Essays in production scheduling with just-in-time related performance measures

215 Pages·1994·8.8 MB·English

by  Krämer, Franz-Josef, 1965-

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