Table Of ContentLecture Notes in Economics
and Mathematical Systems 644
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Julia Drechsel
Cooperative Lot Sizing
Games in Supply Chains
123
Julia Drechsel
Germany
[email protected]
Zugl.: Duisburg-Essen, Univ., Diss., 2009
ISSN 0075-8442
ISBN978-3-642-13724-2 e-ISBN978-3-642-13725-9
DOI10.1007/978-3-642-13725-9
Springer Heidelberg Dordrecht London New York
LibraryofCongressControlNumber: 2010934000
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Preface
ThisbookresultsfrommyworkasaresearchassistantattheUniversitiesofFreiberg
andDuisburg-Essen.Startingthisresearchwasverychallengingdueto the highly
mathematically coined literature regarding game theoretical topics. The literature
showsthatcooperativegametheoryiswellstudiedfromatheoreticalpointofview,
but that fields of application are fairly limited up to now. Hence, the following
researchquestionarose:Whatcanbedonetomakeconceptsfromcooperativegame
theorymoreapplicabletopracticalproblems?
The field of supply chain managementis predestined for many kinds of coop-
eration because companies will be more successful if they consider relationships
tosuppliers,customers,and/orcompetitors.Therefore,thisworkfocusesoncoop-
erationsin supply networks(horizontaland verticalcooperation)concerningjoint
orderingand/orjointproduction.
Problemsoforderingandproductionforsingledecisionmakersarewellstudied
in operations research and are used as a basis to develop cooperative models that
displaysituationsofcooperativedecisionmaking.Beingoneofthemostessential
problemsincooperations,theallocationofjointprofitsorcostsisthefundamental
questionincooperativegametheoryandwillbethebasicquestionansweredbythis
workaswell.
Inbothplacesofmydoctorialstudies,FreibergandDuisburg,Ihadvaluablesup-
portwithoutwhichthisworkwouldnothavebeenpossible.Firstofall,IthankAlf
Kimmsforprovidingmethechancetoworkonthistopic,hisguidance,challenging
discussions,andthepossibilitytocontinuemyworkinDuisburg.
I express my gratitude to Sarah Bretschneider, Demet Çetiner, Michaela Graf,
Klaus-Christian Maassen, and Michael Müller-Bungart for providing me such a
greatresearchandworkingatmosphereinFreibergandDuisburg–allofyouhelped
methroughstruggleswithresearch,programming,orteaching,aswellassometimes
non-work related things, which was always valuable and helpful and made these
fouryearsawonderfultime.Furthermore,IwouldliketothankAnnetteHoffmann
andNadineKrumpholzforwelcomingmeinDuisburg,StefanieKockerolsforpro-
vidingmehelpwiththeliteratureresearchandÇag˘das¸ÖzgürforaconstantLATEX
support.
IamdeeplyindebtedtoIngmarSchaaffforhislove,hispatience,andforkeep-
ingmegroundedwheneveritwasneeded.Finally,Ithankmyparents,Nicoleand
v
vi Preface
StephanDrechsel,formakingsomanythingspossible,fortheirsupport,andtheir
trust.
Duisburg JuliaDrechsel
May2010
Contents
1 Introduction.................................................................... 1
2 SelectedTopicsinCooperativeGameTheory.............................. 5
2.1 HistoryofGameTheory................................................. 5
2.2 BasicsinCooperativeGameTheory.................................... 8
2.2.1 ACooperativeGame........................................... 9
2.2.2 PropertiesofCooperativeGames ............................. 11
2.2.3 Variants and FundamentalApplications
oftheClassicalCooperativeGame............................ 13
2.2.4 Interval-ValuedGames......................................... 15
2.3 AllocatingCooperativeCosts ........................................... 17
2.3.1 MotivationandClassificationofAllocationMethods........ 17
2.3.2 PropertiesofCostAllocations................................. 18
2.3.3 Non-Game-TheoreticalCostAllocationMethods............ 20
2.3.4 TheCore........................................................ 22
2.3.5 AdditiveCoreVariants......................................... 26
2.3.6 MultiplicativeCoreVariants................................... 29
2.3.7 TheSubcoalition-PerfectCore ................................ 32
2.3.8 TheIntervalCore............................................... 36
2.3.9 TheShapleyValue ............................................. 37
2.3.10 Conclusions .................................................... 38
3 AlgorithmicGameTheory ................................................... 41
3.1 Literature................................................................. 42
3.2 ComputingCoreCostAllocations...................................... 43
3.3 TheoreticalBackground................................................. 45
3.4 IncludingFairnessCriteria .............................................. 46
3.5 ComputingCoreVariants................................................ 49
3.6 ComputingIntervalCoreElements..................................... 51
3.7 Conclusions.............................................................. 52
vii
viii Contents
4 CooperationinSupplyChains............................................... 55
4.1 HorizontalversusVerticalCooperation................................. 56
4.2 SupplyChainGamesintheLiterature.................................. 59
5 AnEconomicLotSizingGame .............................................. 63
5.1 CooperativeOrderingSituations........................................ 63
5.1.1 TheUnderlyingProblem....................................... 63
5.1.2 PropertiesoftheELSGame................................... 65
5.2 ComputingCoreCostAllocationsfortheELSGame ................. 65
5.2.1 TheRowGenerationProcedure............................... 66
5.2.2 ANumericalExample ......................................... 69
5.3 ComputationalStudyfortheELSGame ............................... 73
5.4 ExtensionsfortheELSGame........................................... 79
6 ALotSizingGamewithUncertainDemand............................... 81
6.1 TheUnderlyingProblem................................................ 81
6.2 SpecialPhenomenaofIntervalCores................................... 82
6.3 ANewDefinitionoftheIntervalCoreandItsComputation .......... 83
6.4 ComputationalStudyfortheIntervalELSGame...................... 85
7 A CapacitatedLotSizingGamewithTransshipments,
ScarceCapacities,andPlayer-DependentCostCoefficients.............. 87
7.1 CooperativeProductionSituations...................................... 87
7.1.1 TheUnderlyingProblem....................................... 87
7.1.2 TheCLSPGame ............................................... 89
7.1.3 PropertiesoftheCLSPGame ................................. 93
7.2 SolvingtheCooperativeCLSP.......................................... 94
7.2.1 ALagrangeanRelaxationBasedHeuristic ................... 94
7.2.2 AFix-and-OptimizeHeuristic................................. 98
7.3 ComputingCoreCostAllocationsfortheCLSPGame................101
7.3.1 TheRowGenerationProcedure...............................101
7.3.2 ComputingtheSubcoalition-PerfectCore....................104
7.3.3 ComputingtheMinmaxCore .................................106
7.4 ComputationalStudyfortheCLSPGame..............................106
7.4.1 ComputationalStudy:LagrangeanRelaxation
BasedHeuristic.................................................107
7.4.2 ComputationalStudy:Fix-and-OptimizeHeuristic ..........109
7.4.3 ComputationalStudy:Subcoalition-PerfectCore ............114
7.4.4 ComputationalStudy:MinmaxCore..........................115
7.5 ExtensionsfortheCLSPGame.........................................117
8 AMultilevelLotSizingGamewithRestrictedCooperation .............119
8.1 CooperativeSupplySituations..........................................119
8.1.1 TheUnderlyingProblem.......................................119
8.1.2 GameswithRestrictedCooperation...........................124
8.1.3 PropertiesoftheMLCLSPGame.............................129
Contents ix
8.2 ComputingCoreCostAllocationsfortheMLCLSPGame ...........129
8.2.1 TheRowGenerationProcedure...............................130
8.2.2 ANumericalExample .........................................133
8.2.3 ComputingCoreVariants......................................135
8.3 ComputationalStudyfortheMLCLSPGame .........................136
9 ConclusionsandFutureResearch...........................................141
A ComputationalStudyCLSPGame..........................................145
A.1 ComputationalStudy:LagrangeanRelaxationBasedHeuristic.......145
A.2 ComputationalStudy:Fix-and-OptimizeHeuristic....................149
B ComputationalStudyMLCLSPGame .....................................151
Bibliography........................................................................155
