Essays in production scheduling with just-in-time related performance measures PDF
Preview Essays in production scheduling with just-in-time related performance measures
ESSAYSINPRODUCTIONSCHEDULING WITHJUST-IN-TIMERELATEDPERFORMANCEMEASURES BY FRANZ-JOSEFKRAMER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1994 ACKNOWLEDGMENTS TheauthorgratefullyacknowledgesDr. Chung-YeeLee,thecommitteechairman,and Dr. ShermanBai,thecochairman,fortheirexpertguidanceandtheirconstantencour- agementatanytimethroughoutthisstudy. AppreciationisexpressedtotheDepartment ofIndustrialandSystemsEngineeringoftheUniversityofFloridaandinparticulartoits chairmanDr.JackElzingaforthefinancialsupport. Theauthorwishestothankhisfamilyandfriendsfortheircontinuoussupportduring thecourseofthiswork.Specialthanksareduetohisdeceasedfatherwhoinhisearlyyears createdandnurturedhiscommitmenttoeducationandwhohasmadeabundantprovision forhisfamily,andtohislovingmotherwholethimgoacrosstheAtlantictopursuehis goalsanddreams. TheauthoralsowouldliketothankGod. Somanytimesduringthecreativepartof thisresearch,theLordhasprovenHimselffaithfultoHispromise: CommityourworkstotheLord, andyourthoughtsshallbeestablished. (Proverbs16:3). 11 TABLEOFCONTENTS ACKNOWLEDGMENTS ii ABSTRACT v 1 INTRODUCTION 1 2 COMMONDUE-WINDOWSCHEDULING 8 2.1 Introduction 8 2.2 NotationandGeneralProperties 9 2.3 DecisionVariableCase 12 2.4 ParameterCase 18 2.5 FinalRemarksandConclusions 23 3 PARALLELDUEWINDOWSCHEDULING 24 3.1 Introduction 24 3.2 NotationandGeneralProperties 26 3.3 ComplexityandOptimalityProperties 29 33..45 TGehneerTawliozaMtaiconhifnoertChaesemmachinecase 4398 3.6 Summary 53 4 OPTIMALCONTROLOFAPRODUCTIONSYSTEMWITHPERIODICMAIN- TENANCE 54 4.1 Introduction 54 4.2 ProblemDefinition 57 4.3 AOne-CycleProblemanditsOptimalSolution 59 4.3.1 SingularIntervals 62 4.3.2 SwitchingTimesandSwitchingPoints 68 4.3.3 InitialControlLevelu*(0) 73 4.3.4 OptimalSolution 78 4.4 ATwo-CycleProblemanditsOptimalSolution Ill 4.5 4444O....p4444t....i1234malISSOniwpisnittogtiiluacumllhtaaiiClrnoongnISntotPtlrooeuoirttlnvihtaoeLlsnesAvfffeofo-rlorcryPfPcPorrlrroeoobPbbplrllreoeeombmmbllPePPe2m2m2P(2N>3) 111114111126654 4.5.1 SingularIntervalsforProblemPV 144 4.5.2 SwitchingPointsforProblemPjv 144 4.5.3 InitialControlLevelforProblemPjv 144 4.5.4 OptimalSolutionforProblemP^v 144 iii 4.6 TransientStateandSteadyState 155 4.7 Summary 158 APPENDICES A PROOFOFLEMMA1 160 B ALGORITHMFORm=2PARALLELMACHINECASE 165 C CALCULATIONFORAPRODUCTIONSYSTEMWITHPERIODICMAINTE- NANCE 170 C.l Therangeof5i(a;(0))forProblemPi(CaseB4-2) 170 C.2 SwitchingPointforProblemPi(CaseB4-2) 172 C.3 TheSwitchingCurve 173 C.4 TheFirstDerivativeofJJ'^O)) 177 C.5 TheDerivativeofJ{(x(0))(CaseE4) 178 C.6 TheSecondDerivativeofJj*(2(0)) 181 C.7 APropertyofj^jJi(x(0)) 183 C.8 APropertyof^0)^2(x(0)) 187 C.9 AnInitialProductionControlforProblemP2 189 C.10Rewritingaquadraticequation(CaseC4-2) 191 C.ll52(a:(0))ismonotonedecreasing(C4-2-2) 195 C.12EvaluationofS2(d(a- (CaseC4-2-2) 197 REFERENCES 199 BIOGRAPHICALSKETCH 207 IV AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ESSAYSINPRODUCTIONSCHEDULING WITHJUST-IN-TIMERELATEDPERFORMANCEMEASURES By Franz-JosefKramer August1994 Chairman:Dr.Chung-YeeLee Cochairman: Dr.ShermanBai MajorDepartment:IndustrialandSystemsEngineering Just-In-Timeconceptsareofincreasingimportanceintoday’scompetitivemanufactur- ingenvironment.Inthiscontext,westudyseveraldiscreteschedulingproblemsaswellasa productionflowcontrolproblem. Thediscreteschedulingproblemsconsiderthecasewherewehaveacommonduewindow. Jobswhicharecompletedwithintheduewindowincurnopenalty. Completionbeforethe duewindowincursearlinesspenaltyandcompletionaftertheduewindowincurstardiness penalty.Ifthelocationoftheduewindowisadecisionvariable,wecanprovideapolynomial algorithmforthesinglemachinecase.Fortwoparallelmachines,weshowthattheproblem isNP-completeandusedynamicprogrammingtosolveit. Wepresentaheuristicwith polynomialcomplexityformparallelmachinesandshowthattheerrorboundvanishesas thenumberofjobsincreases. Ifthelocationoftheduewindowisagivenparameter,the v problemisNP-completeevenforthesinglemachinecase.Weusedynamicprogrammingto solvethisproblem. Themaincontributionofthisresearchistheextensionofthedue-date concepttothedue-windowconcept.Nopriorresearchhasanalyzedearlinessandtardiness performancemeasuresincombinationwithduewindows. Fortheproductionflowcontrolproblem,weconsiderasystemconsistingofonemachine forwhichmaintenanceisperformedonaperiodicbasis.Thetimeiscontinuousandthetime horizonisfinite. Foreachcycle,boththetimeintervalofoperationandthemaintenance intervalaredeterministic. Duringthemaintenanceinterval,thesystemisshutdownand cannotproduce. Onepart-typeisproducedandthedemandrateisassumedtobeconstant. Thesupplyisassumedtobeunlimited. Inordertomakeon-timedelivery,theobjective istokeeptheproductionasclosetothedemandaspossible. However,themaintenance disruptionsmaketheproductiondeviatefromthedemand. Wemodeltheproblemasan optimalcontrolproblemandsolveitbyPontryagin’sMinimumPrinciple. Wediscussboth thetransient stateandthesteadystate. Themaincontributionistheanalysis ofthe transientstate. vi CHAPTER1 INTRODUCTION Inthe1990s,internationalcompetitionwillincreaseduetoanumberofreasons. One ofthesereasonsisthat advancementsininformationandprocess technologiesoccur at anacceleratedpace. Also,theconsumerdemandsareincreasinglysophisticatedandin- dividualized. Thisresultsintheneedforoperationalmodelswhichreflectlong-termand nonfinancialperformancemeasuressuchascustomersatisfactionwithboththeproduct anditsservice. Revolutionarychangesinthemanagementofproductionsystemsareyet anotherfactorwhichcontributestoanincreasedcompetition. World-Class-Manufacturingrequiresanintegratedmanufacturingstrategy,encompass- ing TotalQuality Control(TQC),Just-In-Time (JIT) Production, FactoryAutomation (FA),andTotalProductiveMaintenance(TPM).Inthisdissertation,wewillfocusour attentiontotheJust-In-Timeproductionstrategy.AccordingtotheJust-In-Timephiloso- phy,theidealsituationistotoeliminateallinventory. Inventoryinitsenlargeddefinition notonlyreferstoeitherpartiallyorfullycompletedproducts,butincludesforexample alsoexcessamountoflaborandmachinetimewhichiscommonlyreferredtoasidletime. TheJust-In-Timephilosophyseekstoavoidinventoryofanykind. Anothermotivationto reduceinventoryisthatinventoryhidesrootcausesofaproblem. Forexample,inventory hidesproblemssuchasrework,machinedowntime,changeorder,longsetups,andvendor delinquencies. Itisthereforedesirabletoidentifyandeliminatewastecomponentssuchas overproduction,waitingtime,transportation,processing,inventory,movement,anddefec- tiveproducts.TheseareimportantstepstowardaJust-In-Timeenvironment.Theprocess 1 2 ofcreatingaJust-In-Timeproductionenvironmentismulti-facetedandinvolvesmanycon- cepts.AveryimportantcomponentinthisprocessistheutilizationofJust-In-Timeoriented productionschedulingtechniques. At this point, wenotethat the areaofboth sequencingand scheduling isofgreat importanceandismotivatedbyproblemsthatariseinproductionplanningandcontrol, andingeneralallsituationsinwhichscarceresourceshavetobeallocatedtoactivities overtime. GeneralreferencesonthesubjectaretheclassicbookbyConway,Maxwelland Miller [30], theintroductorytextsby Baker[17] and French [40],thevolumeedited by Dempster,Lenstra,andRinnooyKan[34],andsurveypaperssuchasbyGraves[51]inthe areaofproductionplanning,andbyLawler,Lenstra,RinnooyKan,andShmoys[75]inthe generalareaofSequencingandScheduling. SequencingandScheduling“isconcernedwith theoptimalallocationofscarceresourcestoactivitiesovertime''’([75]),andmorethanany otherareainoperationsresearch,thetheoryofsequencingandschedulingischaracterized byavirtuallyunlimitednumberofproblemtypes. Inregardtothedeterministicmodels, theproblemscanbeclassifiedaccordingtotheircomplexity. Complexitytheoryprovides amathematicalframeworkwhich canclassify problemsas “easy” oras “hard” (orNP- complete). PioneeringworkinthetheoryofNP-completenesshasbeendonebyCook[28] andKarp[64]and[65].ThetextbyGareyandJohnson[46]isaclassicalguideinthisarea. Formanyyears,schedulingresearchfocusedonsingleperformancemeasuresthatare nondecreasinginjobcompletiontimes. Thesearealsoreferredtoasregularperformance measures. Examplesinclude meanflowtime, meanlateness, numberoftardyjobs,and totaltardiness. Regulardue-date-relatedperformancemeasuresareunabletocaptureany penaltyforearlyjobcompletion. Wethusarerequired tousenonregularperformance measuresinordertomodeltheobjectiveofJust-In-Timeproductionschedulingproblems. Suchperformancemeasurespenalizebothearlinessandtardinesswhereearlinessoccurs whenajobiscompletedbeforetheduedateandtardinessisincurredwhenajobiscompleted aftertheduedate. 3 Inthisstudy,weexamineseveralproductionschedulingproblemswherebothearlyand tardyjobcompletionsarepenalized. Theproblemscanbecategorizedintwogroups:The firstgroupconsistsofproblemswhicharemodelledby discreteschedulingmodels. The secondgroupconsistsofproblemswhicharemodelledbyproductionflowcontrolmodels. Fordiscreteschedulingmodels,weconsidertwoproblemsinChapters2and3.Theper- formancemeasureisdue-dateorientedandpenalizesbothearlinessandtardiness.Accord- ingtotheremarkabove,suchaperformancemeasureisanonregularperformancemeasure. Morespecifically,weconsideradue-datewindowwhichconsistsofanearliestdue-dateand alatestdue-date. Thereareveryfewreferenceswhichallowforazeropenaltywhenthe completiontimefallswithinatimeinterval(seeChapter2),andnoneofthemdevelopthe conceptofdue-windowscheduling. Asurveyofearlinessandtardinessrelatedliteratureis giveninBakerandScudder[20]wherethedue-daterelatedperformancemeasuresinmost casesrefertothesamepointintimefortheearlinesspenaltyaswellasforthetardiness penalty. Whilethismodelisapplicableinmanysituations(forexample,aproductionfine wheredeviationfromapointintimeaffectstheprocessingofthepartatthenextmachine), thereareothersituationswherethereisatimewindowduringwhichnocostisincurred. Consider,forexample,thesituationwhereatruckcanbeloadedovernight. Sothereisa timewindowfromthetimewhenthedriverdeliversthetruckintheeveningtothetime whenthedriverstartsthetripthenextmorningwherethereisnopenaltyfortheproducts whicharecompletedandloadedduringthenightshift.Productswhicharecompletedearly requireholdingandproductswhicharefinishedtardyrequirespecialshipment,forexam- ple. Thus,forsituationswherethedue-dateasonepointintimedoesnotsatisfactorily reflectthesituation,weconsideranintervaloftimeforeachjobduringwhichnopenalty isincurred.Werefertothistimewindowasduewindow. Inthesecondchapter,weconsiderasinglemachineproblemwherenjobshavetobe processed. Wehaveacommonduewindow[e,d]withearliestdue-dateeandlatestdue- datedwhichisthesameforeachjob. Foracompletionbeforetheearliestdue-datee,an 4 earlinesscostisincurred. Completionafterthelatestdue-datedresultsintardinesscost. Thesizeoftheduewindowd—eisagivenparameter.Jobswhicharecompletedwithinthe duewindowdonotincurapenalty. Ourobjectiveistofindaschedulewhichminimizesthe totalpenalty.Wedistinguishbetweentheparametercase,inwhichthelocationofthedue windowisgiven,andthedecisionvariablecase.Inthedecisionvariablecase,thelocation oftheduewindowistobedetermined. Wedonotconsidercostfordue-dateassignment. Forthedecisionvariablecaseandacertaincoststructure,weareabletoprovidean algorithmwhichrequires0(n)time,providedtheindicesaresortedaccordingtotheshortest processingtimeorder. Iftheindicesarenotsorted,thecomplexityis0(nlog(n)). Since eventheeasiestcasewithunitweightsisNP-completeford—e(seeChapter2),thereis norealistichopeinfindingapolynomialalgorithmforthisproblem.Thus,weusedynamic programmingtosolvetheparametercase. Weobservethatinthecasewherethesizeof theduewindowexceedsthelargestprocessingtime,weobtainamoreefficientalgorithm thanforthecorrespondingproblemforthecommonduedatecased=e. Theresultsof thisresearcharetoappearinKramerandLee[72]. Inthethirdchapter,westudytheextensionofthisproblemtoanenvironmentwith parallelmachines. Concerningthelocationoftheduewindow,weconsiderthedecision variablecase.Weshowthattheeasiesttwo-machineproblem(whichhasunitcost)isNP- completeandprovideadynamicprogrammingalgorithmtosolvefortheoptimalsolution. Subsequently,wepresentaheuristicforthetwo-machineandforthem-machinecase. We giveanexpressionfortheupperboundoftheabsoluteerrorandshowthattherelative errorvanishesasthenumberofjobsincreases.Theresultsofthisresearcharetoappearin KramerandLee[73]. Chapter3concludestheproblemswithdiscreteschedulingmodels. Wethenturntothesecondgroupofproblemswhicharemodelledasproductionflow controlmodels. Thetermproductionflowcontrolmodelincludestwoconcepts:production flowandcontrol. Bymodellingtheproductionasaproductionflow,wemodeladiscrete processbyacontinuousproductionrate. Thishasthedisadvantageofintroducingsome
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