferentrawmaterialsandpossiblesources.Theaveragearrivalrateoftypejrawmaterialfromsourcekisdefinedasλjk(j=1,2,...,J;k=1,2,...,K).People,machines,manufacturingprocessesandthematerialhandlingsystemarenecessarytotransformtherawmaterialsintofinishedgoodsforshipmenttoconsumersatthroughputratesθ1,θ2,...θn.Figure1isaλ11θ1λ21Σθ2λjkθnFig.1.Factoryflowdesignparadigm Dilemmasinfactorydesign:paradoxandparadigm5usefulcaricatureoftheflowparadigm.TheΣrepresentsthemathematicalmodelofthequeueingnetworkunderlyingthepeople,resources,productsandtheirflowrelationships.Theprofessionals(especiallytheacademics)wouldliketoknowthesetofunderlyingequationsΣ(noquestionsasked)whichwouldallowthemtodesignthefactorytomaximizetheoverallthroughput(Θ)oftheproductsandalsominimizethework-in-process(WIP)insidetheplant.Thedesiretofindalltheseequations,orlaws[9]assomepeoplewouldliketocharacterizethem,islargelyattributedtothescientificfoundationofIndustrialEngineeringeducationwithastrongphysics,chemistry,andmathematicsback-ground.AsterlingexampleofoneoftheselawsisLittle’sLawL=λWwhichisanextremelyrobust,effectivetooltocalculatenumbersofmachines,throughput,andwaitingtimesinqueueingprocesses[9].Whatwillbeshowninthefollowingisthatthisscientificapproachisdeficient.Theproblemsoffactorydesigncannotbeansweredwithjustascientificbackground,butneedtobeaugmentedwithotherknowledge-basedskills.Thescientificbackgroundisnecessarybutnotsufficienttosolvetheproblem.Inordertorealizethisfactoryflowparadigm,mostIEprofessionalssystem-aticallydefinethemultipleproducts(therecanbehundreds)andtheirinputratesandrawmaterialrequirements,theconstraintrelationshipswiththemachines,peo-ple,resources,andmaterialshandlingequipment,andthefunctionalequationsforachievingtheWIPandthroughputobjectives,utilization,cycletime,lateness,etc.Thisfactoryflowparadigmisoftenrealizedasaseriesofwell-definedstepsorphasessimilartothefollowingtop-downapproach(seeFig.2).Thistop-downapproachisalsoahallmarkofanoperationsresearch(OR)paradigmtypicallyarguedforinORtextbooksfoundintheIndustrialEngineeringcurriculum.Whilethistop-down(“waterfall”)[3]paradigmhasitsmerits,mainlyforprojectmanagement,itwillbearguedinthispaperthatotherparadigmsarewarranted,onesmorerealisticallyappropriatefortreatingFDPs.Akeycriticismofthetop-downapproachisthatnofeedbackloopsoccuratthedetailedstages,whichisclearlyunrealistic.Abottom-upapproach,ontheotherhand,isreallynotmuchbetter,sinceonehasnorealoverallknowledgeofwhatisbeingconstructed.Oneneedsaparadigmthatisparadoxicallytop-downandbottomupatthesametime.Unfortunately,veryfewindividualsarecapableofthisprescientfeat,thusnecessitatingdevelopmentofnewexternalaids.Itwillalsobearguedlateroninthispaper,thattherecommendedparadigmhasstrongimplicationsforchangesintheprofessionandintheeducationofmanufac-turingandindustrialengineers.2.1DefinitionsBeforeweproceedtoofaralong,itwouldbegoodtopositsomeofthekeydefinitionsandnotationutilizedthroughoutthepaper[6].Dilemma:(LateGreek)dilEmmat,dilEmmatos-anargumentpresentingtwoormoreconclusivealternativesagainstanopponent;aprobleminvolvingadifficultchoice;aperplexingpredicament. 6J.MacGregorSmithStep1.0IdentifyProductClasses/SourcesStep2.0ProductRoutingVectorsStep3.0DistanceandFlowMatricesTopologicalNetworkDesignStep4.0(TND)DiagramsOptimalTNDStep5.0AlternativesStochasticFlowStep6.0MatricesEvaluationofStep7.0AlternativesFactoryPlanStep8.0SynthesisSensitivityStep9.0AnalysisStep10.0FactoryPlanImplementationFig.2.FactorydesignprocessparadigmParadox:(Greek)paradoxon,paradoxos-Atenetcontrarytoreceivedopinion.Astatementthatisseeminglycontradictoryoropposedtocommonsense.Paradigm:(Greek)paradeigma,paradeiknynai-Toshowsidebysideapattern-anoutstandinglyclearexampleorarchetype(a.k.a.aphilosophy)ThenotionofadilemmainFactoryDesignisthatweareoftenfacedwithdiffi-cultissuesofwhattodo,and,occasionally,wemustselectbetweentwoalternativesthatarenotnecessarilydesirable.Thenotionofparadoxisimportantbecauseithelpsframetheseeminglycon-tradictoryelementswhicharecontrarytocommonsense.Dilemmasgiverisetoparadoxeswhichinturnunderlyparadigmsforsolution.Paradigmisaparticularlyappropriatewordwhenonethinksofitasa“pattern”,sincethisisoftenwhatweemployinresolvingdesignproblemsbecauseofitsmodularstructure.Allthreeoftheseconceptsarecrucialunderpinningstowhatistofollowandtheyformthebasisofthegeneraldesign“philosophy”purportedinthispaper.ThefactthatthesethreeconceptsarederivedfromtheGreekphilosophersisanindicationoftheirimportance.2.2NotationThefollowingnotationshallbeutilizedtoaidthediscussion: Dilemmasinfactorydesign:paradoxandparadigm7–∆:=Dilemma–χ:=Paradox–δi:=Deonticissue–i:=Causalorexplanatoryissue–ιi:=Instrumentalissue–φi:=Factualissue–πi:=PlanningIssue–FDP:=FactoryDesignProblem–WP:=WickedProblem–TP:=TameProblem–IBIS:=IssueBasedInformationSystem–NI:=Non-Inferiorsetofsolutions3Casestudy:polymerrecyclingprojectInordertoplacethingsinperspective,acasestudywillbeutilizedtocharacterizetheideasandconceptsofthepaper.Oneprojectcompletedeightyearsagostandsoutasacompellingexampleoftheideasinthispaper.ItwasconcernedwiththeFDPofapolymerre-processingplantinWestern,Massachusetts.3.1ProblemdescriptionEssentially,thisplantrepresentedamanufacturing/warehousecapacitydesignprob-lem.Theplantmaintainsadynamicmaterialhandlingsystemwhichoperates3shifts24hoursaday.Theproblemasfirstposedtothefactorydesignteamlargelyrevolvedaroundspacecapacityandequipmentneedssincethebusinesswasgrowingandtherewassomerealconcernabouttheabilityofthepresentsitetoaccommodatefuturegrowthofthebusiness.ThebusinessislargelyconcernedwithmanufacturingessentiallyfourdifferentpolymerproductsPC,PC/ABS,PS,ABSandtheircombinations.Infact,theunitloadoftheplantis1000#gaylords(rawmaterialsandfinishedgoods)filledwithvariousplasticpellets.Aswillunfold,forecastingtheabilityoftheplanttorespondtofluctuationsindemandovertimealsobecameacriticalpartofthestudy.3.2LinkstopaperFigure3illustratestheinitiallayoutoftheplantthatformedthebasisofthelayoutandsystemsmodelabouttobediscussed.Onecanseethe4
×4
gaylordsspreadthroughoutthefacilityinFigure3.Asonecanseeintheplant,thereislittleroomforexpansionandthereisarestrictedmaterialhandlingsystemwheretheforklifttrafficcomingandgoingmusttraversethesameaisles. 8J.MacGregorSmithFig.3.Existingpolymerre-processingplant4DilemmasinfactorydesignThenotionofthedilemmasinfactorydesignstemsfromaseminalpaperofHorstRittelandMelWebber[17]onwickedproblems.Theyoutlinethecharacteristicsofwickedproblemsandgoontorecounthowmanyplanningproblemsareactu-allywickedproblems.Infacttheyarguethatthereareessentiallytwoclassesofproblems:–TameProblems(TPs)–WickedProblems(WPs)Tameproblemsarelikepuzzles:preciselydescribed,withafinite(orcount-ablyinfinite)setofsolutions,althoughperhapsextremelydifficulttosolve.Problemssolvedvianumericalandcombinatorialalgorithmscanbegroupedinthiscategory.TherelationshipofComputationalComplexityanditsclassesP,NP,NP−Complete,andNP−Hardareveryappropriatecharacterizationsfortameproblems.Also,morerecently,designinglargescaleinteractingsystemshasbeenshowntobeNP-complete[5].ItwillbearguedthattheNPComplexityclassificationisausefulwayofchar-acterizingTPs.Ontheotherhand,Wickedproblemsaretheexactoppositeoftameproblems,andwhilenot“evil”inthemselves,presentparticularynastycharacter-isticswhichRittelandWebberfeeljustlytodeservetheapprobation.Theirwicked Dilemmasinfactorydesign:paradoxandparadigm9WPWPNP−HardNPPNP−CompleteFig.4.WickedproblemtameproblemdichotomyproblemframeworkisusefulforcharacterizingtheFDP,sincethecharacteristicsofFDPsasshallbearguedaresimilar.NotallIEsormanufacturingengineersmightagreewiththeequivalencestatement,buttheequivalenceframework,asweshallargue,willbecomethebasisforthenewparadigm.Veryoften,IEsutilizealgorithmicapproachestosolveFDPs,sotheybecomeintegralpartsofthesolutionprocessoffactorydesignproblems,butakeyquestionhereis:CanweutilizesystematicprocedurestosolveFDPs?WhilenoformalclassificationofWPshasbeendevelopedsofar,otherthanwhatisdepictedinFigure4,itappearsthatthedistinctionbetweenonetypeofwickedproblemandanothercanbebasedonthefollowingthreemeasurabledimensions:–x:=#Stakeholders(#personsconcerned,involvedandaffectedbytheproblem)–y:=#Objectivesintheproblem{f1,f2,...fp}–z:=Timeframeorplanninghorizon(inyears)Thedegreeof“wickedness”iscorrelatedwiththecardinalityofthedimensions.Forexample,establishingthesolutionforthedisposalofnuclearwasteisoneofthemostdifficultWPs,sincethetimeframeisthousandsofyears,andtheconsequencesaffectmillionsofpeople.Thereasonforselectingtheseproblemdimensionsshouldbecomeclearerasthepaperunfolds.ProjectmanagementisaclassicexampleofaWP.WeknowthatminimizingthenumberofdummyactivitiesinaPERT/CPMdiagramisactuallyNP-Complete[12],however,thecomplexityofbalancingtime,cost,andqualitytradeoffsinschedulingtheconstructionandlaunchingforexampleofthespaceshuttleisaverywickedproblem.TameProblemsandtheirsolutionsareoftensubsetsofWPsandtheyhavetheirusefulnessespeciallyinprovidingargumentstoconvincepeopleonewayoranotheronresolvingaplanningissue,buttheTPsareinanotherclasscomparedtoWPs.Manyotherresearchershavebeguntorealizetheimportanceandextentofwickedproblemsinotherprofessionsbesidesfactorydesign.Someoftheliteratureonwickedproblemsisrelatedtopublicservicefacilityplanning[22],governmentresourceplanningwithindevelopingcountries[19]softwareengineeringdesignprojects[3],planningandprojectscheduling[20].UnlikeTPs,thefirstcharacteristicofawickedproblemisthat: 10J.MacGregorSmith∆1:Thereisnodefinitiveproblemformulation.Thedilemmaarguesthatfactorydesignproblemscannotbewrittendownonasheetofpaper(likeaquadraticequation),giventosomeone,wheretheythencangooffintoacornerandworkoutthesolution.Studentsarecontinuallydrilledwithtextbookproblems(theauthorisguiltyofthishimself),butthesearenottherealproblems.Recentresearchonthemodularizationofdesignproblemshasshownthatmodularizationavoidstrade-offsindecisionmakingandoftenignoresimportantinteractionsbetweendecisionchoices[5].Ifsomeonestatestheproblemas:“buildanewplant”or“remodeltheexistingfacility”,or“addanotherstorey”,then,i.e.thesolutionandproblemareoneandthesame!Thisisantitheticaltothescientificparadigm.Infact,theentireedificeofNP-Completenessproblems(i.e.TameProblems)iscriticallystructuredaroundthepreciseproblemdefinitione.g.3-satisfiability.ForFDPs,itisimportantwhomyoutalkwithandtheirworldviewbecauseintheensuingdialogthesolutiontotheproblemandtheproblemdefinitionwillemerge.Inthecaseofthepolymerrecyclingplant,whenthefacilitywasfirstexamined,theirreceivingandshippingareaswereco-locatedinthesameareaoftheplant,seethelowerlefthandcornerofFigure3whichresultedinseverematerialhandlingconflictswithforklifttruckmovements,accidents,andspaceutilizationproblems.Itwasobviousthatseparatereceivingandshippingareasweredesirable–thus,theproblemwasthesameasthesolution:“re-layouttheplantandseparatereceivingandshipping.”Thus,wehavethefirstformalparadox:χ1:=Everyformulationofaproblemcorrespondstoastatementofitssolutionandviceversa[14].Thisfirstdilemmaoffactorydesignisamostdifficultone.Onecannotknowaprioritheproblemsinherentinfactorydesign,independentoftheclientandthecontextaroundwhichtheproblemoccurs.Inessence,thefactorydesignprocessisessentiallyinformationdeficient.Many“experts”inmanufacturingandIEpurporttoknowtheanswers,yetonemusttalkwiththeowners,theplantmanager,thelinestaff,andmanyothersinvolvedwiththefacility,beforetheproblemsandtheirsolutionscanbeidentified.Asthepaperproceeds,wewillpostulatetheunderlyingprinciplesofthenewparadigmasPropositions.Infact,theprincipleunderlyingtheparadigmassociatedwiththisfirstdilemmaandparadoxis:Proposition1.TheFDPdesignsystem≡Knowledge/InformationSystem.Whatismeantherebyanknowledge/informationsystem?Theknowl-edge/informationsystemhereisaspecialtypeofinformationsystem,notjustasophisticateddatabasesystem,whereonecollectsdataforthesakeofcollectingdata,butdataiscollectedtoresolvetheplanningissues.Theplanningissuesarethefundamentalunitswithintheinformationsystem[13].ArelatedinformationsystemapproachbasedonthefirstpropositionisthatofPeterCheckland’swork[1],however,theinformationsystemandresultingparadigmdiscussedinthispaperisbasedupondifferentconceptsandisdirectlyrelatedtotheFDP. Dilemmasinfactorydesign:paradoxandparadigm11φiι1πiiδiι2Fig.5.PlanningissueπiWhatarethebuildingblocksofthisknowledge/informationsystem?Thereareessentiallyfourcategoriesofknowledge(issues)neededtohelpformulatetheproblem.ThesefundamentalcategoriesofissuesarebasictotheIBIS[13]:–Factualissue(φi):=Knowledgeofwhatis,was,orwillbethecase.–Deonticissue(δi):=Knowledgeofwhatoughttobeorshouldbethecase.–Explanatoryissue(i):=Knowledgeofwhysomethingisthecase.–Instrumentalissue(ιi):=Knowledgeoftheconditionsandmethodsunderwhichtheproblemcanberesolved.Proposition2.Aplanningissueπiisadiscrepancybetweenwhatisthecaseφiandwhatoughttobethecaseδi[15].Theconflictbetweenφiandδigivesrisetoπi.Deonticknowledgeiscriticaltotheproblemformulationandmightbeconsideredasfactoryplanningprinciples,or“goldenrules.”Theexplanatoryissuesidescribewhytheproblemoccursandtheinstrumentalissuesι1,ι2describealternativewaysofresolvingtheπi.Atleasttwoalternativewaysofresolvinganissuearefelttobeimportantfortheproblemstructureanditscompleteness.Figure5illustratestherelationshipbetweenafactualissue,adeonticissue,theexplanatoryandinstrumentalissues.Eachplanningissueshouldbecomprisedofthesecomponentparts.Theplanningissuestructureisausefulparadigmitselfoftheelementsofproblemformulation.Itbecomesclearhowthecomponentpartsofaproblemshouldbedefined.Italsoprovidesanunambiguousmethodfordefiningaproblem.Eachplanningissueisdynamicbutalsobounded.Abriefexampleofaplanningissueisderivedfromthepolymerrecyclingplant.–FactualIssue(φi):=Thenumberofaccidentsandpotentialconflictswithper-sonnelintheplantatthereceivingandshippingareasisexcessive.–DeonticIssue(δi):=Thenumberofconflictsbetweenplantpersonnelandfork-lifttrucksshouldbeminimized.–PlanningIssue(πi):=Howshouldcongestionbetweenforklifttrucksandplantpersonnelbeavoidedatthereceivingandshippingarea? 12J.MacGregorSmithQuestionsIssuesAnswersPositionsTakenArgumentsHeardDecisionsReachedKnowledgeGainedFig.6.Planningissuesresolutionprocess–ExplanatoryIssue(i):=Thereisnotclearseparationbetweentheforklifttrucksandtheplantpersonnelwithinthereceivingandshippingarea.InstrumentalIssue(ι1):=Ifspaceisavailable,separatereceivingandshippinganddesignthematerialhandlingsystemsintheplantinaU-shapelayout.InstrumentalIssue(ι2):=Ifspaceisunavailable,clearlydemarcatethereceivingandshippingareasandthepathsofthevehiclesandpedestrians.Thereasontheabovearestatedasissuesisthatevidencefortheirsupportmustbebroughtforthtosupportorrefuteeachissue.Peoplemustbeconvincedofthecasebeingmade.Someissuesareeasilyresolvedasquestions,whileothersmaynotbesoeasilyresolved.Noteveryonemightagreewithwhatwemeanby“excessive”trafficinthereceivingandshippingareaofφisosomesupportingdatamaybenecessary.Likewise,eventheinstrumentalissueswilllikelyneedsupportingevidencesuchasispossiblewithsophisticatedsimulationandqueueingmodelstoestimateexpected(maximum)volumeofforklifttraffic,#numberofexpectedgaylordsintheshippingandreceivingareas,etc.WhyaU-shapelayout?iscertainlyarguable.Figure6issuggestiveoftheissueresolutionprocess.Whilethisapproachtoproblemformulationthroughtheplanningissuesparadigmcanbeseenaswell-structured,therecanbemanyplanningissuesinfactorydesign,which,unfortunately,leadstothenextdilemma. Dilemmasinfactorydesign:paradoxandparadigm13C1C2···Cj···Cn−1Cnπ11π1,2π1jπ1,n...πij...πm1πmjπmnFig.7.IBISdynamicprogrammingparadigm∆2:Everyfactorydesignproblemissymptomaticofeveryotherfactorydesignproblem.Theseconddilemmaunderscoresthefactthattherearemanyproblemsnestedtogether,thereisnotsimplyoneisolatedproblemtobesolved.Theparadoxsur-roundingtheseconddilemmaisthat:χ2:=Tacklingtheproblemasformulatedmayleadtocuringthesymptomsoftheproblemratherthantherealproblem-youareneversureyouaretacklingtherightproblemattherightlevel.Oneneedstotackletheproblemsonashighalevelaspossible.Inthepoly-merrecyclingproject,issuesofscheduling,resourceconfigurationandutilization,qualitycontrol,andmanyothersbecamefunctionallyrelatedtotheplantlayoutproblem.Aswillbeshown,theseotherissuesemergedascriticaltotheplantlay-out.Theprincipleneededintheparadigminresponsetotheparadoxofdilemma#2is:Proposition3.Constructanetworkofplanningissues,anIssue-BasedInformationSystem(IBIS).An(IBIS)isneededinordertoidentify,interrelate,andquantify(weightsofimportance)thedifferentplanningissueswithintheFDP.Figure7illustratesonerealizationofanIBISthroughadynamicprogramming(DP)paradigm.AnIBIShasanumberofstagesC1,...Cnwhichserveasusefulwaysoforganizingtheplanningissuesastheyaredefinedandemergeintheplanningprocess.Eachnodewithinastagejrepresentsaplanningissueπij.TheplanningissuesrepresentthestatesoftheDPframework.WithineachstageCallπ
sarejinter-connectedcliques.Therecanbemanylinksfromoneπijtoanotherπiksoitmakesthemostsensethatthedataorganizationwouldbesometypeofrelationaldatabase.However,dependingupontheproblem,otherwaysoforganizingtheissueswouldbepossible,suchasasimplematrix. 14J.MacGregorSmithEachCjrepresentsastageoftheDPparadigmandeachstatehasasetofalternativewaysofresolvingeachplanningissueπijlabelledasalternativekwithineachplanningissuexijkTransitionsbetweenstatesinadjacentstageswouldhaveanassociatedcostfortransitioningorlinkingadjacentstates.Onepossiblerecursivecostfunctionforanadditiveorseparableresourceconstrainedproblemcouldbe[8]:f(π,x)=c+f∗(x)jiijkπxkj+1ijkIngeneral,therecursivecostfunctionneednotbeadditive,yettheadditivesituationwouldbequiteappropriateinmanyresourceconstrainedIBISscenarios.Thegeneralrecursivecostfunctionrelationshipwouldmorelikelybe:fj(π)=max/min{fj(π,xijk)}xijkxijkOnecanconsidertheoverallcostofresolvingasetofplanningissuesasapath/treethroughthestagesandstatesoftheIBISproblem.Eachsuchpathrepre-sentsamorphologicalplansolution.πiπncijπ
cjkckmcmnπjπmFigure8illustratesanotherIBISnetworkthatwasutilizedbytheauthortoapproacharesourceplanningproblemattheUniversityofMassachusetts[20].Inthisstudytherewerefivecategories(stages)ofplanningissues(22issuestotal):–C1:ClientCommunication/Ownership–I2:InformationTrackingofProjects–S3:SchedulingandControlofProjects–G4:GeneralProjectManagement–O5:OutreachtoClientsTheIBISprovidedaviableframeworkwhichresultedinasuccessfulresolutionofthemanagementprocessofsmall-scaleconstructionprojects.Infact,aswespeak,thismanagementstruggleisstillon-goingattheUniversity.Theplanningissueswillsimplynotgoaway.TheobviousimplicationsforthemanufacturingandIEprofessionalsandtheireducationisthatthedesignandanalysisofinformationsystemsarecrucialtotheprofession.Thisisinresponsetodilemmas∆1and∆2.Well,let’sarguethatthesenotionsofplanningissuesandinformationsystemsarereasonable,whatnext? Dilemmasinfactorydesign:paradoxandparadigm15Fig.8.UniversityofMassachusettsIBISproject∆3Thereisnolistofpermissableoperations.Whenoneplayschess,thereareonlyafinitenumberofmovestostartthegame.Inlinearprogramming,oneneedsastartingfeasiblesolutiontobegintheprocess.Infactorydesign,thereisnoonesingleplacetostarttheproblemformulationandsolutionprocess.Forthepolymerrecyclingproject,wecouldhavevisitedotherpolymerpro-cessingplants,travelledtootherlocationsbesidesWesternMassachusetts,readalltheliteratureonpolymerre-processing,carriedoutamailsurvey,talkedwithalltheemployees,andsoon.Weshouldhavedonealltheabove,butalas,itwasnotprac-ticalnorcost-effective.Thisdilemmaisfoundedonthefollowingparadox:χ3:=Ifoneisrational,oneshouldconsidertheconsequencesoftheiractions;however,oneshouldalsoconsidertheconsequencesofconsideringtheconsequences,i.e.ifthereisnowheretostarttoberational,oneshouldsomehowstartearlier[15].TheparadoxindicatesthatagreatdealofknowledgeaboutthesystemunderstudyisneededtoassisttheclientandtheengineersinmakingdecisionsabouttheFDP.Ofcourse,alogicalresponsetothisparadoxisthefollowingprinciple:Proposition4.ConstructasystemrepresentationΣ(analyticalorsimulation)ofthemanufacturingsystemwithinwhichtheFDPissituated.Thisprincipleisveryusefulonebutobviouslycanbeexpensiveintimetocon-struct.Itmakeseminentsenseinthesupply-chainbusinessenvironmentcurrentlypopular,sothemoreoneunderstandsthelogisticsandthemanufacturingsystemsandprocesses,thebetter.Atthispoint,thesystemmodelΣbecomesanintegralpartofthenewparadigm.Adiscrete-eventdigitalsimulationmodelofthepolymerrecyclingplantwasconstructedinordertobetterunderstandthemanufacturingprocessesandthesys-temaswellasthelogisticsoftheproductshipmentstoandfromtheplant.This 16J.MacGregorSmithFig.9.Finalplanforpolymerre-processingplantwasfelttobecrucialbeforesimplyre-layingouttheplantandwillbeshowntobeanextremelyfortunatedecision.Figure9illustratesthelayoutplanarrivedatwithau-shapedcirculationflowtoeliminatetheforkliftconflictsfromthepreviousscheme(Fig.3).Unfortunately,thiswasnottheendofthestory.Thus,fortheManufacturingandIEprofessional,systemmodelssuchassupply-chainnetworks,simulationandqueueingnetworkmodelsarecriticallyimportanttoframethecontextoftheproblem.The“systemsapproach”isstillsageadvice.Relatedto∆3is:∆4:Thereisnostoppingrule.Inchess,youeitherwin,lose,ordraw–gameover!Inlinearprogramming,eitheryoufindtheoptimalsolution,anunboundedone,orfindoutthattheproblemisinfeasible.Infactorydesign,youcanalwaysmakeimprovementstothesystem.Aswesawabove,simplyarrivingatthelayoutdesigninnotenough.Thus,wehavethefollowingparadox:χ4:=Ifoneisrational,oneshouldalsoknowthateveryconsequencehasaconsequence,soonceonestartstoberational,onecannotstop-onecanalwaysdobetter[15]. Dilemmasinfactorydesign:paradoxandparadigm17ClientInformationSchedulingGeneralOutsourceComm.SystemsControlProcessC11I12S13G14O15C21I22S23G24O25C31I32S33G34O35C41I42S43G44I52S53G45Fig.10.PaththroughIBISnetworkInfact,anotherparadoxwhichinterrelates∆3and∆4is:χ5:=Onecannotstarttoberationaland,consequently,onecannotstop[15].ThefinalstepingeneratingplansforFDPshereisthatinfactorydesignandinmostwickedproblems,time,resources,andthefinancesinvolvedindicatethatonemustterminatethedesignprocessandarriveatafinalplan.InthecontextoftheIBISnetwork(seeFig.10),thehighlightedcirclesillustratetheselectedpath/planthroughtheIBISissueswhichisactuallythepaththatwastakenfortheUniversityofMassachusettsproject.Thispathincludedthefollowingprescientissueswhichwasusedtoformulatetheultimatestrategy(andproblem!)forsolution:–C31:Thereisnocustomerfeedbackloop.–I32:Smallconstructionprojectsarenotaswellmanagedaslargercapitalprojects.–S33:Cycletimesforsmallconstructionprojectsarenotsatisfactory.–G34:Thereisnodedicatedprofessionalstaffassignedtosmallprojects.–O35:Outsideprivatecontractors(ratherthanUniversitypersonnel)donothaveaccesstoas-builtdrawingsofUniversityfacilities.Giventheresources,time,andfinancialconstraints,thisselectedpaththroughtheIBISrepresentedareasonablemorphologicalplansolution.Also,theremainingissuenetworkdoesnotdisappearoncethefinalplanisagreedupon.Thisisarealisticassessmentoftheplanningprocessandisalsorelatedtothenextdilemma.∆5:TherearemanyalternativeexplanationsforaplanningissueAsonecanargue,therearemanyexplanationsforeachplanningissue,andthus,therearemanypotentialsolutions,notjustone.RefertoFigure11foranillustrationofthisprocess. 18J.MacGregorSmithι11φiι12i1πii2ι21δiι22ι11φiι12i1ι21i2πiι22δii3ι31ι32Fig.11.ManyexplanationspossibleforπiTheparadoxsurroundingthisdilemma:χ5:=Peopleneedtochooseoneso-lutionasa“best”solution;but,unfortunately,therearemanypotentialsolutions,withcorrespondinglydifficulttradeoffs.Inresponsetothissituation,oneneedsmuchhelptogenerateinnovativeso-lutionstotheunderlyingFDPproblems.Layoutplanningalgorithmswereusedinthepolymerprocessingplanttohelpcometoasolutiontothelayoutproblemandalsowereseenasvehiclestoresolveissuesinthelayoutproblem,notasendsinthemselves.Besidesusingcombinatorialoptimizationalgorithms,oneneedstogenerateaspecialsetofsolutions,infact,theparadigmaticprinciplewhich∆5isbaseduponiscloselyrelatedtothenextdilemmabothinspiritandinpractice.∆6:Thereisnosinglecriterionforcorrectness.InmostTPs,thereareobjectivefunctionswhichclearlydemarcatefeasiblefromoptimalsolutions.ThegapbetweenlinearandnonlinearprogrammingTPscanbequitehuge.Inwickedproblems,thereareamultiplenumberofobjectivefunc-tions,notonlylinearandnonlinearones.Paradoxically,infactorydesignwehave:χ6:=Solutionsareeithergoodorbadnotrightorwrong(trueorfalse).Therearemultiplecriteriaembeddedwithineachplanningissue. Dilemmasinfactorydesign:paradoxandparadigm19∆5and∆6arecloselyrelatedsinceoneofthereasonswhytherearesomanysolutionsisthattherearemultipleobjectivesinFDP.Thus,weneedtogenerateaNon-Inferior(NI)setofsolutions,andthenotionofoptimalitybecomesspuriousbecauseitonlymakessenseinasingleobjectiveenvironment.ItisveryrarethatanFDPhasonlyoneobjective.Inanotherprojectweworkedon,theprojectmanagergaveoutthefollowingdauntinglistofobjectivesbeforewestartedourproject:Optimizeproductflowinorderto:–Reduceprojectcosts;–ReduceWIPinvestment;–IncreaseInventoryturns;–Reducescrapandrework;–Quickerresponsetocustomerneeds;–Improveresponsetimetoqualityproblems;–Improvehousekeeping;–Betterutilizefloorspace;–Improvesafety;–Eliminateforklifttrucks.Thus,wehavethefollowing:Proposition5.GenerateaNon-inferior(NI)setofSolutionsbaseduponthemul-tipleobjectives/criteria{f1(x),f2(x),...,fp(x)}involvedintheFDP.Theimplicationsof∆5and∆6forthemanufacturingandIEprofessionandcurriculumarethatmulti-criteriaandmulti-objectiveprogrammingareessentialmethodologicalconceptsandalgorithmictoolsinmanufacturingsystemsandIE.MCDMconceptsandmethodologieshaveslowlybeenintroducedintoIEcurricu-lumswhichisaverypositivesign.Relatedtothelastdilemmaisthefactthat:∆7:ThereisnoimmediateorultimatetestofasolutionMathematicalprogrammingmodels,analyticalstochastictools,andsimulationmodelsbecomeveryimportantforarguingwhyresolvingacertainissueinacertainwayshouldbecarriedout.Thus,thesystemsmodelsuggestedin∆3arecriticalforresolving∆7.Theparadoxhereis:χ7:=Unlikechessorsolvinganequationsys-tem,thereisnoimmediateorultimatetestofasolution,becausetherearedynamicconsequencesovertime,i.e.agreatdealofuncertainty.∆7iscloselyrelatedto∆4.Bothsimulationandanalyticalstochasticanddynamicmodelsarenecessary.∆8:Everyfactorydesignproblemisaone-shotoperation.Infactorydesignproblems,onedoesn’tgetasecondchance.Onecanplaychessorsolitairemanytimesover.Solvingmathematicalprogrammingprogramsononecomputeroradistributedcomputernetworkisroutine.Markovianqueueingnetworkscanberunforwardsorbackwardsintimeandthisaffordstheirdecom-posability.Theparadoxisthatχ8:=FDPsarenottimereversible.Thereisno 20J.MacGregorSmithtrialanderrorwithfactorydesignproblems,noexperimentationyoucannotbuildaplant,tearitdown,andrebuilditwithoutsignificantconsequences.Thisdilemmaandparadoxareverytroublingbecauseoncethefactorydesignprojectgoestotheconstructionphase,thereisnoturningback.Inmanyscientificdisciplines,repeatedexperimentationtotestanhypothesisisroutineandacceptedpracticebecausethecostsandconsequencesarejustified.Theprinciplerelating∆7and∆8is:Proposition6.DynamicModelsΣ(t)areneededforFDPs.ForthemanufacturingandIEprofession,simulationmodellingisacceptedpracticeandwithgoodreason.Analyticalsystemmodelswithqueueingnetworksarealsobecomingmoreimportantandmanyoftheseanalyticaltoolsareoftenusedinadditiontosimulation.Thepolymerrecyclingprojectismostappropriateasanillustrationofthesedilemmasatthisstage.Inordertotestourfinalfactorydesignlayout,weranthesimulationmodelandcalculatedthenumberofgaylordsinthewarehouseasafunctionofvariationsintheinputdemandλi,∀i,from0%−20%.Figure12illustratestheresultsofthesimulationrunsforthetotalnumberofrawmaterialgaylordspossibleonthey-axisvs.theinputdemandonthex-axis.Thefirst3columnsofFigure12illustratethenumberofrawmaterialgaylordsasafunctionofinputdemand.Thus,asonecanseetheinitialdesignoftheplantwasfairlyrobust.However,Figure13revealedthatastheinputvolumerampedupintheplantto120%(3rdcolumn),aseriousproblemarosewithoneofthekeyresourcesbecauseat120%ofinputdemandtheminimumrawmaterialinputvolumewentnegativeby670gaylords.Essentially,theplantinput-processingofrawmaterialsbasicallyshut-down.Weneededtofindoutwhichresourcewasthebottleneck.Afteradetailedanalysisofthesimulationmodeloutputs,itwasrevealedinthethirdcolumnofFigure14whereitisshownthattheaugerblenderwasoperatingat100%capacityandcouldnothandleanymoreinput.Theaugerblenderwasthebottleneck.Thus,iftheinputdemandwastobegreaterthan20%ofthecurrentdemand,itbecameobviousthataminimumof2augerblenderswereneededasopposedtoonlyone.Fig.12.Totalnumberofrawmaterialgaylords Dilemmasinfactorydesign:paradoxandparadigm21Fig.13.AverageandminimumrawmaterialcapacityFig.14.Blenderutilizationsvs.inputdemandInsubsequentrunsofthesimulationmodel,2augerblenderswereutilized,sothatinviewingthefourthcolumninFigures12,13,and14,theoutputstatisticsinclude2augerblendersoperatingwithintheplant.Finally,Figure15illustratesthatwith2augerblenders,thetotalcapacityoftheplant(#ofgaylordsincludingrawmaterialsandfinishedgoodsintherevisedlayout)isacceptableforthegiveninputlevelsofinputdemand.Additionalrunsofthesimulationmodelrevealedthatiffutureinputdemandweretoincreasebeyond20%,fourextrudersratherthecurrentthreewouldbeneededtohandlethedemand.Thus,thesimulationmodelbecameaninvaluabletooltoidentifytheshiftingbottlenecksandforecasttheconfigurationofresourcesneededwithintheplantasdemandincreasedovertime.Thenextdilemmaisverytroublingforacademics,becauseitarguesthat: 22J.MacGregorSmithFig.15.Finaltotal#ofgaylordswarehousecapacity∆9:Everyfactorydesignproblemisunique.Inacademia,onelearnsgeneralprinciples(deonticknowledge);however,inprac-tice,thesegeneralprinciplesmustbetemperedwiththesurroundingcontext,theclient,theever-changingproblemrequirements,anduncertaintyinmodelling.WitheverynewFDP,onemuststartoveragain.Theparadoxisthat:χ9:=Generalknowl-edgeandrulesareverylimited.Youcannotlearnforthenexttime.Onecannoteasilyusestrategiesthathaveworkedinthepastandexpectthattheywillworkinthefuture[15].Evenwithallthedetailedsimulationmodelsandunderstandingoftheplantpainstakinglydone,whenitcametoexaminingtherelocationofthepolymerpro-cessingplanttwoyearsafterthestudy,everythinghadtobere-donealloveragain,becausethesitewasdifferent,theexistingbuildingswerenotthesame,theinputvolumehadchanged,etc.CertainlyonemightarguethatexperiencedpeoplehavespecialknowledgeoftheissuessurroundingaFDP,butthereisnoguarantee,evenifoneknowstheissues,thatthesolutionsusedinthepasttoresolvethemwillworkinthefuture.Proposition7.Youshouldneverdecidetooearlythenatureofthesolutionandwhetherornotanoldsolutioncanbeusedinanewcontext[15].Finally,wehavethelastdilemma:∆10:=Wehavenorighttobewrong.Thisisalsoverychallengingforprofessionalsaswellasacademics,becausetheprinciplesofscientificresearchcanbecompromised.Sciencecanacceptorrefuteanhypothesis,mathematicianscandisproveconjectures,butrunningabusinesscannotacceptfailure.Compromiseisessential.ThecynicalremarksbyGeorgeBernardShaw[21]mentionedatthebeginningofthispaperunderlythemoraldilemmacapturedbythislastdilemma.Theparadoxsurroundingthislastdilemmaisthat:χ10:=Designcannotbecarriedoutinsolitaryconfinement,theFDPdesignprocessisdemocratic.ThefinalprinciplesummarizesouroverallapproachtoFDP: Dilemmasinfactorydesign:paradoxandparadigm23Proposition8.SolvingFDPsisanargumentativeanddynamicprocessconcernedwithidentifying,explaining,andresolvingoftheplanningissues.Thislastprinciplelinksbackto∆1,sincetheproblemformulationprocessmuststartwithinquiriesandissues,andthusanargumentative,dynamicprocessthroughanIBISiscriticaltotheentireFDPprocess.5ImplicationsfortheprofessionandthecurriculumTobrieflysummarizeandemphasizetheimportanceoftheprecedingdiscussion,thetendifferentdilemmasarere-presentedbelow:∆1:Thereisnodefinitiveproblemformulation.∆2:Everyproblemissymptomaticofeveryotherproblem.∆3:Thereisnolistofpermissableoperations.∆4:Thereisnostoppingrule.∆5:Therearemanyalternativesolutionsforaplanningissue.∆6:Thereisnosinglecriterionforcorrectness.∆7:Thereisnoimmediateorultimatetestofasolution.∆8:Everyfactorydesignproblemisaone-shotoperation.∆9:Everyfactorydesignproblemisunique.∆10:Wehavenorighttobewrong.TheelementalimplicationsforthemanufacturingandIEprofessionareproba-blybestdescribedinasummaryimplicationdiagramcenteredaroundthedilemmas∆1∆2∆3Σ∆4∆5∆6IBIS∆7Σ(t)∆8∆9∆10Fig.16.FinalIBISparadigm 24J.MacGregorSmithandtheIBISwhichmustintegratethemandthemodelsnecessarytoresolvetheissues(seeFig.16).TheIBISisnecessarytoframe∆1andtointerrelatethedifferentissuesandproblemsspawnedby∆2.AsystemsmodelΣisnecessaryfor∆3,∆4.Generatingideasandevaluatingascapturedby∆5and∆6mustrelyoneffectivealgorithmictoolsbutthesemustbetemperedwithacognizanceofthemultipleobjectivesandcriteriainvolvedsothateffectivetradeoffscanbemade.AStochastic/DynamicmodelΣtisnecessarytoaddressthevariability,prediction,andcontrolissuessurrounding∆7,∆8,∆9.Indeed,thedegreeofuncertaintyinmostFDPsmakesthislaststageverychallenging.Finally,theIBISneedstobeanopenanddemocraticsystemthatlinksallaspectsoftheFDPprocess.PerhapstheweakestelementinmostmanufacturingandIEcurriculums,atleastfromtheperspectivesarguedinthispaper,isadequateexposuretoFDPsasWickedProblems.Designproblemswithinacademiawithrealclientsaremostdesirable,whereas,ifthisisnotpossible,projectsderivedfromarealworldsettingwithrealisticcon-straintsandexpectationsshouldbepursued.Inaverypositivesense,manyschoolshavesemesteroryear-longseniordesignprojectswhichcancapturethisaspectoftheFDPproblem.AninterestingdevelopmentinEngineeringeducationistheConceiving-Designing-Implementing-Operatingreal-worldsystemsandproducts(CDIO)collaborativehttp://www.cdio.org/whichunderscoresmuchofwhathasbeenarguedhereintheispaper.ItisorientedtoallofEngineeringratherthanjustIndustrialandManufacturingEngineering,butitsphilosophyissimilar.However,itdoesnotappeartorelyonanIBISapproach,whichasarguedforinthispaper,isverycriticaltosuccessinresolvingreal-worldproblems.ProblemformulationandstructuringforWPsareverydifficulttopicstotreatandteach,buttheIBISframeworkissomethingwhichhasclearparadigmaticandteachableelements.Ofcourse,howtheseelementsareputtogetherintothecurriculumremainstherealwickedproblem.6SummaryandconclusionsTheunderlyingdilemmas,paradoxes,andpossibleparadigmsoffactorydesignhavebeenexpoundedupon.AlltheseconceptsarecloselyintertwinedanditishopedthatilluminatingtherelationshipbetweentheseelementswillshedsomelightonpossibleapproachestoFDPs.AnIBISisproposedtobethevehicleforstructuringthedesignprocessforFDPs.Also,asasidebenefit,possiblechangestothemanufacturingandIEcurriculumshavebeendiscussed,sinceFDPsposeamicrocosmandsynthesisofmanyoftheactivitiesmanufacturingandIEsprofess. Dilemmasinfactorydesign:paradoxandparadigm25References1.ChecklandP(1984)Rethinkingasystemsapproach.In:TomlnsonR,KissI(eds)Re-thinkingtheprocessofoperationalresearchandsystemsanalysis,pp43–65.PergamonPress,NewYork2.CookSA(1971)Thecomplexityoftheorem-provingprocedures.Proc.oftheThirdACMSymposiumonTheoryofComputing,pp4–183.DeGraceP,StahlL(1990)Wickedproblems,righteoussolutions.YourdenPressCom-putingSeries,UpperSaddleRiver,NJ4.DixonJR,PoliC(1995)Engineeringdesignanddesignformanufacturing.FieldStone,Conway,MA5.EthirajSK,LevinthalD(2004)Modularityandinnovationincomplexsystems.Man-agementScience50(2):159–1736.Funk,Wagnalis(1968)Standardcollegedictionary.Harcourt,BraceandWorld,NewYork7.GareyMR,JohnsonDS(1979)Computersandintractability:aguidetothetheoryofNP-completeness.Freeman,SanFrancisco8.HillierF,LiebermanG(2001)Introductiontooperationsresearch.McGraw-Hill,NewYork9.HoppW,SpearmanM(1996)Factoryphysics.McGraw-Hill,NewYork10.KarpRM(1972)Reducibilityamongcombinatorialproblems.In:MillerRE,ThatcherJW(eds)Complexityofcomputercomputations.PlenumPress,NewYork11.KarpRM(1975)Onthecomputationalcomplexityofcombinatorialproblems.Net-works5:45–6812.KrishnamoorthyMS,DeoN(1979)Complexityofminimum-dummy-activitiesprob-leminaPERTnetwork.Networks9:189–19413.KunzW,RittelH(1970)Issuesaselementsofinformationsystems.InstituteforUrbanandRegionalDevelopment,UniveristyofCalifornia,Berkeley,WorkingPaperNo.13114.RittelH(1968)Lecturenotesforarch,vol130.Unpublished,DepartmentofArchitec-ture,UniversityofCalifornia,Berkeley,CA15.RittelH(1972a)Ontheplanningcrisis:systemsanalysisofthe‘firstandsecondgenerations’.BedriftsOkonomen8:390–39616.RittelH(1972b)Structureandusefulnessofplanninginformationsystems.BedriftsOkonomen(8):398–40117.RittelH,WebberM(1973)Dilemmasinageneraltheoryofplanning.PolicySciences4:155–16718.RittelH(1975)Ontheplanningcrisis:systemsanalysisofthefirstandsecondgen-erations.Reprintedfrom:BedriftsØKonmen,No.8,October1972;ReprintNo.107,Berkeley,InstituteofUrbanandRegionalDevelopment19.RobertsN(2001)Copingwithwickedproblems:thecaseofAfghanistan.LearningfromInternationalPublicManagementForum,vol11B,pp353–375.Elsevier,Amsterdam20.RobinsonD,etal.(1999)Achangeproposalforsmall-scalerenovationandconstructionprojects.ProjectTeamReport.InternalUniversityofMassachusettsReport.CampusCommitteeforOrganizationRestructuring,UniversityofMassachusetts,Amherst,MA21.ShawGB(1946)Thedoctorsdilemma.Penguin,NewYork22.SmithJMacGregor,LarsonRJ,MacGilvaryDF(1976)Trialcourtfacility.NationalClearinghouseforCriminalJusticePlanningandArchitecture,MonographB5.In:Guidelinesfortheplanninganddesignofstatecourtprogramsandfacilities.UniversityofIllinois,Champaign,IL SectionII:UnreliableProductionLines Leanbufferinginserialproductionlineswithnon-exponentialmachinesEmreEnginarlar1,JingshanLi2,andSemyonM.Meerkov31DecisionApplicationsDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545,USA2ManufacturingSystemsResearchLaboratory,GMResearchandDevelopmentCenter,Warren,MI48090-9055,USA3DepartmentofElectricalEngineeringandComputerScience,UniversityofMichigan,AnnArbor,MI48109-2122,USA(e-mail:smm@eecs.umich.edu)Abstract.Inthispaper,leanbuffering(i.e.,thesmallestlevelofbufferingneces-saryandsufficienttoensurethedesiredproductionrateofamanufacturingsystem)isanalyzedforthecaseofseriallineswithmachineshavingWeibull,gamma,andlog-normaldistributionsofup-anddowntime.Theresultsobtainedshowthat:(1)theleanlevelofbufferingisnotverysensitivetothetypeofup-anddowntimedis-tributionsanddependsmainlyontheircoefficientsofvariation,CVupandCVdown;(2)theleanlevelofbufferingismoresensitivetoCVdownthantoCVupbutthedifferenceinsensitivitiesisnottoolarge(typically,within20%).Basedontheseobservations,anempiricallawforcalculatingtheleanlevelofbufferingasafunc-tionofmachineefficiency,lineefficiency,thenumberofmachinesinthesystem,andCVupandCVdownisintroduced.Itleadstoareductionofleanbufferingbyafactorofupto4,ascomparedwiththatcalculatedusingtheexponentialassump-tion.Itisconjecturedthatthisempiricallawholdsforanyunimodaldistributionofup-anddowntime,providedthatCVupandCVdownarelessthan1.Keywords:Leanproductionsystems–Seriallines–Non-exponentialmachinereliabilitymodel–Coefficientsofvariation–Empiricallaw1Introduction1.1GoalofthestudyThesmallestbuffercapacity,whichisnecessaryandsufficienttoachievethedesiredthroughputofaproductionsystem,isreferredtoasleanbuffering.In(Enginarlaretal.,2002,2003a),theproblemofleanbufferingwasanalyzedforthecaseofCorrespondenceto:S.M.Meerkov 30E.Enginarlaretal.serialproductionlineswithexponentialmachines,i.e.,themachineshavingup-anddowntimedistributedexponentially.Thedevelopmentwascarriedoutintermsofnormalizedbuffercapacityandproductionsystemefficiency.ThenormalizedbuffercapacitywasintroducedasNk=,(1)TdownwhereNdenotedthecapacityofeachbufferandTdowntheaveragedowntimeofeachmachineinunitsofcycletime(i.e.,thetimenecessarytoprocessonepartbyamachine).ParameterkwasreferredtoastheLevelofBuffering(LB).TheproductionlineefficiencywasquantifiedasPRkE=,(2)PR∞wherePRkandPR∞representedtheproductionrateoftheline(i.e.,theaveragenumberofpartsproducedbythelastmachinepercycletime)withLBequaltokandinfinity,respectively.Thesmallestk,whichensuredthedesiredE,wasdenotedaskEandreferredtoastheLeanLevelofBuffering(LLB).Usingparameterizations(1)and(2),Enginarlaretal.,(2002,2003a)derivedclosedformulasforkEasafunctionofsystemcharacteristics.Forinstance,inthecaseoftwo-machineslines,itwasshownthat(Enginarlaretal.,2002)⎧⎪⎪2e(E−e)⎪⎪,ife2-machineseriallines,thefollowingformulahadbeenderived(Enginarlaretal.,2003a):⎧⎪⎪e(1−Q)(eQ+1−e)(eQ+2−2e)(2−Q)⎪⎪2×⎪⎪Q(2e−2eQ+eQ+Q−2)⎪⎪⎨2expE−eE+eEQ−1+e−2eQ+eQ+Q1k(M≥3)=ln,ife3.Initialresultsonleanbufferingfornon-exponentialmachineshavebeenre-portedin(Enginarlaretal.,2002).Twodistributionsofup-anddowntimehavebeenconsidered(RayleighandErlang).IthasbeenshownthatLLBforthesecasesissmallerthanthatfortheexponentialcase.However,(Enginarlaretal.,2002)didnotprovideasufficientlycompletecharacterizationofleanbufferinginnon-exponentialproductionsystems.Inparticular,itdidnotquantifyhowdifferenttypesofup-anddowntimedistributionsaffectLLBanddidnotinvestigaterelativeeffectsofuptimevs.downtimeonLLB.ThegoalofthispaperistoprovideamethodforselectingLLBinseriallineswithnon-exponentialmachines.WeconsiderWeibull,gamma,andlog-normalreliabilitymodelsundervariousassumptionsontheirparameters.ThisallowsustoplacetheircoefficientsofvariationsatwillandstudyLLBasafunctionofup-anddowntimevariability.Moreover,sinceeachofthesedistributionsisdefinedbytwoparameters,selectingthemappropriatelyallowsustoanalyzetheleanbufferingfor26variousshapesofdensityfunctions,rangingfromalmostdelta-functiontoalmostuniform.ThisanalysisleadstothequantificationofbothinfluencesofdistributionshapesonLLBandeffectsofup-anddowntimeonLLB.Basedoftheseresults,wedevelopamethodforselectingLLBinseriallineswithWeibull,gamma,andlog-normalreliabilitycharacteristicsandconjecturethatthesamemethodcanbeusedforselectingLLBinseriallineswitharbitraryunimodaldistributionsofup-anddowntime.1.2Motivationforconsideringnon-exponentialmachinesThecaseofnon-exponentialmachinesisimportantforatleasttworeasons:First,inpracticethemachinesoftenhaveup-anddowntimedistributednon-exponentially.Astheempiricalevidence(Inman,1999)indicates,thecoefficientsofvariation,CVupandCVdownoftheserandomvariablesareoftenlessthan1;thus,thedistributionscannotbeexponential.Therefore,ananalyticalcharacterizationofkEfornon-exponentialmachinesisoftheoreticalimportance.Second,suchacharacterizationisofpracticalimportanceaswell.Indeed,itexpcanbeexpectedthatkEistheupperboundofkEforCV<1and,moreover,kEexpmightbesubstantiallysmallerthank.ThisimpliesthatasmallerbuffercapacityEisnecessarytoachievethedesiredlineefficiencyEwhenthemachinesarenon-exponential.Thus,selectingLLBbasedonrealistic,non-exponentialreliabilitycharacteristicswouldleadtoincreasedleannessofproductionsystems.1.3Difficultiesinstudyingthenon-exponentialcaseAnalysisofleanbufferinginserialproductionlineswithnon-exponentialmachinesiscomplicated,ascomparedwiththeexponentialcase,bythereasonsoutlinedinTable1.Especiallydamagingisthefirstone,whichpracticallyprecludesanalyticalinvestigation.Theotherreasonsleadtoacombinatoriallyincreasingnumberofcasestobeinvestigated.Inthiswork,wepartiallyovercomethesedifficultiesby 32E.Enginarlaretal.Table1.Difficultiesofthenon-exponentialcaseascomparedwiththeexponentialoneExponentialcaseNon-exponentialcaseAnalyticalmethodsforevaluatingNoanalyticalmethodsforevaluatingPRareavailablePRareavailableMachineup-anddowntimesaredistributedMachineup-anddowntimesmayidentically(i.e.,exponentially).havedifferentdistributions.CoefficientsofvariationofmachineCoefficientsofvariationofmachineup-anddowntimesareidenticalup-anddowntimesmaytakearbitraryandequalto1.positivevaluesandmaybenon-identical.AllmachinesinthesystemhavetheEachmachineinthesystemmayhavesametypeofup-anddowntimedistributionsdifferenttypesofup-anddowntime(i.e.,exponential).distributions.usingnumericalsimulationsandbyrestrictingthenumberofdistributionsandcoefficientsofvariationanalyzed.1.4RelatedliteratureThemajorityofquantitativeresultsonbuffercapacityallocationinserialproduc-tionlinesaddressthecaseofexponentialorgeometricmachines(Buzacott,1967;Caramanis,1987;Conwayetal.,1988;SmithandDaskalaki,1988;JafariandShanthikumar,1989;Park,1993;Seongetal.,1995;GershwinandSchor,2000).Justafewnumerical/empiricalstudiesaredevotedtothenon-exponentialcase.Specifically,two-stagecoaxiantypecompletiontimedistributionsareconsideredbyAltiokandStidham(1983),Chow(1987),HillierandSo(1991a,b),andtheeffectsoflog-normalprocessingtimesareanalyzedbyPowell(1994),PowellandPyke(1998),HarrisandPowell(1999).Thesepapersconsiderlineswithreliablemachineshavingrandomprocessingtime.Anotherapproachistodevelopmethodstoextendtheresultsobtainedforsuchcasestounreliablemachineswithdeterminis-ticprocessingtime(Tempelmeier,2003).Phase-typedistributionstomodelrandomprocessingtimeandreliabilitycharacteristicsareanalyzedbyAltiok(1985,1989),AltiokandRanjan(1989),YamashitaandAltiok(1998),buttheresultingmethodsarecomputationallyintensiveandcanbeusedonlyforshortlineswithsmallbuffers(e.g.,two-machinelineswithbuffersofcapacitylessthansix).Finally,asitwasmentionedintheIntroduction,initialresultsonleanlevelofbufferinginseriallineswithRayleighandErlangmachineshavebeenreportedin(Enginarlaretal.,2002). Leanbufferinginserialproductionlineswithnon-exponentialmachines331.5ContributionsofthispaperThemainresultsderivedinthispaperareasfollows:–LLBisnotverysensitivetothetypeofup-anddowntimedistributionsanddependsmostlyontheircoefficientsofvariation(CVupandCVdown).–LLBismoresensitivetoCVdownthantoCVup,butthisdifferenceinsensi-tivitiesisnottoolarge(typically,within20%).–InseriallineswithMmachineshavingWeibull,gamma,andlog-normaldis-tributionsofup-anddowntimewithCVupandCVdownlessthan1,LLBcanbeselectedusingthefollowingupperbound:kE(M,E,e,CVup,CVdown)max{0.25,CVup}+max{0.25,CVdown}exp≤k(M,E,e),(7)2Eexpwherekisgivenby(5),(6).ThisboundisreferredtoastheempiricalElaw.Itisconjecturedthatthisboundholdsforallunimodalup-anddowntimedistributionswithCVup<1andCVdown<1.–AlthoughforsomevaluesofCVupandCVdown,bound(7)maynotbetootight,itstillleadstoareductionofleanbufferingbyafactorofupto4,ascomparedtoLLBbasedontheexponentialassumption.1.6PaperorganizationInSection2,themodeloftheproductionsystemunderconsiderationisintroducedandtheproblemsaddressedareformulated.Section3describestheapproachofthisstudy.Sections4and5presentthemainresultspertaining,respectively,tosystemswithmachineshavingidenticalandnon-identicalcoefficientsofvariationofup-anddowntime.InSection6,seriallineswithmachineshavingarbitrary,i.e.,general,reliabilitymodelsarediscussed.Finally,inSection7,theconclusionsareformulated.2Modelandproblemformulation2.1ModelTheblockdiagramoftheproductionsystemconsideredinthisworkisshowninFigure1,wherethecirclesrepresentthemachinesandtherectanglesarethebuffers.Assumptionsonthemachinesandbuffers,describedbelow,aresimilartothoseof(Enginarlaretal.,2003a)withtheonlydifferencethatup-anddowntimedistributionsarenotexponential.Specifically,theseassumptionsare:(i)Eachmachinemi,i=1,...,M,hastwostates:upanddown.Whenup,themachineiscapableofprocessingonepartpercycletime;whendown,noproductiontakesplace.Thecycletimesofallmachinesarethesame. 34E.Enginarlaretal.mm11bb22mmmM-2bbM-2M-1M-1MFig.1.Serialproductionline(ii)Theup-anddowntimeofeachmachinearerandomvariablesmeasuredinunitsofthecycletime.Inotherwords,uptime(respectively,downtime)oflengtht≥0impliesthatthemachineisup(respectively,down)duringtcycletimes.Theup-anddowntimearedistributedaccordingtooneofthefollowingprobabilitydensityfunctions,referredtoasreliabilitymodels:(a)Weibull,i.e.,PfW(t)=pPe−(pt)PtP−1,upRfW(t)=rRe−(rt)RtR−1,(8)downwherefW(t)andfW(t)aretheprobabilitydensityfunctionsofup-andupdowndowntime,respectivelyand(p,P)and(r,R)aretheirparameters.(Here,andinthesubsequentdistributions,theparametersarepositiverealnumbers).ThesedistributionsaredenotedasW(p,P)andW(r,R),respectively.(b)Gamma,i.e.,(pt)P−1fg(t)=pe−pt,upΓ(P)(rt)R−1fg(t)=re−rt,(9)downΓ(R)∞whereΓ(x)isthegammafunction,Γ(x)=sx−1e−sds.Thesedistribu-0tionsaredenotedasg(p,P)andg(r,R),respectively.(c)Log-normal,i.e.,2LN1−(ln(t)−p)fup(t)=√e2P2,2πPt2LN1−(ln(t)−r)fdown(t)=√e2R2.(10)2πRtWedenotethesedistributionsasLN(p,P)andLN(r,R),respectively.Theexpectedvalues,variances,andcoefficientsofvariationofdistributions(8)–(10)aregiveninTable2.(iii)Theparametersofdistributions(8)–(10)areselectedsothatthemachineeffi-ciencies,i.e.,Tupe=,(11)Tup+Tdownand,moreover,Tup,Tdown,CVup,andCVdownofallmachinesareidenticalforallreliabilitymodels,i.e.,−11Tup=pΓ1+(Weibull)P Leanbufferinginserialproductionlineswithnon-exponentialmachines35Table2.Expectedvalue,variance,andcoefficientofvariationofup-anddowntimedistri-butionsconsideredGammaWeibullLog-normal−1p+P2/2TupP/ppΓ(1+1/P)e−1r+R2/2TdownR/rrΓ(1+1/R)e222−222p+P2(eP−1)σupP/pp[Γ(1+2/P)−Γ(1+1/P)]e222−222r+R2(eR−1)σdownR/r√r[Γ(1+2/R)−Γ(1+1/R)]eCV2P2up1/PΓ(1+2/P)−Γ(1+1/P)Γ(1+1/P)e−1√CVRΓ(1+2/R)−Γ2(1+1/R)Γ(1+1/R)eR2−1down1/P=(gamma)p2=ep+P/2(log-normal);T=r−1Γ(1+1/R)(Weibull)downR=(gamma)r2=er+R/2(log-normal);Γ(1+2/P)−Γ2(1+1/P)CVup=(Weibull)Γ(1+1/P)1=√(gamma)PP2=e−1(log-normal);Γ(1+2/R)−Γ2(1+1/R)CVdown=(Weibull)Γ(1+1/R)1=√(gamma)RR2=e−1(log-normal).(iv)Bufferbi,i=1,...,M−1isofcapacity0≤N≤∞.(v)Machinemi,i=2,...,M,isstarvedattimetifitisupattimet,bufferbi−1isemptyattimetandmi−1doesnotplaceanyworkinthisbufferattimet.Machinem1cannotbestarved.(vi)Machinemi,i=1,...,M−1,isblockedattimetifitisupattimet,bufferbiisfullattimetandmi+1failstotakeanyworkfromthisbufferattimet.MachinemMcannotbeblocked. 36E.Enginarlaretal.Remark1.–Assumptions(i)–(iii)implythatallmachinesareidenticalfromallpointsofviewexcept,perhaps,forthenatureofup-anddowntimedistributions.Thebuffersarealsoassumedtobeofequalcapacity(see(iv)).Wemaketheseassumptionsinordertoprovideacompactcharacterizationofleanbuffering.–Assumption(ii)implies,inparticular,thattime-dependent,ratherthanoperation-dependentfailures,areconsidered.Thisfailuremodesimplifiestheanalysisandresultsinjustasmalldifferenceincomparisonwithoperation-dependentfailures.2.2NotationsEachmachineconsideredinthispaperisdenotedbyapair[Dup(p,P),Ddown(r,R)]i,i=1,...,M,(12)whereDup(p,P)andDdown(r,R)represent,respectively,thedistributionsofup-anddowntimeofthei-thmachineinthesystem,DupandDdown∈{W,g,LN}.TheseriallinewithMmachinesisdenotedas{[Dup,Ddown]1,...,[Dup,Ddown]M}.(13)Ifallmachineshaveidenticaldistributionofuptimesanddowntimes,thelineisdenotedas{[Dup(p,P),Ddown(r,R)]i,i=1,...,M}.(14)If,inaddition,thetypesofup-anddowntimedistributionsarethesame,thenotationforthelineis{[D(p,P),D(r,R)]i,i=1,...,M}.(15)Finally,ifup-anddowntimedistributionsofthemachinesarenotnecessarilyW,g,orLNbutaregeneralinnature,however,unimodal,thelineisdenotedas{[Gup,Gdown]1,...,[Gup,Gdown]M}.(16)2.3ProblemsaddressedUsingtheparameterizations(1),(2),themodel(i)–(vi),andthenotations(12)–(16),thispaperisintendedto–developamethodforcalculatingLeanLevelofBufferinginproductionlines(13)–(15)undertheassumptionthatthecoefficientsofvariationofup-anddowntime,CVupandCVdown,areidentical,i.e.,CVup=CVdown=CV;–developamethodofcalculatingLLBinproductionlines(13)–(15)forthecaseofCVup=/CVdown;–extendtheresultsobtainedtoproductionlines(16).SolutionsoftheseproblemsarepresentedinSections4–6whileSection3describestheapproachusedinthiswork. Leanbufferinginserialproductionlineswithnon-exponentialmachines373Approach3.1GeneralconsiderationsSinceLLBdependsonlineefficiencyE,thecalculationofkErequirestheknowl-edgeoftheproductionrate,PR,ofthesystem.Unfortunately,asitwasmentionedearlier,noanalyticalmethodsexistforevaluatingPRinseriallineswitheitherWeibull,orgamma,orlog-normalreliabilitycharacteristics.Approximationmeth-odsarealsohardlyapplicablesince,inourexperiences,even1%-2%errorsintheproductionrateevaluation(duetotheapproximatenatureofthetechniques)oftenleadtomuchlargererrors(upto20%)inleanbufferingcharacterization.There-fore,theonlymethodavailableistheMonteCarloapproachbasedonnumericalsimulations.Toimplementthisapproach,aMATLABcodewasconstructed,whichsimulatedtheoperationoftheproductionlinedefinedbyassumptions(i)–(vi)ofSection2.Then,asetofrepresentativedistributionsofup-anddowntimewasse-lectedand,finally,foreachmemberofthisset,PRandLLBwereevaluatedwithguaranteedstatisticalcharacteristics.Eachofthesestepsisdescribedbelowinmoredetail.3.2Up-anddowntimedistributionsanalyzedThesetof26downtimedistributionsanalyzedinthisworkisshowninTable3,wherethenotationsintroducedinSection2.1areused.Thesedistributionsareclassifiedaccordingtotheircoefficientsofvariation,CVdown,whichtakevaluesfromtheset{0.1,0.25,0.5,0.75,1.0}.TheanalysisofLLBforthissetisintendedtorevealthebehaviorofkEasafunctionofCVdown.Toinvestigatetheeffectoftheaveragedowntime,thedistributionsofTable3havebeenclassifiedaccordingtoTdown,whichtakesvalues20and100.AnillustrationofafewofthedowntimedistributionsincludedinTable3isgiveninFigure2forCVdown=0.5.Asonecansee,theshapesofthedistributionsincludedinTable3rangefrom“almost”uniformto“almost”δ-function.Table3.DowntimedistributionsconsideredCVdownTdown=20Tdown=1000.1g(5,100),g(1,100),W(0.048,12.15),LN(2.99,0.1)W(0.01,12.15),LN(4.602,0.1)0.25g(0.8,16),g(0.16,16),W(0.046,4.54),LN(2.97,0.25)W(0.009,4.54),LN(4.57,0.25)0.5g(0.2,4),g(0.04,4),W(0.044,2.1),LN(2.88,0.49)W(0.009,2.1),LN(4.49,0.49)0.75g(0.09,1.8),g(0.018,1.8),W(0.046,1.35),LN(2.77,0.66)W(0.009,1.35),LN(4.38,0.66)1.00LN(2.65,0.83)LN(4.26,0.83) 38E.Enginarlaretal.0.06g,T=20downg,T=100downW,T=20downW,T=1000.05downLN,T=20downLN,T=100down0.04f(t)0.030.020.010050100150200250300350tFig.2.Differentdistributionswithidenticalcoefficientsofvariation(CVdown=0.5)Theuptimedistributions,correspondingtothedowntimedistributionsofTa-ble3,havebeenselectedasfollows:Foragivenmachineefficiency,e,theaverageuptimewaschosenaseTup=Tdown.1−eNext,CVupwasselectedasCVup=CVdown,whenthecaseofidenticalcoef-ficientsofvariationofup-anddowntimewasconsidered;otherwiseCVupwasselectedasaconstantindependentofCVdown.Finally,usingtheseTupandCVup,thedistributionofuptimewasselectedtobethesameasthatofthedowntime,ifthecaseofidenticaldistributionswasanalyzed;otherwiseitwasselectedasanyotherdistributionfromtheset{W,g,LN}.Forinstance,ifthedowntimewasdistributedaccordingtoDdown(r,R)=g(0.018,1.8)andewas0.9,theuptimedistributionwasselectedasg(0.002,1.8)forCVup=CVdown,Dup(p,P)=g(0.0044,4)forCVup=0.5,orLN(6.69,0.47)forCVup=CVdown,Dup(p,P)=LN(2.88,0.49)forCVup=0.5.Remark2.BothCVupandCVdownconsideredarelessthan1because,accordingtotheempiricalevidenceof(Inman,1999),theequipmentonthefactoryflooroftensatisfiesthiscondition.Inaddition,ithasbeenshownbyLiandMeerkov(2005)thatCVupandCVdownarelessthan1ifthebreakdownandrepairratesofthemachinesareincreasingfunctionsoftime,whichoftentakesplaceinreality. Leanbufferinginserialproductionlineswithnon-exponentialmachines393.3ParametersselectedInallsystemsanalyzed,particularvaluesofM,E,andehavebeenselectedasfollows:(a)Thenumberofmachinesinthesystem,M:Since,asitwasshownin(Enginarlarexpetal.,2002),kisnotverysensitivetoMifM≥10,thenumberofmachinesinEthesystemwasselectedtobe10.Forverificationpurposes,weanalyzedalsoseriallineswithM=5.(b)Lineefficiency,E:Inpractice,productionlinesareoftenoperatedclosetotheirmaximumcapacity.Therefore,forthepurposesofsimulation,Ewasselectedtobelongtotheset{0.85,0.9,0.95}.Forthepurposesofverification,additionalvaluesofEanalyzedwere{0.7,0.8}.(c)Machineefficiency,e:Althoughinpracticeemayhavewidelydifferentval-ues(e.g.,smallerinmachiningoperationsandmuchlargerinassembly),toob-tainamanageablesetofsystemsforsimulation,ewasselectedfromtheset{0.85,0.9,0.95}.Forverificationpurposes,weconsiderede∈{0.6,0.7,0.8}.3.4SystemsanalyzedSpecificsystemsoftheform(15)consideredinthisworkare:{[W(p,P),W(r,R)]i,i=1,...,10},{[g(p,P),g(r,R)]i,i=1,...,10},(17){[LN(p,P),LN(r,R)]i,i=1,...,10}.Systemsoftheform(13)havebeenformedasfollows:Foreachmachinemi,i=1,...,10,theup-anddowntimedistributionswerechosenfromtheset{W,g,LN}equiprobablyandindependentlyofeachotherandallothermachinesinthesystem.Asaresult,thefollowingtwolineswereselected:Line1:{(g,W),(LN,LN),(W,g),(g,LN),(g,W),(LN,g),(W,W),(g,g),(LN,W),(g,LN)},Line2:{(W,LN),(g,W),(LN,W),(W,g),(g,LN),(18)(g,W),(W,W),(LN,g),(g,W),(LN,LN)}.WewillusenotationsA∈{(17)},A∈{(18)}orA∈{(17),(18)}toindicatethatlineAisoneof(17),oroneof(18),andoneof(17)and(18),respectively.Lines(17)and(18)areanalyzedinSections4and5forthecasesofCVup=CVdownandCVup=/CVdown,respectively.3.5EvaluationoftheproductionrateToevaluatetheproductionrateinsystems(17)and(18),usingtheMATLABcodeandtheup-anddowntimedistributionsdiscussedinSections3.1–3.3,zeroinitial 40E.Enginarlaretal.conditionsofallbuffershavebeenassumedandthestatesofallmachinesattheinitialtimemomenthavebeenselected“up”.Thefirst100,000cycletimeswereconsideredaswarm-upperiod.Thesubsequent1,000,000cycletimeswereusedforstatisticalevaluationofPR.Eachsimulationwasrepeated10times,whichresultedin95%confidenceintervalsoflessthan0.0005.3.6EvaluationofLLBTheleanbuffering,kE,necessaryandsufficienttoensurelineefficiencyE,wasevaluatedusingthefollowingprocedure:Foreachmodelofserialline(13)–(15),theproductionratewasevaluatedfirstforN=0,thenforN=1,andsoon,untiltheproductionratePR=E·PR∞wasachieved.ThenkEwasdeterminedbydividingtheresultingNEbythemachineaveragedowntime(inunitsofthecycletime).Remark3.Although,asitiswellknown(HillierandSo,1991b),theoptimalallocationofafixedtotalbuffercapacityisnon-uniform,tosimplifytheanalysisweconsideronlyuniformallocations.Sincetheoptimal(i.e.,invertedbowl)allocationtypicallyresultsinjust1−2%throughputimprovementincomparisonwiththeuniformallocation,forthesakeofsimplicityweconsideronlythelattercase.4LLBinseriallineswithCVup=CVdown=CV4.1System{[D(p,P),D(r,R)]i,i=1,...,10}Figures3and5presentthesimulationresultsforproductionlines(17)foralldistributionsofTable3.Thesefiguresarearrangedasmatriceswheretherowsandcolumnscorrespondtoe∈{0.85,0.9,0.95}andE∈{0.85,0.9,0.95},re-spectively.Since,duetospaceconsiderations,thegraphsinFigures3and5arecongestedandmaybedifficulttoread,oneofthemisshowninFigure4inalargerscale.(ThedashedlinesinFigs.3–5willbediscussedinSect.4.3.)Examiningthesedata,thefollowingmaybeconcluded:exp–Asexpected,kEfornon-exponentialmachinesissmallerthankE.Moreover,kEisamonotonicallyincreasingfunctionofCV.Inaddition,kE(CV)isconvex,whichimpliesthatreducinglargerCV’sleadstolargerreductionofkEthanreducingsmallerCV’s.–FunctionkE(CV)seemstobepolynomialinnature.Infact,eachcurveofFigures3and5canbeapproximatedbyapolynomialofanappropriateorder.However,sincetheseapproximationsare“parameter-dependent”(i.e.,differentpolynomialsmustbeusedfordifferenteandE),theyareofsmallpracticalimportance,and,therefore,arenotreportedhere.–Sinceforeverypair(E,e),correspondingcurvesofFigures3and5areidentical,itisconcludedthatkEisnotdependentofTupandTdownexplicitlybutonlythroughtheratioe.Inotherwords,thesituationhereisthesameasinlineswithexponentialmachines(see(5),(6)). Leanbufferinginserialproductionlineswithnon-exponentialmachines41Fig.3.LLBversusCVforsystems(17)withTdown=2010GammaWeibull8log−normalempiricallaw6Ek42000.20.40.60.81CVFig.4.LLBversusCVforsystem{(D(p,P),D(r,R))i,i=1,...,10}withTdown=20,e=0.9,E=0.9–Finally,andperhapsmostimportantly,thebehaviorofkEasafunctionofCVisalmostindependentofthetypeofup-anddowntimedistributionsconsidered.Indeed,letkA(CV)denoteLLBforlineA∈{(17)}withECV∈{0.1,0.25,0.5,0.75,1.0}.ThenthesensitivityofkEtoup-anddown-timedistributionsmaybecharacterizedbykA(CV)−kB(CV)(CV)=maxEE·100%.(19)1AA,B∈{(17)}k(CV)E 42E.Enginarlaretal.Fig.5.LLBversusCVforsystems(17)withTdown=100Fig.6.SensitivityofLLBtothenatureofup-anddowntimedistributionsforsystems(17) Leanbufferinginserialproductionlineswithnon-exponentialmachines43Function1(CV)isillustratedinFigure6.Asonecansee,inmostcasesittakesvalueswithin10%.Thus,itispossibletoconcludethatforallpracticalpurposeskEdependsonthecoefficientsofvariationofup-anddowntime,ratherthanonactualdistributionoftheserandomvariables.4.2System{[D(p,P),D(r,R)]1,...,[D(p,P),D(r,R)]10}Figures7and8presentthesimulationresultsforlines(18),whileFigure9char-acterizesthesensitivityofkEtoup-anddowntimedistributions.Thissensitivityiscalculatedaccordingto(19)withtheonlydifferencethatthemaxistakenoverA,B∈{(18)}.Basedonthesedata,weaffirmthattheconclusionsformulatedinSection4.1holdforproductionlinesofthetype(13)aswell.4.3Empiricallaw4.3.1AnalyticalexpressionSimulationresultsreportedaboveprovideacharacterizationofkEforM=10andEande∈{0.85,0.9,0.95}.HowcankEbedeterminedforothervaluesofM,E,ande?Obviously,simulationsforallvaluesofthesevariablesareimpossible.EvenforparticularvaluesofM,E,ande,simulationstakeaverylongtime:Figures3and5requiredapproximatelyoneweekofcalculationsusing25Sunworkstationsworkinginparallel.Therefore,ananalyticalmethodforevaluatingkEforallvaluesofM,E,e,andCVisdesirable.AlthoughanexactcharacterizationofthefunctionkE=kE(M,E,e,CV)isallbutimpossible,resultsofSections4.1and4.2provideanopportunityforintroducinganupperboundofkEasafunctionofallfourexpexpvariables.Thisupperboundisbasedontheexpressionofk=k(M,E,e),EEgivenby(5),(6),andthefactthatallcurvesofFigures3,5and7,8arebelowtheexplinearfunctionofCVwiththeslopek,if0.25β,itmustbeconcludedthatCVdownhasalargereffectonkEthanCVup.Iftheinequalityisreversed,CVuphasastrongereffect.Finally,if(25)holdsforsomeαandβfrom(24)anddoesnotholdforothers,theconclusionwouldbethat,ingeneral,neitherhasadominanteffect.Toinvestigatewhichofthesesituationstakesplace,weevaluatedfunctions(22)and(23)usingtheapproachdescribedinSection3.SomeoftheresultsforWeibulldistributionareshowninFigure12(wherethebrokenlinesandCVeffwillbedefinedinSect.5.2).Similarresultswereobtainedforgammaandlog-normaldistributionsaswell(seeEnginarlaretal.,2003bfordetails).Fromtheseresults,thefollowingcanbeconcluded:–Forallαandβ,suchthatα>β,inequality(25)takesplace.Thus,CVdownhasalargereffectonkEthanCVup.–However,sinceeachpairofcurves(22),(23)correspondingtothesameαareclosetoeachother,thedifferenceintheeffectsofCVupandCVdownisnottoodramatic.Toanalyzethisdifference,introducethefunctionA(CV|CV=CV=α)3updownkA(CV=CV|CV=α)−kA(CV=CV|CV=α)=EupdownEdownup·100,(26)kA(CVup=CV|CVdown=α)EwhereA∈{W,g,LN}.ThebehaviorofthisfunctionforWeibulldistributionisshowninFigure13(seeEnginarlaretal.,2003bforgammaandlog-normaldistributions).Thus,theeffectsofCVupandCVdownonkEarenotdramaticallydifferent(typicallywithin20%andnomorethan40%). 48E.Enginarlaretal.Fig.12.LLBversusCVforM=10Weibullmachines5.2Empiricallaw5.2.1AnalyticalexpressionSincetheupperbound(20)isnottootight(and,hence,mayaccommodateadditionaluncertainties)andtheeffectsofCVupandCVdownonkEarenotdramaticallydifferent,thefollowingextensionoftheempiricallawissuggested:kE(M,E,e,CVup,CVdown)max{0.25,CVup}+max{0.25,CVdown}exp≤k(M,E,e),2ECVup≤1,CVdown≤1,(27)expwhere,asbefore,kE,isdefinedby(5),(6).IfCVup=CVdown,(27)reducesto(20);otherwise,ittakesintoaccountdifferentvaluesofCVupandCVdown.Thefirstfactorintheright-hand-sideof(27)isdenotedasCVeff:max{0.25,CVup}+max{0.25,CVdown}CVeff=.(28)2Thus,(27)canberewrittenasexpkE≤CVeffkE(M,E,e).(29)Theright-hand-sideof(29)isshowninFigure12bythebrokenlines.Theutilizationofthislawcanbeillustratedasfollows:SupposeCVup=0.1andCVdown=1.ThenCVeff=0.625and,accordingto(27),expkE≤0.625kE(M,E,e). Leanbufferinginserialproductionlineswithnon-exponentialmachines49WFig.13.Function3(CV|CVup=CVdown=α)Table4.∆(10,E,e)forallCVup=CVdowncasesconsideredE=0.85E=0.9E=0.95e=0.850.10160.03860.0687e=0.90.04250.16470.1625e=0.950.04020.04880.1200Toinvestigatethevalidityoftheempiricallaw(27),considerthefollowingfunction:∆(M,E,e)=minmin(30)A∈{(17)}CVup,CVdown∈{(24)}upperboundAkE(M,E,e,CVeff)−kE(M,E,e,CVup,CVdown),upperboundwherekistheright-hand-sideof(29),i.e.,EupperboundexpkE(M,E,e,CVeff)=CVeffkE(M,E,e).Ifforallvaluesofitsarguments,function∆(M,E,e)ispositive,theright-hand-sideofinequality(27)isanupperbound.Thevaluesof∆(10,E,e)forE∈{0.85,0.9,0.95}ande∈{0.85,0.9,0.95}areshowninTable4.Asonecansee,function∆(10,E,e)indeedtakespositivevalues.Thus,theempiricallaw(27)takesplaceforalldistributionsandparametersanalyzed. 50E.Enginarlaretal.Fig.14.Thetightnessoftheempiricallaw(27)Toinvestigatethetightnessofthebound(27),considerthefunction4(CVeff)=maxmax(31)A∈{(17)}CVup,CVdown∈{(24)}kupperbound(M,E,e,CV)−kA(M,E,e,CV,CV)EeffEupdown·100.kA(M,E,e,CVup,CVdown)EFigure14illustratesthebehaviorofthisfunction.ComparingthiswithFigure10,weconcludethatthetightnessofbound(27)appearstobesimilartothatof(20).5.2.2VerificationToevaluatethevalidityoftheupperbound(27),serialproductionlineswithM=5,E∈{0.7,0.8,0.9},e∈{0.6,0.7,0.8},andTup=10weresimulated.Foreachoftheseparameters,systems(17)and(18)havebeenconsidered.(Forsystem(18),thefirst5machineswereselected.)TypicalresultsareshowninFigure15(seeEnginarlaretal.,2003bformoredetails).Thevalidityofempiricallaw(27)forthesecasesisanalyzedusingfunction∆(M,E,e),definedin(30)withtheonlydifferencethatthefirstministakenoverA∈{(17),(18)}.Sincethevaluesofthisfunction,showninTable5,arepositive,weconcludethatempiricallaw(27)isindeedverifiedforallvaluesofM,E,e,andalldistributionsofup-anddowntimeconsidered. Leanbufferinginserialproductionlineswithnon-exponentialmachines51Fig.15.Verification:LLBversusCVforM=5WeibullmachinesTable5.Verification:∆(5,E,e)forallCVup=CVdowncasesconsideredE=0.7E=0.8E=0.9e=0.60.00390.02420.0547e=0.70.01020.02130.0481e=0.80.00840.01620.03556SYSTEM{[Gup,Gdown]1,...,[Gup,Gdown]M}Sofar,serialproductionlineswithWeibull,gamma,andlog-normalreliabilitymodelshavebeenanalyzed.Itisofintereststoextendthisanalysistogeneralprobabilitydensityfunctions.Basedontheresultsobtainedabove,thefollowingconjectureisformulated:Theempiricallaws(20)and(27)holdforserialproductionlinessatisfyingassumptions(i),(iii)–(vi)withup-anddowntimehavingarbitraryunimodalprob-abilitydensityfunctions.Theverificationofthisconjectureisatopicforfutureresearch. 52E.Enginarlaretal.7ConclusionsResultsdescribedinthispapersuggestthefollowingprocedurefordesigningleanbufferinginserialproductionlinesdefinedbyassumptions(i)–(vi):1.Identifytheaveragevalueandthevarianceoftheup-anddowntime,Tup,T,σ2,andσ2,forallmachinesinthesystem(inunitsofmachinedownupdowncycletime).Thismaybeaccomplishedbymeasuringthedurationoftheup-anddowntimesofeachmachineduringashiftoraweekofoperation(dependingonthefrequencyofoccurrence).Iftheproductionlineisatthedesignstage,thisinformationmaybeobtainedfromtheequipmentmanufacturer(however,typicallywithalowerlevelofcertainty).2.Using(5),(6),andTup,Tdown,determinethelevelofbuffering,necessaryandsufficienttoobtainthedesiredefficiency,E,oftheproductionline,iftheexpdowntimeofallmachinesweredistributedexponentially,i.e.,k.E3.Finally,ifCV=σup≤1andCV=σdown≤1,evaluatethelevelofupTupdownTdownbufferingforthelinewithmachinesunderconsiderationusingtheempiricallawmax{0.25,CVup}+max{0.25,CVdown}expkE≤·kE.2Asitisshowninthispaper,thisprocedureleadstoareductionofleanbufferingbyafactorofupto4,ascomparedwiththatbasedontheexponentialassumption.ReferencesAltiokT(1985)Productionlineswithphase-typeoperationandrepairtimesandfinitebuffers.InternationalJournalofProductionResearch23:489–498AltiokT(1989)Approximateanalysisofqueues.In:Serieswithphase-typeservicetimesandblocking.OperationsResearch37:601–610AltiokT,StidhamSS(1983)Theallocationofinterstagebuffercapacitiesinproductionlines.IIETransactions15:292–299AltiokT,RanjanR(1989)Analysisofproductionlineswithgeneralservicetimesandfinitebuffers:atwo-nodedecompositionapproach.EngineeringCostsandProductionEconomics17:155–165BuzacottJA(1967)Automatictransferlineswithbufferstocks.InternationalJournalofProductionResearch5:183–200CaramanisM(1987)Productionlinedesign:adiscreteeventdynamicsystemandgeneral-izedbendersdecompositionapproach.InternationalJournalofProductionResearch25:1223–1234ChowW-M(1987)Buffercapacityanalysisforsequentialproductionlineswithvariableprocessingtimes.InternationalJournalofProductionResearch25:1183–1196ConwayR,MaxwellW,McClainJO,ThomasLJ(1988)Theroleofwork-in-processinven-toryinserialproductionlines.OperationsResearch36:229–241EnginarlarE,LiJ,MeerkovSM,ZhangRQ(2002)Buffercapacitytoaccommodatingma-chinedowntimeinserialproductionlines.InternationalJournalofProductionResearch40:601–624 Leanbufferinginserialproductionlineswithnon-exponentialmachines53EnginarlarE,LiJ,MeerkovSM(2003a)Howleancanleanbuffersbe?ControlGroupReportCGR03-10,DeptartmentofEECS,UniversityofMichigan,AnnArbor,MI;acceptedforpublicationinIIETransactionsonDesignandManufacturing(2005)EnginarlarE,LiJ,MeerkovSM(2003b)Leanbufferinginserialproductionlineswithnon-exponentialmachines.ControlGroupReportCGR03-13,DeptartmentofEECS,UniversityofMichigan,AnnArbor,MIGershwinSB,SchorJE(2000)Efficientalgorithmsforbufferspaceallocation.AnnalsofOperationsResearch93:117–144HarrisJH,PowellSG(1999)Analgorithmforoptimalbufferplacementinreliableseriallines.IIETransactions31:287–302HillierFS,SoKC(1991a)Theeffectofthecoefficientofvariationofoperationtimesontheallocationofstoragespaceinproductionlinesystems.IIETransactions23:198–206HillierFS,SoKC(1991b)Theeffectofmachinebreakdownsandinternalstorageontheperformanceofproductionlinesystems.InternationalJournalofProductionResearch29:2043–2055InmanRR(1999)Empiricalevaluationofexponentialandindependenceassumptionsinqueueingmodelsofmanufacturingsystems.ProductionandOperationManagement8:409–432JafariMA,ShanthikumarJG(1989)Determinationofoptimalbufferstoragecapacityandoptimalallocationinmultistageautomatictransferlines.IIETransactions21:130–134LiJ,MeerkovSM(2005)Onthecoefficientsofvariationofup-anddowntimeinmanufac-turingequipment.MathematicalProblemsinEngineering2005:1–6ParkT(1993)Atwo-phaseheuristicalgorithmfordeterminingbuffersizesinproductionlines.InternationalJournalofProductionResearch31:613–631PowellSG(1994)Bufferallocationinunbalancedthree-stationlines.InternationalJournalofProductionResearch32:2201–2217PowellSG,PykeDF(1998)Bufferingunbalancedassemblysystems.IIETransactions30:55–65SeongD,ChangeSY,HongY(1995)Heuristicalgorithmforbufferallocationinaproductionlinewithunreliablemachines.InternationalJournalofProductionResearch33:1989–2005SmithJM,DaskalakiS(1988)Bufferspaceallocationinautomatedassemblylines.Opera-tionsResearch36:343–357TempelmeierH(2003)Practicalconsiderationsintheoptimizationofflowproductionsys-tems.InternationalJournalofProductionResearch41:149–170YamashitaH,AltiokT(1988)Buffercapacityallocationforadesiredthroughputofproduc-tionlines.IIETransactions30:883–891 AnalysisofflowlineswithCox-2-distributedprocessingtimesandlimitedbuffercapacity
StefanHelberUniversityofHannover,DepartmentforProductionManagement,KonigswortherPlatz1,¨30167Hannover,Germany(e-mail:stefan.helber@prod.uni-hannover.de)Abstract.WedescribeaflowlinemodelconsistingofmachineswithCox-2-distributedprocessingtimesandlimitedbuffercapacities.Atwo-machinesub-systemisanalyzedexactlyandalargerflowlinesareevaluatedthroughadecom-positionintoasetofcoupledtwo-machinelines.OurresultsarecomparedtothosegivenbyBuzacott,LiuandShantikumarfortheir“StoppedArrivalQueueModell”.Keywords:Flowline–Performanceevaluation–Decomposition–Generalpro-cessingtimes–Cox-2-distribution1IntroductionWedescribeanapproximateapproachtodeterminetheproductionrateandin-ventorylevelofaflowlineconsistingofmorethantwomachineswhereadjacentmachinesaredecoupledthroughbuffersoflimitedcapacity.Weassumethatma-chinesarereliableandthatprocessingtimesareCox-2-distributed.Thisallowsustomodelprocessingtimeswithanysquaredcoefficientofvariationc2≥0.5.TheseprocessingtimescanincludetherandomdelayofworkpieceswhichisduetorandomfailuresandrepairsofthemachinesifweusethecompletiontimeconceptproposedbyGaver[17].Severalresearchershavestudiedtransferlinesorassembly/disassembly(A/D)systemswithlimitedbuffercapacity.AcomprehensivesurveyisgivenbyDalleryandGershwin[15].Thisreviewincludestheliteratureonreliabletwo-machinetransferlines,ontransferlineswithoutbuffersaswellaslongerlineswithmorethantwomachinesandA/Dsystems.Earlierreviewsare[7,11],and[28].TransferlinesandA/DsystemsareoftenstudiedusingMarkovchainorprocessmodelstoallowforananalyticsolutionoranaccurateapproximation.Manyofthese
Theauthorthankstheanonymousrefereesfortheirhelpfulcommentsandsuggestions. 56S.HelberTable1.Two-machinemodelsandapproximationapproachesTypeofAnalysisofApproximatedecompositionprocesstwo-machinemodelsapproachesDiscretestate/[2,5,7,24,26,29,34][13,18,20,25]discretetimeDiscretestate/[6,22,31][12,19,25]continuoustimeContinuousstate/[21,23,32,33,35][4,14,16]continuoustimeapproximationsarebasedonadecompositionofthecompletesystemintoasetofsingleserverqueues[27]ortwo-machinetransferlines[18,32,35]whichcanbeevaluatedanalytically.Themainadvantageofanalyticalapproachesasopposedtosimulationmodelsisthattheanalyticaltechniquesaremuchfaster.Thisiscrucialifalargenumberofdifferentsystemshastobeevaluatedinordertofindaconfigurationwhichisoptimalwithrespecttosomeobjective.Whenanalyzingtherelatedworkwithrespecttotwo-machinemodelsanddecompositionapproaches,wecandistinguish[15]–Markovprocesseswithdiscretestateanddiscretetime,–Markovprocesseswithdiscretestateandcontinuoustime,and–Markovprocesseswithmixedstateandcontinuoustime.Inthefirsttwocases,thestateisdiscretesincediscretepartsareproduced.Anadditionalpossiblereasontohavediscretestatesisthatmachinescanbeeitheroperationalorunderrepair.Timeisdividedintodiscreteperiodsinthefirstcaseortreatedascontinuousinthesecond.ThethirdgroupofMarkovprocessesassumesthatcontinuousmaterialisproducedincontinuoustime(whichleadstoacontinuousbufferlevel),butmachinestatesarediscrete.Inthispaperwedescribeadiscrete-state,continuous-timemodelwherethediscretestatesreflectdiscretebufferlevels.Table1givesanoverviewoftwo-machinemodelsanddecompositionap-proachesforthecaseoflimitedbuffercapacity.Inmanyofthesepapersmachinesareassumedtobeunreliable.Textbookscoveringtheseandsimilartechniquesindetailare[1,10,30]aswellas[21]whichgivesathoroughintroductionintohowtoderivethesemodels.Inthispaper,wedevelopatwo-machinetransferlinede-compositionofthediscretestate-continuoustimetype.Weassume,however,thatmachinesarereliableandthatprocessingtimesmayexhibitvariabilitywithanysquaredcoefficientofvariationlargerthan0.5.ThetwopapersmostcloselyrelatedtothisoneareanolderpaperbyBuzacottandKostelski[8]ontheanalysisofaspecifictwo-machinelineandbyBuzacottetal.[9]onparticulardecompositiontechniquesforlongerlineswithlimitedbuffercapacity.Thepaperisstructuredasfollows:InSection2weformallydescribethetypeofflowlinetobeanalyzed.Section3outlinestheexactanalysisofthetwo-machine,one-buffersubsystemthatservesasthebuildingblockofadecompositionand AnalysisofflowlineswithCox-2-distributedprocessingtimes57whichhasalreadybeenanalyzedbyBuzacottandKostelski[8]usingtheMatrixgeometricmethod.Ouranalysisofthetwo-machinesystem,however,followstheapproachfortwo-machinesystemswhichisthoroughlyexplainedbyGershwin[21].ThedecompositionalgorithmisbrieflydescribedinSection4.InSection5wepresentsomepreliminarynumericalresultsbycomparingourresultstothoseobtainedfromthemultistageflowlineanalysiswiththestoppedarrivalqueuemodelasproposedbyBuzacottetal.[9].2ThemodelWeassumethattheflowlineconsistsofMmachinesorstages.TheprocessingtimesatmachineMifollowaCox-2distribution.EachbufferBibetweenmachinesMiandMi+1hasthecapacitytoholduptoCiworkpieceswhichflowfromtheleftmosttotherightmostmachine.AnexampleofsuchaflowlineisdepictedinFigure1.aaa123µµµµµµ1112212231321-a1-a121-a3Fig.1.FlowlinewiththreemachinesTheratesofthetwophasesofstageiareµi1andµi2respectively.Thesecondphaseofstageiwillberequiredaftercompletionofphaseonewithprobabilityai.Therefore,aworkpiececompletesitsserviceatstageiwithprobability1−aiafterthecompletionofthefirstphaseandwithprobabilityaiafterthecompletionofthesecondphase.Notethatthesestatesofamachineorstagedonotrepresentservers:Nomorethanoneworkpiececanbeatamachineatanymomentintime,andifitisthere,itisinoneoutofthetwophasesoftherespectivemachine.EachmachineMiexceptforthefirstandthelastcanbeeitheridle(starved)orblockedoritcanbeprocessingapartinphaseoneortwo.ThestateofmachineMiisdenotedasαi(t).Thepossiblemachinestatesareαi(t)∈{1,2,B,S},representingphaseone,phasetwo,blockingandstarvation.Thebufferleveln(i)isdefinedsuchthatitincludesthepartsbetweenmachinesMiandMi+1,theonepartresidingatmachineMi+1(ifthismachineisnotstarved)andtheonepartwaitingatmachineMiifthismachineisblockedbecausethebufferbetweenMiandMi+1isfull.3Thetwo-maschinesubsystem3.1StatespaceandtransitionequationsInordertoanalyzelargersystemswithmorethantwomachines,wefirststudyatwo-machineline.Thestateofthistwo-machinelineisgivenbythestateofthefirstmachine,thestateofthesecondmachine,andthebufferlevel.Intheanalysistofollow,wedefinethebufferlevelincludeallpartsthatarecurrentlybeingprocessedatthesecondmachine,thatarewaitinginthephysicalbufferbetweenthefirstand 58S.Helberthesecondmachine,andthosepartsthathavebeenprocessedatthefirstmachinebutcannotleaveitbecausethephysicalbufferbetweenthemachinesisfullsothatthefirstmachineisblocked.Thatis,wefollowtheblockageconventionwhichisdescribedin[21,p.95].The(totalorextended)buffercapacityisthereforeNi=Ci+2.Inordertodescribethestatespace,weuseusethetriple(n,α1,α2)wherendenotesthebufferlevel.Theprobabilityofthesystembeinginthisstateisp(n,α1,α2).MachineM1caneitherbeinthefirstphase(α1=1),inthesecondphase(α1=2)oritcanbeblocked(α1=B).ThedownstreammachineM2caneitherbeinthefirstphase(α2=1),inthesecondphase(α2=2)oritcanbestarved(α2=S).Thisleadstothefollowingtransitionequationswhichdifferforstateswithanemptyoralmostemptybuffer,statesforafullaalmostfullbuffer,andthestateswithabufferlevelthatisinbetween:Lowerboundarystates:µ11p(0,1,S)=(1−a2)µ21p(1,1,1)+µ22p(1,1,2)(1)µ12p(0,2,S)=a1µ11p(0,1,S)+(1−a2)µ21p(1,2,1)+µ22p(1,2,2)(2)(µ11+µ21)p(1,1,1)=(1−a1)µ11p(0,1,S)+µ12p(0,2,S)+(1−a2)µ21p(2,1,1)+µ22p(2,1,2)(3)(µ11+µ22)p(1,1,2)=a2µ21p(1,1,1)(4)Intermediatestages:(µ11+µ21)p(n,1,1)=(1−a1)µ11p(n−1,1,1)+µ12p(n−1,2,1)+(1−a2)µ21p(n+1,1,1)+µ22p(n+1,1,2)(for2≤n≤N−2)(5)(µ11+µ22)p(n,1,2)=(1−a1)µ11p(n−1,1,2)+µ12p(n−1,2,2)+a2µ21p(n,1,1)(for2≤n≤N−1)(6)(µ12+µ21)p(n,2,1)=a1µ11p(n,1,1)+µ22p(n+1,2,2)+(1−a2)µ21p(n+1,2,1)(for1≤n≤N−2)(7)(µ12+µ22)p(n,2,2)=a1µ11p(n,1,2)+a2µ21p(n,2,1)(for2≤n≤N−1)(8)Upperboundarystates:µ21p(N,B,1)=(1−a1)µ11p(N−1,1,1)+µ12p(N−1,2,1)(9)µ22p(N,B,2)=a2µ21p(N,B,1)+(1−a1)µ11p(N−1,1,2)+µ12p(N−1,2,2)(10) AnalysisofflowlineswithCox-2-distributedprocessingtimes59(µ11+µ21)p(N−1,1,1)=(1−a1)µ11p(N−2,1,1)+µ12p(N−2,2,1)+(1−a2)µ21p(N,B,1)+µ22p(N,B,2)(11)(µ12+µ21)p(N−1,2,1)=a1µ11p(N−1,1,1)(12)TogetherwiththenormalizationequationN−122p(0,1,S)+p(0,2,S)+p(n,α1,α2)+n=1α1=1α2=1p(N,B,1)+p(N,B,2)=1(13)thisleadstoalinearsystemofequationswhichcanbesolvedinseveralways.Analmostidenticalsystemofequationshasbeenformulatedin[8]andsolvedviathematrixgeometricmethodandarecursivealgorithm.Sincetheirmethodssufferedfromnumericalinstabilities,wedevelopedasolutiontechniqueusingtheideasfortheanalysisoftwo-machinemodelspresentedin[21,pp.105].Itleadstoanumericallystablealgorithmprovidingtheexactvaluesofallthesystemstatesaswellastheperformancemeasuressuchastheproductionrateandtheinventorylevel.3.2IdentitiesConservationofflow.TherateatwhichpartsleavemachineM1istheproductofthesteady-stateprobabilitiesofallstateswhereM1isnotblockedtimestherespectiverateforthisstate:PR1=µ11(1−a1)p(0,1,S)+µ12p(0,2,S)+N−12(µ11(1−a1)p(n,1,α2)+µ12p(n,2,α2))(14)n=1α2=1ThereasoningformachineM2(whichmaynotbestarved)issimilar:N−12PR2=(µ21(1−a2)p(n,α1,1)+µ22p(n,α1,2))+n=1α1=1µ21(1−a2)p(N,B,1)+µ22p(N,B,2)(15)TheConservation-of-Flow-identity(COF)statesthattheratesofpartspassingthroughmachinesM1andM2areequal:PR1=PR2(16)Thereasonisthattheflowofmaterialislinearandpartsareneithercreatednordestroyedateithermachine. 60S.HelberRateofchangesfromphaseonetotwoequalsrateofchangesfromphasetwotoone.ForeachchangeofmachineM1fromphaseonetophasetwotheremustbeachangefromphasetwotophaseoneN−12a1µ11p(0,1,S)+p(n,1,α2)n=1α2=1N−12=µ12p(0,2,S)+p(n,2,α2)(17)n=1α2=1andthesameholdstrueformachineM2:N−12a2µ21p(N,B,1)+p(n,α1,1)n=1α1=1N−12=µ22p(N,B,2)+p(n,α1,2)(18)n=1α1=1Flow-Rate-Idle-Time-Equations.TheFlow-Rate-Idle-Time-Equations(FRIT-Equations)relatethefloworproductionratesoftheup-anddownstreammachinestotheprobabilityoftherespectivemachinebeingblockedorstarved.TheexpectedprocessingtimeE[T1]attheupstreammachineM1ofatwo-machine-lineistheweightedsumoftheexpectedprocessingtime1ifaworkpieceµ11onlygoesthroughphaseone(whichhappenswithprobability(1−a1))andtheexpectedprocessingtime1+1ifitundergoesbothphases(withprobabilityµ11µ12a1):111E[T1]=(1−a1)+a1+(19)µ11µ11µ12ThereasoningfortheexpectedprocessingtimeE[T2]atatthesecond(down-stream)machineofatwo-machine-lineleadstoananalogousresult:111E[T2]=(1−a2)+a2+(20)µ21µ21µ22NowtheproductionratePR1ofmachineM1isthemultiplicativeinverseoftheaverageprocessingtimeofthismachinetimestheprobability1−pB=1−(p(N,B,1)+p(N,B,2))ofnotbeingblocked:1−pB1−pBPR1==(21)E[T1](1−a)1+a1+11µ111µ11µ12Thisleadstoanequationfortheprobabilityofthemachinebeingblocked:111pB=1−PR1(1−a1)+a1+(22)µ11µ11µ12 AnalysisofflowlineswithCox-2-distributedprocessingtimes61ForthedownstreammachinetheFRIT-equation1−pS1−pSPR2==(23)E[T2](1−a)1+a1+12µ212µ21µ22issimilaranditalsoleadstoasimilarequationfortheprobabilityofthedownstreammachinenotbeingstarved:111pS=1−PR2(1−a2)+a2+(24)µ21µ21µ22Whileequations(21)and(23)canbeusedtodeterminetheproductionrateofatwo-machinesystem,theequations(22)and(24)willlaterbeusedinadecom-positionapproachtoanalyzelargerflowlineswithmorethantwomachines.3.3DerivationofthesolutionInthissection,wederiveaspecializedsolutionproceduresimilartotheonegivenin[22].3.3.1AnalysisofinternalstatesFollowingthebasicapproachinGershwinandBerman,weassumethattheinternalequations(5)–(8)haveasolutionoftheformJJp[n,α,α]=cξ(n,α,α)=cXnYα1−1Yα2−1(25)12jj12jj1j2jj=1j=1wherecj,Xj,Y1j,andY2jareparameterstobedetermined.Theanalysisbelowisverysimilartotheonein[22]and[26,Sect.3.2.4].Replacingp(n,α1,α2)byXnYα1−1Yα2−1inEquations(6),(7)and(8),wederivethefollowingnon-linearj1j2jsetofequations:(µ11+µ22)XY2=a2µ21X+(1−a1)µ11Y2+µ12Y1Y2(26)(µ12+µ21)Y1=a1µ11+(1−a2)µ21XY1+µ22XY1Y2(27)(µ12+µ22)Y1Y2=a2µ21Y1+a1µ11Y2(28)Equations(26)and(27)areusedtoeliminateX.Fromtheresultingequationand(28)wecannexteliminateY2.AconsiderablealgebraiceffortleadstothefollowingfourthdegreeequationinY1aµ(µY−aµ)(Y3+sY2+tY+v)=0(29)221121111111withauxiliaryvariabless,t,v,andwdefinedasfollows:w=µ21(a2µ12−µ12−µ22)(30) 62S.Helber1s=(µµ−µ2+aµµ+aµµ(31)1112121112121121w−a1a2µ11µ21−µ12µ21+µ11µ22−µ12µ22−µ21µ22)1t=(a1µ11(−µ11+2µ12+µ21+µ22))(32)w1v=−(a2µ2)(33)111wFromthefirsttermontheleftsideofEquation(29)weseethatonesolutionto(29)isa1µ11Y11=(34)µ12ApplyingthisresulttoEquation(28),wefinda2µ21Y21=(35)µ22andfrom(34)and(35)in(26)or(27)weseethatX1=1.(36)Theremainingthreesolutionsto(29)are1aφsY12=2−cos−(37)333aφ2φsY13=2−cos+−(38)3333aφ4φsY14=2−cos+−(39)3333withauxiliaryvariables1a=3t−s2(40)31b=2s3−9st+27v)(41)27⎛⎞bφ=arccos⎝−⎠(42)−a3227ThecorrespondingvaluesofY22,Y22,andY24areagaindeterminedvia(28).ThevaluesofX2,X3,andX4arenextcomputedfrom(26)or(27).Sincewehavefoundfoursolutionstoequations(26),(27),and(28),thegeneralexpressionforthesteady-stateprobabilitiesoftheinternalstatesisasfollows44p(n,α,α)=cξ(n,α,α)=cXnYα1−1Yα2−1(43)12jj12jj1j2jj=1j=1wherewestillhavetodeterminetheparameterscj.1See[3,Sect.2.4.2.3,p.131] AnalysisofflowlineswithCox-2-distributedprocessingtimes633.3.2AnalysisofboundarystatesThereisatotalof12boundarystatesinthemodel.Thetransitionequationsoffourofthem((1,2,1),(1,2,2),(N−1,1,2),and(N−1,2,2))areofinternalform(6)-(8),i.e.theirsteady-stateprobabilitiescanbecomputedfromequation(43)eventhoughtheyareboundarystates.Sincep(1,2,1)andp(1,2,2)aregivenfrom(43),thecorrespondingequations(7)and(8)relatedtostates(1,2,1)and(1,2,2)constitutealinearsystemoftwoequationsintwounknownsp(1,1,1)andp(1,1,2)withthefollowingsolution:(µ12+µ21)p(1,2,1)−(µ21−a2µ21)p(2,2,1)p(1,1,1)=−a1µ11µ22p(2,2,2)(44)a1µ11(µ12+µ22)p(1,2,2)−a2µ21p(1,2,1)p(1,1,2)=(45)a1µ11Givenp(1,1,1),p(1,1,2),p(1,2,1),andp(1,2,2),Equations(1)and(2)canimmediatelybeusedtodeterminefirstp(0,1,S)andnextp(0,2,S)(inthisorder).Theupperboundarysteady-stateprobabilitiesaredeterminedinexactlythesamewayasnowstates(N−1,1,2)and(N−1,2,2)areofinternalformandwemaycomputep(N−1,1,2)andp(N−1,2,2)from(43),thensolve(6)and(8)forp(N−1,1,1)andp(N−1,2,1)tofind(µ11+µ22)p(N−1,1,2)−(µ11−a1µ11)p(N−2,1,2)p(N−1,1,1)=−a2µ21µ12p(N−2,2,2)(46)a2µ21(µ12+µ22)p(N−1,2,2)−a1µ11p(1,1,2)p(N−1,2,1)=.(47)a2µ21Givenp(N−1,1,1)andp(N−1,2,1),wecannow(inthisorder)computep(N,B,1)fromequation(9)andfinallyp(N,B,2)fromequation(10).Consideragainthesymmetryofupperandlowerboundaryvalues.Sinceboundarystatesarenowexpressedintermsofinternalstates,andsinceinternalstatesareoftheform4p(n,α1,α2)=cjξj(n,α1,α2),(48)j=1theequationsforboundarystatesholdforeachsolutionξj(n,α1,α2)oftheequa-tionsforinternalstates.Theequation(45)correspondingtostate(1,1,2),forexample,leadsto44(µ12+µ22)j=1cjξj(1,2,2)cjξj(1,1,2)=−a1µ11j=14a2µ21j=1cjξj(1,2,1)(49)a1µ11 64S.HelberSimilarequationscanbefoundtodeterminethetermsξj(n,α1,α2)fortheotherboundarystateprobabilities.Thetermsξj(n,α1,α2)correspondingtotransientstatesareallzero.Nowallsteady-stateprobabilitieshavebeenrelatedtoequation(43).Whatremainstobedoneistofindappropriatevaluesofthecoefficientscjin(43).3.3.3DeterminationofcoefficientscjTodeterminefourcoefficientscj,j=1,...,4,alinearsystemoffourequationsinthefourunknownscjcanbesolved.Thefollowingfourequationscanbederivedbyinserting(43)intotheconservationofflowequation(16),thetwoequationsstatingthatforeverytransitionfromphaseonetophasetwothereisonefromphasetwotophaseone((17)and(18)),andthecondition(13)thatallprobabilitiessumuptoone:Conservationofflow44µ11(1−a1)cjξj(0,1,S)+µ12cjξj(0,2,S)+j=1j=1N−1244(µ11(1−a1)cjξj(n,1,α2)+µ12cjξj(n,2,α2))−n=1α2=1j=1j=1N−1244(µ21(1−a2)cjξj(n,α1,1)−µ22cjξj(n,α1,2))−n=1α1=1j=1j=144µ21(1−a2)cjξj(N,B,1)−µ22cjξj(N,B,2)=0(50)j=1j=1Rateofchangesfromphaseoneto2equalsrateofchangesfromphasetwoto1atMaschineM1⎛⎞4N−124a1µ11⎝cjξj(0,1,S)+cjξj(n,1,α2)⎠j=1n=1α2=1j=1⎛⎞4N−124−µ12⎝cjξj(0,2,S)+cjξj(n,2,α2)⎠=0(51)j=1n=1α2=1j=1Rateofchangesfromphaseoneto2equalsrateofchangesfromphasetwoto1atMaschineM2⎛⎞4N−124a2µ21⎝cjξj(N,B,1)+cjξj(n,α1,1)⎠j=1n=1α1=1j=1⎛⎞4N−124−µ22⎝cjξj(N,B,2)+cjξj(n,α1,2)⎠=0(52)j=1n=1α1=1j=1 AnalysisofflowlineswithCox-2-distributedprocessingtimes65Probabilitiessumuptoone⎛⎞44N−1224cjξj(0,1,S)+cjξj(0,2,S)+⎝cjξj(n,α1,α2)⎠+j=1j=1n=1α1=1α2=1j=144cjξj(N,B,1)+cjξj(N,B,2)=1(53)j=1j=1Notethattherighthandsideofthethreeofthefourequationsiszero.Forthisreason,itisrelativelypainlesstosolvethislinearsystemofequationsinthefourunknownscj,j=1..4numerically.3.4Thealgorithmtodeterminesteady-stateprobabilitiesandperformancemeasuresThealgorithmtocomputetherequiredsteady-stateprobabilitiesp[n,α1,α2]andperformancemeasuresPRandnconsistsofthefollowingsteps:1.Computeauxiliaryvariablesw,s,t,v,a,b,andφfrom(30)-(33)and(40)-(42).ComputeY11from(34)andY12...Y14from(37)-(39).ComputeY21...Y24from(28)andX1...X4from(26)or(27).2.Determinethecoefficientscj,j=1,...,4inEquation(43)bysolvingthelinearsystemofequationsgivenby(50)-(53).3.UsethecjfromStep2tocomputetherequiredsteady-stateprobabilitiesp(n,α1,α2)ofstatesofinternalformvia(43)andthoseoftheremainingboundarystatesasdescribedinSection3.3.2.4.Determineperformancemeasures.Determinetheproductionratefrom(14)or(15),thein-processinventoryviaN−122n¯=np(n,α1,α2)+N(p(N,B,1)+p(N,B,2)(54)n=1α1=1α2=1andblockingandstarvationprobabilitiespBandpSviapB=p(N,B,1)+p(N,B,2)(55)pS=p(0,1,S)+p(0,2,S).(56)Thisalgorithmprovedtobenumericallystableanditwasusedasabuildingblockwithinthedecompositionapproachemployedtoanalyzeflowlineswithmorethantwomachines.4Thedecompositionapproach4.1DerivationofdecompositionequationsWhileitispossibletoanalyzeatwo-machinesystemexactly,theexactanalysisoflargersystemsispracticallyimpossibleasthestatespaceofthesystemexplodes 66S.Helberveryquickly.Forthisreasondecompositionapproachesarefrequentlyusedtoanalyzelargersystems.ThebasicideaistodecomposeasystemwithKmachinesandK−1buffersintoK−1two-machinesystemswithvirtualmachinesthatmimictoanobserverinthebuffertheflowofmaterialinandoutofthisbufferasitwouldbeseeninthecorrespondingbufferoftherealsystem.Wefollowedtheideaspresentedingreatdetailin[21]todevelopaniterativedecompositionalgorithmtoanalyzeflowlineswithmorethantwomachines.However,somemodificationswerenecessarywhichwewillnowbrieflyoutline.Whilethemodelsanalyzedin[21]assumedunreliablemachinesandconsequentlyleadtoso-calledinterruption-of-flow-andresumption-of-flow-equations,wearestudyingaflowlinewithreliablemachineswhichcannotfail.Themachinesinoursystem,however,changetheirphasesofoperationasdescribedinSection2.Forthisreason,wederivedthefollowingthreetypesofdecompositionequations:–Phase-One-to-Two(P1t2)-Equation:Thistypeofequationdealswiththeprobabilityofthetransitionofthevirtualmachinefromoperatinginitsfirstphasetoitssecond.–Phase-Two-to-One(P2t1)-Equation:Thistypeofequationdealswiththeprobabilityofthetransitionofthevirtualmachinefromoperatinginitssecondphasetoitsfirst.–Flow-Rate-Idle-Time(FRIT)-Equation:Thisisatypeofequationwhichre-latestheflowofmaterialthroughamachinetoitsisolatedproductionrateanditsprobabilityofbeingblockedandstarved.ThistypeofequationhasalsobeenusedbyGershwinetal.Inthefollowingwewillbrieflydiscussthederivationoftheparametersofthevirtualmachines.ThekeytothederivationoftheP1t2-andP2t1-equationsisthedefinitionofvirtualmachinestates.Westudyavirtualtwo-machinelineL(i)whichisrelatedtothebufferbetweenmachinesMiandMi+1.ThevirtualmachinesoflineL(i)areMu(i)(upstreamofthebuffer)andMd(i)(downstreamofthebuffer).Wewanttodeterminetheparametersau(i),µu1(i),andµu2(i)ofthevirtualmachineMu(i)aswellastheparametersad(i),µd1(i),andµd2(i)ofthevirtualmachineMd(i)inordertobeabletouseourtwo-machinemodelinSection3todetermineperformancemeasuresfortheflowline.Theupstreammachineofatwo-machinelineisneverstarved(andthedown-streammachineisneverblocked).WethereforeassumethatthevirtualmachineMu(i)isinphaseoneiftherealmachineMiisprocessingaworkpieceinphaseoneorwhenitiswaitingforthenextworkpiece:{αu(i,t)=1}iff{αi(t)=1}or{αi(t)=S}(57)MachineMu(i)isinphasetwoifMiisinphasetwo{αu(i,t)=2}iff{αi(t)=2}(58)anditisblockedifMiisblocked:{αu(i,t)=B}iff{αi(t)=B}(59) AnalysisofflowlineswithCox-2-distributedprocessingtimes67ThedefinitionofvirtualmachinestatesformachineMd(i)issymmetric:Ma-chineMd(i)isinphaseoneifthemachineMi+1downstreamofthebuffernumberiisinphaseoneorblocked:{αd(i,t)=1}iff{αi+1(t)=1}or{αi+1(t)=B}(60)ItisinphasetwoifmachineMi+1intherealsystemisinphasetwo{αd(i,t)=2}iff{αi+1(t)=2}(61)andstarvedifMi+1isstarved:{αd(i,t)=S}iff{αi+1(t)=S}(62)Phase-One-to-Two(P1t2)-Equation:ToderivetheP1t2-equationformachineMu(i),weaskfortheprobabilityofobservingatransitionofthevirtualmachineMu(i)fromphaseonetophasetwo.Forthistohappen,wehavetoobserveacompletionofphaseone(withprobabilityµu(i)δt)andtheprocessmustenterthesecondphase,whichhappenswithprobabilityau(i).au(i)µu1(i)δt=Prob[{αu(i,t+δt)=2}|{αu(i,t)=1}](63)Thejointprobabilityau(i)µu1(i)δtcanberelatedtoachangeinthemachinestatesdefinedaboveifweinsertthedefinitionsofthevirtualmachinestatesgivenin(57)and(58):au(i)µu1(i)δt=Prob[{αu(i,t+δt)=2}|{αu(i,t)=1}]=Prob[{αi(t+δt)=2}|{αi(t)=1}or{αi(t)=S}]=Prob[{αi(t+δt)=2}|{αi(t)=1}]·Prob[{αi(t)=1}|{αi(t)=1}or{αi(t)=S}]+Prob[{αi(t+δt)=2}|{αi(t)=S}]·Prob[{αi(t)=S}|{αi(t)=1}or{αi(t)=S}]au(i)µu1(i)≈aiµi1Prob[n(i−1,t)>0](64)Intheabovederivation,theprobabilityofmachineMibeinginphasetwoattimet+δt,giventhatitwasstarvedattimet,iszero.However,therestofthisderivationisstillonlya(possiblycrude)approximationsincetheconditionalprobabilityProb[{αi(t)=1}|{αi(t)=1}or{αi(t)=S}]ofmachineMibeinginphaseonegiventhatitiseitherinphaseoneorstarvedissimplyapproximatedbytheprobabilityProb[n(i−1,t)>0]ofmachineMinotbeingstarved.Thisiscrudesinceifitisnotstarved,incanstillbeinphasetwoorblocked.ThereasoningbehindthiscrudeapproximationisthatifmachineMiisinphaseone,weatleastknowthatitcannotbestarvedandtheprobabilityofthisstateisrelatedtotheprobabilityofmachineMd(i−i)notbeingstarved.Whilethereisnostrongeranalyticaljustificationforthissubstitution,itappearstoworkwellinthenumericalalgorithmtobedescribedbelow.ThebasicapproachtoderivetheprobabilityofatransitionfromphaseonetotwoatthevirtualmachineMd(i)issimilar:ad(i)µd1(i)δt=Prob[{αd(i,t+δt)=2}|{αd(i,t)=1}](65) 68S.HelberWeagaininsertthedefinitionofvirtualmachinestatesandfindad(i)µd1(i)δt=Prob[{αd(i,t+δt)=2}|{αd(i,t)=1}]=Prob[{αi+1(t+δt)=2}|{αi+1(t)=1}or{αi+1(t)=B}]=Prob[{αi+1(t+δt)=2}|{αi+1(t)=1}]·Prob[{αi+1(t)=1}|{αi+1(t)=1}or{αi+1(t)=B}]+Prob[{αi+1(t+δt)=2}|{αi+1(t)=B}]·Prob[{αi+1(t)=B}|{αi+1(t)=1}or{αi+1(t)=B}]ad(i)µd1(i)≈ai+1µi+1,1Prob[n(i+1,t)0}and{ni+1(t)0}and{ni+1(t)1,whilefori=1theycanbeevaluatedusingequations(16).–Distributionofthecalculatedprobabilitiesintothetwopseudo-machinemodelsofFigure4,usingequations(19)to(23),forbothparttypes.–Evaluationofnewlocalfailuresusingequations(24),(25).–Calculationofremotefailuresusingequations(26),(27).–Evaluationofcompetitionfailuresusingequations(29),(30),(31)and(32).–Insertionofcalculatedfailureparametersintoupstreampseudo-machinesofbuildingblocksa(i)andb(i).–Evaluationofaveragethroughput,probabilitiesofblockingandprobabil-itiesofstarvationofblocksa(i)andb(i)usingthebuildingblocksolutionproposedin[5].3.Step2.Fori=K,...,2:failureparametersofmachinesMD(A)(i−1)andMD(B)(i−1)areevaluated:–Unknowntransitionprobabilitiesarecalculatedusingequations(16).–EvaluationofallthestateprobabilitiesoftheflexiblemachineMibyusingthelinearsystemformedbyequations(1)to(14).Starvationprobabilitiesandtransitionstostarvationstatesarederivedfrompreviousiterationsofthealgorithmandtheyareequaltoremotefailuresofupstreampseudo-machinesMU(A)(i)andMU(B)(i),incaseofi2%6,8%4,7%2,9%10%ERROR<1%81,8%66,6%70,6%72,2%MAXERROR2,5%2,38%2,77%3,13%Table7.ErrorinaveragebufferlevelevaluationforcasesofTable4AFig.6.Throughputevaluationwithαiequalforallthemachinesofthelineandvariablecalculatedbyusingthefollowingequations:(nA)−(nA)∆%(nA)=iSIMiCMT·100iNAi(nB)−(nB)∆%(nB)=iSIMiCMT·100iNBi 96M.Colledanietal.Table8.ErrorinthroughputevaluationAsitnormallyhappensindecompositionmethods,errorsinaveragebufferslevelevaluationaremuchhigherthanthoseregardingthroughput.Itisworthnotingthatthetotalthroughputoftheline(thesumofthroughputsofparttypesAandB)isdividedbetweenparttypesAandBdifferentlyfromthevaluesofαAandαBintroducedforthemachinesoftheline.ThisisduetotheiifactthattheoccurrenceofblockingandstarvationisdifferentforparttypeAorBdependingontheirrelativebuffercapacities.InthefollowingtablesαindicatesthevalueofαAandα1indicatestheratiobetweenthroughputofparttypeAandtheitotalthroughput,resultingfromthesimulation.Itwouldbeimportant,asafuturedevelopmentoftheresearch,todevelopamethodabletoassessthevaluesofαAiparametersforeachmachineoftheline,startingwiththeα1valuethatwewanttoeffectivelyobtainfromtheline.InordertostudytheaccuracyofthemethodfordifferentvaluesofαAandiαBinthelineandforeachsinglemachine,somefocusedtestshavebeenrealized.iInparticularwestudiedasixmachinelinewithαAvariableforthebottleneckimachineandequalto0,6foralltheothersmachinesoftheline.ThebehaviorofthesystemiswellapproximatedbythemethodforvaluesofαAsimilartothoseofother5machinesbut,inothercases,themethoddoesn’tevaluateperformancemeasuresofthatlinecorrectly.Thislimitationoftheapplicationfieldofthemethodisnotveryrelevant,becauseinrealautomatedmultiproductflowlinestheαAparameteriisnormallyconstantthroughouttheline.Inthiscasetheproposedmethodcorrectly Performanceevaluationoftwoparttypelines97estimatesaveragethroughputofthetestlineasitisshownin(Fig.6)and(Table8)forawiderangeofvariabilityofparameterαA.iAsitcanbeseenbytheresultsprovidedinthissection,thealgorithmhasproventobereliableandaccurateinallthetestedcaseswithαAandαBparametersequaliiforallthemachinesoftheline.7ConclusionsAnewapproximateanalyticalmethodfortheperformanceevaluationofmultiprod-uctautomatedflowlineswithmultiplefailuremodesandfinitebuffercapacityhasbeenproposed.Themethodhasbeenappliedtothecaseoflinesproducingtwodifferentparttypes,butisamenableofextensiontothecaseofnparttypes.AnalgorithminspiredbytheDDXalgorithmhasbeendevelopedtoevaluatefailureprobabilitiesforallpseudo-machinesofthedecomposedlines.Extensivetestinghasproventheaccuracyofthemethod.Asafuturedevelopment,themethodcouldbeextendedtothecaseofcontinuouslineswithmultipleparttypes.Inaddition,themethodinprinciplecanbeextendedtostudyassembly/disassemblynetworks[1,2]andforkandjoinsystems[3,4].References1.TolioT,MattaA,LevantesiR(2000)Performanceevaluationofassembly/disassemblysystemswithdeterministicprocessingtimesandmultiplefailuremodes.In:ICPR2000InternationalConferenceonProductionResearch,Bangkok,Thailand2.GershwinSB(1991)Assembly/disassemblysystems:anefficientdecompositionalgo-rithmfortreestructurednetworks.IIETransactions23(4):302–3143.HelberS(1999)Performanceanalysisofflowlineswithnonlinearflowofmaterial,vol243.Lecturenotesineconomicsandmathematicalsystems.Springer,BerlinHei-delbergNewYork4.HelberS(2000)Approximateanalysisofunreliabletransferlineswithsplitsintheflowofmaterials.AnnalsofOperationsResearch(93):217–2435.TolioT,GershwinSB,MattaA(2002)Analysisoftwo-machinelineswithmultiplefailuremodes.IIETransactions200234(1):51–626.NemecJE(1999)Diffusionanddecompositionapproximationsofstochasticmodelsofmulticlassprocessingnetworks.PhDthesis,MassachusettsInstituteofTechnology,February7.TolioT,MattaA(1998)Amethodforperformanceevaluationofautomatedflowlines.AnnalsofCIRP47(1):373–3768.LeBihanH,DalleryY(1999)Animproveddecompositionmethodfortheanalysisofproductionlineswithunreliablemachinesandfinitebuffers.InternationalJournalofProductionResearch37(5):1093–1117 AutomatedflowlineswithsharedbufferA.Matta,M.Runchina,andT.TolioPolitecnicodiMilano,DipartimentodiMeccanica,viaBonardi9,20133Milano,Italy(e-mail:{andrea.matta,tullio.tolio}@polimi.it)Abstract.Thepaperaddressestheproblemoffullyusingbufferspacesinman-ufacturingflowlines.Theideaistoexploitrecenttechnologicaldevicestomoveinreasonabletimespiecesfromamachinetoacommonbufferareaofthesystemandviceversa.Insuchawaymachinescanavoidtheirblockingsincetheycansendpiecestothesharedbufferarea.Theintroductionofthebufferareasharedbyallmachinesofthesystemleadstoanincreaseofproductionrateasdemon-stratedbysimulationexperiments.Also,apreliminaryeconomicevaluationonarealcasehasbeencarriedouttoestimatetheprofitabilityofthesystemcomparingtheincreaseofproductionrate,obtainedwiththenewsystemarchitecture,withtherelatedadditionalcost.Keywords:Flowlines–Bufferallocation–Systemdesign–Performanceevalu-ation1IntroductionAmanufacturingflowlineisdefinedinliteratureasaserialproductionsysteminwhichpartsareworkedsequentiallybymachines:piecesflowfromthefirstmachine,inwhichtheyarestillrawparts,tothelastmachinewheretheprocesscycleiscompletedandthefinishedpartsleavethesystem.Whenamachineisnotavailable,partswaitinthebufferimmediatelyupstreamthemachine.Ifthenum-berofpartsflowinginthesystemisconstantduringtheproduction,thesesystemsarealsocalledclosedflowlines(seeFig.1whererectanglesandcirclesrepresentmachinesandbuffersofthesystemrespectively)todistinguishthemfromopenflowlineswherethenumberofpartsisnotmaintainedconstant.Gershwingivesin[4]ageneraldescriptionofflowlinesinmanufacturing.Theproductionrateofflowlinesisclearlyafunctionofspeedandreliabilityofmachines:fasterandmorereliablemachinesareandhighertheproductionrateis.However,sincemachinesCorrespondenceto:A.Matta 100A.Mattaetal.Fig.1.Schemeofclosedflowlinescanhavedifferentspeedsandmaybeaffectedbyrandomfailures,thepartflowcanbeinterruptedatacertainpointofthesystemcausingblockingandstarvationofmachines.Inparticular,blockinginthelineoccurswhenatleastonemachinecannotmovethepartsjustworked(BAS,BlockingAfterService)orstilltowork(BBS,BlockingBeforeService)tothenextstation.Inflowlinestheblockingofamachinecanbecausedonlybyalongprocessingtimeorafailureofadownstreammachine.Analogously,starvationoccurswhenoneormoremachinescannotbeoperationalbecausetheyhavenoinputparttowork;inthiscasethemachinecan-notworkanditissaidtobestarved.Inflowlinesthestarvationofamachinecanbecausedonlybyalongprocessingtimeorafailureofanupstreammachine.Therefore,inflowlinesthestateofamachineaffectstherestofthesystembecauseofblockingandstarvationphenomenathatpropagateupstreamanddownstreamrespectivelythesourceofflowinterruptionintheline.Ifthereisnoareawheretostorepiecesbetweentwoadjacentmachines,thebehaviorofmachinesisstronglycorrelated.Inordertodecreaseblockingandstarvationphenomenainflowlines,buffersbe-tweentwoadjacentmachinesarenormallyincludedtodecouplethemachinesbehavior.Indeed,buffersallowtoadsorbtheimpactofafailureoralongprocess-ingtimebecause(a)thepresenceofpartsinbuffersdecreasesthestarvationofmachinesand(b)thepossibilityofstoringpartsinbuffersdecreasestheblockingofmachines.Therefore,productionrateofflowlinesisalsoafunctionofbuffercapacities;moreprecisely,productionrateisamonotonepositivefunctionofthetotalbuffercapacityofthesystem.Referto[5,7]foralistofworksfocusedonthepropertiesofproductionrateinflowlinesasafunctionofthebuffersize.Traditionally,flowlineshavebeendeeplyinvestigatedinliterature.Re-searchers’effortshavebeendevotedtodevelopnewmodelsforevaluatingtheperformanceofflowlinesandforoptimizingtheirdesignandmanagementinshopfloors.Operationsresearchtechniqueslikesimulationandanalyticalmethodshavebeenwidelyusedtoestimatesystemperformanceparameterssuchasthroughputandworkinprocesslevel.Performanceevaluationmodelsarecurrentlyusedinconfigurationalgorithmsforfindingtheoptimaldesignofflowlinestakingintoaccountthetotalinvestmentcost,operativecostandproductionrateofthesystem.Insynthesis,academicinnovationhasbeenmainlyfocusedonthedevelopmentofperformanceevaluationandoptimizationmethodsofflowlineswithoutenteringintoseveralmechanicaldetails.Seealsothereview[1]ofDalleryandGershwinonadetailedviewofperformanceevaluationmodelsforflowlinesandanupdatedrecentstateoftheartonoptimizationtechniquesappliedinpractice[8].Indeed,mostofworksisatsystemlevelastheydealwithoptimizationofmacrovariablessuchasnumberofmachinesintheline,buffercapacitiesandmachines’speedand Automatedflowlineswithsharedbuffer101efficiency.Ontheotherhand,engineersoffirmshavehadtofacethecomplex-ityduetothefactthatflowlinesaredesignedinpracticewithalltheirmechanicalcomponents.Innovationfrombuildersofmanufacturingflowlineshasbeenmainlydedicatedtoincreasemachinesreliabilityandtoreducesystemcostsbyimprov-ingthedesignofspecificmechanicalcomponentssuchasfeeddrives,spindles,transporters,etc.Therefore,advancementsinflowlineevolutiondonotregardthemainphilos-ophyofthesystem.Partsareloadedintothesystematthefirstmachineand,afterhavingbeenprocessed,theyaremovedintothefirstbufferwaitingfortheavail-abilityofthesecondmachine.Blockingphenomenaislimitedbybuffers,largeristheircapacityandhigherthethroughputofthelineis.However,buffersinflowlinesarededicatedtomachines;thischaracteristicimpliesthatabuffercancon-tainonlypiecesworkedbytheimmediatelyupstreammachine.Therefore,whenalongfailureoccursatamachineoftheline,theportionofthesystemupstreamthefailedmachineisblockedbutupstreammachinescontinuetoworkuntiltheircorrespondingbuffersarefull.Ontheotherhand,theportionofthesystemdown-streamthefailedmachineisstarvedbecausedownstreammachinescannotworksincetheydonothaveanypiecetowork.Inthatcasethebufferareadownstreamthefailedmachinecannotbeusedtostorepartsworkedbymachinesthatareup-streamthefailedmachinesinceemptybuffersarededicatedandcannotbeusedforpiecescomingoutfromothermachines.Itappearsthatbufferspacesarenotfullyexploitedwhenneeded.Theproblemofproperlyusingalltheavailablespaceinflowlinesrepresentstheargumentofthispaper.2Flowlineswithsharedbuffer2.1MotivationThepaperpresentsanewconceptofmanufacturingflowlinecharacterizedbytwodifferenttypesofbuffers:traditionaldedicatedbuffersandacommonbuffersharedbyallthemachinesofthesystem.Thecommonbufferallowstostorepiecesatanypointofthesystemthusincreasingthebuffercapacityofeachmachine(seeFig.2).Themainadvantageisrelatedtothefactthatwhereveraninterruptionofflowisinthesystem,thecommonsharedbuffercanbeusedbyallmachines.Asaconsequence,blockingofmachinesshouldbelowerthanthatofclassicalflowlinesthusallowinganincreaseofproductionrateatconstanttotalbuffercapacity.However,profitabilityofthenewsystemarchitecturedependsoncostsincurredfortheadditionalsharedbuffer.TraditionallythemaingoalinthedesignphaseofflowlinesistofindthesystemconfigurationatminimumcostsconstrainedFig.2.Schemeoftheproposedsystemarchitecture 102A.Mattaetal.toaminimumvalueofproductionrate.Inthiscontext,theintroductionofthesharedbufferinflowlinesispossibleonlyifthetimenecessaryformovingpartsfromsharedbuffertomachinesissmallandtherelativeinvestmentforadditionalmechanicalcomponentsisreasonable.Indeed,inouropinioncostsarethemainreasonforwhichsharedbuffershavenotstillbeadoptedinmanufacturingflowlines.Designingsharedbufferinflowlinesimpliestohaveadditionalcomponents,andthuslargercosts,formovingpiecesfrommachinestothecentralbufferandviceversa.However,technologyisnowmaturetobeusedforthisscopeataffordablecosts.Severalmanufacturerscanprovideatlowcostsawidesetoftransportmodulesforpartmovements.Thesemodulescanbeassembledinaflexiblewaytomovepartsthroughthesystem;actuallythespeedofconveyorsisaround20m/minonaveragedependingontheweightofparts.Partscanfollowlinearpaths,asusualinflowlines,andcircularpathswithsmallrounds.Furthermoreinordertosavefloorspace,partscanbemovedupordownforreachingdifferentheights.Thecostoftransportermodulesisnowaffordableallowingtheirintensiveusageinpracticeatthesameproductivitylevel,definedinthepaperastheamountofoutputobtainedforoneunitofinput.Weconsidertheproductionrateofthesystemastheoutputandthetotalcostofthesystemastheinput.Itisratherdifficulttoincreaseproductivityofmanufacturingsystemssinceaspecificactionthatcanincreasetheproductionrateofasystemisnormallybalancedbytheeffortrequired.Actionsthatcanimprovesystemproductivityshouldreducethetotalcosts(reductionofmachinesandfixturescost,reductionofadaptationcost,etc.)withoutreducingtheproductionrateorshouldincreasetheproductionrate(shortersystemset-uptimes,reductionofunproductivetimes,improvementofsystemavailability,etc.)withoutincreasingcosts.Theproposedsystemcanbeconsideredinterestingforpracticalexploitationifitsproductivityremainsconstantorincreasesincomparisonwithtraditionalsystems.2.2SystemdescriptionTheproposedsystemarchitectureisaflowlinecomposedofKmachinesseparatedbylimitedbuffers.IncaseofopensystemsthenumberofbuffersisequaltoK−1andweassumethatthefirstmachineisneverstarvedandthelastmachineisneverblocked;incaseofclosedsystemsthenumberofbuffersisequaltothenumberofmachines.WedenotewithMiandBi(withi=1,...,K−1,K)thei-thmachineandthei-thdedicatedbufferrespectively.Machinesarenormallyunreliableandtheirefficiencydependsontheirfailureandrepairratesdistributions.TheK−1buffers(orKinclosedflowlines)arededicatedtotheircorrespondingmachines:bufferB1containsonlypiecesalreadyworkedbyfirstmachineM1,bufferB2containsonlypiecesalreadyworkedbysecondmachineM2,andsoon.IfbufferBiisfull,i.e.thebufferlevelhasreachedthebuffercapacity,machineMicansendworkedpiecestothesharedbufferdenotedwithBsthatislocatedinaspecificareaofthesystem,sharedbyallthemachines,whereitispossibletoputpiecesindependentlybytheirprocessstatus.AgenericmachineMiisblockedonlyifbothdedicatedandsharedbuffers,i.e.buffersBiand Automatedflowlineswithsharedbuffer103Bs,arefull.Thepresenceofsharedbufferdecreasesblockingphenomenaintheflowline:ifthededicatedbufferisfull,piecesworkedbymachineMicanbemovedtothesharedbufferuntilthepartflowresumesatmachineMi+1andthelevelofbufferBidecreases.Inmoredetail,apartwhichcannotbestoredinadedicatedbufferstaysinthesharedbufferuntilaplaceinthededicatedbufferbecomesavailable.Thewayinwhichpartsinthesharedbufferarepositioneddependsonthetechnologyusedandthemanagementrulesadopted.Ifthesharedbufferconsistsofasimpleconveyoronwhichpartsflowuntilanewspaceisavailableatdedicatedbuffers,theorderingofpartsdependsonthetheirenteringsequenceintheconveyor.Ifthesharedbufferconsistsofaseriesofracks,theorderingofpartsdependsontheparticularmanagementruleadopted;inthiscaseitisnecessarytohavearesourcelikearobotoracarrierthattakespartsfrommachinesandputthemonracks.TempelmeierandKuhndescribesdifferentmechanismsinthecaseofFlexibleManufacturingSystems(FMS)withcentralbuffer[9].Acertainamountoftimeisnecessaryforphysicallymovingpartsfromaded-icatedbufferareatothesharedbufferareaandviceversa.Therefore,ifblockingdecreaseswiththeintroductionofthesharedbufferthestarvationincreases[9].Intheremainderofthepaperwecallthistimethetraveltimedenotedwithtt.Theprofitabilityofthesystemdependsonthevalueoftraveltimeanditsimpactonsystemperformance.Ifthetraveltimeisreasonablysmall,thenthepenaltytimeincurredforusingthesharedbufferdoesnotdeeplydecreasethesystemperfor-mancesince,aftertheresumptionofflow,thetimespentbypartsforgoingfromthesharedbufferareatothededicatedoneishidden,i.e.coveredbythepiecesalreadypresentinthededicatedareaandprocessedinthemeanwhilebymachineMi+1.Ifthetraveltimeislarge,thenthepenaltytimeincurredforusingthesharedbuffercanstronglydecreasethesystemperformancesincemachinesarefrequentlystarved.Theincreaseofstarvationasaconsequenceofthetransporttimefrom/tothesharedbufferhasbeenpreviouslydescribedbyTempelmeierandKuhn[9]intheanalysisofaspecialFMSconfiguredasaflexibleflowline.Thenextsectionreportsanumericalanalysisforassessingtheproductivityofflowlineswithsharedbufferindifferentsituations.3NumericalevaluationTheobjectiveofthesectionistoevaluatethegainintermsofproductivityduetotheintroductionofsharedbuffersinproductionlines.Todothis,theexperimentationhasbeencarriedoutbysimulatingflowlinesonsimpletestcases,createdadhoctounderstandthesystembehaviorindifferentsituations(Sect.3.1),andonarealflowline(Sect.3.2).3.1TestcasesWeconsideraclosedflowlinecomposedoffivemachines,eachonewithafinitebuffercapacityimmediatelydownstream.Machinesareunreliableandcharacter-izedbythesametypeoffailure.Failuresaretimedependentanddonotdepend 104A.Mattaetal.onprocessingtimesofoperationsatmachines.Failureshavemeantimetofailure(MTTF)andmeantimetorepair(MTTR)exponentiallydistributedwithmeans1000sand100srespectively.TheblockingmechanismistheBAS(BlockingAf-terService)type.Thecycletimeofeachmachineofthesystemisthesameandisdenotedwithtc,i.e.thelineisbalancedbecausemachineshavealsothesameefficiency.ThenumberofpartscirculatinginthesystemismaintainedconstantduringproductionandequaltoP.Forsimplicity,dedicatedbuffershavethesamecapacityNiwithi=1,...,K.5Table1.Testcase:factorlevelsofthe2experimentFactorsLowHighCycletime(tc)5s60sTotalbuffercapacity(NTOT)100125Portionofdedicatedbuffercapacity(α)0.51Traveltime(tt)0s30sNumberofparts(P)7590Thegoaloftheexperimentistoevaluatebymeansofsteadystatesimulationsthesignificancethatpotentialfactorsmayhaveonthemainperformanceindicatorastheproductionrateis.Factorstakenintoconsiderationintheexperimentare:themachinecycletimetc,thetotalbuffercapacityNTOT,theportionofdedicatedbuffercapacityα,thetraveltimettandthenumberofpartsthatcirculateinthesystemP.Thedesignofexperimentsisa25factorialplan;Table1reportsthefac-tors’levelschosenintheexperiment.Theparameterαcanassumevaluesbetween0and1.Noticethatsystemswithα=1correspondtotraditionalflowlinesinwhichthewholebuffercapacityofthesystemisdedicated.Ineachtreatmentofthedesignedfactorialplan15replicationsofsimulationhavebeencarriedoutandstatisticshavebeencollectedafterawarm-upperiodof86400simulatedsecondsand25000finishedpieces.Ineachsimulatedscenariothecapacityofdedicatedbuffersiscalculatedinthefollowingway:NTOT·αNi=,i=1,...,K(1)KwhilethecapacityofthesharedbufferisequaltoNTOT·(1−α).Theanalysisofvariancehasbeenappliedtotestthesignificanceoftheanalyzedfactorsonthesystem’sefficiency,denotedwithEandcalculatedas:XE=(2)X∗whereXistheaverageproductionratecollectedinasimulationrunandX∗isthemaximumproductionratecalculatedwithoutconsideringfailures,blockingandstarvationatmachines.However,sincenormalityassumptionsonresidualsisnotsatisfied,wehavebeenforcedtodividetheanalyzedresponsevaluesintotwodistinctpopulationscorrespondingtolowandhighlevelsofthecycletimefactor. Automatedflowlineswithsharedbuffer105Afterthat,allassumptionsrequiredbytheanalysisofvariancehavebeensatisfiedandthemainresultsarenowpresented.InparticulartheAnderson-DarlingandBartletttestsat95percentconfidencelevelhavebeenusedtotestnormalityandvariancehomogeneityofresidualsrespectively,theindependencewasassuredbytherandomizedexecutionofexperimentsanddifferentseedsusedforgeneratingpseudo-randomnumbers.ResultsfromANOVAarereportedinTables2and3wheresignificantfactorsandinteractionsarerecognizable:asourceissignificantonthesystem’sefficiencyifthep-valueinthelastcolumnislowerthantheBonferroni’salphafamily(chosenequalto0.05)dividedbythenumberofexecutedstatisticaltests(i.e.15inthisexperiment).Asfarasthemaineffectsareconcerned,thenumberofpalletsthatcirculateinthesystem,theportionofdedicatedbuffersandthetotalbuffercapacityaresignificantforboththepopulationswithdifferentcycletimes.Themaineffectofafactoristheaveragechangeintheresponseduetomovingthefactorfromitslowleveltoitshighlevel[6];thisaverageistakenoverallcombinationsofthefactorlevelsinthedesign.Afirstconclusionisthatintheanalyzedsystematraveltimevalueequalto0or30sisnotrelevantforthesystem’sefficiencyduetothevalueschosenforthelevels.Howeverincreasingtoathresholdvalue,greaterthan30s,thetraveltimeleadstobadperformance;wewillseeattheendoftheparagraphthethresholdvaluesafterwhichthetraveltimebecomessignificant.Thefactorαissignificantforbothpopulationsand,furthermore,itispossibletoconclude,bycomparingwiththeTukey’smethodthetwolevels,thattheanalyzedclosedflowlinewithsharedbufferhasstatisticallyanefficiencysuperiorthanthatoftheanalyzedtraditionalflowline.Noticethatthedifferenceofefficiencyinthetwolevelsisaround5%fortc=5sand1%fortc=60s.Thesignificanceofthenumberofpalletsandthetotalbuffercapacityisawellknownresultinliterature[2–4].Asfarastheinteractionseffectsareconcerned,itispossibletostatethattwo-wayinteractionsbetweenP,αandNTOTarestatisticallyrelevant(seealsoFig.3and4).Atwo-wayinteractionissignificantifthecombinedvariationofthetwofactorshasarelevanteffectontheresponse.TheinteractionbetweenPandαshowsthatthesystemperformancedecreaseswhenthenumberofpalletsishighandallbuffersarededicated:inthiscasetheblockingofmachinesisfrequentanditdeeplyaffectsthelineefficiency.Noticethatthesystemwithsharedbufferhasapproximatelythesameefficiencyvalueindependentlybyhowmanypalletcirculateintheline.Onthecontrary,performancedecreasesinthetraditionalsystemwhenthenumberofpalletsisaugmented.TheinteractionbetweenNTOTandαshowsthatsystemperformancedecreaseswhenthetotalbuffercapacityislowandallbuffersarededicated:inthiscasetheblockingofmachinesisfrequentduetothecontemporaryreducedanddedicatedbufferscapacity.TheinteractionbetweenNTOTandPisknowninliterature[3,9]andwedonotcommentmore.ThetripleinteractionamongP,αandNTOTresultstobesignificantonlyforcycletimeequalto60s.Thesamesystemhasbeensimulatedalsoforawidersetofvaluestobetterunderstandtheeffectofthesharedbufferonthesystemperformance.Figure5showstheaveragethroughputfordifferentsharinglevelsofthecentralbufferwhen 106A.Mattaetal.Table2.Testcase:ANOVAresults(tc=5s)SourceDFSeqSSAdjSSAdjMSFPPalletnumber(P)10.0184750.0184750.018475441.970.000Alpha(α)10.1543700.1543700.1543703693.010.000Traveltime(tt)10.0001260.0001260.0001263.010.084Totalbuffercapacity(NTOT)10.0773450.0773450.0773451850.320.000P*α10.0106840.0106840.010684255.590.000P*tt10.0000040.0000040.0000040.090.762P*NTOT10.0174840.0174840.017484418.280.000α*tt10.0000680.0000680.0000681.630.203α*NTOT10.0206730.0206730.020673494.560.000tt*NTOT10.0000630.0000630.0000631.500.222P*α*tt10.0001670.0001670.0001673.990.047P*α*NTOT10.0003580.0003580.0003588.570.004P*tt*NTOT10.0000770.0000770.0000771.850.175α*tt*NTOT10.0000050.0000050.0000050.120.732P*α*tt*NTOT10.0000240.0000240.0000240.580.445Error2240.0093630.0093630.000042Total2390.309285Table3.Testcase:ANOVAresults(tc=60s)SourceDFSeqSSAdjSSAdjMSFPPalletnumber(P)10.00101760.00101760.0010176440.480.000Alpha(α)10.00491180.00491180.00491182126.160.000Traveltime(tt)10.00000650.00000650.00000652.810.095Totalbuffercapacity(NTOT)10.00229270.00229270.0022927992.460.000P*α10.00097860.00097860.0009786423.600.000P*tt10.00000110.00000110.00000110.490.483P*NTOT10.00072920.00072920.0007292315.650.000α*tt10.00000010.00000010.00000010.060.804α*NTOT10.00171080.00171080.0017108740.570.000tt*NTOT10.00000180.00000180.00000180.780.380P*α*tt10.00000140.00000140.00000140.600.438P*α*NTOT10.00029360.00029360.0002936127.100.000P*tt*NTOT10.00000990.00000990.00000994.300.039α*tt*NTOT10.00000090.00000090.00000090.400.525P*α*tt*NTOT10.00000130.00000130.00000130.550.461Error2240.00051750.00051750.0000023Total2390.0124749 Automatedflowlineswithsharedbuffer107Fig.3.Testcase:interactionplot(tc=5s)Fig.4.Testcase:interactionplot(tc=60s)thetraveltimeisequalto5s.Whenthenumberofpalletsissmallthesharedbufferisneverusedandthedifferentsystemshavethesameperformance.Whenthenumberofpalletsincreases,thestarvationdecreasesandthethroughputincreases;howeverasthenumberofpalletsincreasestheblockingoccursmorefrequentlyandthesystemswithsharedbufferperformbetterthanthetraditionalone(i.e.α=1).Inmoredetailhigherthesharingpercentageisandbetteristheperformance.Afteracertainvalueofpalletsinthesystemtheblockingpenalizesthesystemperformance 108A.Mattaetal.Fig.5.Testcase:averageproductionrate(±1.2part/hour)vsPwhenNTOT=50andtt=5sfordifferentvaluesofαFig.6.Testcase:averageproductionrate(±1.2part/hour)vsttwhenNTOT=50andP=30fordifferentvaluesofαandthethroughputdecreases[2,3,9].Thisinversionvalueislargerinsystemswithsharedbufferthanintraditionalsystems.Inparticulartheinversionpointincreasesasthepercentagesharingofbuffersincreases.Thus,inordertoincreasethethroughputthesystem’susercouldmoveaportionofthebuffercapacityfromdedicatedtobufferandcontemporarytoincreasethenumberofpallets.Figure6showstheeffectofthetraveltimeontheaveragethroughput.Thesystemperformancestaysstableforvaluesoftheparametertraveltimeinferiorto Automatedflowlineswithsharedbuffer109Fig.7.Testcase:averageproductionrate(±1.2part/hour)vsttwhenNTOT=50andα=0.75fordifferentvaluesofPathresholdvalueanddeterioratesafterit.Inthisexperimentthethresholdvalueisequalto40s,25sand15sforvaluesofαequalto0.75,0.5and0.25respec-tivelywhenthenumberofpalletsis30.Thethresholdvalueofthetraveltimedecreasesasthepercentageofsharedbufferincreasesbecausetherearelesspalletsinthededicatedbuffersandstarvationoccursmorefrequently.Thiseffectcouldbecompensatedbyincreasingthenumberofpallets.Figure7showstheeffectofthetraveltimeforthesystemwithα=0.5anddifferentvaluesofpallets.Noticethatthelossofproductionafterthethresholdvalueofthetraveltimeislargerforsmallnumberofpallets.Itisworthwhiletonoticethattheresultsreportedinthissectionarevalidforclosedflowlines.Openflowlinesaremoredifficulttomanageduetothelargenumberofpartswhichmayenterfromthefirstmachine.Indeed,ifthereisnolimittothenumberofpartsenteringintothesystemandthefirstmachineisveryefficient,ithappensthatallthepartsjustenteredandprocessedbythefirstmachinefillsthesharedbufferthuslimitingthepossibilitytotheothermachinesofrecurringtothesharedbuffer.Thus,specificrulesformanagingtheenteringofpartsshouldbedesigned.3.2RealcaseInthisparagraphweconsiderarealassemblylinecomposedoffivemachinesseparatedbybufferswithlimitedcapacity.Palletsareemptiesbeforeenteringintothefirstmachine;thencomponentsareloadedonpalletsuntiltheassembledfinalproductisobtainedatthelastmachineofthesystem.Thecomponentsarestoredoncontainerslocatedateachmachineandarenotmodelledascustomers,thusthesystemcanbeviewasaflowlinecrossedbyparts(i.e.thepallets)thatvisitmachinesinafixedsequence.Thenumberofpalletsinthesystemremainsconstant 110A.Mattaetal.Fig.8.Realcase:lay-outoftherealsystemFig.9.Realcase:lay-outofalternative1duringtheproduction.Machinesareunreliableandcharacterizedbydifferenttypesoffailures.InparticularmachinesM1,M2andM5canfailinthreedifferentwayswhileM3andM4inonlyoneway.MTTFandMTTRforeachfailuretypeareexponentiallydistributedwithvaluesasreportedinTable4andcalculatedbythefirm.Asintherealsystem,amachinecanfailonlywhenitisoccupiedbyapart.Thefirstfailuretypeofeachmachinemodelsmechanicalandelectronicfailuresinanaggregatedway,andthesecondandthirdfailuretypesofM1,M2andM5modeltheemptyingofcomponentcontainers.Table4reportsthefailureparametersandthedeterministicprocessingratesofmachines.TheBAScontrolruleisconsidered,thatismachinesmayenterinablockingstateonlyafterthecompletionoftheprocess.Aphysicalconstraintinthelay-outdoesnotallowchangesintheportionofthesystembetweenM5andM1,i.e.thebufferB5isdedicatedandcannotbemodifiedinitscapacity.Thelay-outofthesystemisshowninFigure8.Therealsystemalreadyusesinthetraditionalwayflexibletransportmodulesformovingpartsthroughthelineataconstantspeedof17.6m/min.Amongalargesetoffeasiblesolutions,twoalternativereasonablesystemswithsharedbufferareconsideredinthecomparisonwiththerealone.Thefirstalternativehasasharedbuffer,locatedatthecenteroftheline,inadditiontothededicatedbuffersoftherealsystem.Theincreaseoftotalbuffercapacityisaround31%correspondingtoanincreaseofapproximately44kEuroofthetotalinvestmentcost(thisvaluehasbeenestimatedonthebasisofadditionalconveyors,sensors,enginesandcontrolsystem).Thesecondalternativehasbeendesigned Automatedflowlineswithsharedbuffer111Fig.10.Realcase:lay-outofalternative2Fig.11.Input/outputinto/fronmthesharedbufferwithinvestmentcostequaltothatoftherealline;thetotalbuffercapacityislowerthanthatofthereallinebecauseoftheadditionalcostsofsensorsandengines.TotalbuffercapacityisreportedinTable5whilethelay-outsofalternativeswithsharedbufferareshowninFigures9and10.Intheproposedalternativeseachmachineisblockedonlyifitsdedicatedbufferandthecommonbufferarefull.Themechanismofinput/outputinto/fromthesharedbufferisnowdescribedreferringtoFigure11.LetusconsidertheportionofthesystembetweenmachinesM1andM2.BeforethemachineM1releasesaprocessedpart,thesystemcontrolstheavailabilityofspaceintheportionofconveyorbetweenM1andM2,i.e.inthededicatedbufferwithsizeN1.IfthereisspaceinthededicatedbufferthemachinereleasesthepartwhichthenmovestowardsmachineM2,otherwisethesystemcontrolsifthepartcanbeintroducedinthesharedbuffer.Ifthereisspaceavailableinthesharedbufferthemachinereleasesthepartthatentersinthesharedbuffer,otherwisethemachineisblockeduntilanewspacebecomesavailableinthededicatedbufferorsharedbuffer.ThisreactionTable4.Realcase:processingrates[part/min]andMTTFsandMTTRs[min]ofmachinesMachineProcessingMTTFMTTRMTTFMTTRMTTFMTTRnumberratetype1type1type2type2type3type3120.05.640.81499.174.00143.997.16217.32.901.0894.785.1669.235.16316.55.610.57––––415.721.280.51––––516.010.600.63274.435.1629.935.00 112A.Mattaetal.Table5.Realcase:buffercapacitiesofrealsystemandalternativeswithsharedbufferSystemN1N2N3N4N5NSharedTotalDedicatedSharedReal1106610770830436100%0%Alternative11104392528319257261%39%Alternative2442429378314936647%53%Table6.Realcase:comparisonbetweenrealsystemandalternativeswithsharedbufferSystemMaxaverageproductionInvestmentAverageproductivityrate[part/h]cost[kEuro]indexReal650.9±3.622500.289Alternative1677.7±3.922940.295Alternative2667.3±3.122500.297ofthemachinehasbeencalledas”block-and-recirculate”strategybyTempelmeierandKuhnintheirbook[9].ThepointdenotedwithAinFigure11isthetransferpointatwhichpartscanchangeconveyor.Thetransferpointisbi-directional,thatisapartisswitchedfromtheoutputconveyorofthemachinetothesharedconveyorandviceversa.Switchingdevicesareavailableinthemarketataffordablecostsandallowthemachinetoavoidtheblockingstate.AnexampleofswitchingmechanismisshowninFigure12.Whenanewspacebecomesavailableinthesharedbufferacontrolrulemustbedefinedtodecidewhichpart,ifany,willaccesstothecommonarea.Intheproposedsystemstheprecedenceisgiventothemachinethathasjustmadefreetheplaceinthesharedbuffer.Eachtimeapartmustleavethesharedbufferitisnecessarythatthepartreachesthetransferpoint.Ifthesharedbufferislargethetimetoreachthetransferpointcanbesohighthatthededicatedbufferemptiesandstarvationthusoccurs.Inordertodecreasethistime,whichisaportionoftheabovedefinedtraveltime,twoinneralternativepathshavebeenintroducedinthefirstalternative(seeFig.9).Inthesecondalternativemachinesarecloserandthetraveltimeisnotcritical.Thetransfertimeinwhichthepartleaveschangestheconveyorfromthesharedtothededicatedbuffertakesfewseconds.Theperformanceofsystemshasbeencalculatedbyterminatingsimulationsofoneproductiondaybecauseattheendoftheshiftthesystemisalwaysemptied.Thesimulationmodelhasbeenvalidatedontherealproductionratesofsevendays.Statisticshavebeencollectedandconfidenceintervalsonproductionratehavebeencalculatedat95%confidencelevel.Figure13showstheaverageproductionratefortheanalyzedsystemswhichdependsonthenumbersofcustomersthatcirculateinthesystem[2,3,9].AsshowninTable6,boththeproposedalternativesystemshaveproductivityindex(calculatedasaverageproductionrateoverinvestmentcost)greaterthanthatoftherealsystem.Inparticular,atequalinvestmentcost,thesecondalternativehasanaverageproductionrategreater2.5%thanthatintherealcase.Figures13–18reportthedetailedstatesofmachinesfortherealsystemand Automatedflowlineswithsharedbuffer113Fig.12.SwitchingmechanismFig.13.Realcase:averageproductionratevsnumberofpallets(±4part/h)thealternative2.Itcanbenoticedhowtheblockingofmachinesdecreasesfromtherealsystemtothesystemwiththesharedbufferandstarvationincreasesduetothefactthatpartscirculateinthesharedbuffer.Howeverthedisadvantagesduetotheincreaseofstarvationdonotcompensatetheadvantagesduetothereductionofblockinginthesystem.AlongterminvestmentanalysismustconsidertheNetPresentValue(NPV)relatedtotheinvestment,i.e.thesumofalldiscountedcashflowsduringthelifeofthesystem.TheNPVconsiderstheinitialinvestmentcost(fundamentallyma-chines,buffers),thefuturediscountedcashflowsduringthenormalrunningofthesystem(revenuesandproductioncosts)andtheresidualvalueofthesystemaftertheplanninghorizonoftheinvestment.Intheinvestmentanalysisonlythediscrim-inatingvoiceshavetobeconsidered.Inthiscasethethreealternativesystemsdifferinthebuffercapacitysincemachinesremainthesame.Thustheinvestmentcostofmachinesisnotdifferentialandisnotconsidered.Theinvestmentcostofbuffersisdifferentonlyforthealternative1.Revenuesarediscriminantiftheadditionalproductioncapacityoftheproposedalternativesisconvertedinadditionalsales. 114A.Mattaetal.Fig.14.Realcase:Machine1statesvsnumberofpalletsforrealsystemandalternative2Areasonableassumptionisthattheunitproductioncostdoesnotdifferbecausetheprocesstechnologyisthesameforeachalternative.Again,iftheadditionalcapacityisexploitedinnewsalesthetotalvariableproductioncostsareadiffer-entialitem.Theresidualvalueofaproductionsystemattheendoftheplanninghorizonisverydifficulttoestimate.However,thedifferencebetweentheinitialinvestmentwiththerealsystemislimitedto44kEuroforthefirstalternativeandnullforthesecondoneandwecanimaginethatthedifferencebetweentheresidualvaluewillbesmaller,sowecanneglecttheresidualvaluevoiceintheanalysis.Thesecondalternativehasthesameinvestmentcostoftherealsystemwithahigherproductionrate,andinthiscasetheNPVanalysisisnotnecessarybecausetherealsystemisdominatedbythealternative.Forthefirstalternativewehavetoimpose Automatedflowlineswithsharedbuffer115Fig.15.Realcase:Machine2statesvsnumberofpalletsforrealsystemandalternative2someassumptionstocompensatethelackofinformationonrevenuesandcosts.Assumingayearlydiscountrateof0.1,aplanninghorizonof5years,8hoursofproductionperday,andthatalltheadditionalcapacityissoldinthemarket,themarginalgain(i.e.thedifferencebetweenpriceandvariablecostoftheproduct)theproductmusthavetocompensatetheadditionalinvestmentofthefirstalternativeisequalto0.17euro/part,correspondingapproximatelyto0.6%ofthemarketpriceoftheproduct.Wethinkthatthemarginalgainontheproductismuchlargerthanthecalculatedthresholdvalueandthereforetheproposedalternative1seemstobeprofitable.Obviously,iftheadditionalcapacitywillnotbeexploitedtherealsystemisclearlymoreprofitablethanthefirstalternative,howeverinthissituationthesecondalternativedominatestheothers. 116A.Mattaetal.Fig.16.Realcase:Machine3statesvsnumberofpalletsforrealsystemandalternative24PracticalconsiderationsInordertointroducemanufacturingflowlineswithsharedbufferinrealshopfloorsseveralaspects,bothtechnologicalandeconomic,havetobeclarified.Firstofall,anecessaryconditionforthecommonbufferexploitationisthat,inordertobeabletodispatchpartstothedifferentmachines,partsmustbetrackedduringtheirmovementsinthesystem.Todothis,severaltechnologiesareavailableatlowcosts.Lasersmarkerscansculpturecodes,easilyreadablebyopticaldevices,onmetalcomponentsinafastandcheapway.Standarddeviceslikechipscansaveinforma-tionandexchangeitwiththesystemsupervisor.Alsoradiofrequencytechnologyisnowreadytobeusedinshopfloorstoexchangeinformationwithoutthelimiting Automatedflowlineswithsharedbuffer117Fig.17.Realcase:Machine4statesvsnumberofpalletsforrealsystemandalternative2constraintofdesigningcontrolpointsinthesystem.Therefore,traceabilityofpartsinthesystemdoesnotseemtobeanobstacleinfutureapplications.Conveyorsseemtobeagoodandconsolidatedsolutiontomovepartsfrommachinestothecommonbufferandviceversa.However,otherdevicesshouldbeinvestigatedsuchasrobotmanipulatorsthatcanmovepartsthroughthesystem.Themainadvantageofmanipulatorsistheirflexibilitysincetheycanbeadaptedtodifferentsituations(e.g.adaptationtoreacttochangesinthelay-outofthesystem)bysimplyre-programmingthem.Themaindrawbackofmanipulatorsisrelatedtotheirinvestmentcostandtheskillsneededtoinstructthem.ShuttlesandAGV(AutomatedGuidedVehicle)representsatraditionalsolutiontopartmovementin 118A.Mattaetal.Fig.18.Realcase:Machine5statesvsnumberofpalletsforrealsystemandalternative2FlexibleManufacturingSystems,whichrepresentthefirstcaseofsharedbufferinautomatedmanufacturingsystems.Anotherimportantaspectthatisnormallytakenintoconsiderationinthedesignphaseofaflowlineisthefloorspaceoccupiedbythesystem.Theoretically,itisnecessaryanadditionalspacetolocatethecommonbufferinamanufacturingflowline.However,itisalsotruethatthespaceoccupiedbydedicatedbuffersdecreasesandconsequentlythemachinesofthelinearecloser.Therefore,itisnotpossibleaprioritosayanythingabouttheeffectsofthesharedbufferontheoccupiedfloorspacesincethisaspectiscloselyconnectedtothelay-outoftherealsystem. Automatedflowlineswithsharedbuffer1195ConclusionsandfuturedevelopmentsThepaperaddressestheproblemoffullyusingbufferspacesinflowlines.Theideaistoexploitrecenttechnologicaldevicestomoveinreasonabletimepiecesfromamachinetoacommonbufferareaofthesystemandviceversa.Insuchawaymachinescanavoidtheirblockingmovingpiecestothesharedbufferarea.Thedecreaseofblockinginflowlineshasapositiveimpactontheirproductionrate.Thenumericalanalysisreportedinthepaperdemonstratesthevalidityoftheideapointingoutalsothefactorsthataffecttheimprovementoftheproposedsystemarchitectureintermsofproductivity.Inconclusion,severalpracticalaspectshavetobeinvestigatedbeforetostatethatsharedbufferscanbesuccessfullyadoptedinrealmanufacturingflowlines,howeverthefirstresultsshowninthispaperandthetechnologiesnowavailablemotivatefurtherresearchinthisdirection.Ongoingresearchisdedicatedtoidentifythepotentialsectorsforpracticalapplicationsofthenewconceptsproposedinthispaper.Then,furtherresearchwillfocusonnewkey-issuesneveraddressedinliteratureandintroducedbythearchitecturewiththesharedbuffer:–Allocationofdedicatedandsharedbuffers.Traditionallyonlycapacitiesofdedicatedbuffershavebeenconsideredinthedesignphaseofmanufacturingflowlines.Inouropinionthebufferallocationprobleminthecaseofsharedbufferwillbeeasierincomparisonthanthetraditionalonebecausethenewsystemarchitectureismorerobust,i.e.thesystemperformanceisstableinseveralconditionsanddonotdecayaftersomechangesinthedesignoftheline.–Performanceevaluationofflowlineswithsharedbuffer.Newanalyticalmeth-odsarenecessarytoestimateperformanceofnewsystemarchitectures.ThemethodofTempelmeieretal.[10],originallydevelopedtoevaluatetheper-formanceofFlexibleManufacturingSystemswithblocking,couldbeadoptedalsoforflowlineswithsharedbuffer.Thismethodisbeingtestedintermsofaccuratenessofprovidedresults.–Managementofflowlineswithsharedbuffer.Newdispatchingrulescouldbenecessarytoavoiddeadlockinthenewsystemarchitecturewhenpiecesconvergetothesameareacomingfromdifferentpositions.References1.DalleryY,GershwinSB(1992)Manufacturingflowlinesystems:areviewofmodelsandanalyticalresults.QueueingSystems12:3–942.DalleryY,TowsleyD(1991)Symmetrypropertyofthethroughputinclosedtandemqueueingnetworkswithfinitecapacity.OperationsResearch10(9):541–5473.FreinY,CommaultC,DalleryY(1996)Modelingandanalysisofclosed-loopproduc-tionlineswithunreliablemachinesandfinitebuffers.IIETransactions28:545–5544.GershwinSB(1994)Manufacturingsystemsengineering.PTRPrenticeHall,NewJersey5.GershwinSB,SchorJE(2000)Efficientalgorithmsforbufferspaceallocation.AnnalsofOperationsResearch93:91–116 120A.Mattaetal.6.LawAM,KeltonWD(2000)Simulationmodellingandanalysis.McGraw–Hill,NewYork7.ShantikumarJG,YaoDD(1989)Queueingnetworkswithfinitebuffers.In:PerrosHG,AltiokT(eds)chapterMonotonicityandconcavitypropertiesincyclicqueueingnetworkswithfinitebuffers,pp.325–344.NorthHolland,Amsterdam8.TempelmeierH(2003)Practicalconsiderationsintheoptimizationofflowproductionsystems.InternationalJournalofProductionResearch41(1):149–1709.TempelmeierH,KuhnH(1993)Flexiblemanufacturingsystems–Decisionsupportfordesignandoperation.Wiley,NewYork10.TempelmeierH,KuhnH,TetzlaffU(1989)Performanceevaluationofflexiblemanu-facturingsystemswithblocking.InternationalJournalofProductionResearch,27(11):1963–1979 Integratedqualityandquantitymodelingofaproductionline
JongyoonKimandStanleyB.GershwinDepartmentofMechanicalEngineering,MassachusettsInstituteofTechnology,Cambridge,MA02139-4307,USA(e-mail:gershwin@mit.edu)Abstract.Duringthepastthreedecades,thesuccessoftheToyotaProductionSys-temhasspurredmuchresearchinmanufacturingsystemsengineering.Productivityandqualityhavebeenextensivelystudied,butthereislittleresearchintheirinter-section.Thegoalofthispaperistoanalyzehowproductionsystemdesign,quality,andproductivityareinter-relatedinsmallproductionsystems.WedevelopanewMarkovprocessmodelformachineswithbothqualityandoperationalfailures,andweidentifyimportantdifferencesbetweentypesofqualityfailures.Wealsodevelopmodelsfortwo-machinesystems,withinfinitebuffers,buffersofsizezero,andfinitebuffers.Wecalculatetotalproductionrate,effectiveproductionrate(ie,theproductionrateofgoodparts),andyield.Numericalstudiesusingthesemodelsshowthatwhenthefirstmachinehasqualityfailuresandtheinspectionoccursonlyatthesecondmachine,therearecasesinwhichtheeffectiveproductionratein-creasesasbuffersizesincrease,andtherearecasesinwhichtheeffectiveproductionratedecreasesforlargerbuffers.Weproposeextensionstolargersystems.Keywords:Quality,Productivity,Manufacturingsystemdesign1Introduction1.1MotivationDuringthepastthreedecades,thesuccessoftheToyotaProductionSystemhasspurredmuchresearchinmanufacturingsystemsdesign.Numerousresearchpa-pershavetriedtoexploretherelationshipbetweenproductionsystemdesignand
WearegratefulforsupportfromtheSingapore-MITAlliance,theGeneralMotorsRe-searchandDevelopmentCenter,andPSAPeugeot-Citroen.¨Correspondenceto:S.B.Gershwin 122J.KimandS.B.Gershwinproductivity,sothattheycanshowwaystodesignfactoriestoproducemoreprod-uctsontimewithlessresources(suchaspeople,material,andspace).Ontheotherhand,topicsinqualityresearchhavecapturedtheattentionofpractitionersandre-searcherssincetheearly1980s.TherecentpopularityofStatisticalQualityControl(SQC),TotalQualityManagement(TQM),andSixSigmahavedemonstratedtheimportanceofquality.Thesetwofields,productivityandquality,havebeenextensivelystudiedandreportedseparatelybothinthemanufacturingsystemsresearchliteratureandthepractitionerliterature,butthereislittleresearchintheirintersection.TheneedforsuchworkwasrecentlydescribedbyauthorsfromtheGMCorporationbasedontheirexperience[13].Allmanufacturersmustsatisfythesetworequirements(highproductivityandhighquality)atthesametimetomaintaintheircompetitiveness.ToyotaProductionSystemadvocatesadmonishfactorydesignerstocombineinspectionswithoperations.IntheToyotaProductionSystem,themachinesaredesignedtodetectabnormalitiesandtostopautomaticallywhenevertheyoccur.Also,operatorsareequippedwithmeansofstoppingtheproductionflowwhenevertheynoteanythingsuspicious.(Theycallthispracticejidoka.)ToyotaProductionSystemadvocatesarguethatmechanicalandhumanjidokapreventthewastethatwouldresultfromproducingaseriesofdefectiveitems.Thereforejidokaisameanstoimprovequalityandincreaseproductivityatthesametime[23],[24].Butthisstatementisarguable:qualityfailuresareoftenthoseinwhichthequalityofeachpartisindependentoftheothers.Thisisthecasewhenthedefecttakesplaceduetocommon(orchanceorrandom)causesofvariations[16].Inthiscase,thereisnoreasontostopamachinethathasmadeabadpartbecausethereisnoreasontobelievethatstoppingitwillreducethenumberofbadpartsinthefuture.Inthiscase,therefore,stoppingtheoperationdoesnotinfluencequalitybutitdoesreduceproductivity.Ontheotherhand,whenqualityfailuresarethoseinwhichonceabadpartisproduced,allsubsequentpartswillbebaduntilthemachineisrepaired(duetospecialorassignableorsystematiccausesofvariations)[16],catchingbadpartsandstoppingthemachineassoonaspossibleisthebestwaytomaintainhighqualityandproductivity.Non-stockorleanproductionisanotherpopularbuzzwordinmanufacturingsystemsengineering.Someleanmanufacturingprofessionalsadvocatereducinginventoryonthefactoryfloorsincethereductionofwork-in-process(WIP)revealstheproblemsintheproductionlines[3].Thus,itcanhelpimproveproductquality.Itistrueinsomesense:lessinventoryreducesthetimebetweenmakingadefectandidentifyingthedefect.Butitisalsotruethatproductivitywoulddiminishsignificantlywithoutstock[5].Sincethereisatradeoff,theremustbeoptimalstocklevelsthatarespecifictoeachmanufacturingenvironment.Infact,Toyotarecentlychangedtheirviewoninventoryandaretryingtore-adjusttheirinventorylevels[9].Whatismissingindiscussionsoffactorydesign,quality,andproductivityisaquantitativemodeltoshowhowtheyareinter-related.Mostoftheargumentsaboutthisarebasedonanecdotalevidenceorqualitativereasoningthatlackasoundsci-entificquantitativefoundation.Theresearchdescribedheretriestoestablishsuchafoundationtoinvestigatehowproductionsystemdesignandoperationinfluence Integratedqualityandquantitymodelingofaproductionline123productivityandproductqualitybydevelopingconceptualandcomputationalmod-elsoftwo-machine-one-buffersystemsandperformingnumericalexperiments.1.2Background1.2.1Qualitymodels.Therearetwoextremekindsofqualityfailuresbasedonthecharacteristicsofvariationsthatcausethefailures.Inthequalityliterature,thesevariationsarecalledcommon(orchanceorrandom)causevariationsandassignable(orspecialorunusual)causevariations[18].Figure1showsthetypesofqualityfailuresandvariations.Commoncausefailuresarethoseinwhichthequalityofeachpartisindependentoftheothers.Suchfailuresoccuroftenwhenanoperationissensitivetoexternalperturbationslikedefectsinrawmaterialorwhentheoperationusesanewtechnologythatisdifficulttocontrol.Thisisinherentinthedesignoftheprocess.SuchfailurescanberepresentedbyindependentBernoullirandomvariables,inwhichabinaryrandomvariable,whichindicateswhetherornotthepartisgood,ischoseneachtimeapartisoperatedon.Agoodpartisproducedwithprobabilityπ,andabadpartisproducedwithprobability1−π.Theoccurrenceofabadpartimpliesnothingaboutthequalityoffutureparts,sonopermanentchangescanhaveoccurredinthemachine.Forthesakeofclarity,wecallthisaBernoulli-typequalityfailure.Mostofthequantitativeliteratureoninspectionallocationassumesthiskindofqualityfailure[21].Inthiscase,ifbadpartsaredestinedtobescrapped,itisusefultocatchthemassoonaspossiblebecausethelongerbeforetheyarescrapped,themoretheyconsumethecapacityofdownstreammachines.However,thereisnoreasontostopamachinethathasproducedabadpartduetothiskindoffailure.Thequalityfailuresduetoassignablecausevariationsarethoseinwhichaqualityfailureonlyhappensafterachangeoccursinthemachine.Inthatcase,itisverylikelythatonceabadpartisproduced,allsubsequentpartswillbebaduntilthemachineisrepaired.Here,thereismuchmoreincentivetocatchdefectivepartsandstopthemachinequickly.Inadditiontominimizingthewasteofdownstreamcapacity,thisstrategyminimizesthefurtherproductionofdefectiveparts.Forthiskindofqualityfailure,thereisnoinherentmeasureofyieldbecausethefractionsPersistent-typequalityfailureBernoulli-typequalityfailureRepairtakesplaceUpperSpecificationLimitMeanLowerRandomVariationSpecificationLimitAssignableVariation(toolbreakage)takesplaceFig.1.Typesofqualityfailures 124J.KimandS.B.Gershwinofpartsthataregoodandbaddependonhowsoonbadpartsaredetectedandhowquicklythemachineisstoppedforrepair.Inthispaper,wecallthisapersistent-typequalityfailure.MostquantitativestudiesinStatisticalQualityControl(SQC)arededicatedtofindingefficientinspectionpolicies(samplinginterval,samplesize,controllimits,andothers)todetectthistypeofqualityfailure[26].Inreality,failuresaremixturesofBernoulli-typequalityfailuresandpersistent-typequalityfailures.ItcanbearguedthatthequalitystrategyoftheToyotaProduc-tionSystem[17],inwhichmachinesarestoppedassoonasabadpartisdetected,isimplicitlybasedontheassumptionofthepersistent-typequalityfailure.Inthispaper,wefocusonpersistentfailures.1.2.2Systemyield.Systemyieldisdefinedhereasthefractionofinputtoasys-temthatistransformedintooutputofacceptablequality.Thisisanimportantmetricbecausecustomersobservethequalityofproductsonlyafterallthemanufacturingprocessesaredoneandtheproductsareshipped.Thesystemyieldisacomplexfunctionofhowthefactoryisdesignedandoperated,aswellasofthecharacteristicsofthemachines.Someofinfluencingfactorsincludeindividualoperationyields,inspectionstrategies,operationpolicies,buffersizes,andotherfactors.Compre-hensiveapproachesareneededtomanagesystemyieldeffectively.Thisresearchaimstodevelopmathematicalmodelstoshowhowthesystemyieldisinfluencedbythesefactors.1.2.3Qualityimprovementpolicy.Systemyieldisacomplexfunctionofvariousfactorssuchasinspection,individualoperationyields,buffersize,operationpoli-cies,andothers.Therearemanywaystoaffectthesystemyield.Inspectionpolicyhasreceivedthemostattentionintheliterature.Researchoninspectionpoliciescanbedividedintooptimizinginspectionparametersatasinglestationandtheinspectionstationallocationproblem.Theformerissuehasbeeninvestigatedex-tensivelyintheSQCliterature[26].Here,optimalSQCparameterssuchascontrollimits,samplingsize,andfrequencyaresoughtforanoptimalbalancebetweentheinspectioncostandthecostofquality.Thelatterresearchlooksfortheoptimaldistributionsofinspectionstationsalongproductionlines[21].Improvingindividualoperationyieldisanotherimportantwaytoincreasethesystemyield.Manystudiesinthisfieldtrytostabilizetheprocesseitherbyfindingrootcausesofvariationandeliminatingthemorbymakingtheprocessinsensitivetoexternalnoise.TheformertopichasnumerousqualitativeresearchpapersinthefieldsofTotalQualityManagement(TQM)[2]andSixSigma[19].Quantitativeresearchismoreorientedtowardthelattertopic.Robustengineering[20]isanareathathasgainedsubstantialattention.Ithasbeenarguedthatinventoryreductionisaneffectivemeanstoimprovesystemyield.Manyleanmanufacturingspecialistshaveassertedthatlessinventoryonthefactoryfloorrevealsproblemsinthemanufacturinglinesmorequicklyandhelpsqualityimprovementactivities[1,17].Therealsohavebeeninvestigationstoexplaintherelationshipbetweenplantlayoutdesignandquality[7].TheyarguethatU-shapedlinesarebetterthanstraightlinesforproducinghigherqualityproductssincetherearemorepointsofcontactbetweenoperators.Thereisalsolessmaterialmovement,andthereareotherrea-sons. Integratedqualityandquantitymodelingofaproductionline125Therearemanywaystoimprovesystemyield,butusingonlyasinglemethodwillgivelimitedgains.Theeffectivenessofeachmethodisgreatlydependentonthedetailsofthefactory.Thus,thereisneedtodeterminewhichmethodorwhichcombinationofmethodsismosteffectiveineachcase.Thequantitativetoolsthatwillbedevelopedfromthisresearchcanhelpfulfillthisneed.1.3OutlineInSection2weintroducethestructureofthemodelingtechniquesusedinthispaper.Wepresentmodeling,solutiontechniques,andvalidationofthe2-machine-1-finitebuffercaseinSection3.DiscussionsonthebehaviorofaproductionlinebasedonnumericalexperimentsareprovidedinSection5.AfutureresearchplanisshowninSection6.Parametersofmanyofthesystemsstudiednumericallyhere,anddetailsoftheanalyticalsolutionofthetwo-machineline,canbefoundintheappendices.2Mathematicalmodels2.1SinglemachinemodelTherearemanypossiblewaystocharacterizeamachineforthepurposeofsimul-taneouslystudyingqualityandquantityissues.Here,wemodelamachineasadiscretestate,continuoustimeMarkovprocess.Materialisassumedcontinuous,andµiisthespeedatwhichMachineiprocessesmaterialwhileitisoperatingandnotconstrainedbytheothermachineorthebuffer.Itisaconstant,inthatµidoesnotdependontherepairstateoftheothermachineorthebufferlevel.Figure2showstheproposedstatetransitionsofasinglemachinewithpersistent-typequalityfailures.Inthemodel,themachinehasthreestates:∗State1:Themachineisoperatingandproducinggoodparts.∗State-1:Themachineisoperatingandproducingbadparts,buttheoperatordoesnotknowthisyet.∗State0:Themachineisnotoperating.Themachinethereforehastwodifferentfailuremodes(i.e.transitiontofailurestatesfromstate1):∗Operationalfailure:transitionfromstate1tostate0.Themachinestopspro-ducingpartsduetofailureslikemotorburnout.∗Qualityfailure:transitionfromstate1tostate-1.Themachinestopsproducinggoodparts(andstartsproducingbadparts)duetoafailurelikeasuddentooldamage.Whenamachineisinstate1,itcanfailduetoanon-qualityrelatedevent.Itgoestostate0withtransitionprobabilityratep.Afterthatanoperatorfixesit,andthemachinegoesbacktostate1withtransitionrater.Sometimes,duetoanassignablecause,themachinebeginstoproducebadparts,sothereisatransition 126J.KimandS.B.GershwinpgfState1State-1State0rFig.2.Statesofamachinefromstate1tostate-1withaprobabilityrateg.HeregisthereciprocaloftheMeanTimetoQualityFailure(MTQF).AmorestableoperationleadstoalargerMTQFandasmallerg.Themachine,whenitisinstate-1,canbestoppedfortworeasons:itmayexperiencethesamekindofoperationalfailureasitdoeswhenitisinstate1;andtheoperatormaystopitforrepairwhenhelearnsthatitisproducingbadparts.Thetransitionfromstate-1tostate0occursatprobabilityratef=p+hwherehisthereciprocaloftheMeanTimeToDetect(MTTD).AmorereliableinspectionleadstoashorterMTTDandalargerf.(Thedetectioncantakeplaceelsewhere,forexampleataremoteinspectionstation.)Notethatthisimpliesthatf>p.Here,forsimplicity,weassumethatwheneveramachineisrepaired,itgoesbacktostate1.Alltheindicatedtransitionsareassumedtofollowexponentialdistributions.Singlemachineanalysis.Todeterminetheproductionrateofasinglemachine,wefirstdeterminethesteady-stateprobabilitydistribution.Thisiscalculatedbasedontheprobabilitybalanceprinciple:theprobabilityofleavingastateisthesameastheprobabilityofenteringthatstate.Wehave(g+p)P(1)=rP(0)(1)fP(−1)=gP(1)(2)rP(0)=pP(1)+fP(−1)(3)Theprobabilitiesmustalsosatisfythenormalizationequation:P(0)+P(1)+P(−1)=1(4)Thesolutionof(1)–(4)is1P(1)=(5)1+(p+g)/r+g/f(p+g)/rP(0)=(6)1+(p+g)/r+g/fg/fP(−1)=(7)1+(p+g)/r+g/f Integratedqualityandquantitymodelingofaproductionline127Thetotalproductionrate,includinggoodandbadparts,is1+g/fPT=µ(P(1)+P(−1))=µ(8)1+(p+g)/r+g/fTheeffectiveproductionrate,theproductionrateofgoodpartsonly,is1PE=µP(1)=µ(9)1+(p+g)/r+g/fTheyieldisPEP(1)f==(10)PTP(1)+P(−1)f+g2.22-machine-1-buffercontinuousmodelAflow(ortransfer)lineisamanufacturingsystemwithaveryspecialstructure.Itisalinearnetworkofservicestationsormachines(M1,M2,...,Mk)separatedbybufferstorages(B1,B2,...,Bk−1).MaterialflowsfromoutsidethesystemtoM1,thentoB1,thentoM2,andsoforthuntilitreachesMk,afterwhichitleaves.Figure3depictsaflowline.Therectanglesrepresentmachinesandthecirclesrepresentbuffers.MBMBMBMBM112233445Fig.3.Five-machineflowline2-machine-1-buffer(2M1B)modelsshouldbestudiedfirst.Thenadecompo-sitiontechnique,thatdividesalongtransferlineintomultiple2-machine-1-buffermodels,couldbedeveloped.(See[14].)Amongthevariousmodelingtechniquesforthe2M1Bcase,includingdeterministic,exponential,andcontinuousmodels,thecontinuousmateriallinemodelisusedforthisresearchbecauseitcanhandledeterministicbutdifferentoperationtimesateachoperation.Thisisanextensionofthecontinuousmaterialseriallinemodelingof[10]byaddinganothermachinefailurestate.Figure4showsthe2M1Bcontinuousmodelwherethemachines,bufferanddiscretepartsarerepresentedasvalves,atank,andacontinuousfluid.M1BMMBM212Fig.4.Two-machine-one-buffercontinuousmodelWeassumethataninexhaustiblesupplyofworkpiecesisavailableupstreamofthefirstmachineintheline,andanunlimitedstorageareaispresentdownstream 128J.KimandS.B.Gershwinofthelastmachine.Thus,thefirstmachineisneverstarved,andthelastmachineisneverblocked.Also,failuresareassumedtobeoperationdependent(ODF).Finally,weassumethateachmachineworksonadifferentfeature.Forexample,thetwomachinesmaybemakingtwodifferentholes.Wedonotconsidercaseswherethebothmachinesworkonthesamehole,inwhichthefirstmachinedoesaroughingoperationandtheseconddoesafinishingoperation.Thisallowsustoassumethatthefailuresofthetwomachinesareindependent.2.3InfinitebuffercaseAninfinitebuffercaseisaspecial2M1Blineinwhichthesizeofthebuffer(B)isinfinite.Thisisanextremecaseinwhichthefirstmachine(M1)neversuffersfromblockage.Toderiveexpressionsforthetotalproductionrateandtheeffectiveproductionrate,weobservethatwhenthereisinfinitebuffercapacitybetweentwomachines(M1,M2),thetotalproductionrateofthe2M1BsystemisaminimumofthetotalproductionratesofM1andM2.Thetotalproductionrateofmachineiisgivenby(8),sothetotalproductionrateofthe2M1Bsystemis!"∞µ1(1+g1/f1)µ2(1+g2/f2)PT=min,(11)1+(p1+g1)/r1+g1/f11+(p2+g2)/r2+g2/f2TheprobabilitythatmachineMidoesnotaddnon-conformitiesisPi(1)fiYi==(12)Pi(1)+Pi(−1)fi+giSincethereisnoscrapandreworkinthesystem,thesystemyieldbecomesf1f2(13)(f1+g1)(f2+g2)Asaresult,theeffectiveproductionrateis∞f1f2∞PE=PT(14)(f1+g1)(f2+g2)Theeffectiveproductionrateevaluatedfrom(14)hasbeencomparedwithadiscrete-event,discrete-partsimulation.Table1showsgoodagreement.Thepa-rametersforthesecasesareshowninAppendixB.AsindicatedinSection2.1,thedetectionofqualityfailuresduetomachineM1neednotoccuratthatmachine.Forexample,theinspectionofthefeaturethatM1worksoncouldtakeplaceataninspectionstationatM2,andthisinspectioncouldtriggerarepairofM1.(Wecallthisqualityinformationfeedback.SeeSection4.)Inthatcase,theMTTDofM1(andthereforef1)willbeafunctionoftheamountofmaterialinthebuffer.WereturntothisimportantcaseinSection4.2.4ZerobuffercaseThezerobuffercaseisoneinwhichthereisnobufferspacebetweenthemachines.Thisistheotherextremecasewhereblockageandstarvationtakeplacemostfrequently. Integratedqualityandquantitymodelingofaproductionline129Table1.Validationofinfinitebuffercase∞∞Case#PE(Analytic)PE(Simulation)%Difference10.7620.7610.1720.7080.7080.0030.6570.6570.0040.5770.580−0.5050.5270.530−0.4260.7450.7450.0170.7620.7600.3081.5241.5220.1490.7620.7620.00101.5241.526−0.13Inthezero-buffercaseinwhichmachineshavedifferentoperationtimes,when-everoneofthemachinesstops,theotheroneisalsostopped.Inaddition,whenbothofthemareworking,theproductionrateismin[µ1,µ2].ConsideralongtimeintervaloflengthTduringwhichM1failsm1timesandM2failsm2times.IfweassumethattheaveragetimetorepairM1is1/r1andtheaveragetimetorepairMis1/r,thenthetotalsystemdowntimewillbeclosetoD=m1+m2.22r1r2Consequently,thetotaluptimewillbeapproximatelym1m2U=T−D=T−(+)(15)r1r2Sinceweassumeoperation-dependentfailures,theratesoffailurearereducedforthefastermachine.Therefore,bmin(µ1,µ2)bmin(µ1,µ2)bmin(µ1,µ2)pi=pi,gi=gi,andfi=fi(16)µiµiµiThereductionofpiisexplainedindetailin[10].Thereductionsofgiandfiaredoneforthesamereasons.Table2liststhepossibleworkingstatesα1andα2ofM1andM2.Thethirdcolumnistheprobabilityoffindingthesystemintheindicatedstate.Thefourthandfifthcolumnsindicatetheexpectednumberoftransitionstodownstatesduringthetimeintervalfromeachofthestatesincolumn1.Table2.Zero-bufferstates,probabilities,andexpectednumbersofeventsα1α2Probabilityπ(α1,α2)Em1(α1,α2)Em2(α1,α2)fbfb1112pbUπ(1,1)pbUπ(1,1)fb+gbfb+gb121122fbgb1−112pbUπ(1,−1)fbUπ(1,−1)fb+gbfb+gb121122gbfb−1112fbUπ(−1,1)pbUπ(−1,1)fb+gbfb+gb121122gbgb−1−112fbUπ(−1,−1)fbUπ(−1,−1)fb+gbfb+gb121122 130J.KimandS.B.GershwinFromTable2,theexpectationsofm1andm2are11Ufb(pb+gb)111Em1=Em1(α1,α2)=bb(17)α1=−1α2=−1f1+g111Ufb(pb+gb)222Em2=Em2(α1,α2)=bbα1=−1α2=−1f2+g2Bypluggingthemintoequation(15),wefindtotalproductionrate:0min[µ1,µ2]P=(18)Tfb(pb+gb)fb(pb+gb)1+111+222r1(fb+gb)r2(fb+gb)1122TheeffectiveproductionrateisfbfbP0=12P0(19)E(fb+gb)(fb+gb)T1122ThecomparisonwithsimulationisshownininTable3.TheparametersofthecasesareshowninAppendixB.Table3.Zerobuffercase00Case#PE(Analytic)PE(Simulation)%Difference10.6570.662−0.7320.6200.627−1.1530.6140.621−1.0340.5290.534−0.9950.4800.484−0.7760.6470.651−0.5770.7060.712−0.9181.3771.406−2.1090.7060.711−0.77101.3771.380−0.2232-machine-1-finite-bufferlineThetwo-machinelineisthesimplestnon-trivialcaseofaproductionline.Intheexistingliteratureontheperformanceevaluationofsystemsinwhichqualityisnotconsidered,two-machinelinesareusedindecompositionapproximationsoflongerlines(see[10]).WedefinethemodelhereandshowthesolutiontechniqueinAppendixA. Integratedqualityandquantitymodelingofaproductionline1313.1StatedefinitionThestateofthe2M1Blineisdefinedas(x,α1,α2)where∗x:thetotalamountofmaterialinbufferB,0≤x≤N,∗α1:thestateofM1.(α1=−1,0,or1),∗α2:thestateofM2.(α2=−1,0,or1)TheparametersofmachineMiareµi,ri,pi,fi,giandthebuffersizeisN.3.2Modeldevelopment3.2.1Internaltransitionequations.WhenbufferBisneitheremptynorfull,itslevelcanriseorfalldependingonthestatesofadjacentmachines.Sinceitcanchangeonlyasmallamountduringashorttimeinterval,itisreasonabletouseacontinuousprobabilitydensityf(x,α1,α2)anddifferentialequationstodescribeitsbehavior.Theprobabilityoffindingbothmachinesatstate1withastoragelevelbetweenxandx+δxattimet+δtisgivenbyf(x,1,1,t+δt)δx,wheref(x,1,1,t+δt)={1−(p1+g1+p2+g2)δt}f(x+(µ2−µ1)δt,1,1)(20)+r2δtf(x−µ1δt,1,0)+r1δtf(x+µ2δt,0,1)+o(δt)Exceptforthefactorofδx,thefirsttermistheprobabilityoftransitionfrombetween(x+(µ2−µ1)δt,1,1)and(x+(µ2−µ1)δt+δx,1,1)attimettobetween(x,1,1)and(x+δx,1,1)attimet+δt.Thisisbecause∗Theprobabilityofneithermachinefailingbetweentandt+δtis{1−(p1+g1)δt}{1−(p2+g2)δt}{1−(p1+g1+p2+g2)δt}(21)∗Iftherearenofailuresbetweentandt+δtandthebufferlevelisbetweenxandx+δxattimet+δt,thenitcouldonlyhavebeenbetweenx+(µ2−µ1)δtandx+(µ2−µ1)δt+δxattimet.Theotherterms,whichrepresenttheprobabilitiesoftransitionfrom(1)machinestates(1,0)withbufferlevelbetweenx−µ1δtandx−µ1δt+δxand(2)machinestates(0,1)withbufferlevelbetweenx+µ2δtand(x+µ2δt+δxcanbefoundsimilarly.Noothertransitionsarepossible.Afterlinearizing,andlettingδt→0,thisequationbecomes∂f(x,1,1)∂f(x,1,1)=(µ2−µ1)−(p1+g1+p2+g2)f(x,1,1)∂t∂x+r2f(x,1,0)+r1f(x,0,1)(22)∂fInsteadystate=0.Then,wehave∂tdf(x,1,1)(µ2−µ1)−(p1+g1+p2+g2)f(x,1,1)+r2f(x,1,0)dx+r1f(x,0,1)=0(23) 132J.KimandS.B.GershwinInthesameway,theeightotherinternaltransitionequationsfortheprobabilitydensityfunctionaredf(x,1,0)p2f(x,1,1)−µ1−(p1+g1+r2)f(x,1,0)+f2f(x,1,−1)dx+r1f(x,0,0)=0(24)df(x,1,−1)g2f(x,1,1)+(µ2−µ1)−(p1+g+f2)f(x,1,−1)dx+r1f(x,0,−1)=0(25)df(x,0,1)p1f(x,1,1)+µ2−(r1+p2+g2)f(x,0,1)+r2f(x,0,0)dx+f1f(x,−1,1)=0(26)p1f(x,1,0)+p2f(x,0,1)−(r1+r2)f(x,0,0)+f2f(x,0,−1)+f1f(x,−1,0)=0(27)df(x,0,−1)p1f(x,1,−1)+g2f(x,0,1)−(r1+f2)f(x,0,−1)+µ2dx+f1f(x,−1,−1)=0(28)df(x,−1,1)g1f(x,1,1)−(p2+g2+f1)f(x,−1,1)+(µ2−µ1)dx+r2f(x,−1,0)=0(29)df(x,−1,0)g1f(x,1,0)−µ1−(r2+f1)f(x,−1,0)+p2f(x,−1,1)dx+f2f(x,−1,−1)=0(30)df(x,−1,−1)g1f(x,1,−1)+g2f(x,−1,1)+(µ2−µ1)dx−(f1+f2)f(x,−1,−1)=0(31)3.2.2Boundarytransitionequations.Whiletheinternalbehaviorofthesystemcanbedescribedbyprobabilitydensityfunctions,thereisanonzeroprobabilityoffindingthesystemincertainboundarystates.Forexample,ifµ1<µ2andbothmachinesareinstate1,thelevelofstoragetendstodecrease.Ifbothmachinesremainoperationalforenoughtime,thestoragewillbecomeempty(x=0).Oncethesystemreachesstate(0,1,1),itwillremainthereuntilamachinefails.Thereare18probabilitymassesforboundarystates(P(N,α1,α2)andP(0,α1,α2)whereα1=−1,0or1,andα2=−1,0or1)and22boundaryequationsfortheµ1=µ2case.Toarriveatstate(0,1,1)attimet+δtwhenµ1=µ2,thesystemmayhavebeeninoneoftwostatesattimet.Itcouldhavebeeninstate(0,1,1)withoutanyofoperationalfailuresandqualityfailuresforbothofmachines.Itcouldhavebeeninstate(0,0,1)witharepairofthefirstmachine.(Thesecondmachinecouldnothavefailedsinceitwasstarved).Ifthesecondordertermsareignored,P(0,1,1,t+δt)={1−(p+g+pb+gb)δt}P(0,1,1)+rP(0,0,1)(32)11221 Integratedqualityandquantitymodelingofaproductionline133Aftertheusualanalysis,(32)becomes∂P(0,1,1)bb=(p1+g1+p2+g2)P(0,1,1)+r1P(0,0,1)(33)∂tInsteadystate−(p+g+pb+gb)P(0,1,1)+rP(0,0,1)=0(34)11221Thereare21otherboundaryequationsderivedsimilarlyforµ1=µ2[14]:P(0,1,0)=0(35)gbP(0,1,1)−(p+g+fb)P(0,1,−1)+rP(0,0,−1)=0(36)21121p1P(0,1,1)−r1P(0,0,1)+µ2f(0,0,1)+f1P(0,−1,1)+r2P(0,0,0)=0(37)−(r1+r2)P(0,0,0)=0(38)p1P(0,1,−1)−r1P(0,0,−1)+µ2f(0,0,−1)+f1P(0,−1,−1)=0(39)gP(0,1,1)−(f+pb+gb)P(0,−1,1)=0(40)1122P(0,−1,0)=0(41)gP(0,1,−1)+gbP(0,−1,1)−(f+fb)P(0,−1,−1)=0(42)1212−(pb+gb+p+g)P(N,1,1)+rP(N,1,0)=0(43)11222p2P(N,1,1)−r2P(N,1,0)+µ1f(N,1,0)+f2P(N,1,−1)+r1P(N,0,0)=0(44)gP(N,1,1)−(pb+gb+f)P(N,1,−1)=0(45)2112P(N,0,1)=0(46)−(r1+r2)P(N,0,0)=0(47)P(N,0,−1)=0(48)gbP(N,1,1)−(fb+g+p)P(N,−1,1)+rP(N,−1,0)=0(49)11222−r2P(N,−1,0)+µ1f(N,−1,0)+f2P(N,−1,−1)+p2P(N,−1,1)=0(50)gbP(N,1,−1)+gP(N,−1,1)−(fb+f)P(N,−1,−1)=0(51)1212µf(0,1,0)=rP(0,0,0)+pbP(0,1,1)+fbP(0,1,−1)(52)1122µf(0,−1,0)=pbP(0,−1,1)+fbP(0,−1,−1)(53)122µf(N,0,1)=rP(N,0,0)+pbP(N,1,1)+fbP(N,−1,1)(54)2211µf(N,0,−1)=pbP(N,1,−1)+gP(N,0,1)+fbP(N,−1,−1)(55)21213.2.3Normalization.Inadditiontothese,alltheprobabilitydensityfunctionsandprobabilitymassesmustsatisfythenormalizationequation:#$%Nf(x,α1,α2)dx+P(0,α1,α2)+P(N,α1,α2)=1(56)α1=−1,0,1α2=−1,0,10 134J.KimandS.B.Gershwin3.2.4Performancemeasures.Afterfindingallprobabilitydensityfunctionsandprobabilitymasses,wecancalculatetheaverageinventoryinthebufferfrom⎡⎤$Nx=⎣xf(x,α1,α2)dx+NP(N,α1,α2)⎦(57)α1=−1,0,1α2=−1,0,10ThetotalproductionrateisP=P1=TT⎡⎤$Nµ1⎣{f(x,−1,α2)+f(x,1,α2)}dx+P(0,1,α2)+P(0,−1,α2)⎦α2=−1,0,10+µ2{P(N,1,−1)+P(N,1,1)+P(N,−1,−1)+P(N,−1,1}(58)TherateatwhichmachineM1producesgoodpartsis$NP1=µ[f(x,1,α)dx+P(0,1,α)]E122α2=−1,0,10+µ2{P(N,1,−1)+P(N,1,1)}(59)Theprobabilitythatthefirstmachineproducesanon-defectivepartisthenY1=P1/P.Similarly,theprobabilitythatthesecondmachinefinishesitsoperationETwithoutaddingabadfeaturetoapartisY=P2/P,where2ET$NP2=µ[f(x,α,1)dxE21α1=−1,0,10+P(N,α1,1)]+µ1{P(0,−1,1)+P(0,1,1)}(60)Therefore,theeffectiveproductionrateisPE=Y1Y2PT(61)3.3ValidationThe2M1Bsystemswiththesamemachinespeed(µ1=µ2)aresolvedinAp-pendixA.Aswehaveindicated,werepresentdiscretepartsinthismodelasacontinuousfluidandtimeasacontinuousvariable.Wecompareanalyticalandsimulationresultsinthissection.Inthesimulation,bothmaterialandtimearediscrete.Detailsarepresentedin[14].Figure5showsthecomparisonoftheeffectiveproductionrateandtheaverageinventoryfromtheanalyticmodelandthesimulation.50casesaregeneratedbychangingmachinesandbufferparametersand%errorsareplottedinthevertical Integratedqualityandquantitymodelingofaproductionline13510.00%10.00%8.00%8.00%6.00%6.00%4.00%4.00%E2.00%2.00%0.00%0.00%14714710131619222528313437404346491013161922252831343740434649-2.00%-2.00%%errorofP%errorofInv-4.00%-4.00%-6.00%-6.00%-8.00%-8.00%-10.00%-10.00%CaseNumberCaseNumberFig.5.Validationoftheintermediatebuffersizecaseaxis.TheparametersforthesescasesaregiveninAppendixB.The%errorintheeffectiveproductionrateiscalculatedfromPE(A)−PE(S)PE%error=×100(%)(62)PE(S)wherePE(A)andPE(S)aretheeffectiveproductionratesestimatedfromtheanalyticalmodelandthesimulationrespectively.Butthe%errorintheaverageinventoryiscalculatedfromInvE(A)−InvE(S)InvE%error=×100(%)(63)0.5×NwhereInvE(A)andInvE(S)areaverageinventoryestimatedfromtheanalyticalmodelandthesimulationrespectivelyandNisabuffersize1.Theaverageabsolutevalueofthe%errorintheeffectiveproductionrateesti-mationis0.76%anditis1.89%foraverageinventoryestimation.4QualityinformationfeedbackFactorydesignersandmanagersknowthatitisidealtohaveinspectionaftereveryoperation.However,itisoftencostlytodothis.Asaresult,factoriesareusuallydesignedsothatmultipleinspectionsareperformedatasmallnumberofstations.Inthiscase,inspectionatdownstreamoperationscandetectbadfeaturesmadebyupstreammachines.Wecallthisqualityinformationfeedback.Asimpleexampleofthequalityinformationfeedbackin2M1BsystemsiswhenM1producesdefectivefeaturesbutdoesnothaveinspectionandM2hasinspectionanditcandetectbadfeaturesmadebyM1.Inthissituation,aswedemonstratebelow,theyieldofalineisafunctionofthesizeofbuffer.Thisisbecausewhenbuffergetslarger,morematerialcanaccumulatebetweenanoperation(M1)andtheinspectionofthatoperation(M2).Allsuchmaterialwillbedefectiveifapersistentqualityfailure1Thisisanunbiasedwaytocalculatetheerrorinaverageinventory.Ifitwerecalculatedinthesamewayastheerrorintheeffectiveproductionrate,theerrorwoulddependontherelativespeedsofthemachines.Thisisbecausetherewillbealowererrorwhenthebufferismostlyfull(ie,whenM1isfasterthanM2)andahighererrorwhenthebufferisempty(whenM1isfasterthanM2). 136J.KimandS.B.Gershwintakesplace.Inotherwords,ifbufferislarger,theretendstobemorematerialinthebufferandconsequentlymorematerialisdefective.Inadditionittakeslongertohaveinspectionsafterfinishingoperations.WecancapturethisphenomenonwiththeadjustmentofatransitionprobabilityrateofM1fromstate-1tostate0.qLetusdefinef1asatransitionrateofM1fromstate-1tostate0whenthereisaqualityinformationfeedbackandf1asthetransitionratewithoutthequalityinformationfeedback.TheadjustmentcanbedoneinawaythattheyieldofM1isgZthesameas1whereZg+Zb11∗Zb:theexpectednumberofbadpartsgeneratedbyMwhileitstaysinstate-1.11g∗Z1:theexpectednumberofgoodpartsproducedbyM1fromthemomentwhenM1leavesthe-1statetothenexttimeitarrivesatstate-1.From(10),theyieldofM1isqP(1)f1=q(64)P(1)+P(−1)f1+g1SupposethatM1hasbeeninstate1foraverylongtime.ThenallpartsinthebufferBarenon-defective.SupposethatM1goestostate-1.Defectivepartswillthenbegintoaccumulateinthebuffer.Untilallthepartsinthebufferaredefective,theonlywaythatM1cangotostate0isduetoitsowninspectionoritsownoperationfailure.Therefore,theprobabilityofatransitionto0beforeM1finishesapartisf1≡χ11µ1EventuallyallthepartsinthebufferarebadsothatdefectivepartsreachM2.Then,thereisanotherwaythatM1canmovetostate0fromstate-1:qualityinformationfeedback.TheprobabilitythattheinspectionatM2detectsanonconformitymadebyM1ish21χ21≡µ2where1/h21isthemeantimeuntiltheinspectionatM2detectsabadpartmadebyM1afterM2receivesthebadpart.TheexpectedvalueofthenumberofbadpartsproducedbyM1beforeitisstoppedbyeitheroperationalfailuresorqualityinformationfeedbackisZb=[χ+2χ(1−χ)+3χ(1−χ)2+...+wχ(1−χ)w−1]111111111111111+[(w+1)(1−χ)wχ+(w+2)(1−χ)w+1χ(1−χ)+...](65)1121112121wherewisaverageinventoryinthebufferB.ThisisanapproximateformulasincewesimplyusetheaverageinventoryratherthanaveragingtheexpectednumberofbadpartsproducedbyM1dependingondifferentinventorylevelswi.Aftersomemathematicalmanipulation,1−(1−χ)wZb=11−w(1−χ)w111χ11(1−χ)wχ[(w+1)−w(1−χ)(1−χ)]11211121+(66)[1−(1−χ11)(1−χ21)]2 Integratedqualityandquantitymodelingofaproductionline137gOntheotherhand,Z1isgivenasgµ1p1µ1p12µ1µ1Z1=++()()...=(67)p1+g1p1+g1p1+g1p1+g1p1+g1g1qgfZBysetting1=1wehavefq+ggb1Z+Z111qµ1f=(68)11−(1+wχ11)(1−χ11)w(1−χ11)wχ21[1+w(χ21+χ11−χ21χ11)]χ11+[1−(1−χ11)(1−χ21)]2qqSincetheaverageinventoryisafunctionoff1andf1isdependentontheaverageinventory,aniterativemethodisrequiredtodeterminethesevalues.15.00%15.00%10.00%10.00%5.00%5.00%E0.00%0.00%1591591317212529333713172125293337%errorofP-5.00%%errorofInv-5.00%-10.00%-10.00%-15.00%-15.00%CaseNumberCaseNumberFig.6.ValidationofthequalityinformationfeedbackformulaFigure6showsthecomparisonoftheeffectiveproductionrateandtheaverageinventoryfromtheanalyticmodelandthesimulation.50casesaregeneratedbyselectingdifferentmachineandbufferparametersand%errorsareplottedinthey-axis.TheparametersforthesescasesaregiveninAppendixB.%errorsintheeffectiveproductionrateandaverageinventoryarecalculatedusingequations(62)and(63)respectively.Theaverageabsolutevalueofthe%errorinPEandxestimationsare1.01%and3.67%respectively.5InsightsfromnumericalexperimentationInthissection,weperformasetofnumericalexperimentstoprovideintuitiveinsightintothebehaviorofproductionlineswithinspection.TheparametersofallthecasesarepresentedinAppendixB.5.1Beneficialbuffercase5.1.1Productionrates.Havingqualityinformationfeedbackmeanshavingmoreinspectionthanotherwise.Therefore,machinestendtostopmorefrequently.Asaresult,thetotalproductionrateofthelinedecreases.However,theeffectiveproductionratecanincreasesinceaddedinspectionspreventthemakingofdefectiveparts.ThisphenomenonisshowninFigure7.NotethatthetotalproductionratePTwithoutqualityinformationfeedbackisconsistentlyhigherthanPTwithqualityinformationfeedbackregardlessofbuffersizeandtheoppositeistrueforthe 138J.KimandS.B.GershwineffectiveproductionratePE.Alsoitshouldbenotedthatinthiscase,boththetotalproductionrateandtheeffectiveproductionrateincreasewithbuffersize,withorwithoutqualityinformationfeedback.0.80.8withoutfeedbackwithfeedback0.750.75withoutfeedbackwithfeedbackTotalProductionRateEffectiveProductionRate0.70.70.650.650510152025303540455005101520253035404550BufferSizeBufferSizeFig.7.Productionrateswith/withoutqualityinformationfeedback5.1.2Systemyieldandbuffersize.Eventhoughalargerbufferincreasesbothtotalandeffectiveproductionratesinthiscase,itdecreasesyield.AsexplainedinSection4,thesystemyieldisafunctionofthebuffersizeifthereisqualityinformationfeedback.Figure8showssystemyielddecreasingasbuffersizeincreaseswhenthereisqualityinformationfeedback.Thishappensbecausewhenthebuffergetslarger,morematerialaccumulatesbetweenanoperationandtheinspectionofthatoperation.Allsuchmaterialwillbedefectivewhenthefirstmachineisatstate-1buttheinspectionatthefirstmachinedoesnotfindit.Thisisacaseinwhichasmallerbufferimprovesquality,whichiswidelybelievedtobegenerallytrue.Ifthereisnoqualityinformationfeedback,thenthesystemyieldisindependentofthebuffersize(andissubstantiallyless).5.2Harmfulbuffercase5.2.1Productionrates.Typically,increasingthebuffersizeleadstohigheref-fectiveproductionrate.ThisisthecaseinFigure7.Butundercertainconditions,theeffectiveproductionratecanactuallydecreaseasbuffersizeincreases.Thiscanhappenwhen∗Thefirstmachineproducesbadpartsfrequently:thismeansg1islarge.∗Theinspectionatthefirstmachineispoorornon-existentandinspectionatthesecondmachineisreliable:thismeansh1h2orf1−p1f2−p2.∗Thereisqualityinformationfeedback.∗Theisolatedproductionrateofthefirstmachineishigherthanthatofthesecondmachine:1+g1/f11+g2/f2µ1>µ21+(p1+g1)/r1+g1/f11+(p2+g2)/r2+g2/f2Figure9showsacaseinwhichabuffersizeincreaseleadstoalowereffectiveproductionrate.Notethateveninthiscase,totalproductionratemonotonicallyincreasesasbuffersizeincreases. Integratedqualityandquantitymodelingofaproductionline1390.970.960.95withoutfeedbackwithfeedback0.940.93SystemYield0.920.910.90.8905101520253035404550BufferSizeFig.8.Systemyieldasafunctionofbuffersize1.51.511WithoutfeedbackWithfeedbackWithoutfeedbackWithfeedbackTotalProductionRate0.50.5EffectiveProductionRate005101520253035404550005101520253035404550BufferSizeBufferSizeFig.9.Totalproductionrateandeffectiveproductionrate10.90.80.70.6WithoutfeedbackWithfeedback0.5SystemYield0.40.30.20.1005101520253035404550BufferSizeFig.10.Systemyieldasafunctionofbuffersize 140J.KimandS.B.Gershwin5.2.2Systemyield.ThesystemyieldforthiscaseisshowninFigure10.Notethattheyielddecreasesdramaticallyasthebuffersizeincreases.Inthiscase,thedecreaseofthesystemyieldismorethantheincreaseofthetotalproductionratesothattheeffectiveproductionratemonotonicallydecreasesasbuffersizegetsbigger.5.3HowtoimprovequalityinalinewithpersistentqualityfailuresTherearetwomajorwaystoimprovequality.Oneistoincreasetheyieldofin-dividualoperationsandtheotheristoperformmorerigorousinspection.Havingextensivepreventivemaintenanceonmanufacturingequipmentandusingrobustengineeringtechniquestostabilizeoperationshavebeensuggestedastoolstoin-creaseyieldofindividualoperations.BothapproachesincreasetheMeanTimetoQualityFailure(MTQF)(i.e.decreaseg).Ontheotherhand,theinspectionpolicyaimstodetectbadpartsassoonaspossibleandpreventtheirflowtowarddown-streamoperations.Morerigorousinspectiondecreasesthemeantimetodetect(MTTD)(i.e.increaseshandthereforeincreasesf).Itisnaturaltobelievethatusingonlyonekindofmethodtoachieveatargetqualitylevelwouldnotgivethemostcostefficientqualityassurancepolicy.Figure11indicatesthattheimpactofindividualoperationstabilizationonthesystemyielddecreasesastheoperationbecomesmorestable.Italsoshowsthateffectofimprovinginspection(MTTD)onthesystemyielddecreases.Therefore,itisoptimaltouseacombinationofbothmethodstoimprovequality.110.90.90.80.80.70.70.60.60.50.5SystemYield0.4SystemYield0.40.30.30.20.20.10.1005010015020025030035040045000.10.20.30.40.50.60.70.80.91MTQFf=p+hFig.11.Qualityimprovement5.4HowtoincreaseproductivityImprovingthestand-alonethroughputofeachoperationandincreasingthebufferspacearetypicalwaystoincreasetheproductionrateofmanufacturingsystems.Ifoperationsareapttohavequalityfailures,however,theremaybeotherwaystoincreasetheeffectiveproductionrate:increasingtheyieldofeachoperationandconductingmoreextensiveinspections.Stabilizingoperations,thusimprovingtheyieldofindividualoperations,willincreaseeffectivethroughputofamanufacturing Integratedqualityandquantitymodelingofaproductionline141systemregardlessofthetypeofqualityfailure.Ontheotherhand,reducingthemeantimetodetect(MTTD)willincreasetheeffectiveproductionrateonlyifthequalityfailureispersistentbutitwilldecreasetheeffectiveproductionrateifthequalityfailureisBernoulli.ThisisbecausethequalityofeachpartisindependentoftheotherswhenthequalityfailureisBernoulli.Therefore,stoppingthelinedoesnotreducethenumberofbadpartsinthefuture.Inasituationinwhichmachinesproducedefectivepartsfrequentlyandin-spectionispoor,increasinginspectionreliabilityismoreeffectivethanincreasingbuffersizetoboosttheeffectiveproductionrate.Figure12showsthis.Also,inothersituationsinwhichmachinesproducedefectivepartsfrequentlyandinspectionisreliable,increasingmachinestabilityismoreeffectivethanincreasingbuffersizetoenhanceeffectiveproductionrate.Figure13showsthisphenomenon.0.80.75MTTD=20MTTD=10MTTD=20.70.650.6EffectiveProductionRate0.550.50510152025303540BufferSizeFig.12.Meantimetodetectandeffectiveproductionrate0.90.80.7MTQF=200.6MTQF=100MTQF=5000.5EffectiveProductionRate0.40.30510152025303540BufferSizeFig.13.Qualityfailurefrequencyandeffectiveproductionrate 142J.KimandS.B.Gershwin6FutureresearchThe2-Machine-1-Buffer(2M1B)modelwithµ1=/µ2isanalyzedin[14].Thiscaseismorechallengingbecausethenumberofrootsoftheinternaltransitionequationsdependsonparametersofmachine.Amoregeneral2M1Bmodelwithmultiple-yieldqualityfailures(amixtureofBernoulli-andpersistent-typequalityfailures)shouldalsobestudied.Alonglineanalysisusingdecompositionisunderthedevelopment.RefertoKim[14]formoredetailedinformation.AppendixASolutiontechniqueItisnaturaltoassumeanexponentialformforthesolutiontothesteadystatedensityfunctionssinceequations(23)–(31)arecoupledordinarylineardifferentialequa-tions.AsolutionoftheformeλxKα1Kα2workedsuccessfullyinthecontinuous12materialtwo-machinelinewithperfectquality[10].Therefore,asolutionofaformf(x,α,α)=eλxG(α)G(α)(69)121122isassumedhere.Thisformsatisfiesthetransitionequationsifallofthefollowingequationsaremet.Equations(23)–(31)become,aftersubstituting(69),{(µ2−µ1)λ−(p1+g1+p2+g2)G1(1)G2(1)}+r2G1(1)G2(0)+r1G1(0)G2(1)=0(70)−{µ1λ+(p1+g1+r2)}G1(1)G2(0)+p2G1(1)G2(1)+f2G1(1)G2(−1)+r1G1(0)G2(0)=0(71){(µ2−µ1)λ−(p1+g1+f2)}G1(1)G2(−1)+g2G1(1)G2(1)+r1G1(0)G2(−1)=0(72){µ2λ−(r1+p2+g2)}G1(0)G2(1)+p1G1(1)G2(1)+r2G1(0)G2(0)+f1G1(−1)G2(1)=0(73)p1G1(1)G2(0)+p2G1(0)G2(1)−(r1+r2)G1(0)G2(0)+f2G1(0)G2(−1)+f1G1(−1)G2(0)=0(74){µ2λ−(r1+f2)}G1(0)G2(−1)+p1G1(1)G2(−1)+g2G1(0)G2(1)+f1G1(−1)G2(−1)=0(75){(µ2−µ1)λ−(p2+g2+f1)}G1(−1)G2(1)+g1G1(1)G2(1)+r2G1(−1)G2(0)=0(76)−{µ1λ+(r2+f1)}G1(−1)G2(0)+g1G1(1)G2(0)+p2G1(−1)G2(1)+f2G1(−1)G2(−1)=0(77){(µ2−µ1)λ−(f1+f2)}G1(−1)G2(−1)+g1G1(1)G2(−1)+g2G1(−1)G2(1)=0(78)Thesearenineequationsinsevenunknowns(λ,G1(1),G2(0),G1(−1),G2(1),G2(0),andG2(−1)).Thus,theremustbesevenindependentequationsandtwodependentones. Integratedqualityandquantitymodelingofaproductionline143Ifwedivideequations(70)–(78)byG1(0)G2(0)anddefinenewparametersGi(1)Gi(−1)Γi=pi−ri+fi(79)Gi(0)Gi(0)Gi(0)Ψi=−pi−gi+ri(80)Gi(1)Gi(1)Θi=−fi+gi(81)Gi(−1)thenequations(70)–(78)canberewrittenasΓ1+Γ2=0(82)−µ2λ=Γ1+Ψ2(83)µ1λ=Γ2+Ψ1(84)(µ1−µ2)λ=Ψ1+Ψ2(85)(µ1−µ2)λ=Θ1+Θ2(86)µ1λ=Γ2+Θ1(87)−µ2λ=Γ1+Θ2(88)(µ1−µ2)λ=Ψ2+Θ1(89)(µ1−µ2)λ=Ψ1+Θ2(90)Fromequations(82)–(90),itisclearthatonlysevenequationsareindependent.Aftermuchmathematicalmanipulation[14],theseequationsbecome{(M+r)(µN−1)−f}21110=(f1−p1)(µ1N−1){(p1+g1−f1)+r1(µ1N−1)}{(M+r1)(µ1N−1)−f1}−−r1(91)(f1−p1)(µ1N−1){(−M+r)(µN−1)−f}22220=(f2−p2)(µ2N−1){(p2+g2−f2)+r2(µ2N−1)}{(−M+r2)(µ2N−1)−f2}−−r2=0(92)(f2−p2)(µ2N−1)whereG1(1)G1(−1)G2(1)G2(−1)p1−r1+f1=−p2−r2+f2=M(93)G1(0)G1(0)G2(0)G2(0)⎛⎞11⎝1+⎠=µ1G1(1)/G1(−1)/G1(0)+G1(0)⎛⎞11⎝1+⎠=N(94)µ2G2(1)/G2(−1)/G2(0)+G2(0)Nowalltheequationsandunknownsaresimplifiedintotwounknownsandtwoequations.Bysolvingequations(91)and(92)simultaneouslywecancalculate 144J.KimandS.B.GershwinMandN.AnexampleoftheseequationsisplottedinFigure14.Equation(91)isrepresentedwithlighterlinesandequation(92)isshownasdarkerlines.Theintersectionsofthetwosetsoflinesarethesolutionsoftheequations.321N0−1−2−3−3−2−10123MFig.14.Plotofequations(91)and(92)Thesearehighorderpolynomialequationsforwhichnogeneralanalyticalsolutionexists.Anumericalapproachisrequiredtofindtherootsoftheequations.Aspecialalgorithmtofindthesolutionshasbeendeveloped[14]basedonthecharacteristicsoftheequations.Oncewefindrootsofequations(91)and(92),Gi(1)Gi(−1)wecangetratiosand(i=1,2)fromequation(94).BysettingGi(0)Gi(0)G1(0)=G2(0)=1,wecancalculateG1(1),G1(−1),G2(1),andG2(−1).Aftersomemathematicalmanipulation,wefindthatλcanbeexpressedas−p1−g1+r1/G(1)−p1G1(1)+r1−f1G1(−1)1λ=(95)µ1Therefore,wecangetaprobabilitydensityfunctionf(x,α1,α2)correspondingtoan(M,N)pair.Thenumberofrootsinequations(91)and(92)dependsonmachineparameters.Thereareonly3rootswhenµ1=µ2regardlessofotherparameters.Therefore,ageneralexpressionoftheprobabilitydensityfunctioninthiscaseisf(x,α1,α2)=c1f1(x,α1,α2)+c2f2(x,α1,α2)+c3f3(x,α1,α2)(96)wheref1(x,α1,α2),f2(x,α1,α2),f3(x,α1,α2)aretherootsoftheequations(91)and(92).Theremainingunknowns,includingc1,c2,c3andprobabilitymassesattheboundaries,canbecalculatedbysolvingboundaryequations((34)–(55))andthenormalizationequation(56)withfi(x,α1,α2)givenbyequation(96). Integratedqualityandquantitymodelingofaproductionline145BMachineparametersfornumericalandsimulationexperimentsTable4.MachineparametersforinfinitebuffercaseandzerobuffercaseCase#µ1µ2r1r2p1p2g1g2f1f211.01.00.10.10.010.010.010.010.20.221.01.00.30.30.0050.0050.050.050.50.531.01.00.20.050.010.010.010.010.20.241.01.00.10.10.050.0050.010.010.20.251.01.00.10.10.010.010.050.0050.20.261.01.00.10.10.010.010.010.010.50.172.01.00.10.10.010.010.010.010.50.183.02.00.10.10.010.010.010.010.20.291.02.00.10.10.010.010.010.010.20.2102.03.00.10.10.010.010.010.010.20.2Table5.MachineparametersforFigures7and8µ1µ2r1r2p1p2g1g2f1f21.01.00.10.10.010.010.010.010.10.9Table6.MachineparametersforFigures9and10µ1µ2r1r2p1p2g1g2f1f22.02.00.50.10.0050.050.50.0050.020.9Table7.MachineparametersforFigure11µ1µ2r1r2p1p2g1g2f1f21.01.00.10.10.010.010.010.010.20.2Table8.MachineparametersforFigure12µ1µ2r1r2p1p2g1g21.01.00.10.10.010.010.010.01Table9.MachineparametersforFigure13µ1µ2r1r2p1p2f1f21.01.00.10.10.010.010.20.2 146J.KimandS.B.GershwinTable10.MachineparametersforintermediatebuffercasevalidationCase#µ1µ2r1r2p1p2g1g2f1f2N11.01.00.10.10.010.010.020.010.10.23021.01.00.10.10.010.010.010.010.20.2531.01.00.10.10.010.010.010.010.20.21041.01.00.10.10.010.010.010.010.20.21551.01.00.10.10.010.010.010.010.20.22061.01.00.10.10.010.010.010.010.20.22571.01.00.10.10.010.010.010.010.20.23581.01.00.10.10.010.010.010.010.20.24091.01.00.10.10.010.010.010.010.20.245100.50.50.10.10.010.010.010.010.20.230111.51.50.10.10.010.010.010.010.20.230122.02.00.10.10.010.010.010.010.20.230132.52.50.10.10.010.010.010.010.20.230143.03.00.10.10.010.010.010.010.20.230151.01.00.010.010.010.010.010.010.20.230161.01.00.050.050.010.010.010.010.20.230171.01.00.20.20.010.010.010.010.20.230181.01.00.50.50.010.010.010.010.20.230191.01.00.80.80.010.010.010.010.20.230201.01.00.10.10.0010.0010.010.010.20.230211.01.00.10.10.0050.0050.010.010.20.230221.01.00.10.10.020.020.010.010.20.230231.01.00.10.10.050.050.010.010.20.230241.01.00.10.10.10.10.010.010.20.230251.01.00.10.10.010.010.0010.0010.20.230261.01.00.10.10.010.010.0050.0050.20.230271.01.00.10.10.010.010.020.020.20.230281.01.00.10.10.010.010.050.050.20.230291.01.00.10.10.010.010.100.100.20.230301.01.00.10.10.010.010.010.010.020.0230311.01.00.10.10.010.010.010.010.050.0530321.01.00.10.10.010.010.010.010.10.130331.01.00.10.10.010.010.010.010.50.530341.01.00.10.10.010.010.010.010.950.9530351.01.00.50.10.010.010.010.010.20.230361.01.00.010.10.010.010.010.010.20.230371.01.00.10.50.010.010.010.010.20.230381.01.00.10.010.010.010.010.010.20.230391.01.00.10.10.10.010.010.010.20.230401.01.00.10.10.0010.010.010.010.20.230411.01.00.10.10.010.10.010.010.20.230421.01.00.10.10.010.0010.010.010.20.230431.01.00.10.10.010.010.10.010.20.230441.01.00.10.10.010.010.0010.010.20.230451.01.00.10.10.010.010.010.10.20.230461.01.00.10.10.010.010.010.0010.20.230471.01.00.10.10.010.010.010.010.90.230481.01.00.10.10.010.010.010.010.050.230491.01.00.10.10.010.010.010.010.20.930501.01.00.10.10.010.010.010.010.20.0530 Integratedqualityandquantitymodelingofaproductionline147Table11.MachineparametersforqualityinformationfeedbackvalidationCase#µ1µ2r1r2p1p2g1g2f1f2N11.01.00.10.10.010.010.010.010.011.01021.01.00.10.10.010.010.010.010.011.0031.01.00.10.10.010.010.010.010.011.0541.01.00.10.10.010.010.010.010.011.02051.01.00.10.10.010.010.010.010.011.03061.01.00.010.010.010.010.010.010.011.01071.01.00.050.050.010.010.010.010.011.01081.01.00.40.40.010.010.010.010.011.01091.01.00.80.80.010.010.010.010.011.010101.01.00.10.10.0010.0010.010.0010.011.010111.01.00.10.10.0050.0050.010.0050.011.010121.01.00.10.10.020.020.010.010.021.010131.01.00.10.10.10.10.010.010.11.010141.01.00.10.10.010.010.0010.0010.011.010151.01.00.10.10.010.010.0050.0050.011.010161.01.00.10.10.010.010.020.020.011.010171.01.00.10.10.010.010.050.050.011.010180.50.50.10.10.010.010.010.010.011.010191.51.50.10.10.010.010.010.010.011.010202.02.00.10.10.010.010.010.010.011.010211.01.00.10.10.010.010.010.010.051.010221.01.00.10.10.010.010.010.010.21.010231.01.00.10.10.010.010.010.010.51.010241.01.00.10.10.010.010.010.010.81.010251.01.00.50.10.010.010.010.010.011.010261.01.00.010.10.010.010.010.010.011.010271.01.00.10.50.010.010.010.010.011.010281.01.00.10.010.010.010.010.010.011.010291.01.00.10.10.10.010.010.010.11.010301.01.00.10.10.0010.010.010.010.0011.010311.01.00.10.10.010.10.010.010.011.010321.01.00.10.10.010.0010.010.010.011.010331.01.00.10.10.010.010.050.010.011.010341.01.00.10.10.010.010.0010.010.011.010351.01.00.10.10.010.010.010.050.011.010361.01.00.10.10.010.010.010.0010.011.010371.01.00.10.10.010.010.010.010.51.010381.01.00.10.10.010.010.010.010.21.010391.01.00.10.10.010.010.010.010.010.810401.01.00.10.10.010.010.010.010.010.210References1.AllesM,AmershiA,DatarS,SarkarR(2000)InformationandincentiveeffectsofinventoryinJITproduction.ManagementScience46(12):1528–15442.BesterfieldDH,Besterfield-MichnaC,BesterfieldG,Besterfield-SacreM(2003)Totalqualitymanagement.PrenticeHall,EnglewoodCliffs3.BlackJT(1991)Thedesignofthefactorywithafuture.McGraw-Hill,NewYork4.BonvikAM,CouchCE,GershwinSB(1997)Acomparisonofproductionlinecontrolmechanisms.InternationalJournalofProductionResearch35(3):789–804 148J.KimandS.B.Gershwin5.BurmanM,GershwinSB,SuyematsuC(1998)Hewlett-Packardusesoperationsre-searchtoimprovethedesignofaprinterproductionline.Interfaces28(1):24–266.BuzacottJA,ShantikumarJG(1993)Stochasticmodelsofmanufacturingsystems.Prentice-Hall,EnglewoodCliffs7.ChengCH,MiltenburgJ,MotwaniJ(2000)TheeffectofstraightandUshapedlinesonquality.IEEETransactionsonEngineeringManagement47(3):321–3348.DalleryY,GershwinSB(1992)Manufacturingflowlinesystems:areviewofmodelsandanalyticalresults.QueuingSystemsTheoryandApplications12:3–949.FujimotoT(1999)TheevolutionofamanufacturingsystemsatToyota.OxfordUni-versityPress,Oxford10.GershwinSB(1994)Manufacturingsystemsengineering.PrenticeHall,EnglewoodCliffs11.GershwinSB(2000)Designandoperationofmanufacturingsystems–thecontrol-pointpolicy.IIETransactions32(2):891–90612.GershwinSB,SchorJE(2000)Efficientalgorithmsforbufferspaceallocation.AnnalsofOperationsResearch93:117–14413.InmanRR,BlumenfeldDE,HuangN,LiJ(2003)Designingproductionsystemsforquality:researchopportunitiesfromanautomotiveindustryperspective.InternationalJournalofProductionResearch41(9):1953–197114.KimJ(2004)Integratedqualityandquantitymodelingofaproductionline.Mas-sachusettsInstituteofTechnologyPhDthesis(inpreparation)15.LawAM,KeltonDW,KeltonWD,KeltonDM(1999)Simulationmodelingandanal-ysis.McGraw-Hill,NewYork16.LedolterJ,BurrillCW(1999)Statisticalqualitycontrol.Wiley,NewYork17.MondenY(1998)Toyotaproductionsystem–anintegratedapproachtojust-in-time.EMPBooks,Norcross18.MontgomeryDC(2001)Introductiontostatisticalqualitycontrol,4thedn.Wiley,NewYork19.PandeP,HolppL(2002)Whatissixsigma?McGraw-Hill,NewYork20.PhadkeM(1989)Qualityengineeringusingrobustdesign.PrenticeHall,EnglewoodCliffs21.RazT(1986)Asurveyofmodelsforallocatinginspectioneffortinmultistageproduc-tionsystems.JournalofQualityTechnology18(4):239–24622.ShinWS,MartSM,LeeHF(1995)Strategicallocationofinspectionstationsforaflowassemblyline:ahybridprocedure.IIETransactions27:707–71523.ShingoS(1989)AstudyoftheToyotaproductionsystemfromanindustrialengineeringviewpoint.ProductivityPress,Portland24.ToyotaMotorCorporation(1996)TheToyotaproductionsystem25.WeinL(1988)Schedulingsemiconductorwaferfabrication.IEEETransactionsonsemiconductormanufacturing1(3):115–13026.WooddallWH,MontgomeryDC(1999)Researchissuesandideasinstatisticalprocesscontrol.JournalofQualityTechnology31(4):376–386 Stochasticcyclicflowlineswithblocking:MarkovianmodelsYoung-DooLeeandTae-EogLeeDepartmentofIndustrialEngineering,KoreaAdvancedInstituteofScienceandTechnology,373-1,Kuseong-Dong,Yuseong-Gu,Taejon305-701,Korea(e-mail:{ydlee,telee}@kaist.ac.kr)Abstract.Weconsideracyclicflowlinemodelthatrepetitivelyproducesmultipleitemsinacyclicorder.Weexamineperformanceofstochasticcyclicflowlinemod-elswithfinitebuffersofwhichprocessingtimeshaveexponentialorphase-typedistributions.Wedevelopanexactmethodforcomputingatwo-stationmodelbymakinguseofthematrixgeometricstructureoftheassociatedMarkovchain.Wepresentacomputationallytractableapproximateperformancecomputingmethodthatdecomposesthelinemodelintoanumberoftwo-stationsubmodelsandparam-eterizingthesubmodelsbypropagatingthestarvationandblockingprobabilitiesthroughtheadjacentsubmodels.Wediscussperformancecharacteristicsincludingcomparisonwithrandomorderprocessingandeffectsofthejobvariationandthejobprocessingsequence.Wealsoreporttheaccuracyofourproposedmethod.Keywords:Cyclicflowline–Stochastic–Blocking–Performance–Decompo-sition1IntroductionCyclicproductionisawayofproducingmultipleitemssimultaneouslyinashop.Itrepetitivelyproducesanidenticalsetofitemsinthesameloadingandsequenceateachstation.Forinstance,wehaveproductionrequirementof100,200,and300itemsforitemtypesa,b,andc,respectively.Then,theminimalsetof1a,2b’s,and3c’sisproduced100timesinthesameproductionmethod.Dependingonthevisitsequenceoftheitemsthroughthestations,theshopcanbeajobshoporaflowline.Inacyclicflowline,eachitemflowsthroughthestationsinthesamesequence.Eachstationprocessestheitemsintheorderoffirstcomefirstservice.Therefore,eachstationrepeatsanidenticalcyclicsequenceofprocessingtheitems,forinstance,Correspondenceto:T.-E.Lee,373-1Gusung-Dong,Yusong-Gu,Daejon305-701,Korea 150Y.-D.LeeandT.-E.Leea,b,b,c,c,c,whichisthesameasthereleasesequenceoftheitemsintotheline.Cyclicflowlinesarewidelyusedforassemblylinesorserialprocessinglineswheremultipletypesofitemsaresimultaneouslyproducedandthesetuptimesarenotsignificant.Advantagesofcyclicproductionoverconventionalbatchproductionorrandomorderproductionincludebetterutilizationofthemachines,simplifiedflowcontrol,continuousandsmoothsupplyofcompletepartsetsfordownstreamassembly,timelydelivery,andreducedwork-in-progressinventory[14].Therehavebeenstudiesoncyclicshops.Essentialissuescanbefoundin[1,7,10,11,14–18,22].Cyclicflowlinesareoftenusedforprintedcircuitboardas-semblyandelectronicsorotherhomeapplianceassembly,andintegratedwithanaccumulation-typeconveyorsystem.Suchaconveyorsystemallowsonlyafewpartstowaitbeforeeachstation.Suchcyclicflowlineswithblockinghavebeenexamined[1,18,22].Theydealwithschedulingissuesforthecaseswhereprocesstimesarecompletelyknown.However,cyclicshopsaresubjecttorandomdisrup-tionssuchastooljammingandrecovery,retrialsofanassemblyoperation,etc.Thesetendtocontributetorandomvariationinjobprocessingtimes.Schedulingmodelsoftenneglecttransporttimesortendtoincreasetheprocessingtimesbythetransporttimes.Suchapproximatemodelingsimplifiestheschedulingmodel,butaddsrandomnessofthetransporttimestothecombinedprocesstimes.Thereareafewworksonstochasticcyclicshopmodels.RaoandJackson[20]developanapproximatealgorithmtocomputetheaveragecycletimeforacyclicjobshopwithgeneralprocessingtimedistributions,whichmakesuseofClark’sapproximationmethodforstochasticPERTnetworks.BowmanandMuckstadt[2]deliberatelydevelopafiniteMarkovchainmodelforacyclicjobshopwithexpo-nentialprocessingtimesandcomputetheaveragecycletime,butdonotdiscussthequeuelength.ZhangandGraves[23]findtheschedulesthatareleastdis-turbedbymachinefailuresinare-entrantcyclicflowshop.Forcyclicflowlines,SeoandLee[21]examinethequeuelengthdistributionsofthecasesthathaveexponentialprocessingtimesandinfinitebuffers.Stochasticcyclicflowlineswithlimitedbuffershavedistinctperformancecharacteristicsandrequireadifferentper-formanceanalysismethodduetoblocking.Therefore,itisnecessarytoexamineperformanceofstochasticflowlineswithblocking.KarabatiandTan[13]proposeaheuristicprocedureforschedulingstochasticcyclictransferlinesthatmovejobsbetweenthestationssynchronously.Stochasticcyclicflowlinesarecomparabletoconventionaltandemqueueswithmultiplecustomerclasses.Whiletheformerproducesdifferenttypesofitemsinacyclicorder,thelatterprocessestheitemsinrandomorder.Therefore,stochas-ticflowlinesrequireadistinctperformanceanalysismethod.Nonetheless,itisexpectedthatsomeideasforanalyzingtandemqueuesalsowillbeusefulforex-aminingstochasticcyclicflowlines.Animportanttechniqueforanalyzingatandemqueuemodelistodecomposethemodelintomultipletwo-stationmodels,eachofwhichismodeledbyanappro-priatelyparameterizedsingle-queuemodel,andapproximatetheperformanceofthetandemqueuefromtheperformanceestimatesofthedecomposedsingle-queuemodels[4–6,8].WhileDalleryetal.[4,5]andGershwin[8]proposesuchdecom-positiontechniquefortransferlineswithunreliablemachinesandfinitebuffers, Stochasticcyclicflowlineswithblocking:Markovianmodels151thetechniqueispopularlyusedfortandemqueues.Variousdecomposedsingle-queuemodelsanddifferentapproximationschemescanbefoundinthesurveyonmodelingandanalysisoftandemqueues[6].Wenotethatmostworksontandemqueuesassumesinglecustomerclasswhileastochasticcyclicflowlineprocessesmultiplecustomerclassessimultaneously.Itisexpectedthatstochasticcyclicflowlineswithblockingrequireyetanotherdecompositionandapproximationmethod.Inthispaper,weexamineperformanceofcyclicflowlinemodelsthathavefinitebuffersandprocessingtimesofexponentialorphase-typedistributions.Phase-typedistributionsaremorerealisticformodelingprocessingtimedistributionssinceanydistributioncanbearbitrarilycloselyapproximatedbyphase-typedistributions.Whilesuchacyclicflowlinemodelwouldbemodeledbyafinitecontinuous-timeMarkovchain,thenumberofstatestendstoexplodeandthechaineasilybecomescomputationallyintractableasthenumberofstations,thebuffercapac-ities,thenumberofjobtypes,andthenumberofphasesintheprocessingtimedistributionsincrease.Therefore,wepresentacomputationallytractableperfor-manceapproximationmethodthatdecomposesthelinemodelintoanumberoftwo-stationsubmodelsandappropriatelyparameterizesthemeanprocessingtimesofthedecomposedsubmodels.Weexaminetheperformancecharacteristicsandreporttheexperimentalaccuracyoftheproposedalgorithm.Wealsocomparetheperformanceofcyclicproductionwiththatofrandomorderproduction.Theeffectofthejobprocessingsequenceisalsodiscussed.2StochasticcyclicflowlinemodelswithfinitebuffersWefirstexplainstochasticcyclicflowlinemodels.Weconsideracyclicflowlinethatconsistsof(K+1)stations(S0,S1,...,SK).Eachstationhasasinglemachine.ThefirststationS0hasanunlimitedbuffer.EachsubsequentstationSi(i=1,...,K)hasaninputbufferofcapacityB−1(thatis,eachstationhascapacityB).Eachstationcanprocessthenextjobinthebufferafterthepreviousonecompletesandleavesthestation.Ajobcompletedatastationimmediatelyleavesthestationandenterstheinputbufferofthenextstation.Whenthenextinputbufferisfull,thejobcannotleavethestationandwaitsuntilthenextbufferisavailable.Suchwaitingiscalledblocking,morespecificallyblockingafterservice(BAS).Whenthereisnojobavailableattheinputbuffer,thestationisidle.Thisiscalledstarvation.Thetransporttimesofjobsbetweenthestationsarenegligibleorincludedintheprocessingtimes.Thejobsinaninputbufferareprocessedintheorderoffirstcomeandfirstservice.Ajobbeingprocessedatastationcannotbepreempted.Weassumethatthestationsareallreliableandthereisnobreakdown.ThereareenoughjobsavailableandhencenoshortageorstarvationatthefirststationS0.ThelaststationSKhasnoblockingsincethereisnonextstation.TheJtypesofjobsarerepetitivelyloadedintothelineinapredefinedcyclicorder.Therefore,eachstationrepeatstheidenticalcyclicorderofprocessingthejobs.Weassumethattheprocessingtimesofthejobsatastationhaveexponentialorphase-typedistributions.Thesetuptimesarenegligible. 152Y.-D.LeeandT.-E.Lee3Two-stationmodelsWefirstexamineperformanceofatwo-stationmodelthathasexponentialprocess-ingtimedistributions.Weintroduceparametersforthetwo-stationmodel.λiandµiaretheprocessingratesofjobiatstation1and2,respectively,andBisthecapacityofstation2.Thestateofthelinemodelisthendenotedby(m1,m2,n),wheren=thenumberofjobsatstation2includingthejobinprogress,andmi=thestateofstationi.miusuallyindicatesthejobtypebeingprocessedatstationi.However,whenstation1isbeingblocked,itsstateisindicatedbym1=b.m2=smeansthatstation2isstarving.Forexample,(1,1,3)indicatesthatbothstations1and2areprocessingjob1,and3jobsatstation2(2inthebufferand1inprogress).(b,2,2)representsthatstation1isblockedafterprocessingajobsincestation2hascapacity2.Wenotethatifweknowthenumberofjobsatthebufferandthejobtypeinprogressatastation,thejobtypeinprogressorjustcompletedatanotherstationiseasilydetermined.Byexaminingtheoperationalbehaviorofthelinemodel,thestatetransitiondiagramisobtainedasinFigure1.Sincealleventoccurrences1s02s03s0Js0λ1λ2λ3λJµ1µ2µ3µJ2113214311J1λ2λλµJλ341µµJ1µ2µJµ1µ2µ3J1(J-1)12(J-1)23(J-1)(J-1)J(J-1)λλ1λ2λJ−1JµµµµJ12311J22J33JJJJµJλ1λ2λ3λJµ1µ2µµJ321(J+1)32(J+1)43(J+1)1J(J+1)λ2λλµJλ341µ1µ2µJµ1µ2µ3b1Bb2Bb3BbJBFig.1.Statetransitiondiagramofatwo-stationmodel Stochasticcyclicflowlineswithblocking:Markovianmodels153aregovernedbyexponentialprocessingtimes,thestatetransitionprocessformsacontinuous-timeMarkovchain.WeobservethatthediagramrepeatsanidenticalstructureeachmultipleofJforstatevariablen.Thetransitionratesaremarkedonthecorrespondingarcs.WethereforeexpectthatthegeneratoroftheMarkovchainhasarepeat-ingpattern.Definer≡B.Weletπ(m,m,n)denotetheprobabilitythatJ12thelineisatstate(m1,m2,n)inthesteadystate.Forexpositionconvenience,weexplainthecaseofJ=2.Definethesteadystatedistributionvectorasπ≡(πs,π0,π1,...,πk,...,πr−1,πb),whereπs≡(π(1,s,0),π(2,s,0)),πk≡(π(2,1,2k+1),π(1,2,2k+1),π(1,1,2k+2),π(2,2,2k+2)),k=0,1,...,r−1,andπb≡(π(b,1,B),π(b,2,B)).Fromthestatetransitiondiagram,wehavethegeneratormatrixQthatisrepresentedbyblockmatriceswithspecialstructuresas⎛⎞S1S2···⎜⎟⎜⎟⎜S3S4A0···⎟⎜⎟⎜⎟⎜A2A1A0···⎟⎜⎟⎜.⎟Q=⎜..⎟,⎜⎟⎜⎟⎜⎟⎜A2A1A0⎟⎜⎟⎜⎟⎜A2B4B3⎟⎝⎠B1B2⎛⎞−λ10λ10⎜⎟⎜⎜0−λ20λ2⎟⎟whereS1=⎜⎟,⎜⎝0µ1−(µ1+λ2)0⎟⎠µ200−(µ2+λ1)⎛⎞0000⎜⎟⎜0000⎟⎜⎟S2=A0=⎜⎟,⎜⎝λ2000⎟⎠0λ100⎛⎞−(µ1+λ1)0λ10⎜⎟⎜⎜0−(µ2+λ2)0λ2⎟⎟S4=B4=A1=⎜⎟,⎜⎝0µ1−(µ1+λ2)0⎟⎠µ200−(µ2+λ1) 154Y.-D.LeeandT.-E.Lee⎛⎞000µ1⎜⎟⎜⎜00µ20⎟⎟000µ1−µ10S3=A2=⎜⎟,B1=,B2=,⎜⎝0000⎟⎠00µ200−µ20000⎛⎞00⎜⎟⎜00⎟⎜⎟andB3=⎜⎟.⎜⎝λ20⎟⎠0λ1GeneratorQforJ>2thathasthesameblockstructurealsocanbesimilarlyidentified.Thesizeofeachblockmatrixisdeterminedbythenumberofjobtypes,J,andthestationcapacity,B.AisaJ2×J2squarematrixregardlessofB.iItiseasilyseenthatQisirreducibleandthefiniteMarkovchainispositiverecurrentandhenceergodic.Therefore,thesteadystateprobabilityπisthesolutionofthebalanceequationπQ=0.ThebalanceequationcanbeefficientlysolvedsinceQhasaspecialstructurecalledageneralizedbirth-and-deathprocess.Suchastructureisalsoaspecialcaseofthegeneralmatrixgeometricstructure.Therearetwogenerallyknownstrategiesforsolvingthebalanceequationforsuchastructuredgenerator,matrixgeometrictechnique[19]andrecursivetechnique[3].BuzacottandKostelski[3]reportthatthereisnosignificantdifferencebetweenthetwomethodsintheiraccuracy,buttherecursivemethodismoreefficientthanthematrixgeometricalgorithm.Therecanbedifferentimplementationsoftherecursivemethoddependingonthedetailedmatrixstructure.WeadapttherecursivealgorithmofHongetal.[12]thatisusedfortwo-stationtandemqueueswithrandomfailuresofstations.FromπQ=0,weobtainthefollowingequations.πsS1+π0S3=0,(1)πsS2+π0S4+π1A2=0,(2)πk−1A0+πkA1+πk+1A2=0,k=1,2,...,r−2,(3)πr−2A0+πr−1B4+πbB1=0,and(4)πr−1B3+πbB2=0.(5)From(1),(2),(3),and(4),wederivefollowingrelationships.From(1)and(2),−1πs=π0T0,whereT0=−S3S1,and(6)π=πT,whereT=−A(S+TS)−1.(7)01112402From(3),wecanderiveπ=πT,whereT=−A(A+TA)−1,k=1,...,r−2.(8)kk+1k+1k+121k0From(4),wehaveπ=πT,whereT=−B(B+TA)−1.(9)r−1bbb14r−10 Stochasticcyclicflowlineswithblocking:Markovianmodels155Fromtheinitialvalueofπb,T0,T1,T2,...,Tr−1,andTbaresuccessivelyobtainedandvectorπiscomputedfromthenormalizingcondition:r−1πs1+πk1+πb1=1,(10)k=0where1isthecolumnvectorof(1,1,...,1)withanappropriatedimension.Theprocedureforcomputingtheperformanceissummarizedasfollows.Algorithm:Two-stationStep1.Setπb=1andSAVE=πb=1.Step2.Computeπr−1from(9).Step3.Computeπbfrom(5).Step4.If||SAVE−πb||≤,gotoStep5.ElseSAVE=πb,andgotoStep2.Step5.Computeπr−1,πr−2,...,π0,πsfrom(6)–(9).Step6.Normalizethesteadystatedistributionvectorπ.Vectorsπb,πk,andπsarecomputedrecursivelybymatrixoperations.Aftertakingsomeinitialvalueofπb,computeπk’sandπbrecursivelyuntilπbvaluecon-vergestoafinitevalue.Then,πvectorisnormalized.Thesteadystatequeuelengthdistributions,thestarvationprobability,theblockingprobability,thethroughputrate,andthemeanqueuelengtharecomputed,respectively,asJJpn=π(m1,m2,n),n=1,...,B,m1=1m2=1Jps=π(m1,s,0),m1=1Jpb=π(b,m2,B),m2=1JBJT=µj[π(m1,j,y)]+µJπ(b,J,B),andj=1y=1m1=1BJJJL=yπ(m1,m2,y)+Bπ(b,m2,B).y=1m1=1m2=1m2=1Notethat1/Tisthemeancycletimeforalltypesofitems.ThemeancycletimeofjobsetsisJ/T.4ModelswithmorethantwostationsInordertoanalyzetheperformanceofalinemodelwithmorethantwostations,weextendthedecompositiontechniquethathasbeenusedfortandemqueues[8].The 156Y.-D.LeeandT.-E.LeeSS0B1S1B2S2B3S3B4S4S(1)udS(1)B(1)S(1)S(2)udS(2)B(2)S(2)S(3)udS(3)B(3)S(3)S(4)udS(4)B(4)S(4)Fig.2.Two-stationdecompositionprocedureisoutlinedasfollows.First,thelinemodelisdecomposedintoKtwo-stationsubmodelsasshowninFigure2.Eachtwo-stationsubmodelS(i)consistsofupstreamstationSu(i),downstreamstationSd(i),andbufferB(i)betweenthemwiththesamecapacityB−1asintheoriginallinemodelS.StationsSu(i)andSd(i)areparameterizedtohavetheperformancesclosetothoseofstationsSi−1andSi,respectively,intheoriginalline,whicharesubjecttostarvationandblocking.4.1ExponentialmodelsWefirstexaminethedecompositionmethodforthecasewhereallprocessingtimesareexponentiallydistributed.Lettj(i)denotethemeanprocessingtimeofjobjatstationiintheoriginallinemodel.Lett(i)≡(t(i),...,t(i))
,i=0,...,K,1JbethemeanprocessingtimevectoratstationS.Lettu(i)≡(tu(i),...,tu(i))
i1Jandtd(i)≡(td(i),...,td(i))
,i=1,...,K,bethemeanprocessingtimesof1JSu(i)andSd(i)insubmodelS(i),respectively.Theprocessingcapacityateachstationofadecomposedtwo-stationsubmodelisparameterizedtobeascloseaspossibletotheeffectiveprocessingcapacityofthecorrespondingstationintheoriginalline.Theprocessingcapacityofastationintheoriginallineisreducedduetostarvationorblockingatthestation.Therefore,theprocessingtimestu(i)oftheupstreamstationSu(i)ofeachsubmodelS(i)areextendedasmuchasthedelaysduetostarvationofthecorrespondingstationSi−1intheoriginalline.Similarly,theprocessingtimestd(i)ofthedownstreamstationSd(i)ofeachsubmodelS(i)areextendedasmuchasthedelaysduetoblockingofthecorrespondingstationsSiintheoriginalline.However,forthefirstsubmodelS(1),theprocessingtimesofstationSu(1)arekeptsameasthoseofSintheoriginalline.Itisbecausethefirst0stationS0isneverstarved.Similarly,forthelastsubmodelS(K),theprocessingtimesofSd(K)arekeptsameasthoseofSintheoriginallinebecausethelastK Stochasticcyclicflowlineswithblocking:Markovianmodels157stationSKisneverblocked.Therefore,wehavethefollowingboundaryconditions:tu(1)=t(0)andtd(K)=t(K).(11)ConsiderasubmodelS(i)suchthat1≤i1/2,thentherateandcoefficientofvariationoftheCoxiandistributiond,i2withdensity−µ1tµ1µ2−µ2t−µ1tf(t)=(1−q)µ1e+qe−e,t≥0,µ1−µ2matcheswithµd,iandcd,i,providedtheparametersµ1,µ2andqarechosenas(cf.Marie[14]):1µ1=2µd,i,q=2,µ2=µ1q.(1)2cd,iIf1/k≤c2≤1/(k−1)forsomek>2,thentherateandcoefficientofvariationd,ioftheErlangk−1,kwithdensitytk−2tk−1f(t)=pµk−1e−µt+(1−p)µke−µt,t≥0,(k−2)!(k−1)!matcheswithµd,iandcd,iiftheparametersµandparechosenas(cf.Tijms[22]):kc2−k(1+c2)−k2c2d,id,id,ip=2,µ=(k−p)µd,i.(2)1+cd,iOfcourse,alsootherphase-typedistributionsmaybefittedontherateandcoef-ficientofvariationofDi,butnumericalexperimentssuggestthatotherdistributionsonlyhaveaminoreffectontheresults,asshownin[10].TheservicetimesAiofthearrival-serversinsubsystemLiaremodelledsimi-larly.Insteadofbi,jwenowusesi,jdefinedastheprobabilitythatjustafterservicecompletionofaserverinserver-groupMi,exactlyjserversofMiarestarved.Thismeansthat,withprobabilitysi,j,aserverinserver-groupMihastowaitone 174M.vanVuurenetal.Fig.3.RepresentationoftheservicetimeAiofanarrival-serverofsubsystemLiresidualinter-departuretimeandj−1fullinter-departuretimesfromthepreced-ingserver-groupMi−1.Figure3displaysarepresentationoftheservicetimeofanarrival-serverofsubsystemLi.4SpectralanalysisofasubsystemByfittingCoxianorErlangdistributionsontheservicetimesAiandDi,subsystemLicanbemodelledasafinitestateMarkovprocess;belowwedescribethisMarkovprocessinmoredetailforasubsystemwithmaarrivalservers,mddepartureserversandabufferofsizeb.Toreducethestatespacewereplacethearrivalanddepartureserversbysuperserverswithstate-dependentservicetimes.Theservicetimeofthesuperarrivalserveristheinter-departuretimeoftheserviceprocessesofthenon-blockedarrivalservers.Ifthebufferisnotfull,allarrivalserversareworking.Inthiscase,theinter-departuretime(orsuperservicetime)isassumedtobeCoxianldistributed,wherephasej(j=1,...,l)hasparameterλjandpjistheprobabilitytoproceedtothenextphase(notethatErlangdistributionsareaspecialcaseofCoxiandis-tributions).Ifthebufferisfull,oneormorearrivalserversmaybeblocked.ThenthesuperservicetimeisCoxiandistributed,theparametersofwhichdependonthenumberofactiveservers(andfollowfromtheinter-departuretimedistributionoftheactiveserviceprocesses).Theservicetimeofthesuperdepartureserverisdefinedsimilarly.Inparticular,ifnoneofthedepartureserversisstarved,thesuperservicetimeistheinter-departuretimeoftheserviceprocessesofallmddepartureservers.Thisinter-departuretimeisassumedtobeCoxianndistributedwithpa-rametersµjandqj(j=1,...,n).So,thetimespendinphasejisexponentiallydistributedwithparameterµjandtheprobabilitytoproceedtothenextphaseisqj.NowthesubsystemcanbedescribedbyaMarkovprocesswithstates(i,j,k).Thestatevariableidenotesthetotalnumberofcustomersinthesubsystem.Clearly,iisatmostequaltomd+b+ma.Notethat,ifi>md+b,theni−md−bactually Performanceanalysisofmulti-servertandemqueues175indicatesthenumberofblockedarrivalservers.Thestatevariablej(k)indicatesthephaseoftheservicetimeofthesuperarrival(departure)server.Ifi≤md+b,thentheservicetimeofthesuperarrivalserverconsistsoflphases;thenumberofphasesdependsonifori>md+b.Similarly,thenumberofphasesoftheservicetimeofthesuperdepartureserverisnfori≥md,anditdependsoniforimd+bcanbedeterminedfromtheequilibriumequationsfori≤mdandi≥md+bandthenormalizationequation.5IterativealgorithmWenowdescribetheiterativealgorithmforapproximatingtheperformancechar-acteristicsoftandemqueueL.ThealgorithmisbasedonthedecompositionofLinM−1subsystemsL1,L2,...,LM−1.BeforegoingintodetailinSection5.2,wepresenttheoutlineofthealgorithminSection5.1.5.1Outlineofthealgorithm•Step0:DetermineinitialcharacteristicsoftheservicetimesDiofthedepartureserversofsubsystemLi,i=M−1,...,1.•Step1:ForsubsystemLi,i=1,...,M−1:1.DeterminethefirsttwomomentsoftheservicetimeAiofthearrivalservers,giventhequeue-lengthdistributionandthroughputofsubsystemLi−1.2.Determinethequeue-lengthdistributionofsubsystemLi.3.DeterminethethroughputTiofsubsystemLi.•Step2:DeterminethenewcharacteristicsoftheservicetimesDiofthedepartureserversofsubsystemLi,i=M−1,...,1.•RepeatStep1and2untiltheservicetimecharacteristicsofthedepartureservershaveconverged.5.2DetailsofthealgorithmStep0:Initialization:Thefirststepofthealgorithmistosetbi,j=0foralliandj.Thismeansthatweinitiallyassumethatthereisnoblocking.ThisalsomeansthattherandomvariablesDiareinitiallythesameastheservicetimesSi.Step1:Evaluationofsubsystems:WenowknowtheservicetimecharacteristicsofthedepartureserversofLi,butwealsoneedtoknowthecharacteristicsoftheservicetimesofitsarrivalservers,beforeweareabletodeterminethequeue-lengthdistributionofLi.(a)ServicetimesofarrivalserversForthefirstsubsystemL1,thecharacteristicsofA1arethesameasthoseofS0,becausetheserversofM0cannotbestarved.Fortheothersubsystemsweproceedasfollows.ByapplicationofLittle’slawtothearrivalservers,itfollowsthatthethroughputofthearrivalserversmultipliedwiththeservicetimeofanarrivalserverisequaltomeannumberofactive(i.e. 178M.vanVuurenetal.non-blocked)arrivalservers.Theservicetimeofanarrivalserverofsubsystemiisequalto1/µa,iandthemeannumberofactiveserversisequalto⎛⎞mi−1mi−1⎝1−pi,mi+bi+j⎠mi−1+pi,mi+bi+j(mi−1−j).j=1j=1So,wehaveforthethroughputTiofsubsystemLi,⎛⎞mi−1mi−1Ti=⎝1−pi,mi+bi+j⎠mi−1µa,i+pi,mi+bi+j(mi−1−j)µa,i,(18)j=1j=1wherepi,jdenotestheprobabilityofjcustomersinsubsystemLi.Bysubstituting(n)(n−1)theestimateTi−1forTiandpi,ni+jforpi,ni+jwegetasnewestimatefortheservicerateµa,i,(n)(n)Ti−1µa,i=mi−1(n−1)mi−1(n−1),(1−j=1pi,mi+bi+j)mi−1+j=1pi,mi+bi+j(mi−1−j)wherethesuperscriptsindicateinwhichiterationthequantitieshavebeencalcu-lated.Toapproximatethecoefficientofvariationca,iofAiweusetherepresentationforAiasdescribedinSection3(whichisbasedonsi−1,j,Si−1,RSAi−1andSAi−1).(b)AnalysisofsubsystemLiBasedonthe(new)characteristicsoftheservicetimesofbotharrivalanddepartureserverswecandeterminethesteady-statequeue-lengthdistributionofsubsystemLi.TodosowefirstfitCoxian2orErlangk−1,kdistributionsonthefirsttwomo-mentsoftheservicetimesofthearrival-serversanddeparture-serversasdescribedinSection3.Thenwecalculatetheequilibriumprobabilitiespi,jbyusingthespectralexpansionmethodasdescribedinSection4.(c)ThroughputofsubsystemLiOncethesteady-statequeuelengthdistributionisknown,wecandeterminethe(n)newthroughputTiaccordingto(cf.(18))⎛⎞mi−1mi−1T(n)=⎝1−p(n)⎠mµ(n−1)+p(n)jµ(n−1).(19)ii,jid,ii,jd,ij=0j=1Wealsodeterminenewestimatesfortheprobabilitiesbi−1,jthatjserversofserver-groupMi−1areblockedafterservicecompletionofaserverinserver-groupMi−1andtheprobabilitiessi,jthatjserversofserver-groupMiarestarvedafterservicecompletionofaserverinserver-groupMi.WeperformStep1foreverysubsystemfromL1uptoLM−1. Performanceanalysisofmulti-servertandemqueues179Step2:Servicetimesofdepartureservers:Nowwehavenewinformationaboutthedepartureprocessesofthesubsystems.Sowecanagaincalculatethefirsttwomomentsoftheservicetimesofthedeparture-servers,startingfromDM−2downtoD1.NotethatDM−1isalwaysthesameasSM−1,becausetheserversinserver-groupMM−1canneverbeblocked.Anewestimatefortherateµd,iofDiisdeterminedfrom(cf.(18))(n)(n)Ti+1µd,i=mi−1(n)mi−1(n)(20)(1−j=0pi,j)mi+j=1pi,jjThecalculationofanewestimateforthecoefficientofvariationcd,iofDiissimilartotheoneofAi.Convergencecriterion:AfterStep1and2wecheckwhethertheiterativealgo-rithmhasconvergedbycomparingthedepartureratesinthe(n−1)-thandk-thiteration.Wedecidetostopwhenthesumoftheabsolutevaluesofthediffer-encesbetweentheseratesislessthanε;otherwisewerepeatStep1and2.SotheconvergencecriterionisM−1(n)(n−1)µd,i−µd,i<ε.i=1Ofcourse,wemayuseotherstop-criteriaaswell;forexample,wemayconsiderthethroughputinsteadofthedeparturerates.Thebottomlineisthatwegoonuntilallparametersdonotchangeanymore.Remark.Equalityofthroughputs.Itiseasilyseenthat,afterconvergence,thethroughputsinallsubsystemsare(n)(n−1)equal.Letusassumethattheiterativealgorithmhasconverged,soµ=µd,id,iforalli=1,...,M−1.Fromequations(19)and(20)wefindthefollowing:⎛⎞mi−1mi−1T(n)=⎝1−p(n)⎠mµ(n−1)+p(n)jµ(n−1)ii,jid,ii,jd,ij=0j=1⎛⎞mi−1mi−1=⎝1−p(n)⎠mµ(n)+p(n)jµ(n)i,jid,ii,jd,ij=0j=1(n)=Ti+1.Hencewecanconcludethatthethroughputsinallsubsystemsarethesameafterconvergence.Complexityanalysis:Thecomplexityofthismethodisasfollows.Withintheiterativealgorithm,solvingasubsystemconsumesmostofthetime.InoneiterationasubsystemissolvedMtimes.Thenumberofiterationsneededisdifficulttopredict,butinpracticethisnumberisaboutthreetoseveniterations.Thetimeconsumingpartofsolvingasubsystemissolvingtheboundaryequa-tions.ThiscanbedoneinO((m+m)(kk)3)time,wherekisthenumberadada 180M.vanVuurenetal.ofphasesofthedistributionofonearrivalprocessandkdisthenumberofphasesofthedistributionofonedepartureprocess.Then,thetimecomplexityofoneit-erationbecomesO(Mmax((m+m)(kk)3)).Thismeansthatthetimeiii−1ii−1complexityispolynomialanditdoesn’tdependonthesizesofthebuffers.6NumericalresultsInthissectionwepresentsomenumericalresults.Toinvestigatethequalityofourmethodwecompareitwithdiscreteeventsimulation.Afterthat,wecompareourmethodwiththemethoddevelopedbyTahilramanietal.[21],whichisimplementedinQNAT[25].6.1ComparisonwithsimulationInordertoinvestigatethequalityofourmethodwecomparethethroughputandthemeansojourntimewiththeonesproducedbydiscreteeventsimulation.Weareespeciallyinterestedininvestigatingforwhichsetofinput-parametersourmethodgivessatisfyingresults.Eachsimulationrunissufficientlylongsuchthatthewidthsofthe95%confidenceintervalsofthethroughputandthemeansojourntimearesmallerthan1%.Inordertotestthequalityofthemethodweuseabroadsetofparameters.WetesttwodifferentlengthsMoftandemqueues,namelywith4and8server-groups.Foreachtandemqueuewevarythenumberofserversmiintheserver-groups;weusetandemswith1serverperserver-group,5serversperserver-groupandwiththesequence(4,1,2,8).Wealsovarythelevelofbalanceinthetandemqueue;everyserver-grouphasamaximumtotalrateof1andthegrouprightafterthemiddlecanhaveatotalrateof1,1.1,1.2,1.5and2.Thecoefficientofvariationoftheservicetimesvariesbetween0.1,0.2,0.5,1,1.5and2.Finallywevarythebuffersizesbetween0,2,5and10.Thisleadstoatotalof720test-cases.TheresultsforeachcategoryaresummarizedinTable1upto5.Eachtableliststheaverageerrorinthethroughputandthemeansojourntimecomparedwiththesimulationresults.Eachtablealsogivesfor4error-rangesthepercentageofthecaseswhichfallinthatrange.Theresultsforaselectionof54casescanbefoundinTables6and7.Table1.OverallresultsfortandemqueueswithdifferentbuffersizesBufferErrorinthroughputErrorinmeansojourntimesizes(bi)Avg.0–5%5–10%10–15%>15%Avg.0–5%5–10%10–15%>15%05.7%55.0%35.0%4.4%5.6%6.8%42.8%35.0%14.4%7.8%23.2%76.1%22.8%1.1%0.0%4.7%57.2%35.0%7.2%0.6%52.1%90.6%9.4%0.0%0.0%4.5%60.6%32.2%7.2%0.0%101.4%95.6%4.4%0.0%0.0%5.1%53.3%34.4%12.2%0.0% Performanceanalysisofmulti-servertandemqueues181Table2.OverallresultsfortandemqueueswithdifferentbalancingratesRatesErrorinthroughputErrorinmeansojourntimeunbalancedAvg.0–5%5–10%10–15%>15%Avg.0–5%5–10%10–15%>15%server-group(miµp,i)1.03.3%76.4%20.8%1.4%1.4%3.4%74.3%22.2%2.1%1.4%1.13.1%78.5%18.1%2.1%1.4%4.0%68.1%27.1%3.5%1.4%1.23.0%79.2%18.8%0.7%1.4%4.6%59.7%34.7%4.2%1.4%1.53.0%81.3%16.0%1.4%1.4%6.5%38.2%43.1%16.7%2.1%2.03.1%81.3%16.0%1.4%1.4%7.9%27.1%43.8%25.0%4.2%Table3.OverallresultsfortandemqueueswithdifferentcoefficientsofvariationoftheservicetimesCoefficientsErrorinthroughputErrorinmeansojourntimeofvariationAvg.0–5%5–10%10–15%>15%Avg.0–5%5–10%10–15%>15%2(cp,i)0.14.4%54.2%44.2%1.7%0.0%3.1%77.5%21.7%0.8%0.0%0.22.6%88.3%11.7%0.0%0.0%3.4%75.8%22.5%1.7%0.0%0.52.2%90.8%9.2%0.0%0.0%4.5%60.8%32.5%6.7%0.0%1.01.5%93.3%2.5%4.2%0.0%4.1%64.2%30.0%5.0%0.8%1.53.0%82.5%13.3%0.0%4.2%7.5%25.8%54.2%15.0%5.0%2.04.8%66.7%26.7%2.5%4.2%9.1%16.7%44.2%32.5%6.7%Table4.Overallresultsfortandemqueueswithadifferentnumberofserversperserver-groupNumberofErrorinthroughputErrorinmeansojourntimeservers(mi)Avg.0-5%5–10%10–15%>15%Avg.0–5%5–10%10–15%>15%All12.9%83.8%9.2%2.9%4.2%5.9%46.3%39.2%10.0%4.6%All53.8%68.3%30.8%0.8%0.0%4.6%60.0%29.2%10.8%0.0%Mixed2.6%85.8%13.8%0.4%0.0%5.3%54.2%34.2%10.0%1.7%Wemayconcludethefollowingfromtheaboveresults.First,weseeinTable1thattheperformanceoftheapproximationbecomesbetterwhenthebuffersizesincrease.Thismaybeduetolessdependenciesbetweentheservers-groupswhenthebuffersarelarge.Wealsonoticethattheperformanceisbetterforbalancedlines(Table2);forunbalancedlines,especiallytheestimateforthemeansojourntimeisnotasgoodasforbalancedlines.Ifwelookatthecoefficientsofvariationoftheservicetimes(Table3),wegetthebestapproximationsforthethroughputwhenthecoefficients 182M.vanVuurenetal.Table5.Overallresultsfortandemqueueswith4and8server-groupsNumberofErrorinthroughputErrorinmeansojourntimeserver-Avg.0–5%5–10%10–15%>15%Avg.0–5%5–10%10–15%>15%groups(M)42.3%87.2%12.2%0.6%0.0%4.7%57.5%32.8%9.7%0.0%83.9%71.4%23.6%2.2%2.8%5.8%49.4%35.6%10.8%4.2%Table6.Detailedresultsforbalancedtandemqueues2miMcp,iBuffersTApp.TSim.Diff.SApp.SSim.Diff.140.100.7350.771−4.7%4.704.631.5%820.9060.926−2.2%16.1415.990.9%4100.9810.985−0.4%19.2219.031.0%81.000.4880.44310.2%11.7313.43−12.7%420.7030.7000.4%9.099.25−1.7%8100.8550.8550.0%49.5249.81−0.6%41.500.5040.4736.6%5.826.27−7.2%820.6070.5814.5%21.9423.52−6.7%4100.8340.835−0.1%22.3822.310.3%540.100.7890.856−7.8%22.4821.783.2%820.8270.926−10.7%52.3549.715.3%4100.9270.983−5.7%36.8835.244.7%81.000.6930.697−0.6%49.2049.140.1%420.7970.808−1.4%26.3726.170.8%8100.8670.882−1.7%83.0983.96−1.0%41.500.7420.7242.5%22.9923.90−3.8%820.7590.7373.0%54.6357.27−4.6%4100.8670.874−0.8%37.9738.86−2.3%Mixed40.100.7460.793−5.9%16.1916.28−0.6%820.8450.921−8.3%39.9038.962.4%4100.9560.984−2.8%31.6130.055.2%81.000.6190.6042.5%37.9038.55−1.7%420.7560.757−0.1%20.1520.140.0%8100.8630.871−0.9%71.6771.74−0.1%41.500.6330.6192.3%16.7818.01−6.8%820.7050.6784.0%43.3846.32−6.3%4100.8500.856−0.7%31.4332.37−2.9% Performanceanalysisofmulti-servertandemqueues183Table7.Detailedresultsforunbalancedtandemqueues2miMcp,iBuffersTApp.TSim.Diff.SApp.SSim.Diff.180.100.7180.751−4.4%8.909.27−4.0%420.9600.9580.2%6.186.41−3.6%8100.9800.983−0.3%38.4543.22−11.0%41.000.5940.5615.9%4.845.28−8.3%820.6900.6703.0%18.8120.31−7.4%4100.9180.9120.7%16.2017.41−7.0%81.500.4820.40917.8%11.2613.79−18.3%420.7140.6913.3%8.038.60−6.6%8100.8300.8191.3%46.7550.16−6.8%580.100.7810.851−8.2%43.0342.650.9%420.9020.958−5.8%21.6321.500.6%8100.9220.983−6.2%71.8973.95−2.8%41.000.8010.7940.9%20.7921.13−1.6%820.7890.7870.3%51.5253.49−3.7%4100.9270.929−0.2%30.3732.61−6.9%81.500.7300.6925.5%44.4347.95−7.3%420.8500.8282.7%21.9523.70−7.4%8100.8640.8620.2%74.6981.01−7.8%Mixed80.100.7440.790−5.8%30.9632.41−4.5%40.120.9200.953−3.5%16.7217.14−2.5%80.1100.9450.983−3.9%61.0062.54−2.5%41.000.7140.7021.7%16.2216.43−1.3%81.020.7500.7421.1%39.6442.20−6.1%41.0100.9260.9190.8%25.9927.60−5.8%81.500.6280.5886.8%32.6837.66−13.2%41.520.7870.7731.8%17.5218.93−7.4%81.5100.8440.8430.1%61.8269.32−10.8%ofvariationare1,andalsotheestimateforthemeansojourntimeisbetterforsmallcoefficientsofvariation.Thequalityoftheresultsseemstoberatherinsensitivetothenumberofserversperserver-group(Table4),inspiteofthesuper-serverapproximationusedformulti-servermodels.FinallywemayconcludefromTable5thattheresultsarebetterforshortertandemqueues.Mostcrucialtothequalityoftheapproximationofthethroughputappearstobethebuffer-size.Forthesojourntimethisappearstobethecoefficientofvariationoftheservicetime.InFigures4and5wepresentascatter-plotofsimulationresultsversusapproximationresultsforthethroughputandmeansojourntimes;theplottedcasesarethesameasinTables6and7.Theresultsofthethroughputaresplit-up 184M.vanVuurenetal.Fig.4.Scatter-plotofthethroughputof54casessplitupbybuffer-sizeaccordingtothebuffer-size;theoneforthesojourntimesaresplit-upaccordingtothesquaredcoefficientofvariationoftheservicetimes.Overallwecansaythattheapproximationproducesaccurateresultsinmostcases.Inthemajorityofthecasestheerrorofthethroughputiswithin5%ofthesimulationandtheerrorofthemeansojourntimeiswithin10%ofthesimulation(seealsoTables6and7).Theworstperformanceisobtainedforunbalancedlineswithzerobuffersandhighcoefficientsofvariationoftheservicetimes.Butthesecasesareunlikely(andundesired)tooccurinpractice.Thecomputationtimesareveryshort.Onamoderncomputerthecomputationtimesaremuchlessthanasecondinmostcases,onlyincaseswithservicetimeswithlowcoefficientsofvariationand1serverperserver-groupthecomputationtimesincreasetoafewseconds.Therefore,forthedesignofproductionlines,thisisaveryusefulapproximationmethod.6.2ComparisonwithQNATWealsocomparethepresentmethodwithQNAT,amethoddevelopedbyTahilra-manietal.[21].Weuseatandemqueuewithfourserver-groups.Itwasonlypossibletotestcaseswherethefirstserver-groupconsistsofasingleexponentialserver.Thereasonisthatthetwomethodsassumeadifferentarrivalprocesstothesystem.Bothprocesses,however,coincideforthespecialcaseofasingleexponentialserveratthebeginningoftheline.Wevariedthenumberofserversperserver-groupandthesizeofbuffers.Table8showstheresults. Performanceanalysisofmulti-servertandemqueues185Fig.5.Scatter-plotofthemeansojourntimeof54casessplitupbycoefficientofvariationTable8.ComparisonofourmethodwithQNATTPTPOurTPQNATSoj.Soj.OurSoj.QNATmibiSim.App.errorQNATErrorSim.App.errorQNATerror(1,1,1,1)00.5150.537−4.3%0.5002.9%5.955.615.7%––(1,1,1,1)20.7020.703−0.1%0.750−6.8%9.259.101.7%8.1711.7%(1,1,1,1)100.8790.8760.3%0.917−4.3%21.4321.410.1%18.5513.5%(1,5,5,5)00.7110.717−0.8%0.16776.5%17.8717.671.1%––(1,5,5,5)20.7910.7880.3%0.800−1.1%20.5320.450.4%––(1,5,5,5)100.8980.8841.6%0.8950.3%32.2732.59−1.0%22.8829.1%(1,4,2,8)00.6770.692−2.3%0.20070.5%16.5916.281.9%––(1,4,2,8)20.7750.7740.1%0.800−3.2%19.2919.150.7%––(1,4,2,8)100.8930.8860.8%0.902−1.0%31.0330.860.6%23.0425.7%WeseethatthepresentapproximationmethodismuchmorestablethanQNATandgivesinalmostallcasesbetterresults.Especiallytheapproximationofthemeansojourntimeismuchbetter;inanumberofcasesQNATisnotabletoproduceanapproximationofthemeansojourntime.Ofcourse,oneshouldbecarefulwithdrawingconclusionsfromthislimitedsetofcases.Table8onlygivesanindicationofhowthetwomethodsperform. 186M.vanVuurenetal.6.3IndustrialcaseTogiveanindicationoftheperformanceofourmethodinpractice,wepresenttheresultsofanindustrialcase.Thecaseinvolvesaproductionlinefortheproductionoflightbulbs.Theproductionlineconsistsof5productionstageswithbuffersinbetween.Eachstagehasadifferentnumberofmachinesvaryingbetween2and8.Themachineshavedeterministicservicetimes,buttheydosufferfrombreakdowns.Inthequeueingmodelweincludedthebreakdownsintothecoefficientofvariationoftheservicetimes,yieldingeffectiveservicetimeswithcoefficientsofvariationlargerthan0.InTable9theparametersoftheproductionlineareshown.Table9.Parametersfortheproductionlinefortheproductionofbulbs2Stagemiµp,icp,ibi125.730.96−281.530.0921343.430.80114132.180.57345416.120.9619Weonlyhavedataofthethroughputandnotofthemeansojourntimeoftheline,sowecanonlytesttheapproximationforthethroughput.Theoutputoftheproductionlinebasedonthemeasureddatais11.34productspertimeunit.Ifwesimulatethisproductionline,weobtainathroughputof11.41productspertimeunit.Thethroughputgivenbyourapproximationmethodis11.26,sointhiscasetheapproximationisagoodpredictionfortheactualthroughput.7ConcludingremarksInthispaperwedescribedamethodfortheapproximateanalysisofamulti-servertandemqueuewithfinitebuffersandgeneralservicetimes.Wedecomposedthetandemqueueinsubsystems.Weusedaniterativealgorithmtoapproximatethearrivalsanddeparturesatthesubsystemsandtoapproximatesomeperformancecharacteristicsofthetandemqueue.Eachmulti-serversubsystemisapproximatedbyasingle(super)serverqueuewithstate-dependentinter-arrivalandservicetimes,thesteady-statequeuelengthdistributionofwhichisdeterminedbyaspectralexpansionmethod.Thismethodisrobustandefficient;itprovidesagoodandfastalternativetosimulationmethods.Inmostcasestheerrorsforperformancecharacteristicsasthethroughputandmeansojourntimearewithin5%ofthesimulationresults.Numer-icalresultsalsogiveanindicationoftheperformanceofthemethodcomparedwithQNAT.Themethodcanbeextendedinseveraldirections.Onemaythinkofmore Performanceanalysisofmulti-servertandemqueues187Fig.6.Phasediagramofanarbitraryinter-departuretimegeneralconfigurations,likesplittingandmergingofstreamsorthepossibilityoffeedback.Otherpossibilitiesforextensionareforexampleunreliablemachinesandassembly/disassembly(see[24]).Possibilitiesforimprovingthequalityoftheap-proximationare,forexample,usingamoredetaileddescriptionofthearrivaltoanddeparturesfromthesubsystems(e.g.includingcorrelationsbetweenconsecutivearrivalsanddepartures)orimprovingthesubsystemanalysisbyusingadescriptionoftheserviceprocessthatismoredetailedthanthesuper-serverapproach.Appendix:SuperpositionofserviceprocessesLetusconsidermindependentserviceprocesses,eachofthemcontinuouslyser-vicingcustomersoneatatime.Theservicetimesareassumedtobeindependentandidenticallydistributed.Weareinterestedinthefirsttwomomentsofanarbi-traryinter-departuretimeofthesuperpositionofmserviceprocesses.BelowwedistinguishbetweenCoxian2servicetimesandErlangk−1,kservicetimes.A.1Coxian2servicetimesWeassumethattheservicetimesofeachserviceprocessareCoxian2distributedwiththesameparameters.Therateofthefirstphaseisµ1,therateofthesecondphaseisµ2andtheprobabilitythatthesecondphaseisneededisq.Thedistributionofanarbitraryinter-departuretimeofthesuperpositionofmserviceprocessescanbedescribedbyaphase-typedistributionwithm+1phases,numbered0,1,...,m.Inphaseiexactlyiserviceprocessesareinthesecondphaseoftheservicetimeandm−iserviceprocessesareinthefirstphase.Aphasediagramofthephase-typedistributionofanarbitraryinter-departuretimeisshowninFigure6.Theprobabilitytostartinphaseiisdenotedbyai,i=0,...,m−1.ThesojourntimeinphaseiisexponentiallydistributedwithrateR(i),andpiistheprobabilitytocontinuewithphasei+1aftercompletionofphasei.Nowweexplainhowtocomputetheparametersai,R(i)andpi.Theprobabilityaicanbeinterpretedasfollows.Itistheprobabilitythatiserviceprocessesareinphase2justafteradeparture(i.e.,servicecompletion).Thereisatleastoneprocessinphase1,namelytheonethatgeneratedthedeparture.Sincetheserviceprocessesaremutuallyindependent,thenumberofserviceprocessesinphase2isbinomiallydistributedwithm−1trialsandsuccessprobabilityp. 188M.vanVuurenetal.Thesuccessprobabilityisequaltothefractionoftimeasingleserviceprocessisinphase2,soqµ1p=.qµ+µ2Hence,fortheinitialprobabilityaiwegetim−1−im−1qµ1µ2ai=(21)iqµ1+µ2qµ1+µ2TodeterminetherateR(i),notethatinstateithereareiprocessesinphase2andm−iinphase1,sothetotalrateatwhichoneoftheserviceprocessescompletesaservicephaseisequaltoR(i)=(m−i)µ1+iµ2(22)Itremainstofindpi,theprobabilitythatthereisnodepartureafterphasei.Inphaseithreethingsmayhappen:–Case(i):Aserviceprocesscompletesphase1andimmediatelycontinueswithphase2;–Case(ii):Aserviceprocesscompletesphase1andgeneratesadeparture;–Case(iii):Aserviceprocesscompletesphase2(andthusalwaysgeneratesadeparture).Clearly,piistheprobabilitythatcase(i)happens,soq(m−i)µipi=(23)R(i)Nowtheparametersofthephase-typedistributionareknown,wecandetermineitsfirsttwomoments.LetXidenotethetotalsojourntime,giventhatwestartinphasei,i=0,1,...,m.Startingwith122EXm=,EXm=2,R(m)R(m)thefirsttwomomentsofXicanbecalculatedfromi=m−1downtoi=0byusing1EXi=+piEXi,(24)R(i)222EXi+12EXi=2+pi+EXi+1.(25)R(i)R(i)Thentherateµsandcoefficientofvariationcsofanarbitraryinter-departuretimeofthesuperpositionofmserviceprocessesfollowfromm11qµ−1=aEX=+,(26)siimµ1µ2i=0mc2=µ2aEX2−1(27)ssiii=0 Performanceanalysisofmulti-servertandemqueues189A.2Erlangk−1,kservicetimesNowtheservicetimesofeachserviceprocessareassumedtobeErlangk−1,kdistributed,i.e.,withprobabilityp(respectively1−p)aservicetimeconsistsofk−1(respectivelyk)exponentialphaseswithparameterµ.Clearly,thetimethatelapsesuntiloneofthemserviceprocessescompletesaservicephaseisexponentialwithparametermµ.Thenumberofservicephasescompletionsbeforeoneoftheserviceprocessesgeneratesadeparturerangesfrom1uptom(k−1)+1.Sothedistributionofanarbitraryinter-departuretimeofthesuperpositionofmserviceprocessesisamixtureofErlangdistributions;withprobabilitypiitconsistsofiexponentialphaseswithparametermµ,i=1,...,m(k−1)+1.Figure7depictsthephasediagram.Belowweshowhowtodeterminetheprobabilitiespi.Anarbitraryinter-departuretimeofthesuperpositionofmserviceprocessesistheminimumofm−1equilibriumresidualservicetimesandonefullservicetime.Bothresidualandfullservicetimehavea(different)mixedErlangdistribution.Inparticular,theresidualserviceconsistswithprobabilityriofiphaseswithparameterµ,where1/(k−p),i=1,2,...,k−1;ri=(1−p)/(k−p),i=k.TheminimumoftwomixedErlangservicetimeshasagainamixedErlangdistribu-tion;belowweindicatehowtheparametersofthedistributionoftheminimumcanbedetermined.ThenrepeatedapplicationofthisprocedureyieldstheminimumofmmixedErlangservicetimes.LetX1andX2betwoindependentrandomvariableswithmixedErlangdis-tributions,i.e.,withprobabilityqk,itherandomvariableXk(k=1,2)consistsofiexponentialphaseswithparameterµk,i=1,...,nk.ThentheminimumofX1Fig.7.Phasediagramofanarbitraryinder-departuretime 190M.vanVuurenetal.andX2consistsofatmostn1+n2−1exponentialphaseswithparameterµ1+µ2.Tofindtheprobabilityqithattheminimumconsistsofiphases,weproceedasfollows.Defineqi(j)astheprobabilitythattheminimumofX1andX2consistsofiphasestransitions,wherej(≤i)transitionsareduetoX1andi−jtransitionsareduetoX2.Obviouslywehavemin(i,n1)qi=qi(j),i=1,2,...,n1+n2−1.j=max(0,i−n2)Todetermineqi(j)notethattheithphasetransitionoftheminimumcanbeduetoeitherX1orX2.IfX1makesthelasttransition,thenX1clearlyconsistsofexactlyjphasesandX2ofatleasti−j+1phases;theprobabilitythatX2makesi−jtransitionsbeforethejthtransitionofX1isnegative-binomiallydistributedwithparametersjandµ1/(µ1+µ2).TheresultissimilarifX2insteadofX1makesthelasttransition.Hence,weobtain⎛⎞ji−jn2i−1µ1µ2⎝⎠qi(j)=q1,jq2,kj−1µ1+µ2µ1+µ2k=i−j+1⎛⎞ji−jn1i−1µ1µ2⎝⎠q+q1,k2,i−j,jµ1+µ2µ1+µ2k=j+11≤i≤n1+n2−1,0≤j≤i,wherebyconvention,q1,0=q2,0=0.Byrepeatedapplicationoftheaboveprocedurewecanfindtheprobabilitypithatthedistributionofanarbitraryinter-departuretimeofthesuperpositionofmErlangk−1,kserviceprocessesconsistsofexactlyiservicephaseswithparametermµ,i=1,2,...,m(k−1)+1.Itisnoweasytodeterminetherateµsandcoefficientofvariationcsofanarbitraryinter-departuretime,yielding−11p(k−1)(1−p)kk−pµs=+=,mµµmµand,byusingthatthesecondmomentofanEkdistributionwithscaleparameterµisk(k+1)/µ2,m(k−1)+1m(k−1)+122i(i+1)1cs=µspi(mµ)2−1=−1+(k−p)2pii(i+1).i=1i=1A.3Equilibriumresidualinter-departuretimeTodeterminethefirsttwomomentsoftheequilibriumresidualinter-departuretimeofthesuperpositionofmindependentserviceprocessesweadoptthefollowingsimpleapproach.LettherandomvariableDdenoteanarbitraryinter-departuretimeandletRdenotetheequilibriumresidualinter-departuretime.ItiswellknownthatE(D2)E(D3)E(R)=,E(R2)=.2E(D)3E(D) Performanceanalysisofmulti-servertandemqueues191IntheprevioussectionswehaveshownhowthefirsttwomomentsofDcanbedeterminedincaseofCoxian2andErlangk−1,kservicetimes.ItsthirdmomentisapproximatedbythethirdmomentofthedistributionfittedonthefirsttwomomentsofD,accordingtotherecipeinSection3.References1.BertsimasD(1990)AnanalyticapproachtoageneralclassofG/G/squeueingsystems.OperationsRearchearch1:139–1552.BuzacottJA(1967)Automatictransferlineswithbufferstock.InternationalJournalofProductionResearch5:183–2003.CruzFRB,MacGregorSmithJ(2004)AlgorithmforanalysisofgeneralizedM/G/C/Cstatedependentqueueingnetworks.http://www.compscipreprints.com/comp/Preprint/fcruzfcruz/20040105/14.CruzFRB,MacGregorSmithJ,QueirozDC(2004)ServiceandcapacityallocationinM/G/C/Cstatedependentqueueingnetworks.Computers&OperationsResearch(toappear)5.DalleryY,DavidR,XieX(1989)Approximateanalysisoftransferlineswithunreliablemachinesandfinitebuffers.IEEETransactionaonAutomaticControl34(9):943–9536.DalleryY,GershwinB(1992)Manufacturingflowlinesystems:areviewofmodelsandanalyticalresults.QueueingSystems12:3–947.GershwinSB,BurmanMH(2000)Adecompositionmethodforanalyzinginhomoge-neousassembly/disassemblysystems.AnnalsofOperationResearch93:91–1158.HillierFS,SoKC(1995)Ontheoptimaldesignoftandemqueueingsystemswithfinitebuffers.QueueingSystemsTheoryApplication21:245–2669.JainS,MacGregorSmithJ(1994)OpenfinitequeueingnetworkswithM/M/C/Kparallelservers.ComputersOperationsResearch21(3):297–31710.JohnsonMA(1993)Anempiricalstudyofqueueingapproximationsbasedonphase-typedistributions.CommunicationStatistic-StochasticModels9(4):531–56111.KerbacheL,MacGregorSmithJ(1987)Thegeneralizedexpansionmethodforopenfinitequeueingnetworks.TheEuropeanJournalofOperationsResearch32:448–46112.KouvatsosD,XeniosNP(1989)MEMforarbitraryqueueingnetworkswithmultiplegeneralserversandrepetative-serviceblocking.PerformanceEvaluation10:169–19513.LiY,CaiX,TuF,ShaoX(2004)Optimizationoftandemqueuesystemswithfinitebuffers.Computers&OperationsResearch31:963–98414.MarieRA(1980)Calculatingequilibriumprobabilitiesforλ(n)/Ck/1/Nqueue.Pro-ceedingsPerformance’80,Toronto,pp117–12515.MickensR(1987)Differenceequations.VanNostrand-Reinhold,NewYork16.MitraniI,MitraD(1992)Aspectralexpansionmethodforrandomwalksonsemi-infinitestrips.In:BeauwensR,deGroenP(eds)Iterativemethodsinlinearalgebra,pp141–149.North-Holland,Amsterdam17.PerrosHG(1989)Abibliographyofpapersonqueueingnetworkswithfinitecapacityqueues.PerfEval10:255–26018.PerrosHG(1994)Queueingnetworkswithblocking.OxfordUniversityPress,Oxford19.PerrosHG,AltiokT(1989)Queueingnetworkswithblocking.North-Holland,Ams-terdam20.MacGregorSmithJ,CruzFRB(2000)Thebufferallocationproblemforgeneralfinitebufferqueueingnetworks.http://citeseer.nj.nec.com/smith00buffer.html21.TahilramaniH,ManjunathD,BoseSK(1999)ApproximateanalysisofopennetworkofGE/GE/m/Nqueueswithtransferblocking.MASCOTS’99,pp164–172 192M.vanVuurenetal.22.TijmsHC(1994)Stochasticmodels:analgorithmicapproach.Wiley,Chichester23.TolioT,MattaA,GershwinSB(2002)Analysisoftwo-machinelineswithmultiplefailuremodes.IIETransactions34:51–6224.vanVuurenM(2003)Performanceanalysisofmulti-servertandemqueueswithfinitebuffers.Master’sThesis,UniversityofTechnologyEindhoven,TheNetherlands25.http://poisson.ecse.rpi.edu/hema/qnat/ AnanalyticalmethodfortheperformanceevaluationofechelonkanbancontrolsystemsSteliosKoukoumialosandGeorgeLiberopoulosDepartmentofMechanicalandIndustrialEngineering,UniversityofThessaly,Volos,Greece(e-mail:skoukoum@mie.uth.gr;glib@mie.uth.gr)Abstract.Wedevelopageneralpurposeanalyticalapproximationmethodfortheperformanceevaluationofamulti-stage,serial,echelonkanbancontrolsystem.Thebasicprincipleofthemethodistodecomposetheoriginalsystemintoasetofnestedsubsystems,eachsubsystembeingassociatedwithaparticularechelonofstages.Eachsubsystemisanalyzedinisolationusingaproduct-formapproximationtechnique.Aniterativeprocedureisusedtodeterminetheunknownparametersofeachsubsystem.Numericalresultsshowthatthemethodisfairlyaccurate.Keywords:Production/inventorycontrol–Multi-stagesystem–Echelonkanban–Performanceevaluation1IntroductionIn1960,ClarkandScarf[10]initiatedtheresearchonthecoordinationofmulti-stage,serial,uncapacitatedinventorysystemswithstochasticdemandandconstantleadtimes.Theirworkreceivedconsiderableattentionintheyearsthatfollowedandspawnedalargeamountoffollow-onresearch.Muchofthatresearchevolvedaroundvariantsofthebasestockcontrolsystem.Researchonthecoordinationofmulti-stage,serial,production/inventorysystemshavingnetworksofstationswithlimitedcapacity,ontheotherhand,hasbeendirectedmostlytowardsvariantsofthekanbancontrolsystem.Inthispaper,wedevelopananalyticalapproximationmethodfortheperformanceevaluationofanechelonkanbancontrolsystem,usedforthecoordinationofproductioninamulti-stage,serialproduction/inventorysystem.Wetestthebehaviorofthismethodwithseveralnumericalexamples.Theterm“echelonkanban”wasintroducedin[19].Thebasicprincipleoftheoperationoftheechelonkanbancontrolsystemisverysimple:WhenapartleavesCorrespondenceto:G.Liberopoulos 194S.KoukoumialosandG.Liberopoulosthelaststageofthesystemtosatisfyacustomerdemand,anewpartisdemandedandauthorizedtobereleasedintoeachstage.Itisworthnotingthattheechelonkanbancontrolsystemisequivalenttotheintegralcontrolsystemdescribedin[8].Theechelonkanbancontrolsystemdiffersfromtheconventionalkanbancontrolsystem,whichisreferredtoasinstallationkanbancontrolsystemorpolicyin[19],inthatintheconventionalkanbancontrolsystem,anewpartisdemandedandauthorizedtobereleasedintoastagewhenapartleavesthisparticularstageandnotwhenapartleavesthelaststage,asisthecasewiththeechelonkanbancontrolsystem.Thisimpliesthatintheconventionalkanbancontrolsystem,theplacementofademandandanauthorizationfortheproductionofanewpartintoastageisbasedonlocalinformationfromthisstage,whereasintheechelonkanbancontrolsystem,itisbasedonglobalinformationfromthelaststage.Thisconstitutesapotentialadvantageoftheechelonkanbancontrolsystemovertheconventionalkanbancontrolsystem.Moreover,theechelonkanbancontrolsystem,justliketheconventionalkanbancontrolsystem,dependsononlyoneparameterperstage,thenumberofechelonkanbans,aswewillseelateron,andisthereforesimplertooptimizeandimplementthanmorecomplicatedkanban-typecontrolsystemsthatdependoftwoparametersperstage,suchasthegeneralizedkanbancontrolsystem[7]andtheextendedkanbancontrolsystem[12].Thesetwoapparentadvantagesoftheechelonkanbancontrolsystemmotivatedourefforttodevelopanapproximationmethodforitsperformanceevaluation.Kanban-typeproduction/inventorysystemshaveoftenbeenmodeledasqueue-ingnetworksintheliterature.Consequently,mostofthetechniquesthathavebeendevelopedfortheanalysisofkanban-typeproduction/inventorysystemsarebasedonmethodsfortheperformanceevaluationofqueueingnetworks.Exactanalyticalsolutionsexistforaclassofqueueingnetworksknownasseparable,inwhichthesteady-statejointprobabilitieshaveaproduct-formsolution.Jackson[18]wasthefirsttoshowthatthesteady-statejointprobabilityofanopenqueueingnetworkwithPoissonarrivals,exponentialservicetimes,probabilisticrouting,andfirst-come-first-served(FCFS)servicedisciplineshasaproduct-formsolution,whereeachstationofthenetworkcanbeanalyzedinisolationasanM/M/1queue.ForclosedqueueingnetworksoftheJacksontype,GordonandNewell[17]showedthatananalytical,product-formsolutionalsoexists.Theperformanceparametersofsuchnetworkscanbeobtainedusingefficientalgorithms,suchasthemeanvalueanaly-sis(MVA)algorithm[22]andtheconvolutionalgorithm[9].TheBCMPtheorem[1]summarizesextensionsofproduct-formnetworksthatincorporatealternativeservicedisciplinesandseveralclassesofcustomers.Sincetheclassofqueueingnetworksforwhichanexactsolutionisknown(separablenetworks)istoorestrictiveformodelingandanalyzingrealsystems,muchworkhasbeendevotedtothedevelopmentofapproximationmethodsfortheanalysisofnon-separablenetworks.Whitt[27]presentedanapproximationmethodfortheanalysisofageneralopenqueueingnetworkthatisbasedondecomposingthenetworkintoasetofGI/GI/1queuesandanalyzingeachqueueinisolation.Inthecaseofclosedqueueingnetworks,theapproximationmethodsareforthemostpartbasedontwoapproaches.ThefirstapproachreliesonheuristicextensionsoftheMVAalgorithm(e.g.[23]).Thesecondapproachreliesonapproximatingthe Ananalyticalmethodfortheperformanceevaluation195performanceoftheoriginalnetworkbythatofanequivalentproduct-formnetwork.Spanjersetal.[24]developedamethodthatisbasedonthesecondapproachforaclosed-loop,two-indenture,repairable-itemsystem.Interestingly,theirsystemisequivalenttoanechelonkanbancontrolsystemwithafinitepopulationofexter-naljobs.Theirmethodaggregatesseveralstatesoftheunderlyingcontinuous-timeMarkovchainandadjustssomeserviceratesusingNorton’sTheoremforclosedqueueingnetworkstoobtainaproduct-formsolution.Amongthedifferentmethodsthatrelyonthesecondapproach,Marie’smethod[20]hasattractedconsiderableat-tention.ExtensionsandcomparativestudiesofMarie’smethodhavebeenproposedforavarietyofqueueingnetworks[2–5],and[11].DiMascolo,FreinandDallery[14,16]developedapproximationmethodsbasedonMarie’smethodfortheperfor-manceevaluationoftheconventionalkanbancontrolsystemandthegeneralizedkanbancontrolsystem.TheapproximationmethodthatwedevelopinthispaperfortheperformanceevaluationoftheechelonkanbancontrolsystemreliesonMarie’smethod.Todevelopourmethod,wefirstmodelthesystemasanopenqueueingnetworkwithsynchronizationstations.Byexchangingtherolesofjobs(parts)andresources(echelonkanbans)intheopennetwork,weobtainanequivalent,multi-class,nested,closedqueueingnetwork,inwhichthepopulationofeachclassisequaltothejobcapacityornumberofechelonkanbansoftheechelonofstagesassociatedwithaparticularstage.Theechelonofstagesassociatedwithaparticularstageisthestageitselfandallitsdownstreamstages.Wethendecomposetheclosednetworkintoasetofnestedsubsystems,eachsubsystembeingassociatedwithaparticularclass.Thismeansthatwehaveasmanysubsystemsasthenumberofthestages.EachsubsystemisanalyzedinisolationusingMarie’smethod.Eachsubsysteminteractswithitsneighboringsubsystemsinthatitincludesitsdownstreamsubsystemintheformofasingle-serverstationwithload-dependent,exponentialservicerates,anditreceivesexternalarrivalsfromitsupstreamsubsystem.Afixed-point,iterativeprocedureisusedtodeterminetheunknownparametersofeachsubsystembytakingintoaccounttheinteractionsbetweenneighboringsubsystems.Therestofthispaperisorganizedasfollows.InSection2,wedescribetheexactoperationoftheechelonkanbancontrolsystembymeansofasimpleexam-ple.InSection3wepresentthequeueingnetworkmodeloftheechelonkanbancontrolsystemandtheperformancemeasuresofthesystemthatweareinterestedinevaluating.InSection4,wedescribethedecompositionoftheoriginalsystemintomanysubsystems.InSection5,wepresenttheanalysisinisolationofeachsub-system,andinSection6wedeveloptheanalysisoftheentiresystem.InSection7,wepresentnumericalresultsontheeffectsandoptimizationoftheparameters.Finally,inSection8,wedrawconclusions.Theanalysisofthesynchronizationsta-tionsthatappearinthequeueingnetworkmodelsofeachsubsystemispresentedinAppendicesAandB,andatableofthenotationusedinthepaperisgiveninAppendixC. 196S.KoukoumialosandG.LiberopoulosRawManufacturingOutputFinishedPartsProcess1Buffer1PartsM1M2M3M4M5M6M7M8M9Stage1Stage2Stage3CustomerDemandsFig.1.Aserialproductionsystemdecomposedintothreestagesinseries2TheechelonkanbancontrolsystemInthissection,wegiveaprecisedescriptionoftheoperationoftheechelonkan-bancontrolsystembymeansofasimpleexample.Inthisexample,weconsideraproductionsystemthatconsistsofM=9machinesinseries,labeledM1toM9,producesasingleparttype,anddoesnotinvolveanybatching,reworkingorscrappingofparts.Eachmachinehasarandomprocessingtime.AllpartsvisitsuccessivelymachinesM1toM9.TheproductionsystemisdecomposedintoN=3stages.Eachstageisaproduction/inventorysystemconsistingofamanufactur-ingprocessandanoutputbuffer.Theoutputbufferstoresthefinishedpartsofthestage.Themanufacturingprocessconsistsofasubsetofmachinesoftheoriginalmanufacturingsystemandcontainspartsthatareinserviceorwaitingforserviceonthemachines.Thesepartsrepresenttheworkinprocess(WIP)ofthestageandareusedtosupplytheoutputbuffer.Intheexample,eachstageconsistsofthreemachines.Morespecifically,thesetsofmachines{M1,M2,M3},{M4,M5,M6}and{M7,M8,M9}belongtostages1,2and3,respectively.ThedecompositionoftheproductionsystemintothreestagesisillustratedinFigure1.Eachstagehasassociatedwithitanumberofechelonkanbansthatareusedtodemandandauthorizethereleaseofpartsintothisstage.Anechelonkanbanofaparticularstagetracesaclosedpaththroughthisstageandallitsdownstreamstages.ThenumberofechelonkanbansofstageiisfixedandequaltoKi.Theremustbeatleastoneechelonkanbanofstageiavailableinordertoreleaseanewpartintothisstage.Ifsuchakanbanisavailable,thekanbanisattachedontothepartandfollowsitthroughthesystemuntiltheoutputbufferofthelaststage.SinceanechelonkanbanofstageiisattachedtoeverypartinanystagefromitoN,thenumberofpartsinstagesitoNislimitedbyKi.PartsthatareintheoutputbufferofstageNarethefinishedpartsoftheproductionsystem.Thesepartsareusedtosatisfycustomerdemands.Whenacustomerdemandarrivestothesystem,ademandforthedeliveryofafinishedpartfromtheoutputbufferofthelaststagetothecustomerisplaced.Iftherearenofinishedpartsintheoutputbufferofthelaststage,thedemandcannotbeimmediatelysatisfiedandisbackordereduntilafinishedpartbecomesavailable.Ifthereisatleastonefinishedpartintheoutputbufferofthelaststage,thispartisdeliveredtothecustomerafterreleasingthekanbansofallthestages(1,2,and3,intheexample)thatwereattachedtoit,hencethedemandisimmediatelysatisfied.Thereleasedkanbansareimmediatelytransferredupstreamtotheircorrespondingstages.Thekanbanofstageicarrieswithitademandfortheproductionofanew Ananalyticalmethodfortheperformanceevaluation197stage−ifinishedpartandanauthorizationtoreleaseafinishedpartfromtheoutputbufferofstagei−1intostagei.Whenafinishedpartofstagei−1istransferredtostagei,thestage-ikanbanisattachedtoitontopofthekanbansofstages1toi−1,whichhavealreadybeenattachedtothepartatpreviousstages.Withthisinmind,wecanjustaswellassumethatKi≥Ki+1,i=1,...,N−1.(1)3QueueingnetworkmodeloftheechelonkanbancontrolsystemInordertodeveloptheapproximationmethodfortheperformanceevaluationoftheechelonkanbancontrolsystem,wefirstmodelthesystemasanopenqueueingnetworkwithsynchronizationstations.Figure2showsthequeueingnetworkmodeloftheechelonkanbancontrolsystemwiththreestagesinseries,consideredinSection2.Themanufacturingprocessofeachstageismodeledasasubnetworkinwhichthemachinesofthemanufacturingprocessarerepresentedbysingle-serverstations.ThesubnetworkassociatedwiththemanufacturingprocessofstageiisdenotedbyLi,andthesingle-serverstationsrepresentingmachinesM1,...,M9aredenotedbyS1,...,S9,respectively.ThenumberofstationsofsubnetworkLiisdenotedbymi.Intheexample,mi=3,i=1,2,3.Theechelonkanbancontrolmechanismismodeledviathreesynchronizationstations,denotedbyJi,attheoutputofeachstagei,i=1,2,3.Asynchronizationstationisamodelingelementthatisoftenusedtomodelassemblyoperationsinqueueingnetworks.Itcanbethoughtofasaserverwithinstantservicetimes.Thisserverisfedbytwoormorequeues(inourcasebytwo).Whenthereisatleastonecustomerineachofthequeuesthatfeedtheserver,thesecustomersmoveinstantlythroughandoutoftheserver.Thisimpliesthat,atanytime,atleastoneofthequeuesthatfeedtheserverisempty.Customersthatentertheserver,exittheserverafterpossiblyhavingbeensplitintomoreormergedintofewercustomers.Inourcase,thequeuesineachsynchronizationstationcontaineitherpartsordemandscombinedwithkanbans.Toillustratetheoperationofthesynchronizationstations,letusfirstfocusonanysynchronizationstationJi,exceptthatofthelaststage.Thissynchronizationstationrepresentsthesynchronizationbetweenastage-ifinishedpartandastage-(i+1)freekanban.LetPAiandDAi+1denotethetwoqueuesofJi.PAirepresentsL1J1L2J2L3J3S1S2S3PA1S4S5S6PA2S7S8S9PA3DA2DA3D4CustomerK3DemandsK2K1Fig.2.QueueingnetworkmodeloftheecholonkanbancontrolsystemofFigure1 198S.KoukoumialosandG.Liberopoulostheoutputbufferofstageiandcontainsstage-ifinishedparts,eachofwhichhasattachedtoitakanbanfromeachstagefrom1toi.DAi+1containsdemandsfortheproductionofnewstage-(i+1)parts,eachofwhichhasattachedtoitastage-(i+1)kanban.Thesynchronizationstationoperatesasfollows.AssoonasthereisoneentityineachqueuePAiandDAi+1,thestage-ifinishedpartengagesthestage-(i+1)kanbanwithoutreleasingthekanbansfromstages1toithatwerealreadyattachedtoit,andjoinsthefirststationofstagei+1.Notethatatstage1,assoonasastage-1kanbanisavailable,anewpartisimmediatelyreleasedintostage1sincetherearealwaysrawpartsattheinputofthesystem.LetusnowconsiderthelastsynchronizationstationJN(J3intheexample).JNsynchronizesqueuesPAN,andDN+1.PANrepresentstheoutputbufferofstageNandcontainsstage-Nfinishedparts,eachofwhichhasattachedtoitakanbanfromeachstagefrom1toN.DN+1containscustomerdemands.Whenacustomerdemandarrivestothesystem,itjoinsDN+1,therebydemandingthereleaseofafinishedpartfromPANtothecustomer.IfthereisafinishedpartinqueuePAN,itisreleasedtothecustomerandthedemandissatisfied.Inthiscase,thefinishedpartinPANreleasesthekanbansthatwereattachedtoit,andthesekanbansaretransferredupstreamtoqueuesDAi(i=1,...,N).Thekanbanofstageicarriesalongwithitademandfortheproductionofanewstage-i(i=1,...,N)finishedpartandanauthorizationforthereleaseofafinishedpartfromqueuePAi−1intostagei.IftherearenofinishedpartsinqueuePAN,thecustomerdemandremainsonholdinDN+1asabackordereddemand.Animportantspecialcaseoftheechelonkanbancontrolsysteminthecasewheretherearealwayscustomerdemandsforfinishedparts.Thiscaseisknownasthesaturatedechelonkanbancontrolsystem.Itsimportanceliesinthefactthatitsthroughputdeterminesthemaximumcapacityofthesystem.Inthesaturatedsystem,whentherearefinishedpartsatstageN,theyareimmediatelyconsumedandanequalnumberofpartsenterthesystem.Asfarasthequeueingnetworkcorrespondingtothismodelisconcerned,thesynchronizationstationJNcanbeeliminatedsincequeueDN+1isneveremptyandcanthereforebeignored.Inthesaturatedechelonkanbancontrolsystem,whentheprocessingofapartiscompletedatstageN,thispartisimmediatelyconsumedafterreleasingthekanbansofstages1,...,NthatwereattachedtoitandsendingthembacktoqueuesDAi(i=1,...,N).Itisworthnotingthattheechelonkanbancontrolsystemcontainsthemake-to-stockCONWIPsystem[23]asaspecialcase.Inthemake-to-stockCONWIPsystem,assoonasafinishedpartleavestheproductionsystemtobedeliveredtoacustomer,anewpartentersthesystemtobeginitsprocessing.AnechelonkanbancontrolsystemwithK1≤Ki,i/=1,behavesexactlylikethemake-to-stockCONWIPsystem.Thedynamicbehavioroftheechelonkanbancontrolsystemdependsonthemanufacturingprocesses,thearrivalprocessofcustomerexternaldemands,andthenumberofechelonkanbansofeachstage.Amongtheperformancemeasuresthatareofparticularinterestaretheaverageworkinprocess(WIP)andtheaveragenumberoffinishedpartsineachstage,theaveragenumberofbackordered(notimmediatelysatisfied)demands,andtheaveragewaitingtimeandpercentageof Ananalyticalmethodfortheperformanceevaluation199backordereddemands.Inthecaseofthesaturatedechelonkanbancontrolsystem,themainperformancemeasureofinterestisitsproductionrate,Pr,i.e.theaveragenumberoffinishedpartsleavingtheoutputbufferofstageNperunitoftime.Prrepresentsthemaximumrateatwhichcustomerdemandscanbesatisfied.Withthisinmind,theaveragearrivalrateofexternalcustomerdemandsintheunsaturatedsystem,sayλD,mustbestrictlylessthanPrinorderforthesystemtomeetallthedemandsinthelongrun.Inotherwords,thestabilityconditionfortheunsaturatedsystemisλDn)andconcernsthesituationwherewehavetheprerogativetointroduceadelayinfillingorders,whichisequivalenttoauthorizingdemandstowait.Specifically,Pruptisthemarginalstationaryprobabilityofhavingnofinishedpartsinthelastsynchronizationstation,whichisgivenbyequation(18)inAppendixA.Similarly,P(Q>n)isthestationaryprobabilityofhavingmorethanncustomerswaitingandcanbecomputedfromthefollowingexpression:∞nP(Q>n)=P(Q=x)=1−P(Q=y),(10)x=n+1y=0whereP(Q=n)isgivenby(seeAppendixA):nNNλDP(Q=n)=pO(0,n)=pO(0,0)N.(11)λ(0)OThestationarydistributionpN(0,0)thatisneededtoevaluatebothPandOruptP(Q>n)isgivenbythefollowingexpression:N1pO(0,0)=.(12)KNx,−11+(1λN(i))1−λDλxONx=1Di=0λ(0)O Ananalyticalmethodfortheperformanceevaluation215Thecostfunctionthatwewanttominimizeisthelong-run,expected,averagecostofholdinginventory,NCtotal=hiE[WIPi+FPi],(13)i=1wherehiistheunitcostofholdingWIPi+FPiinventoryperunittimeinstagei.Intheremainingofthissection,weoptimizetheechelonkanbansofanechelonkanbancontrolsystemconsistingofN=5stages,whereeachstagecontainsasinglemachinewithexponentiallydistributedservicetimeswithmeanequalto1,fordifferentcombinationsofinventoryholdingcostrates,hi,i=1,...,5,anddemandarrivalrateλD=0.5.Inallcasesweassumethatthereisvalueaddedtothepartsateverystagesothattheinventoryholdingcostincreasesasthestageincreases,i.e.h1n)≤0.02,forn=2,5,10.FromtheresultsinTable9,weseethatthehigherthenumberofbackordereddemandsninthequalityofservicedefinition,P(Q>n),thelowertheoptimalnumberofechelonkanbans,andhencetheinventoryholdingcost.Asthedifferencebetweentheholdingcostrateshi,i=1,...,5,increases,thedifferencebetweentheoptimalvaluesofKi,i=1,...,5,alsoincreases,sincethebehavioroftheechelonkanbancontrolsystemdivertsfurtherfromthatofthemake-to-stockCONWIPsystem.Whentherelativedifferencebetweentheholdingcostrateshi,i=1,...,5,islow,thebehavioroftheechelonkanbancontrolsystemtendstothatofthemake-to-stockCONWIPsystem.Table10showstheoptimaldesignparameterK1andassociatedminimuminventoryholdingcostforλD=0.5anddifferentqualityofserviceconstraintsandinventoryholdingcostratesh1,...,h5,forthemake-to-stockCONWIPsystem.ThelastcolumnofTable10showstherelativeincreaseincostoftheoptimalmake-to-stockCONWIPsystemcomparedtotheoptimalechelonkanbancontrolsystem.ComparingtheresultsbetweenTables9and10,wenotethattheoptimalmake-to-stockCONWIPsystemperformsconsiderablyworsethantheoptimalechelonkanbancontrolsystem,particularlywhentherelativedifferencebetweentheholdingcostrateshi,i=1,...,5,ishighand/orthenumberofbackordereddemandsninthequalityofservicedefinition,P(Q>n),ishigh,indicatingthatthequalityofserviceislow. 216S.KoukoumialosandG.LiberopoulosTable9.OpimalconfigurationandassociatedcostssforλD=0.5anddifferentvaluesofh1,...,h5,fortheechelonkanbancontrolsystemDesigncriterionK1K2K3K4K5Costh1=1,h2=2,h3=3,h4=4,h5=5Prupt≤0.0215131210855.885P(Q>2)≤0.021311108746.555P(Q>5)≤0.0210876231.120P(Q>10)≤0.027653120.253h1=3,h2=8,h3=9,h4=10,h5=12Prupt≤0.02151312108144.314P(Q>2)≤0.0213111096121.161P(Q>5)≤0.0210876284.074P(Q>10)≤0.027653157.360h1=1,h2=2,h3=4,h4=11,h5=12Prupt≤0.0215141398121.288P(Q>2)≤0.021413107698.890P(Q>5)≤0.0210985267.383P(Q>10)≤0.028643139.483h1=1,h2=6,h3=11,h4=16,h5=21Prupt≤0.02171311108218.702P(Q>2)≤0.0215111085178.162P(Q>5)≤0.02108762115.601P(Q>10)≤0.028653176.523h1=1,h2=11,h3=21,h4=31,h5=41Prupt≤0.02171311108420.405P(Q>2)≤0.0215111085341.324P(Q>5)≤0.02108762221.203P(Q>10)≤0.0286531145.047h1=1,h2=2,h3=4,h4=8,h5=16Prupt≤0.0217151297143.879P(Q>2)≤0.0214131175112.442P(Q>5)≤0.0210876265.843P(Q>10)≤0.028653139.934h1=1,h2=3,h3=9,h4=27,h5=81Prupt≤0.02191714106633.178P(Q>2)≤0.0217151284471.867P(Q>5)≤0.021210861231.446P(Q>10)≤0.0286531139.066 Ananalyticalmethodfortheperformanceevaluation217Table10.OptimalconfigurationandassociatedcostsforλD=0.5anddifferentvaluesofh1,...,h5,fortheCONWIPsystemDesigncriterionK1CostRelativecostincreaseh1=1,h2=6,h3=11,h4=16,h5=21Prupt≤0.0214244.16310.43%P(Q>2)≤0.0212202.41511.98%P(Q>5)≤0.0210161.00628.2%P(Q>10)≤0.028120.30736.39%h1=1,h2=11,h3=21,h4=31,h5=41Prupt≤0.0214474.32611.37%P(Q>2)≤0.0212392.83013.11%P(Q>5)≤0.0210312.01229.1%P(Q>10)≤0.028232.61337.64%h1=1,h2=2,h3=4,h4=8,h5=16Prupt≤0.0214175.16017.86%P(Q>2)≤0.0212143.40721.59%P(Q>5)≤0.0210111.98641.2%P(Q>10)≤0.02881.26050.86%h1=1,h2=3,h3=9,h4=27,h5=81Prupt≤0.0214850.92725.59%P(Q>2)≤0.0212690.35831.65%P(Q>5)≤0.0210531.71556.47%P(Q>10)≤0.028377.10263.12%8ConclusionsWedevelopedananalytical,decomposition-basedapproximationmethodfortheperformanceevaluationoftheechelonkanbancontrolsystemandtesteditonseveralnumericalexamples.Thenumericalexamplesshowedthatthemethodisquiteaccurateinmostcases.Theyalsoshowedthattheechelonkanbancontrolsystemhassomeadvantagesovertheconventionalkanbancontrolsystem.Specifically,whenthetwosystemshavethesamevalueofK,theechelonkanbancontrolsystemhashigherproductioncapacity,loweraveragenumberofbackordereddemands,butonlyslightlyhigheraverageWIPandeitherslightlyhigherorslightlylowerFPthantheconventionalkanbancontrolsystem.Thenumericalresultsalsoshowedthatasthevariabilityoftheservicetimedistributionincreases,theproductioncapacityoftheechelonkanbancontrolsystemandtheaccuracyoftheapproximationmethoddecrease.Finally,weknowthattheoptimizedechelonkanbancontrolsystemalwaysperformsatleastaswellastheoptimizedmake-to-stockCONWIPsystemsincethelattersystemisaspecialcaseofthefirstsystem.Thenumericalresultsshowedthatinfactthesuperiorityinperformanceoftheechelonkanbancontrolsystemoverthatofthemake-to-stockCONWIPsystemcanbequitesignificant,particularlywhentherelativeincreaseininventoryholdingcostsfromonestagetothenextdownstreamstageishighand/orthequalityofserviceislow. 218S.KoukoumialosandG.LiberopoulosAppendixA–AnalysisofsynchronizationstationONONisasynchronizationstationfedbyacontinuous-timeMarkovarrivalprocesswithstate-dependentarrivalrateλN(nN),0≤nN0(15)ODOODDDThemarginalprobabilitiesPN(nN)arethensimplygivenbyOOPN(nN)=pN(nN,0)fornN=1,...,K,(16)OOOOON∞PN(0)=pN(0,n).(17)OODnD=0From(15)and(17)weget∞nDNNλDN1PO(0)=pO(0,0)N=pO(0,0)λD.(18)nD=0λO(0)1−λN(0)OTheconditionalthroughputsofsubsystemONareobtainedfrom(5),(14)and(16),asfollows:vN(nN)=λfornN=2,...,K(19)OODONFrom(5),(14),(16)and(18),wealsogetN1vO(1)=11.(20)λD−λN(0)O Ananalyticalmethodfortheperformanceevaluation219AppendixB–AnalysisofsynchronizationstationIiIi,i=2,...,N,isasynchronizationstationfedbytwocontinuous-timeMarkovarrivalprocesseswithstate-dependentarrivalrates:λi(ni),0≤ni≤K,andIIIiλi(ni),0≤ni≤K.Theunderlyingcontinuous-timeMarkovchainisshowni−1inFigure6.ThestateofthisMarkovchainis(ni,ni),whereniisthenumberIuIoffreekanbansandniisthenumberofexternalresources(finishedpartsofstageui−1)currentlypresentinsubsystemIi.Recallthatnicanbeobtainedfromnianduniusing(3).Thesteady-stateprobabilitiespi(ni,ni)canbederivedassolutionsIIIuoftheunderlyingbalanceequationsandaregivenby:⎡⎤in-Iλi(n−1)pi(ni,0)=⎣I⎦pi(0,0),(21)IIλi(K−n)Iin=1in,uλi(K+n−1)ipi(0,ni)=n=1pi(0,0).(22)IuniIλi(0)uIThemarginalprobabilities,Pi(ni),canthenbederivedbysumminguptheprob-IIabilitiesaboveasfollows:⎡⎤in-Iλi(n−1)Pi(ni)=⎣I⎦pi(0,0)forni=1,...,K,(23)IIλi(K−n)IIiin=1⎡i⎤n,uλi(K+n−1)⎢Ki−1−Kii⎥i⎢n=1⎥iPI(0)=⎢1+ni⎥pI(0,0).(24)⎣λi(0)u⎦ni=1IuTheestimationoftheconditionalthroughputsofsubsystemIicanthenbeobtainedbysubstitutingtheaboveprobabilitiesinto(5),asfollows:vi(ni)=λi(K−ni)forni=2,...,K,(25)IIiIIiiiFig.6.Continuous-timeMarkovchaindescribingthestate(nI,nu)ofsynchronizationsta-itionI 220S.KoukoumialosandG.Liberopoulos⎡i⎤n,uλi(K+n−1)⎢Ki−1−Kii⎥ii⎢n=1⎥vI(1)=λ(Ki−1)⎢1+ni⎥.(26)⎣λi(0)u⎦ni=1IuAppendixC–TableofnotationNNumberofstagesKiNumberofechelonkanbansofstageiLiSubnetworkassociatedwiththemanufacturingprocessofstageimiNumberofstationsofsubnetworkLiJiSynchronizationstationattheoutputofstageiλDAveragearrivalrateofexternalcustomerdemandsintheunsaturatedsystemPrMaximumrateatwhichcustomerdemandscanbesatisfiedRQueueingnetworkoftheechelonkanbancontrolsystemRiSubsystemassociatedwithstageiIiUpstreamsynchronizationstationofsubsystemRiONDownstreamsynchronizationstationofsubsystemRNSˆDownstreamsingle-serverpseudo-stationofsubsystemRiiniStateofsubsystemRiλi(ni)State-dependentarrivalrateofstage-irawpartsattheupstreamsynchro-nizationstationIiofsubsystemRivi(ni)ConditionalthroughputofsubsystemRik∈MIndexdenotingthestationswithinsubsystemRi,whereM=i1{1,...,m1,Sˆ},Mi={I,1,...,mi,Sˆ}fori=2,...,N−1,andMN={I,1,...,mN,O}niStateofstationkinsubsystemRikµi(ni)Load-dependentservicerateofstationkinsubsystemRikkµ(n)Sameasµi(ni)withindexidroppedkkkkTiOpensystemrepresentingstationkinsubsystemRikTSameasTiwithindexidroppedkkλi(ni)Rateofstate-dependentPoissonarrivalprocessatTikkkλ(n)Sameasλi(ni)withindexidroppedkkkkvi(ni)ConditionalthroughputofTikkkv(n)Sameasvi(ni)withindexidroppedkkkkPi(ni)Steady-stateprobabilityofTikkkpBProportionofbackordereddemandsQDAveragenumberofbackordereddemandsWBAveragewaitingtimeofbackordereddemands Ananalyticalmethodfortheperformanceevaluation221References1.BaskettF,ChandyKM,MuntzRR,Palacios-GomezF(1975)Open,closedandmixednetworksofqueueswithdifferentclassesofcustomers.JournalofACM22:248–2602.BaynatB,DalleryY(1993)Aunifiedviewofproduct-formapproximationtechniquesforgeneralclosedqueueingnetworks.PerformanceEvaluation18(3):205–2243.BaynatB,DalleryY(1993)Approximatetechniquesforgeneralclosedqueueingnet-workswithsubnetworkshavingpopulationconstraints.EuropeanJournalofOpera-tionalResearch69:250–2644.BaynatB,DalleryY(1996)Aproduct-formapproximationmethodforgeneralclosedqueueingnetworkswithseveralclassesofcustomers.PerformanceEvaluation24:165–1885.BaynatB,DalleryY,RossK(1994)Adecompositionapproximationmethodformul-ticlassBCMPqueueingnetworkswithmultiple-serverstations.AnnalsofOperationsResearch48:273–2946.BruellSC,BalboG(1980)Computationalalgorithmsforclosedqueueingnetworks.ElsevierNorth-Holland,Amsterdam7.BuzacottJA(1989)QueueingmodelsofkanbanandMRPcontrolledproductionsys-tems.EngineeringCostsandProductionEconomics17:3–208.BuzacottJA,ShanthikumarJG(1993)Stochasticmodelsofmanufacturingsystems.Prentice-Hall,EnglewoodCliffs,NJ9.BuzenJP(1973)Computationalalgorithmsforclosedqueueingnetworkswithexpo-nentialservers.Comm.ACM16(9):527–53110.ClarkA,ScarfH(1960)Optimalpoliciesforamulti-echeloninventoryproblem.Man-agementScience6:475–49011.DalleryY(1990)Approximateanalysisofgeneralopenqueueingnetworkswithre-strictedcapacity.PerformanceEvaluation11(3):209–22212.DalleryY,CaoX(1992)Operationalanalysisofstochasticclosedqueueingnetworks.PerformanceEvaluation14(1):43–6113.DalleryY,LiberopoulosG(2000)Extendedkanbancontrolsystem:combiningkanbanandbasestock.IIETransactions32(4):369–38614.DiMascoloM,FreinY,DalleryY(1996)Ananalyticalmethodforperformanceeval-uationofkanbancontrolledproductionsystems.OperationsResearch44(1):50–6415.DuriC,FreinY,DiMascoloM(2000)Comparisonamongthreepullcontrolpolicies:kanban,basestockandgeneralizedkanban.AnnalsofOperationsResearch93:41–6916.FreinY,DiMascoloM,DalleryY(1995)Onthedesignofgeneralizedkanbancontrolsystems.InternationalJournalofOperationsandProductionManagement15(9):158–18417.GordonWJ,NewellGF(1967)Closedqueueingnetworkswithexponentialservers.OperationsResearch15:252–26718.JacksonJR(1963)Jobshop-likequeueingsystems.ManagementScience10(1):131–14219.LiberopoulosG,DalleryY(2002)Comparativemodelingofmulti-stageproduction-inventorycontrolpolicieswithlotsizing.InternationalJournalofProductionResearch41(6):1273–129820.MarieR(1979)Anapproximateanalyticalmethodforgeneralqueueingnetworks.IEEETransactionsonSoftwareEngineering5(5):530–53821.MarieR(1980)Calculatingequilibriumprobabilitiesforλ(n)/Ck/1/Nqueues.Perfor-manceEvaluationReview9:117–12522.ReiserM,LavenbergSS(1980)Meanvalueanalysisofclosedmultichainqueueingnetworks.JournalofACM27(2):313–322 222S.KoukoumialosandG.Liberopoulos23.SchweitzerPJ(1979)Approximateanalysisofmulticlassclosednetworksofqueues.ProceedingsoftheInternationalConferenceonStochasticControlandOptimization,Amsterdam24.SpanjersL,vanOmmerenJCW,ZijmWHM(2005)Closedlooptwo-echelonreparableitemsystems.ORSpectrum27(2–3):369–39825.SpearmanML,WoodruffDL,HoppWJ(1990)CONWIP:apullalternativetokanban.InternationalJournalofProductionResearch28:879–89426.StewartWJ,MarieR(1980)Anumericalsolutionfortheλ(n)/Ck/r/Nqueue.EuropeanJournalofOperationalResearch5:56–6827.WhittW(1983)Thequeueingnetworkanalyser.BellSystemsTechnologyJournal62(9):2779–2815 Closedlooptwo-echelonrepairableitemsystemsL.Spanjers,J.C.W.vanOmmeren,andW.H.M.ZijmFacultyofElectricalEngineering,MathematicsandComputerScience,UniversityofTwente,P.O.Box217,7500AEEnschede,TheNetherlands(e-mail:w.h.m.zijm@utwente.nl)Abstract.Inthispaperweconsiderclosedlooptwo-echelonrepairableitemsys-temswithrepairfacilitiesbothatanumberoflocalservicecenters(calledbases)andatacentrallocation(thedepot).Thegoalofthesystemistomaintainanumberofproductionfacilities(oneateachbase)inoptimaloperationalcondition.Eachproductionfacilityconsistsofanumberofidenticalmachineswhichmayfailinci-dentally.Eachrepairfacilitymaybeconsideredtobeamulti-serverstation,whileanytransportfromthedepottothebasesismodeledasanampleserver.Atallbasesaswellasatthedepot,ready-for-usespareparts(machines)arekeptinstock.Onceamachineintheproductioncellofacertainbasefails,itisreplacedbyaready-for-usemachinefromthatbase’sstock,ifavailable.Thefailedmachineiseitherrepairedatthebaseorrepairedatthecentralrepairfacility.Inthecaseoflocalrepair,themachineisaddedtothelocalsparepartsstockasaready-for-usema-chineafterrepair.Ifarepairatthedepotisneeded,thebaseordersamachinefromthecentralsparepartsstocktoreplenishitslocalstock,whilethefailedmachineisaddedtothecentralstockafterrepair.Ordersaresatisfiedonafirst-come-first-servedbasiswhileanyrequirementthatcannotbesatisfiedimmediatelyeitheratthebasesoratthedepotisbacklogged.Incaseofabacklogatacertainbase,thatbase’sproductioncellperformsworse.Todeterminethesteadystateprobabilitiesofthesystem,wedevelopaslightlyaggregatedsystemmodelandproposeaspecialnear-product-formsolutionthatprovidesexcellentapproximationsofrelevantperformancemeasures.Thedepotrepairshopismodeledasaserverwithstate-dependentservicerates,ofwhichtheparametersfollowfromanapplicationofNorton’stheoremforClosedQueuingNetworks.AspecialadaptationtoageneralMulti-ClassMarginalDistributionAnalysis(MDA)algorithmisproposed,onwhichtheapproximationsarebased.AllrelevantperformancemeasurescanbecalculatedwitherrorswhicharegenerallyCorrespondenceto:W.H.M.Zijm 224L.Spanjersetal.lessthanonepercent,whencomparedtosimulationresults.Theapproximationsareusedtofindthestocklevelswhichmaximizetheavailibilitygivenafixedconfigurationofmachinesandserversandacertainbudgetforstoringitems.Keywords:Multi-echelonsystems–Repairableitems–Sparepartsinventory–Closedqueueingnetworks–Near-productformsolutions1IntroductionRepairableinventorytheoryinvolvesdesigninginventorysystemsforitemswhicharerepairedandreturnedtouseratherthandiscarded.Theitemsarelessexpensivetorepairthantoreplace.Suchitemscanforexamplebefoundinthemilitary,avi-ation,copyingmachines,transportationequipmentandelectronics.Therepairableinventoryproblemistypicallyconcernedwiththeoptimalstockingofpartsatbasesandacentraldepotfacilitywhichrepairsfailedunitsreturnedfrombaseswhilepro-vidingsomepredeterminedlevelofservice.Differentperformancemeasuresmaybeused,suchascost,backordersandavailability.Overthepast30yearstherehasbeenconsiderableinterestinmulti-echeloninventorytheory.MuchofthisworkoriginatesfromamodelcalledMETRIC,whichwasfirstreportedintheliteraturebySherbrooke[9].ThemodelwasdevelopedfortheUSAirForceattheRandCorporationforamulti-echelonrepairable-iteminventorysystem.Inthismodelanitematfailureisreplacedbyaspareifoneisavailable.Ifnoneareavailableaspareisbackordered.Ofthefaileditemsacertainproportionisrepairedatthebaseandtherestatarepairdepot,therebycreatingatwo-echelonrepairable-itemsystem.Itemsarereturnedfromthedepotusingaone-for-onereorderingpolicy.TheMETRICmodeldeterminestheoptimallevelofsparestobemaintainedateachofthebasesandatthedepot.AshortfalloftheMETRICmodelisthatitassumesthatfailuresarePoissonfromaninfinitesourceandthattherepaircapacityisunlimited.Therefore,othershavecontinuedtheresearchtogainresultsmoreusefulforreallifeapplications.Gross,KioussisandMiller[5],AlbrightandSoni[1]andAlbright[2]focusedtheirattentiononclosedqueuingnetworkmodels,therebydroppingtheassumptionofPoissonfailuresfromaninfinitesource.Theintensitybywhichmachinesentertherepairshopsdependsonthenumberofmachinesoperatingintheproductioncell.Incaseofabacklogatabase,thisintensityisthereforesmallerthanintheoptimalcasewherethemaximumnumberofmachinesisoperatingintheproductioncell.AlsotheassumptionofunlimitedrepaircapacityisdroppedinGrossetal.[5]andAlbright[2].Thispaperdealswithsimilarmodels.Ithandlesclosedqueuingnetworkmod-elswithlimitedrepair.However,theapproximationmethoddiffersconsiderably.TheapproximationmethodbuildsonthemethodbyAvsarandZijm[3].AvsarandZijmconsideredanopenqueuingnetworkmodelwithlimitedrepair.Byasmallaggregationstep,thesystemischangedintoasystemwithaspecialnear-product-formsolutionthatprovidesanapproximationforthesteadystatedistribution.Fromthesteadystatedistributionallrelevantperformancemeasurescanbecomputed. Closedlooptwo-echelonrepairableitemsystems225Wewillperformasimilaraggregationstepinthispaperandagainaspecialnear-product-formsolutionwillbeobtained.However,asopposedtoopensystems,inasystemwithfinitesources,thedemandratestothedepotalsobecomestatedepen-dent;moreover,thesedemandratesareclearlyinfluencedbytheefficiencyofthebaserepairstations.Nevertheless,weareabletodeveloprelativelysimpleapproxi-mationalgorithmstoobtaintherelevantperformancemeasures.Theseperformancemeasurescanultimatelybeusedwithinanoptimizationmodeltodeterminesuchquantitiesastheoptimalrepaircapacitiesandtheoptimalinventorylevels.Theorganizationofthispaperisasfollows:Inthenextsectionweconsideraverysimpletwo-echelonsystem,consistingofonebase,abaserepairshopandacentralrepairshop.Therepairshopsaremodeledassingleservers.Thismodelmainlyservestoexplaintheessentialelementsoftheaggregationstep.Wepresentthemodifiedsystemwithnear-product-formsolutionandnumericalresultstoshowtheaccuracyoftheapproximation.Next,inSection3,weturntomoregeneralre-pairableitemnetworkstructures,containingmultiplebasesandtransportlinesfromthedepottothebases.Therepairshopsaremodeledasmulti-servers.Theapprox-imationmethodleadingtoanadaptedMulti-ClassMDAalgorithmispresentedandsomenumericalresultsarediscussed.InSection4,anoptimizationalgorithmbasedonthisapproximationmethod,isgivenwhichfindsthestocklevelsthatmax-imizethe(weighted)availibilityunderagivencostconstraint.Inthelastsection,wesummarizeourresultsanddiscussanumberofextensionsthatarecurrentlybeinginvestigated.2Analysisofasimpletwo-echelonsystemwithsingleserverfacilitiesInthissectionasimplifiedrepairableitemsystemisdiscussedtoexplainhowaslightmodificationturnsthissystemintoanear-productformnetworkthatcanbeanalyzedcompletely.Inthenextsectionweturntomorecomplexsystems.2.1ThesinglebasemodelwithouttransportationConsiderthesystemasdepictedinFigure1.Thesystemconsistsofasinglebaseandadepot.AtthebaseamaximumofJ1machinescanbeoperationalintheproductioncell.Operationalmachinesfailatexponentialrateλ1andarereplacedbyamachinefromthebasestock(ifavailable).Bothatthebaseandatthedepotthereisarepairshop.Failedmachinesarebase-repairablewithprobabilityp1andconsequentlydepot-repairablewithprobability1−p1.Therepairshopsaremodeledassingleserverswithexponentialservicerateµ0forthedepotandexponentialservicerateµ1forthebase.InadditiontotheJ1machinesanothergroupofS1machinesisdedicatedtothebasetoactasspares.Whenamachinefails,thefailedmachinegoestoarepairshopwhileatthesametimearequestissenttoplaceasparemachinefromthebasestockintheproductioncell.Thisrequestiscarriedoutimmediately,ifpossible.Incasenosparemachinesareatthebase,abacklogoccurs.Assoonasthereisarepairedmachineavailable,itbecomesoperational.AnumberofS0 226L.Spanjersetal.Productioncell1−p1λ1λp11Baserepairλµ11J1machinesDepotrepairS1µ0S0Fig.1.Thesinglebaserepairableitemsystemmachinesisdedicatedtothedepottoactasspares.Whenafailedmachinecannotberepairedatthebaseandhenceissenttothedepot,asparemachineisshippedfromthedepottothebasetoreplenishthebasestock,or-incaseofabacklog-tobecomeoperationalimmediately.Whennosparesareavailableatthedepot,abackorderiscreated.Inthatcase,assoonasamachineisrepairedatthedepotrepairshop,itissenttothebase.Inthissimplemodel,transporttimesfromthebasetothedepotandviceversaarenottakenintoaccount.InFigure1(andsubsequentfigures),requestsareindicatedbydottedlines.Thematchingofarequestandaready-for-usemachineismodeledasasynchronizationqueue,bothatthebaseandatthedepot.Atthebasehowever,somereflectionrevealsthatthesynchronizationqueuecanbeseenasanormalqueuewheremachinesarewaitingtobemovedintotheproductioncell.Thisisonlypossiblewhentheproductioncelldoesnotcontainthemaximumnumberofmachines,thatis,ifamachineintheproductioncellhasfailed.ThisleadstothemodelinFigure2.Productioncell1−p1λ1λp11Baserepairµλ1k1m11machinesj1Depotrepairm12operationalµ0n1n2Fig.2.ThemodifiedsinglebasesystemInthisfigurethevariablesn1,n2,k,m11andm12indicatethelengthsofthevariousqueuesinthesystem.Thenumberofmachinesin(orawaiting)depotrepairisdenotedbytherandomvariablen1,thenumberofsparemachinesatthedepotisdenotedbytherandomvariablen2andthebacklogofmachinesatthedepotisdenotedbyk.Atthebasetherearem11machineswaitingforrepairorbeingrepairedandm12machinesareactingasspares.Intheproductioncelljmachines1areoperational.Asaresultoftheoperatinginventorycontrolpolicies,forn1=n1,n2=n2,k=k,m11=m11,m12=m12andj1=j1thefollowingequations Closedlooptwo-echelonrepairableitemsystems227musthold:n1+n2−k=S0,(1)n2·k=0,(2)k+m11+m12+j1=S1+J1,(3)m12·(J1−j1)=0,(4)whereEquations(2)and(4)followfromthefactthatitisimpossibletohaveabacklogandtohavesparemachinesavailableatthesametime.Ifsparemachinesareavailable,arequestissatisfiedimmediately.Incaseofabacklog,arequestisnotsatisfieduntilarepaircompletion.Therepairedmachineismergedwiththelongestwaitingrequest.Fromtheserelationsitfollowsimmediatelythatn1andm11completelydeter-minethestateofthesystem,includingthevaluesofn2,k,m12andj1.Therefore,thesystemcanbemodeledasacontinuoustimeMarkovchainwithstatedescription(n1,m11).ThecorrespondingtransitiondiagramisdisplayedinFigure3.m11S1+J1(J+S−m)pλ111111µ0(J+S−m)(1−p)λ111111µ1(J+S+S−m−n)pλII10111111S1µ0(J+S+S−m−n)(1−p)λ10111111µ1JpλJpλ111111µµ00J(1−p)λJ(1−p)λ111111µµ1I1IIIIVS0S0+S1S0+S1+J1n1Fig.3.Transitiondiagramforstatedescription(n1,m11)LetP(n1,m11)=P(n1=n1,m11=m11)bethesteadystateprobabilityofbeinginstate(n1,m11).Thissteadystateprobabilitycanbefoundbysolvingtheglobalbalanceequationsofthesystem.Thesecanbededucedfromthetransitiondiagram.Nevertheless,itisnotpossibletofindanalgebraicexpressionforthesteadystateprobabilities.Moreover,forlargersystemswithe.g.multiplebases,thecomputationaleffortbecomesprohibitive.Thereforethesystemwillbeslightlyadjustedinthenextsubsection,inordertoarriveatanear-productformnetwork.Notethattheanalysispresentedinthispaper,ispartlysimilartotheonegiveninAvsarandZijm[3],wheretheequivalentopentwo-echelonnetworkisconsidered.Forthisopennetwork,analgebraicandeasilycomputableproductformapproxi-mationisfound.Inthecurrentpaper,aclosednetworkisconsidered,andaneasilycomputablealgebraicapproximationcouldnotbefound.However,theaggregated 228L.Spanjersetal.networkhasaproductformsteadystatedistribution,andwecanuseMDA-likealgorithmstofindnumericalapproximationsforperformancemeasures.Analternativeapproachistomodelthenumberofmachinesatthedepotandthebasesasaleveldependentquasibirthdeathprocess.Thismethodmayyieldanalgebraicsolutionbut,heretoo,thefinitestatespacemakestheanalysismorecom-plex.Moreover,thetransitionratesinagivenstate,donotonlydependonthephasebutalsoonthelevel.Together,thismakesthealternativemethodcomputationallyverydemanding,ifnotintractable.2.2ApproximationAfirststeptowardsanapproximationforthesteadystateprobabilitiesistoaggregatethestatespace.ThemostdifficultpartsofthetransitiondiagramareregionsIandII,thatis,thepartswithn1≤S0or,equivalently,thepartswithk=0.Thepartswithk>0areequivalenttothestateswithn1=k+S0.Anaturalaggregationofthesystemisadescriptionthroughthestates(k,m11).Thestates(n1,m11)withn1=0,1,...,S0arethenaggregatedintoonestate(0,m11).DenotethesteadystateprobabilitiesforthenewmodelbyP˜thenthefollowingholdsforanym11:S0P˜(k=0,m11=m11)=P(n1=n1,m11=m11),(5)n1=0P˜(k=k,m11=m11)=P(n1=S0+k,m11=m11).(6)ThetransitiondiagramcorrespondingtothealternativestatespacedescriptionisdisplayedinFigure4.TheratesonlydifferfromthetransitiondiagraminFigure3forthecasek=0.Letq(m11)bethesteadystateprobabilitythatanarrivingrequestforamachineatthedepothastowait,giventhatitfindsnootherwaitingrequestsinfrontofit(k=0)andm11=m11.Giventhe(aggregated)state(0,m11),thestatedoesnotchangeincaseofanarrivingrequestwithprobability1−q(m11),becausesparesareavailable.Withprobabilityq(m11)nosparesareavailableandthestatechangesinto(1,m11).Thetransitionratefrom(0,m11)to(1,m11)equalsj1(1−p1)λ1q(m11).Todetermineq(m11)oneneedsq(m11)=P(n1=S0|n1≤S0,m11=m11).(7)However,tocomputethis,oneneedstoknowthesteadystatedistributionoftheoriginalsystem,whichisexactlywhatweattempttoapproximate.Therefore,weapproximatetheq(m11)’sbytheirweightedaverage,i.e.wefocusontheconditionalprobabilityqdefinedbyq=q(m11)P(m11=m11|n1≤S0)=P(n1=S0|n1≤S0)(8)m11andforeverym11wereplaceq(m11)inthetransitiondiagrambythisq.InthenextsectionitwillbeexplainedhowareasonableapproximationforthisqcanbeobtainedbymeansofanapplicationofNorton’stheorem. Closedlooptwo-echelonrepairableitemsystems229m11S1+J1(J+S−m)pλ111111(J+S−m)(1−p)λq(m)µ111111111(J+S−m−k)pλ111111S1J1p1λ1µ0(J+S−m−k)(1−p)λ111111µ1J(1−p)λq(m)11111µ1Jpλ111µ0J(1−p)λ111µ10SS+Jk111Fig.4.Transitiondiagramforstatedescription(k,m11)Lemma1Thesteadystateprobabilitiesforthemodelwithstatedescription(k,m11)andtransitionratesasdenotedinFigure4withq(m11)replacedbyarbitraryqhaveaproductform.Proof.Tofindthesteadystateprobabilities,considerboththeoriginalmodelinFigure2andthealternativemodelinFigure5.Productioncell1−p1λ1λp11Baserepairµλ11m11Depotrepairmachinesj1µ/∞m12operational0kFig.5.Typical-serverClosedQueuingNetwork(TCQN)InFigure5thedepotrepairshopwithsynchronizationqueueisreplacedbyatypicalserver.Forjobsthatfindtheserveridletheserverhasinfiniteserviceratewithprobability1−q(thecasesparesareavailable)andservicerateµ0withprobabilityq(thecasenosparesareavailable).Letb1betherandomvariableequaltom12+j,thenbylookingatthesystemwiththetypicalserver,andconditioning1onthefactthatthenetworkcontainsexactlyJ1+S1jobs,itiseasilyverifiedthatthefollowingexpressionforP˜(k=k,m11=m11,b1=b1)satisfiesthebalance 230L.Spanjersetal.equationsoftheTCQN:⎧b1⎪⎪⎪⎪m11k1⎪⎪Gq˜p11−p1λ1⎪⎪µµb1−J1,b1>J1,k>0⎪⎪10J1!J1⎪⎪b1⎪⎪m11k1⎪⎪Gq˜p11−p1λ1⎪⎪⎪⎨,b1≤J1,k>0µ1µ0b1!P˜(k,m,b)=b1111⎪⎪m111⎪⎪G˜p1λ1⎪⎪µb1−J1,b1>J1,k=0⎪⎪1J1!J1⎪⎪b1⎪⎪m111⎪⎪G˜p1λ1⎪⎪⎪⎪,b1≤J1,k=0⎪⎩µ1b1!(9)withk+m11+b1=J1+S1andG˜thenormalizationconstant.Expressedintermsofthestatevariables(k,m11),thisresultimmediatelyleadsto:Lemma2Thesteadystatedistributionfortheaggregatemodelisgivenby⎧m11k⎪⎪⎪⎪Gqp1λ1(1−p1)λ1,⎪⎪J!JS1−k−m11µ1µ0⎪⎪11⎪⎪⎪⎪k+m11≤S1,k>0⎪⎪⎪⎪m11k⎪⎪Gqp1λ1(1−p1)λ1⎪⎪,⎪⎪⎪⎨(S1+J1−k−m11)!µ1µ0k+m11>S1,k>0P˜(k,m11)=(10)⎪⎪Gpλm11⎪⎪11⎪⎪S,⎪⎪J!J1−m11µ1⎪⎪11⎪⎪m11≤S1,k=0⎪⎪⎪⎪m11⎪⎪Gp1λ1⎪⎪,⎪⎪⎪⎩(S1+J1−m11)!µ1m11>S1,k=0withG=Gλ˜−(J1+S1)thenormalizationconstant.1Thepreviouslemmagivesanexplicitexpressionforthesteadystateprobabili-ties.ForlargesystemsitmaybedifficulttocalculatethenormalizationconstantG.However,sincewearedealingwithaproductformnetwork,MarginalDistributionAnalysis(seee.g.BuzacottandShanthikumar[4])canbeusedtocalculatetheappropriateperformancemeasuresdirectly.Theresultspresentedsofarholdtrueforanyvalueofq∈[0,1].Inthederivationofthelemmasabovetheinterpretationofqastheconditionalprobabilitythata Closedlooptwo-echelonrepairableitemsystems231requestatthedepothastowaitgiventhatitfindsnootherrequestsinfrontofit(see(8)),hasnotbeenused.Thereforeanyq∈[0,1]willdo,butitisexpectedthatagoodapproximationwillbeobtainedbyusingaqthatdoescorrespondtothisinterpretation.InthenextsubsectionNorton’stheoremwillbeusedtofindaqwithameaningfulinterpretationthatgivesgoodresults.2.3ApplyingNorton’stheoremtoapproximateqAlthoughwehavestatedintheprevioussectionthattheproductformdoesnotdependonq,itisstillneededtofindaqthatgivesagoodapproximationfortheperformancemeasures.Inthissection,thebasicideaofNorton’stheorem(seeHarrisonandPatel[6]foranoverview)isusedtofindanapproximationforqthatgivesgoodresults.Thisbasicideaisthataproductformnetworkcanbeanalyzedbyreplacingsubnetworksbystatedependentservers.Norton’stheoremstatesthatthejointdistributionsforthenumbersofcustomersinthesubnetworksandthequeuelengthsatthereplacingstatedependentserversarethesame.Tousethisidea,firstrecalltheoriginalmodelasshowninFigure2.Wewanttofindq,theconditionalprobabilitythatarequestcorrespondingwithamachinefailurefindsnosparepartsinstockatthedepot,althoughtherewasnobacklogsofar.Thebase,consistingoftheproductioncellandthebaserepairshop,istakenapartandreplacedbyastatedependentserver.Productioncell1−p1λ1λp11Baserepairµλ1k1m11TH(i)machinesj1DepotrepairTH(i)1m12operational1µ0n1n2abFig.6.aThenewnetworkwithstatedependentserver.bTheshortcircuitednetworkThenewnetworkwiththestatedependentserverisdisplayedinFigure6a.Inordertofindtheserviceratesforthisstatedependentserver,theoriginalnetworkisshortcircuitedbysettingtheservicerateatthedepotrepairfacilitytoinfinity.ThisshortcircuitednetworkisalsodepictedinFigure6b.Theservicerateforthenewstatedependentserverwithijobspresentisequaltothethroughputoftheshortcircuitednetworkwithijobspresent,denotedbyTH1(i).Theevolutionofn1=n1,thenumberofmachinesinorawaitingdepotre-pair,canbedescribedasabirth-deathprocess.ThetransitiondiagramisshowninFigure7. 232L.Spanjersetal.TH1(J1+S1)TH1(J1+S1)TH1(J1+S1)TH1(J1+S1)TH1(J1+S1−1)TH1(2)TH1(1)012S0−1S0S0+1S0+S1+J1−1S0+S1+J1µ0µ0µ0µ0µ0Fig.7.Transitiondiagramforn1NotethatthisisjustanapproximationduetothefactthatNorton’stheoremisonlyvalidforproductformnetworks.IncaseS0=0,wewouldhaveaproductformnetworkandtheresultswouldbeexact.FromthediagramonecanobservethatP(n=n)TH(J+S−(n−S)+)=P(n=n+1)µ(11)1111110110forn1=0,...,J1+S1+S0−1.Inprincipleonecanderiveanapproximationofthedistributionofn1fromthis.However,bythedefinitionofq(see(8)),weonlyneedtostudythebehaviorforn1≤S0.Forthesestates,theservicerateofthestatedependentserverisequaltoTH1(J1+S1).Letδ=TH1(J1+S1)/µ0.From(11)weobservethatP(n=n)=δn1P(n=0)forn=0,...,Sso11110P(n=S)δS0P(n=0)δS0P(n=0)δS0q=10=1=1=P(n≤S)S0S0n11−δS0+110n1=0P(n1=n1)n1=0δP(n1=0)1−δS01−δ=δ.(12)1−δS0+1ItremainstofindthethroughputoftheshortcircuitednetworkinFig-ure6bwithJ1+S1jobspresent.AsimpleobservationrevealsthatP(b1=b1)min(b1,J1)λ1p1=P(b1=b1−1)µ1forb1=1,...,J1+S1fromwhichthesteadystateprobabilitiesofb1areimmediatelydeduced.Moreover,thethroughputsatisfiesJ1+S1TH1(J1+S1)=(1−p1)P(b1=b1)min(b1,J1)λ1b1=11−p1=µ1(1−P(b1=J1+S1)).(13)p1Wecandetermineqwith(12)and(13).Thisqcanbeusedtoapproximatethesteadystatedistributionusing(10)orusingMarginalDistributionAnalysis.Resultsofthisapproximationarepresentedinthenextsection.2.4ResultsInthissectionnumericalresultsobtainedbytheapproximationdescribedabovewillbepresented.Tobeabletojudgetheapproximationtheresultsarecomparedtoexactresults.Theexactresultsareobtainedbysolvingthebalanceequationsfortheoriginalmodel.Theperformancemeasuresweareinterestedinaretheavailability,i.e.theprobabilitythatthemaximumnumberofmachinesisworkingintheproduction Closedlooptwo-echelonrepairableitemsystems233cell,denotedbyA,andtheexpectednumberofmachinesoperatingintheproductioncell(Ej).Thesearedefinedasfollows:1A=P(j1=J1)=P(b1≥J1)=P(k+m11≤S1),(14)Ej=E(J−[k+m−S]+)=(J−[k+m−S]+)P(k,m).(15)11111111111k,m11TheperformancemeasuresarecomputedforseveralvaluesofJ1,S0,S1,p1,λ1,µ0andµ1.TheresultsaregiveninTable1andinTables5and6intheAppendix.Also,thepercentagedeviationisgiven.Thenumbersrevealthatinthesesystems,theapproximationgivesanerrorofatmost1%.Inallothercasesthatwetested,wegotsimilarresults.Thelargesterrorsareattainedinthecaseswithonlyasmallnumberofspares(S0>0)inthesystem.ForthecaseS0=0theresultsareexact.3Generaltwo-echelonrepairableitemsystemsInthissectionthesimplesystemfromSection2willbeextendedtoamorerealisticone.Thesystemwillcontainmultiplebasesandtransportlines.Furthermore,thesingleserversthatareusedintherepairshopsarereplacedbymultipleparallelservers.Theseadjustmentswillmaketheanalysisofthesystemmorecomplicated.Nevertheless,thebasicideaoftheaggregationstepwillbethesame.3.1Themulti-basemodelwithtransportationThesysteminthissectionconsistsofmultiplebases,wherethenumberofbasesisdenotedbyL.AgraphicalrepresentationofthesystemisgiveninFigure8forthecaseL=2.Asinthesimplesystemdescribedbefore,atbase=1,...,LatmostJmachinesareoperatingintheproductioncell.Themachinesfailatexponentialrateλandarealwaysreplacedbyamachinefromthecorrespondingbasestock(ifavailable).Failedmachinesfrombasearebase-repairablewithprobabilitypanddepot-repairablewithprobability1−p.Incontrasttothesimplemodeldescribedbefore,therepairshopsaremodeledasmulti-servers.Thatis,attherepairshopofbase=1,...,LRrepairmenareworking,eachatexponentialrateµ.AtthedepotrepairshopR0repairmenareworkingatexponentialrateµ0.ConsistentwiththesimplemodelSmachinesarededicatedtobasetoactassparesandS0sparemachinesarededicatedtothedepot.Brokenmachinesatacertainbasethatarebase-repairablearesenttothebaserepairshop.Afterrepairtheyfillupthesparesbufferatbaseor,incaseofabacklogatthatbase,becomeoperationalimmediately.Brokenmachinesfrombasethatareconsidereddepot-repairablearesenttothedepotrepairshop.Whendepotsparesareavailable,aspareisimmediatelysenttothestockofbase.Incasetherearenosparesavailableabacklogoccurs.Machinesthathavecompletedrepairaresenttothebasethathasbeenwaitingthelongest.Thatis,anFCFSreturnpolicyisused.Inthismodelthetransportationfromthedepotto 234L.Spanjersetal.Table1.Resultsforthesimplesinglebasemodel,p1=0.5,λ1=1,µ0=2J1,µ1=J1J1S0S1AexactAappr%devEjexactEjappr%dev113100.56510.56740.41852.42252.42460.08533300.58890.58920.05432.45722.45760.01453500.59010.59010.00412.45892.45900.00123110.79450.79520.09342.72832.72860.00983310.81100.81110.01542.75062.75070.00363510.81200.81200.00142.75182.75180.00043130.95060.95060.00122.93492.93480.00573330.95540.95540.00002.94122.94120.00063530.95570.95570.00002.94162.94160.00003140.97550.97540.00122.96772.96760.00363340.97790.97790.00042.97092.97090.00053540.97810.97810.00002.97112.97110.00005100.53690.53870.33144.31474.31600.03185300.56250.56280.04614.35814.35840.00645500.56390.56390.00374.36044.36040.00065110.77590.77650.07614.67034.67040.00065310.79400.79410.01274.69784.69790.00125510.79500.79500.00124.69944.69940.00025130.94530.94530.00124.91984.91960.00415330.95060.95060.00004.92764.92760.00055530.95100.00000.00004.92814.92810.00005140.97270.97270.00094.96014.96000.00255340.97550.97550.00034.96414.96400.00045540.97570.97570.00004.96434.96430.000010100.50910.51020.21789.18309.18370.007310300.53630.53650.03289.23759.23770.001710500.53790.53790.00289.24069.24060.000210110.75650.75690.05079.59799.59770.001610310.77620.77620.00879.63219.63210.000010510.77740.77740.00089.63419.63410.000010130.93950.93950.00069.90069.90040.002010330.94550.94550.00019.91049.91040.000310530.94580.94580.00009.91109.91100.000010140.96980.96980.00079.95049.95030.001210340.97280.97280.00029.95549.95540.000210540.97300.97300.00009.95579.95570.0000 Closedlooptwo-echelonrepairableitemsystems235Productioncell1−p1λ1λp11Base1repairµ1λ1µ1J1machinesTransportγ1S1Depotrepairµ0TransportdepottobasesµSTransportγ2S200Productioncellµ2λ2Base2repairµ2λ2p2λ21−p2J2machinesFig.8.Themulti-baserepairableitemsystemforL=2
1−p1λ1λp11

µ1m11λm112µ1kt1
j1
γ1


µ0 

n1n
γµ02t22
µ2m22λm221

µ2λ2p2λ21−p2
j2
Fig.9.Themodifiedmulti-basesystemforL=2thebasesistakenintoaccountexplicitly.Thetransportlinesaremodeledasampleserverswithexponentialservicerateγforthetransporttobase=1,...,L.Thenumberofmachinesintransporttobaseisdenotedbytherandomvariablet.Thetransportfromthebasestothedepotisnottakenintoaccount.Asinthesimplemodel,thesynchronizationqueuesatthebasescanbereplacedbyordinaryqueuesasisdepictedinFigure9.Thevectorm1=(m11,m21,...,mL1)denotesthenumberofmachinesinbaserepair(=1,...,L)andthevectorm2=(m12,m22,...,mL2)denotesthenumberofsparesatthebases(=1,...,L).Thevariablen1standsforthenumberofmachinesindepotrepairandn2isthenumberofsparemachinesatthe 236L.Spanjersetal.depot.Thevectork0=(k01,k02,...,k0L)denotesthebackordersatthedepot,originatingfrombase(=1,...,L).ThetotalnumberofbackordersatthedepotLequalsk==1k0.Themachinesintransittothebasesaregivenbythevectort=(t1,t2,...,tL)andthenumbersofmachinesoperatingintheproductioncellsareexpressedinvectorj=(j1,j2,...,jL).Thesumofthenumberofmachinesinbasestockandthenumberofmachinesoperatingintheproductioncellisdenotedinthevectorb=(b1,b2,...,bL),whereb=m2+j.Asaresultoftheoperatinginventorycontrolpolicies,forn1=n1,n2=n2,k0=k0,t=t,m1=m1,m2=m2andj=jthefollowingequationsmusthold:n1+n2−k=S0,(16)n2·k=0,(17)andfor=1,2,...,L:k0+t+m1+m2+j=S+J,(18)m2·(J−j)=0.(19)Fromtheserelationsitfollowsimmediatelythatk0,n1,tandm1completelydeterminethestateofthesystem.Therefore,thesystemcanbemodeledasacontinuoustimeMarkovchainwithstatedescription(k0,n1,t,m1).Remark3Inthevectorthatdenotesthenumberofbackordersoriginatingfromthebases,k0=(k01,k02,...,k0L),itisnottakenintoaccountthattheorderofthebackordersmatters.SinceanFCFSreturnpolicyisassumed,thisordershouldbeknown.Nevertheless,inthismodelallstateswithsimilarnumbersofbackordersperbase,areaggregatedintoonestate.Thisaggregationstepwillnothaveabiginfluenceontheresults,butitwillconsiderablysimplifytheanalysis.3.2ApproximationIncorrespondencewiththesimplemodelasdescribedinSection2asimilaraggre-gationstepisperformedtotacklethisextendedmodel.Oncemore,allstateswith0≤n1≤S0areaggregatedintoonestate.TheaggregationstepisperformedasfollowsP(k0=0,k=0,t=t,m1=m1)S0=P(k0=0,n1=n1,t=t,m1=m1)(20)n1=0P(k0=k0,k=k,t=t,m1=m1)=P(k0=k0,n1=S0+k,t=t,m1=m1)(21)Theaggregatedsystemcanbedescribedby(k0,k,t,m1).Furthermore,becauseLk==1k0thestatespacecanalsobedescribedby(k0,t,m1).Defineqasbefore,thatisqistheconditionalprobabilitythatanarrivingrequestatthedepotcannotbefulfilledimmediately,giventhattherearenootherrequests Closedlooptwo-echelonrepairableitemsystems237waiting.Inaformulaitsaysq=P(n1=S0|n1≤S0).So,giventhereisnobacklogatthedepot,anarrivingrequesthastowaitwithprobabilityq.Thewaitingtimedependsonthenumberofsparesalreadyinthequeue.Productioncell1−p1λ1λp11Base1repairµ1m11λm1µ121t1machinesj1Transportγ1operationalDepotrepairmin(R0,S0+k)µ0/∞TransportdepottobaseskTransportγ2t2Productioncellµ2m22λm221Base2repairµ2λ2p2λ21−p2machinesj2operationalFig.10.TheTypical-serverClosedQueuingNetworkThefirstsparethatfinishesrepairwillfulfillthejustarrivedrequest.Withprobability1−qsparesareavailableandthearrivingrequestdoesnothavetowait.ThisaggregatednetworkisdepictedasaTypical-serverClosedQueuingNetworkinFigure10.Thedepotrepairshopismodeledasatypicalserver.Incaseofnobacklog(k=0)theservicerateequalsinfinitywithprobability1−qandequalsmin(S0,R0)µ0withprobabilityq.Inallothercases(k>0)theservicerateequalsmin(k+S0,R0)µ0.TodetermineqNorton’stheoremisusedoncemore.AsinSubsection2.3eachbase(thetransportline,thebaserepairshopandtheproductioncell)isreplacedbyastatedependentserver.Todeterminethetransitionrateofthisstatedependentserver,eachbase-partofthenetworkisshortcircuitedanditsthroughputiscalculated.Thisthroughputoperatesastheservicerateofthestatedependentserver.ThenewnetworkwiththestatedependentserversandtheshortcircuitednetworksaredepictedinFigure11.Onceagaintheevolutionofn1canbedescribedasabirth-deathprocess.The(approximated)transitiondiagramforn1=0,...,S0isgiveninFigure12.LetTH(i)bethethroughputofthesubnetworkreplacingbase(=1,...,L)withijobspresent.Asinthesimplemodelonlythebehaviorforn1≤S0needsto 238L.Spanjersetal.Productioncell1−p1λ1λp11Base1repairµ1m11λm112µ1kTH1(i)t1machinesj1TH1(i)Transportγ1operationalDepotrepairµ0n1nTH(i)Transportγµ022TH(i)t222Productioncellµ2m22λm221Base2repairµ2λ2p2λ21−p2machinesj2operationalabFig.11.aThenewnetworkwithstatedependentservers.bTheshortcircuitednetworks∑THl(Jl+Sl)∑THl(Jl+Sl)∑THl(Jl+Sl)∑THl(Jl+Sl)llll012ii+1S0−1S0µ02µ0min(i,R0)µ0min(i+1,R0)µ0min(S0−1,R0)µ0min(S0,R0)µ0Fig.12.Transitiondiagramforn1bestudiedtodetermineq.Takeδ=TH(J+S)/µ0,thenn11P(n1=n1)=δ,n1P(n1=0)forn1=0,...,S0k=1min(k,R0)(22)andP(n1=S0)P(n1=S0)q==P(n1≤S0)S0P(n=n)n1=011δS01P(n=0)
S01min(k,R0)=k=1S0nδ1
1P(n=0)n1=0n1min(k,R10)k=1δS01
S0k=1min(k,R0)=.(23)S0δn1
1n=0n11min(k,R0)k=1ThethroughputscanbeobtainedbyapplyingastandardMDAalgorithm(see[4])ontheshortcircuitedproductformnetworksasshowninFigure11.Thesteadystatemarginalprobabilitiesaswellasthemainperformancemea-suresfortheaggregatedsystemcanbefoundbyusinganadaptedMulti-ClassMarginalDistributionAnalysisalgorithm(seeBuzacottandShanthikumar[4]forordinaryMulti-ClassMDA).Toseethis,introducetokensofclasswith Closedlooptwo-echelonrepairableitemsystems239=1,...,Lthateitherrepresentmachinespresentatbase(intheproduc-tioncell,inthebaserepairshop,inthebasestockorintransittothisbase)orrepresentrequeststothedepotstockemergingfromafailureofamachineatbasethatcannotberepairedlocally.Recallthatmachinesthathavetoberepairedinthedepotrepairshop,infactlosetheiridentity,i.e.aftercompletiontheyareplacedinthedepotstock,fromwhichtheycaninprinciplebeshippedtoanyarbitrarybase.However,therequestarrivingjointlywiththatbrokenmachineatthedepot,maintainsitsidentity,meaningthatitismatchedwiththefirstsparemachineavail-able,afterwhichthecombinationistransportedtothebasetherequestoriginatedfrom.Therefore,atokencanbeseenasconnectedtoamachineaslongasthatmachineisatthebase(inanystatus)andconnectedwiththecorrespondingrequestassoonasthemachineissenttothedepot.Thisrequestmatcheswithanavailablemachinefromstock(whichgenerallyisdifferentfromtheonesenttothedepot,unlessS0=0)andthecombinationreturnstothebasethatgeneratedtherequest.Hence,inthisway,amulti-classnetworkarisesinanaturalway.Theadaptedalgorithmisgivenbelow.AnimportantaspectofanMDAalgo-rithmisthecomputationoftheexpectedsojourntimeinthestations.Sincethedepotrepairshopismodeledasatypicalserver,thestandardsojourntimeasdescribedin[4]willnotdoforthisstation.Asdenotedbefore,incaseofnobacklog(k=0)theservicerateequalsinfinitywithprobability1−qandequalsmin(S0,R0)µ0withprobabilityq.Inallothercases(k>0)theservicerateequalsmin(k+S0,R0)µ0.Theexpectedsojourntimeofanarrivingrequestisthetimeittakesuntilallre-questsinfrontofit(k)arefulfilledandtherequestitselfisfulfilled.Thatis,thetimeuntilk+1machinescomeoutofrepair.Incasek=0withprobability1−qthesojourntimeequals0becauseasparefulfillstherequest.Theadaptationstothesojourntimerevealthemselvesinthealgorithminstep4.Anotheradaptationtotheordinaryalgorithmisfoundinstep6.Thetransitionratesfromthestateswith0machinesindepotrepairtothestateswith1machineindepotrepairnowequalqtimesthethroughput,insteadofjustthethroughput.Algorithm4Thedepotrepairshopisdefinedasstation0andallotherstationsaredefinedasstationi,wheredenotesthenumberofthebase(=1,...,L)andidenotesthespecificstationassociatedwiththatbase.Theproductioncellisdenotedbyi=b,thebaserepairshopbyi=mandthetransportlinefromthedepottothebasebyi=t.(r)LetVjbethevisitratioofstationjforclassrtypemachines.Letzdenotethenumberofmachinesinthesystemandz=(z1,...,zr,...,zL)thevectordenotingthestatethatindicatesthenumberofmachinesperclass.Thesteadystateprobabilitythatymachinesareinstationj,givenvectorzisdenotedbypj(y|z).Theexpectedsojourntimefortypermachinesarrivingatstationjgiventhatz(r)(r)machinesarewanderingthroughthesystemisgivenbyEWj(z)andTHj(z)denotesthethroughputoftypermachinesgivenstatez.Thealgorithmisexecutedasfollows:1.(Initialization)For=1,...,LsetV()=1,V()=1,V()=pand0lb1−plm1−p()(r)V=1.For=1,...,L,r=1,...,L,r/=,i∈{b,m,t}setV=0.lt0iSetz=0andpj(0|0)=1forj∈{lb,lm,lt}∪{0}. 240L.Spanjersetal.2.z:=z+1.3.Forallstatesz∈{z|Lz()=zandz()≤J+S}executesteps4=1through6.4.Computethesojourntimesfor=1,...,Lforwhichz()>0from:z−1()k+1EW0(z)=p0(k|z−e)min(R0,S0+k+1)µ0k=1q+p0(0|z−e),min(R0,S0+1)µ0z−1()b−J+11EWlb(z)=plb(b|z−e)+,Jλλb=Jz−1()m1−R+11EWlm(z)=plm(m1|z−e)+,Rµµm1=R()1EW(z)=.ltγ5.ComputeTH()(z)for=1,...,Lifz()>0from:0z()()TH0(z)=()()()(),V0EW0+i∈{b,m,t}ViEWiandifz()=0thenTH()(z)=0.ComputeTH()(z)for=1,...,Land0ii∈{b,m,t}from:()()()THi(z)=ViTH0(z).6.Computethemarginalprobabilitiesforallstationsfrom:L()µ0min(R0,S0+1)p0(1|z)=TH0(z)qp0(0|z−e),=1L()µ0min(R0,S0+k)p0(k|z)=TH0(z)p0(k−1|z−e)fork=2,...,z,=1andfor=1,...,Lfrom:()λmin(J,b)plb(b|z)=THlb(z)plb(b−1|z−e)forb=1,...,z,()µmin(R,m1)plm(m1|z)=THlm(z)plm(m1−1|z−e)form1=1,...,z,()γtplt(t|z)=THlt(z)plt(t−1|z−e)fort=1,...,z.0Computepj(0|z)forj∈{lb,lm,lt}∪{0}from:zpj(0|z)=1−pj(y|z).y=1 Closedlooptwo-echelonrepairableitemsystems241L7.Ifz==1J+Sthenstop;elsegotostep2.WiththeadaptedMulti-ClassMDAalgorithmpresentedabove,themarginalprobabilitiesofthesystemaswellasthethroughputsandthesojourntimescanbeapproximated.Fromthese,variousperformancemeasurescanbecomputed.Inthenextsectionsomeresultsobtainedbythealgorithmwillbecomparedwithresultsfromsimulation.Remark5OppositetothesimpleproblemdiscussedinSection2(whichmerelyservedtoillustratethebasicstepsoftheaggregationprocedure),anexactsolutionapproachforthecurrentextendedproblemalreadyprovestobecomputationallyintractable,duetothecurseofdimensionality.Theaggregationprocedure,ontheotherhand,yieldsnoessentialcomputationaldifficulties.Thisisduetotworeasons.Firstofall,theaggregationandsubsequentsmallchangesonsomebordertransitionratesallowustocomeupwithanear-productformsolutionfortheapproximatedsystem.Second,asaresultofthat,weareabletoapplyNorton’stheorem,whichallowsforanexactdecompositionoftheremainingapproximatedmodel.AlthoughforlargeproblemstheadaptedMulti-ClassMDAalgorithmbecomesslower,stan-dardapproximationtechniquesformulti-classsystemsareavailabletospeedupthesealgorithmsfurther,withoutlosingmuchaccuracy(seealsoourfinalremarksinSect.5).3.3ResultsInthissectionresultsobtainedbytheadaptedMulti-ClassMDAalgorithmfromtheprevioussectionwillbepresented.Theywillbecomparedtoresultsobtainedbysimulation.Foreachbaseweareinterestedintheavailability,thatistheprobabilitythatthemaximumnumberofmachinesisoperatingintheproductioncell.ForbasethisisdenotedbyAfor=1,...,L.Furthermoreweareinterestedintheexpectednumberofmachinesoperatingintheproductioncell,denotedbyEjforbase=1,...,L.For=1,...,LtheperformancemeasurescanbecomputedbyA=P(j=J)=P(b≥J)=P(k0+m1≤S),(24)Ej=E(J−[k+m−S]+)01=(J−[k+m−S]+)P(k,m).(25)0101k0,m1InTable2andTable7intheAppendix,theparametersettingsforsomerepre-sentativetestproblemsaregiven.Inthissection,weconsiderdualbasesystems(L=2);intheappendixwealsohaveexamplesofsystemswiththree(L=3)andfourbases(L=4).TheotherparametersinthiscasearegiveninTable2withJ,themaximumnumberofworkingmachinesatbase,S,themaximumnumberofstoreditemsatbase(oratthedepot),λ,thebreakdownrateofindividualmachinesatbase,µ,therepairrateofindividualmachinesatbase(oratthedepot),R,thenumberofrepairmenatbase(oratthedepot), 242L.Spanjersetal.p,theprobabilitythatamachinecanberepairedatbase,γ,thetransportationratetobaseandρ,thetrafficintensityatthebase(oratthedepot).Thefirst10modelsaresymmetric,thatisthesameparametervaluesapplytobothbases.Theother10problemsconcernasymmetriccases.Itisobviousthatalargenumberofinputparametersisrequiredtospecifyagivenproblem.Thismakesitdifficulttovarytheseparametersinatotallysystematicmanner.InAlbright[2]itisshownthattrafficintensitiesaregoodindicatorsofwhetherasystemwillworkwell(minimalbackorders)andarebetterindicatorsthanthestocklevels.Thereforeweselectedmostofthetestproblemparametersettingsbyselectingvaluesofthetrafficintensities,usuallywelllessthan1,andthenselectingparameterstoachievethesetrafficintensities.Forthebaserepairfacility,thetrafficintensityρisdefinedasρ=Jλp/Rµ,(26)themaximumfailureratedividedbythemaximumrepairrate.Similarly,thedepottrafficintensityρ0isdefinedasLρ0=Jλ(1−p)/R0µ0.(27)=1TheresultsaregiveninTable3andTable8intheAppendix.Thesimulationleadsto95%confidenceintervals.ThesimulationmethodwasthesocalledreplicationdeletionmethodwherethewarmupperiodwasfoundbyWelch’sgraphicalproce-dure(cf.LawandKelton[7]).Tocomparetheapproximationswiththesimulationresults,thedeviationfromtheapproximationtothemidpointoftheconfidenceintervaliscalculated.Thesepercentagedeviationsaregivenaswell.Fromtheresultsitcanbeconcludedthattheapproximationsareveryaccurate.Themaximumdeviationfortheavailabilityaswellastherelativedeviationfortheexpectednumberofworkingmachines,iswelllessthan1%andallapproximat-ingvaluesliewithintheconfidenceintervals.Furthermore,alltypesofproblemsexhibitedsimilarlevelsofaccuracy.4OptimizationInthepreceedingsectionsanaccurateapproximationforseveralperformancemea-suresofclosedtwo-echelonrepairableitemsystemshasbeenobtained.Theseap-proximationmethodscanbeusedtofindanoptimalallocationofsparesinthesystem,inordertoachievethebestperformance.Inthissectionwegivealgoritmstofindtheoptimalallocation.Atfirst,weformulatetheoptimizationproblem.Subsequently,wepresentafastbutreliablegreedyapproximationschemefortheoptimizationproblem.Thesectionisconcludedwithsomenumericalresults. Closedlooptwo-echelonrepairableitemsystems243Table2.Parametersettingsfortestproblemsmulti-basemodelwithtransportation(1)depotProblemS0µ0R0ρ0baseJSλµRpγρ112010.51/210211010.5∞0.5211010.51/2521510.5∞0.5311020.251/2521520.5∞0.25411010.51/2521510.5100.5511010.51/2521510.520.5611020.251/2521520.520.2571250.51/2521150.520.587250.51/2521150.520.595610.831/2551310.5∞0.831051010.51/2551510.5∞0.51112010.5110211010.5∞0.521021310.5∞1.671212010.38110211010.5∞0.521021310.75∞2.51312010.38110211010.510.5210212010.75∞0.3751422010.25110511010.520.5210212010.520.251521010.5110111210.5∞0.4221041340.5∞0.421611010.51521510.5∞0.52521510.53000.51711010.51521510.5∞0.52521510.550.51811010.51521510.5∞0.52521510.520.5191611.6711021510.5∞121021510.521201101111021510.5∞121021510.5214.1TheoptimizationproblemTheaimistomaximizetheoverallperformanceofthesystemunderabudgetcon-straintforstockingcosts.Fortheoverallperformanceofthetwo-echelonrepairableitemsystem,thetotalavailabilityAtot,definedbyL=1JλAAtot=L,=1Jλ 244L.Spanjersetal.Table3.ResultsfortestproblemsfromTable2ProblembaseAsimAappr%devEjsimEJappr%dev11/2(0.8529,0.8563)0.85420.05(9.7533,9.7615)9.75620.0121/2(0.8638,0.8750)0.86830.13(4.7957,4.8161)4.80430.0331/2(0.9695,0.9714)0.97010.04(4.9626,4.9655)4.96330.0241/2(0.8311,0.8403)0.83530.04(4.7461,4.7640)4.75430.0251/2(0.6548,0.6639)0.66050.18(4.4542,4.4737)4.46720.0761/2(0.7490,0.7539)0.75140.00(4.6463,4.6545)4.65210.0471/2(0.2938,0.3008)0.29780.17(3.6284,3.6497)3.64450.1581/2(0.3781,0.3883)0.38000.83(3.8866,3.9096)3.89070.1991/2(0.8165,0.8361)0.82340.34(4.6622,4.7032)4.67700.12101/2(0.9854,0.9894)0.98750.01(4.9785,4.9851)4.98170.00111(0.8631,0.8703)0.86630.05(9.7672,9.7864)9.77820.012(0.0739,0.0830)0.07971.54(5.8615,5.9716)5.90360.22121(0.8733,0.8785)0.87530.07(9.7915,9.8028)9.79400.032(0.0078,0.0100)0.00828.06(3.9778,4.0665)3.99650.64131(0.1252,0.1367)0.13030.49(7.5031,7.5736)7.53900.012(0.9423,0.9455)0.94520.14(9.9154,9.9219)9.92080.02141(0.8466,0.8565)0.85120.04(9.7283,9.7524)9.74080.002(0.4846,0.4995)0.48950.52(9.0411,9.0802)9.06020.00151(0.4413,0.4647)0.43823.27(8.6368,8.7362)8.63870.552(0.7007,0.7231)0.70121.50(9.3273,9.3925)9.33750.24161(0.8617,0.8694)0.86930.43(4.7934,4.8076)4.80430.082(0.8625,0.8717)0.86730.02(4.7938,4.8113)4.80280.00171(0.8644,0.8734)0.86910.03(4.7985,4.8139)4.80570.012(0.7899,0.8005)0.79570.06(4.6831,4.7016)4.69280.01181(0.8690,0.8752)0.87070.16(4.8049,4.8158)4.80820.052(0.6514,0.6618)0.65790.20(4.4494,4.4689)4.46200.06191(0.0742,0.0837)0.07692.60(6.2112,6.2972)6.23540.302(0.0277,0.0338)0.03012.15(5.5644,5.6682)5.59460.39201(0.3298,0.3430)0.33540.29(8.0845,8.1484)8.10020.202(0.1417,0.1492)0.14721.20(7.2625,7.3228)7.29670.06istaken.Itcanbeconsideredastheweightedaverageoftheavailabilitiesperbase.ThetotalavailabilityisconsideredasafunctionofthemaximalstocksizesS0,S1,···,SL;theotherparametersthatinfluencethetotalavailabilityaregiven.TheconstraintfortheoptimizationproblemisanupperboundCforthetotalstockingcosts.Thestockingcostsarelinearinthemaximumstocksizes.Letcbethestoragecostforkeepingonespareatstockpoint.The(non-linear)optimization Closedlooptwo-echelonrepairableitemsystems245problemcannowbeformulatedas:maxAtot(S0,...,SL),Ls.t.cS≤C,=0S≥0,for=0,...,L.InthenextsubsectionagreedyapproximationschemewillbegiventoapproximatetheoptimalvaluesforS0,...,SL.4.2OptimizationalgorithmThemoststraightforwardsolutionmethodtofindoptimalstocklevels,isthebruteforcemethod.Thismethodsimplychecksallfeasibleallocationsandpickstheonewhichgivesthehighesttotalavailability.ByassumingthatAtotisanincreasingfunction,thebruteforcecanbeimprovedbyconsideringonlyallocationsontheboundaryofthefeasibleregion,thatisthoseallocationwhereaddinganothersparepartwouldleadtoaninfeasibleallocation.Eventhisimprovedbruteforceapproachturnsouttoberathertimeconsuming.InZijmandAvsar[10],agreedyapproximationprocedureisgiventofindtheoptimalallocationofstocksforanopentwo-indenturemodel.Thismethodcanalsobeappliedonclosedtwo-echelonrepairableitemsystems.Atthestartoftheheuristicalgorithmnosparesareallocated.Onerepeatedlyallo-catesonesparetothelocationthatleadstothemaximumincreaseintotalavailabilityperunitofmoneyinvested,undertheconstraintthattheallocationisfeasible.Theheuristiccontinuesaslongasthismaximumincreaseispositive;itcanbepresentedasfollows:Algorithm6Approximativeoptimizationmethod(greedyapproach)1.(Initialization)SetSˆ=0,for=0,1,...,L,andsetCˆ=0.2.(Repetition)Define∆for=0,1,...,L,by⎧⎪⎨Atot(Sˆ0,...,Sˆ+1...,SˆL)−Atot(Sˆ0,...,Sˆ...,SˆL)ifCˆ+c≤C,∆=c⎪⎩0,otherwise.Letˆ=argmax∆.If∆ˆ≤0thenstop;otherwiserepeatthisstepaftersettingSˆˆ=Sˆˆ+1andCˆ=Cˆ+cˆ.3.(Solution)Theresultingstockallocation(Sˆ0,Sˆ1,...,SˆL)istheapproximativesolutiontotheoptimizationproblem.ThegreedyheuristicpresentedabovebuildsontheobservationthatAtot(S0,...,SL)tendstobehaveasanincreasingmulti-dimensionalconcavefunc-tion,inparticularfornottoosmallvaluesofSi,i=1,...,L.Thisobservation 246L.Spanjersetal.Table4.OptimalstocksizesfortestproblemsProblembaseJλµRpγcCAtot,bfS,bfAtot,greedyS,greedy1depot511100.751320.75132151510.510144251510.5101442depot511200.866860.86627151510.510177251510.5101763depot511200.832890.83289151510.510233251510.5101554depot511200.797780.79778151510.510233251510.5102335depot511200.648740.64382151510.51245251510.512446depot521200.971640.97164151510.510244271520.5102447depot522200.998700.99870151510.51011010271520.510110108depot531200.914440.914441101520.5102442101520.5102449depot531200.632760.623461102540.5102542101520.51022310depot321200.697620.69762131310.21233271320.81266ofconcavityisstronglysupportedbyempiricalevidence.Inaddition,wenotethatintheuncapacitatedcase,aformalproofoftheconcavityoftheavailabilityfunc-tioncanbegiven,basedonconvexitypropertiesofbackorderprobabilitiesasafunctionofthebasestocklevels(seee.g.Rustenburgetal.[8]),atleastwhenthevaluesofSi,i=1,...,L,exceedcertain(low)thresholds.Inotherwords:alawofdiminishingaddedvalueisvalidhere,andisagainverylikelytoholdinthecapacitatedcaseaswell.IfAtotisanincreasingfunction,theheuristicwillstopwhentheboundaryofthefeasibleregionisreached.Inthenextsectionthegreedyapproachisnumericallycomparedwiththebruteforceapproach. Closedlooptwo-echelonrepairableitemsystems247Inthissectionresultsareobtainedforseveraltestproblems.Theresultsob-tainedbythebruteforceapproacharecomparedtotheresultsfoundbythegreedyapproach.Evenwhenthegreedyapproachgivesadifferentallocationforspareitems,thetotalavailabilityonlydecreasesslightly.InTable4severaltestproblemsarepresented.TheparametersinthiscaseareJ,themaximumnumberofworkingmachinesatbase,λ,thebreakdownrateofindividualmachinesatbase,µ,therepairrateofindividualmachinesatbase(oratthedepot),R,thenumberofrepairmenatbase(oratthedepot),p,theprobabilitythatamachinecanberepairedatbase,γ,thetransportationratetobase,c,thecoststostoreanitematbase(oratthedepot),C,theavailablebudgetforstoringitems.NotethatthethemaximalstocksizesS0,S1,...,SLandthetotalavailabilityAtotarenotgivenbutcomputedbyeitherthebruteforceapproach(Atot,bfandS,bf)orbyHeuristic6(Atot,greedyandS,greedy).Thenumericalresultsindicatethatthegreedyapproachyieldsgoodresults.5SummaryandpossibleextensionsInthispaperwehaveanalyzedaclosedlooptwo-echelonrepairableitemsystemwithafixednumberofitemscirculatinginthenetwork.Thesystemconsistsofseveralbasesandacentralrepairfacility(depot).Eachbaseconsistsofaproductioncellandabaserepairshop.Therearetransportlinesleadingfromthedepottothebases.Transportfrombasestothedepotisnottakenintoaccount.Therepairshopsaremodeledasmulti-serversandthetransportlinesasampleservers.Repairshopsatthedepotaswellasatthebasesareabletokeepanumberofready-for-useitemsinstock.Machinesthathavefailedintheproductioncellofacertainbaseareimmediatelyreplacedbyaready-for-usemachinefromthatbase’sstock,ifavailable.Thefailedmachineissenttoeitherthebaserepairfacilityortothedepotrepairfacility,inthelattercaseasparemachineissentfromthedepottothebase,todepletethebase’sstockofready-for-useitems.Oncethemachineatthedepotisrepaired,itisaddedtothecentralstock.Ordersaresatisfiedonafirst-come-first-servedbasiswhileanyrequirementthatcannotbesatisfiedimmediatelyeitheratabaseoratthedepotisbacklogged.Incaseofabacklogatacertainbase,thatbase’sproductioncellperformsworse.Thisalsomeansthattheexpectedtotalrateatwhichmachinesfailattheproductioncellissmallerthaninthecaseofnobacklog.TheexactanalysisofaMarkovchainmodelforthissystemwithmultiplebasesandmanymachinesorwithlargeinventories,isdifficulttohandle.Therefore,weaggregatedanumberofstatesandadjustedsomeratestoobtainaspecialnear-product-formsolution.ThenewsystemcanbeobservedasaTypical-serverClosedQueuingNetwork(TCQN).Thenotiontypicalcomesfrommodelingthecentralrepairfacilitytogetherwiththesynchronizationqueue,asatypicalserverwithstatedependentservicerates.Thesestatedependentserviceratesfollowfromanappli-cationofNorton’stheoremforClosedQueuingNetworks.AnadaptedMulti-Class 248L.Spanjersetal.MarginalDistributionAnalysisalgorithmisdevelopedtocomputethesteadystateprobabilities.Fromthesesteadystateprobabilitiesseveralperformancemeasurescanbeobtained,suchastheavailabilityandtheexpectednumberofmachinesop-eratingintheproductioncells.Numericalresultsshowthattheapproximationsareextremelyaccurate,whencomparedtosimulationresults.Theapproximationsareusedinanoptimizationheuristictodetermineinventorylevelsatboththecentralandlocalfacilitieswithamaximaltotalavailabilityunderacostconstraint.AdisadvantageoftheadaptedMulti-ClassMarginalDistributionAnalysisal-gorithmisthecomputationalslowness.Especiallyforlargesystemswithmultiplebases,manymachinesandlargeinventories,thealgorithmisnotveryfast.Here,furtheraggregationstepsmayspeedupthesystemevaluationconsiderably,unfor-tunatelyatthecostofsomeaccuracy.Furthermore,themodelconsideredisquitearealisticmodel.However,itcouldbemorerealisticbyincludingtransportfromthebasestothedepotandtoallowformorecomplicatednetworksintherepairfacilities.Inthemodeldescribedinthispaper,eachrepairshopismodeledasamulti-server.Aninterestingextensiontothis,istoconsidertherepairfacilitytobeajobshopandmodelitasalimitedcapacityopenqueuingnetwork,ashasbeendonein[3]forthecaseofanopenmulti-echelonrepairableitemsystem.Then,itiseasytoincludetransporttothedepotrepairfacilityasjustanadditionalnodeinthejobshop.Lastbutnotleast,itisinterestingtofindaheuristictooptimizeinventorylevelsatthecentralandlocalfacilitiesincombinationwithoptimalrepaircapacities.Thiswillbethesubjectoffutureresearch.References1.AlbrightSC,SoniA(1988)Markovianmulti-echelonrepairableinventorysystem.NavalResearchLogistics35(1):49–612.AlbrightSC(1989)Anapproximationtothestationarydistributionofamultieche-lonrepairable-iteminventorysystemwithfinitesourcesandrepairchannels.NavalResearchLogistics36(2):179–1953.AvsarZM,ZijmWHM(2002)Capacitatedtwo-echeloninventorymodelsforrepairableitemsystems.In:GershwinSBetal.(eds)Analysisandmodelingofmanufacturingsystems,pp1–36.Kluwer,Boston4.BuzacottJA,ShanthikumarJG(1993)Stochasticmodelsofmanufacturingsystems.Prentice-Hall,EnglewoodCliffs,NJ5.GrossD,KioussisLC,MillerDR(1987)Anetworkdecompositionapproachforap-proximatesteadystatebehaviorofMarkovianmulti-echelonrepairableiteminventorysystems.ManagementScience33:1453–14686.HarrisonPG,PatelNM(1993)Performancemodellingofcommunicationnetworksandcomputerarchitectures.AddisonWesley,NewYork7.LawAM,KeltonWD(2000)Simulationmodelingandanalysis,3rdedn.McGraw-HillHigherEducation,Singapore8.RustenburgWD,vanHoutumGJ,ZijmWHM(2000)Sparepartsmanagementfortechnicalsystems:resupplyofsparepartsunderlimitedbudgets.IIETransactions32:1013–10269.SherbrookeCC(1968)METRIC:amulti-echelontechniqueforrecoverableitemcon-trol.OperationsResearch16:122–14110.ZijmWHM,AvsarZM(2003)Capacitatedtwo-indenturemodelsforrepairableitemsystems.InternationalJournalofProductionEconomics81–82:573–588 Closedlooptwo-echelonrepairableitemsystems249AppendixInthisappendix,numericalresultsaregivenforvariousparametersettingsinourmodel.Inmostcases,theavailabilityishighasdesiredinpracticalsituations.InTable5andTable6thefocusisonthesinglebasemodel.MultiplebasemodelsareconsideredinTable7andTable8.Table5.Resultsforthesimplesinglebasemodel,p1=0.5,λ1=1,µ0=J1,µ1=J1J1S0S1AexactAappr%devEjexactEjappr%dev113100.50560.51000.85752.31782.32250.20373300.57490.57710.37842.43382.43680.12273500.58740.58800.10662.45442.45530.03793110.73220.73400.25162.63312.63450.05453310.79480.79610.15902.72642.72790.05313510.80820.80870.05782.74632.74690.02173130.91710.91720.01142.88752.88730.00613330.94650.94660.01062.92872.92870.00053530.95350.95360.00552.93852.93850.00143140.95380.95380.00082.93762.93740.00583340.97220.97220.00012.96302.96290.00223540.97660.97660.00062.96912.96910.00035100.46900.47220.69474.16544.16880.08175300.54520.54700.32634.32244.32500.05955500.56020.56070.09874.35294.35380.02095110.70450.70590.20704.54074.54160.01875310.77480.77580.13184.66434.66540.02375510.79050.79090.04864.69154.69200.01085130.90680.90690.00944.85734.85700.00595330.94030.94040.00784.91114.91100.00165530.94840.94840.00404.92404.92400.00015140.94800.94800.00074.92074.92050.00455340.96890.96890.00064.95374.95360.00225540.97400.97400.00024.96174.96170.000510100.43180.43390.47038.96588.96760.020610300.51500.51620.23639.18199.18360.018210500.53290.53330.07569.22799.22860.007310110.67460.67560.14079.41759.41770.002310310.75350.75420.09079.58429.58480.005710510.77180.77210.03369.62259.62280.003110130.89530.89530.00589.81659.81610.003610330.93350.93350.00439.88809.88790.001710530.94280.94280.00219.90549.90530.000410140.94140.94140.00099.89809.89780.002410340.96520.96520.00119.94159.94140.001510540.97110.97110.00029.95229.95220.0004 250L.Spanjersetal.Table6.Resultsforthesimplesinglebasemodel,p1=0.25,λ1=1,µ0=2J1,µ1=J1J1S0S1AexactAappr%devEjexactEjappr%dev113100.53480.53830.66122.34022.34360.14753300.67430.67830.58782.57262.57770.19783500.72820.73100.37962.66192.66580.14683110.72010.72080.09132.59512.59560.01763310.83840.83940.11942.77462.77570.04053510.89060.89140.09582.85372.85480.03813130.87050.87050.00072.81102.81090.00073330.93110.93110.00192.89992.89990.00013530.96130.96130.00232.94422.94430.00073140.90750.90750.00012.86492.86490.00033340.95050.95050.00002.92782.92780.00023540.97260.97260.00022.96022.96020.00005100.49000.49230.45154.14934.15140.04835300.64290.64550.40154.46414.46750.07625500.70660.70850.26214.59464.59750.06205110.68140.68180.06054.45584.45600.00425310.81470.81540.07744.69834.69900.01425510.87610.87670.06174.80984.81050.01495130.84770.84770.00024.73714.73700.00055330.91820.91820.00074.85974.85960.00035530.95400.95400.00114.92194.92190.00015140.89040.89040.00024.81064.81060.00025340.94090.94090.00024.89804.89800.00025540.96720.96720.00004.94364.94360.000110100.43900.44010.25038.84818.84890.009410300.60510.60640.21989.28909.29060.017610500.68070.68170.14159.48919.49060.015810110.63380.63400.03169.22829.22820.000010310.78430.78460.03879.57039.57060.002410510.85740.85760.03009.73649.73670.003210130.81770.81770.00019.61129.61120.000310330.90070.90070.00019.78989.78980.000210530.94400.94400.00029.88289.88280.000110140.86750.86750.00029.71709.71700.000110340.92770.92770.00029.84609.84600.000110540.95970.95970.00019.91469.91460.0001 Closedlooptwo-echelonrepairableitemsystems251Table7.Parametersettingsfortestproblemsmulti-basemodelwithtransportation(2)ProblemDepotBaseJSλµRpγρS0µ0R0ρ02151010.51/2551510.5100.52231010.81/2521210.2∞0.52331020.41/2521220.2∞0.252431020.41/2521220.250.252525111/2511510.5∞0.5262330.561/2531230.550.422742100.251/2521510.5100.5288530.331/2521510.5100.5298180.631/2521510.5100.53031011.051/2731510.25∞0.353131010.751511510.5∞0.5210311010.5∞0.5323510.681211210.5∞0.52831810.7∞0.73311010.61521510.5∞0.52721510.5∞0.7348530.51/2/3521510.5100.5351480.231/2/3511230.5100.42363480.251212310.550.672511230.5100.423711530.5100.23375370.91751330.5100.392752330.2100.313753370.8100.8385521.051701520.5100.352751520.5100.3537101520.5100.35392520.451321510.550.32321520.550.153321530.550.1402540.51/2/3/4521520.5100.25 252L.Spanjersetal.Table8.ResultsfortestproblemsfromTable7ProblemBaseAsimAappr%devEjsimEJappr%dev211/2(0.9826,0.9854)0.98400.00(4.9737,4.9792)4.97650.00221/2(0.8151,0.8266)0.81920.20(4.7062,4.7304)4.71290.11231/2(0.9720,0.9742)0.97310.01(4.9650,4.9690)4.96690.00241/2(0.8559,0.8607)0.85630.23(4.8108,4.8181)4.81180.05251/2(0.5462,0.5522)0.55260.61(4.1962,4.2112)4.20610.06261/2(0.8487,0.8510)0.84930.06(4.7788,4.7833)4.78040.01271/2(0.8567,0.8614)0.85940.04(4.7882,4.7982)4.79310.00281/2(0.8704,0.8752)0.87140.16(4.8103,4.8195)4.81130.08291/2(0.8526,0.8606)0.85550.13(4.7798,4.7945)4.78510.04301/2(0.6480,0.6813)0.66080.58(6.2491,6.3370)6.28060.20311(0.7250,0.7325)0.73050.25(4.5786,4.5933)4.58840.052(0.8776,0.8871)0.88130.12(9.7689,9.7944)9.77830.03321(0.7985,0.8068)0.80190.09(1.7560,1.7676)1.76070.062(0.7977,0.8020)0.79940.05(7.6077,7.6190)7.60960.05331(0.8511,0.8587)0.85610.14(4.7765,4.7885)4.78490.052(0.6898,0.6971)0.69330.02(6.4198,6.4366)6.42370.07341/2(0.8676,0.8718)0.87110.25(4.8051,4.8128)4.81090.08351/2/3(0.5041,0.5130)0.51090.46(4.2436,4.2618)4.25580.07361(0.6456,0.6538)0.65250.43(1.5538,1.5658)1.56380.262(0.5754,0.5821)0.57900.04(4.3828,4.3952)4.38830.023(0.7056,0.7089)0.70700.04(6.5989,6.6031)6.60160.01371(0.9577,0.9617)0.95990.02(6.9352,6.9433)6.94000.012(0.7820,0.7903)0.78590.03(6.5683,6.5911)6.57780.033(0.4492,0.4542)0.45100.15(5.8151,5.8303)5.81960.05381(0.1745,0.1807)0.17660.59(5.0709,5.1143)5.08590.132(0.8530,0.8624)0.85750.03(6.7166,6.7389)6.72800.003(0.9845,0.9862)0.98480.05(6.9731,6.9760)6.97380.01391(0.9430,0.9450)0.94430.04(2.9308,2.9337)2.93260.012(0.9649,0.9671)0.96700.10(2.9595,2.9624)2.96220.043(0.9674,0.9686)0.96860.07(2.9627,2.9644)2.96440.03401/2/3/4(0.9250,0.9273)0.92680.07(4.9047,4.9083)4.90740.02 Aheuristictocontrolintegratedmulti-productmulti-machineproduction-inventorysystemswithjobshoproutingsandstochasticarrival,set-upandprocessingtimes
P.L.M.VanNyen1,J.W.M.Bertrand1,H.P.G.VanOoijen1,andN.J.Vandaele21TechnischeUniversiteitEindhoven,DepartmentofTechnologyManagement,DenDolech2,Pav.F-14,P.O.Box513,5600MBEindhoven,TheNetherlands(e-mail:{p.v.nyen,j.w.m.bertrand,h.p.g.v.ooijen}@tm.tue.nl)2UniversityofAntwerp,Antwerpen,Belgium(e-mail:nico.vandaele@ua.ac.be)Abstract.Thispaperinvestigatesamulti-productmulti-machineproduction-inventorysystem,characterizedbyjobshoproutingsandstochasticdemandin-terarrivaltimes,set-uptimesandprocessingtimes.Theinventorypointsandtheproductionsystemarecontrolledintegrallybyacentralizeddecisionmaker.Wepresentaheuristicthatminimizestherelevantcostsbymakingnear-optimalpro-ductionandinventorycontroldecisionswhiletargetcustomerservicelevelsaresatisfied.Theheuristicistestedinanextensivesimulationstudyandtheresultsarediscussed.Keywords:Production-inventorysystem–Queueingnetworkanalyser–Produc-tioncontrol–Inventorycontrol–Performanceanalysis1IntroductionThispaperaddressestheproblemofdeterminingoptimalinventoryandproductioncontroldecisionsforanintegratedproduction-inventory(PI)systeminwhichmul-tipleproductsaremade-to-stockthroughafunctionallyorientedshop.AscanbeseeninFigure1,inventoryiscarriedtoservicecustomerdemand.Thecustomersrequirethattheirordersareservicedwithatargetfillrate.Unsatisfieddemandisbackordered.Customersarriveaccordingtoarenewalprocessthatischaracterisedbyinterarrivaltimeswithaknownmeanandsquaredcoefficientofvariation(scv).
Theauthorswouldliketothanktwoanonymousrefereesandtheeditorformanyvaluablesuggestions. 254P.L.M.VanNyenetal.Fig.1.Integratedproduction-inventorysystemwithjobshoproutingsTheinventorypointsgeneratereplenishmentordersthatare,inthisintegratedPIsystem,equivalenttoproductionorders.Thereisafixedcostincurredeverytimeaproductionorderisgenerated.Inthispaper,allinventorypointsarecontrolledusingperiodicreview,order-up-to(R,S)inventorypolicies.Otherinventorypoliciescanbeembeddedintheframework,ifdesired.Theproductionordersaremanufacturedthroughtheshop.Weassumeamplesupplyofrawmaterial.Theproductionsystemconsistsofmultiplefunctionallyorientedworkcentersthroughwhichaconsider-ablenumberofdifferentproductscanbeproduced.Eachoftheproductscanhaveaspecificserialsequenceofproductionsteps,whichresultsinajobshoproutingstructure.Theproductionordersfordifferentproductscompeteforcapacityatthedifferentworkcenters,wheretheyareprocessedinorderofarrival(FCFSpriority).Beforestartingtheproductionofanorder,aset-upthattakesacertaintimeandcostisperformed.Thetotalproductiontimeofaproductionorderdependsonitssize.Whentheproductionoftheentireorderisfinished,itismovedtotheinventorypointwheretheproductsaretemporarilystoreduntiltheyarerequestedbyacustomer.Weconsidersituationsinwhichtheaveragedemandforendproductsisrelativelyhighandstationary.Sincetheproductionsystemischaracterizedbyconsiderableset-uptimesandcosts,theproductsareproducedinbatches.Weassumethatacentralizeddecisionmakercontrolsboththeinventorypointsandtheproductionsystem.Then,theobjectiveofthecentralizeddecisionmakeristominimizethesumoffixedset-upcosts,work-in-processholdingcostsandfinalinventoryholdingcosts.Moreover,weimposethatcustomerdemandhastobeservicedwithatargetfillrate.Thedecisionvariablesthatcanbeinfluencedbythedecisionmakerarethereviewperiodsandtheorder-up-tolevelsoftheproducts.Typically,integratedPIsystemswithjobshoproutingstructureandstochasticdemandinterarrivaltimes,set-uptimesandprocessingtimescanbefoundinmetalorwoodworkingcompanies,e.g.thesuppliersoftheautomotiveoraircraftbuildingindustry.Inparticular,theadventofVendorManagedInventory(VMI)hasforcedmanufacturingcompaniestointegrateproductionandinventorydecisions.ItappearsthatitisimpossibletoanalysethisintegratedPIsystemexactlyandconsequently,itisverydifficult(ifpossible)tosolvetheoptimizationproblemfaced Controlofmulti-productmulti-machineproduction-inventorysystems255bythecentralizeddecisionmakertooptimality.Therefore,weproposeaheuristicmethodthatallowsustodeterminethenear-optimalproductionandinventorycon-troldecisions.Tothisend,weapplyandintegrateaspectsfromproductioncontrolandinventorytheory.WefollowthebasicideaofZipkin[34]:weusestandardinventorymodelstorepresenttheinventorypointsofeveryproductandstandardresultsonopenqueueingnetworkstorepresenttheproductionsystem.Next,welinkthesesubmodelstogetherappropriately.OurmodeldiffersfromZipkin’smodelinseveralways.Firstly,weconsiderperiodicreview(R,S)policies,insteadofcon-tinuousreview(s,Q)policies.Secondly,weusethefillrateasmeasureofcustomerservice,insteadofabackordercost.Thirdly,ourmulti-workcenterproductionfa-cilityismodeledasageneralopenqueueingnetwork,insteadofanopenJacksonnetwork.Thegeneralopenqueueingnetworkmodelallowsustoaccountmoreaccuratelyfortheeffectofbatchingonthearrivalandproductionprocesses.Inourheuristic,theproductioncapacityisexplicitlymodeledbecausetheinventorycontrolandtheproductioncontrolareinterrelatedanddependbothontheproduc-tioncapacity.Theinventorypolicydeterminesthereviewperiods,whichontheirturndeterminetheorderthroughputtimesintheproductionsystem.Basedonthesethroughputtimes,thesafetystockcanbesetattheinventorypoints.Thisreasoningshowshowtheproductionandinventorysystemformoneintegratedsystem.Ifbothsubsystemsarecontrolledinisolation,asequentialcontrolapproachcanbeused(seee.g[24]).Thereviewperiodsaresetwithoutassessingtheimpactontheproductionsystem(e.g.usingtheeconomicorderquantity).Next,oneobservesthethroughputtimesthatresultfromtheselectedreviewperiods.Basedontheob-servedthroughputtimes,thesafetystockscanbeset.Thecostsresultingfromthissequentialdecision-makingapproachtypicallyexceedthecostsoftheintegratedcontrolapproachsincetreatingthesubsystemsinisolationleadstosuboptimality.Theintegratedproduction-inventorycontrolapproachproposedinthispaperisasimplifiedversionofthehierarchicalproductioncontrolapproachadvocatedbyLambrechtetal.[16]forthecontrolofreal-lifemake-to-orderjobshops.Theirhierarchicalcontrolapproachconsistsofthreeimportantdecisions:(i)lotsizingdecisions;(ii)determinationofthereleasedatesofproductionorders;and(iii)sequencingdecisions.Thecontrolapproachproposedinthispaperfocusesonthedecisionsatthehighestlevelofthecontrolhierarchy:thelotsizingdecisions.Inourcontrolapproach,thelotsizesaredeterminedbysettingthereviewperiodsforallproductsinthePIsystem.Notethattheproductionsystemischaracterizedbyconsiderableset-uptimes.Therefore,bysettingthereviewperiodsthedecisionmakerallocatestheavailableproductioncapacitytothedifferentproducts.Weproposeanapproximateanalyticalmodelthatallowsdeterminingthereviewperiodsthatminimizethetotalrelevantcosts.Theapproximateanalyticalmodelisanextensionofthelotsizingprocedurebasedonopenqueueingnetworksdevelopedin[16].Theirlotsizingprocedureisusedtominimizetheexpectedleadtimesinamake-to-orderjobshop.WeproposeanapproximateanalyticalmodelthattakesintoaccountmanyofthecharacteristicsoftheintegratedPIsystem.Mostimportantly,itexplicitlydealswiththeinteractionbetweentheproductionordersforthedifferentproductsinthejobshop,assumingaFCFSpriorityruleforthesequencingoftheproductionorders.Inthisway,theapproximateanalyticalmodeltakesintoaccount 256P.L.M.VanNyenetal.theinfluenceofthereviewperiodsonthecapacityutilizationoftheworkcentersandontheorderthroughputtimes(thusdealingwiththemulti-productaspectofthePIsystemunderstudy).Similarlyto[13,15]and[16],weobserveaconvexrelationshipbetweenthereviewperiodsandtheorderthroughputtimes.ThecontroldecisionssituatedatthelowerlevelofthehierarchicalframeworkofLambrechtetal.[16],thedeterminationofthereleasedatesandthesequencingdecisions,aredealtwithinastraightforwardway.Firstly,theproductionordersarereleasedimmediately.Secondly,thesequencingofallordersisdoneusingtheFCFSpriorityrule.TheuseoftheFCFSruleallowsforthequeueingnetworktobeanalyzedusingstandardqueueingnetworkanalysers.However,fromproductioncontroltheoryitisknownthatothersequencingpolicies(priorityrulesorschedulingmethods)mayleadtosubstantialtimeandcostsavings.Anoverviewofpriorityrulesandschedulingmethodscanbefoundin[21].Webelievethatsequencingpolicies,otherthantheFCFSpriorityrule,canbeembeddedinthehierarchicalcontrolframeworkinthesamewayasLambrechtetal.[16]incorporatetheshiftedbottleneckprocedure[1]fortheschedulingofproductionorders.Observethatbychoosingreasonable–notnecessarilyoptimal–reviewperiodsorlotsizes,thesequencingdecisionssituatedlowerinthecontrolhierarchybecomesignificantlyeasiertomake.Tothebestofourknowledge,thispaperistheonlyresearchthatstudiesthespecificproblemofminimizingthetotalrelevantcostsinintegratedmulti-productmulti-machinePIsystemswithjobshoproutingstructureandstochasticdemandinterarrivaltimes,set-uptimesandprocessingtimes.Forrelatedproblems,however,solutionmethodshavebeendeveloped.First,forthedeterministicversionofthisproblem,Ouennicheetal.[19,20]presentheuristicsthatarebasedoncyclicalproductionplans.Thecost-optimalcyclicalplansaregeneratedusingmathematicalprogrammingtechniques.WebelievethatitisnotpossibletoapplytheresultsofOuennicheetal.directlyinoursettingbecausetheinfluenceofvariabilityinthedemandandmanufacturingprocessesmakestheproposedproductionplansinfeasible.Asecondrelevantcontributionistheliteraturereviewonthestochasticlotschedulingproblem(SLSP)bySoxetal.[25].Thispapergivesanextensiveoverviewofcurrentresearchonthecontrolofmulti-productPIsystemsinwhichtheproductionsystemconsistsofasingleworkcenter.Thecriticalassumptioninthisbodyofresearchisthattheperformanceoftheproductionsystemisdeter-minedbyasinglebottleneckprocess.Althoughthisassumptionmaybevalidinsomesituations,itiscertainlynotinothers.Theheuristicproposedinthispaperexplicitlyconsiderssituationsinwhichtheproductionsystemconsistsofmultipleworkcenters.Thirdly,Benjaafaretal.[4,5]studymanagerialissuesrelatedtoPIsystems,suchastheeffectofproductvarietyandthebenefitsofpooling.Benjaafaretal.[4]proposesamethodtojointlyoptimizebatchsizesandbase-stocklevelsforamulti-productsingleworkcenterintegratedPIsystemcontrolledbycontinuousreview(s,Q)policies.Fourthly,AminandAltiok[3]andAltiokandShiue[2],studyproductionallocationissuesinmulti-productPIsystems.Morespecifically,theyaddresssuchissuesas“whentoswitchproductiontoanewproduct”and“whatproducttoswitch Controlofmulti-productmulti-machineproduction-inventorysystems257to”.Theyproposetohandlethefirstissuewithacontinuousreviewinventorycontrolpolicy.Thesecondissueisresolvedbyusingswitchingpoliciesthatarebasedonprioritystructures.AminandAltiok[3]usesimulationtocompareanumberofmanufacturingstrategiesandswitchingpoliciesforaserialmulti-stageproductionsystemwithfiniteinterstagebufferspace.AltiokandShiue[2]developanapproximateanalyticalmodeltocomputetheaverageinventorylevelsunderdifferentpriorityschemesforasinglemachineproductionsystem.Fifthly,RubioandWein[22]studyaPIsystemwheretheproductionsystemismodeledasaJacksonqueueingnetwork.Underthisassumption,theyderiveaformulacharacterizingtheoptimalbasestocklevel.Themaindifferencewithourapproachisthattheydonothavebatchingissuesintheirmodelbecauseoftheabsenceofset-uptimesandset-upcosts.Notethatitispreciselythebatchingissuethatmakestheproblemveryhardtosolve.Asasixthcontribution,wementiontheworkofLambrechtetal.[16]whostudythecontrolofastochasticmake-to-orderjobshop.Theydescribealotsizingprocedurethatminimizestheexpectedleadtimesandthus,theexpectedwork-in-processcosts.Tothisend,theymodeltheproductionenvironmentasageneralopenqueueingnetwork.Vandaeleetal.[28]successfullyimplementedthemethoddescribedinLambrechtetal.[16]tosolvelotsizingproblemswiththemedium-sizedmetalworkingcompanySpicerOff-Highway.Theirresearchshowsthatthelotsizingprocedureiscapableofsolvingrealistic,large-scaleproblems.OurresearchbuildsontheworkofLambrechtetal.Weextendtheirmake-to-ordermodelbyincludinginventorypointssothatweobtainanintegratedPIsysteminwhichproductsaremade-to-stockinsteadofmade-to-order.Seventhly,BowmanandMuckstadt[7]presentaproductioncontrolapproachforcyclicallyscheduledproductionsystemsthatarecharacterizedbydemandandprocessvariability.Theproductionmanagementcandelaythereleaseofmaterialtothefloorandincreaseproductioninacycletoanticipateondemandandtoavoidovertimeinfuturecycles.ThecontrolapproachusesestimatesforthecycletimeandthetaskcriticalitythatareobtainedfromaMarkovchainmodel.Usingtheseestimates,thecontrolapproachseeksatrade-offbetweeninventoryholdingcostsandovertimecosts.Finally,LiberopoulosandDallery[18]proposeaunifiedmodelingframeworkfordescribingandcomparingseveralproduction-inventorycontrolpolicieswithlotsizing.Thecontrolmethodusedinthispaperisbasedon(R,S)inventorypoliciesandiscomparabletotheReorderPointPolicies(RPPs)describedin[18].MoreinsightsonRPPscanbefoundintheirpaper.Ourworkdiffersfromtheirworkinseveralaspects.First,theystudyaN-stageserialPIsystemthroughwhichasingleproductismanufacturedwhilewefocusonasinglestagePIsysteminwhichmultipleproductsareproduced.Secondly,LiberopoulosandDalleryusequeueingnetworkrepresentationstodefine(notanalyzeoroptimize)severalproduction-inventorycontrolpoliciesthatdecidewhentoplaceandreleasereplenishmentordersateachstage.Ourworkfocusesona(R,S)inventoryrulebasedcontrolpolicyforwhichwenotonlydefinethecontrolpolicyinplace,butalsopresentaheuristictomaketheproductionandinventorycontroldecisionsthatminimizetherelevantcosts. 258P.L.M.VanNyenetal.Theremainderofthispaperisorganizedasfollows:Section2presentsthefor-malproblemstatement;Section3proposesaheuristictodeterminereviewperiodsandorder-up-tolevelsthatminimizetherelevantcosts;inSection4theperfor-manceoftheheuristicistestedinanextensivesimulationstudy;inSection5,theresultsofthesimulationstudyarediscussed;andfinally,Section6summarizesthemajorfindingsofourpaper.2FormalproblemstatementFirst,weintroducethenotationusedinthispaper.Then,wederiveformulasforthedifferentcostcomponents.Afterthis,aformalproblemdefinitionisgiven.2.1NotationGeneralinputvariables:–P:numberofproducts;–M:numberofworkcentersintheproductionsystem;–AD:interarrivaltimesofdemandforproducti(stochasticvariable);i–α∗:targetfillrateforproducti;iCostrelatedinputvariables:–ci:fixedset-upcostsincurredforoneproductionorderofproducti;–vij:echelonvalueofoneitemofproductiatworkcenterj;–vi:endvalueofproducti;–r:inventoryholdingcostperunitoftime;Tacticalcontrolvariables:–Ri:reviewperiodofproducti;–Si:order-up-tolevelofproducti;–ssi:safetystockforproducti;Performancerelatedoutputvariables:–Tij:throughputtimeofproductionordersforproductiatworkcenterj(stochas-ticvariable);–αi:realizedfillrateforproducti;Mathematicaloperators:–E[.]:expectationofastochasticvariable;–σ2[.]:varianceofastochasticvariable;–c2[.]:squaredcoefficientofvariationofastochasticvariable. Controlofmulti-productmulti-machineproduction-inventorysystems2592.2ModelingthecostcomponentsInaperiodicreviewpolicy,areplenishmentorderforproductiisplacedeveryRitimeunits.Consequently,thenumberofordersplacedpertimeunitisgivenby−1Ri,sothatthetotalset-upcostspertimeunitforproductiaregivenby:−1SCi=ciRi.WeuseLittle’slawtocomputethattheaveragenumberofitemsofproductiatE[Tij]workcenterjequalsD.Multiplyingtheaveragework-in-processatamachineE[A]iwiththeholdingcostandsummingoverallmachinesleadstothetotalwork-in-processcostforproducti:ME[Tij]WIPCi=vijr.EADj=1iThefinalinventorycostforproductiisgivenbytheformulabelow[24].Thetermbetweenbracketsgivestheaverageamountoffinalinventoryatinventorypointi,whichconsistsofhalftheaverageorderquantityplusthesafetystock.RiFICi=D+ssivir.2EAiThetotalcostforproductiissimplythesumofitscomponents.Clearly,thetotalcostTCforthewholePIsystemisgivenbythesumoverallproductsofthetotalcostsforeachproduct:PTC=(SCi+WIPCi+FICi)(1)i=12.3FormalproblemstatementInthissystem,wehaveonecentralizeddecisionmakerwhowantstominimizethetotalcostsofthePIsystem.Asstatedintheintroduction,thetotalcostsconsistofset-upcosts,finalinventoryholdingcostsandwork-in-processholdingcosts.Thedecisionmakerhastoensurethatthetargetfillratesaresatisfiedandthatthereviewperiodsarepositive.Consequently,themathematicalformulationoftheoptimizationproblemcanbestatedas:⎡⎤PME[T]Rmin⎣cR−1+ijvr+i+ssvr⎦(2)iiDijDiiRiEA2EAi=1j=1iisubjectto:1.α≥α∗fori=1,...,Pii2.Ri>0fori=1,...,PObservethatonecaneasilycomputemostofthecostcomponentsifthereviewperiodsforalltheproductsaregiven.However,twovariables–thethroughputtime 260P.L.M.VanNyenetal.intheworkcentersTijandthesafetystockssi–cannotbecomputedanalytically.InordertofindanexpressionforthethroughputtimesTijintheproductionsystem,ageneralopenqueueingnetworkshouldbesolved.Unfortunately,noexactresultsforthethroughputtimesinsuchaqueueingsystemareknown.Consequently,itisalsoimpossibletofindanexactexpressionforthesafetystockssisincessidependsontheaverageandthevarianceofthethroughputtimes.Inconclusion,itisimpossibletoderiveexactexpressionsforthesevariablesandthisimpliesthatourobjectivefunctionisanalyticallyintractable.Therefore,wehavetorelyonestimatestoevaluatethecostofagivensolution.3Heuristictodeterminereviewperiodsandorder-up-tolevelsInthissection,wepresentathree-phaseheuristicthatallowsfindingnear-optimalreviewperiodsandorder-up-tolevels.Theheuristicisbasedonanintegratedviewoftheinventoryandproductionsystemandtakesintoaccountallrelevantcosts,includingwork-in-processandsafetystockcosts.Moreover,theheuristicsimul-taneouslyconsiderscostandcapacityaspects.Thisresultsinasolutionthatisnear-optimalintermsofcostsandfeasiblewithrespecttoproductioncapacity.Giventhattheexactanalyticalevaluationoftheobjectivefunctionismathe-maticallyintractable,wehavetouseestimationmethodstoevaluateandoptimizetheobjectivefunction.TwoestimationmethodscanbeusedtoestimatetherelevantcostsinthePIsystemunderstudy:simulationandapproximateanalyticalmodels.Simulationisanaccurateestimationmethodandtherefore,simulation-basedop-timizationtechniquescanbeusedtoaccuratelysolvetheoptimizationproblem.Fordetailsonsimulation-basedoptimization,seee.g.[17].Unfortunately,thesetechniquesareveryexpensiveintermsofcomputationtime.Thisoftenprohibitstheuseofsimulationbasedoptimizationtechniques,evenformedium-sizedprob-lems.Alternatively,itispossibletooptimizeourproblemusinganapproximateanalyticalmodel.Themainadvantageofanapproximateanalyticalmodelisthelowamountofcomputationtimerequired.Obviously,thepriceofthegaininspeedisacertaindegreeofinaccuracy.Tohavethebestofbothworlds,weproposeathree-phaseheuristicthatcombinesanapproximateanalyticalmodelwithsimu-lationtechniques.Thesimulationtechniquesareusedtocircumventsomeoftheinaccuraciesduetotheapproximateanalyticalmodel.OurheuristicispresentedgraphicallyinFigure2.Intheoptimizationphase,theheuristicdeterminesnear-optimalreviewperiodsandinitialorder-up-tolevelsbasedonanapproximateanalyticalmodel.TheapproximateanalyticalmodelisdesignedsothatitcapturesthemostessentialcharacteristicsofthePIsystemwhileitcanbeoptimizedfastusingagreedysearchalgorithm.Unfortunately,theuseofanapproximatemodelmayresultinsuboptimalcontroldecisions.Also,theinitialorder-up-tolevelscomputedbytheapproximatemodelmaybeinsufficienttomeetthetargetfillrates.Therefore,thesecondphaseoftheheuristicusessimulationtechniquestofine-tunetheorder-up-tolevels.Finally,inthethirdphaseoftheheuristic,thecostsandoperationalcharacteristics(fillrates,throughputtimes,etc.)areestimatedaccuratelywithsimulation.Theremainderofthissectiondiscussesthethreephasesoftheheuristicinmoredetail. Controlofmulti-productmulti-machineproduction-inventorysystems261Fig.2.Outlineofthree-phaseheuristic3.1OptimizationphaseInthefirstphaseoftheheuristic,near-optimaltacticalinventoryandproductioncontroldecisionsaredetermined.Theoptimizationtoolconsistsoftwomainele-ments:(i)ananalyticalmodelthatapproximatestherelevantcostsgivenavectorofreviewperiodsand(ii)agreedysearchmethodthatfindsthevectorofreviewperiodsthatminimizestherelevantcosts.Inthesubsectionsbelow,bothelementsaredescribedinmoredetail.TheapproximateanalyticalmodelfollowsthesamebasicideaasZipkin[34].First,weuseastandardinventorymodeltorepresentthe(R,S)inventorypolicy 262P.L.M.VanNyenetal.ofeveryproduct,seee.g.[24].Next,weusestandardresultsonopenqueueingnetworkstorepresenttheproductionsystem.OuropenqueueingnetworkissolvedusingtheQueueingNetworkAnalyserdevelopedbyWhitt[32].Similarlyto[16],theexpressionsfortheperformancemeasuresofthequeueingnetworkarewrittenasafunctionoftheproductionlotsize.Finally,welinkbothsubmodelstogetherusingconceptsfromrenewaltheory;see[8].TheresultinganalyticalmodelisanapproximationtotherealPIsystem.SimilarlytotheapproachproposedbyZipkin,wesacrificeaccuracyforthesakeofsimplicityandcomputationaltractability.3.1.1Approximateanalyticalmodel.InthissectionwepresentanapproximateanalyticalmodeltoestimatetherelevantcostsinthePIsystemunderstudy,givenasetofreviewperiodsR=(R1,...,Ri,...,RP).Theanalyticalmodelconsistsoffoursuccessivesteps.Step1:DeterminecharacteristicsofproductionordersWestartbyanalysingthegenerationofreplenishmentordersbytheinventorypoints.NotethatinthePIsystemunderstudy,generatingareplenishmentorderataninventorypointresultsinplacingaproductionordertotheproductionsystem.Byanalysingthecharacteristicsofthereplenishmentorders,wethereforeimplicitlyanalysethecharacteristicsoftheproductionordersthatarrivetotheproductionsystem.Inourapproximationmodel,wefocusontwomaincharacteristicsoftheproductionorders:thetimebetweenthearrivalsoftwosuccessiveproductionordersofproducti,referredtoastheinterarrivaltimeAP,andtheprocessingtimeofaiproductionorderforproductionmachinej,denotedasPP.WelimitourselvestoijthedeterminationoftheexpectationE[.]andvarianceσ2[.]oftheinterarrivaltimesandtheprocessingtimes.Inthecaseofan(Ri,Si)-inventorypolicy,aproductionorderofvariablesizeisgeneratedateveryreviewmoment,i.e.everyRitimeunits.Therefore,theexpectationandvarianceoftheinterarrivaltimesofproductionordersaregivenby:E[AP]=Randσ2[AP]=0.Theproductionordersforproductiiiiareofvariablesize,whichwedenoteherebyNi.Byapplyinglimitingresultsfromrenewaltheory[8]weobtainthatthenumberofarrivalsNiinareviewperiodRisapproximatelynormallydistributedwithmeanE[N]=RE−1[AD]andiiiivarianceσ2[N]≈Rc2[AD]E−1[AD].Notethatthenormalapproximationforiiiithenumberofarrivalsinareviewperiodisonlyjustifiedifthereviewperiodisrelativelylongorthearrivalrateisrelativelyhigh,sincetheremustbeasignificantnumberofarrivalswithinareviewperiod.Sinceweareconcernedwithsituationsinwhichtheaveragedemandforendproductsisrelativelyhigh,theuseofthenormalapproximationisacceptablehere.Fromtheseexpressions,wecanderivethemeanandvarianceoftheproductionorderprocessingtimes,butfirstweintroducesomeadditionalnotation:Pij:processingtimeofoneitemofproductiatmachinej;Lij:set-uptimeofproductionordersofproductiatmachinej.Theexpectedprocessingtimeofanorderofproductionmachinejisgivenbytheexpectedtotalprocessingtimeplustheexpectedset-uptime,i.e.E[PP]=ijRE−1[AD]E[P]+E[L].Weassumethattheprocessingtimesofsingleunitsiiijij Controlofmulti-productmulti-machineproduction-inventorysystems263areindependentandidenticallydistributed(i.i.d.)andindependentoftheset-uptime.Then,thevarianceofthenetprocessingtimes,excludingset-uptime,equalsthevarianceofthesumofavariablenumberofvariableprocessingtimes,whichcanbecomputedwithaformulagivenbye.g.[24].Consequently,thevarianceoftheprocessingtimesoftheordersofproductiatmachinejisgivenby:σ2PP=E[N]σ2[P]+E2[P]σ2[N]+σ2[L]ijiijijiij(3)=RE−1ADσ2[P]+E2[P]Rc2ADE−1AD+σ2[L]iiijijiiiijStep2:AnalysequeueingnetworkThesecondstepintheapproximateanalyticalmodelusesthecharacterizationoftheproductionorderstocomputeperformancemeasuresoftheproductionsystem.Basedonthecharacterizationoftheproductionorders,wecanmodeltheproductionsystemasageneralopenqueueingnetworkinwhichthearrivalandproductionpro-cessesoftheordershaveknownfirstandsecondmoments.Fromthelateseventieson,extensiveresearchhasbeenexecutedontheestimationofperformancemea-suresinsuchqueueingsystems.OurprocedureisbasedonthequeueingnetworkanalyserdevelopedbyWhitt[32].ThelotsizingprocedureproposedinLambrechtetal.[16]isalsobasedonWhitt[32].However,ourapproachdiffersfromtheworkofLambrechtetal.intwoways:(i)theyuseasimplifiedexpressionforthescvoftheaggregatedarrivalprocess,whereasweuseWhitt’soriginalapproximation;(ii)weuseanimprovedexpressionforthescvofthedepartureprocessesthatisdueto[33],whereasLambrechtetal.useanadaptedversionofShantikumarandBuzacott[23].VanNyenetal.[29]presentsimulationresultsontheestimationperformanceofthequeueingnetworkanalyzer,whichindicatethatseriousestimationerrorsmayoccur.Thequeueingnetworkanalyserallowsustofindapproximationsfortheexpectationandvarianceofthethroughputtimesofproductiatthedifferentmachinesjintheproductionsystem,i.e.E[T]andσ2[T].Inordertoapprox-ijijimatethethroughputtimesintheentireproductionsystem,Whitt[32]assumesthatthethroughputtimesatdifferentmachinesareindependent.Then,theexpec-tationandvarianceofthetotalthroughputtimeofaproductionorderaregivenby:M2M2E[Ti]=j=1E[Tij]andσ[Ti]=j=1σ[Tij].Formoredetailsontheuseofthequeueingnetworkanalyser,see[26],[30]or[31].Step3:Calculateorder-up-tolevelsandsafetystocksInthethirdstepoftheapproximateanalyticalmodel,wedeterminetheorder-up-tolevelsS=(S1,...,Si,...,SP)usingstandardinventorytheory.Thereorderlevelissetsuchthatthecustomerdemandissatisfiedwithatargetfillrateα∗.WeneedaicharacterisationofthecustomerdemandDTi+RiduringthethroughputtimeTandiithereviewperiodRitodeterminetheappropriateorder-up-tolevel.NotethattheTi+RicustomerdemandDiisrelatedtothetimebetweensuccessivedemandarrivalsAt.Usingrenewaltheory,weobtainexpressionsforE[DTi+Ri]andσ[DTi+Ri].iiiSee[31]formoredetails.Then,theorder-up-tolevelSicanbedeterminedby:S=E[DTi+Ri]+kσ[DTi+Ri](4)iiii 264P.L.M.VanNyenetal.wherekiistheso-calledsafetyfactorforproductithatdependsonthetargetfillrateα∗.Givenatargetfillrate,[24]presentsaveryaccuraterationalapproximationiforkiinthecaseofnormallydistributeddemand.Finally,thesafetystockssiforTi+Riproducticanbecomputedas:ssi=kiσ[Di].Thisstepresultsinavectoroforder-up-tolevelsS=(S1,...,Si,...,SP)thatcorrespondtothevectorofreviewperiodsR=(R1,...,Ri,...,RP).Step4:EstimatecostsIntheprevioussteps,wepresentedanapproachtoapproximatetheexpectedthroughputtimesE[Tij]andthesafetystocksssi,givenavectorR=(R1,...,Ri,...,RP).UsingtheexpressionsforthedifferentcostcomponentsgiveninSection2.1wecannowcomputethetotalcostsTCcorrespondingtoasolutionR.3.1.2Optimizationoftacticalcontrolparameters.Intheprevioussubsection,wepresentedanapproximateanalyticalmodeltoestimatethecostofagivensetofreviewperiodsR=(R1,...,Ri,...,RP).Inthissubsection,wetrytofindthevec-torofreviewperiodsR∗=(R∗,...,R∗,...,R∗)thatminimizesthetotalrelevant1iPcosts.Unfortunately,wecannotprovetheunimodalityofthetotalcostfunctionintermsofthereviewperiodsR.However,extensivetestsallowustopostulatethattheobjectivefunction,whenestimatedwiththeapproximateanalyticalmodel,isunimodalinthereviewperiods.Wegroundthisunimodality-postulateontheob-servationthat(i)usingagreedysearchalgorithmwithdifferentstartingsolutionsalwaysresultedinthesamefinalsolution;(ii)usingsimulatedannealing,ageneraloptimizationtechniquefornon-convexfunctions(seee.g.[9]),neverresultedinabettersolutionthanthegreedysearchalgorithm.Moreover,ourpostulateisconsis-tentwith[27]whereitispostulatedthattheexpectedthroughputtimesareconvexinthelotsizes.Basedontheunimodality-postulate,weproposetouseasimplegreedysearchalgorithmfortheminimizationoftherelevantcosts,calledunivariantsearchparal-leltotheaxes.Thisapproachfixesallreviewperiodsbutoneandperformsadirectsearchalongthisvariableuntiltheminimumoftheobjectivefunctioninthecurrentdirectionhasbeenfound.Thisminimumisthenusedasthestartingpointforthenextiteration.Again,allreviewperiodsbutonearefixedandadirectsearchisperformed.Thisprocessisrepeateduntilthevalueoftheobjectivefunctioncannotbefurtherimproved.ThefinalsolutionR∗cannotbeimprovedinanydirectionparalleltotheaxesandisthesolutionproposedbythegreedysearchheuristic.Theperformanceofthegreedysearchheuristichasbeentestedagainstasimulatedannealingalgorithm.Forallinstancesinthetestbed,thegreedysearchalgorithmoutperformedsimulatedannealing.3.2TuningphaseInthefirstphaseoftheheuristic,weuseanapproximateanalyticalmodeltode-terminethenear-optimalreviewperiodsR∗andinitialsettingsfortheorder-up-tolevelsS.Becauseoftheuseofapproximations,therealizedfillratesmaybelower Controlofmulti-productmulti-machineproduction-inventorysystems265thanthetargetfillrates.Giventheconstraintsonthefillrateintheformalproblemstatementthismayresultintheinfeasibilityofthesolution.Also,itmayhappenthattherealizedfillratesarehigherthanthetargetfillrates.Obviously,thisleadstoasolutionthatisunnecessarilyexpensive.Forthesereasons,weaddasecondsteptotheheuristicinwhichtheorder-up-levelsarefine-tuned.Inthissecondstepoftheheuristic,weuseaprocedureproposedbyGudumanddeKok[10].Theirprocedurebuildsonthefollowingobservation.Giventhattheinventorypointsarecontrolledbyperiodicreview(R,S)policieswithfullbacko-rdering,thesizeandthetimingofreplenishmentordersisdeterminedbythereviewperiodsonly.InanintegratedPIsystem,thisimpliesthatthearrivalsofproductionorderstotheproductionsystem,aswellastheprocessingtimesoftheproductionordersatthedifferentworkcentersaredeterminedbythereviewintervalsandnotbytheorder-up-to-levels.Therefore,thethroughputtimesofthereplenishmentordersarecompletelydeterminedbythereviewintervals.Thisalsoimpliesthatachangeintheorder-up-tolevelsonlyinfluencesthecustomerservicelevels.Inconclusion,thisobservationstatesthatagivenselectionofthereviewperiodsfullydeterminesthestochasticbehaviorofthePIsystemandthattheorder-up-tolevelscanbeadjustedtoachieveacertaincustomerservicelevelwithoutaffectingthebehaviorofthePIsystem.Formoredetailsonthisobservation,see[10].Inourheuristic,thefine-tuningphasestartsbysimulatingtheinitialsolutiontoestimatetherealizedfillratescorrespondingtothesolution(R*,S).Basedonatraceoftheinventorylevelsinthefirstsimulationrun,aproceduredevelopedin[10]isusedtofine-tunetheorder-up-tolevels.Notethatwedonotadjustthereviewperiods.Theproceduremakesuseoftheobservationabove,statingthatchangesintheorder-up-tolevelsonlyinfluencethefillrates.Thisprocedureallowsustosettheorder-up-tolevelstothelowestpossiblelevelsthatsatisfythefillrateconstraints,denotedasS∗.Thisresultsinthesolution(R∗,S∗).Fromacomputationalpointofview,theproceduredevelopedin[10]isattractivebecauseitusesonlyonesimulationruninsteadofiterativesimulationruns.3.3EstimationphaseInthethirdandfinalphaseoftheheuristic,asimulationexperimentisusedtoestimatethecostsandoperationalperformancecharacteristicscorrespondingtothesolution(R∗,S∗).Simulationisusedbecauseofitsestimationaccuracy.AllthecoststhatarelistedinSection2.2areestimated.Theoperationalperformancecharacteristicsthatareestimatedincludethefillrates,thethroughputtimesatthedifferentworkcentersandtheutilizationoftheworkcenters.4TestingtheheuristicinasimulationstudyTheheuristicusesanapproximateanalyticalmodeltodeterminethetacticalpro-ductionandinventorydecisions,whichmayresultinsuboptimaldecisions.Inthissectionwetesttheperformanceofourheuristicinanextensivesimulationstudy.Aspecificprobleminstanceisstudiedindetailinordertogainunderstandingof 266P.L.M.VanNyenetal.themechanismsembeddedintheoptimizationtool.Furthermore,wecompareourheuristictotwosimulationbasedoptimizationmethods.Sincesimulationbasedoptimizationtechniquesallowfortheaccurateoptimizationoftheobjectivefunc-tion,thecomparisonenablesustoassessthequalityofourheuristic.Thissectionisorganizedasfollows.Wefirstpresenttheexperimentaldesignofoursimulationstudy.Then,aspecificprobleminstanceisstudiedintodetail.Finally,wecomparetheperformanceofourheuristicwiththesimulationbasedoptimizationmethods.4.1ExperimentaldesignofthesimulationstudyInthesimulationstudy,weconsideranintegratedPIsystemwith10productsand5workcenters.WeassumethatthecustomerdemandsarriveaccordingtoaPois-sonprocess.Furthermore,theset-uptimesandprocessingtimesareexponentiallydistributed,leadingtophase-typeproductionorderprocessingtimes.Thisassump-tionallowsincorporatingallkindsofvariabilitythatarepresentinrealproductionsystems:operatorinfluences,workcenterdefects,etc.Theparametervaluesinourexperimentaldesignarebasedondatafromtwomedium-sizedmetalworkingcom-panies.Inthesimulationstudy,wevaryfourfactorsoverseverallevels:–netutilizationoftheworkcentersρnet(0.65,0.75,0.85);–set-uptimesLij(randomlygeneratedintheintervals[30,60]min.or[90,180]min.);–set-upcostsci(randomlygeneratedintheintervals
[0,0],
[6.67,13.33],
[20,40]or
[60,120]);–targetfillratesα∗(0.90,0.98).iThenumberofcombinationsthatcanbemadewiththelevelsofthefourfactorsequals3×2×4×2=48combinations.WeuseaprocedurepresentedinAppendix1togeneratefiverandominstancesforeachcombinationofthelevelsofthefourfactors.Therefore,thetotalsimulationstudyconsistsof5×48=240instances.Inordertoreducethecomputationtimerequiredfortheoptimizationphaseoftheheuristic,werestrictthevalueofthereviewperiodstomultiplesof100minutes.Sincetheobjectivefunctionisflataroundtheoptimum,thisrestrictionhasanegligibleimpactonthetotalcostofasolution.Inthesecondandthirdstepofourheuristic,weuseasimulationmodelthatisbuiltinSimula.Simulaisageneral-purposesimulationlanguage,formoredetailssee[6].Weusethebatch-meansmethodwith10subrunstofindperformanceestimates.Thelengthofthesubrunsischosensothatatleast100,000customerordersforeachproductarrive.Thereviewmomentsareinitializedbylettingthemstartatarandommomentintheinterval[0,Ri]fori=1,...,P.Thisensuresthatnospecialpatternsarebuiltintotheordergenerationprocessandintothearrivalprocessoforderstotheproductionsystem. Controlofmulti-productmulti-machineproduction-inventorysystems2674.2MechanismsembeddedintheapproximateanalyticalmodelInthissection,wediscusshowtheapproximateanalyticalmodelusesthereviewperiodstominimizethetotalrelevantcosts.Themechanismsbehindtheselectionofthereviewperiodsareillustratedbycomparingthedetailedoutputoftheopti-mizationtoolwiththeoutputofasimpleheuristicmethodforsettingthereviewperiods.Morespecifically,weusetheeconomicorderquantity(EOQ)expressedasatimesupplytosetthereviewperiods,seee.g.[24]:12ciEADiRi=(5)virTheEOQformulacompletelyignorestheimpactofthelotsizingdecisionontheproductionsystem.Hence,itdoesnottakeintoaccountthecoststhatarerelatedtocapacityutilizationandthroughputtimes,i.e.work-in-processandsafetystockcosts.WeusetheuncapacitatedEOQapproachtosolveonerandomsetof48probleminstancesoftheexperimentaldesign.Below,theresultsforall48probleminstancesaresummarized,butfirstwestudyindetailonespecificprobleminstance.Thisprobleminstanceisselectedbecauseitclearlydemonstrateshowourheuristicworks.Inthisway,thereadercangainunderstandingofthemechanismsthatareembeddedintheheuristic.Theselectedprobleminstanceischaracterizedby:ρnet=0.85;L∈[90,180]min.;c∈[6.67,13.33]
∗iji;αi=0.98.InTables1upto3,wepresentthedetailedoutputofthesimulationoftheheuristic(HEU)andtheuncapacitatedEOQmethodforthisprobleminstance.Fromtheanalysisofthedecisionvariablesandthecorrespondingperformancemeasures,welearnhowtheoptimizationtoolworksandhowittriestoachievetheminimaltotalrelevantcosts.Table1displaysthedecisionvariables(reviewperiodsandorder-up-tolevels)andtheresultingthroughputtimes,characterizedbytheirexpectationE[T]andstandarddeviationσ[T].ItcanbeseenfromTable1thattheheuristicproposesconsiderablysmallerreviewperiodsthantheuncapacitatedEOQmethod.Theorder-up-tolevelsareloweredaccordingly.Remarkthatthereviewperiodsofthedifferentproductsaredecreasedinanon-proportionalwayinordertoaccountforthespecificprocessingcharacteristicsofeveryproduct.Theimpactofthesmallerreviewperiodsontheexpectationandstandarddeviationofthethroughputtimesishigh:theexpectedthroughputtimesdecreaseby36.5%onaveragewhilethestandarddeviationsofthethroughputtimesdecreaseby42.4%onaverage.Table2showstheimpactofthechangesinthedecisionvariablesontherelevantcosts.Sincethesolutionoftheheuristicusesconsiderablysmallerreviewperiods,theset-upcostsaresubstantiallyhighercomparedtotheuncapacitatedEOQsolu-tion(+58.4%).However,sincethereviewperiodschosenbytheheuristicleadtoshorterandmorereliablethroughputtimes,thework-in-processcostsandthefinalinventoryholdingcostsdeclinesignificantly(−35.8%and−36.5%).Overall,thisleadstoacostdecreaserealizedbytheheuristicversustheuncapacitatedEOQapproachof9.0%.Finally,Table3givesinsightintothemechanismsembeddedintheoptimizationtool.Weuseelementaryinsightsfromqueueingtheorytoillustratethetrade-offs 268P.L.M.VanNyenetal.Table1.Decisionvariablesandthroughputtimesforoneprobleminstance:heuristicvs.uncapacitatedEOQmethodProd.HEUEOQRSE[T]σ[T]RSE[T]σ[T]137006653784.2653.1815713266718.31362.6248006083844.4865.761788275385.91307.7346006575107.2974.5744910287812.71574.1454005362291.0628.963596793062.31012.5550008073639.5857.0570310925424.31481.8640008605028.4923.8760514908259.61539.9751005612781.4782.075348254045.01185.1870005742422.7673.672897033461.81329.2943006624763.3852.6700010617518.31472.91037004113455.2694.880958466780.51452.2Avg.4760634.13711.7790.67136.9987.75846.91371.8Table2.Costcomponentsforoneprobleminstance:heuristicvs.uncapacitatedEOQmethodProd.HEUEOQOCFICWIPCOCFICWIPC18867.42526.52337.84021.75475.64183.723790.33230.42183.12945.14416.53079.635571.83242.32867.63440.85123.04410.643277.12849.11257.72783.03681.21681.354325.44521.52909.13792.26161.74257.868224.63579.13793.24325.66332.56192.074709.42842.71409.53188.04203.92050.283393.53581.21340.63259.24523.11960.094820.32670.32648.02960.64331.34224.1106096.11877.11555.82786.53929.43055.4Tot.53076.030920.222302.333502.648178.135094.6Overall106298.4116775.2thataremadebytheheuristic.FromelementaryqueueingtheoreticalresultsfortheGI/G/1queue,e.g.HoppandSpearman[12],welearnthattherearefourmainelementsaffectingtheexpectationofthethroughputtimesE[T]onthemachines:(i)utilizationρ;(ii)variationofthearrivalsc2;(iii)averageprocessingtimet;ap Controlofmulti-productmulti-machineproduction-inventorysystems269Table3.Operationalcharacteristicsofproductionsystemforoneprobleminstance:heuristicvs.uncapacitatedEOQmethodMach.nr.HEUEOQ2222ρcatpcpρcatpcp10.910.7015582.50.0460.890.6735867.70.08220.930.6834547.90.0470.900.7526807.20.08230.890.61461022.60.0240.880.59641563.90.05040.900.4963806.30.0450.880.65331256.10.18050.920.691636.20.0530.890.79671070.70.133Avg.0.910.637719.10.0430.890.6951113.120.105(iv)variationoftheprocessingtimesc2.ThisinsightisbasedontheKingmanpapproximationfortheexpectationofthethroughputtimesinaGI/G/1queue[14]:c2+c2apρE[T]=tp+tp(6)21−ρFromTable3,itcanbeseenthattheoptimizationtooladaptsthereviewperiodssothatthevariationofthearrivalsandtheexpectationandthevariationoftheprocessingtimesarereduced.However,thishappensattheexpenseofincreasedutilizationlevels.Wemayconcludethattheoptimizationtool‘harmonizes’thereviewperiodsofthedifferentproductssoastoobtainthebestbalancebetweenutilizationandvariability.Inthejobshopproductionsystemunderstudy,thedepartureprocessofama-chineisthearrivalprocesstothenextmachineintheroutingofaproduct.Anelementaryapproximation,duetoHoppandSpearman[12],forthescvofthedepartureprocessleavingaqueueis:c2=ρ2c2+1−ρ2c2(7)dpaFromthisapproximation,itcanbeobservedthatwhentheutilizationofthema-chinesishigh,itisimportanttoachievelowvariationintheprocessingtimesinordertoobtainanarrivalprocesswithlowvariabilitytothenextmachine.Table3showsthattheheuristicrealizesalowvariationintheprocessingtimes,whiletheutilizationlevelsarehigh(around90%).FromTable1,itcanbeseenthattheactionstakenbytheheuristic,basedonthemechanismspresentedabove,resultinshorterandlessvariablethroughputtimes.Notethatthemechanismsdescribedaboveareembeddedintheproposedapproximateanalyticalmodelusingadvancedqueueingtheoreticalresultsdevelopedin[32,33].Now,webrieflypresenttheresultsforthe48instancesthatweresolvedusingtheuncapacitatedEOQmethod.In14outof48probleminstances,theunca-pacitatedEOQapproachresultedintoasolutionthatisinfeasiblewithrespecttoproductioncapacity.Forthe34feasibleinstances,theuncapacitatedEOQsolutionisonaverage5.2%moreexpensivethanthesolutionoftheheuristic.Themaxi-mumcostincreasereportedonthissetofexperimentsis10.1%.Theconclusionof 270P.L.M.VanNyenetal.theseexperimentsisthattheuncapacitatedEOQmethodmayworkrelativelywellcomparedtotheheuristic,butsincetheuncapacitatedEOQapproachdoesnottakeintoaccountcapacityissues,itmayresultinunnecessarilyexpensivesolutionsorinsolutionsthatareinfeasiblewithrespecttoproductioncapacity(andthatrequirecapacityexpansionintheformofoverwork,outsourcing,etc.).Inordertoavoidthatinfeasiblesolutionsareobtained,onecanaddcapac-ityrestrictionstotheuncapacitatedEOQmethod.Doingso,weobtainthefol-lowingmathematicalprogrammingproblem,whichwecallthecapacitatedEOQapproach:#%Pcivirmin+RiRiRi2EADi=1isubjectto:#%(8)PE[Pij]E[Lij]max1.D+≤ρjforj=1,...,Mi=1EAiRi2.Ri>0fori=1,...,PTheobjectivefunctionofthismathematicalprogramisidenticaltothecostfunctionoftheuncapacitatedEOQmethod.Thefirstsetofconstraintsimposesthatthemachineutilizationmustbelowerthanamaximumallowableutilizationlevelρmax.Thesecondsetofconstraintsstatesthatthereviewperiodsshouldbestrictlyjpositive.NotethattheobjectivefunctionandtheconstraintsareconvexinthereviewperiodsRi.ThisconvexprogrammingproblemcaneasilybesolvedtooptimalityusingthecommerciallyavailableCONOPTalgorithm.TheCONOPTalgorithmattemptstofindalocaloptimumsatisfyingtheKarush-Kuhn-Tuckerconditions.Itiswellknownthatforconvexprogrammingproblemsalocaloptimumisalsotheglobaloptimum(seee.g.[11]).ThemaindifficultythatariseswiththiscapacitatedEOQmethodisthechoiceofthemaximumallowableutilizationlevelρmax.Fordeterministicproblemsρmaxjjisusuallychosensothatallproductioncapacityisutilized,i.e.ρmax=1.Clearly,jinstochasticsettingsρmaxshouldbelowerthan1forreasonsofstability.Itis,jhowever,notobvioushowtheprecisevalueofρmaxshouldbechosen.Ifρmaxisjjchosentoolow,thisleadstolongreviewperiods(inordertoreducethecapacityutilizationduetoset-ups)andthustohighcyclestocks.Onthecontrary,ifρmaxisjchosentoohigh,thisresultsinhighcongestion,largeamountsofwork-in-process,longthroughputtimesandhighsafetystocks.Apriori,theEOQmodelisnotabletopredictwhichvalueofρmaxleadstothelowesttotalrelevantcosts.Therefore,jinourexperimentswevaryρmaxoverarangeofreasonablevaluesandobservethejresultingtotalrelevantcosts.WeusethecapacitatedEOQmethodtosolvethe48probleminstancesthatwerealsosolvedusingtheuncapacitatedEOQmethod.Thevalueofρmaxissetjto0.90,0.95and0.99.LetusdefinethedeviationinthetotalcostsbetweenthecapacitatedEOQmethodandourheuristicas:TCeoq−TCheu∆=×100%.TCheu Controlofmulti-productmulti-machineproduction-inventorysystems271Table4.Summarystatisticsfor∆,therelativedeviationintotalcostsbetweenthecapacitatedEOQmethodandtheproposedheuristic(instanceswithset-upcosts>0)maxmaxmaxρj=0.90ρj=0.95ρj=0.99min.1.91.91.9netρ=0.65avg.3.73.73.7max.5.45.45.4min.1.81.71.7netρ=0.75avg.5.05.05.0max.7.57.57.5min.5.74.23.8netρ=0.85avg.25.27.86.7max.80.210.710.1Table5.Summarystatisticsfor∆,therelativedeviationintotalcostsbetweenthecapacitatedEOQmethodandtheproposedheuristic(instanceswithset-upcosts=0)maxmaxmaxρj=0.90ρj=0.95ρj=0.99min.3.37.1121.2netρ=0.65avg.3.89.0157.6max.4.611.0224.5min.17.51.871.5netρ=0.75avg.19.23.184.5max.21.24.596.2min.106.815.011.7netρ=0.85avg.114.617.913.8max.119.120.416.0Tables4and5givetheminimum,averageandmaximumof∆,respectivelyfortheinstanceswithset-upcosts>0andtheinstanceswithset-upcosts=0.Theresultsareshownforthedifferentlevelsofthenetutilizationofthemachinesρnet.TheresultsinTables4and5showthattheproposedheuristicalwaysoutper-formsthecapacitatedEOQmethod.ThecapacitatedEOQapproachmayworkreasonablywell,providedthatagoodchoiceismadeforρmax:thelowest∆ob-jservedinthissetofinstancesis1.7%.However,onecanalsoobservethataninappropriatechoiceofρmaxmayresultinaverypoorperformance:themaximumjof∆inthissetofinstancesis224.5%.Asmentionedbefore,theEOQapproachdoesnotprovideanyguidelineforchoosingthevalueofρmax.jFortheinstanceswithset-upcosts>0andρmax=0.99,theperformanceofjthecapacitatedEOQmethodseemsreasonable:theaverageof∆is5.1%witha 272P.L.M.VanNyenetal.maximumof10.1%.Itappearsthatinthemajorityoftheseinstancesthecapacityconstraintsarenon-bindingsothatthesolutionofthecapacitatedEOQmethodisidenticaltothesolutionoftheuncapacitatedEOQmethod.Whenρmaxislowered,jtheperformanceofthecapacitatedEOQmethoddegradesfortheinstanceswithhighρnet.Whenρnet=0.85andρmax=0.90,theaverageof∆is25.2%withjamaximumof80.2%.Forthemajorityoftheinstanceswithlowandmoderateρnet,thecapacityconstraintsarealsonon-bindingwhenρmax=0.90and0.95.jTherefore,intheseinstancestheperformanceofthecapacitatedEOQmethodissimilartothatoftheuncapacitatedEOQmethodandthecapacitatedEOQmethodwithρmax=0.99.jFortheinstanceswithset-upcosts=0,itseemstobeevenmoreimportanttose-lecttheappropriatevalueofρmaxthaninthecaseofset-upcosts>0.Forexample,jwhenρnet=0.65,thecapacitatedEOQmethodworkswellwhenρmax=0.90:theaverageof∆is3.8%withamaximumof4.6%.Ifρmaxischosentoohigh,theperformanceofthecapacitatedEOQmethoddegradesstrongly:forρmax=0.99,theaverageof∆is157.6%withamaximumof224.5%.Similarresultsholdfortheinstanceswithρnet=0.75.Fortheinstanceswithρnet=0.85,thecapacitatedEOQapproachperformsratherpoorlyforallchoicesofρmax:theminimumof∆reportedonthissetofinstancesis11.7%.ThemainconclusionfromtheseexperimentsisthatthecapacitatedEOQap-proachisverysensitivetothechoiceofρmax.Sincetheappropriatevalueofρmaxdependsonthespecificcharacteristicsoftheprobleminstance,itisdifficulttodevelopageneralruleofthumbforselectingρmax.Theheuristicproposedinthispaperdoesnotsufferfromthisproblem.Theapproximateanalyticalmodelembed-dedintheheuristicexplicitlymodelstheimpactofthereviewperiodsoncapacityutilizationandoncongestionphenomena,takingintoconsiderationthespecificcharacteristicsoftheprobleminstance.Therefore,ourheuristicisamorerobustandreliablemethodtosetthedecisionvariables.4.3TestingthequalityoftheheuristicInthissection,wetesttheoptimizationqualityoftheheuristic.Themethodologyusedforthistestwarrantssomediscussion.First,notethattheoptimalsolutionfortheproblemunderstudyisunknown.Furthermore,nohigh-qualityboundsontheoptimalcostsareavailable,mainlyduetothedifficultytofindboundsonthewaitingtimesintheproductionsystem.Moreover,tothebestofourknowledge,noothercontrolapproacheshavebeendevelopedfortheintegratedPIsystemwithjobshoproutingsandstochasticarrival,processingandset-uptimes.Consequently,theheuristicsolutioncannotbecomparedtothetrueoptimum,nortoagoodbound,nortoanothercontrolapproachreportedintheliterature.Inshort,thereexistsnogoodbenchmarktotestthequalityofourheuristic.Therefore,weconstructedourownbenchmarksinordertotesttheperformanceoftheheuristic.First,wetestthepredictionqualityoftheapproximateanalyticalmodel.Ifthepredictionqualityoftheapproximatemodelissatisfactory,thenonemayexpectthattheoptimizationqualityofthetoolisgood.However,whentheapproximateanalyticalmodelwronglyestimatestheabsolutevalueofthecosts,butcorrectly Controlofmulti-productmulti-machineproduction-inventorysystems273Fig.3.Frequencydiagramforrelativedifferencebetweentotalcostsapproximateanalyticalmodel(AAM)andsimulation(SIM)capturestherelativebehaviorofthecostsinfunctionofthereviewperiods,theoptimizationprocessmaystillperformwell.InFigure3,wepresenttherelativedifferencebetweenthecostestimatesoftheapproximateanalyticalmodel(AAM)andsimulation(SIM)forthesolutionsproposedbytheheuristicforthe240in-stances.Sinceforoptimizationpurposesmainlytheabsolutevalueoftherelativedeviationisrelevant,wegroupthepositiveandnegativeintervalswiththesameabsolutevalue.Thefrequencyofnegativeandpositivevaluesoftherelativedevi-ationisshownindistinctivecolors.FromFigure3,itcanbeseenthattherelativeapproximationerrorisrathersmall,lyingintherangeof−17%to12%onthissetof240experiments.Fromthisfigure,italsocanbeseenthatforthevastmajorityoftheinstances(morethan92%)theabsoluterelativeapproximationerrorislowerthan10%.Negativerelativedifferencesoccur,butinmorethan70%ofthecasestherelativeerrorispositive.Fromtheseresults,weconcludethattheestimationqualityoftheapproximateanalyticalmodelissatisfactory.Secondly,inordertotesttheoptimizationqualityoftheheuristicwedeveloptwosimulationbasedoptimizationmethodstosolveseveralinstancesofouropti-mizationproblem.Simulationbasedoptimizationisknowntobeanaccuratebuttime-consumingoptimizationmethod,seee.g.LawandKelton[17].Theperfor-manceofthesimulationbasedoptimizationmethodsiscomparedtotheperfor-manceofourheuristicintermsoftheoptimizationqualityaswellastherequiredcomputationtime.Weusetwodifferentsimulationbasedoptimizationmethods:–AmodificationofthegreedysearchalgorithmpresentedinSection3.1.2.Weusethreedifferentstepsizestoperformthesearchalongtheaxes.Foreachofthestepsizes,weusethegreedysearchalgorithm.Thesolutionofonephaseisusedasaninputforthenextphase.Thestepsizesforthereviewperiodsare2500,500and100minutes.–OptQuest,acommerciallyavailablesoftwarepackagedevelopedbyGloveretal.OptQuestcombineselementsofscattersearch,taboosearchandneuralnetworkstofindsolutionsfornon-convexoptimizationproblems[17].Welimitthereviewperiodstotheinterval[0.5,2]timesthereviewperiodsR∗proposed 274P.L.M.VanNyenetal.byourheuristic.Moreover,weuseastepsizeof100minutesforthereviewperiods.BothoptimizationtechniquessuggestnewvectorsR
thatneedtobeevaluatedusingsimulation.Inourresearch,weusethesecondandthethirdphaseofourheuristictoevaluatethevectorR
.TheevaluationofavectorR
consistsofatuningphase,seeSection3.2,inwhichthecorrectorder-up-tolevelsarecomputedbasedonasimulationrun.Next,thetotalcostofthesolutionR
isestimatedusingasecondsimulationrunintheevaluationphasepresentedinSection3.3.Inordertoreducethecomputationtimerequired,weusethesolutionoftheheuristicasinitialsolution.Thesimulationbasedoptimizationmethodsarethenusedtoimprovethisinitialsolution.Furthermore,welimitthenumberofsimulationsubrunsto5.However,evenwiththesemeasures,thesimulationbasedoptimizationtechniquestakeverylargeamountsofcomputationtime.Forthisreason,itisimpossibletousethesimulationbasedoptimizationtechniquesforall240instancesintheexperimentaldesign.Instead,weselect15worst-caseinstancesforwhichweapplysimulationbasedoptimizationtechniques.Theselectionofthe15worst-caseinstanceswarrantssomediscussion.LetusfirstintroducealowerboundforthetotalcostsinthePIsystemunderstudy.Thelowerboundneglectstheimpactofthevariabilityinthesystemaswellastheinteractionbetweendifferentproducts.MoredetailsonthelowerboundcanbefoundinAppendix2.Thecomputationofthelowerboundisonlypossibleforthe180instanceswithset-upcostsci>0.Wecomputetherelativedeviatione1betweenthetotalcostofourheuristicandthelowerbound.Onthesetof180instances,e1usuallyliesintheinterval10–30%withanaverageof20%.Inthispaper,weusee1asanindicatorforthepotentialimprovementthatcanberealisedwhensimulationbasedoptimizationtechniquesareused.Weselect10instanceswiththelargeste1tobeoptimizedusingsimulationbasedoptimizationtechniques.Sincetheseinstanceshavethelargestpotentialforimprovement,wecallthemworst-caseinstances.Weensurethatonlyoneofthefiverandominstancesofthesamecombinationoflevelsofthefourfactorsisselectedasaworst-caseinstance.Forthecaseofset-upcosts=0,wecannotcomputethelowerboundpresentedabove.Therefore,weuseanothercriteriontoselecttheinstances.SincetheoptimizationqualityoftheheuristicdependsontheaccuracyoftheapproximateanalyticalmodelpresentedinSection3.1.1,weselectthe5instanceswiththelargestdeviationbetweencostestimateoftheapproximatemodelandthetotalcostsestimatedwithsimulationfortheoptimalsolutionR∗proposedbytheheuristic.Theselectedinstancesareworst-caseinstancesonthecriterionofapproximationperformance.Again,weensurethatonlyoneofthefiverandominstancesofthesamecombinationoflevelsischosen.Inthisway,15worst-caseinstancesareselected.Table6summarizesthefindingsofthesimulationbasedoptimizationexper-imentsforthe15worst-caseinstances.ItpresentstheoptimaltotalcostsofourheuristicTCheu,thegreedysearchalgorithmTCgsandtheOptQuestalgorithmTCoptq.Alsothe90%confidenceintervalonthetotalcostsisgiven.Finally,Table6presentstherelativeimprovementofthegreedysearchandOptQuestalgorithmwithregardtothesolutionoftheheuristic,definedas: Controlofmulti-productmulti-machineproduction-inventorysystemsTable6.ResultsofsimulationbasedoptimizationexperimentsheugsoptqExp.nr.TCCI(90%)TCCI(90%)i1(%)TCCI(90%)i2(%)max(i1,i2)(%)1172843.8426.6169700.5722.71.8170462.7335.51.41.82166937.9288.5164988.8403.11.2163182.4153.92.22.23180776308.7179496.93050.7175361.6391.33.03.04308863.3508.5305763.6759.41.0303366.35871.81.85193689.6650.5187546.9838.73.2191255.4790.81.33.26119521.1592.6118594.79380.8118768.7657.50.60.87103428.2270.3101259.92402.1101309.4240.72.02.18139734.6717.5135119.111773.3138166.8622.91.13.39292435.2566.0290025.9558.70.8286411.8424.32.12.11098308.4143.095414.6100.62.995801.9129.42.52.91121316.695.021027.7151.21.420899.5127.92.02.01226897.1158.026503.7119.71.526105.3123.72.92.9136883.825.06755.418.41.96752.222.81.91.9148690.843.7851037.72.18665.632.70.32.11513561.9102.21272972.46.113189.471.32.76.1Average2.01.92.5275 276P.L.M.VanNyenetal.TCheu−TCgsTCheu−TCoptqi1=×100%andi2=×100%.TCheuTCheuOnthesetof15experiments,thegreedysearchheuristicachievesa2.0%improvementonaverage.TheOptQuesttechniqueachievesa1.9%improvementonaverage.Takingthemaximumimprovementforeachinstance,weseethatthesolutionofourheuristiccanbeimprovedby2.5%onaverageusingsimulationbasedoptimizationtechniques.Onthissetof15worst-caseinstances,themaximumimprovementis6.1%.Basedontheseresults,weclaimthattheoptimizationqualityofourheuristicissatisfactory.Next,wecomparetheresultsofthegreedysearchalgorithmandtheOptQuestalgorithm.Table6showsthatin60%oftheinstancesthegreedysearchalgorithmoutperformstheOptQuestalgorithm,whilein40%oftheinstancesOptQuestout-performsthegreedysearchalgorithm.Thedifferenceinoptimizationperformancebetweenbothsearchmethodsdoesnotexhibitaclearpatternandwewerenotabletorelateittoanyofthefactorsinthestudy.Inouropinion,thedifferenceinper-formanceisduetothefactthatbothsimulationbasedoptimizationtechniquesareheuristicswithoutanyperformanceguarantee.Bothsimulationbasedoptimizationtechniquescangetstuckintonon-optimalsolutions,ascanbeobservedinTable6.Thesimulationbasedoptimizationtechniquesrequirelargeamountsofcompu-tationtime:theOptQuestalgorithmisstoppedafter1000iterations,resultinginthesolutionspresentedinTable6.ThegreedysearchalgorithmusedavariablenumberofiterationstoachievetheresultsinTable6:onaverage123,withaminimumof80andamaximumof185.Dependingontheprobleminstance,oneiterationtakesabout2.5to4minutesonanIntelPentium4–2.00GHz.processor.Onaverage,theOptQuestalgorithmtakesabout54hourstofindthesolutionspresentedinTable6.Thegreedysearchalgorithmgivescomparableresultsinamuchshorteramountoftime:about6.5hoursonaverage.Ourheuristic,however,ismanytimesfasterthanthesimulationbasedtechniques:ittakesabout8minutesonaveragetofindsolu-tionsthatareonlyslightlyworsethanthesolutionsfoundbythesimulationbasedoptimizationtechniques.Inconclusion,wecanstatethatourheuristicperformssatisfactory:ittakesafractionthetimerequiredbysimulationbasedoptimizationtechniquestofindsolutionsthatareonlyslightlyworseintermsoftotalcosts.Table7presentstheaverageofthereviewperiods,order-up-tolevelsandma-chineutilizationforeveryprobleminstancesolvedusingsimulationbasedoptimiza-tion.Thefirstcolumngivestheinstancenumber.Thesecondandthirdcolumnspresenttherelativedifferencebetweentheaverageofthereviewperiodsoverallproductsforthesimulationbasedtechniquesandtheheuristic.Thefourthandfifthcolumnsgivetherelativedifferencebetweentheaverageorder-up-tolevelsforthesimulationbasedoptimizationmethodsandtheheuristic.Thesixthandsev-enthcolumnsgivetherelativedifferenceintheaverageoftheutilizationofthemachinesintheproductionsystembetweenthesimulationoptimizationandtheheuristic.Finally,theeightandninthcolumnsrepeattherelativeimprovementi1andi2thatisobtainedbyusingthesimulationbasedoptimizationtechniquesversustheheuristic.FromtheanalysisofTable7,wetrytogainunderstandingofhowthethreeop-timizationtechniqueswork.Surprisingly,noclearpatternsappearinthenumerical Controlofmulti-productmulti-machineproduction-inventorysystemsTable7.Relativedifferenceinaveragereviewperiods,order-up-tolevels,utilizationandtotalcostsforsimulationbasedoptimizationvs.heuristicExp.nr.%diff.inavg.%diff.inavg.%diff.inavg.%improvementreviewperiodsorder-up-tolevelsutilizationlevelintotalcostsRx−RheuSx−Sheuρx−ρheuTCheu−TCxRheu×100%Sheu×100%ρheu×100%TCheu×100%gsoptqgsoptqgsoptqgsoptq12.46.52.23.3−0.5−0.51.81.424.96.85.66.3−0.7−1.01.22.233.36.33.93.9−0.1−0.30.7344.610.15.27.7−0.2−0.211.852.68.7−1.04.0−0.3−0.63.21.36−3.8−1.4−2.3−1.20.40.20.80.677.78.74.75.1−1.2−1.62.128−8.4−1.7−6.0−1.60.70.13.31.199.55.38.62.7−0.6−0.40.82.1101.71.5−2.6−1.40.0−0.32.92.511−3.42.2−1.7−0.91.7−0.71.4212−1.52.6−1.2−1.60.8−0.51.52.9134.14.50.00.0−0.6−1.11.91.914−0.82.0−1.90.10.1−0.62.10.315−2.43.0−4.8−1.41.0−0.66.12.7277 278P.L.M.VanNyenetal.resultsinTable7.Therelativeimprovementdoesnotseemtobedirectlyrelatedtotherelativedifferenceintheaveragereviewperiodsandorder-up-tolevels.Takee.g.instance4wheretheOptQuestalgorithmincreasestheaveragereviewperi-odsby10.1%,whilethegreedysearchalgorithmproposesanincreaseof4.6%.ForthisinstancetherelativeimprovementrealizedbytheOptQuestalgorithmis1.8%whiletherelativeimprovementforthegreedysearchalgorithmis1.0%.Onemayconcludethatlargerdifferencesinthereviewperiodsleadtolargercostim-provements.However,thisconclusioniscontradictedbye.g.instances9and15.Ininstance9,thegreedysearchalgorithmincreasesthereviewperiodsby9.5%leadingtoacostsavingof0.8%,whiletheOptQuestalgorithmincreasesthelotsizesbyonly5.3%leadingtoalargercostsavingof2.1%.Ininstance15,thegreedysearchalgorithmobtainsacostimprovementof6.1%,thelargestobservedonthissetofexperiments.Inordertoachievethiscostimprovement,thereviewperiodsarereducedby2.4%onaverage.Furthermore,itcanbeobservedthateveniftheutilizationremainsalmostunchanged,stillasubstantialcostimprovementcanberealizedbytheharmonizationofthereviewperiods.Thiscane.g.beseenfrominstancenr.10wheretheutilizationdoesnotchangeforthegreedysearchalgorithm,butthecostsimproveby2.9%.Fromthisanalysis,weconcludethatthereisnodirectrelationbetweentheaveragereviewperiods,order-up-tolevels,machineutilizationandtherelativecostimprovementthatcanbeobtainedfromtheuseofsimulationbasedoptimizationtechniques.Thisconclusionleadstotheobviousquestionhowtherelativecostimprovementisthenrealizedbythesimulationbasedoptimizationtechniques.WebelievethattheanswerliesinthemechanismsthatweredescribedinSection4.2.Similarlytotheapproximateanalyticalmodel,thesimulationbasedoptimizationtechniques‘harmonize’thereviewperiodsofthedifferentproductssoastoobtainthebestbalancebetweenutilizationandvariability.Bothsimulationbasedopti-mizationtechniquesseekthebestpossibletrade-offbetweentheutilizationofthemachines,variationofthearrivals,averageprocessingtimesandvariationoftheprocessingtimes.Inthisway,itispossiblethatrelativelysmalldifferencesintheaveragereviewperiodscanleadtorelativelylargecostsavings.5DiscussionofthesimulationresultsInthissectionweanalyseandinterprettheresultsoftheheuristicforthe240instancesinthesimulationstudy.Thisallowsustoverifythesoundnessofthesolutionsproposedbytheheuristic.5.1MaineffectsoffactorsFigures4–6showtheimpactofthefourfactorsontheaveragecosts,reviewperiodsandorder-up-tolevels.Theimpactofeveryfactorisdiscussedsuccessively.Theimpactofincreasesinthenetutilizationonthetotalcostsisabout17%and19%forchangesfrom0.65to0.75andfrom0.75to0.85respectively.Theincreaseofthenetutilizationfrom0.65to0.75causesthereviewperiodstodecrease,while Controlofmulti-productmulti-machineproduction-inventorysystems279abcdFig.4a–d.Impactoffactoronaveragecosts:anetutilization–bfillrate–cavg.set-uptimes–davg.set-upcostsabcdFig.5a–d.Impactoffactoronaveragereviewperiods:anetutilization–bfillrate–cavg.set-uptimes–davg.set-upcoststheyincreasewhenthenetutilizationisincreasedfrom0.75to0.85.Thereasonforthispatternistobefoundinthemechanismsembeddedintheheuristic,asdescribedinSection4.2.Whenthenetutilizationgoesfrom0.65to0.75,thereviewperiodsaredecreasedinordertoreducethroughputtimes,work-in-processcostsandfinalinventoryholdingcosts.However,afurtherincreaseinthenetutilizationrequiresthatthereviewperiodsbeslightlyincreasedinordertoreducetheimpactofthe 280P.L.M.VanNyenetal.abcdFig.6a–d.Impactoffactoronaverageorder-up-tolevels:anetutilization–bfillrate–cavg.set-uptimes–davg.set-upcostsset-uptimesonthecapacityutilization.Theorder-up-tolevelsonthecontrary,increasesteadilywhentheutilizationisincreased.Thiscanbeexplainedbytwoeffects.First,intheexperiments,theriseinthenetutilizationiscausedbyincreasesinthedemandratefortheproducts.Theincreaseddemandrateresultsinhigherdemandduringareviewperiodandincreasedcyclestock.Secondly,theriseinthenetutilizationincreasesthecongestioninthesystem,leadingtolongerorderthroughputtimesandhighersafetystocks.Whenthefillratesincreasefrom90%to98%,theaveragetotalcostsincreasewith11%.Inordertoaccountfortheincreaseinthefillrate,theorder-up-tolevelsareincreased.However,theincreaseintheorder-up-tolevelsisfairlysmall.Thisisduetothefactthattheheuristicproposessmallerreviewperiods,whichleadtolowercyclestock.Moreover,theorderthroughputtimesarereducedsothatlesssafetystockisrequired.Theincreaseoftheaverageset-uptimesfrom45to135leadstoanincreaseinthetotalcostsof12%.Thereviewperiodsareraisedtolimittheimpactonthecapacityutilization.Theincreasedreviewperiodsleadtolowerset-upcosts,butalsotolongerthroughputtimesandhigherwork-in-processcosts.Becauseoftheincreasesinthereviewperiodsandthethroughputtimes,theorder-up-tolevelsandthefinalinventorycostsincrease.Theset-upcostsappeartobethedominantfactorinourexperimentaldesign:whentheaverageset-upcostsincreasefrom0to10,theaveragetotalcostriseswith156%.Furtherincreasesintheaverageset-upcostsresultincostincreasesof64%and70%.Thereviewperiodsandtheorder-up-tolevelsincreaseinasimilarfashion.Fromtheseresults,itappearsthateffortstocutset-upcosts(asadvocatedbye.g.theJust-In-Timephilosophy)mayeffectivelyresultinlargesavingsintheoverallcosts.Figure4-dpresentsthedivisionofthetotalcosts(TC)overthedifferent Controlofmulti-productmulti-machineproduction-inventorysystems281Fig.7.Frequencydiagramofallocationoffreecapacityforset-ups(instanceswithset-upcosts=0)Fig.8.Frequencydiagramofallocationoffreecapacityforset-ups(instanceswithset-upcosts>0)costcomponents.Fortheinstanceswithset-upcosts=0,theheuristicproposessolutionsinwhichthefinalinventorycosts(FIC)andthework-in-processcosts(WIPC)arealmostbalanced,theformerbeingslightlydominant.Fortheinstanceswithset-upcosts>0,theheuristicseeksabalancebetweenthefixedset-upcosts(SC)andthefinalinventoryandwork-in-processholdingcosts.Remarkably,thisresultissimilartothewell-knownEconomicOrderQuantitymodelforwhichtheholdingcostsandfixedorderingcostsarethesameiftheeconomicorderquantityisordered.Similarlytotheinstanceswithset-upcost=0,thefinalinventorycostsdominatethework-in-processcosts.5.2Allocationofcapacityforset-upsNowwetakealookatthefractionofthefreecapacity,computedas1−ρnet,thatisallocatedforperformingset-ups.FromFigure7weseethatfortheinstanceswithset-upcosts=0,theallocatedfractionoffreecapacityliesintheinterval60–74%,theaveragebeing65%.Asaruleofthumb,itappearsthatinthecaseofset-upcosts=0about2/3ofthefreecapacityshouldbeallocatedforset-uptimes.However,fromtheanalysisofresultsofsimulationbasedoptimizationtechniquesinTable7itappearsthatitisnotonlyimportanttoselecttherightlevelofcapacityutilization. 282P.L.M.VanNyenetal.ThenumericalresultsinTable7indicatethatsubstantialsavingscanberealisedbyadjustingthereviewperiods,butkeepingthecapacityutilizationmoreorlessatthesamelevel.Indeed,theharmonizationofthereviewperiodsisofhighimportanceinthemulti-productsystemunderstudy.Figure8showsthecaseofnon-zeroset-upcostsforwhichawiderangeofallocationoffreecapacityisobserved(3–69%).Ingeneral,theallocatedfractionoffreecapacityforset-upsislowerthaninthecaseofzeroset-upcosts.Moreover,ourexperimentsindicatethatthefractionofallocatedcapacitydecreaseswhentheset-upcostsincrease.Thissoundbehaviorisduetoincreasesinthereviewperiodsbecauseofrisingset-upcosts.Obviously,theincreasesinthereviewperiodsleadtolowercapacityallocationforset-ups.5.3Behaviorofreviewperiodswhenset-uptimesarechangedNext,weturnourattentiontothebehavioroftheaveragereviewperiods,order-up-tolevelsandcapacityutilizationwhentheset-uptimesaretripledfrom[30,60]to[90,180].TheresultsaredisplayedinFigures9and10.Whenset-upcostsarezeroandtheset-uptimesaretripled,ourheuristicincreasesthereviewperiodsandorder-up-tolevelswithalmostthesamefactorastheset-uptimes([2.7,3.2]inoursetofexperiments).Inthisway,thefractionoffreecapacityallocatedforset-upsremainsalmostunchanged.Fortheinstanceswithset-upcosts>0,asubstantialchangeinset-uptimeshasvirtuallynoimpactontheaveragereviewperiodsandorder-up-tolevels.Inthemainpartoftheinstances,theproportionliesintheinterval[0.9,1.3].Therefore,thefractionoffreecapacityusedforset-uptimesincreaseswiththesamefactorastheincreaseinset-uptimes.Fromthisobservation,weconcludethatthecostaspectdominatesthecapacityaspectintheoptimizationprocesswhentheset-upcosts>0.Finally,weobservethattheseresultsfadewhenset-upcostsarerelativelysmall,i.e.Lij∈[6.67,13.33].Especiallywhenthenetutilizationishigh(0.85),anincreaseintheset-uptimesleadstorisesinthereviewperiodsandtheFig.9.Frequencydiagramofproportionofaveragereviewperiodsforset-uptimes[90,180]over[30,60](instanceswithset-upcosts=0) Controlofmulti-productmulti-machineproduction-inventorysystems283Fig.10.Frequencydiagramofproportionofreviewperiodsforset-uptimes[90,180]over[30,60](instanceswithset-upcosts>0)correspondingorder-up-tolevels.Again,thisindicatesthatouroptimizationtoolworkssoundly.Fromtheresultsinthissubsectionandtheprevioussubsection,itappearsthatbehaviouroftheoptimizedcontrolparametersisratherdifferentinthecasewhereset-upcostsareequaltozeroandthecasewheretheset-upcostsarelargerthanzero.Inthecasethatset-upcostsarezero,theoptimizationtoolfocusesmoreonthecapacityutilizationaspectwhereaswhentheset-upcostsarelargerthanzero,thetoolismainlyconcernedwiththecostaspect.Thelessonthatcanbelearnedfromtheseobservationsisthatitisveryimportanttotakeintoaccountbothcostandcapacityissueswhenmakingproductionandinventorycontroldecisions.Decisionsupportsystemsthatfocussolelyononeoftheseissuesarecursedtomakeerrorsthatcanresultinsubstantialcostincreases.Unfortunatelyitisthecasethatmostdecisionsupportsystemsdofocuseitheronthecapacityaspectoronthecostaspect.Finally,theseobservationsillustratetheimportanceofagoodknowledgeofthecoststructureofthePIsystem,sothattheapplicationofsoundmanagementaccountingtechniquesshouldbegivenhighpriority.6ConclusionsWeproposeathree-stepheuristictocoordinateproductionandinventorycontrolde-cisionsinanintegratedmulti-productmulti-machineproduction-inventorysystemcharacterizedbyjobshoproutingsandstochasticdemand,set-upandprocessingtimes.Ourheuristicminimizesthesumofset-upcosts,work-in-processholdingcostsandfinalinventoryholdingcostswhilestochasticcustomerdemandissatis-fiedwithatargetfillrate.Thefirststepusesanapproximateanalyticalmodelandagreedysearchalgorithmtofindnear-optimalcontrolparameters.Severalapproxi-mationsareusedinthisstep.Sincethismayresultincustomerservicelevelsthataretoolowortoohigh,theorder-up-tolevelsarefine-tunedinthesecondstep.Thisstepensuresthatallcustomerservicelevelrequirementsaresatisfied.Inthethirdstep,theperformancecharacteristicsofthesystemareaccuratelyestimatedusingsimulation. 284P.L.M.VanNyenetal.Wetestedourheuristicinanextensivesimulationstudy,consistingof240instances.Weselectedasubsetof15worst-caseinstancesthatwereoptimizedusingourheuristicandtwosimulationbasedoptimizationtechniques.Thecomparisonoftheperformanceofourheuristictothesimulationbasedoptimizationtechniquesallowedustoconcludethatourheuristicperformssatisfactory,bothintermsofoptimizationqualityandrequiredcomputationtime.Thedetailedanalysisofoneprobleminstancehelpedtogainunderstandingofthemechanismsthatareembeddedintheheuristic.Itappearedthatouroptimizationtoolharmonizesthereviewperiodsofthedifferentproductssothatthevariabilityinthearrivalandproductionprocesses,theaverageprocessingtimesandtheutilizationoftheworkcentersarebalanced.Basedontheresultsofoursimulationstudy,weconcludethattheset-upcostisthedominantfactorinthestudy.Theimpactoftheset-upcostsonthetotalcostsismanytimeshigherthanthatoftheotherfactorsinthestudy:utilization,fillrateandset-uptimes.Theresultssupporttheinsightthatsubstantialcostsavingscanberealizedbyset-upcostreductionprograms,asadvocatedbytheJust-In-Timephilosophy.Whencomparedtotheotherrelevantcostscomponents,i.e.work-inprocessandfinalinventoryholdingcosts,theset-upcostsaredominant.Fortheinstanceswithset-upcosts>0,about50%ofthetotalcostsareduetothefixedset-upcosts.Thework-in-processcostsareslightlydominatedbythefinalinventoryholdingcosts,bothintheinstanceswithandwithoutset-upcosts.Whenset-upcostsareabsent,reviewperiodsarechosensothatabout2/3ofavailablecapacityisallocatedtoset-ups.Whenset-upcostsareconsiderablyhigh,itseemsthatthereviewperiodsarechosenbasedoncostconsiderationsonly.Instancesthatareinbetweenthebothextremesshowatrade-offbetweencostandcapacityconsiderations.Theseresultsindicatethatagoodknowledgeofthecoststructureoftheproduction-inventorysystemisofthehighestimportanceinordertomakecontroldecisionsthatminimizethetotalrelevantcosts.Unlikeotherapproaches,ourheuristicintegratesbothcapacityandcostaspects.Therefore,ourheuristicisabletomakerobustdecisionsforeveryinstance,regardlessofthevaluesofthedifferentcostparameters.Moreover,theheuristiccapturestheinteractionbetweendifferentproductsanddifferentworkcentersandtheirimpactonthecongestionphenomenaintheproductionsystem.Finally,ourheuristiccombinesthespeedofaqueueingnetworkmodelwiththeaccuracyofasimulationexperiment.Therefore,itworksfastandaccurately.Someinterestingdirectionsforfurtherresearchmaybetoimprovetheapprox-imationtechniquesforopenqueueingnetworks,toextendtheheuristictootherinventorypoliciesandtotesttheheuristicinreal-lifecasestudies.Furthermore,itwouldbeinterestingtocomparetheperformanceofthereorderpointpolicybasedcontrolmethodpresentedinthispapertoothercontrolmethods.Othercon-trolmethodscane.g.bebasedoncyclicalproductionplansorpull-typecontrol.Finally,itcanbeworthwhiletoinvestigatetheinfluenceofdifferentpriorityrulesontheperformanceoftheproduction-inventorysystem. Controlofmulti-productmulti-machineproduction-inventorysystems285Appendix1:generationofprobleminstancesWeusethefollowingproceduretogeneratetheprobleminstances.1.Randomlygenerateasetofroutings.Theroutingstructuresarechosensothattheaveragenumberofoperationsperproductequals3andthenumberofoperationsperproductliesintheinterval[2,4].Furthermore,thenumberofproductsperworkcenterliesintheinterval[4,8];2.Allocatetoeveryproductiarelativeshareofcapacityutilizationofworkcenterj,denotedasrscuij.WeusethecapacityutilizationprofilesthatarepresentedinTableA.1.Theseprofilesdependonthenumberofproductsthatareproducedataworkcenter,denotedasNj;3.Randomlydrawthedemandforproductiλ=E−1ADfromtheinter-iivalλLB,λUB.Thisintervalischosensothattheexpecteditemproduc-tiontimeE[P]variesbetweenPLB=1min.andPUB=5min.ThenijnetλLBandλUBaregivenby:λLB=ρmaxi,j(rscuij)=0.06ρnetandPUBnetλUB=ρmini,j(rscuij)=0.1ρnet.Ifρnet=0.85,thentheyearlydemandPLBliesintheinterval[26805;44676].4.Calculatetheaverageitemprocessingtimeforeveryproductiandeverywork-netrscuijρcenterj:E[Pij]=λi;5.Generaterandomly:r∈[0.15,0.25]
∗/(
year);6.Generaterandomlyforeveryi:–ci∈
[0,0]or
[6.67,13.3]or
[20,40]or
[60,120];–vi∈
[10,15];–Costofrawmaterialasafraction[0.35,0.50]ofvi;–Thedifferencebetweenviandthecostofrawmaterialistheaddedvalue.Tofindtheechelonvaluevijofaproduct,wedistributetheaddedvalueequallyoverthedifferentproductionsteps.7.Generaterandomlyforeveryiandeveryj:–Lij∈[30,60]min.or[90,180]min.8.Thelengthofasimulationsubrunischosenas100,000∗maxEAD.iiAppendix2:lowerboundontotalcostsInthisappendixweproposealowerboundforthetotalcostsinaproduction-inventorysystemwithjobshoproutingsandstochasticarrivalandprocessingtimes.Thelowerboundneglectstheimpactofthevariabilityinthesystemaswellastheinteractionbetweendifferentproducts.Thelowerboundconsistsofthreeterms:1.finalinventorycosts.Thelowerboundisbasedonaninventorymodelcharac-terizedbydeterministicdemandandzeroreplenishmentleadtimes;2.work-in-processcostsduetoprocessingtimesandset-uptimes(waitingtimesareexcluded);3.fixedset-upcosts(whichareknownexactly). 286P.L.M.VanNyenetal.TableA1.Capacityutilizationprofiles↓profilenr.Nj→4567810.300.300.250.200.1720.280.250.200.170.1530.250.200.170.160.1540.170.150.150.140.1250.100.130.120.1160.100.110.1070.100.1080.10Total11111WeformulatealowerboundforthetotalcostsforproductiforagivenreviewperiodRi:M−1vijrRiE[Pij]LBTCi(Ri)=ciRi+DD+E[Lij](9)j=1EAiEAi(α∗)2Rvriii+2EADiBasedontheformulaforLBTC(R),thereviewperiodR∗thatminimizesiiiLBTCiequals:2332E2ADciR∗=3i(10)i3M42E[P]vr+(α∗)2EADvrijijiiij=1Thecomputationofthelowerboundbecomesinfeasibleifci=0.Furthermore,thelowerboundforthetotalcostsofthewholesystemisgivenbythesumofthelowerboundsofthedifferentproductssincetheinteractionbetweenthedifferentproductsisignored.References1.AdamsJ,BalasE,ZawackD(1988)Theshiftingbottleneckprocedureforjob-shopscheduling.ManagementScience34:391–4012.AltiokT,ShiueGA(2000)Pull-typemanufacturingsystemswithmultipleproducttypes.IIETransactions32:115–1243.AminM,AltiokT(1997)Controlpoliciesformulti-productmulti-stagemanufacturingsystems:anexperimentalapproach.InternationalJournalofProductionResearch35:201–2234.BenjaafarS,KimJS,VishwanadhamN(2004)Ontheeffectofproductvarietyinproduction-inventorysystems.AnnalsofOperationsResearch126:71–101 Controlofmulti-productmulti-machineproduction-inventorysystems2875.BenjaafarS,CooperWL,KimJS(2003)Onthebenefitsofpoolinginproduction-inventorysystems.Workingpaper,UniversityofMinnesota,Minneapolis.ManagementScience(toappear)6.BirtwistleGM,DahlOJ,NijgaardK(1984)Simulabegin.Studentenlitteratur,Lund7.BowmanRA,MuckstadtJA(1995)Productioncontrolofcyclicscheduleswithdemandandprocessvariability.ProductionandOperationsManagement4:145–1628.CoxDR(1962)Renewaltheory.Methuen,London9.EgleseRW(1990)Simulatedannealing:atoolforoperationalresearch.EuropeanJournalofOperationalResearch46:271–28110.GudumCK,deKokAG(2002)Asafetystockadjustmentproceduretoenabletar-getservicelevelsinsimulationofgenericinventorysystems.Workingpaper,BETAResearchSchool,TheNetherlands11.HillierFS,LiebermanGJ(2005)IntroductiontoOperationsResearch,8thedn.McGraw-Hill,Boston12.HoppWJ,SpearmanML(1996)Factoryphysics.Irwin,Chicago13.KarmarkarUS(1987)Lotsizes,leadtimesandin-processinventories.ManagementScience33:409–41914.KingmanJFC(1961)Thesingleserverqueueinheavytraffic.ProceedingsoftheCambridgePhilosophicalSociety57:902–90415.LambrechtMR,VandaeleNJ(1996)Ageneralapproximationforthesingleproductlotsizingmodelwithqueueingdelays.EuropeanJournalofOperationalResearch95:73–8816.LambrechtMR,IvensPL,VandaeleNJ(1998)ACLIPS:Acapacityandleadtimeintegratedprocedureforscheduling.ManagementScience44:1548–156117.LawAM,KeltonWD(2000)Simulationmodelingandanalysis,3rdedn.McGraw-Hill,Boston18.LiberopoulosG,DalleryY(2003)Comparativemodellingofmulti-stageproduction-inventorycontrolpolicieswithlotsizing.InternationalJournalofProductionResearch41:1273–129819.OuennicheJ,BoctorF(1998)Sequencing,lotsizingandschedulingofseveralproductsinjobshops:thecommoncycleapproach.InternationalJournalofProductionResearch36:1125–114020.OuennicheJ,BertrandJWM(2001)Thefinitehorizoneconomiclotsizingprobleminjobshops:themultiplecycleapproach.InternationalJournalofProductionEconomics74:49–6121.PinedoM,ChaoX(1999)Operationsschedulingwithapplicationsinmanufacturingandservices.McGraw-Hill,London22.RubioR,WeinLM(1996)Settingbasestocklevelsusingproduct-formqueueingnetworks.ManagementScience42:259–26823.ShantikumarJG,BuzacottJA(1981)Openqueueingnetworkmodelsofdynamicjobshops.InternationalJournalofProductionResearch19:255–26624.SilverEA,PykeDF,PetersonR(1998)Inventorymanagementandproductionplanningandscheduling.Wiley,NewYork25.SoxCR,JacksonPL,BowmanA,MuckstadtJA(1999)Areviewofthestochasticlotschedulingproblem.InternationalJournalofProductionEconomics62:181–20026.SuriR,SandersJL,KamathM(1993)Performanceevaluationofproductionnetworks.In:GravesSCetal(eds)HandbooksinOperationsResearchandManagementScience,vol4,pp199–286.Elsevier,Amsterdam27.VandaeleNJ(1996)Theimpactoflotsizingonqueueingdelays:multiproduct,multimachinemodels.Ph.D.thesis,DepartmentofAppliedEconomics,KatholiekeUniver-siteitLeuven,Belgium 288P.L.M.VanNyenetal.28.VandaeleNJ,LambrechtMR,DeSchuyterN,CremmeryR(2000)SpicerOff-HighwayProductsDivision-Bruggeimprovesitslead-timeandschedulingperformance.Inter-faces30:83–9529.VanNyenPLM,VanOoijenHPG,BertrandJWM(2004)SimulationresultsontheperformanceofAlbinandWhitt’sestimationmethodforwaitingtimesinintegratedproduction-inventorysystems.InternationalJournalofProductionEconomics90:237–24930.VanNyenPLM,BertrandJWM,VanOoijenHPG,VandaeleNJ(2004)Thecontrolofanintegratedmulti-productmulti-machineproduction-inventorysystem.Workingpaper,BETAResearchSchool,TheNetherlands31.VanNyenPLM(2005)Theintegratedcontrolofproductioninventorysystems.Ph.D.thesis,DepartmentofTechnologyManagement,TechnischeUniversiteitEindhoven,TheNetherlands(toappear)32.WhittW(1983)Thequeueingnetworkanalyzer.BellSystemTechnicalJournal62:2779–281533.WhittW(1994)Towardsbettermulti-classparametric-decompositionapproximationsforopenqueueingnetworks.AnnalsofOperationsResearch48:221–22434.ZipkinP(1986)Modelsfordesignandcontrolofstochasticmulti-itembatchproductionsystems.OperationsResearch34:91–104 PerformanceanalysisofparallelidenticalmachineswithageneralizedshortestqueuearrivalmechanismG.J.vanHoutum1,I.J.B.F.Adan2,J.Wessels2,andW.H.M.Zijm1,31FacultyofTechnologyManagement,EindhovenUniversityofTechnology,P.O.Box513,5600MBEindhoven,TheNetherlands(e-mail:G.J.v.Houtum@tm.tue.nl)2FacultyofMathematicsandComputingScience,EindhovenUniversityofTechnology,Eindhoven,TheNetherlands3FacultyofAppliedMathematics,UniversityofTwente,Twente,TheNetherlandsReceived:February8,2000/Accepted:November28,2000Abstract.Inthispaperwestudyaproductionsystemconsistingofagroupofparallelmachinesproducingmultiplejobtypes.Eachmachinehasitsownqueueanditcanprocessarestrictedsetofjobtypesonly.Onarrivalajobjoinstheshortestqueueamongallqueuescapableofservingthatjob.UndertheassumptionofPoissonarrivalsandidenticalexponentialprocessingtimeswederiveupperandlowerboundsforthemeanwaitingtime.Theseboundsareobtainedfromso-calledflexibleboundmodels,andtheyprovideapowerfultooltoefficientlydeterminethemeanwaitingtime.Theboundsareusedtostudyhowthemeanwaitingtimedependsontheamountofoverlap(i.e.commonjobtypes)betweenthemachines.Keywords:Queueingsystem–Shortestqueuerouting–Performanceanalysis–Flexibility–Truncationmodel–Bounds1IntroductionInthispaperweconsideraqueueingsystemconsistingofagroupofparallelidenticalserversservingmultiplejobtypes.Eachserverhasitsownqueueandiscapableofservingarestrictedsetofjobtypesonly.JobsarriveaccordingtoaPoissonprocessandonarrivaltheyjointheshortestfeasiblequeue.Theservicetimesareexponentiallydistributed.WewillrefertothisqueueingmodelastheGeneralizedShortestQueueSystem(GSQS).ThismodelismotivatedbyasituationencounteredintheassemblyofPrintedCircuitBoards(PCBs).Thisisexplainedinmoredetailbelow.Figure1showsatypicallayoutofanassemblysystemforPCBs.Itconsistsofthreeparallelinsertionmachines,eachwithitsownlocalbuffer.AninsertionCorrespondenceto:G.J.vanHoutum 290G.J.vanHoutumetal.Fig.1.Aflexibleassemblysystemconsistingofthreeparallelinsertionmachines,onwhichthreetypesofPCBsaremademachinemountsverticalcomponents,suchasresistorsandcapacitators,onaPCBbytheinsertionhead.Thecomponentsaremountedinacertainsequence,whichisprescribedbyaNumericalControlprogram.Theinsertionheadisfedbythesequencer,whichpickscomponentsfromtapesandtransportsthemintherightordertotheinsertionhead.Eachtapecontainsonlyonetypeofcomponents.Thetapesarestoredinthecomponentmagazine,whichcancontainatmost80tapes,say.EachPCBneedsonaverage60differenttypesofcomponents.ToassembleaPCBallrequiredcomponentshavetobeavailableinthecomponentmagazine.Hence,thesetofcomponentsavailableinthemagazinedeterminesthesetofPCBtypesthatcanbeprocessedonthatmachine.ThesysteminFigure1hastoassemblethreePCBtypes,labeledA,BandC.Themachinesarebasicallysimilar,butduetothefactthattheyareloadedwithdifferenttypesofcomponents,thesetsofPCBtypesthatcanbehandledbythemachinesaredifferent.MachineM1canhandletheAandBtypes,machineM2theAandC,andmachineM3theBandC.WhenthemountingtimesforallPCBtypesareapproximatelythesame,itisreasonabletosendarrivingPCBstotheshortestfeasiblequeue.SincetheassemblyofPCBsisoftencharacterizedbyrelativelyfewjobtypes,largeproductionbatchesandsmallmountingtimes(seeZijm[16]),theuseofaqueueingmodelseemsappropriatetopredictperformancecharacteristicssuchasthemeanwaitingtime.Animportantissueistheassignmentoftherequiredcomponentstothemachines.Ideally,eachmachineshouldgetallcomponentsneededtoprocessallPCBtypes.However,sincethecomponentmagazineshaveafinitecapacity,theycancontainthecomponentsneededfora(small)subsetofPCBtypesonly.Inthispaperwewillinvestigatehowmuchoverlap(i.e.commoncomponents)betweenthemachinesisrequiredsuchthatthesystemnearlyperformsasintheidealsituationwherethemachinesareequippedwithallcomponents.TheGSQSisalsorelevantformanyotherpracticalsituations;e.g.,forparallelmachinesloadedwithdifferentsetsoftools,computerdisksloadedwithdifferent Performanceanalysisofparallelidenticalmachines291informationfiles,oroperatorsinacallcenterhandlingrequestsfromdifferentcustomers.Nevertheless,theliteratureontheGSQSislimited.Schwartz[12](seealsoRoque[11])consideredasystemrelatedtotheGSQS,butwithaspecificserverhierarchy.Hederivedsomeexpressionsforthemeanwaitingtimes.Adan,WesselsandZijm[2]derivedroughapproximationsforthemeanwaitingtimesinaGSQS.Green[7]constructedatruncationmodelforarelatedsystemwithtwotypesofjobsandtwotypesofservers:serverswhichcanservebothjobtypesandserverswhichcanonlyservejobsofthesecondtype.Forthepresentmodelwithgeneral(i.e.nonexponential)arrivals,Sparaggis,CassandrasandTowsley[13]showedthatthegeneralizedshortestqueueroutingisoptimalwithrespecttotheoverallmeanwaitingtimeforsymmetriccases(seeTheorem3.1in[13];seealsoSubsection2.3).Formoregeneralsystems,FossandChernova[6]usedafluidapproximationapproachtoestablishergodicitycondi-tions(seealsotheremarksattheendofSection2.2).TheissueofergodicityhasalsobeenconsideredinarecentreportbyFoleyandMcDonald[5].Theirmaincontribution,however,consistsofresultsontheasymptoticbehaviorofaGSQSwithtwoexponentialserverswithdifferentservicerates.Finally,HassinandHa-viv[8]havestudiedasymmetricGSQSwithtwoserversandanadditionalpropertycalledthresholdjockeying.Theyfocusonthedifferenceinwaitingtimebetweenjobswhichcanchoosebetweenbothserversandjobswhichcannotchoose.TheGSQScanbedescribedbyacontinuous-timeMarkovprocesswithmulti-dimensionalstateswhereeachcomponentdenotesthequeuelengthatoneoftheservers.Onlyinveryspecialcasesexactanalyticalsolutionscanbefound(seee.g.[3]).Therefore,todeterminethemeanwaitingtimes,wewillconstructtruncationmodelswhich:(i)areflexible(i.e.thesizeoftheirstatespacecanbecontrolledbyoneormoretruncationparameters);(ii)canbesolvedefficiently;(iii)provideupperandlowerboundsforthemeanwaitingtimes.Suchmodelsarecalledsolvableflexibleboundmodels.Theyarederivedbyusingtheso-calledprecedencerelationmethod.Thisisasystematicapproachfortheconstructionofboundmodels,whichhasbeendevelopedin[14,15].Inthispaperwewillconstructalowerandupperboundmodelforthemeanwaitingtimes.Thesetwomodelsconstitutethecoreofapowerfulnumericalapproach:thetwoboundmodelsaresolvedforincreasingsizesofthetruncatedstatespaceuntilthemeanwaitingtimesaredeterminedwithinagiven,desiredaccuracy.Thispaperisorganizedasfollows.InSection2,wedescribetheGSQSandwediscussconditionsunderwhichtheGSQSisergodicandbalanced.Next,inSection3,weconstructtheflexibleboundmodelsandweformulateanumericalapproachtodeterminethemeanwaitingtimes.Finally,inSection4,weinvestigatehowthemeanwaitingtimesfortheGSQSdependontheamountofoverlap(i.e.commonjobtypes)betweentheservers.Thisisdonebynumericallyevaluatingseveralscenarios.2ModelThissectionconsistsofthreesubsections.Inthefirstsubsection,wedescribetheGSQS.InSubsection2.2wepresentasimpleconditionthatisnecessaryandsuf- 292G.J.vanHoutumetal.Fig.2.AGSQSwithc=2serversandthreejobtypesficientforergodicity.Inthelastsubsection,wepresentarelatedconditionunderwhichtheGSQSissaidtobebalancedandwebrieflydiscusssymmetricsystems.2.1ModeldescriptionTheGSQSconsistsofc≥2parallelserversservingmultiplejobtypes.Eachserverhasitsownqueueandiscapableofservingarestrictedsetofjobtypesonly.Allservicetimesareexponentiallydistributedwiththesameparameterµ>0.ThearrivalstreamofeachjobtypeisPoissonandanarrivingjobjoinstheshortestqueueamongallqueuescapableofservingthatjob(tiesarebrokenwithequalprobabilities).Figure2showsaGSQSwithc=2serversandthreejobtypes:typeA,BandCjobsarrivewithintensityλA,λBandλC,respectively.TheAjobscanbeservedbybothservers,theBjobscanonlybeservedbyserver1,andtheCjobsmustbeservedbyserver2.Weintroducethefollowingnotations.Theserversarenumberedfrom1,...,candthesetIisdefinedbyI={1,...,c}.ThesetofalljobtypesisdenotedbyJ.Thearrivalintensityoftypej∈Jjobsisgivenbyλj≥0,andλ=j∈Jλjisthetotalarrivalintensity.Foreachj∈J,I(j)denotesthesetofserversthatcanservethejobsoftypej.Weassumethateachjobtypecanbeservedbyatleastoneserverandeachservercanhandleatleastonejobtype;so,I(j)=∅forallj∈J,and∪j∈JI(j)=I.Withoutlossofgenerality,wesetµ=1.Thentheaverageworkloadperserverisgivenbyρ=λ/c.Obviously,therequirementρ<1isnecessaryforergodicity.ThebehavioroftheGSQSisdescribedbyacontinuous-timeMarkovprocesswithstates(m1,...,mc),wheremidenotesthelengthofthequeueatserveri,i∈I(jobsinserviceareincluded).So,thestatespaceisequaltoM={m|m=(m1,...,mc)withmi∈IN0foralli∈I}.(1)Weassumethatj∈Jλj1{i∈I(j)}>0forallserversi∈I(here,1{G}istheindicatorfunction,whichis1ifGistrueand0otherwise),i.e.,thatallservershaveapositivepotentialarrivalrate.ThisguaranteesthattheMarkovprocessis Performanceanalysisofparallelidenticalmachines293Fig.3.ThetransitionratediagramfortheGSQSinFigure2irreducible.Thetransitionratesaredenotedbyqm,n.Figure3showsthetransitionratesfortheGSQSinFigure2.TherelevantperformancemeasuresarethemeanwaitingtimesW(j)foreachofjobtypej∈J,andtheoverallmeanwaitingtimeW,whichisequaltoλW=jW(j).(2)λj∈JItisobviousthatforanergodicsystem,W(j)=minmπ(m,...,m),j∈J,(3)i1ci∈I(j)(m1,...,mc)∈Mwhereπ(m1,...,mc)denotesthesteady-stateprobabilityforstate(m1,...,mc).2.2ErgodicityBystudyingthejobrouting,weobtainasimple,necessaryconditionfortheergod-icityoftheGSQS.ForeachsubsetJ
⊂J,J
=∅,jobsoftypej∈J
arrivewithanintensityequaltoj∈J
λjandtheymustbeservedbytheservers∪j∈J
I(j).Thisimmediatelyleadstothefollowinglemma.Lemma1TheGSQScanonlybeergodicif

λj<|∪j∈J
I(j)|forallJ⊂J,J=∅.(4)j∈J
294G.J.vanHoutumetal.NotethatforJ
=J,thisinequalityisequivalenttoρ<1.FortheGSQSinFigure2,condition(4)statesthatforergodicityitisnecessarythattheinequalitiesλB<1,λC<1andλ<2(or,equivalently,ρ<1)aresatisfied.Itappearsthatcondition(4)isalsosufficientforergodicity.Toshowthis,weconsiderso-calledcorrespondingstaticsystems.AcorrespondingstaticsystemisasystemthatisidenticaltotheGSQS,butwithstatic(random)routinginsteadofdynamicshortestqueuerouting.Thestatic(j)routingisdescribedbydiscretedistributions{xi}i∈I(j),j∈J,whereforeach(j)j∈Jandi∈I(j),thevariablexidenotestheprobabilitythatanarrivingjoboftypejissenttoserveri.Understaticrouting,itholdsforeachj∈JthatthePoissonstreamofarrivingtypejjobsissplitupintoPoissonstreamswithintensities(j)xj,i=λjxi,i∈I(j),fortypejarrivalsjoiningserveri.Hencethequeuesi∈IconstituteindependentM/M/1queueswithidenticalmeanservicetimesequaltoµ=1andarrivalintensitiesj∈J(i)xj,i,whereJ(i)={j∈J|i∈I(j)}.Asaresult,weobtainasimplenecessaryandsufficientconditionfortheergodicityofacorrespondingstaticsystem,viz.xj,i<1foralli∈I.j∈J(i)Lemma2ForaGSQS,thereexistsacorrespondingstaticsystemthatisergodic,ifandonlyifcondition(4)issatisfied.Proof.Thereexistsacorrespondingstaticsystemthatisergodicifandonlyifthereexistsanonnegativesolution{xj,i}(j,i)∈A,withA={(j,i)|j∈J,i∈Iandi∈I(j)},ofthefollowingequationsandinequalities:xj,i=λjforallj∈J,xj,i<1foralli∈I;(5)i∈I(j)j∈J(i)theequalitiesin(5)guaranteethatthesolution{xj,i}(j,i)∈Acorrespondstodiscrete(j)distributions{xi}i∈I(j)whichdescribeastaticrouting,andtheinequalitiesin(5)mustbesatisfiedforergodicity.Itiseasilyseenthat(5)hasnosolutionifcondition(4)isnotsatisfied.Now,assumethatcondition(4)issatisfied.Toprovethatthereexistsanonnega-tivesolution{xj,i}(j,i)∈Aof(5),weconsideratransportationproblemwithsupplynodesVˆ1=J∪{0},demandnodesVˆ2=I,andarcsAˆ=A∪{(0,i)|i∈I}(supplynode0denotesanextratypeofjobs,whichcanbeservedbyallservers).Definethesuppliesaˆjbyaˆj=λjforallj∈Vˆ1{0}andaˆ0=c−λ−c,where|∪j∈J
I(j)|−j∈J
λj:=minJ
⊂J|∪j∈J
I(j)|J
=∅(from(4),itfollowsthat>0,andaˆ≥0sincebytakingJ
=Jweobtain0theinequality≤(c−λ)/c).Further,wedefinethedemandsˆbibyˆbi=1−foralli∈Vˆ2;notethatj∈Vˆ1aˆj=i∈Vˆ2ˆbi.Itmaybeverifiedthatthistrans-portationproblemsatisfiesanecessaryandsufficientconditionfortheexistenceofafeasibleflow;seeLemma5.4of[14]anditsproofisbasedonatransformation Performanceanalysisofparallelidenticalmachines295toamaximum-flowproblemfollowedbytheapplicationofthemax-flowmin-cuttheorem(seee.g.[4]).So,thereexistsafeasibleflowforthetransportationproblem,i.e.,thereexistsanonnegativesolution{xˆj,i}(j,i)∈Aˆoftheequationsxˆj,i=ˆajforallj∈Vˆ1,xˆj,i=ˆbiforalli∈Vˆ2.i∈Vˆ2j∈Vˆ1(j,i)∈Aˆ(j,i)∈AˆItiseasilyseenthatthenthesolution{xj,i}(j,i)∈Adefinedbyxj,i=ˆxj,iforall(j,i)∈A,isanonnegativesolutionof(5),whichcompletestheproof.Insituationswithmanyjobtypesshortestqueueroutingwillbalancethequeuelengthsmorethananystaticrouting.Soifthereisacorrespondingstaticsystemthatisergodic,thentheGSQSwillalsobeergodic.TogetherwithLemma2,thisinformallyshowsthatthefollowingtheoremholds.Theorem1TheGSQSisergodicifandonlyifcondition(4)issatisfied.Foraformalproofofthistheorem,thereaderisreferredtoFossandChernova[6]orFoleyandMcDonald[5].Inthelatterpaper,ageneralizationofcondition(4)isprovedtobenecessaryandsufficientforthe(moregeneral)modelwithdiffer-entservicerates.Theirproofalsoexploitstheconnectionwithacorrespondingstaticsystem.FossandChernova[6]useafluidapproximationapproachtoderivenecessaryconditionsforamodelwithgeneralarrivalsandgeneralservicetimes.2.3BalancedandsymmetricsystemsItisdesirablethattheshortestqueuerouting,asreflectedbythesetsI(j),balancestheworkloadamongtheservers.Formally,wesaythataGSQSisbalancedifthereexistsacorrespondingstaticsystemforwhichallqueueshavethesameworkload.(j)Thismeansthattheremustexistdiscretedistributions{xi}i∈I(j)suchthatforeachserveri∈I,thearrivalintensityj∈J,(j,i)∈Axj,iisequaltoλ/c=ρ,wherethexj,iandthesetAaredefinedasbefore.Suchdiscretedistributionsexistifandonlyifthereexistsanonnegativesolution{xj,i}(j,i)∈Aoftheequationsλxj,i=λjforallj∈J,xj,i=foralli∈I.(6)ci∈Ij∈J(j,i)∈A(j,i)∈ATheseequationsarepreciselytheequationswhichmustbesatisfiedbyafeasibleflowforthetransportationproblemwithsupplynodesV1=J,demandnodesV2=I,arcsA,suppliesaj=λjforallj∈V1anddemandsbi=λ/cforalli∈V2.Applyingthenecessaryandsufficientconditionfortheexistenceofsuchafeasibleflow(see[14])leadstothefollowinglemma.Lemma3AGSQSisbalancedifandonlyifλ
λj≤|∪j∈J
I(j)|forallJ⊂J.(7)cj∈J
296G.J.vanHoutumetal.NotethatforJ
=∅andJ
=J,condition(7)holdsbydefinition.Further,itfollowsthatabalancedGSQSsatisfiescondition(4)ifandonlyifρ<1.So,forabalancedGSQS,thesimpleconditionρ<1isnecessaryandsufficientforergodicity.ForabalancedGSQStheworkloadsundertheshortestqueueroutingarenotnecessarilybalanced.ThiscanbeseenbyconsideringtheGSQSinFigure2.Ac-cordingtocondition(7),thisGSQSisbalancedifandonlyifλB≤λ/2andλC≤λ/2,i.e.ifandonlyifλB≤λA+λCandλC≤λA+λB.ThisconditionisobviouslysatisfiedifwetakeλC=λA+λB.Inthiscase,equalworkloadsforbothserverscanonlybeobtainedifalljobsoftypeAaresenttoserver1.But,undertheshortestqueuerouting,itwillstilloccurthatjobsoftypeAaresenttoserver2,andthereforeserver2willhaveahigherworkloadthanserver1.Nevertheless,onemayexpectthatforabalancedGSQS,theshortestqueueroutingatleastensuresthattheworkloadswillnotdiffertoomuch.Asubclassofbalancedsystemsarethesymmetricsystems.AGSQSissaidtobesymmetric,ifλ(I1)=λ(I2)forallI1,I2⊂Iwith|I1|=|I2|,(8)whereλ(I
):=λ,I
⊂I.jj∈JI(j)=I
So,aGSQSissymmetric,ifforallsubsetsI
⊂Iwiththesamenumberofservers|I
|,thearrivalintensityλ(I
)forthejobswhichcanbeservedbypreciselytheserversofI
,isthesame.TheGSQSinFigure2issymmetricifλ=λ.BCForasymmetricGSQS,allqueuelengthshavethesamedistribution,whichimpliesthatallservershaveequalworkloads.Forsuchasystem,itfollowsfromSparaggisetal.[13],thattheshortestqueueroutingminimizesthetotalnumberofjobsinthesystemandhencetheoverallmeanwaitingtimeW.Inparticular,thisimpliesthattheoverallmeanwaitingtimeinasymmetricGSQSislessthaninthecorrespondingsystemconsistingofNindependentM/M/1queueswithworkloadρ.3FlexibleboundmodelsInthissectionweconstructtwotruncationmodelswhicharemucheasiertosolvethantheoriginalmodel.Onetruncationmodelproduceslowerboundsforthemeanwaitingtimes,andtheotheroneupperbounds.Attheendofthissectionwedescribeanumericalmethodforthecomputationofthemeanwaitingtimeswithinagiven,desiredaccuracy.Thetruncationmodelsexploitthepropertythattheshortestqueueroutingcausesadrifttowardsstateswithequalqueuelengths.ThestatespaceM
ofthetwomodelsisobtainedbytruncatingtheoriginalstatespaceMaroundthediagonal,i.e.,M
={m∈M|m=(m,...,m)andm≤min(m)+Tforalli∈I},(9)1cii Performanceanalysisofparallelidenticalmachines297wheremin(m):=mini∈ImiandT1,...,Tc∈INareso-calledthresholdparam-eters;thecorrespondingvectorTˆ:=(T1,...,Tc)iscalledthethresholdvector.Sostatem∈MalsoliesinM
ifandonlyifforeachi∈IthelengthofqueueiisatmostTigreaterthanthelengthofanyotherqueue.LateroninthissectionwediscusshowappropriatevaluesforTˆcanbeselected.TherearetwotypesoftransitionspointingfromstatesinsideM
tostatesoutsideM
:(i)instatem=(m,...,m)∈M
withmin(m)>0andI
={i∈I|m=1cimin(m)+Ti}=∅,ataserverk∈Iwithmk=min(m)aservicecompletionoccurswithrate1andleadstoatransitionfrommtostaten=m−e∈M
;k(ii)instatem=(m,...,m)∈M
withI
={i∈I|m=min(m)+1ciT}=∅,ataserveri∈I
anarrivalofanewjobleadstoatransi-itionfrommtothestaten=m+e∈M
;thistransitionoccurswithirate|I(j;m)|−1λ1,wherethesetI(j;m)isdefinedbyj∈Jj{i∈I(j;m)}I(j;m)={i∈I(j)|mi=mink∈I(j)mk}(notethatthisratemaybeequalto0).Inthelower(upper)boundmodel,thetransitionstostatesnoutsideM
areredi-rectedtostatesn
withless(more)jobsinsideM
.Inthelowerboundmodel,thetransitionin(i)isredirectedton
=m−e−k
.Thismeansthatthedepartureofajobatanon-emptyshortesti∈I
ei∈Mqueueisaccompaniedbykillingonejobateachofthequeuesi∈I
,whicharealreadyTigreaterthantheshortestqueue.Thetransitionin(ii)isredirectedtomitself,i.e.,anewjobarrivingatoneoftheserversi∈I
isrejected.ThelowerboundmodelisthereforecalledtheThresholdKillingandRejection(TKR)model.Intheupperboundmodel,thetransitionin(i)isredirectedtomitself.ThismeansthatifatleastonequeueisalreadyTigreaterthantheshortestqueue,thefinishedjobintheshortestqueueisnotallowedtodepart,butisservedoncemore;thisisequivalenttosayingthattheserversattheshortestqueuesareblocked.Transition(ii)isredirectedton
=m+e+e∈M
,withI={k∈ik∈IsqksqI|mk=min(m)}.ThismeansthatanarrivalofanewjobatoneofthequeueswhichisalreadyTigreaterthantheshortestqueue,isaccompaniedbytheadditionofoneextrajobateachoftheshortestqueues.TheupperboundmodelisthereforecalledtheThresholdBlockingandAddition(TBA)model.Notethatthismodelmaybenon-ergodicwhiletheoriginalmodelisergodic.However,thelargerthevaluesofthethresholdsTithemoreunlikelythissituation.InFigure4,weshowtheredirectedtransitionsinthelowerandupperboundmodelfortheGSQSofFigure3.ItisintuitivelyclearthatthequeuesintheTKRmodelarestochasticallysmallerthanthequeuesintheoriginalmodel.Hence,foreachj∈J,theTKRmodelyieldsalowerboundforthemeanlengthoftheshortestqueueamongthequeuesi∈I(j),andthusalsoforthemeanwaitingtimeoftypejjobs(cf.(3)).Denotethesteady-stateprobabilitiesintheTKRmodelbyπTKR(m1,...,mc)andlet(j)WTKR(Tˆ)=minmiπTKR(m1,...,mc),j∈J.i∈I(j)(m1,...,mc)∈M
298G.J.vanHoutumetal.Fig.4.TheredirectedtransitionsintheTKRandTBAmodelfortheGSQSdepictedinFigure2.Forbothmodels,Tˆ=(T1,T2)=(3,3)Thenwehaveforeachj∈JthatW(j)(Tˆ)≤W(j),andthus(cf.(2))TKRλj(j)WTKR(Tˆ)=WTKR(Tˆ)λj∈JyieldsalowerboundfortheoverallmeanwaitingtimeW.Thelowerbounds(j)WTKR(Tˆ)monotonicallyincreaseasthethresholdsT1,...,Tcincrease.Similarly(j)theTBAmodelproducesmonotonicallydecreasingupperboundsW(Tˆ),j∈TBAJ,andWTBA(Tˆ).Theboundsandthemonotonicitypropertiescanberigorouslyprovedbyusingtheprecedencerelationmethod,see[14].ThismethodisbasedonMarkovrewardtheoryandithasbeendevelopedin[14,15].Thetruncationmodelscanbesolvedefficientlybyusingthematrix-geometricapproachdescribedin[10].Sincethetruncationmodelsexploitthepropertythatshortestqueueroutingtriestobalancethequeues,onemayexpectthattheboundsaretightforalreadymoderatevaluesofthethresholdsT1,...,Tc.Wewillnowformulateanumericalmethodtodeterminethemeanwaitingtimeswithanabsoluteaccuracyabs.ThemethodrepeatedlysolvestheTKRandTBAmodelforincreasingthresholdvectorsTˆ=(T1,...,Tc).Foreachvec-torTˆweuse(W(j)(Tˆ)+W(j)(Tˆ))/2asanapproximationforW(j)andTKRTBA∆(j)(Tˆ)=(W(j)(Tˆ)−W(j)(Tˆ))/2asanupperboundfortheerror;wesim-TBATKRilarlyapproximateWby(WTKR(Tˆ)+WTBA(Tˆ))/2wheretheerrorisatmost∆(Tˆ)=(WTBA(Tˆ)−WTKR(Tˆ))/2.Theapproximationsanderrorboundsaresetequalto∞iftheTBAmodelisnotergodic(whichmaybethecaseforsmallthresholds).Thecomputationprocedurestopswhenallerrorboundsarelessthanorequaltoabs;otherwiseatleastoneofthethresholdsisincreasedby1andnewapproximationsarecomputed.ThedecisiontoincreaseathresholdTiisbasedontherateofredirectionsrrd(i).Thisisexplainedinthenextparagraph. Performanceanalysisofparallelidenticalmachines299Thevariablerrd(i),i∈I,denotestherateatwhichredirectionsoccurintheboundarystatesm=(m1,...,mc)withmi=min(m)+Tiofthetruncatedstatespace.IfforgivenTˆonlytheTKRmodelisergodic,thenrrd(i)denotestheratefortheTKRmodel,otherwiserrd(i)denotesthesumoftheratefortheTKRandTBAmodel.Theratesrrd(i)canbecomputeddirectlyfromthesteady-statedistributionsoftheboundmodels.Thehighertheraterrd(i),thehighertheexpectedimpactofincreasingTi.ThecomputationprocedureincreasesallthresholdsTiforwhichrrd(i)=maxk∈Irrd(k).Thenumericalmethodissummarizedbelow.Algorithm(todeterminethemeanwaitingtimesfortheGSQS)Input:ThedataofanergodicinstanceoftheGSQS,i.e.,c,J,I(j)forallj∈J,andλjforallj∈J;theabsoluteaccuracyabs;theinitialthresholdvectorTˆ=(T1,...,Tc).Step1.DetermineW(j)(Tˆ),W(j)(Tˆ)and∆(j)(Tˆ)forallj∈J,TKRTBAandWTKR(Tˆ),WTBA(Tˆ)and∆(Tˆ),andrrd(i)foralli∈I.Step2.If∆(j)(Tˆ)>forsomej∈Jor∆(Tˆ)>,absabsthenTi:=Ti+1foralli∈Iwithrrd(i)=maxk∈Irrd(k),andreturntoStep1.Step3.W(j)=(W(j)(Tˆ)+W(j)(Tˆ))/2forallj∈J,TKRTBAandW=(WTKR(Tˆ)+WTBA(Tˆ))/2.NotethatforasymmetricGSQSitisnaturaltostartwithathresholdvectorTˆwithequalcomponents.Thenineachiterationallratesrrd(i)willbeequal,andhenceeachTiwillbeincreasedby1.SothecomponentsofTˆwillremainequal.4NumericalstudyoftheGSQSInthissectionweconsiderthreescenarios.InSubsection4.1wedistinguishtwotypesofjobs:commonjobsandspecialistjobs.Thecommonjobscanbeservedbyallserversandtheotheronescanbeservedbyonlyonespecificserver.WefocusonthebehavioroftheoverallmeanwaitingtimeWasafunctionofthefractionofworkduetocommonjobs.Thehigherthisfraction,themorebalancedthequeuesandthebettertheperformance.SoWwillbedecreasingasthenumberofcommonjobsincreases.Inoneextremecase,viz.whenalljobsarespecialistjobs,theGSQSreducestoindependentM/M/1queues,andWismaximal.Intheotherextremecase,viz.whenalljobsarecommonjobs,theGSQSisidenticaltoapureSymmetricShortestQueueSystem(SSQS),andWisminimal.InSubsection4.1weinvestigatehowWbehavesinbetweenthesetwoextremes.InSubsection4.2weconsiderasymmetricGSQSwithc=3servers,and,besidescommonandspecialistjobs,wealsohavesemi-commonjobs.Thesejobscanbeservedbytwoservers.Wecomparetwosituations:(i)aGSQSwithagivenfractionofcommonjobs(andnosemi-commonjobs);(ii)aGSQSwithtwicethis 300G.J.vanHoutumetal.fractionofsemi-commonjobs(andnocommonjobs).Inbothcasestheaveragenumberofserverscapableofservinganarbitraryjobisthesame.InSubsection4.3weevaluateaseriesofbalanced,asymmetricsystems.Weinvestigatehowthemeanwaitingtimesdeteriorateduetotheasymmetry.Finally,inSubsection4.4,themainconclusionsaresummarized.4.1TheimpactofcommonjobsWedistinguishc+1jobtypes,numbered1,...,c,c+1.Typejjobsarespecialistjobs,whichcanonlybeservedbyserverj,j=1,...,c.Thetypec+1jobsarecommonjobs,whichcanbeservedbyallservers.Thetotalarrivalintensityisequaltoλ=cρ,withρ∈(0,1).Thecommonjobsconstituteafractionp,p∈[0,1],ofthetotalarrivalstream,whileeachofthestreamsofspecialistjobsconstitutesanequalpartoftheremainingstream.Soλc+1=pλandλj=(1−p)λ/cforj=1,...,c.Table1liststhemeanwaitingtimesforspecialistjobs(=W(1)=...=W(c)),commonjobs(=W(c+1)),andanarbitraryjob(=W)asafunctionofpforasystemwithc=2andc=3servers,respectively,andaworkloadp=0.9.Forp=0therearenocommonjobs;thenW(c+1)isdefinedasthelimitingvalueofthewaitingtimeofcommonjobsasp↓0.Forp=1asimilarremarkholdsforthemeanwaitingtimesW(1)=···=W(c).Table1alsoliststherealizedreductionrr(p).ThisisdefinedasWM/M/1−Wrr(p)=,(10)WM/M/1−WSSQSwhereWM/M/1andWSSQSdenotethemeanwaitingtimeinanM/M/1systemandSSQS,respectively,bothwiththesameworkloadρ=0.9andmeanservicetimeµ=1asfortheGSQS.ThemeanwaitingtimeWM/M/1isrealizedwhenp=0,andWSSQSisrealizedwhenp=1.Clearly,rr(0)=0andrr(1)=1bydefinition.ForallcasesinTable1,WM/M/1=9andWSSQS=4.475forc=2andWSSQS=2.982forc=3.ThemeanwaitingtimesintheSSQShavebeendeterminedwithanabsoluteaccuracyof0.0001byusingtheboundmodelsin[1].ThemeanwaitingtimesinTable1havebeendeterminedbyusingthealgorithmdescribedinSection3withanabsoluteaccuracyabs=0.005.InTable1weseethattheoverallmeanwaitingtimeW=pW(c+1)+(1−p)W(1)sharplydecreasesforsmallvaluesofp;seealsoFigure5.Already73%ofthemaximalreductionisrealizedwhen20%ofthejobsiscommonand91%ofthemaximalreductionisrealizedwhen50%ofthejobsiscommon.Asurprisingresultisthattherealizedreductionrr(p)isalmostthesameforc=2andc=3servers.FurthernotethatforlargepthemeanwaitingtimeW(1)forspecialistjobsisonlyalittlebitlargerthanthemeanwaitingtimeW(c+1)forcommonjobs.Thisisduetothebalancingeffectofthecommonjobs.ThebehavioroftheoverallmeanwaitingtimeWisfurtherinvestigatedinTable2fordifferentvaluesofp,ρandc.Themeanwaitingtimesareagaindeterminedwithanabsoluteaccuracyabs=0.005(and0.0001forWSSQS).Onlyforlowworkloads(i.e.,ρ≤0.4),themeanwaitingtimehasbeendeterminedevenmore Performanceanalysisofparallelidenticalmachines301Table1.Meanwaitingtimesasafunctionofpandcc=2c=3(1)(c+1)(1)(c+1)pWWWrr(p)WWWrr(p)0.09.004.269.000.0%9.002.699.000.0%0.16.804.366.5654.0%6.072.825.7554.1%0.26.044.405.7272.6%5.062.884.6372.7%0.35.664.435.2982.0%4.562.914.0682.0%0.45.434.445.0487.6%4.252.933.7287.7%0.55.284.454.8691.4%4.052.953.5091.4%0.65.174.464.7494.1%3.902.963.3494.1%0.75.094.464.6596.1%3.792.973.2196.1%0.85.024.474.5897.7%3.712.973.1297.7%0.94.974.474.5299.0%3.642.983.0499.0%1.04.934.484.48100.0%3.582.982.98100.0%Fig.5.GraphicalrepresentationofthemeanwaitingtimesWlistedinTable1accuratelyinordertoobtainsufficientlyaccurateestimatesforrr(p).TheresultsinTable2showthatforeachcombinationofpandc,thevaluesforWforvaryingworkloadsρarenotthatfarawayfromthevaluesforWSSQS;inparticular,theabsolutedifferencesaresmallforsmallworkloadsρandtherelativedifferencesaresmallforhighworkloadsρ.Theresultsalsosuggestthattherr(p)isinsensitivetothenumberofserversc.However,rr(p)stronglydependsonρ;itisrathersmallforlowworkloadsandlargeforhighworkloads(itseemsthatrr(p)↑1asρ↑1).4.2Commonversussemi-commonjobsInSubsection4.1wedistinguishedtwojobtypesonly,specialistandcommonjobs.ForGSQSswithmorethantwoservers,onemayalsohavejobsinbetween, 302G.J.vanHoutumetal.Table2.Meanwaitingtimesasafunctionofp,ρandcc=2c=3pρWM/M/1WWSSQSrr(p)WWSSQSrr(p)0.250.20.250.190.0732.1%0.180.0230.8%0.40.670.510.2639.6%0.460.1338.6%0.61.501.100.6849.2%0.970.4248.9%0.84.002.671.9664.8%2.241.2964.8%0.99.005.474.4777.9%4.312.9878.0%0.9519.0010.699.4987.3%7.946.3387.4%0.9849.0025.8624.4994.4%18.1716.3594.4%0.500.20.250.140.0758.7%0.120.0257.2%0.40.670.400.2666.4%0.320.1365.5%0.61.500.890.6874.5%0.700.4274.3%0.84.002.271.9684.7%1.701.2984.8%0.99.004.864.4791.4%3.502.9891.4%0.9519.009.939.4995.4%6.926.3395.4%0.9849.0024.9724.4998.1%16.9816.3598.1%i.e.,jobsthatcanbeservedbytwoormore,butnotallservers.InthissubsectionweinvestigatewhichjobtypesleadtothelargestreductionofW:commonorsemi-commonjobs?WeconsideraGSQSwithc=3serversandatotalarrivalrateλ=3ρwithρ∈(0,1).Thefollowingtwocasesaredistinguishedforthedetailedarrivalstreams.ForcaseI,wecopythesituationinSubsection4.1.Inthiscasethereare4jobtypes.Thetype4jobsarecommonjobs;theyarrivewithintensityλ4=pλwithp∈[0,0.5](thereasonwhypmaynotexceed0.5followsbelow).Typejjobs,j=1,2,3arespecialistjobswhichonlycanbeservedbyserverj;theyarrivewithintensityλj=(1−p)λ/3.Sothemeannumberofserverscapableofservinganarbitraryjobisequalto1+2p.IncaseIIwehave6jobtypes.Thetypejjobs,j=1,2,3,areagainspecialistjobswhichcanonlybeservedbyserverj.Thetype4,5and6jobsaresemi-commonjobs;thetype4jobscanbeservedbytheservers1and2,thetype5jobsby1and3,andthetype6jobsby2and3.Toguaranteethatthemeannumberofserverscapableofservinganarbitraryjobremainsthesame(i.e.,equalto1+2p),thearrivalintensityλjissetequaltoλj=2pλ/3forj=4,5,6andλj=(1−2p)λ/3forj=1,2,3(toavoidnegativeintensities,pmustbelessthanorequalto0.5).Table3liststheoverallmeanwaitingtimeWfordifferentvaluesofpandρ.TheresultsforcaseIarecopiedfromTable2.WecanconcludethattheabsolutedifferencebetweenthemeanwaitingtimeWincaseIandIIisrathersmallineachsituation.ThissuggeststhatWismainlydeterminedbythemeannumberofserverscapableofservinganarbitraryjob;itdoesnotmatterwhetherthismeannumberisrealizedbycommonorby(twiceasmany)semi-commonjobs.Nevertheless,theresultsinTable3alsoshowthatineachsituationcaseIIyieldsasmallerWthan Performanceanalysisofparallelidenticalmachines303Table3.MeanwaitingtimesasafunctionofpandρWDiff.(I−II)pρCaseICaseIIAbs.Rel.0.250.20.180.140.0423.4%0.40.460.380.0818.1%0.60.970.830.1514.9%0.82.241.970.2711.9%0.94.313.920.388.9%0.957.947.460.486.0%0.9818.1717.620.553.0%0.500.20.120.050.0755.1%0.40.320.220.1032.5%0.60.700.560.1420.1%0.81.701.510.1911.2%0.93.503.270.226.4%0.956.926.670.253.6%0.9816.9816.720.261.5%caseI.Thismaybeexplainedasfollows.Letusconsiderthesituationwithp=0.5.IncaseI,λ1=λ2=λ3=λ/6andλ4=λ/2.Hence,foreachgroupof6arrivingjobs,onaverage4jobsjointheshortestqueue,1jobjoinstheshortestbutonequeue,and1jobjoinsthelongestqueue.IncaseII,however,λ1=λ2=λ3=0andλ4=λ5=λ6=λ/3.Thusforeachgroupof6arrivingjobs,onaverage4jobsjointheshortestqueueand2jobsjoinstheshortestbutonequeue.SoincaseIIthebalancingofqueueswillbeslightlystronger,andthusWwillbeslightlysmaller.4.3BalancedasymmetricsystemsInthissubsectionwestudytheGSQSwithc=2serversandthreejobtypesasdepictedinFigure2.Theparametersarechosenasfollows:ρ=0.9,λ=2ρ=1.8,λA=λ/2=0.9,λB=ˆpλ/2=0.9ˆp,λC=(1−pˆ)λ/2=0.9(1−pˆ)wherepˆ∈[0,0.5].Soonehalfofthejobsarecommon(typeA)jobsandtheotherhalfarespecialist(typeBandC)jobs.Butthespecialistjobsarenotequallydividedovertheservers.Thefractionpˆofspecialistjobswhichmustbeservedbyserver1(i.e.,thetypeBjobs)islessthanorequaltothefraction1−pˆofspecialistjobswhichmustbeservedbyserver2(i.e.thetypeCjobs).Onlyforpˆ=0.5wehaveasymmetricsystem.Forallpˆ∈[0,0.5)wehaveanasymmetric,butbalancedsystem;astaticsystemwithequalworkloadsforbothserversisobtainedwhenafraction1−pˆofthetypeAjobsissenttoserver1andafractionpˆtoserver2.Table4showsthemeanwaitingtimesW(A),W(B),W(C)foreachjobtypeandtheoverallmeanwaitingtimeWforpˆ=0,0.1,...,0.5.Thesewaitingtimeshaveagainbeencomputedwithanabsoluteaccuracyabs=0.005.Inthelastcolumnof 304G.J.vanHoutumetal.Table4.Meanwaitingtimesasafunctionofpˆ(A)(B)(C)pWˆWWWrr(ˆp)0.04.284.3413.058.667.5%0.14.374.528.526.2560.8%0.24.424.686.935.4578.5%0.34.444.846.125.0986.5%0.44.455.035.624.9290.3%0.54.455.285.284.8691.4%Table4welisttherealizedreductionrr(ˆp)definedby(10),whereWM/M/1=9andW=4.475forρ=0.9.TheresultsinTable4showthatW(A)isfairlySSQSconstantforallvaluesofpˆ.Asexpected,W(B)decreasesandW(C)increasesaspˆdecreases.AstrikingobservationisthatW(C)sharplyincreasesforpˆcloseto0;andthusalsoW=(W(A)+ˆpW(B)+(1−pˆ)W(C))/2.Forpˆ=0wehaveλA=λC=0.9andλB=0,andtheoverallmeanwaitingtimeWisequalto8.66.ThisisclosetoWM/M/1=9,whichisrealizedwhenalltypeAjobswouldbesenttoserver1.4.4ConclusionThemainconclusionfromthenumericalexperimentsisthattheoverallmeanwait-ingtimemayalreadybereducedsignificantlybycreatingalittlebitof(semi-)commonwork.Furthermore,thisreductionismainlydeterminedbytheamountofoverlap,i.e.,themeannumberofserverscapableofhandlinganarbitraryjob.Finally,thebeneficialeffectof(semi-)commonjobsmayvanishforhighlyasym-metricsituations.References1.AdanIJBF,VanHoutumGJ,VanderWalJ(1994)Upperandlowerboundsforthewaitingtimeinthesymmetricshortestqueuesystem.AnnalsofOperationsResearch48:197–2172.AdanIJBF,WesselsJ,ZijmWHM(1989)Queueinganalysisinaflexibleassemblysystemwithajob-dependentparallelstructure.In:OperationsResearchProceedings,pp551–558.Springer,BerlinHeidelbergNewYork3.AdanIJBF,WesselsJ,Zijm,WHM(1990)Analysisofthesymmetricshortestqueueproblem.StochasticModels6:691–7134.AhujaRK,MagnantiTL,OrlinJB(1993)Networkflows:theory,algorithms,andapplications.Prentice-Hall,EnglewoodCliffs,NJ5.FoleyRD,McDonaldDR(2000)Jointheshortestqueue:Stabilityandexactasymp-totics.TheAnnalsofAppliedProbability(toappear)6.FossS,ChernovaN(1998)Onthestabilityofapartiallyaccessiblemulti-stationqueuewithstate-dependentrouting.QueueingSystems29:55–73 Performanceanalysisofparallelidenticalmachines3057.GreenL(1985)Aqueueingsystemwithgeneral-useandlimited-useservers.OperationsResearch33:168–1828.HassinR,HavivM(1994)Equilibriumstrategiesandthevalueofinformationinatwolinequeueingsystemwiththresholdjockeying.StochasticModels10:415–4359.LatoucheG,RamaswamiV(1993)Alogarithmicreductionalgorithmforquasi-birth-deathprocesses.JournalofAppliedProbability30:650–67410.NeutsMF(1981)Matrix-geometricsolutionsinstochasticmodels.JohnsHopkinsUniversityPress,Baltimore11.RoqueDR(1980)Anoteon“Queueingmodelswithlaneselection”.OperationsRe-search28:419–42012.SchwartzBL(1974)Queueingmodelswithlaneselection:anewclassofproblems.OperationsResearch22:331–33913.SparaggisPD,CassandrasCG,TowsleyD(1993)Optimalcontrolofmulticlasspar-allelservicesystemswithandwithoutstateinformation.InProceedingsofthe32ndConferenceonDecisionandControl,SanAntonio,pp1686–169114.VanHoutumGJ(1995)Newapproachesformulti-dimensionalqueueingsystems.Ph.D.Thesis,EindhovenUniversityofTechnology,Eindhoven15.VanHoutumGJ,ZijmWHM,AdanIJBF,WesselsJ(1998)Boundsforperformancecharacteristics:Asystematicapproachviacoststructures.StochasticModels14:205–224(SpecialissueinhonorofM.F.Neuts)16.ZijmWHM(1991)OperationalcontrolofautomatedPCBassemblylines.In:FandelG,ZaepfelG(eds)Modernproductionconcepts:theoryandapplications,pp146–164.Springer,BerlinHeidelbergNewYork Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategiesJohnGeraghty1andCathalHeavey21SchoolofMechanicalandManufacturingEngineering,DublinCityUniversity,Glasnevin,Dublin9,Ireland(e-mail:john.geraghty@dcu.ie)2DepartmentofManufacturingandOperationsEngineering,UniversityofLimerick,Limerick,Ireland(e-mail:cathal.heavey@ul.ie)Abstract.InordertoovercomethedisadvantagesofKanbanControlStrategy(KCS)innon-repetitivemanufacturingenvironments,tworesearchapproacheshavebeenfollowedintheliteratureinpasttwodecades.Thefirstapproachhasbeenconcernedwithdevelopingnew,orcombiningexisting,pull-typeproductioncontrolstrategiesinordertomaximisethebenefitsofpullcontrolwhileincreasingtheabilityofaproductionsystemtosatisfydemand.ThesecondapproachhasfocusedonhowbesttocombineJust-In-Time(JIT)andMaterial-Requirements-Planning(MRP)philosophiesinordertomaximisethebenefitsofpullcontrolinnon-repetitivemanufacturingenvironments.Thispaperprovidesareviewoftheresearchactivitiesinthesetwoapproaches,presentsacomparisonbetweenaPro-ductionControlStrategy(PCS)fromeachapproach,andpresentsacomparisonoftheperformanceofseveralpull-typeproductioncontrolstrategiesinaddressingtheServiceLevelvs.WIPtrade-offinanenvironmentwithlowvariabilityandalight-to-mediumdemandload.Keywords:HybridPush/Pull–CONWIP/Pull–EKCS–BSCS–Kanban–Markovdecisionprocess–Discreteeventsimulation–Simulatedannealingopti-mizationalgorithm1IntroductionTheselection,implementationandmanagementofanappropriateProductionCon-trolStrategyisanimportanttooltoanyorganisationaimingtoadoptaLeanManu-facturingPhilosophy.ProductioncontrolstrategiesthatpushproductsthroughthesystembasedonforecastedcustomerdemandsareclassifiedasPush-typeproduc-tioncontrolstrategies.SuchstrategiesaimtomaximisethethroughputofthesystemCorrespondenceto:J.Geraghty 308J.GeraghtyandC.Heaveysoastominimiseshortageinsupplyandtendtoresultinexcesswork-in-progressinventory,WIP,thatmasksflawsinthesystem.ProductioncontrolstrategiesthatpullproductsthroughthesystembasedonactualcustomerdemandsattheendofthelineareclassifiedasPull-typeproductioncontrolstrategies.SuchstrategiestendtominimiseWIPandunveilflawsinthesystemattheriskoffailuretosatisfydemand.TheadvantagesanddisadvantagesofpushsystemssuchasMRPandpullsystemssuchaskanbancontrolledJust-In-Timehavebeenwelldocumentedintheliterature[11,23,24,31].InordertoovercomethedisadvantagesofKanbanControlStrategy(KCS),tworesearchapproacheshavebeenfollowedinthelasttwodecades.Thefirstapproachhasbeenconcernedwithdevelopingnew,orcombiningexisting,pull-typeproductioncontrolstrategiesinordertomaximisethebenefitsofpullcontrolwhileincreasingtheabilityofaproductionsystemtosatisfydemand.ThesecondapproachhasfocusedonhowbesttocombineJITandMRPphilosophiesinordertomaximisethebenefitsofpullcontrolinnon-repetitivemanufacturingenvironments.Ahybridproductionsystemcouldbecharacterisedasaproductionsystemthatcombineselementsofthetwophilosophiesinordertominimiseinven-toryandunmaskflawsinthesystemwhilemaintainingtheabilityofthesystemtosatisfydemand.Theseresearchapproachesarenotmutuallyexclusiveasthereareintersectionsbetweentheseapproaches.Forinstance,weclassifyCONWIPasapull-typeproductioncontrolstrategy,howeverCONWIPcouldalsobeconsideredasahybridPush/Pullproductioncontrolstrategythatutilisesapull-typecontrolstrategytolimittheamountofinventoryinthelineandapush-typecontrolcontrolstrategywithinthelinetospeedtheprogressofinventorytowardthefinished-goodsbuffer.Inaddition,GeraghtyandHeavey[15]showedthatundercertainconditionstheHorizontallyIntegratedHybridProductionControlStrategy,HIHPS,favouredbyHodgsonandWang[20,21]isequivalenttothepull-typeproductioncontrolstrategyhybridKanban-CONWIPintroducedbyBonvikandGershwin[3],Bonviketal.[2].Inthispaperwefirstlypresentabriefreviewoftheresearcheffortsinthede-signanddevelopmentofbothPull-typePCSandHorizontallyIntegratedHybridSystems,HIHS(seeSects.2and3respectively).Section4comparestheperfor-manceofonepopularmodelofHIHSwiththePull-typePCSknownasExtendedKanbanControlStrategy,EKCS.Section5presentsanexperimenttoexplorethecomparativeperformanceofseveralPull-typePCSinaddressingtheServiceLevelvs.WIPtrade-off.FinallySection6presentsadiscussionofthemainresultsoftheexperimentsandresearchpresentedinthispaper.2Pull-typeproductioncontrolstrategiesTheKanbanControlStrategy,KCS,developedbyToyotaallowspartflowinaJust-In-Time,JIT,linetobecontrolledbybasingproductionauthorisationsonend-itemdemands.KCSisoftenreferredtoasa‘Pull’productioncontrolstrategysincepartdemandstravelupstreamandpullproductsdownthelinebyauthorisingproductionbasedonthepresenceofKanbancardswhicharelimitedinnumberandcirculatedbetweenproductionstages.KCShasbeenthefocusofconsiderableresearcheffortsincetheearly1980’s.Inparticular,optimisingthenumberanddistributionof Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies309Kanbanshasreceivedalotofattention.However,inpractice,Kanbandistributionstendtobedeterminedbyimplementingrulesofthumborsimpleformulae[2].Berkley[1]providesareviewofKanbancontrolliterature,whileMuckstadtandTayur[26]provideareviewofKCSmechanismsthathavebeendeveloped.TheBasestockControlStrategy,BSCS,istheoldestpull-typeproductioncon-trolstrategy.ThedefinitivepaperonBSCS[7]waspublishedin1960.InaBSCSlinetheinventorypointsofeachstageareinitialisedtopredefinedlevels.Whenademandeventoccurs,demandcardsaretransmittedtoeachproductionstage.Thesedemandcardsarematchedwithapartinthestage’sinputbuffertoauthoriseproductionandaredestroyedonceproductionbegins.LiberopoulosandDallery[25]demonstratedthatBSCSisequivalenttotheHedgingPointControlSystemwhichhasitsoriginsintheworkofKimemiaandGershwin[22].TheprimaryadvantageofBSCSisthatitrespondsquicklytodemandevents.Everystageisinformedinstantlyofdemandevents,unlikeKCSwheredemandinformationmustpassslowlyupstream.However,BSCShasbeencriticisedfortheloosecoordina-tionprovidedbetweenstagesandthefactthatitdoesnotprovideanyguaranteetolimitthenumberofpartsthatmayenterthesystem[25].Everydemandeventauthorisesthereleaseofnewpartsintothesystem.GeneralisedKanbanControlStrategy,GKCS,andExtendedKanbanControlStrategy,EKCS,arebothbasedontheintegrationofKCSandBSCS.GKCSwasfirstproposedby[4,36]andEKCSwasproposedby[9,10].InbothsystemstheinventorypointsareinitialisedtoapredefinedlevelasinBSCSanddemandinformationiscommunicatedtoeachstageintheline.ThemovementofpartsbetweenstagesiscoordinatedbyKanbansasinKCS.Thedifferencebetweenthecontrolstructuresemployedinbothsystemsisverysubtleandhastodowithhowdemandinformationiscommunicatedtotheindividualproductionstages.InGKCSwhenademandeventoccursinformationaboutthedemandiscommunicatedtothefinalstageintheformofdemandcards.EachdemandcardmustbematchedwithafreeKanban.Whenthismatchoccurs,ademandcardissenttothestage’simmediatepredecessorandproductionatthestageisauthorisedifthedemand-Kanbanmatchcanbematchedwithapart.Therefore,demandinformationisnotnecessarilytransferredinstantlytoallproductionstages.ThearrivalofdemandinformationatastagecanbedelayedifdownstreamstagesfailtomatchthedemandcardswithKanbansinstantly.InanEKCSgovernedlinedemandinformationiscommunicatedinstantlytoallproductionstages.Productionisauthorisedwhenademandcard,aKanbanandapartareavailable.TheadvantagesofEKCSoverGKCSare,firstly,itscomparativesimplicityandsecondly,theseparationoftheroleofthebasestockandKanbanparametersisclearlydistinguishable,whereasinaGKCSsystemitisnot[25].ConstantWorkInProcessorCONWIPhasreceivedalotofresearchattentionsinceitwasfirstproposedby[29,30].InitiallyCONWIPwasproposedasaPullalternativetoKCSandoftenreferredtoasanOrderReleaseMechanismasopposedtoaProductionControlStrategy.CONWIPwaspurportedtobringtheadvantagesofpull-controltonon-repetitivemanufacturingenvironments[29,30].ThemechanismutilisedbyCONWIPisverysimple.AlimitknownastheWIPCapisplacedontheamountofinventorythatmaybeinthesystematanygivenperiodoftime. 310J.GeraghtyandC.HeaveyOncethislevelofinventoryhasbeenachieved,inventorymaynotenterthesystemuntilademandeventremovesacorrespondingamountofinventoryfromtheline.Withonlyoneparametertooptimise,i.e.theWIPCap,CONWIPisverysimpletoimplementandmaintain.ThemainreasonwhyCONWIPlinesoutperformKCSlinesisthatdemandinformationisinstantlycommunicatedtotheinitialstageandthereleaserateisadjustedtomatchthedemandrate.InaKCSline,ashasbeenstatedearlier,demandinformationhastotravelupstreamfromtheend-iteminventorypointtotheinitialstage.Thelongerthelineandthemoredelaysencounteredatindividualproductionstages(e.g.processingtime,breakdown/repairtime,set-uptimeetc.)thelongertheinformationdelayencountered.AdisadvantageofCONWIPisthatinventorylevelsarenotcontrolledattheindividualstages,whichcanresultinhighinventorylevelsbuildingupinfrontofbottleneckstages.ChangandYih[6]introducedaPullPCSnamedGenericKanbanSystem(GKS)applicabletodynamic,multi-product,non-repetitivemanufacturingenvironments.KCSrequiresinventoriesofsemi-finishedproductsofeachproducttypetobemaintainedateachproductionstage.Inmulti-productenvironmentstheamountofsemi-finishedinventorymaintainedinthelinecouldbeprohibitivelylarge[6].GKSoperatesbyprovidingafixednumberofKanbansateachworkstationthatcanbeacquiredbyanypart.Apart/jobcanonlyenterthesystemifitacquiresaKanbanfromeachoftheworkstationsinthesystem.GKSreducestoCONWIPifanequalnumberofKanbansaredistributedtoallworkstations.Comparisonwithapush-typeproductioncontrolstrategywasfavourablewithGKSshowntobelesssusceptibletothepositionofthebottleneck.GKSoutperformedKCSintermsofWIPrequiredtoachieveadesiredCycleTime.ComparisonwithCONWIPwasfavourableandGKSwasshowntobemoreflexibleinthatbymanipulatingthenumberofKanbansateachworkstationtheperformanceofGKScouldbeimprovedbeyondthatachievedbyCONWIP.ChangandYih[5]presentedasimulatedannealingalgorithmfordeterminingtheoptimalKanbandistributionforaGKSline.Inordertoovercomethedisadvantagesofloosecoordinationbetweenpro-ductionstagesinaCONWIPlineBonvikandGershwin[3]andBonviketal.[2]proposedanalternativestrategy,hybridKanban-CONWIP.InhybridKanban-CONWIP,asinCONWIP,anoverallcapisplacedontheamountofinventoryallowedintheproductionsystem.Inaddition,inventoryiscontrolledusingKan-bansinallstagesexceptthelaststage.CONWIPcanbeconsideredasspecialcaseofhybridKanban-CONWIPinwhichthereisaninfinitenumberofKanbansdistributedtoeachproductionstage[2].AcomparisonofKCS,minimalblockingKCS,BSCS,CONWIPandhybridKanban-CONWIPwaspresentedin[2].ThedifferentPCSwerecomparedinafour-stagetandemproductionlineusingsimula-tion.EachofthePCSwerecomparedusingconstantdemandanddemandthathadasteppedincrease/decrease.ItwasfoundthatthehybridKanban-CONWIPstrategydecreasedinventoriesby10%to20%overKCSwhilemaintainingthesameservicelevels(percentageofdemandsinstantaneouslymatchedwithafinishedproduct).TheperformanceofbasestockandCONWIPstrategiesfellbetweenthoseofKCSandhybridKanban-CONWIP.TwopapersthatgeneralizehybridKanban-CONWIPare[14]and[13].Thesepapersproposeagenericpullmodelthat,aswellasencapsulatingthethreebasic Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies311pullcontrolstrategies,KCS,CONWIPandBSCS,alsoallowscustomizedpullcontrolstrategiestobedeveloped.Simulationandanevolutionaryalgorithmwereusedtostudythegenericmodel.Detailsoftheevolutionaryalgorithmaregivenin[14]whileresultsonextensiveexperimentationontheeffectoffactors(i.e.,lineim-balance,machinereliability)ontheproposedgenericpullmodelaregivenin[13].GauryandKleijnen[12]notedthatOperationsResearchhastraditionallycon-centratedonoptimisationwhereaspractitionersfindtherobustnessofaproposedsolutionmoreimportant.Amethodologywaspresentedin[12]thatwasastagewisecombinationoffourtechniques:(i)simulation,(ii)optimization,(iii)riskorun-certaintyanalysis,and(iv)bootstrapping.GauryandKleijnen[12]illustratedtheirmethodologythroughaproduction-controlstudyforthefour-stage,singleproductproductionlineutilisedby[2].Robustnesswasdefinedin[12]asthecapabilitytomaintainshort-termservice,inavarietyofenvironments;i.e.theprobabilityoftheshort-termfill-rate(servicelevel)remainingwithinapre-specifiedrange.Be-sidessatisfyingthisprobabilisticconstraint,thesystemminimisedexpectedlong-termWIP.Foursystemswerecomparedin[12],namelyKanban,CONWIP,hybridKanban-CONWIP,andGeneric.Theoptimalparametersfoundin[2]wereusedforKCS,CONWIPandhybridKanban-CONWIP.GauryandKleijnen[12]usedaGeneticAlgorithmtodeterminetheoptimalparametersfortheGenericpullsys-tem.Fortheriskanalysisstep,seventeeninputswereconsidered;themeanandvarianceoftheprocessingtimeforeachofthefourproductionstages,meantimebetweenfailuresandmeantimetorepairperproductionstage,andthedemandrate.Theinputswerevariedoverarangeof±5%aroundtheirbasevalues.GauryandKleijnen[12]concludedthatinthisparticularexample,hybridKanban-CONWIPwasbestwhenriskwasnotignored;otherwiseGenericwasbestandtherefore,riskconsiderationscaninfluencetheselectionofaPCS.Eachofthepull-typeproductioncontrolstrategiesdiscussedabove,withtheexceptionofGKCS,haveoneimportantadvantageoverKCSthatensuresthattheyaremorereadilyapplicabletonon-repetitivemanufacturingenvironments.ThatadvantagestemsfromthemannerinwhichdemandinformationiscommunicatedincomparisontoKCS.InKCS,demandinformationisnotcommunicateddirectlytoproductionstagesthatreleaseparts/jobsintothesystem.Ratheritiscommunicatedsequentiallyupthelinefromthefinishedgoodsbufferaswithdrawalsaremadebycustomerdemands.Thiscommunicationdelaymeansthatthepaceoftheproductionlineisnotadjustedautomaticallytoaccountforchangesinthedemandrate.ThearrivalofdemandinformationtotheinitialstagesinaGKCSlinemightbedelayedifthedemandcardsataproductionstageinthelinearenotinstantaneouslymatchedwithKanbancards.BSCS,EKCS,CONWIP,GKSandhybridKanban-CONWIPall,however,communicatethedemandinformationinstantaneouslytotheinitialstagesallowingthereleaseratetobepacedtotheactualdemandrate.Forinstance,Bonviketal.[2]showedthatifthedemandratedecreasesunexpectedlytheimpactonaCONWIPstrategyandhybridKanban-CONWIPstrategywouldbeforthefinished-goodsbuffertoincreasetowardtheWIPCapwithallintermediatebufferstendingtowardempty.Theimpact,however,onaKCSlinewouldbethatalltheintermediatebufferswouldincreasetowardtheirmaximumpermissiblelimits. 312J.GeraghtyandC.HeaveyTherefore,theKCSlinewouldhavesemi-finishedinventorydistributedthroughouttheline.3HybridproductioncontrolstrategiesHybridcontrolstrategiescanbeclassifiedintotwocategories:verticallyintegratedhybridsystems(VIHS)orhorizontallyintegratedhybridsystems(HIHS)[8].VIHSconsistoftwolevels,usuallyanupperlevelpush-typePCSandalowerlevelpull-typePCS.Forexample,SynchroMRPutilisesMRPforlongrangeplanningandKCSforshopfloorexecution[17].ThemaindisadvantageofVIHSisthatMRPcalculationsmustbeperformedforeachstageintheproductionsystem.ThismakesVIHScomplextoimplementandmaintainandaccountsfortheirrelativelackofuseinindustry[20].HIHSconsistofonelevelwheresomeproductionstagesarecontrolledbypush-typePCSandotherstagesbypull-typePCS.OnlyHIHSareconsideredinthediscussionthatfollows.HodgsonandWang[20,21]developedaMarkovDecisionProcess(MDP)modelforHIHS.Themodelwassolvedusingbothdynamicprogrammingandsimulationforseveralproductionstrategies,includingpurepushandpurepullproductionstrategiesandstrategiesbasedontheintegrationofpushandpullcontrol.Inthispush/pullintegrationstrategyeachindividualstagemaypushorpull.ThistypeofcontrolstrategyisdenotedasHybridPush/Pullin[20,21].Initiallyin[20],theresearchwasappliedtoafour-stagesemi-continuousproductionironandsteelworks(seeFig.1),withthefirsttwostagesinparallelandtheremainingstagesasserialproductionstages.Inordertosimplifytheanalysisthemodelassumesthattheproductionprocessisadiscretetimeprocessandthatdemandperperiodandtheamountofinventoryarebothintegermultiplesofaunitsize.Theresearchwaslaterextendedtoafive-stageproductionsystem[21].Forboththefourandfivestageproductionsystems,astrategywhereproductionstages1and2(P1andP2inFig.1)pushandallotherstagespullwasdemonstratedtoresultinthelowestaveragegain(averagesystemcost).HodgsonandWang[21]statedthattheyhadobservedsimilarresultsforaneight-stagesystemandconcludedthatthisstrategywouldbetheoptimalhybridintegrationstrategyforaJ-stagesystem.Subsequentpapersthatusethemodelin[20,21]orextensionsofitare[11],[28]and[35].Deleersnyderetal.[11]consideredthatthecomplexityofthecontrolstructurerequiredforthesuccessfulimplementationofSynchroMRPresultedinitbeinglargelyignoredbyindustry.SynchroMRPrequiresMRPcontroltobelinkedintoeverystageintheproductionlinewhileutilisinglocalkanbancontroltoauthoriseproductionateachstage.Deleersnyderetal.[11]developedahybridproductionstrategythatlimitedthenumberofstagesintowhichMRPtypeinformationisaddedinordertoreducethecomplexityofthehybridstrategyincomparisontoSynchroMRP,whilerealizingthebenefitsofintegratingpushandpulltypecontrolstrategies.Themodeldevelopedin[11]issimilartothatpresentedin[20,21]andcomparableresultswereobtainedforaserialproductionline.PandeyandKhokhajaikiat[28]extendedthemodelin[20,21]toallowfortheinclusionofrawmaterialconstraintsateachstage.Themodifiedmodelalsoallowedforastagetorequiremorethanoneitemofinventoryand/ormorethanone Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies313Fig.1.Parallel/SerialfourstageproductionsystemmodelledbyHodgsonandWang[20]itemofrawmaterialtoproduceapart.PandeyandKhokhajaikiat[28]presentedresultsfromtwosetsofexperiments.Inthefirstsettheymodelledafour-stageparallel/serialproductionlinesimilartothesystemshowninFigure1.Theinitialproductionstages(P1andP2inFig.1)operatedunderrawmaterialavailabilityconstraints,haddifferentorderpurchasinganddeliverydistributionsbuthadiden-ticalproductionunreliability.Sixteenintegrationstrategieswereconsidered.Inthesecondexperimentalsettheauthorsappliedtherawmaterialavailabilityconstrainttoallstagesoftheproductionline.Theauthorsconcludedthatthehybridstrategyinwhichtheinitialstages(P1andP2)operateunderpushcontrolandtheremainingstagesoperateunderpullcontrolisthebeststrategywhenrawmaterialconstraintsapplyonlytotheinitialstages.Whentherawmaterialavailabilityconstraintisappliedtoallstagesthepushstrategybecomestheoptimalcontrolstrategy.Forsystemswithlargevariabilityindemandnoneofthestrategiesdominated.WangandXu[35]presentedanapproachthatfacilitatedtheevaluationofawiderangeoftopologiesthatutilizehybridpush/pull.Theyusedastructuremodeltodescribeamanufacturingsystem’stopology.Theirmethodologywasusedtoinvestigatefour45-stagemanufacturingsystems:(i)Asingle-materialserialprocessingsystem;(ii)Amulti-materialserialprocessingsystem;(iii)Amulti-partprocessingandassemblysystem,and(iv)Amulti-partmulti-componentprocessingandassemblysystem.WangandXu[35]comparedpurepullandpushstrategiesagainsttheoptimalhybridstrategyfoundin[20,21],wheretheinitialstagespushandallotherstagespull.Theirresultssuggestthattheoptimalhybridstrategyout-performspurepushorpullstrategies.Othermodelsthatimplementhybridpush/pullcontrolstrategiessimilarto[20,21]havebeendeveloped.Takahashietal.[32]definedpush/pullintegrationasasysteminwhichthereisasinglejunctionpointbetweenpushstagesandpullstages.InTakahashietal.[32]amodelwaspresentedtoevaluatethiscontrolstrategy.Two 314J.GeraghtyandC.Heaveysubsequentpapers,[33]and[34]furtherdevelopedandexperimentedwiththismodel.Hirakawaetal.[19]andHirakawa[18]developedamathematicalmodelforahybridpush/pullcontrolstrategythatallowseachproductionstagetoswitchbetweenpushandpullcontroldependingonwhetherdemandcanbeforecastedreliablyornot.CochranandKim[8]presentsaHIHSwithamovablejunctionpointbetweenapushsub-systemandapullsub-system.Thecontrolstrategypresentedhadthreedecisionvariables:(i)thejunctionpoint,i.e.,thelastpushstageintheHIHS;(ii)thesafetystocklevelatthejunctionpoint;(iii)thenumberofkanbansforeachstageinthepullsub-system.Simulationcombinedwithsimulatedannealingwasusedtofindtheoptimaldecisionvariablesforthecontrolstrategy.4ComparisonofEKCSandthepush-typePCSmodelledbyHodgsonandWangSeveralcomparisonsofPull-PCShavebeenreportedintheliterature,forexample[2,12,13,25].TherehasalsobeenseveralcomparisonsbetweenHIHSandKCS,forexample[8,27,32,33,34,20,21,11,28,35].ComparisonsbetweenHIHSandotherPull-TypePCSarerareintheliterature.OrthandCoskunoglu[27]includedCONWIP,inadditiontoKCS,inthecomparisonanalysis.Inapreviouspaper[15]wedemonstratedthattheoptimalHIHSselectedbyHodgsonandWang[20,21]whereinitialstagesemploypushcontrolandallotherstagesemploypullcontrol,isequivalenttoaPull-TypePCS,namelyhybridKanban-CONWIP[3,2].Aswellasconsideringseveralalternativeintegrationstrategies,HodgsonandWang[20,21]alsoincludedaPush-TypeandaPull-TypePCSintheiranalysis,whichtheyreferredtoas‘PurePush’and‘PurePull’PCS.Afterexaminingtheequationsusedin[20,21]tomodelthe‘PurePush’PCSwefeltthatthereweresimilaritiestothecontrolstructureimplementedbythePull-TypePCSknowasEKCS[9,10].Thereforeinthissectionweexplorethesecomparisons.ThenotationusedintheremainderofthispaperisshowninTable1.InHodgsonandWang’s‘Pure-Push’PCS,productionisauthorisedwhen(i)suf-ficientspaceexistsintheoutputbufferofthestage,(ii)sufficientinventoryexistsintheinputbufferofthestage,(iii)sufficientproductioncapacityexistsatthestage,and(iv)downstreaminventorylevelshavedecreasedbelowforecastedrequire-mentsnecessarytomeetexpecteddemand.TheMDPmodelpresentedin[20,21]requiredtheevaluationoftwoequations(i.e.theproductiontrigger,Aj(n),andtheproductionobjective,POj(n)inordertodetermineproductionauthorisationsforastageinperiodn,).Forthepurposesofthediscussionpresentedherewehavecombinedtheseequationstoformasingleequationforthenumberofproductionauthorisations,PAj(n),availabletoastageinperiodn.Thiswasachievedwithoutmakinganysimplifyingassumptions.PAj(n)forasystemcontrolledbyHodgsonandWang’s‘Pure-Push’PCScanbemodelledbyEq.(1)where1≤j≤J−1andbyEq.(2)forthefinalproductionstage. Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies315Table1.NotationusedinmodelspresentedNotationDescriptionAj(n):Productiontriggerforstagejinperiodn.POj(n):Productionobjectiveforstagejinperiodn.maxIj:Maximumcapacityofinventorypointj.SS:DesiredsafetystockleveloffinishedproductNSj:Thenumberofstagesthatsucceedstagej(i.e.,numberofstagesthatcompo-nentsproducedatstagejtraverseafterstagejbeforereachingthecustomer.D(n):Forecasteddemandinperiodn.j,J:Uniquenumberidentifyingaproductionstagewhere1≤j≤J.n:Productionperiod.d(n):Theactualdemandquantityinperiodn.PAj(n):TheProductionAuthorisationforstagejinperiodn.minPj:TheminimumproductioncapacityofstagejmaxPj:ThemaximumproductioncapacityofstagejPj(n):Productionquantityforstagejinperiodn.q:Theproductionreliabilityofastage,whichismodelledbyaProbabilityMassFunctionIj:Theoutputbufferofstagej.Ij(n):Theamountofinventoryheldintheoutputbufferofproductionstagejinperiodn.{Bj(n)}:Thesetofinventoriesheldintheoutputbuffersoftheimmediatepredecessorsofstagejinperiodncj(n):Thesumofinventoriesheldintheoutputbuffersofstagesparallelto,butwithstagenumbergreaterthan,productionstagejKj:ThenumberofKanbansallocatedtoproductionstagej.CC:ThecapontotalinventoryallowedinCONWIPandhybridKanban-CONWIPlines.DCj(n):NumberofdemandcardsheldatstagejinperiodninBSCSandEKCSlines.Sj:TheinitialisationstocklevelforstagejinBSCSandEKCSlinesminSj:TheminimuminitialisationstocklevelforstagejinBSCSandEKCSlinesPj:ProductioncenteratstagejPA(n)=minImax−I(n−1),maxPmin,SS+(NS+1)×D(n)jjjjj⎛⎞⎤⎫J⎬−⎝I(n−1)−c(n−1)⎠⎦,{B(n−1)},Pmax,(1)ijjj⎭i=j∀j≤J−1PA(n)=minImax−I(n−1)+D(n),maxPmin,SS+D(n)JJJJ−I(n−1)],{B(n−1)},Pmax}(2)JJJItispossibleforthetermIJ(n−1)tobecomenegative.Thisoccursintheeventofashortageinperiodn−1,i.e.afailuretosatisfydemand.Thereforetheterm 316J.GeraghtyandC.HeaveyIJ(n−1)isnotonlyusedtorecordtheinventoryinthefinishedgoodsbufferinperiodn−1butalsothebackloginperiodn−1.Ifabacklogoccurs,i.e.IJ(n−1)isnegative,Eq.(2)wouldeffectivelyresultinatemporaryincreaseofthemaximumcapacityofthefinishedgoodsinventorybuffer.Equation(3)modelsthenumberofproductionauthorisationsavailabletothefinalstagewhereshortagesarenotpermittedtotemporarilyincreasethemaximumcapacityofthefinishedgoodsinventorybuffer.PA(n)=minImax−max[0,I(n−1)]+D(n),maxPmin,SSJJJJ+D(n)−I(n−1)],{B(n−1)},Pmax}(3)JJJNoticethatHodgsonandWang’smodelofPushcontrolincludesalimitoninventoryintheoutputbufferofastage,Imax.Sinceaproductionstagedoesnotbecomejawareofachangeinstateofthebufferuntilthesubsequentproductionperiod,n+1,andwiththeexceptionofthefinalstagedoesnotattempttopredicttheremovalofinventoryfromtheoutputbufferbyimmediatesuccessors,thislimitbehavessimilarlytoKanbancontrol.However,thePushstrategymodelledby[20,21]isnotequivalenttoKCSsinceeachstagehasinformationaboutthestatusoflineJdownstreamfromitsoutputbuffer,throughthetermi=jIi(n−1)−cj(n−1)inEq.(1)above.Sinceonlyademandeventcanchangethestateofthedownstreamsectionoftheline,intermsoftotalWIP,thisisequivalenttodemandcardsbeingpassedtoeachstageintheline.Therefore,itwouldappearthatthereissomesimilaritybetweenthe‘Pure-Push’PCSmodelledby[20,21]andEKCS.InanEKCSsystem,aproductionstagehasauthorisationtoproduceapartwhen:(i)inventoryisavailableinitsbuffer;(ii)aKanbancardisavailableand(iii)ademandcardisavailable.ThenumberofdemandcardsavailabletoaproductionstageinperiodnisgivenbyEq.(4).Inperiodnthenumberofproductionau-thorisationsatstagej,PAj(n),isgivenbyEq.(5).ForanEKCSsystemwheretheintroductionoftemporaryKanbansintheeventofshortagesisnotpermittedPAj(n)ismodelledbyEq.(6)forthefinalproductionstage.DCj(n)=DCj(n−1)−Pj(n−1)+d(n),∀j≤J(4)PA(n)=minK−I(n−1),maxPmin,DC(n),jjjjj{B(n−1)},Pmax,∀j≤J−1(5)jjPA(n)=minK−max[0,I(n−1)],maxPmin,DC(n),JJJJJ{B(n−1)},Pmax}(6)JJLetusassumeaserialproductionlinewithJstages.Thestatetransitionequa-tionsfortheinventorylevelsofthebuffers,foreithermodel,canbedeterminedfromEqs.(7)and(8):Ij(n)=Ij(n−1)+Pj(n)−Pj+1(n)∀j≤J−1(7)IJ(n)=IJ(n−1)+PJ(n)−d(n)(8)Fromexaminingtheequationsforthetwomodels,forbothdynamicandstaticKanbandistributions,itisclearthatinorderforthePCStobeequivalentthreeconditionsmustbesatisfied:(i)theinventorylevelofabufferinthe‘Pure-Push’ Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies317modelmustequatetotheinventorylevelofthesamebufferintheEKCSmodelinagivenproductionperiodn.(ii)theKanbandistributioninEKCSmustbeequaltothebuffercapacitylimitsinthe‘Pure-Push’model.K=Imax∀j≤J−1(9)jjK=Imax+D(n)(10)JJand(iii)thefollowingtwoequalitiesmusthold:DCJ(n)=SS+D(n)−IJ(n−1)(11)⎡⎛⎞⎤JDCj(n)=⎣SS+(NSj+1)×D(n)−⎝Ii(n−1)−cj(n−1)⎠⎦(12)i=j∀j≤J−1SubstitutingEq.(11)intoEq.(4)yields:SS+D(n)−IJ(n−1)=SS+D(n)−IJ(n−2)−PJ(n−1)+d(n)(13)Re-writingEq.(13)yields:IJ(n−1)=IJ(n−2)+PJ(n−1)−d(n)(14)Equation(14)isnotequivalenttoEq.(8)andthereforethetwoPCSarenotequiv-alent.TheprimarydifferencebetweenthetwoPCSstemsfromthemethodinwhichdemandinformationiscommunicatedtoeachstage.InEKCSdemandin-formationisinstantlycommunicatedtoallstages.Inthe‘Pure-Push’PCSthecommunicationofdemandinformationisdelayedbyoneperiod.Infact,the‘Pure-Push’PCSdescribedby[20,21]couldmoreaccuratelybedescribedasaVerticallyIntegratedHybridSystem,inwhicheachproductionstagedevelopsaforecastofproductionrequirementsfortheproductionperiodthroughthetermJSS+(NSj+1)×D(n)−(i=jIi(n−1)−cj(n−1))andutiliseskanbanstoimplementtheproductionplanontheshop-floorplan.Itwould,therefore,bemoreaccuratetorefertothisPCSasSynchroMRPthan‘Pure-Push’.However,itisworthnotingthatthetwoPCScanbemadeequivalentifthedemandinformationiscommunicatedinstantlyinbothPCSordelayedforoneperiodinbothPCS.Forinstance,communicationofdemandinformationinthe‘Pure-Push’PCScanbemadeinstantaneousbyadjustingEqs.(1),(2)and(3)byincludingacomponenttoadjustthedownstreaminventorylevelsbythedemandquantityd(n)asshownbyEqs.(15),(16)and(17)below.PA(n)=minImax−I(n−1),maxPmin,SS+(NS+1)×D(n)jjjjj⎛⎞⎤⎫J⎬−⎝I(n−1)−c(n−1)−d(n)⎠⎦,{B(n−1)},Pmax,ijjj⎭i=j∀j≤J−1(15)PA(n)=minImax−I(n−1)+D(n),maxPmin,SS+D(n)JJJJ 318J.GeraghtyandC.Heavey−(I(n−1)−d(n))],{B(n−1)},Pmax}(16)JJJPA(n)=minImax−max[0,I(n−1)]+D(n),maxPmin,SSJJJJ+D(n)−(I(n−1)−d(n))],{B(n−1)},Pmax}(17)JJJThePCSwillbeequivalentifthefirsttwoconditionsstatedearlieraremetandthefollowingtwoequalitieshold:DCJ(n)=SS+D(n)−(IJ(n−1)−d(n))(18)⎡⎛⎞⎤JDCj(n)=⎣SS+(NSj+1)×D(n)−⎝Ii(n−1)−cj(n−1)−d(n)⎠⎦i=j∀j≤J−1(19)SubstitutingEq.(18)intoEq.(4)yields:SS+D(n)−(IJ(n−1)−d(n))=SS+D(n)−(IJ(n−1)−d(n−1))−PJ(n−1)+d(n)(20)Re-writingEq.(20)yields:IJ(n−1)=IJ(n−2)+PJ(n−1)−d(n−1)(21)Equation(21)isequivalenttoEq.(8)andthereforerequiresnofurtherproof.SubstitutingEq.(19)intoEq.(4)yields:⎡⎛⎞⎤J⎣SS+(NSj+1)×D(n)−⎝Ii(n−1)−cj(n−1)−d(n)⎠⎦=i=j⎡⎛⎞⎤J⎣SS+(NSj+1)×D(n)−⎝Ii(n−2)−cj(n−2)−d(n−1)⎠⎦i=j−Pj(n−1)+d(n)(22)Re-writingEq.(22)yields:⎛⎞⎛⎞JJ⎝Ii(n−1)−cj(n−1)⎠=⎝Ii(n−2)−cj(n−2)⎠i=ji=j+Pj(n−1)−d(n−1)(23)Wecanusethestatetransitionequations(i.e.Eqs.(7)and(8))toproveEq.(23)asshownbelow.Notethatinaseriallinethetermscj(n−1)andcj(n−2)arebothzero. Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies319Ij(n−1)=Ij(n−2)+Pj(n−1)−Pj+1(n−1)Ij+1(n−1)=Ij+1(n−2)+Pj+1(n−1)−Pj+2(n−1)Ij+2(n−1)=Ij+2(n−2)+Pj+2(n−1)−Pj+3(n−1).......=.+Pj+3(n−1)−.............................=.+.−PJ−1(n−1)IJ−1(n−1)=IJ−1(n−2)+PJ−1(n−1)−PJ(n−1)IJ(n−1)=IJ(n−2)+PJ(n−1)−d(n−1)JJIi(n−1)=Ii(n−2)+Pj(n−1)−d(n−1)i=ji=jTheinitialisationstocklevelsforthebuffersforbothmodelscanbedeterminedfromEqs.(4)and(19).Forinstance,assumethattheinitialnumberofdemandcards,DCj(0),theproductioninthepreviousperiod,Pj(0),andthedemandinthepreviousperiodd(0)areallzero.Therefore,fromEq.(4)thenumberofdemandcardsavailabletoeachproductionstageinthefirstperiod,DCj(1),willbezero.TheinitialisationstocksforbothmodelscanbecalculatedfromEq.(19)asfollows:⎡⎛⎞⎤J0=⎣SS+(NSj+1)×D(n)−⎝Ii(0)−cj(0)⎠⎦(24)i=jJIi(0)−cj(0)=SS+(NSJ+1)×D(n)(25)i=jTherefore,forthefinalproductionstagetheinitialisationstocklevelshouldbe:IJ(0)−cJ(0)=SS+(NSj+1)×D(n)(26)Sinceitisthefinalproductionstage,thetermcJ(0)ontheLHSwillequalzeroandontheRHSthetermNSJwillequalzero.Therefore:IJ(0)=SS+D(n)(27)ForstageJ−1theinitialisationstocklevelwillbe:IJ(0)+IJ−1(0)−cJ−1(0)=SS+(NSJ−1+1)×D(n)(28)GiventhatcJ−1(0)=0,NSJ−1=1andsubstitutinginEq.27,then:SS+D(n)+IJ−1(0)=SS+2×D(n)(29)IJ−1(0)=SS+2×D(n)−SS−D(n)(30)IJ−1(0)=D(n)(31)Infactitcanbeshownthatfor1≤j≤J−1theappropriatechoiceforinitialisationstockisIj(0)=D(n).Ofcourse,iftheinitialnumberofdemandcardsavailabletoproductionstagesisnotzerothenappropriateinitialisationstocklevelsforbothmodelscanbedeterminedinasimilarmannerfromEq.(19).Therefore,EKCSandHodgsonandWang’s‘Pure-Push’PCSareequivalentif: 320J.GeraghtyandC.Heavey(1)Thedemandeventiscommunicatedinstantaneouslyinthe‘Pure-Push’PCSordelayedbyoneproductionperiodintheEKCSPCS,(2)TheKanbandistributionforEKCSandbuffercapacitylimitsforthe‘Pure-Push’PCSareequivalent,(3)TheinitialisationstocksofbothmodelsareequivalentandcalculatedfromEq.(19)forallstages,i.e.j=1,...,J,and(4)Theforecasteddemandquantity,D(n),inthe‘Pure-Push’PCSisconstantforallvaluesofn.5Comparisonofpull-typePCSWenowturnourattentiontowardexaminingthecomparativeperformanceofseveralPull-TypePCS.ThePCSexaminedareKCS,CONWIP,hybridKanban-CONWIP,BSCSandEKCS.ThestudypresentedherediffersfromBonviketal.[2]andGauryandKleijnen[12]asEKCSisincludedintheanalysis,variabledemandisusedandunsatisfieddemandisbackloggedratherthanbeingtreatedasalostopportunity.Thesystemmodelledforthepurposesoftheseexperimentswasthefive-stageparallel/seriallinedescribedbyHodgsonandWang[21].Thelineproducesasingleproducttypeproducedfromtwocomponents.Intheline,stages1and2operateinparalleltoinputthetwocomponentstothesystemthatareassembledonaone-to-oneratioatstage3.Stages3,4and5areinseries.Theoutputbufferofstage5isthefinishedgoodsbuffer,fromwhichalldemandsmustbesatisfied.Demandinagivenperiod,n,iseither3or4unitswithequalprobability.Forthepurposesoftheexperimentalworkpresentedhereitisassumedthatminimumproductionlevel(Pmin)ofstagejinperiodniszero.ThereliabilityofstagejinperiodnjwasmodelledbytheProbabilityMassFunctiongiveninTable2.NotingthatqandPAj(n)areindependent,theprobabilitythatstagejproducesqunitsinperiodngiventhattheproductionauthorisationisPAj(n),i.e.Pr[Pj(n)=q|PAj(n)],isgivenby:Pr[Pj(n)=q|PAj(n)]=Pr[Pj(n)=q],q=0,1,...,PAj(n)−1(32)Pr[Pj(n)=PAj(n)|PAj(n)]=Pr[Pj(n)=PAj(n)]+Pr[Pj(n)=PAj(n)+1]+...+PrP(n)=Pmaxq≥PA(n)(33)jjjTable2.Probabilitymassfunctionforreliabilityinproductionofindividualstagesq345Pr[q]0.20.60.2TheremainderofthissectiondetailsthemodelsthatweredevelopedforeachPCSexamined,theexperimentdesignandtheresultsfromtheexperiment.The Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies321modelshavebeendevelopedbytheauthorswithreferencetothenotationandmethodologiesemployedby[20,21],and[28].5.1KanbancontrolstrategyInaKCSsystem,productionatstagejisauthorisedbythepresenceofKanbancardsandparts.Whenstagejbeginsproductiononapart,aKanbancardisattachedtothepartandtravelsdownstreamwiththepart.WhenthesucceedingstagebeginsproductionontheparttheKanbancardisremovedandpassedbacktostagejtobeavailabletoauthoriseproductionofanewpart.TheProductionAuthorisationforperiodnforKCSstagej,where1≤j≤J−1,isobtainedfromEq.(34).TheProductionAuthorisationforthefinalstageisobtainedfromEq.(35)andisdifferentfromthemodelin[20,21]inthatthenumberofKanbansavailabletothefinalstagecannotbeincreasedtemporarilyinresponsetoashortage.PA(n)=minK−I(n−1),{B(n−1)},Pmax,∀j≤J−1(34)jjjjjPA(n)=min[K−max[0,I(n−1)−d(n)],{B(n−1)},Pmax](35)JJJJJ5.2CONWIPcontrolstrategyForCONWIPsystems,PAj(n)foraninputstage(j=1,2)wasmodelledbyEq.(36).PAj(n)foraninputstageisconstrainedbyacap(CC)onthetotalinventoryinthesystem,thenumberofcomponentsavailableintherawmaterialbuffersandthemaximumproductioncapacityofthestage.Forthepurposesoftheexperimentsconductedrawmaterialwasassumedtobealwaysavailable.Forthissituationtheterm{Bj(n−1)}wouldbeinfinitelylarge.PAj(n)forallotherstagesisonlyconstrainedbythemaximumamountofunitsthatthestagecanproduceinaproductionperiodandtheavailabilityofcomponentsinthestage’sinputbuffer.Therefore,Eq.(37)wasusedtomodelPAj(n)forallstagesthatarenotinputstages,i.e3≤j≤J,.⎡⎛⎛⎞⎞JPAj(n)=min⎣CC−⎝⎝Ii(n−1)⎠−cj(n−1)−d(n)⎠,i=j{B(n−1)},Pmax,1≤j≤2(36)jjPA(n)=min{B(n−1)},Pmax,3≤j≤J(37)jjj5.3HybridKanban-CONWIPcontrolstrategyProductionAuthorisationsforproductionstagesinahybridKanban-CONWIPsystemweredeterminedbycombiningtheequationsusedtomodelPAj(n)forKCSandCONWIP.ForaninputstageofahybridKanban-CONWIPsystem(j=1,2),PAj(n)wasmodelledbyEq.(38).ThisequationwasdevelopedbyfurtherconstrainingEq.(35)suchthatsufficientKanbansmustalsobeavailableatthe 322J.GeraghtyandC.Heaveystagetoauthoriseproduction.Forstagej,where3≤j≤J−1,PAj(n)wasmodelledbyEq.(34).ForthefinalstagePAJ(n)wasmodelledbyEq.(37)wherej=J.⎡⎛⎛⎞⎞JPAj(n)=min⎣CC−⎝⎝Ii(n−1)⎠−cj(n−1)−d(n)⎠,(38)i=jK−I(n−1),{B(n−1)},Pmax,1≤j≤2jjjj5.4BasestockcontrolstrategyandextendedKanbancontrolstrategyInasystememployingBSCS,productionatstagejinperiodnisauthorisedbythepresenceofdemandcardsattheproductionstage.Whenademandoccurstheequivalentnumberofdemandcardsaredispatchedtoeachproductionstagetoauthorisetheproductionofnewparts.Whenthestagebeginsproductionofanewpartthedemandcardisdestroyed.Thenumberofdemandcardsavailabletoproductionstagejinperiodn,DCj(n),wasdeterminedfromEq.(39).PAj(n)foraBSCSsystemwasdeterminedbyemployingEq.(40).DCj(n)=DCj(n−1)−Pj(n−1)+d(n),∀j≤J(39)PA(n)=minDC(n),{B(n−1)},Pmax,∀j≤J(40)jjjjTheproductioninperiodnofstagejinanEKCSsystemisconstrainedbytheavailabilityofKanbanandDemandcards.Whenademandoccurs,aswithBSCS,theequivalentnumberofdemandcardsaredispatchedtoeachproductionstagetoauthorisetheproductionofnewparts.However,beforeproductioncanbeautho-risedbythepresenceofademandcard,thedemandcardmustbematchedwithaKanbancardandanavailablepart.AdemandcardisdestroyedwhenstagejbeginsproductiononthepartwhiletheassociatedKanbancardisattachedtothepartandtravelsdownstreamwiththepart.WhenthesucceedingstagebeginsproductionontheparttheKanbancardisremovedandpassedbacktostagejtobeavailabletoauthoriseproductionofanewpart.ForanEKCSsystemthenumberofdemandcardsavailabletoproductionstagejinperiodn,DCj(n),wasalsomodelledbyEq.(39)whilePAj(n)foranEKCSsystemwasdeterminedbyemployingEq.(41)for1≤j≤J−1andEq.(42)forthefinalproductionstage,i.e.j=J.PA(n)=minDC(n),K−I(n−1),{B(n−1)},Pmax,jjjjjj∀j≤J−1(41)PAJ(n)=min[DCJ(n),KJ−max[0,IJ(n−1)−d(n)],{B(n−1)},Pmax](42)JJ5.5ExperimentalconditionsThemodelsjustdescribedforeachPCSweretranslatedintodiscreteeventsimu-lationmodelsineM-Plant,anobject-orientedsimulationsoftwaretooldeveloped Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies323byTecnomatixTechnologiesLtd.ThepowerfuldebuggingenvironmentwithineM-Plantwasutilisedtoconductastep-by-stepwalkthroughofeachsimulationmodelinordertoverifythetimingandaccuracyofthecalculationsofthecon-ceptualmodelshadbeencorrectlyencapsulatedinthemodels.Inordertovalidatethesimulationmodels,wefirstlydevelopedsimulationmodelsofthevariousPCSexploredby[20,21].Thesemodelswerevalidatedagainsttheresultspublishedin[20,21]andtheresultswerepresentedinGeraghtyandHeavey[15].InordertovalidatetheindividualsimulationmodelsdevelopedforthePCSexaminedinthissectionweconductedthefollowing:(1)TheoutputofoursimulationmodelofKCSwascomparedtotheoutputofourvalidatedsimulationmodelofHodgsonandWang’sconceptualmodelofKCS.TheresultswereidenticalwhentheassumptionthatabacklogcouldnottemporarilyincreasethenumberofkanbansavailabletothefinalstagewasincorporatedintooursimulationmodelofHodgsonandWang’sconceptualmodelofKCS.(2)InGeraghtyandHeavey[15]weshowedmathematicallythattheoptimalHIHSidentifiedby[20,21]isequivalenttohybridKanban-CONWIP,undercertainconditions.WealsodemonstratedthisequivalencebycomparingtheoutputsofoursimulationmodelofhybridKanban-CONWIPwithourvalidatedsimu-lationmodelofHodgsonandWang’soptimalHIHS.(3)InthispaperwehavedemonstratedmathematicallythatEKCSandHodgsonandWang’s‘Pure-Push’PCSwillgiveequivalentresultsiftheoccurrenceofdemandeventsarecommunicatedatthesametimeinbothPCSandothercon-ditionsdetailedearlieraremet.Resultsarepresentedin[16]thatdemonstratethatoursimulationmodelofEKCSachievesthesameresultsasourvalidatedsimulationmodelofHodgsonandWang’s‘Pure-Push’PCSwhenallconditionsforequivalencearemet.(4)ItwasnotpossibletovalidateourmodelsofCONWIPandBSCS.However,thesePCSaresimplificationsofhybridKanban-CONWIPandEKCS,respec-tively,inwhichkanbansarenotdistributed.Therefore,sincewehavebeenabletovalidateoursimulationmodelsofhybridKanban-CONWIPandEKCS,weassumethatoursimulationmodelsofCONWIPandBSCSarevalid.Forthepurposesoftheexperimentalprocessthesimulationrun-timeoverwhichstatisticswerecollectedwas10,000periodswithawarm-upperiodof1,000peri-ods.Tenreplicationsofeachsimulationwereconducted.ThePCSwerecomparedbyconductingapartialenumerationofthesolutionspacefortheircontrolparame-ters.AdetaileddescriptionofthesolutionspacesevaluatedforeachPCSforeachdemanddistributionisdescribedbelow.Thecomparisonofthestrategieswasachievedbyconductingapartialenumer-ationofthecontrolparametersofthefivePCSexamined.TheminimumvaluesfortheKanbanallocationsfortheKCS,EKCSandhybridKanban-CONWIPmodelswereeightforeachstage.Thiswasselectedsincepreliminaryworkindicatedthatvaluesbelowthislevelsignificantlydegradedthesolution.ForinstancesettingtheKanbanlevelsoftheinputstages(j=1,2)equalto7alwaysresultedinaService 324J.GeraghtyandC.HeaveyLevelof0regardlessofthenumberofKanbansallocatedtotheremainingstagesforbothKCSandhybridKanban-CONWIP.CONWIPCap,CC,valuesbelow16resultedinservicelevelsoflessthan10%forbothCONWIPandhybridKanban-CONWIP.Aminimumoffourpartswasselectedfortheinitialisationstocks(Sj)forbothBSCSandEKCS.Thisvaluewasselectedbecause(i)thenatureofthecontrolstrategiesimpliesthattheinitialisationstocksmustbegreaterthanzeroand(ii)meandemandwas3.5anditwasdesirabletoinitialisethebufferssuchthattheycouldsatisfythemeandemand.ForKCSandhybridKanban-CONWIPthemaximumvalueforthenumberofKanbansconsideredfordistributiontoworkstations1to3was16eachwithamaximumof20toworkstation4.ForKCSworkstation5hadanupperboundof20Kanbans.ForCONWIPandhybridKanban-CONWIPthemaximumvalueconsideredforCCwas50.ForBSCStheupperboundsfortheinitialisationstocksofworkstations1to5were12,12,12,16and50respectively.ForeachsimulationrunthemodelsoftheindividualPCSwereinitialisedwithinventoryasdescribedbyTable3.FortheEKCSmodelitwouldhavebeenimpossibletoconductapartialenu-merationofthesolutionspaceforallparameters(i.e.allpossiblecombinationsofKanbanandinitialisationstocklevels).Theamountofcomputertimerequiredwouldnothavebeenfeasible.Forinstance,supposeapartialenumerationofthesolutionspacefortheEKCSmodelwereconductedwithminimumvaluesasde-scribedaboveandmaximumvaluesforKjandSjequaltothemaximumvaluesforKifortheKCSmodel.Over90,000,000hoursofCPUtimewouldhavebeenrequiredtoconductthisexperiment(basedon5.3secondsperreplicationand10replicationsperiterationona1.8GHzIntelPentium4DellPCwith256MbofRAM).Therefore,inordertominimisethetimerequirementsamethodhadtobefoundtopredeterminetheKanbandistributionortheinitialisationstocklevels.DalleryandLiberopoulos[10]notedthattheproductioncapacityoftheEKCSonlydependsonKjandnotonSj;i=1,...,J.TheysuggestedthatareasonabledesignprocedurefortheEKCScouldbetofirstdesignparametersKjtoobtainadesirableproductioncapacitylevel,andsubsequentlydesignparametersSjtoobtainadesirablecustomersatisfactionlevel.ItseemedthatareasonabledesignfortheKanbanallocationfortheEKCSmodelmightbetheallocationthatachieved100%ServiceLevelforthehybridKanban-CONWIPmodel.Therefore,itisnotclaimedthatEKCSwascomparedforoptimalitywiththeotherPCS.JustthatareasonabledesignforEKCSwascompared.UnderhybridKanban-CONWIPKanbansarenotallocatedtothefinalstagesincethemaximumamountofinventorythatcanbeintheoutputbufferofthefinalstageinanyperiodisCC.Therefore,ifitisdesiredtodesigntheKanbanallocationfortheEKCSsuchthatithasatmosttheequivalentamountofinventoryasahybridKanban-CONWIPlinethenthenumberofKanbanstoallocatetothefinalstagefortheEKCSmodelwouldbethemaximuminventoryfromhybridKanban-CONWIPminustheminimuminventorytobeallocatedtotheinternalbuffersin Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies325theEKCSdesign,i.e.CC−121.TheKanbanallocationforEKCSthereforewas10,10,15,9and13forworkstations1to5respectively.Thesevalueswerealsothesetasthemaximuminitialisationstocklevels,Smax,foreachworkstation.jTable3.InitialisationlevelsforeachBufferundereachPCSStrategyI1I2I3I4I5KCSK1K2K3K4K5CONWIP0000CCHybridKanban-CONWIP0000CCBSCSS1S2S3S4S5EKCSS1S2S3S4S55.6ExperimentalresultsOfthefivePCSexamined,KCSwasconsistentlytheworstperformerintermsofaddressingtheServiceLevelvs.WIPtrade-off.Table4illustratesthisbygivingthepercentagereductioninminimumWIPrequiredbyeachPCStoachieveatargetedServiceLevelwhencomparedtoKCS.hybridKanban-CONWIPwasconsistentlythebestperformer,requiring9%to15.5%lessWIPthanKCStoachieveatargetedServiceLevel.Apaired-ttestdemonstratedthattheperformanceofhybridKanban-CONWIPwasstatisticallysignificantlybetterthanCONWIPatboth95%and99%significancelevels.BSCSandEKCSrequiredonaverage8%to13.5%lessWIPthanKCStoachieveatargetedServiceLevel.Apaired-ttestdemonstratedthattheperformanceofEKCSwasstatisticallysignificantlybetterthanBSCSatboth95%and99%significancelevelsforalltargetedservicelevelswiththeexceptionofatargetedServiceLevelof96%.Tables5and6illustratetheinventoryplacementpatternsachievedbyeachPCSfortargetedservicelevelsof100%and99.9%respectively.KCSrequiredmoresemi-finishedinventorythantheotherfourPCSandasimilaramountofend-iteminventoryasCONWIPandhybridKanban-CONWIPtoachieveatargetedServiceLevel.WhilethedifferencesbetweentheotherfourPCSintermsoftotalWIPwassmall,theinventoryplacementpatternsofthePCSweredifferent.CONWIPandhybridKanban-CONWIPtendedtomaintainlessWIPinsemi-finishedinventoryandmoreintheend-itembufferthanBSCSandEKCS.6DiscussionInthelasttwodecadesresearchershavefollowedtwoapproachestodevelopingpro-ductioncontrolstrategiestoovercomethedisadvantagesofKCSinnon-repetitive1CCisacomponentbasedinventorycap,thereforetheinternalinventoryforacomponentinthisparallel/serialmodelinperiodnisI1(n)+I3(n)+I4(n)orI2(n)+I3(n)+I4(n)minminminminminminandthevalue12isarrivedatasS1+S3+S4orS2+S3+S4 326J.GeraghtyandC.HeaveyTable4.PercentagereductionoverKCSinminimuminventoryrequiredbyeachPCStoachieveatargetedservicelevelSL≥100%99.9%99%98%97%96%CONWIP8.9%14.0%14.3%15.1%14.2%10.2%HybridKanban-CONWIP9.0%14.5%14.8%15.5%14.7%13.0%BSCS7.8%12.8%12.8%13.3%12.5%12.8%EKCS7.9%12.9%12.9%13.5%12.6%8.5%Table5.InventoryplacementsunderoptimalparametersforeachPCSfortargetedservicelevelof100%KCSCONWIPHybridKanban-CONWIPBSCSEKCSI14.31183.94923.93654.26444.2500I24.31133.94733.93464.26254.2480I34.89173.77853.82834.06944.1127I45.08083.77943.74996.67724.0277I59.08339.75959.73416.25558.8573Internal18.595615.454415.449319.273616.6384Total27.679025.213925.183325.529125.4958Table6.InventoryplacementsunderoptimalparametersforeachPCSfortargetedservicelevelof99.9%KCSCONWIPHybridKanban-CONWIPBSCSEKCSI14.31183.94863.90754.26444.2500I24.31133.94693.90534.26254.2480I34.89173.77793.77084.06944.1127I45.08083.77943.77464.05584.0277I56.08435.76085.73134.87844.8593Internal18.595615.452815.358116.652216.6384Total24.679921.213621.089521.530521.4977manufacturingenvironments.Thefirstapproachhasbeentodevelopneworcom-bineexistingPull-typePCSwhilethesecondapproachhasbeentodevelophybridPCSbasedoncombiningelementsofPushandPullPCS.Inapreviouspaper[15]itwasdemonstratedthattheoptimalHIHSselectedbyHodgsonandWang[20,21]whereinitialstagesemploypushcontrolandallotherstagesemploypullcontrol,isequivalenttoaPull-typePCS,namelyhybridKanban-CONWIP[3,2].Hereitwasshownthatthe‘Pure-Push’PCSmodelledbyHodgsonandWang[20,21]would Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies327bemoreaccuratelydescribedasaverticalintegrationproductioncontrolstrategy,sinceeachproductionstageforecastsitsproductionrequirementsandutiliseskan-banstocontrolshop-floorproductionforeachproductionperiod.However,itwasalsoshownthatbyensuringthatdemandinformationinthe‘Pure-Push’PCSiscommunicatedtheinstantitoccursratherthanbeendelayedforoneperiodthe‘Pure-Push’PCSisequivalenttoEKCS.UsingthemodelpresentedinHodgsonandWang[21]acomparativestudyofKCS,CONWIP,hybridKanban-CONWIP,BSCSandEKCSwascarriedout.ThecriterionusedinthestudywastheServiceLevelvs.WIPtrade-off.KCSperformedworstintermsofaddressingthistrade-offinthatKCSconsistentlyrequiredmoreinventorythantheotherfourPCStoachieveatargetedServiceLevel.ThereasonforthepoorperformanceofKCSisduetotheinformationdelaythatoccursinaKCSline.Whenademandeventoccursthisinformationisonlycommunicatedtothefinalproductionstagetoauthoriseproductionofreplacementparts.Thelongerthelineandmoredelaysthatoccurinthesystemsuchasdowntimeduetomachineunreliability,thelongerthedelayincommunicatingthedemandinformationtoinitialstages.Therefore,thereleaserateisnoteasilyadjustedtomatchchangesinthedemandrate.CONWIPandhybridKanban-CONWIPemploylimitsoninventoryinthesystemandoncethislimithasbeenreachedonlytheoccurrenceofademandeventcanauthorisethereleaseofapartintothesystem.Thereleaserateisthereforepacedtomatchthedemandrate.BSCSandEKCSusedemandcardsthatareinstantlycommunicatedtoeachproductionstagetopacetheproductionrateofthelinetothedemandrate.Forthesystemmodelled,thedemandratewas3.5partsperproductionperiod,whichwas87.5%oftheisolatedproductionrateofastage(4partsperperiod).Thecoefficientofvariationofthedemanddistributionwasapproximately14%andthecoefficientofvariationoftheproductionrateofastageinisolationwasapproximately16%.Thisthereforeisasystemwithlowvariabilityandalight-to-mediumdemandload.ForthissystemtherewasminimaldifferencebetweentheperformancesofthevariousPCSexamined,withtheexceptionofKCS.Astatisticalanalysisofthedatahoweverrevealedthesedifferencestobestatisticallysignificant.Forthissystem,hybridKanban-CONWIPperformedthebestinaddressingtheServiceLevelvs.WIPtrade-off.EKCStendedtomaintainsimilaroverallinventorylevelsasCONWIP,hybridKanban-CONWIPandBSCS.However,EKCStendedtomaintainmoreofthisinventoryinternallyintheline,i.e.inasemi-finishedstate,thanCONWIPandhybridKanban-CONWIP.Thismaybeeitheranadvantageordisadvantageandwilldependonthemanufacturingobjectivesoftheorganisation.ThestrategyoftheorganisationmightbetomaintainasmuchaspossibleoftheWIPinafinishedstateandtherebyprovidetheorganisationwithgreaterflexibilitytorespondtounexpecteddemands.IfthisisthestrategyoftheorganisationthenhybridKanban-CONWIPisthepreferablePCSforthemanufacturingsystem.OntheotherhandthestrategyoftheorganisationmightbetomaintainWIPinsemi-finishedstatesclosetothecompletionstateallowingtheorganisationtorespondtochangesincustomerdemandsbyreassigningWIPtoothercustomersoralteringtheWIPto 328J.GeraghtyandC.Heaveymeetnewcustomerspecifications.IfthisisthestrategyoftheorganisationthenEKCSisthepreferablePCSforthemanufacturingsystem.Finally,ashasbeenstated,theexperimentpresentedheretoexaminethecom-parativeperformanceofvariousPull-typePCSwasforamanufacturingsystemwithmoderatevariabilityandalight-to-mediumdemandload.FutureplannedworkistoexaminethecomparativeperformanceofthefivePCSfurtherbyexamininghowthePCSrespondasthecoefficientofvariationofthedemanddistributionincreasesandasthemeanofthedemanddistributionapproachesthemaximumcapacityofthemanufacturingsystem.References1.BerkleyBJ(1992)Areviewofthekanbanproductioncontrolresearchliterature.ProductionandOperationsManagement1(4):393–4112.BonvikAM,CouchCE,GershwinSB(1997)Acomparisonofproduction-linecontrolmechanisms.InternationalJournalofProductionResearch35(3):789–8043.BonvikAM,GershwinSB(1996)Beyondkanban–creatingandanalyzingleanshopfloorcontrolpolicies.In:ProceedingsofManufacturingandServiceOperationsMan-agementConference,DartmouthCollege,TheAmosTuckSchool,Hanover,NH,USA4.BuzacottJA(1989)QueueingmodelsofkanbanandMRPcontrolledproductionsys-tems.EngineeringCostandProductionEconomics17:3–205.ChangT-M,YihY(1994a)Determiningthenumberofkanbansandlotsizesinagenerickanbansystem:Asimulatedannealingapproach.InternationalJournalofProductionResearch32(8):1991–20046.ChangT-M,YihY(1994b)Generickanbansystemsfordynamicenvironments.Inter-nationalJournalofProductionResearch32(4):889–9027.ClarkAJ,ScarfH(1960)Optimalpoliciesforthemulti-echeloninventoryproblem.ManagementScience6(4):475–4908.CochranJK,KimS-S(1998)Optimumjunctionpointlocationandinventorylevelsinserialhybridpush/pullproductionsystems.InternationalJournalofProductionResearch36(4):1141–11559.DalleryY,LiberopoulosG(1995)Anewkanban-typepullcontrolmechanismformulti-stagemanufacturingsystems.In:Proceedingsofthe3rdEuropeanControlConferencevol4(2),pp3543–354810.DalleryY,LiberopoulosG(2000)Extendedkanbancontrolsystem:combiningkanbanandbasestock.IIETransactions32(4):369–38611.DeleersnyderJL,HodgsonTJ,KingRE,O’GradyPJ,SavvaA(1992)IntegratingkanbantypepullsystemsandMRPtypepushsystems:InsightsfromaMarkovianmodel.IEETransactions24(3):43–5612.GauryEGA,KleijnenJPC(2003)Short-termrobustnessofproductionmanagementsystems:Acasestudy.EuropeanJournalofOperationalResearch148:452–46513.GauryEGA,KleijnenJPC,PierrevalH(2001)Amethodologytocustomizepullcontrolsystems.JournaloftheOperationalResearchSociety52(7):789–79914.GauryEGA,PierrevalH,KleijnenJPC(2000)Anevolutionaryapproachtoselectapullsystemamongkanban,CONWIPandhybrid.JournalofIntelligentManufacturing11(2):157–16715.GeraghtyJ,HeaveyC(2004)AcomparisonofhybridPush/PullandCONWIP/Pullproductioninventorycontrolpolicies.InternationalJournalofProductionEconomics91(1):75–90 Areviewandcomparisonofhybridandpull-typeproductioncontrolstrategies32916.GeraghtyJE(2003)Aninvestigationofpull-typeproductioncontrolmechanismsforleanmanufacturingenvironmentsinthepresenceofvariabilityinthedemandprocess.PhD,UniversityOfLimerick,Ireland17.HallRW(1983)Zeroinventories.DowJones-Irwin,Homewood,IL18.HirakawaY(1996)Performanceofamultistagehybridpush/pullproductioncontrolsystem.InternationalJournalofProductionResearch44:129–13519.HirakawaY,HoshinoK,KatayamaH(1992)Ahybridpush/pullproductioncontrolsystemformultistagemanufacturingprocesses.InternationalJournalofOperationsandProductionManagement12(4):69–8120.HodgsonTJ,WangD(1991a)Optimalhybridpush/pullcontrolstrategiesforaparallelmulti-stagesystem:PartI.InternationalJournalofProductionResearch29(6):1279–128721.HodgsonTJ,WangD(1991b)Optimalhybridpush/pullcontrolstrategiesforaparallelmulti-stagesystem:PartII.InternationalJournalofProductionResearch29(7):1453–146022.KimemiaJ,GershwinSB(1983)Analgorithmforthecomputercontrolofaflexiblemanufacturingsystem.IIETransactions15(4):353–36223.KrajewskiLJ,KingBE,RitzmanLP,WongDS(1987)Kanban,MRP,andshapingthemanufacturingenvironment.ManagementScience33(1):39–5724.LeeLC(1989)Acomparativestudyofthepushandpullproductionssystems.Inter-nationalJournalofOperationsandProductionManagement9(4):5–1825.LiberopoulosG,DalleryY(2000)Aunifiedframeworkforpullcontrolmechanismsinmulti-stagemanufacturingsystems.AnnalsofOperationsResearch93:325–35526.MuckstadtJA,TayurSR(1995)Acomparisonofalternativekanbancontrolmecha-nisms.IIETransactions27:140–16127.OrthMJ,CoskunogluO(1995)Comparisonofpush/pullhybridmanufacturingcontrolstrategies.In:ProceedingsofIndustrialEngineeringResearch,Nashville,TN,pp881–890.IIE,Norcross,GA28.PandeyPC,KhokhajaikiatP(1996)Performancemodellingofmultistageproductionsystemsoperatingunderhybridpush/pullcontrol.InternationalJournalofProductionEconomics43(1):17–2829.SpearmanML(1988)Ananalyticalcongestionmodelforclosedproductionsystems.TechnicalReport88-23,Dept.ofIndustrialEngineeringandManagementSciences,NorthwesternUniversity,Evanston,IL30.SpearmanML,WoodruffDL,HoppWJ(1990)CONWIP:Apullalternativetokanban.InternationalJournalofProductionResearch28(5):879–89431.SpearmanML,ZazanisMA(1992)Pushandpullproductionsystems:Issuesandcomparisions.OperationsResearch40(3):521–53232.TakahashiK,HirakiS,SoshirodaM(1991)Integrationofpushtypeandpulltypepro-ductionorderingsystems.In:Proceedingsof1stChina-JapanInternationalSymposiumonIndustrialManagement,Beijing,China,pp396–40133.TakahashiK,HirakiS,SoshirodaM(1994)Push-pullintegrationinproductionorderingsystems.InternationalJournalofProductionEconomics33(1–3):15534.TakahashiK,SoshirodaM(1996)Comparingintegrationstrategiesinproductionorderingsystems.InternationalJournalofProductionEconomics44(1–2):83–8935.WangD,XuCC(1997)Hybridpush/pullproductioncontrolstrategysimulationanditsapplications.ProductionPlanningandControl8(2):142–15136.ZipkinP(1989)Akanban-likeproductioncontrolsystem:analysisofsimplemodels.Workingpaper89-1,GraduateSchoolofBusiness,ColumbiaUniversity,NewYork SectionIV:StochasticProductionPlanningandAssembly PlanningorderreleasesforanassemblysystemwithrandomoperationtimesSvenAxsater¨DepartmentofIndustrialManagementandLogistics,LundUniversity,Sweden(e-mail:Sven.Axsater@iml.lth.se)Abstract.Amulti-stageassemblynetworkisconsidered.Anumberofenditemsshouldbedeliveredatacertaintime.Otherwiseadelaycostisincurred.Enditemsandcomponentsthataredeliveredbeforetheyareneededwillcauseholdingcosts.Alloperationtimesareindependentstochasticvariables.Theobjectiveistochoosestartingtimesfordifferentoperationsinordertominimizethetotalexpectedcosts.Wesuggestanapproximatedecompositiontechniquethatisbasedonrepeatedapplicationofthesolutionofasimplersingle-stageproblem.Theperformanceofourapproximatetechniqueiscomparedtoexactresultsinanumericalstudy.Keywords:Multi-stageproduction/inventorysystems–Decomposition1IntroductionInthispaperweconsidertheplanningofinterrelatedassemblyoperationswithindependentstochasticoperationtimes.Oneormoreenditemsshouldaccordingtoagivencontractbedeliveredatacertaintime.Thedeliverycannottakeplaceuntilallenditemsareready.Incasethegivendeliveryrequirementcannotbesatisfiedthereisadelaycostthatisproportionaltothelengthofthedelay.Iftheenditemsarereadyatdifferenttimesthedelaycostisbasedonthetimewhenallitemsareready.Furthermore,ifenditemsarereadyearlierthanthedeliverytime,holdingcostsareincurred.Afinalassemblyoperationcannormallynotstartunlessasetofprecedingoperations,alsowithstochasticdurations,hasbeencompleted.Delaysforsuchprecedingoperationswillnotresultinanydirectdelaycostsbutmayindirectlyresultinadditionaldelaycostsfortheenditems.Ifsuchprecedingoperationsarefinishedbeforethecorrespondingfinaloperationsstart,holdingcostsareincurred.Theoperationsprecedingthefinaloperationscan,inturn,haveprecedingoperationsandsoon.Ourpurposeistofindstartingtimesforthedifferent 334S.Axsater¨operationsthatminimizethetotalexpectedcosts,i.e.,inotherwordswearelookingforoptimalsafetytimes.Theconsideredproblemwithseveralstagesis,ingeneral,toodifficulttobesolvedexactly.Wethereforesuggestaheuristicthatisbasedonsuccessiveappli-cationsofthesolutionofasimplerone-stageproblem.Differentversionsofsuchsimplerproblemswithoneortwostageshavebeenstudiedinseveralpapersbefore.ExamplesofthisresearchareYano(1987a),Kumar(1989),HoppandSpearman(1993),Chuetal.(1993),andShore(1995).Songetal.(2000)considerstochasticoperationtimesaswellasstochasticdemand.InarecentoverviewSongandZipkin(2003)consideramoregeneralclassofstochasticassemblyproblems.Therearealsoanumberofpapersdealingwithsimilarproblemsforothertypesofsystems.Gongetal.(1994)consideraserialsystemandshowthattheproblemofchoos-ingoptimallead-timesisequivalenttothewell-knownmodelinClarkandScarf(1960).Yano(1987b)dealsalsowithaserialsystem,whileYano(1987c)consid-ersadistribution-typesystem.Examplesofpapersanalyzingrelatedproblemsforsingle-stagesystemsareBuzacottandShanthikumar(1994),HariharanandZipkin(1995),Chen(2001),andKaraesmenetal.(2002).Theoutlineofthispaperisasfollows.Wefirstgiveadetailedproblemfor-mulationinSection2.InSection3weconsiderthesimplersingle-stagesystemthatisthebasisforourheuristic.TheapproximateprocedureisthendescribedinSection4.InSection5weapplyourtechniquetotwosetsofsampleproblems,andfinallywegivesomeconcludingremarksinSection6.2ProblemformulationWeconsideranassemblynetwork(seeFig.1).Thearcsrepresenttheoperations.Thenodewhereoperationistartsisdenotednodei.Theoperationtimesarein-dependentrandomvariableswithcontinuousdistributions.Ourpurposeistoplanproductionsothattheexpectedholdinganddelaycostsareminimized.Letusintroducethefollowingnotation:ti=startingtimeforoperationi,τi=stochasticdurationtimeofoperationi,fi(x)=densityforτi,Fi(x)=cumulativedistributionfunctionforτi,(ItisassumedthatFi(x)<1foranyfinitex.)td=requesteddeliverytimefortheassembly,ei=positiveechelonholdingcostassociatedwithoperationi,h=sumofallechelonholdingcosts,i.e.,holdingcostforallenditems,b=positivedelaycostpertimeunit.Thedelaycostsatnode0areobtainedasb(t+τ−t)+,wherex+=11dmax(x,0).Ifthereareseveralenditems,thedelaycostisbasedonthemaximumdelay.Therearenodelaycostsassociatedwithothernodes.However,delaysatothernodesmayaffectthedelayatnode0.Considernodei,whichisthestartingpointforoperationi.Notefirstthatwemusthaveti≥max(tj+τj,tk+τk).Afterstartingoperationi,theechelonholdingcosteiisincurreduntilthefinaldelivery, Planningorderreleasesforanassemblysystemwithrandomoperationtimes335lmjtiniτik10τ1t1Fig.1.Assemblynetworkwhichwilltakeplaceattherequesteddeliverytimetd,orlaterincaseofadelay.However,wedisregardtheholdingcostsduringtheoperations,becausetheyarenotaffectedbywhentheoperationsarecarriedout.Itisassumedthatrawmaterialcanbeobtainedinstantaneouslyfromanoutsidesupplier.Thismeansthatinitialoperationslikel,m,andninFigure1canstartatanytime.3Single-stagesystemWeshallderiveanapproximatesolutionbysuccessivelyapplyingtheexactsolutionforasingle-stagesystem.ConsiderthereforefirstthesysteminFigure2.1.i.0..NFig.2.Single-stagesystemWecanexpressthedelaydasd=max(t+τ−t)+.(1)iid1≤i≤N 336S.Axsater¨TheaveragecostsCcanthenbeexpressedasNC=eiE(td+d−ti−τi)+bE(d)i=1NN=ei(td−ti−E(τi))+ei+bE(d).(2)i=1i=1WeshalloptimizeCwithrespecttothestartingtimesti(i=1,2,...,N).ThisproblemwassolvedbyYano(1987a),whoalsoshowedthatCisaconvexfunctionofthestartingtimesforN=2.ItiseasytoseethatCisconvexinthestartingtimesalsoforlargervaluesofN.Considertheright-handsideof(2).ThefirsttermislinearsoweonlyneedtoshowthatE(d)isconvex.Becausetheoperationtimesareindependentitisenoughtodemonstratethatdin(1)isconvexinthestartingtimesforgivenoperationtimes.Notethatx+isconvex.Let0≤α≤1,andt
andt

betwoset-upsofstartingtimes.Wehavemax(αt
+(1−α)t

+τ−t)+iiid1≤i≤N≤maxα(t
+τ−t)++(1−α)(t

+τ−t)+iidiid1≤i≤N≤αmax(t
+τ−t)++(1−α)max(t

+τ−t)+.(3)iidiid1≤i≤N1≤i≤NFurthermore,itisevidentthatC→∞asti→∞orti→−∞.ItfollowsthatChasauniqueminimumforfinitevaluesofthestartingtimesti.Seealsoe.g.,HoppandSpearman(1993)andSongetal.(2000).LetG(x)bethecumulativedistributionfunctionofthedelayd.Wehave-NG(x)=P(d≤x)=Fi(x+td−ti).(4)i=1WecannowexpressCas$NN∞-NC=ei(td−ti−E(τi))+ei+b1−Fi(x+td−ti)dx.(5)i=1i=10i=1ConsequentlywegetthepartialderivativeofCwithrespecttotias$∂CN∞-=−ei+ei+bfi(x+td−ti)Fj(x+td−tj)dx.(6)∂ti0i=1j=i/Whenevaluating∂C/∂tinumericallyweneedtocarryoutanumericalintegration.Assumenowfirstthattherearenoconstraintsonthevariablesti.Wewillthenobtaintheminimumastheuniquesolutionof∂C/∂ti=0.NotethatforN=1theproblemdegeneratestothefamiliarNewsboyproblem,andweobtaintheoptimumfromtheconditionbF1(td−t1)=.(7)b+e1Considerthegeneralcaseagain.Forgivenvaluesoftheothertjitiseasytofindthetigiving∂C/∂ti=0.Duetotheconvexity∂C/∂tiisincreasingsowecan Planningorderreleasesforanassemblysystemwithrandomoperationtimes337applyasimplesearchprocedure.Ifwestartwithsomeinitialvaluesofthestartingtimes,e.g.,ti=td−E(τi),andcarryoutoptimizationswithrespecttoonetiatatimethecostsarenonincreasingandwewillultimatelyreachtheoptimalsolutionduetotheconvexity.Assumenowthatwehavelowerboundsforthestartingtimesti.ti0=lowerboundforti.Wecanthenobtaintheoptimalsolutioninalmostthesameway.Westartwithfeasiblevaluesofti,e.g.,ti=max(td−E(τi),ti0).Thenweoptimizewithrespecttoonetiatatime.Consideralocaloptimizationofti.Wefirstcheckwhetherti=ti0isoptimal.Thisisthecaseif∂C/∂ti>0forti=ti0.Otherwisewedeterminethetigiving∂C/∂ti=0asbefore.Againthecostsarenonincreasingandwewillultimatelyreachtheoptimalsolutionduetotheconvexity.4Approximateprocedureforamulti-stagesystemConsidernowageneralassemblysystem.Weshalldescribeourapproximateplan-ningprocedure.LetP(i)=setofoperationsthatareimmediatepredecessorsofnodei.Considerfirsttheimmediatepredecessorsofthefinalnode,i.e.,P(0).InFigure1wehaveasinglepredecessor,butinamoregeneralcasewemayhavemultiplepredecessorslikeinFigure2.AssumethatthereareNoperationsinP(0).Assumealsothattheoperationsthatmustprecedetheseoperationsarefinished,i.e.,theNoperationsinP(0)arereadytostart.Considerfirstalloperations,whichdonotbelongtoP(0).TheholdingcostsassociatedwiththeseoperationsbeforetimetdcannotbeaffectedbythestartingtimesfortheoperationsinP(0).Sowedisregardtheseholdingcostsinthefirststep.Therearealsoholdingcostsassociatedwiththeseoperationsduringapossibledelay,whichshouldbeincludedbecausetheyareaffectedbythestartingtimesfortheoperationsinP(0).FortheoperationsinP(0)boththeholdingcostsfromtheirrespectivestartingtimestotd,andduringapossibledelayareaffectedbythestartingtimesfortheoperationsinP(0).Therearealsodelaycostsassociatedwithadelay.ThisleadstothefollowingcostexpressionfortheoperationsinP(0).⎛⎞N$∞-NC(0)=ej(td−tj−E(τj))+(h+b)⎝1−Fj(x+td−tj)⎠dx.(8)j=10j=1Notethattheonlydifferencecomparedto(5)isthathincludesalsoholdingcostsduringapossibledelayforoperationsprecedingP(0).UsingthealgorithminSection3(withoutlowerboundsforthestartingtimes)weoptimize(8)inourfirststepandgetthecorrespondingstartingtimest∗fortheoperationsinP(0).iAssumethenthatiisoneoftheoperationsinP(0),andconsideritsimmediatepredecessors,i.e.,theoperationsinP(i).Weshallnowconsiderthesinglestagesystemconsistingoftheseoperationsandtherebyinterprett∗asarequesteddeliveryitime.ForanoperationinP(i)thestartingtimewillaffecttheholdingcostsbefore 338S.Axsater¨t∗.Letusalsoconsideradelaycostbˆthatreplacesh+bin(8).TheresultingiproblemistominimizeN$∞-NC(i)=e(t∗−t−E(τ))+bˆ(1−F(x+t∗−t))dx.(9)jijjjijj=10j=1Notethatalthoughwe,forsimplicity,areusingasimilarnotationin(8)and(9),theconsideredoperationsarenotthesame.SothenumberofoperationsN,theholdingcostsej,andthedistributionfunctionsFjarenormallydifferent.Itremainstodeterminebˆ.Ifthereisadelay,thisdelaywillaffectthestartingtimesoftheoperationsinP(0).ThiswillincreasethecostsC(0)in(8).ConsidersomegivenstartingtimesfortheoperationsinP(i)andletthecorrespondingstochasticdelayrelativetot∗beδ.Areasonableapproximatedelaycostisi89b=EˆdC(0)(t∗+δ,t∗+δ,...,t∗+δ)δ>012Ndδ89dC(0)∗∗∗=Eδ(t1+δ,t2+δ,...,tN+δ)/Pr(δ>0).(10)dδThesecondequalityin(10)followsbecausedC(0)/dδ=0forδ=0duetotheoptimalityoft∗.BecauseofourassumptionconcerningthedistributionsoftheioperationtimesweknowthatPr(δ>0)>0.In(10)itisimplicitlyassumedthatalloperationsinP(0)arestartedδtimeunitslatercomparedtotheoptimalsolution,i.e.,notonlyoperationi.Thisisareasonableassumptionandwillalsosimplifythecomputations.Usingthatd-Nd-NFj(x+td−tj−δ)=−Fj(x+td−tj−δ),(11)dδdxj=1j=1wegetfrom(10)dC(0)∗∗∗(t1+δ,t2+δ,...,tN+δ)dδ⎛⎞N$∞d-N=−ej+(h+b)⎝Fj(x+td−tj−δ)⎠dx0dxj=1j=1⎛⎞N-N=−ej+(h+b)⎝1−Fj(td−tj−δ)⎠(12)j=1j=1soitisrelativelyeasytoevaluatebˆaccordingto(10).Recallthatwegetthedistributionofthedelayfrom(4).Becausebˆin(10)dependsonthestartingtimesoftheoperationsinP(i),itisunknown.Itis,however,stilleasytodetermineanoptimalsolutioncorrespondingtothedelaycost(10).Notefirstthatthebˆfrom(10)isboundedfrombelowby0Nandfromabovebyh+b−j=1ej.Theupperboundiseasytoseefrom(12).Itisalsoclearthattheupperboundwillleadtoafiniteoptimalsolutionof(9).(Recallthathisthesumofallechelonholdingcosts.)Assumenowthatwestartwitha Planningorderreleasesforanassemblysystemwithrandomoperationtimes339certainbˆinfromtheconsideredinterval.Therearenowtwopossibilities.Ifˆbinissufficientlysmallthereisnofiniteoptimumof(9).Theresultingbˆoutfrom(10)willapproachtheupperbound.If,ontheotherhand,thereisafinitesolutionweknowthatbˆoutisbetweenthelowerandupperbounds.Clearly,bˆoutisacontinuousfunctionofbˆin.Consequently,itfollowsfromBrouwer’sfixedpointtheoremthatthereexistsafixedpointˆbout=bˆin,i.e.,asolutioncorrespondingtothedelaycost(10).Itiseasytofindsuchafixedpointbyaone-dimensionalsearch.Remark.Normallybˆoutisadecreasingfunctionofbˆin.Inthatcaseitisveryeasytofindtheuniquefixedpoint.WecanthenhandlethepredecessorsoftheoperationsinP(i)inthesameway,etc.LetjbeoneoftheoperationsinP(i).WhendealingwithP(j)welett∗bethejrequesteddeliverytimeforthesingle-stagesystem.In(10)weareusingC(i)insteadofC(0).Adifferencehereisthattheupperboundforbˆwillnotnecessarilyleadtoafinitesolution.Thismeansinthatcasethatthedelayδwillapproachinfinityand,asaconsequence,alsotheoperationsinP(0)willbedelayed.Consequentlyitisreasonabletousethecostsintheprecedingstep,i.e.,inthiscaseC(0)insteadofC(i).Ifnecessarywecangoonestepfurther,andsoon.ThiswillalwaysworkbecauseC(0)willprovideanupperboundleadingtoafiniteoptimalsolution.Wewillendupwithstartingtimest∗foralloperationsanddelaycostsforallisingle-stagesystems.Whenimplementingthesolutionwewillsticktotheobtainedstartingtimesaslongastheyarepossibletofollow.However,delaysmayenforcechanges.Consider,forexample,operationjinFigure1.Assumethatoperationsl,m,andnarefinishedatsometimet>t∗.Wethenderiveanewsolutionforj0jtheoperationsinP(i).(Thedelaycostatnodeiisnotchanged.)Inthesolutionweapplytheconstrainttj≥tj0.Startingtimesthathavealreadybeenimplementedareregardedasgiven.Ifoperationkhasnotyetstarted,itsstartingtimemayincreasebutcannotdecrease.Toseethisconsider(6)andnotethat∂C/∂tiisnonincreasingifsomeothertisincreasing,i.e.,t∗isnondecreasing.jiLetussummarizeourapproximateprocedure:1.OptimizeC(0)asgivenby(8).LetKbethenumberofstages.Setk=−1.2.Setk=k+1.3.Foralloperationsiwithksucceedingoperations,optimize(9)fortheoperationsofP(i)undertheconstraint(10)withthecostfunctionC(i).Ifk>0itmayoccurthatnofiniteoptimumexists.Ifthisisthecaseusethecostfunctionforthesuccessorofi.Ifnecessarygotothesuccessorofthesuccessor,etc.4.Ifk0itisoptimaltousethesolution12obtainedwiththeconstraintt≥t=t∗+d.Recallthatthisleadstot≥t∗.Let110122c(d)bethecorrespondingexpectedcostsforthesingle-stagenetworkaccordingto(8).Weobtainthetotalcostsas5C(ts)=ei(td−ts−E(τ))+Ed{c(d)}.(13)i=3Thisisthecasebothfortheexactandtheapproximatesolution.Theonlydiffer-encebetweentheexactandapproximatesolutionisthedeterminationofts.IntheapproximateprocedureweusetheproceduredescribedinSection4.Intheoptimalsolutionweoptimize(13)withrespecttots.Allechelonholdingcostsarekeptequal,ei=1,whileweconsideredthreedifferentdelaycostsb=5,25,and50.FurthermoretheexpectedoperationtimeE(τ)=1inallconsideredcases.Twodifferenttypesofdistributionsfortheoperationtimewereconsidered.Foreachdistributionweconsideredthestandarddeviationsσ=0.2,0.5,and1.Bothdistributionsareconstructedasα+(1−α)X,whereαisaconstantbetween0and1andXisastochasticvariablewithitsmeanequalto1.Distribution1isobtainedbylettingXhaveanexponentialdistributionwithmean(andstandarddeviation)equalto1.Distribution2issimilarlyobtained Planningorderreleasesforanassemblysystemwithrandomoperationtimes341Table1.OptimalparametersandcostsforProblemset1Distri-Stand.DelayOptimalpolicyApprox.policyCostin-butiondev.σcostbtsCoststsCostscrease%10.25−2.162.06−2.222.081.010.225−2.503.48−2.583.521.110.250−2.664.21−2.754.261.110.55−2.395.15−2.545.190.810.525−3.258.71−3.458.811.110.550−3.6610.52−3.8810.641.1115−2.7910.30−3.0910.401.01125−4.5017.43−4.9017.621.11150−5.3221.04−5.7521.271.120.25−2.092.07−2.172.101.420.225−2.493.74−2.573.770.820.250−2.694.64−2.784.680.920.55−2.305.20−2.435.230.620.525−3.259.38−3.439.450.720.550−3.7211.56−3.9511.681.0215−2.6110.40−2.8610.470.72125−4.4918.75−4.8718.900.82150−5.4323.12−5.9023.351.0bylettingXbethesquareofanormallydistributedrandomvariablewithmean0andstandarddeviation1.Inotherwords,Xhasaχ2-distributionwithonedegree√offreedom.ThismeansthatXhasmean1andstandarddeviation2.Inbothcasesthemeanisequalto1foranyvalueofα,andbyadjustingαwecanobtaintheconsideredstandarddeviations.TheresultsareshowninTable1.Therelativecostincreasewhenusingourapproximatetechniqueisquitesmallinall18cases.Themaximumerroris1.4%.Therelativeerrorsarefairlyinsensitivetothedistributiontype,thedelaycost,andthestandarddeviationoftheoperationtimes.TheapproximatemethodresultsforProblemset1alwaysinanearlierstartingtimetsfortheinitialoperations,i.e.,theneededsafetytimesareoverestimated.Problemset2Oursecondproblemsetconcernsamorecomplicatednetworkwiththreestages(seeFig.4).Operations1,2,3,5,and7havethesamestochasticoperationtimeτ,whilethetimesofoperations4and6are2τ.Alltimesareindependent.Thetimeτhasthesamedistributionasdistribution1inProblemset1withE(τ)=1andthestandarddeviationsσ=0.2,0.5,and1.Furthermore,asforProblemset1all 342S.Axsater¨425td=0100637Fig.4.NetworkforProblemset2Table2.OptimalparametersandcostsforProblemset2Stand.DelayOptimumbysimulationApproximatepolicyCostin-dev.σcostbts1ts2tmCoststs1ts2tmCostscrease%0.25−4.3−3.1−2.33.68−4.20−3.20−2.263.947.10.225−5.0−3.5−2.56.46−4.57−3.53−2.536.866.20.250−5.3−3.6−2.77.82−4.76−3.70−2.688.6911.10.55−4.6−3.2−2.69.18−4.49−3.49−2.669.867.40.525−5.8−3.9−3.416.16−5.42−4.32−3.3317.578.70.550−7.1−4.4−3.819.12−5.90−4.75−3.7022.2916.615−5.2−3.3−3.317.89−4.98−3.99−3.3120.0412.0125−8.3−4.7−4.631.85−6.84−5.64−4.6734.418.0150−9.9−5.5−5.438.79−7.80−6.49−5.4143.6912.6echelonholdingcostsareequal,ei=1,andweconsiderthedelaycostsb=5,25,and50.Table2showstheresults.TheapproximatepolicyinTable2isobtainedasdescribedinSection4.Thecostsfortheapproximatepolicyareobtainedbysimulation.Thestandarddeviationislessthan0.02.Thetimets1isthestartingtimeofthelongeroperations4and6,andts2isthestartingtimeofoperations5and7.Thetimetmisthestartingtimeforoperations2and3ifbothoperationsarereadytostart.Theoptimumbysimulationisobtainedbyacombinationofsimulationandana-lyticaltechniques.Weomitthedetails.Alltimes,bothstartingtimesandstochastictimeswereforsimplicityroundedtomultiplesof0.1.Thisdoesnotaffectthere-sultsmuch.Moreimportantisthatinthedeterminationoftheoptimalpolicywecarriedoutminimizationsoverseveralsimulatedcosts.Thismeansthatthecostsaresomewhatunderestimated.Byconsideringthecostsforthestartingtimessug-gestedbytheapproximatepolicywecould,however,concludethattheerrorisnotverysignificant.ThecostincreaseoftheapproximatepolicyinTable2maybeoverestimatedby1−2%buthardlymore.WemustconcludethattheapproximationerrorsforProblemset2aremuchlargerthanforProblemset1.Theaveragecostincreaseisnearly10%.NotethattheintermediatetimetmisveryaccuratealsoinTable2.Wenotethatour Planningorderreleasesforanassemblysystemwithrandomoperationtimes343approximationunderestimatestheneededsafetytimesforthelongoperations4and6whileitoverestimatesthesafetytimesfortheshorteroperations5and7.Toexplainthelargedifferenceinerrors,considerfirstthenetworkforProblemset1inFigure3.Thedelayatnode1willdeterminethestartingtimest1andt2becausewecaninitiateoperation2atanytime.Alineardelaycostisthenreasonable.ConsiderthenthenetworkforProblemset2inFigure4andthedelayatnode2.Asmalldelaymaythennotbethatseriousbecausetheremayanywaybeadelayatnode3.Alongdelaymay,however,causealongdelayalsoforoperation3.Thisindicatesthatourlineardelaycostmaybelessappropriateinthiscase.Italsoexplainswhythelongermorestochasticoperationsarestartingsoearlyintheoptimalsolution,i.e.,wewishtoavoidlongdelays.Ageneral,notverysurprising,conclusioncanbethatourapproximationworksbetterfornetworkswithasingle“criticalpath”.6ConclusionsWehaveconsideredtheproblemofminimizingexpecteddelayandholdingcostsforacomplexassemblynetworkwheretheoperationtimesareindependentstochasticvariables.Anapproximatedecompositiontechniqueforsolvingtheproblemhasbeensuggested.Thetechniquemeansrepeatedapplicationsofthesolutionofasimplersingle-stageproblem.Theapproximatemethodhasbeenevaluatedfortwoproblemsets.Theresultsareverygoodforthefirstsetoftwo-stageproblemsandtherelativecostincreaseduetotheapproximationisonlyabout1%.Forthesecondsetofthree-stageproblemstheerrorsareabout10%andcannotbedisregarded.Althoughthenumericalresultsshowsomepromise,furtherresearchisneededforevaluationoftheapplicabilityofthesuggestedtechniqueinmoregeneralsettings.Becauseitisdifficulttoderiveexactsolutionsforproblemsofrealisticsizeitmaybemostfruitfultocomparedifferentheuristicsforlargerproblems.ReferencesBuzacottJA,ShanthikumarJG(1994)SafetystockversussafetytimeinMRPcontrolledproductionsystems.ManagementScience40:1678–1689ChenF(2001)Marketsegmentation,advanceddemandinformation,andsupplychainper-formance.Manufacturing&ServiceOperationsManagement3:53–67ChuC,ProthJM,XieX(1993)Supplymanagementinassemblysystems.NavalResearchLogistics40:933–950ClarkAJ,ScarfH(1960)Optimalpoliciesforamulti-echeloninventoryproblem.Manage-mentScience6:475–490GongL,deKokT,DingJ(1994)Optimalleadtimesplanninginaserialproductionsystem.ManagementScience40:629–632HariharanR,ZipkinP(1995)Customer-orderinformation,leadtimes,andinventories.Man-agementScience41:1599–1607HoppW,SpearmanM(1993)Settingsafetyleadtimesforpurchasedcomponentsinassemblysystems.IIETransactions25:2–11KaraesmenF,BuzacottJA,DalleryY(2002)Integratingadvanceorderinformationinmake-to-stockproductionsystems.IIETransactions34:649–662 344S.Axsater¨KumarA(1989)Componentinventorycostsinanassemblyproblemwithuncertainsupplierlead-times.IIETransactions21:112–121ShoreH(1995)Settingsafetylead-timesforpurchasedcomponentsinassemblysystems:ageneralsolutionprocedure.IIETransactions27:634–637SongJ-S,YanoCA,LerssrisuriyaP(2000)Contractassembly:Dealingwithcombinedsupplyleadtimeanddemandquantityuncertainty.Manufacturing&ServiceOperationsManagement2:287–296SongJ-S,ZipkinP(2003)Supplychainoperations:Assemble-to-ordersystems,Ch.11.In:GravesSC,DeKokT(eds)Handbooksinoperationsresearchandmanagementscience,Vol11.Supplychainmanagement:design,coordinationandoperation.Elsevier,AmsterdamYanoC(1987a)Stochasticleadtimesintwo-levelassemblysystems.IIETransactions19:371–378YanoC(1987b)Settingplannedleadtimesinserialproductionsystemswithtardinesscosts.ManagementScience33:95–106YanoC(1987c)Stochasticleadtimesintwo-leveldistribution-typenetworks.NavalResearchLogistics34:831–843 Amultiperiodstochasticproductionplanningandsourcingproblemwithservicelevelconstraints
Is¸ılYıldırım1,Barıs¸Tan2,andFikriKaraesmen11DepartmentofIndustrialEngineering,Koc¸University,RumeliFeneriYolu,Sariyer,Istanbul,Turkey(e-mail:isil.yildirim@insead.edu.tr;fkaraesmen@ku.edu.tr)2GraduateSchoolofBusiness,Koc¸University,RumeliFeneriYolu,Sariyer,Istanbul,Turkey(e-mail:btan@ku.edu.tr)Abstract.Westudyastochasticmultiperiodproductionplanningandsourcingproblemofamanufacturerwithanumberofplantsand/orsubcontractors.Eachsource,i.e.eachplantandsubcontractor,hasadifferentproductioncost,capacity,andleadtime.Themanufacturerhastomeetthedemandfordifferentproductsaccordingtotheservicelevelrequirementssetbyitscustomers.Thedemandforeachproductineachperiodisrandom.Wepresentamethodologythatamanu-facturercanutilizetomakeitsproductionandsourcingdecisions,i.e.,todecidehowmuchtoproduce,whentoproduce,wheretoproduce,howmuchinventorytocarry,etc.Thismethodologyisbasedonamathematicalprogrammingapproach.Therandomnessindemandandrelatedprobabilisticservicelevelconstraintsarein-tegratedinadeterministicmathematicalprogrambyaddinganumberofadditionallinearconstraints.Usingarollinghorizonapproachthatsolvesthedeterministicequivalentproblembasedontheavailabledataateachtimeperiodyieldsanap-proximatesolutiontotheoriginaldynamicproblem.Weshowthatthisapproachyieldsthesameresultasthebasestockpolicyforasingleplantwithstationarydemand.Forasystemwithdualsources,weshowthattheresultsobtainedfromsolvingthedeterministicequivalentmodelonarollinghorizongivessimilarresultstoathresholdsubcontractingpolicy.Keywords:Stochasticproductionplanning–Servicelevelconstraints–Subcon-tracting
TheauthorsaregratefultoYvesDalleryforhisideas,commentsandsuggestionsontheearlierversionsofthispaper.Correspondenceto:F.Karaesmen 346I.Yıldırımetal.1IntroductionandmotivationInthisstudy,weconsideramanufacturerthatsuppliesproductstoaretailer.Themanufacturerhasanumberofproductionsourcesthatareeitheritsownplantsoritssubcontractors.Eachsourcehasadifferentproductioncost,capacity,andleadtime.Thedemandforeachproductineachperiodisrandom.Themanufacturerhastomeetthedemandformultipleproductstakingintoaccounttheservicelevelrequirementssetbytheretailer.Intheproductionplanningandthesourcingproblem,themanufacturer’sdeci-sionvariablesarehowmuchtoproduce,whentoproduce,wheretoproduce,andhowmuchinventorytocarryineachperiod.Theobjectiveistominimizeitstotalproductionandinventorycarryingcostsduringtheplanninghorizonsubjecttotheservicelevelrequirementsandotherpossibleconstraints.Thisproblemismotivatedbytheproblemsfacedbysuppliersofleanretailersinthetextile-apparel-retailchannel(Abernathyetal.,1999).Namely,adoptionofleanretailingpracticesforcesuppliersofleanretailerstoadoptnewstrategiestorespondquicklytochangingdemandeffectively.Usingsubcontractorsemergeasaviablealternativetoincreaseproductioncapacitytemporarilywhenitisneeded.Additionalcostofsubcontractingcanbejustifiedbyloweringinventoriesandim-provingtheservice.However,decidingonwheretoproduceandhowmuchtoproduceisachallengingtaskespeciallywhenthedemandisvolatile.AqualitativediscussionofthisproblemcanbefoundinAbernathyetal.(2000).Figure1belowdepictsthesystemwhichmotivatesthisstudy.Weproposeasolutionmethodologythatisbasedonsolvingadeterministicmathematicalproblemateachtimeperiodonarollinghorizonbasis.RandomnessintheproblemthatcomesfromuncertaindemandandservicelevelconstraintsareintegratedinadeterministicmathematicalprogrambyaddinganumberofadditionallinearconstraintssimilartotheapproachproposedbyBitranandYanasse(1984).WeproposeusingthisapproachtoaddressthemorerelevantbutalsomorePlant1RetailerordersProduct1PlantDistributionCenterRetailerSubcontractorInventory1ProductMSalesdataSubcontractorNDecisionandControlProductionFig.1.Amanufacturerwithmultipleplantsthatsellsmultipleproductstoaretailer Amultiperiodstochasticproductionplanningandsourcingproblem347difficultdynamicproblemwheredecisionscanbeupdatedovertime.Sincetheequivalentdeterministicproblemisawell-structuredmathematicalprogrammingproblem,theproposedmethodologycaneasilybeintegratedwiththeAdvancedPlanningandOptimizationtools,suchastheproductsofi2,Manugistics,etc.,thatarecommonlyusedinpractice.Theorganizationoftheremainingpartsofthepaperisasfollows:InSection2,wereviewtheliteratureonmathematical-programming-basedstochasticproduc-tionplanningmethodologies.TheparticularstochasticproductionplanningandsourcingproblemweinvestigateisintroducedinSection3.Section4presentstheproposedsolutionmethodologythatisbasedonsolvingthedeterministicequiva-lentproblemateachtimesteponarollinghorizonbasis.TheperformanceoftherollinghorizonapproachisevaluatedbyconsideringanumberofspecialcasesinSection5.Finally,conclusionsarepresentedinSection6.2LiteraturereviewTheclassicaldeterministicproductionplanningproblem,itsmathematicalpro-grammingformulationsandsolutionmethodologieshavereceivedalotofattentionformanyyears(seeHaxandCandea,1984foranumberofwell-knownmodels).Inthissection,weonlyreviewtheliteraturedirectlyrelatedtomathematicalpro-grammingbasedapproachesforstochasticproductionplanningproblems.BitranandYanasse(1984)dealwithasimilarstochasticproductionplanningproblemwithaservicelevelrequirement.Theyprovidenon-sequential(static)anddeterministicequivalentformulationsofthemodelandproposeerrorboundsbetweentheexactsolutionandtheproposedapproach.Theirmainfocusisonthesolutionofthestaticproblem,i.e.,thesolutionattimezeroforthewholeplanninghorizon.Bitran,HaasandMatsudo(1986)presentamodelthatismotivatedbyacaseintheconsumerelectronicsandtextileandapparelindustry.Inthismodel,thestochasticproblemistransformedintoadeterministiconebyreplacingtherandomdemandwiththeiraveragevalues.Then,thesolutionofthetransformedproblemprovidesanswerstothequestionsofwhattoproduceandwhentoproduce.Thecompletesolutionisobtainedbydetermininghowmuchtoproducefromanewsboy-typeformulationbasedonthesolutionofthedeterministicproblem.FeiringandSastri(1989)focusonproductionsmoothingplanswithrollinghorizonstrategiesandconfidencelevelsforthedemand,whicharesetbythepro-ductionplanners.Theprobabilisticconstraintsinthedemand-drivenschedulingmodelarerevisedbyBayesianproceduresandaretransformedintodeterministicconstraintsbyinversetransformationsofnormallydistributeddemand.Zapfel(1996)claimsthatMRPIIsystemscanbeinadequateforthesolutionof¨productionplanningproblemswithuncertaindemandbecauseoftheinsufficientlysupportedaggregation/disaggregationprocess.ThepaperthenproposesaproceduretogenerateanaggregateplanandaconsistentdisaggregateplanfortheMasterProductionSchedule.Kelle,ClendenenandDardeau(1994)extendtheeconomiclotschedulingprob-lemforthesingle-machine,multi-productcasewithrandomdemands.Theirob- 348I.Yıldırımetal.jectiveistofindtheoptimallengthofproductioncyclesthatminimizesthesumofset-upcostsandinventoryholdingcostsperunitoftimeandsatisfiesthedemandofproductsattherequiredservicelevels.ClayandGrossman(1997)focusonatwo-stagefixed-recourseproblemwithstochasticRight-Hand-Sidetermsandstochasticcostcoefficientsandproposeasensitivity-basedsuccessivedisaggregationalgorithm.SoxandMuckstadt(1996)presentamodelforthefinite-horizon,discrete-time,capacitatedproductionplanningproblemwithrandomdemandformultipleprod-ucts.Theproposedmodelincludesbackordercostintheobjectivefunctionratherthanenforcingservicelevelconstraints.AsubgradientoptimizationalgorithmisdevelopedforthesolutionoftheproposedmodelbyusingLagrangianrelaxationandsomecomputationalresultsareprovided.BeyerandWard(2000)reportaproductionandinventoryproblemofHewlett-Packard’sNetworkServerDivision.TheauthorsproposeamethodtoincorporatetheuncertaintiesindemandinanAdvancedPlanningSystemutilizedbyHewlett-Packard.Albritton,ShapiroandSpearman(2000)studyaproductionplanningproblemwithrandomdemandandlimitedinformationandproposeasimulationbasedop-timizationmethod.QuiandBurch(1997)studyahierarchicalproductionplanningandschedulingproblemmotivatedbythefibreindustryandproposeanoptimizationmodelthatuseslogicofexpertsystems.VanDelftandVial(2003)considermultiperiodsupplychaincontractswithoptions.Inordertoanalyzethecontracts,theyproposeamethodologytoformulatethedeterministicequivalentproblemfromthebasedeterministicmodelandfromaneventtreerepresentationofthestochasticprocessandsolvethestochasticlinearprogrambydiscretizingdemandunderthebacklogassumption.Forthetextile-apparel-retailproblemdiscussedinAbernathyetal.(2000),asimulationmodelhasalsobeendeveloped(Yangetal.,1997).Thenasimulation-basedoptimizationtechniquethatisreferredasordinaloptimization,hasbeenusedtodeterminetheparametersofaproductionandinventorycontrolpolicythatgivesagood-enoughsolutionapproximately(Yangetal.,1997;Lee,1997).However,oneneedstosetaspecificproductionandinventorycontrolpolicyinthesimulation.Inadditiontothedifficultyofsettingaplausiblepolicyinacomplicatedcase,asthenumberofsourcesandproductsincrease,thenumberofparameterstobeoptimizedalsoincreases.Asaresult,findinganapproximatesolutionrequiresaconsiderabletime.Simplifiedversionsofthesourcingproblemstudiedinthispaperhavebeeninvestigatedinthepastbyusingstochasticoptimalcontrol(Bradley,2002;TanandGershwin,2004;Tan,2001).Bradley(2002)considersasystemwithaproducerandasubcontractoranddiscreteflowofgoods.InanM/M/1settingwithouttheservicelevelrequirements,heprovesthattheoptimalcontrolpolicystructureisadual-basestockpolicy.Inthispolicywhenthenumberofcustomersinthequeuereachesacertainlevel,thennewincomingcustomersaresenttothesubcontractor.Whentherearenocustomerswaitinginthequeue,thentheproducercontinuesproductionuntilacertainthresholdisreached. Amultiperiodstochasticproductionplanningandsourcingproblem349InTan(2001)andTanandGershwin(2004),aproducerwithasinglesub-contractorisformulatedwithcontinuousflowofgoodswithouttheservicelevelrequirements.Theyalsoshowthatathreshold-typepolicyisoptimaltodecidewhenandhowtouseasubcontractor.Inthethresholdpolicy,thesubcontractorisusedwhentheinventoryorthebacklogisbelowacertainthresholdlevel.Ourpaperusestheideaofincorporatingrandomnessinadeterministicmath-ematicalprogramthatisusedinmanyoftheabovestudiesindifferentformats.WeutilizetheapproachproposedbyBitranandYanasee(1984)thatshowstheequivalenceforthestaticproblem.Incontrasttothisstudywherethemainobjec-tiveisdeterminingerrorboundsfortheoptimalcostinthenon-sequentialcase,ourmainfocusisgeneratingaproductionandsourcingplan,i.e.determiningthevaluesofthedecisionvariablesinthesequential(dynamic)problemwheresourc-ingdecisionsaremade(orupdated)dynamicallyovertime.Wealsocomparetheapproximatesolutionofthedynamicproblemwithcertainbenchmarkpolicies.Sincetheexactoptimalsolutionofthedynamicproblemisnotknown,weusetwodifferentbenchmarks.Itisproventhatforasinglesourcewithleadtime,theproposedapproachyieldsthesameproductionpolicyastheoptimalbasestockpolicy.Foradual-source,e.g.aproducerwithasubcontractor,athreshold-typesubcontractingpolicysuggestedbyBradley(2002),Tan(2001),TanandGershwin(2004)isutilizedasabenchmark.Afteradoptingthethresholdpolicytoamoregeneralizedcasewithleadtimeandservice-levelrequirements,itisobservedthattheproposedapproachyieldsverysimilarresultstothethreshold-basedbenchmarkinthenumericalexamplesconsidered.3StochasticmultiperiodsourcingproblemwithservicelevelconstraintsAssumethatthereisasingleproductandNdifferentproductionsources(plantsandsubcontractors).Thedemandforthisspecificproductattimet,dtisrandom.Themaindecisionvariablesaretheproductionquantitiesateachproductionsourceattimet,Xi,t,i=1,...,N.TheinventorylevelattheendoftimeperiodtisdenotedbyIt.ThenumberofperiodsintheplanninghorizonisT.Theinventoryholdingcostperunitperunittimeishtandtheproductioncostatproductionsourceiattimetisci,t.Constraintsontheperformance(relatedtobackorders)ofthesystemareim-posedbyrequiringservicelevels.ThefrequentlyusedType1ServiceLevelisdefinedtobethefractionofperiodsinwhichthereisnostockout.Itcanbeviewedastheplant’sno-stock-outfrequency.Thisservicelevelmeasureswhetherornotabackorderoccursbutisnotconcernedwiththesizeofthebackorder.Inthisstudy,weconsideraModifiedType1ServiceLevelrequirement.TheModifiedType1ServiceLevelforcestheprobabilityofhavingnostockouttobegreaterthanorequaltoaservicelevelrequirementineachperiod.Theservicelevelrequirementinperiodtisdenotedbyαt.TheStochasticProductionPlanningandSourcingProblem(SP)isdefinedas:#%TNZ∗(SP)=MinEh(I)++cXtti,ti,tt=1i=1 350I.Yıldırımetal.subjecttoNIt=It−1+Xi,t−dt,t=1,...,T;(1)i=1P{It≥0}≥αt,t=1,...,T.(2)Xi,t≥0,i=1,...,Nt=1,...,T.(3)where(I)+=Max{0,I},t=1,...,T.ttTheobjectiveoftheproblemistominimizethetotalexpectedcost,whichistheexpectedvalueofthesumoftheinventoryholdingandproductioncostsintheplanninghorizon.Thefirstconstraintsetdefinestheinventorybalanceequationsforeachtimeperiod.Thenextconstraintimposestheservicelevelrequirementforeachperiod.Finally,thelastconstraintstatesthattheproductionquantitiescannotbenegative.Thisformulationcaneasilybeextendedtomultipleproductsandproductionsourceswithleadtimes.Moreoverdifferentserviceleveldefinitionscanalsobeconsideredbyfollowingthesameapproach.4AnapproximatesolutionprocedurebasedonarollinghorizonprocedureThesolutionoftheaboveproblemattime0fortheplanninghorizon[0,T]isreferredasthestaticsolution.Thestaticsolutionisobtainedbyusingtheavailableinformationaboutthedistributionofdemandinthefutureperiodsandtheinitialinventory.Apolicythatsets(orupdates)thefutureproductionquantitiesXi,tattimetbasedontheinformationavailableatthattime,e.g.,demandrealizations,demanddistributionsinthefutureperiods,andcurrentinventorylevels,isreferredtoasthedynamicsolution.Intheory,theoptimalpolicywhichdeterminesproductionquantitiesbasedonactualstateinformationmaybeobtainedbysolvingthestochasticdynamicpro-gramassociatedwiththisproblem.Inpractice,however,thereareseveralproblemswiththestochasticdynamicprogrammingsolution.First,thewell-knowncurseofdimensionalitymakesnumericalsolutionschallengingevenforrelativelysmallproblems.Second,itisdifficulttointegrateconstraintsonthetrajectoryoftheunderlyingstochasticprocessessuchasservicelevelrequirementsininventorymodels.Therefore,weproposearolling-horizonapproachthatisbasedonsolvingthestaticproblemateachtimeperiodbasedontheavailableinformation.This,how-ever,requiressolvingthestaticproblemrepeatedlywhichrequiresatransformationexplainedbelow.4.1DeterministicequivalentformulationforthestaticsolutionAlthoughobtainingtheoptimaldynamicsolutionis,ingeneral,nottractable,thestaticsolutioncanrelativelyeasilybeobtainedbyusingdeterministicmathematicalprogrammingassuggestedbyBitranandYanasse(1984). Amultiperiodstochasticproductionplanningandsourcingproblem351Inparticular,BitranandYanasseshowthatthe(ModifiedType1)servicelevelconstraintcanbetransformedintoadeterministicequivalentconstraintbyspecify-ingcertainminimumcumulativeproductionquantitiesthatdependontheservicelevelrequirements.Tosummarizethisapproach,letltdenotethe(deterministicequivalent)mini-mumcumulativeproductionquantityinperiodtwhichiscalculatedbysolvingtheprobabilisticinequality:tPdτ≤lt=αt,t=1,...,Tforlt(t=1,...,T)τ=1thatyields−1lt=Ft(αt),t=1,...,TtwhereFt(.)isthecumulativedistributionfunctionoftherandomsumτ=1dτ.ThentheprobabilisticconstraintP{It≥0}≥αt,t=1,...,Tcanbeexpressedequivalentlyby:tNXi,τ+I0≥lt,t=1,...,T(4)τ=1i=1Now,thedeterministicequivalentproblemwithservicelevelconstraintsthathasbeenmentionedintheprevioussectionscanbemodeledasbelow(BitranandYanasse,1984):DeterministicEquivalentProblem(DEP):TtNNZ∗(DEP)=Minh(I+X)+cXt0i,τi,ti,tt=1τ=1i=1i=1subjecttotNXi,τ+I0≥lt,t=1,...,T(5)τ=1i=1Xi,t≥0,i=1,...,Nt=1,...,T.(6)TheoptimaldecisionvariablevaluesinDEParethesameastheonesinthesolutionofSPattime0.Thestaticsolutionisobtainedbytransformingthestochasticproblemintoadeterministiconeandthensolvingtheresultingmathematicalprogram.TherollinghorizonapproachrepeatsthisprocedurebyusingtheavailableinformationateachtimeperioduntiltimeT.5PerformanceoftherollinghorizonsolutionItisknownthattherolling-horizonapproachyieldsgoodresultsforanumberofdynamicoptimizationproblems.Insomespecialcases,therollinghorizonmethodmayevenyieldtheoptimalsolution.Inthissection,weevaluatetheperformanceoftheproposedmethodbycomparingittocertainbenchmarkpoliciesintwocommonlyencounteredspecialcasesinproductionplanning. 352I.Yıldırımetal.5.1AsinglesourceproblemwithstationarydemandWestartwiththespecialcaseofasingleproductionsource.Whenthereisonlyonesource,theobjectivefunctionincludesonlytheholdingcost(sincetheexpectedtotalproductioncostsmustequalthetotalexpecteddemandovertheplanninghorizon).Inthiscase,weusethebasestockpolicyasthebenchmarkpolicy.Thebasestockpolicyiswidelyknownandutilizedinmanyapplications.Inaddition,itisknowntobeoptimalinanumberofrelatedinventoryproblems.It,therefore,constitutesanaturalbenchmarkforcomparison.Thebasestockpolicyhasasingleparameterwhichisareorderlevelandabaselotsizeofoneunit.Itaimstomaintainapre-specifiedtargetinventorylevel.Underthispolicy,thesequenceofeventsisasfollows:thesystemstartswithapre-specifiedbasestocklevelinthefinishedgoodsinventory.Thearrivalofthecustomerdemandtriggerstheconsumptionofanend-itemfromtheinventoryandissuingofareplenishmentordertotheproductionfacility.Usingthispolicy,anorderisplaced(orthemanufacturingfacilityoperates)ifandonlyiftheinventoryleveldropsbelowthebasestocklevel.Thecomparisonofthesetwomodelsisperformedfortwocaseswithandwithoutaleadtime.5.1.1SinglesourcewithoutleadtimeInthisfirstscenario,thereisasingleproducttobeproducedbyasingleproductionfacility.Itisassumedthatthedemandofthisspecificproductstaysstationaryovertheplanninghorizon.Weproposethatsolvingthedeterministicequivalentmodelwithmodifiedservicelevelconstraintsonarollinghorizonbasisisequivalenttooperatingthesystemunderthebasestockpolicy.Thenextpropositionestablishesthisequivalence:Proposition1.Whentheproductionfacilityhasnoleadtimeandthedemandisstationary,usingabasestockpolicyisequivalenttosolvingthedeterministicequivalentmodelwithservicelevelconstraintsonarollinghorizonbasis(eitherModifiedType1orModifiedType2)inthefollowingway:assumethatthebasestocklevelinthebasestockpolicyequalsI0(BS)=S1andtheinitialinventorylevelinthedeterministicequivalentproblemequalsI0(DEP)=l1.IfS1=l1,thentheequivalentbasestockpolicygivesthesametotalexpectedcostvalue,yieldsthesameproductionplanandresultsinthesameservicelevelwiththedeterministicequivalentmodelwithmodifiedservicelevelconstraintssolvedonarollinghorizonbasis.Sincethiscaseisaspecialcaseofthenextonewithleadtime,theproofofProposition1isnotgivenherebutreportedin(Yıldırım,2004).Corollary1.Theoptimalbasestocklevelisequaltol1.Equivalently,thebasestocklevelS1=l1ensuresthattheresultingproductionplansatisfiestherequiredservicelevels.Proof.IftheinitialinventorylevelissettobeS1=l1,theresultingproductionplanisthesamewiththatofthebasestockpolicywhichstartswithabasestocklevelofS1=l1.Althoughthebasestockpolicydoesnotguaranteetheassuranceoftheservicelevels,sinceweknowthatthedeterministicequivalentmodelsatisfiesthe Amultiperiodstochasticproductionplanningandsourcingproblem353requiredservicelevelsandthetwopoliciesareequivalent,wecansaythatthebasestocklevelS1=l1ensuresthattheresultingproductionplansatisfiestherequiredservicelevels.NotethatS1=l1mustbeoptimalbecausedecreasingthebasestocklevelfroml1leadstoaninfeasiblesolutionandincreasingitabovel1wouldleadtohigheraverageinventorycostsandthereforecannotbeoptimal.Eventhoughaformalproofislacking,itishighlylikelythatthebasestockpolicy(withastationarybasestocklevel)isoptimalforthesingle-plantsingle-productprobleminaninfinitehorizonsetting.Theorem1andCorollary1establishthatforthisproblem,therollinghorizonapproachyieldsthesamesolutionsastheoptimalbasestockpolicyleadingustoconcludethattherollinghorizonprocedureperformsoptimallyinthiscase.5.1.2SinglesourcewithleadtimeThedeterministicequivalentmodelwithservicelevelconstraints(DEP)canbeextendedtoacaseinwhichtheproductionfacilityhasaproductionleadtime.AssumethatthereisaproductionleadtimeofLTperiodsandtheinitialscheduledreceiptsaredenotedbySRt,t=1,...,LT.Then,theproblemcanbemodeledinthefollowingway:DeterministicEquivalentProductionPlanningProblemincludingLeadTime(DEPLT):LTtZ∗(DEPLT)=Minh(I+SR)t0τt=1τ=1TLTtN+ht(I0+SRτ+Xi,τ−LT)t=LT+1τ=1τ=LT+1i=1subjecttotLTXτ−LT+SRτ+I0≥lt,t=(LT+1),...,T;(7)τ=LT+1τ=1Xt≥0,t=1,...,T.(8)Ourmainresultisasfollows:Proposition2.Whentheproductionfacilityhasanon-negativeleadtimeLT,thedemandisstationaryandtherearenoscheduledreceiptsinitially,usingabasestockpolicyisequivalenttosolvingthedeterministicequivalentmodelwithservicelevelconstraintsonarollinghorizonbasisinthefollowingmanner:assumethatthebasestocklevelinthebasestockpolicyincludingleadtimeequalsI0(BSLT)=S2andtheinitialinventorylevelinthedeterministicequivalentmodelincludingleadtimeequalsI0(DEPLT)=lLT+1.IfS2=lLT+1,thentheequivalentbasestockpolicygivesthesametotalexpectedcostvalue,yieldsthesameproductionplanandresultsinthesameservicelevelwiththedeterministicequivalentmodelwithservicelevelconstraintssolvedonarollinghorizonbasis.Proof.TheproofofProposition2isgivenintheAppendix. 354I.Yıldırımetal.5.2AdualsourceproblemwithstationarydemandSincetheoptimalsolutionofourdynamicproblemisnotknown,aplausiblebench-markisusedtoevaluatetheperformanceoftheproposedapproach.Weproposeathresholdsubcontractingmodelsuggestedinanumberofstudiesinthelitera-ture(Bradley,2002;Tan,2001;TanandGershwin,2004).Althoughthethresholdpolicyisonlyshowntobeoptimalunderspecificassumptionsincludingzeroleadtime,stationarydemand,noservicelevelrequirements,etc.,wethinkthatitisareasonablebenchmarkpolicyforourproblem.5.2.1AthresholdsubcontractingpolicyNowweexplaintheoperationofthethresholdpolicyforourbenchmarkcase.Weconsideradualsourcesystemwithanin-houseproductionfacilityandasubcontrac-tor.Weassumethatthein-housefacilityhasacapacityofCbutthesubcontractorhasaninfinitecapacity.Thereisaleadtimeofoneperiod.Thatis,productionquantitiesscheduledattimetbecomeavailableattimet+1.ThethresholdpolicyischaracterizedbytwothresholdlevelsSandZ.Thein-houseproductionfacilityoperateswhentheinventorylevelisbelowS.Thatis,itstartsproducingwhentheinventoryleveldropsbelowthetargetlevelSandstopsproducingwhentheinventorylevelagainreachesS.ThesubcontractorisusedwhentheinventoryleveldecreasestoathresholdlevelofZ.WhentheinventorylevelisbelowS,butisstillaboveZ,thein-housefacilityproducestocovertheshortfallwithrespecttoS.Ifthereisnotsufficientproductioncapacitytocoverthewholeshortfall,thein-housefacilityoperatesatfullcapacityandtheportionofdemandthatcannotbesatisfiedisbackloggedforthenextperiod.LetX1,tandX2,tdenotetheproductionamountsofthein-housefacilityandthesubcontractorinperiodtrespectively.Then,theproductionamountsofeachproductionfacilityineachtimeperiodcanbedeterminedforthethresholdsubcon-tractingmodelinthefollowingway:X1,t=Min{S−Z,S−It−1,C},t=1,...,T;(9)X2,t=Max{0,Z−It−1},t=1,...,T.(10)ThefollowingfigureshowstheevolutionofX1,t,X2,tandItunderthispolicyforaPoissonarrivalofdemandwithrate10andS=15,Z=7,andC=8.5.2.2ComparisonoftheperformanceofthethresholdpolicyandtherollinghorizonapproachThedeterministicequivalentmodelforthiscaseissolvedforarollinghorizonof10periodsrepeatedlythroughoutaplanninghorizonof1000periods.5000sampledemandstreamsaregeneratedandtherealizedinventorylevelsareintegratedinthemodelaccordingly.Theproductionplansandtherealizedcostvaluesbetweenperiods451and550areobserved.Allcostvaluesarecalculatedonaperperiodbasis.TheoptimalvaluesofthethresholdvaluesSandZaredeterminedbyusingadirectsimulation-basednumericalsearch.Itisassumedthatthereare1000periods Amultiperiodstochasticproductionplanningandsourcingproblem355Fig.2.Samplerealizationofdt,X1,t,X2,tandItunderthethresholdpolicyS=15,Z=7,C=8intheplanninghorizonandthesame5000sampledemandstreamsareutilized.Theservicelevelrequirementisrelaxedwiththeone-sided95%confidenceintervalofthesimulationresult.Thatiswheneverupperconfidenceleveloftheobservedservicelevelreachesthedesiredone,thiscaseisacceptedassatisfyingtheservicelevelrequirement.Theunderlyingreasoningbehindmakingthismodificationinservicelevelsisthat,thesamplesizeweutilizemightnotbesufficientenoughtomaketherealizedservicelevelequalexactlytotherequiredone.Amongthebase 356I.Yıldırımetal.Table1.ThepossiblescenariosforwhichcomparisonsaremadeSubcontractingHoldingIn-house(Subcontracting(Holding(In-housecostcostproductioncost)/(in-housecost)/(in-houseprod.capacity)/capacityprod.cost)prod.cost)(meandemand)4168140.841612141.2416201426181.50.250.861121.50.251.261201.50.2526481.510.864121.511.264201.512stockandthresholdlevelsthatsatisfytherelevantservicelevelrequirements,themodelaimstofindtheonewithminimumtotalcost.Thecalculationsareperformedforperiodsbetween451and550.Forthenumericalexamplesreportedbelow,theorderarrivalsaregovernedbyaPoissonprocesswithrate10productsperperiod.Theproductioncostisassumedtobe$4perproductforthein-housefacility.Theinitialinventorylevelofthespecificproductissettobezero.Theservicelevelrequirementissettobe95%.Thecomparisonbetweenthedeterministicequivalentmodelandthethresholdsubcontractingmodelisperformedforninecombinationsofsubcontractingcosttoin-houseproductioncost,holdingcosttoin-houseproductioncostandcapacitytomeandemandratios.Thecombinationsofsubcontractingcosts,holdingcostsandthein-houseproductioncapacitiesandtherefore,thecombinationsofrelevantsubcontractingcosttoin-houseproductioncost,holdingcosttoin-houseproductioncostandcapacitytomeandemandratiosforwhichthecomparisonsaremadecanbeobservedinTable1.Foreachoftheproblemsettings,thebasestockandthresholdlevelsobservedinthethresholdsubcontractingmodelarereportedinTable2.Notethat,insomeofthecases,thebasestockandthresholdpairsareobservedtobethesame.Thereasoningbehindthisis,thesepairsleadtothesameaverageinventorylevelsandminimumcostvaluesinthesesettings.Whilecomparingthetwomodels,totalexpectedcost,averageproductioncost,averageinventoryholdingcostvaluesandtheassignmentofproductiontotheplants(inpercentages)arethekeyelementswefocuson.Table3summarizesthetotalexpectedcostvaluesofthedeterministicequivalentmodel(DEM)andthethresholdsubcontractingmodel(TSM)fortheninedifferentscenariosforeachmodifiedserviceleveltype.Thebelowtablesdisplaythatthedeterministicequivalentmodelgivesveryclosesolutionswhencomparedwiththethresholdsubcontractingmodelforbothtypesofthemodifiedlevels.Thedeterministicequivalentmodelresultsintotalexpectedcostvaluesequaltooralittlebitlargerthanthoseofthethresholdsubcontracting Amultiperiodstochasticproductionplanningandsourcingproblem357Table2.BasestockandthresholdlevelsobservedineachscenarioSubcontractingHoldingIn-houseCriticallevelscostcostproductionBasestockThresholdcapacity4168157416121534162015−∞6181776112160612015−∞6481576412153642015−∞Table3.ThecomparisonoftotalexpectedcostvaluesobservedineachscenarioSubcontractingHoldingIn-houseTotalexpectedcostcostcostproductionDEMTSMPercentagecapacitydifference4168121.66121.660.0041612121.66121.660.0041620121.66121.620.0361849.9749.890.16611246.1645.651.12612045.1045.100.0264865.3365.330.00641261.4761.470.00642060.4260.400.03model.Foroursetofnumericalexperiments,thedeterministicequivalentmodelgivescloseresultstothethresholdsubcontractingmodelwhentheservicelevelrequirementisofModifiedType1.Tables4and5displaythecomparisonofaverageproductionandholdingcostvalues.Ascanbeseen,thedeterministicequivalentmodelgivessimilarresultstothethresholdsubcontractingmodel.Table6summarizesthepercentageofproductionassignedtothein-housepro-ductionfacilityforboththedeterministicequivalentmodelandthethresholdsub-contractingmodel.Theresultssuggestthattheproductionassignmentsofthede-terministicmodelfollowasimilarpatternwiththebenchmarkchosen.Basedonthesefigures,wecanconcludethattheproposeddeterministicequiv-alentmodelsolvedonarollinghorizonbasisperformsaswellasthethresholdsubcontractingmodelsolvedonasimulation-basedoptimizationtechniqueforthe 358I.Yıldırımetal.Table4.ThecomparisonofaverageproductioncostvaluesobservedineachscenarioSubcontractingHoldingIn-houseAverageproductioncostcostcostproductionDEMTSMPercentagecapacitydifference416839.9939.990.004161239.9939.990.004162039.9939.990.0061844.0644.36−0.68611241.0540.242.03612040.0039.970.0664844.9144.910.00641241.0541.050.00642040.0039.990.01Table5.ThecomparisonofaverageholdingcostvaluesobservedineachscenarioSubcontractingHoldingIn-houseAverageholdingcostcostcostproductionDEMTSMPercentagecapacitydifference416881.6781.670.004161281.6781.670.004162081.6781.630.056185.925.536.9161125.105.41−5.6761205.105.100.0564820.4220.420.00641220.4220.420.00642020.4220.410.05ModifiedType1servicelevel.Thetotalexpectedcostvaluesofdeterministicequiv-alentmodelsforallninedifferentcasesareequaltooralittlebitlargerthanthoseofthethresholdsubcontractingmodel.However,wecannotreachthesameconclusionfortheaverageproductionandholdingcostvalues.Thedeterministicequivalentmodelperformseitherworseforsomecasesorbetterforsomeothercaseswhenthecomparisonisbasedonaverageproductionorholdingcostvalues.However,thesumofthesetwoterms,thetotalexpectedcost,isequaltoalittlebitlargerthanthatofthethresholdsubcontractingmodel.Moreover,theproportionofproductionassignedtothein-housefacilityinthedeterministicequivalentmodelresemblesthatinthesimulationbasedthresholdsubcontractingmodel.Itisworthmentioningthatthesamplesizeutilizedintheabovenumericalcomparisons,5000,mightnotbelargeenoughtosatisfytheservicelevelrequire-mentsineachtimeperiodthatthemodifiedserviceleveldefinitionsnecessitate. Amultiperiodstochasticproductionplanningandsourcingproblem359Table6.Thepercentageofproductionassignmentstothein-houseproductionfacilityob-servedineachscenarioSubcontractingHoldingIn-house%In-houseproductioncostcostproductionBasestockThresholdcapacity416875.4575.404161294.7394.704162099.97100.0061879.7678.17611294.7398.78612099.97100.0064875.4575.40641294.7394.70642099.97100.00Thecoefficientofvariationintherealizedservicelevelvaluesmightbelargerthanexpected.Tohandlethisproblematicissue,weintroducedone-sidedconfidenceintervals.Althoughthethresholdsubcontractingmodelconstitutesalowerboundintermsoftotalexpectedcostvaluesforoursetofnumericalexamples,itcannotbegeneralizedfromourexamplesthatthedeterministicequivalentmodelalwaysgivessolutionsworsethanthoseofthethresholdsubcontractingmodel.Never-theless,theproposedapproachseemstogiveextremelypromisingresultsinthisparticularcaseaswell.6ConclusionsInmanypracticalsituations,mathematicalmodelsofproductionplan-ning/outsourcingproblemshavetodealwiththerandomnessindemand.Wepresentasystematicapproachthatenablestherandomnessindemandandthedesiredser-vicelevelstobeincorporatedinamathematicalprogrammingframework.Weshowthatsolvingthedeterministicequivalentproblemonarolling-horizonbasisgivessimilarresultstotheperformanceofthebenchmarks.Althoughthethreshold-typepoliciesareconceptuallyquiteintuitive,itisverychallengingtodeterminetheoptimalthresholdlevelsbyusingsimulation.Theproposedalgorithmiseasiertoimplementandoptimizebyusingavailablesolvers.Thisstudycanbeextendedinanumberofways.Thesameapproachcanbeusedtoderiveresultsfordifferentserviceleveldefinitions.Yıldırım(2004)reportspreliminaryresultsforType2andModifiedType2servicelevels.Theformulationofthemulti-productcaseisalsostraightforward.Theeffectsofdemandvariability,productioncost,andtheleadtimeontheproductionandsourcingplansneedfurtherinvestigation.Sincetheoptimalsolutiontothegeneralproblemisnotknownforthedynamiccase,investigationofthestatic 360I.Yıldırımetal.caseorastylizedmodelcanyieldinsightsregardingtheinteractionofdemandvariability,cost,andtheleadtime.AppendixProofofProposition2Weuseinductiontoshowthati.Iftheinventorylevelsatthebeginningofthefirstperiodareequal,I0(BSLT)=I0(DEPLT)=lLT+1,thenproductionquantitiesinthefirstperiodandtheinventoryattheendoffirstperiodforbothpoliciesbecomeequal,i.e.X1(BSLT)=X1(DEPLT)=0andI1(BSLT)=I1(DEPLT)=lLT+1−d1;ii.Iftheinventorylevelsattheendofperiodt1suchthatt1≤LTareequal,t1It1(BSLT)=It1(DEPLT)=lLT+1−τ=1dτ,thentheproductionquantitiesinperiod(t1+1)andtheinventorylevelsattheendofperiod(t1+1)forbothpoliciesbecomeequal;i.e.Xt1+1(BSLT)=Xt1+1(DEPLT)=dt1andt1+1It1+1(BSLT)=It1+1(DEPLT)=lLT+1−τ=1dτ.andiii.Iftheinventorylevelsattheendofperiod(LT+1)areequal,LT+1ILT+1(BSLT)=ILT+1(DEPLT)=lLT+1−τ=1dτ,thenproductionquanti-tiesinperiod(LT+2)andtheinventorylevelsattheendofperiod(LT+2)forbothpoliciesbecomeequal,i.e.XLT+2(BSLT)=XLT+2(DEPLT)=dLT+1LT+2andILT+2(BSLT)=ILT+2(DEPLT)=lLT+1−τ=2dτ;iv.Iftheinventorylevelsattheendofperiodt2suchthatt2≥LTareequal,t2It2(BSLT)=It2(DEPLT)=lLT+1−τ=t2−LTdτ,thentheproductionquantitiesinperiod(t2+1)andtheinventorylevelsattheendofperiod(t2+1)forbothpoliciesbecomeequal;i.e.Xt2+1(BSLT)=Xt2+1(DEPLT)=dt2t2+1andIt2+1(BSLT)=It2+1(DEPLT)=lLT+1−τ=t2+1−LTdτ.AssumethattheinitialinventorylevelsareequalsuchthatI0(BSLT)=S2,I0(DEPLT)=lLT+1andS2=lLT+1.Inthebasestockpolicy,eachde-mandobservedisproducedinthenextperiod;thereforethereisnoproductioninthefirstperiod,X1(BSLT)=0.Inthedeterministicequivalentapproach,theproductionquantityinthefirstperiodisdeterminedaccordingtotheconstraintLTX1(DEPLT)+τ=1SRτ(DEPLT)+I0(DEPLT)=X1(DEPLT)+0+lLT+1≥lLT+1andtherefore,X1(DEPLT)≥0.Sincetheproblemisofminimizationtype,theproductionquantityinthefirstperiodequalszero,i.e.X1(DEPLT)=0.Next,acustomerdemandofd1arrives.Theendofperiodinventoryforthebasestockpol-icybecomesI1(BSLT)=I0(BSLT)+SR1(BSLT)−d1=S2+0−d1=S2−d1andtheendofperiodinventoryforthedeterministicequivalentapproachbecomesI1(DEPLT)=I0(DEPLT)+SR1(DEPLT)−d1=lLT+1+0−d1=lLT+1−d1.SinceweknowthatS2=lLT+1,I1(BSLT)=I1(DEPLT).Inthesecondperiod,thebasestockpolicyproducesthedemandofthefirstperiod,i.e.X2(BSLT)=d1.Atthebeginningofthesecondperiod,thedeter-ministicequivalentmodelisrerunsinceitissolvedonarollinghorizonbasis. Amultiperiodstochasticproductionplanningandsourcingproblem361Thedemandisassumedtobestationaryovertheplanninghorizon.Althoughsolvingthemodelonarollinghorizonbasisthroughouttheplanninghorizonre-quiresintegrationoftheminimumcumulativeproductionquantitesforthenum-berofperiodsintherollinghorizonintothemodel,onlytheminimumcumula-tiveproductionquantityofperiod(LT+1),lLT+1,isfullyutilized.Theproduc-tionquantityofthedeterministicequivalentmodelinthesecondperiodisdeter-LT+1minedbyX2(DEPLT)+τ=2SRτ(DEPLT)+I1(DEPLT)=X2(DEPLT)+X1(DEPLT)+I1(DEPLT)=X2(DEPLT)+0+lLT+1−d1≥lLT+1;therefore,X2(DEPLT)≥d1.Inordertominimizetheproductioncosts,theproductionquan-tityinthesecondperiodequalsthedemandofthefirstperiod,i.e.X2(DEPLT)=d1.Afterthearrivalofacustomerdemandofd2,theendofperiodinventoryforthebasestockpolicybecomesI2(BSLT)=I1(BSLT)+SR2(BSLT)−d2=S2−d1−d2andtheendofperiodinventoryforthedeterministicequivalentapproachbe-comesI2(DEPLT)=I1(DEPLT)+SR2(DEPLT)−d2=lLT+1−d1−d2.SinceS2=lLT+1,wecansaythatI2(BSLT)=I2(DEPLT).Sincedemandduringleadtimecannotbesatisfiednosoonerthan(LT+1)peri-odsoftime,theinventorylevelsattheendofanyperiodt1suchthatt1≤(LT−1)canbewrittenasI(BSLT)=S−t1d,I(DEP)=l−t1dandt12τ=1τt1LT+1τ=1τS2=lLT+1.Inperiod(t1+1),thebasestockpolicyproducesXt1+1(BSLT)=dt1.Inthedeterministicequivalentapproach,theproductionquantityisdeterminedt1+LTbytheconstraintXt1+1(DEPLT)+τ=t1+1SRτ(DEPLT)+It1(DEPLT)=t1Xt1+1(DEPLT)+τ=1Xτ(DEPLT)+It1(DEPLT)=Xt1+1(DEPLT)+t1−1t1τ=1dτ+lLT+1−τ=1dτ≥lLT+1;therefore,Xt1+1(DEPLT)≥dt1.Sincetheproblemisofminimizationtype,Xt1+1(DEPLT)=dt1.Then,acustomerde-mandofdt1+1isobserved.Theendofperiodinventoryforthebasestockpolicyt1becomesIt1+1(BSLT)=It1(BSLT)+SRt1+1(BSLT)−dt1+1=S2−τ=1dτ−t1+1dt1+1=S2−τ=1dτandtheendofperiodinventoryforthedeterministicequiv-alentapproachbecomesIt1+1(DEPLT)=It1(DEPLT)+SRt1+1(DEPLT)−t1t1+1dt1+1=lLT+1−τ=1dτ−dt1+1=lLT+1−τ=1dτ.SinceS2=lLT+1,It1+1(BSLT)=It1+1(DEPLT).Similarly,dLT+1isproducedbythebasestockpolicyinperiod(LT+1),i.e.XLT+1=dLT+1.TheconstraintXLT+1(DEPLT)+2LTLTτ=LT+1SRτ(DEPLT)+ILT(DEPLT)=XLT+1(DEPLT)+τ=1Xτ+LT−1LTILT(DEPLT)=XLT+1(DEPLT)+τ=1dτ+lLT+1−τ=1dτ≥lLT+1;i.e.XLT+1(DEPLT)≥dLTdeterminestheproductionquantityofthedeterminis-ticequivalentmodelinperiod(LT+1).Then,XLT+1(DEPLT)=dLT.Next,acustomerdemandofdLT+1arrives.TheendofperiodinventoryforthebasestockpolicybecomesILT+1(BSLT)=ILT(BSLT)+SRLT+1(BSLT)−dLT+1=LTLTS2−τ=1dτ+X1(BSLT)−dLT+1=S2−τ=1dτ+0−dLT+1=LT+1S2−τ=1dτandtheendofperiodinventoryforthedeterministicequivalentap-proachbecomesILT+1(DEPLT)=ILT(DEPLT)+SRLT+1(DEPLT)−dLT+1=LTLTlLT+1−τ=1dτ+X1(DEPLT)−dLT+1=lLT+1−τ=1dτ+0−dLT+1=LT+1lLT+1−τ=1dτ.SinceS2=lLT+1,ILT+1(BSLT)=ILT+1(DEPLT). 362I.Yıldırımetal.Inperiod(LT+2),thebasestockpolicyproducesXLT+1(BSLT)=dLT+2.Forthedeterministicequivalentapproach,weknowthatXLT+2(DEPLT)+2LT+1LT+1τ=LT+2SRτ(DEPLT)ILT+1(DEPLT)=XLT+2(DEPLT)+τ=2Xτ+LTLT+1ILT+1(DEPLT)=XLT+2(DEPLT)+τ=1dτ+lLT+1−τ=1dτ≥lLT+1;i.e.XLT+2(DEPLT)≥dLT+1andthen,XLT+2(DEPLT)=dLT+1.AfterthearrivalofdLT+2,thefollowingendofperiodinventorylevelsareobservedLT+1ILT+2(BSLT)=ILT+1(BSLT)+SRLT+2(BSLT)−dLT+2=S2−τ=1dτ+LT+1LT+2X2(BSLT)−dLT+2=S2−τ=1dτ+d1−dLT+2=S2−τ=2dτandILT+2(DEPLT)=ILT+1(DEPLT)+SRLT+2(DEPLT)−dLT+2=lLT+1−LT+1LT+1τ=1dτ+X2(DEPLT)−dLT+2=lLT+1−τ=1dτ+d1−dLT+2=lLT+1−LT+2τ=2dτ.SinceweknowthatS2=lLT+1,ILT+2(BSLT)=ILT+2(DEPLT).Nowassumethatattheendofanyperiodt2suchthatt2≥t2(LT+1),It2(BSLT)=S2−τ=t2−LTdτ,It2(DEPLT)=lLT+1−t2τ=t2−LTdτandS2=lLT+1.Inperiod(t2+1),Xt2+1(BSLT)=dt2andXt2+1(DEPLT)isdeterminedbytheconstraintXt2+1(DEPLT)+t2+LTSR(DEPLT)+I(DEPLT)=X(DEPLT)+t2X+τ=t2+1τt2t2+1τ=1τt2−1t2It2(DEPLT)=Xt2+1(DEPLT)+τ=1dτ+lLT+1−τ=1dτ≥lLT+1;Xt2+1(DEPLT)≥dt2andsincethemodelisofminimizationtypeXt2+1(DEPLT)=dt2.Next,acustomerdemandofdt2+1arrives.Theendofpe-riodinventorylevelsforbothpoliciesbecomeIt2+1(BSLT)=It2(BSLT)+t2SRt2+1(BSLT)−dt2+1=S2−τ=t2−LTdτ+Xt2+1−LT(BSLT)−t2t2+1dt2+1=S2−τ=t2−LTdτ+dt2−LT−dt2+1=S2−τ=t2+1−LTdτandIt2+1(DEPLT)=It2(DEPLT)+SRt2+1(DEPLT)−dt2+1=S2−t2t2τ=t2−LTdτ+Xt2+1−LT(DEPLT)−dt2+1=S2−τ=t2−LTdτ+dt2−LT−t2+1dt2+1=S2−τ=t2+1−LTdτ.SinceweknowthatS2=lLT+1,It2+1(BSLT)=It2+1(DEPLT).Thisprovesourproposition.ReferencesAbernathyFH,DunlopJT,HammondJH,WeilD(1999)Astitchintime.OxfordUniversityPress,NewYorkAbernathyFH,DunlopJT,HammondJH,WeilD(2000)Controlyourinventoryinaworldofleanretailing.HarvardBusinessReviewNov-Dec:169–176AlbrittonM,ShapiroA,SpearmanM(2000)Finitecapacityproductionplanningwithran-domdemandandlimitedinformation.StochasticProgrammingE-PrintSeriesBeyerRD,WardJ(2000)NetworkserversupplychainatHP:acasestudy.HPLabsTechReport2000-84BitranGR,YanasseHH(1984)Deterministicapproximationstostochasticproductionprob-lems.OperationsResearch32:999–1018BitranGR,HaasEA,Maatsudo(1986)Productionplanningofstylegoodswithhighsetupcostsandforecastrevisions.OperationsResearch34(2):226–236BradleyJR(2002)OptimalcontrolofadualservicerateM/M/1production-inventorymodel.EuropeanJournalofOperationalResearch(2002)(forthcoming)CandeaD,HaxAC(1984)Productionandinventorymanagement.Prentice-Hall,NewJerseyClayRL,GrossmanIE(1997)Adisaggregationalgorithmfortheoptimizationofstochasticplanningmodels.ComputersandChemicalEngineering21(7):751–774 Amultiperiodstochasticproductionplanningandsourcingproblem363FeiringBR,SastriT(1989)Ademand-drivenmethodforschedulingoptimalsmoothpro-ductionlevels.AnnalsofOperationsResearch17:199–216KelleP,ClendenenG,DardeauP(1994)Economiclotschedulingheuristicforrandomdemands.InternationalJournalofProductionEconomics35:337–342LeeLH(1997)Ordinaloptimizationanditsapplicationinapparelmanufacturingsystems.Ph.D.Thesis,HarvardUniversity,Cambridge,MAQiuMM,BurchEE(1997)Hierarchicalproductionplanningandschedulinginamulti-product,multi-machineenvironment.InternationalJournalofProductionResearch35(11):3023–3042SoxCR,MuckstadtJA(1996)Multi-item,multi-periodproductionplanningwithuncertaindemand.IIETransactions28:891–900TanB,GershwinSB(2004)Productionandsubcontractingstrategiesformanufacturerswithlimitedcapacityandvolatiledemand.AnnalsofOperationsResearch(SpecialvolumeonStochasticModelsofProduction/InventorySystems)125:205–232TanB(2002)Managingmanufacturingrisksbyusingcapacityoptions.JournaloftheOp-erationalResearchSociety53(2):232–242VanDelftC,VialJ-PH(2003)Apracticalimplementationofstochasticprogramming:anapplicationtotheevaluationofoptioncontractsinsupplychains.Automatica(toappear)YangMS,LeeLH,HoYC(1997)Onstochasticoptimizationanditsapplicationstomanu-facturing.LecturesinAppliedMathematics33:317–331YıldırımI(2004)Stochasticproductionplanningandsourcingproblemswithservicelevelconstraints.M.S.Thesis,Koc¸University,IndustrialEngineering,Istanbul,TurkeyZapfelG(1996)Productionplanninginthecaseofuncertainindividualdemandextension¨foranMRPIIconcept.InternationalJournalofProductionEconomics46–47:153–164