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Demand Management Models for Two-Echelon Supply Chains

Permanent Link: http://ufdc.ufl.edu/UFE0021196/00001

Material Information

Title: Demand Management Models for Two-Echelon Supply Chains
Physical Description: 1 online resource (174 p.)
Language: english
Creator: Bakal, Ismail Serdar
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: pricing
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In the majority of classical supply chain management and inventory theory literature, demand arises from exogenous sources upon which the firm has little or no control. In many practical contexts, however, enterprises have numerous tools to generate, shape and manipulate the demand they face. This phenomenon introduces a new dimension to supply chain planning problems; the interaction and coordination between the demand and supply side of the supply chain. In our study, we introduce and study demand management tools that are integrated into classical supply chain planning models. We present a nonlinear, combinatorial optimization model to address planning decisions in both deterministic and stochastic supply chain settings, where a firm constructs a demand portfolio from a set of potential markets having price-sensitive demands. We first consider a pricing strategy that dictates a single price throughout all markets and provide an efficient algorithm for maximizing total profit. We also analyze the model under a market-specific pricing policy and describe its optimal solution. An extensive computational study characterizes the effects of key system parameters on the optimal value of expected profit, and provides some interesting insights on how a given market's characteristics can affect optimal pricing decisions in other markets. We analyze the implications of order timing decisions in multi-retailer supply systems in a single period, newsvendor setting. Specifically, we investigate a supply chain with multiple retailers and a single supplier where one of the retailers is considered a preferred or primary customer of the supplier. We compare the expected supplier and retailer profits under two order commitment strategies and specify conditions under which a particular commitment scheme benefits the supplier, the primary retailer, and the entire system. We compare our findings to a single-retailer system, and investigate the effects of capacitated supply. Observing that the outcome of the strategic interaction between the supplier and his primary customer is not in alignment with the supplier's preference, we propose and evaluate a number of demand management tools that the supplier can utilize in order to achieve his desired order commitment scheme. These tools include a capacity limit on the production quantity of the supplier, reallocation of the leftovers to the primary customer after demand realizations, and offering a discounted wholesale price. We also perform a comparative analysis of these tools and assess their effectiveness under various settings through a computational study.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ismail Serdar Bakal.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Geunes, Joseph P.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2008-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021196:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021196/00001

Material Information

Title: Demand Management Models for Two-Echelon Supply Chains
Physical Description: 1 online resource (174 p.)
Language: english
Creator: Bakal, Ismail Serdar
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: pricing
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In the majority of classical supply chain management and inventory theory literature, demand arises from exogenous sources upon which the firm has little or no control. In many practical contexts, however, enterprises have numerous tools to generate, shape and manipulate the demand they face. This phenomenon introduces a new dimension to supply chain planning problems; the interaction and coordination between the demand and supply side of the supply chain. In our study, we introduce and study demand management tools that are integrated into classical supply chain planning models. We present a nonlinear, combinatorial optimization model to address planning decisions in both deterministic and stochastic supply chain settings, where a firm constructs a demand portfolio from a set of potential markets having price-sensitive demands. We first consider a pricing strategy that dictates a single price throughout all markets and provide an efficient algorithm for maximizing total profit. We also analyze the model under a market-specific pricing policy and describe its optimal solution. An extensive computational study characterizes the effects of key system parameters on the optimal value of expected profit, and provides some interesting insights on how a given market's characteristics can affect optimal pricing decisions in other markets. We analyze the implications of order timing decisions in multi-retailer supply systems in a single period, newsvendor setting. Specifically, we investigate a supply chain with multiple retailers and a single supplier where one of the retailers is considered a preferred or primary customer of the supplier. We compare the expected supplier and retailer profits under two order commitment strategies and specify conditions under which a particular commitment scheme benefits the supplier, the primary retailer, and the entire system. We compare our findings to a single-retailer system, and investigate the effects of capacitated supply. Observing that the outcome of the strategic interaction between the supplier and his primary customer is not in alignment with the supplier's preference, we propose and evaluate a number of demand management tools that the supplier can utilize in order to achieve his desired order commitment scheme. These tools include a capacity limit on the production quantity of the supplier, reallocation of the leftovers to the primary customer after demand realizations, and offering a discounted wholesale price. We also perform a comparative analysis of these tools and assess their effectiveness under various settings through a computational study.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ismail Serdar Bakal.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Geunes, Joseph P.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2008-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021196:00001


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Thanksareduersttomysupervisor,Dr.JosephGeunes,forhisgreatinsights,perspectives,guidanceandsupport.IthankDr.ElifAkcal,Dr.AydnAlptekinogluandDr.EdwinRomeijnforservingasmydissertationcommitteemembersandfortheircontributionsandhelpfulsuggestions.Iextendmysinceregratitudeto_IbrahimKarakayal,Dr.PelinBayndr,andDr.TevhideAltekinfortheirendlesssupportandunderstanding.Itwouldhavebeenmuchmoredicultformetogetthroughthisprocesswithouttheirendlesssupportandunderstanding.IcannotpossiblyexpresshowgratefulIamtomybestfriend,Dr.GuvencSahin.Hehasbeenbymysideforyears,evenwhenheisthousandsofmilesaway,andIcouldnothaveaccomplishedthiswork,amongmanyotherthings,withouthim.IthankmyparentsOmerandFatmaBakal,andmybelovedsistersRukenandEsenfortheirunconditionalloveandsupport.Lastly,Ithankmyancee,Pnar,forhersupportandencouragementtocompletemydegree.Itwasthebiggestmotivationforthisworkthatwewouldbetogetherforeveruponthecompletionofmydegree. 3

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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 2LITERATUREREVIEW .............................. 17 2.1PricingandMarketSelection ......................... 17 2.2OrderTimingTradeos ............................ 22 3MARKETSELECTIONDECISIONSFORINVENTORYMODELSWITHPRICE-SENSITIVEDEMANDS .......................... 29 3.1ProblemDescriptionandAssumptions .................... 31 3.1.1NewsvendorModelwithMarketSelectionandPricing ........ 32 3.1.2EOQModelwithMarketSelectionandPricing ............ 35 3.2SolutionAlgorithmsfor(MSP) ........................ 36 3.2.1MarketSelectionwithaSinglePrice{(MSP-S) ........... 36 3.2.2MarketSelectionwithMarket-SpecicPrices{(MSP-MS) ..... 40 3.2.3ApplicationoftheAlgorithms:ANewsvendorExample ....... 41 3.3ComputationalAnalysis ............................ 44 3.3.1DeterministicSingle-PeriodModel ................... 45 3.3.2NewsvendorModel ........................... 47 3.3.2.1Lineardemandmodel .................... 48 3.3.2.2Iso-elasticdemandmodel .................. 53 3.3.3EconomicOrderQuantityModel .................... 55 3.3.3.1LinearDemandModel .................... 55 3.3.3.2Iso-elasticDemandModel .................. 57 3.4Conclusion .................................... 58 4ANALYSISOFORDERTIMINGTRADEOFFSINMULTI-RETAILERSUPPLYSYSTEMS ....................................... 60 4.1ProblemDescriptionandModelAnalysis ................... 65 4.1.1DelayedCommitment .......................... 66 4.1.2EarlyCommitment ........................... 68 4.2ComparisonoftheStrategies .......................... 69 4.2.1TheSupplier ............................... 70 4.2.2PrimaryRetailer'sProt ........................ 74 4

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.......................... 76 4.3ComparisonwiththeSingleRetailerSystem ................. 78 4.4ComputationalAnalysis ............................ 80 4.5OrderTimingInteractionbetweentheRetailers ............... 86 4.6OrderTimingDecisionsinaCapacitatedSetting .............. 90 4.6.1TheSupplier ............................... 90 4.6.2ThePrimaryRetailer .......................... 96 4.6.3SystemPerspective ........................... 96 4.7ConcludingRemarks .............................. 101 5STRATEGICTOOLSFORDEMANDMANAGEMENT ............. 103 5.1AnalysisofStrategicOrderTiming ...................... 105 5.1.1PrimaryRetailer'sLead ......................... 106 5.1.2Supplier'sLeadwithObservableCapacity ............... 108 5.1.3Supplier'sLeadwithUnobservableCapacity ............. 118 5.2EarlyCommitmentwithRecourse ....................... 123 5.2.1Primaryretailer'sLead ......................... 124 5.2.2Supplier'sLeadwithCredibleInformationSharing .......... 130 5.2.3SuppliersLead:UnreliableSupplierCase ............... 136 5.3WholesalePricing ................................ 140 5.3.1HomogenousWholesalePrice ...................... 141 5.3.2DierentWholesalePrices ....................... 144 5.4ComparativeAnalysisandConcludingRemarks ............... 147 6CONCLUSION .................................... 150 APPENDIX AAPPENDIXFORCHAPTER3 ........................... 153 A.1EquivalenceofAdditiveandMultiplicativeRandomness ........... 153 A.2ProofofConcavity:(MSP-S) ......................... 153 A.3ProofofConcavity:(MSP-MS) ........................ 154 A.4ProofofProposition 3.1 ............................ 155 A.5DierentStandardDeviationFunctionsfortheNewsvendorModel ..... 155 BAPPENDIXFORCHAPTER4 ........................... 159 B.1EvaluationoftheSupplier'sProt ....................... 159 B.1.1[QDS]U[QES]U:kmTk11+km2 159 B.1.2[QDS]U>[QES]U:kmT>k11+km2 160 B.2EvaluationoftheTotalSystemProt ..................... 161 B.2.1ProofofLemma 4.1 ........................... 161 B.2.2[QDS]U[QES]U:kmTk11+km2 162 B.2.3[QDS]U>[QES]U:kmT>k11+km2 164 CAPPENDIXFORCHAPTER5:DISTRIBUTIONOFRANDOMINSTANCES 166 5

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....................................... 167 BIOGRAPHICALSKETCH ................................ 174 6

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Table page 3-1SummaryofthealgorithmfortheNewsvendorexample. ............. 42 3-2OptimalsolutionsfortheNewsvendorexample. .................. 43 3-3MarketselectiondecisionsfortheNewsvendormodel. ............... 49 3-4Eectsofuncertaintyonmarketselectiondecisions. ................ 51 3-5AveragepercentagedierencesinprotsfortheNewsvendormodel. ....... 52 3-6MarketselectiondecisionsfortheEOQmodel. ................... 55 3-7AveragepercentagedierencesinprotsfortheEOQmodel. ........... 56 4-1Parameters. ...................................... 80 4-2Strategicinteractionbetweentheretailers. ..................... 87 4-3Illustrationofrisk-dominance. ............................ 89 5-1Parameters. ...................................... 108 5-2Outcomeofstrategicordertiming .......................... 113 5-3Strategicuseofcapacityoverk1andk2. ...................... 117 5-4Strategicuseofcapacityovercv1andcv2. ..................... 118 5-5Strategicuseofcapacityover1and2. ...................... 119 5-6Strategicgamebetweenthesupplierandtheprimaryretailer ........... 121 5-7Example(i)forthestrategicgamewhenR1prefersearlycommitment. ..... 122 5-8Example(ii)forthestrategicgamewhenR1prefersearlycommitment. ..... 123 5-9Strategicuseofleftoversoverk1andk2. ...................... 134 5-10Strategicuseofleftoversovercv1andcv2. ..................... 135 5-11Strategicuseofleftoversover1and2: 136 5-12Strategicuseofwholesalepriceoverk1andk2. ................... 146 5-13Strategicuseofwholesalepriceovercv1andcv2. .................. 146 5-14Strategicuseofwholesalepriceover1and2. .................. 147 C-1Distributionofinstancesoverk1andkm. ...................... 166 7

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..................... 166 C-3Distributionofinstancesover1and2. ...................... 166 8

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Figure page 1-1Integratingdemandandsupplychainmanagement ................ 14 3-1Deterministicdemandanalysis:protsasafunctionofa2. ............ 46 3-2Deterministicdemandanalysis:percentagedierenceinprots. ......... 46 3-3Stochasticdemandanalysis:protasafunctionofa2. .............. 48 3-4Stochasticdemandanalysis:percentagedierenceinprots. ........... 48 4-1PercentincreaseinR1'sexpectedprot:eectsof1 82 4-2PercentincreaseinR1'sexpectedprot:eectsof2 82 4-3PercentincreaseinR1'sexpectedprot:eectsofcv1 83 4-4PercentincreaseinR1'sexpectedprot:eectsofcv2 84 4-5[S]asafunctionofKwhen[QDS]U<[QES]U. .................. 93 4-6[S]asafunctionofKwhen[QDS]U>[QES]U:sampledata(i). ......... 94 4-7[S]asafunctionofKwhen[QDS]U>[QES]U:sampledata(ii). ........ 95 4-8[T]asafunctionofKwhen[QDS]U<[QES]U:sampledata(i) ......... 99 4-9[T]asafunctionofKwhen[QDS]U<[QES]U:sampledata(ii) ......... 99 4-10[T]asafunctionofKwhen[QDS]U>[QES]U. .................. 100 5-1Sequentialgameundertheprimaryretailer'slead. ................. 107 5-2Percentincreaseinthesupplier'sexpectedprot:eectsof1and2. ..... 115 5-3PercentdecreaseinR1'sexpectedprot:eectsof1and2. .......... 115 5-4Percentincreaseinthesupplier'sexpectedprot:eectsofcv1andcv2. ..... 116 5-5PercentdecreaseinR1'sexpectedprot:eectsofcv1andcv2. .......... 116 5-6Sequentialgamewithimperfectinformation .................... 120 5-7Responsefunctionsforthestrategicgame ..................... 139 A-1Standarddeviationfunctionforlineardemand. .................. 156 A-2Standarddeviationfunctionforiso-elasticdemand. ................ 157 A-3Linearstandarddeviationfunctionforlineardemand:I. ............. 158 9

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Supplychainmanagementisasetofapproachesutilizedtoecientlyintegratesuppliers,manufacturers,warehouses,andstores,sothatthemerchandiseisproduced,anddistributedattherightquantities,totherightlocations,andattherighttime,inordertominimizethesystemwidecostswhilesatisfyingservicelevelrequirements.Inmanypracticalcontextsontheotherhand,demandisnottrulyexogenousassuppliersareequippedwithtoolstoinuenceandshapecertaincharacteristicsofdemand.Suchtoolsinclude,butarenotlimitedto,pricing,promotions,discounts,productmix,shelfmanagementandleadtime.Demandmanagementinvolvescarefullyselectingamongthesetoolsandworkingcloselywithcustomerssothattheoverallincomingdemandfortheenterpriseandthesupplychainwillgiverisetomaximumvaluesforthepartiesinvolved(Lee(2001)).Thisrequiresbalancingthecustomerrequirementswiththerm'ssupplycapabilities,andentailsattemptingtodeterminewhatandwhenthecustomerswillpurchase.Thenecessityofincorporatingdemandmanagementintotraditionalsupplychainmanagementpracticesisbecomingmoreevidentamongpractitioners.Forinstance,theCouncilofSupplyChainManagementProfessionalsprovidesthefollowingdenition: SupplyChainManagementencompassestheplanningandmanagementofallactivitiesinsourcingandprocurement,conversion,andalllogisticsmanagementactivities.Importantly,italsoincludescoordinationandcollaborationwithchannelpartners,whichcanbesuppliers,intermediaries,third-partyserviceproviders,andcustomers.Inessence,supplychainmanagementintegratessupplyanddemandmanagementwithinandacrosscompanies. 13

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Integratingdemandandsupplychainmanagement(Lee(2001)) Thefailuretointegratedemandmanagementintosupplychainmanagementmayresultinseriousinecienciesforthesupplychainmembers(seeFigure 1-1 ).Lee(2001)discussesseveralindustrypracticesthatexemplifytheseineciencies.Forinstance,Volvosueredfromlackofpropercommunicationandcoordinationbetweenitssupplychainplanningandmarketinggroupsinthemid-90's.Inordertocuttheexcessiveinventoryofgreencars,themarketingdepartmentoereddeepdiscountsandaggressivedealstothedistributors.However,theincreasedsaleswereinterpretedbythesupplychaingroupasthelatesuccessofthegreencar,andtheyproducedevenmore,resultinginalargerinventory.Failingtoconsidertheoperationalimplicationsofthedemandmanagementtoolsonthesupplychainmayalsoresultinnetlossesforthecompany,althoughitmayboostsales.CampbellSoup'schickennoodlesoupexperienceissuchanexample(ClarkandMcKenney(1994)).Campbellpromotedtheproductexcessivelyaroundthewinterseason,whendemandalreadypeaked.Withthefurtherincreaseddemand,thecompanyhadtopreparefortheseasoninadvance,whichresultedinexcessivestorageinthespring.Productioncapacitywasallocatedtothisproductduringthewinter,whichrequiredotherproductstobemanufacturedinadvanceresultinginhigherinventoryandstorageneeds. 14

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4 ),andanalyzeanumberofdemandmanagementtoolsthatthesuppliercanutilizeinordertoinducehispreferredordertimingscheme(seeChapter 5 ).Inthissection,weprovideareviewofthestudiesintheliteraturethatexaminesimilarsettings.Keyelementsgoverningsupplier-buyerrelationsinasupplychainarethesupplycontractsthatspecifytheconditionsandparametersofthetransactionswithinthesupplychain.Lariviere(1999)andTsayetal.(1999)provideexcellentreviewsoftheliteratureonsupplychaincontracts.Forareviewoftheliteratureoncoordinatingcontracts,thereadermayrefertoCachon(2003).Thefactorsshapingthesecontracts,suchasthelengthoftheplanninghorizonandthetimingandexibilityoftheprocurementdecisions,havereceivedconsiderableattentionintheliteratureinrecentyears.Severalstudiesexistintheoperationsmanagementliteraturethatanalyzethetimeframeofordercommitmentdecisionsbetweenupstreamanddownstreammembersofthesupplychain.Topreserveconsistency,weuse`retailer'and`supplier'forthedownstreamandupstreammembersrespectively.IyerandBergen(1997)investigatetheeectsofa`quickresponse'modelontheprotsofthesupplychainmembers,wheretheretailerisallowedtodelayherorderuntilhavingbetterinformationaboutdemand.Theyinvestigatevariousmechanismsthatprovideincentivesforthesuppliertobeinvolved.Contrarytotheirassumptionthatthesupplieralwaysprovidestheretailerwithitsorderrequest,Ferguson(2003)modelsa`strategic'supplierthatgiveslittlecredibilitytoanorderquantitywithoutarmcommitment.Heexaminesaretailer'schoiceofwhentocommittoanorderquantityfromitspartssupplier.Boththesupplierandtheretailerhaveproductionlead-timesandasignalaboutdemandbecomesavailabletotheretailer 22

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2 foradetaileddiscussion)thataddressmultiplemarketsorlocations,thesealsoassumethatthermdoesnothavetheabilitytochoosewhetherornottoserveamarket.Aconsiderablebodyofliteratureproposesextensionsaddressingthelimitationsofthesemodels.Thevastmajorityofthesestudiesassumeeitherexogenousdemandandrevenue,orapredeterminedmarketportfolio.Inthischapter,weintroduceanoptimizationmodelthatrelaxestheseassumptions.Ourmodelappliestoandgeneralizesrelatedstudiesintheliteraturebothinthedeterministic(EconomicOrderQuantity[EOQ]model)andthestochastic(Newsvendormodel)settings.Weconsideraprot-maximizingrmoeringasingleproduct.Asetofpotentialmarketsexists,andthermmustdecidewhetherornottoserveeachmarket(althoughweconsiderdistinctmarkets,thesettingalsoappliestoasingle-marketproblemwithdierentcustomerclasses).Revenueineachmarketisafunctionofthepriceoered.Thermmustdeterminethemarketsitwillserve,theprice(orthepricesineachmarket),andaprocurementquantitythatwillbeusedtosupplytheselectedmarkets.Theresultingprotmaximizationproblemisquitedierentfromstandardinventorycontrolproblems.Inthetraditionalsettings,theoptimalinventorycontrolpolicyparametervalue(s)dependsonapredeneddemandrate(deterministicsetting),oranexogenousprobability 29

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2 ,ourmodelnotonlydiersfromthisliteraturebymodelingoperationalcostsininventorysystems,butitincorporatesexplicitmarketselectiondecisionsandprovidesanecientalgorithmforsolvingmulti-marketproblemsaswell,whichhasnotbeendoneintheeconomicsliteraturetoourknowledge.Undermildassumptionsontherevenueandcostfunctions,weprovideapolynomial-timesolutionforthesingle-pricestrategyandcharacterizetheoptimalsolutionforthemarket-specicpricingstrategy.Thesemodelscanbeappliedasbenchmarksformakingmarketselection,pricing,andprocurementquantitydecisionsinstochasticenvironmentswithashortsellingseason,anddeterministicenvironmentswithcontinuousandstationarydemand.Usingthesemodels,weperformanextensivecomputationalanalysistodemonstratetheeectsthatdierentcriticalparametersettingshaveontheoptimalvalueof(expected)prot.Theresultsofthisanalysisprovidesomeinterestingand,insomecases,unexpectedinsightsonhowamarket'scharacteristicscanaectpricingdecisionsinothermarkets.Theremainderofthischapterisorganizedasfollows:inSection 3.1 ,weintroduceageneralproblemframeworkandkeymodelingassumptions.Section 3.2 proposessolutionapproachesforthe`single-price'and`market-specicpricing'strategies.WeprovideanextensivecomputationalstudyandpresentourmainndingsinSection 3.3 .WeconcludeinSection 3.4 bysummarizingourwork. 31

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3.1.1 )andeconomies-of-scaleinprocurementcostsindeterministicsettings(suchastheEOQcontextdiscussedinSection 3.1.2 ).Themarketselectionproblemwithpricing(MSP)canthenbeconstructedasfollows:maxG(p;y)=nXi=1Ri(pi)yiSvuut 2 ),demandistypicallydenedaseitherq(p)+X(additivedemandmodel),orq(p)X(multiplicativedemandmodel),whereq(p)isadecreasingfunctionofprice,p,andXisarandom 32

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A.1 ,wedemonstratethatthemultiplicativeandadditivemodelscanbeequivalentlyrepresentedbyoneanotherinoursettingwhentheXisareindependent,normallydistributedrandomvariables.Hence,werestrictouranalysistotheadditiverandomnesscase,notingthatsimilarargumentsandresultsarealsovalidforthemultiplicativecase.Next,wediscusstheassumptionsandtheirimplications,underwhich(MSP)canhandletheNewsvendorproblemwithmarketselectionandpricing. 3.1 statesthatstockisallocatedamongselectedmarketsaftertheuncertaintyisresolved.Thisisquitereasonablewhenthemarketsareclosetoeachotherorwhenthermoerstheproductonaship-to-orderbasis.Inventorypoolingnaturallyfollowsifindividualmarketsrepresentdierentcustomersegmentsinasinglemarket.Notethattheproblemwouldbetrivialifinventorywasnotpooled,sinceeachmarketwouldbeconsideredseparatelyintermsofinventory,pricingandselectiondecisions. 33

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Pni=12i(pi)yi. 3.3 canbeinterpretedashavingcustomerswhoarewillingtowaituntiltheendofthesellingseasontoreceivetheproductwhenthesupplierfacesashortage.Assumption 3.3 ,togetherwithAssumption 3.2 ,resultsinseparateinventoryandpricingdecisionsasfollows:recallthat,byAssumption 3.2 ,theaggregatedemandisnormallydistributedwithmeanqy(p)=Pni=1qi(pi)yiandstandarddeviationy(p)=p Pni=12i(pi)yi.Hence,giventhemarketselectionandpricevectors,theinventorydecisionisequivalenttoaclassicalNewsvendorproblem.Letz=(Qqy(p))=y(p),whereQistheprocurementquantityfromanexternalsupplier.Then,theexpectedshortagesandleftoversaregivenbyy(p)E[z]+andy(p)E[z]+,respectively,whereisastandardnormalrandomvariable.ByAssumption 3.2 ,theexpectedsalesaredirectly 34

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3.1 andsortingmarketsinnondecreasingorderoftheratioofthe(expected)revenue(Ri(p))tothecostcontribution([Ci(p)]2),anoptimalsolutiontothemarketselectionproblemwithaxedpricelevelcanbefoundbyselectingthebestofncandidatesolutions,wherecandidatesolution`selectsmarkets1to`(seeTaaeetal.(2006)fordetails).Notethatthesortingmechanismworksinfavorofmarketsthathavegreaterrevenueandlesscostcontribution,whichsatisesintuition.For(MSP-S),however,priceisadecisionvariableandtheorderingofmarketsmaydieratdierentpricelevels.Toovercomethisproblem,thesortingschemecharacterizedabovecanbeutilizedtodividethefeasibleregioninpriceintoasetofcontiguous,non-overlappingintervals,wherethepreferenceorderofmarketsdoesnotchangewithinaninterval.Hence,withineachinterval,wecanutilizeProperty1withaslightmodicationtoobtainanoptimalsetofmarketsforthepriceinterval.Inparticular,foreachcandidatesolutionineachpriceinterval,weneedtomaximizetheobjectivefunctionwithrespecttopwiththeconstraintthatpfallsinthespeciedinterval.Thepriceintervalsthatenablethisapproachcanbegeneratedasfollows.LetPijdenotethesetofcriticalpricelevelswherethepreferenceratiosformarketsiandjareequal,i.e.,thethresholdpricesbeyondwhichtheorderofthesemarketsisreversedinthesortingscheme.Forall(i;j)pairssuchthati
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3{3 ).Assumingthatthetotalnumberofcriticalpricelevelsisnite,wereindexthesecriticalpricelevelssuchthatc=p0ijPijj+n<1.Asweillustratewithsomeexampleslater,thesetsPijoftencontainatmostoneelement,whichwouldleadtom=O(n2).Thepreferenceorderofmarketsisthesamewithinapriceinterval,p2(pk1;pk).Fortwoconsecutivepriceintervals,(pk1;pk)and(pk;pk+1),therankingwillbethesameexceptthatmarketsiandjwillswitchplacesinorderingsequenceifpk2Pij.Hence,wedonotneedtospecicallyrankorderallmarketsforeachpriceinterval.Instead,wesimplyrankorderthemonce,anddeterminewhichmarketsswitchplacesateachpricebreakpoint.Foreachpriceintervalindexedbyk=1;:::;m,wesolvenmaximizationproblemsofthefollowingform:maxp2(pk1;pk)8<:`Xi=1Ri(p)Svuut 3{4 )foreachpriceintervalandforeach`=1;2;:::;nwithineachinterval,theoptimalsolutionischaracterizedbythesolutionto( 3{4 )thatresultsinthehighestoptimalobjectivevalue.Notethat,inanygiveninterval,wemaydiscardthemarketsattheendoftherankorderingwithzerodemandsincetheywillnotbeselectedanymore.Therunningtimeoftheabovealgorithmdependsonthenumberofthresholdpricelevels,m,andonhowfastwecansolvethemaximizationsubproblemintheinnerloop.NotethatifeachsetPijcontainsatmostoneelementandeachmarkethasapricewheredemandbecomeszero,thereexistm=O(n2+n)=O(n2)priceintervals.HencetherunningtimeofthealgorithmbecomesO(Tn3)whereTdenotesthetimerequiredtosolve( 3{4 ).InAppendix A.2 ,weshowthattheobjectivefunction( 3{4 )isconcaveifRi(p)isconcaveandCi(p)isconvexforpp0iforallmarkets.Inthiscase,wecanutilizerstorderconditionstosolvethesubproblemseciently.UndertheNewsvendorstructure,withboththelinear(qi(p)=aibip)andiso-elastic(qi(p)=ipi)demandmodels,eachsetPijcontainsatmostoneelementwheneither 38

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A.3 .ThisleadstoProposition 3.1 Proof. A.4 Proposition 3.1 impliesthateitherthepriceineachmarketwillequalp0ioritwillbestrictlylessthanthat;thatis,eitheralldemandsarezeroorallmarketshavestrictlypositivedemand.Whenthecostcontributionofeachmarketisindependentofprice(e.g.,whenthestandarddeviationofdemandintheNewsvendormodelisindependentofprice),Proposition 3.1 doesnotholdsincesuchacaseviolatestheassumptionthatCi(pi)convergestozerowhen(expected)revenuetermRi(pi)iszero,i.e.,thereexistsapricelevel,p0isuchthatRi(p0i)=0andCi(p0i)=0.However,theobjectivefunctionofthisproblemisseparablebymarketsandwecansolvefortheoptimalpriceofeachmarketindividually.Letpidenotetheoptimalpriceformarketi.Sincethemarketselectionvariableiszeroforanunselectedmarket,thepricesinsuchmarketscanbesetarbitrarilyandwecanreformulatetheproblemasmaxnXi=1Ri(pi)yiSvuut 41

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SummaryofthealgorithmfortheNewsvendorexample. RankOrderR1R1,R2R1,R2,R3 1(40,77.5)M2M3M177.502,32377.506,53677.507,3292(77.5,88.75)M2M1M388.752,65285.203,19788.758,2503(88.75,100)M1M2M388.7540588.753,17196.388,4314(100,150)M2M3-100.002,727109.308,564||5(150,200)M3--150.005,227|||| 3.1.1 fordetails.Wenextdiscusstheapplicationofouralgorithmforthe(MSP-S)case(p1=p2=p3=p),andthenconsidertheresultsforthe(MSP-MS)case.Werstgeneratethecriticalpricelevelsforeachpairofmarketsinthe(MSP-S)case.Notethatthisexampleconsidersalinearexpecteddemandandaconstantcoecientofvariationforeachmarket.Hence,weknowthatthereisatmostonesolutiontoEquation( 3{3 )foreachpairofmarkets.SolvingEquation( 3{3 )foreachpair,wegetP12=f88:75g,P13=f77:5g,andP23=?.Includingthep0ivalues,wereindexthecriticalpricelevelsasfollows:p0=c=40,p1=77:5,p2=88:75,p3=100,p4=150,andp5=200.Weonlyrankorderthemarketsatp0,whichcorrespondstotheinterval(p0;p1).Atsubsequentcriticalpricelevels,eithertwomarketsswitchoroneofthemarketsisdroppedsincetheexpectedrevenuebecomeszero.Table 3-1 summarizesthesolutionoftheexample.Withineachinterval,wesolveatmost3subproblems.Toillustrate,letusconsiderinterval2,i.e.,(77:5;88:75).Therankorderofmarketsisgivenas(R1;R2;R3)=(M2;M1;M3).Therstsubprobleminthisintervalconsidersselectingthemarketthatisrankedrst,i.e.,M2.Theoptimalpriceandtheassociatedexpectedprotforthissubproblemare88:75and2,652,respectively.Thesecondandthirdsubproblemsselect(R1;R2)=(M2;M1),and(R1;R2;R3)=(M2;M1;M3),respectively.Havingsolvedallsubproblems,theoptimalsolutionforthisintervalisp=88:75andthecorresponding 42

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OptimalsolutionsfortheNewsvendorexample. (MSP-MS)73.4896.29121.88887.903023.336396.46617.529690.16(MSP-S)-109.30-2820.446285.56542.258563.75 expectedprotis8,250.Notethatthisisalocaloptimalsolutionforthegeneralproblem.Theglobaloptimalisthelargestofthelocaloptimalsolutionscalculatedforallintervals.Inthisexample,itisp=109:3,whichcorrespondstotheoptimalsolutionofinterval4.Theassociatedexpectedprotis8,564.Animportantcharacteristicofthealgorithmforthe(MSP-S)problemisthatitprovidesasetofadditional(suboptimal)solutionsfortheentirefeasibleregionintermsofthepricevariable,anditcaneasilybemodiedtocaptureadditionalconstraintsonthepriceleveloftheproduct.Forinstance,let'sassumethatthermdoesnotwanttochargemorethan90forthisparticularexample.Then,theoptimalsolutionisfoundbyconsideringtherstthreeintervalsonly,wherethethirdintervalismodiedtobe(88:75;90).Anotherrestrictionthatcanbehandledcanbeexplainedasfollows:assumethatthermwantstoservespecicmarkets.Then,weonlyneedtoconsidertheintervalsandassociatedsubproblemsthatselectthesemarkets.Forinstance,sayM1mustbeservedintheaboveexample.Insuchacase,onlysubproblem3ofinterval1,subproblems2and3ofinterval2,andallsubproblemsofinterval3shouldbeconsidered.Theassociatedoptimalsolutionisp=96:38.Wenowsolvethesameexampleforthe(MSP-MS)case,allowingdierentpricesindierentmarkets.Recallthatwecaneliminatethemarketselectionvariables,andtheresultingformulationisaconcavemaximizationproblemsinceRi(pi)=(pic)(aibipi)isconcaveandCi(pi)=cvi(aibipi)isconvex.TheoptimalsolutionisprovidedinTable 3-2 alongwiththesolutionforthe(MSP-S)case.Incomparisonto(MSP-S),(MSP-MS)notonlyselectstherstmarketinadditiontotheothers,butalsogeneratesmoreprotinmarkets1and2duetotheexibilitytosetdierentprices.Asaresultitprovides13.15%higherprotsthan(MSP-S). 43

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3.1.1 obtainedbysettingi(pi)=0,8i.Westartouranalysiswiththelineardemandmodel,qi(pi)=aibipi.Inordertoanalyzetheeectofmarketsizesonmarketselectiondecisions,wetreata1asxedandderivethresholdvaluesofa2thatresultindierentqualitativedecisions.Later,weperformthesameanalysisforpricesensitivities,bi.Recallthatweonlyconsidervaluessuchthata2bc>0.Inthemarket-specicpricingmodel,bothmarketswillbeselectedforalla2valuessincetheyhavepositivedemands.Theoptimalpricesarea1 3-1 and 3-2 illustratetheprotsandthepercentagedierenceinprotsbetweenthemarket-specicpricing(MSP-MS)andsingle-price(MSP-S)cases.Notethattheprotinthe(MSP-MS)caseisstrictlyincreasingina2.Forthe(MSP-S)case,itisconstantuptoa02sinceM2isnotselectedifa2
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Deterministicdemandanalysis:protsasafunctionofa2. Figure3-2 Deterministicdemandanalysis:percentagedierenceinprots. alsoequal.Whena2>a002,onlyM2isselectedinthe(MSP-S)caseandthedierenceinprotsisconstantthereafter,andequaltotheprotgeneratedinM1inthe(MSP-MS)case.Thisanalysisillustratesthevaluethattheexibilityofmarket-specicpricingprovidesasmarketsizesdier.Italsoillustratesthefactthatathresholdvalueexistsforamarket'ssizeatwhichpointthemarketbecomesanattractivemarkettosupplyunderasingle-pricestrategy.Moreinterestingly,ifsomemarket(M1inthiscase)maintainsaconstantsizeandanothermarket(M2)grows,athresholdmarketsizeexistsforthegrowingmarketatwhichpointitbecomesattractivetodropthemarketwithaconstantsize(again,assumingasingle-pricestrategy). 46

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3-3 and 3-4 depicttheexpectedprotsandthepercentagedierenceinexpectedprotsbetweenthe(MSP-MS)and(MSP-S)cases.Notethatbothguresaresimilartothoseforthedeterministiccase.ThesmallestsizeatwhichM2isselectedis412.72forthisexample.Thecorrespondingvalueinthedeterministiccaseforthesameparametersis382:84,whichisintuitivelyreasonablesinceuncertaintyresultsinhighercosts,andagreaterexpecteddemandisrequiredtoenteramarketwithuncertainty.Likewise,thethresholdvaluea2valueatwhichM1isnolongerselectedis711:15inthestochasticcase,whereasitis782:84whendemandisdeterministic.Notealsothatthehighestpercentagedierencesinprotalsooccuratthesethresholdvaluesforbothmodels. 47

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Stochasticdemandanalysis:protasafunctionofa2. Figure3-4 Stochasticdemandanalysis:percentagedierenceinprots. WenextperformathoroughcomputationalanalysistostudythemarketselectionandpricingdecisionsinaNewsvendorsetting,andtohighlightthedierencesbetweenthe`singlepricing'and`market-specicpricing'strategies. 48

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MarketselectiondecisionsfortheNewsvendormodel. 221/11/11/01/11/11/1231/01/01/01/11/01/1241/01/01/01/01/01/0320/11/11/11/11/11/1331/11/01/01/11/11/1341/01/01/01/11/01/1420/11/11/10/11/10/1430/11/11/01/11/11/144*/*1/01/01/11/11/1 3.1 and 3.2 applyundergeneralstandarddeviationfunctions,weassumei(p)=qi(p)=cvitoemployasingleparameterfortheuncertaintyinthesystem,sinceourprimarygoalistoanalyzetheeectsofsuchparametersonmarketselectiondecisions.Appendix A.5 containsabriefdiscussionoftheimplicationsofusingdierentfunctionalformsfortherepresentationofqi(p)andi(p).Table 3-3 providesthemarketselectiondecisionsfor(MSP-S)underdierentmarketsizesandpricesensitivitieswheretheentriesareoftheform(y1/y2).Forallcasesexceptwhena1=a2=450andb1=b2=4,theselectiondecisionsareconsistentacrossallremainingparametersettings,whereasforthisparticularcase,theselectiondecisionsarenotconsistentacrossallotherparametersettings.Inthiscase,however,theexpectedprotofthermissolowthatthemarketselectiondecisionsarerelativelyunimportant.Hence,wedonotconsiderthiscaseforfurtheranalysis.

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3.2 indicatesthatunder(MSP-S),customersinamarketarenegativelyaectedbyachangeinanothermarket,whichcallsintoquestiontheactualfairnessofasingle-pricestrategy. 3.2 )impliesthattheexpecteddemandsinbothmarketsdecrease.Thisdecreasemaycauseasmallermarkettobedroppedbecausethisprovidesthermanopportunitytofurtherincreasepriceinlargermarketsandpossiblygenerategreaterprot.Theparametersetgivenatthebeginningofthissectiondoesnotprovideanexampleforthecasewhereamarketisdroppedduetoanincreaseinthecoecientofvariation.Inordertoanalyzethisphenomenonmoreclosely,weconsiderthefollowingexample.Let(a1;a2)=(446;410),(b1;b2)=(3:2;2:2),e=300,c=100andv=50.Theselectionandpricingdecisionsof(MSP-S)forthisexamplearereportedinTable 3-4 .Notethattherstmarketisdroppedwhenthecoecientofvariationofeithermarketincreases. 50

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Eectsofuncertaintyonmarketselectiondecisions. Interestingly,M1ismorevulnerabletochangesincv2thanitistocv1.Forinstance,whencv2=1=7,M1isnotdroppedevenwhencv1isaslargeas1=3.Ontheotherhand,evenifcv1=1=7,itisdroppedwhencv2=1=3,whichcanbeexplainedasfollows.SincetheexpecteddemandinM1isfarlessthanM2,anincreaseincv1doesnothaveasubstantialeectontheaggregatestandarddeviationseenbythesupplier.Hence,thesuppliercanstillaordtoselectM1.However,anincreaseincv2wouldresultinconsiderablylargeraggregatestandarddeviation.Inthiscase,thermmaydropM1tofurtherincreasethepriceandbalancetheincreaseinstandarddeviation.Anotherparameterthataectsselectionandpricingdecisionsistheshortagecost.Asshortagecostincreases,thecostofthermduetouncertaintyincreases.Hence,weexpectthatthermwouldincreasethepricetodecreasetheaggregatevariation.Similartotheresultsforanincreaseincoecientofvariation,increasingshortagecostalsoaectstheselectiondecisionsunder(MSP-S);thatis,anincreaseinshortagecostmayforcethermtodropacurrentlyselectedmarketunder(MSP-S).Considertheexampleabovewithcv1=1=7andcv2=1=5.Bothmarketsareselectedwhentheshortagecostis$300.Whenitincreasesto$400,thesupplierincreasesthepriceasanattempttodecreasetheuncertainty,whichcausestherstmarkettobedropped.Wenowcomparethesingle-pricestrategytothemarket-specicpricingstrategy.Table 3-5 reportstheaveragepercentagedierenceinprotsfordierentmarketsizesandpricesensitivities.Whenattemptingtointerprettrendsinthesepercentagedierences, 51

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AveragepercentagedierencesinprotsfortheNewsvendormodel. 220.1911.120.0615.424.010.032325.677.5914.373.6019.719.87241.990.6115.400.297.3323.993225.670.2614.374.981.209.87331.4220.880.228.017.100.093410.661.6517.240.6516.298.20421.9919.7815.400.3115.8023.994310.664.9017.2417.480.238.204420.625.140.861.4015.570.24 wemustkeepinmindthatthecorrespondingmarketselectiondecisionsmaychangeaswechangeparametervalues(seeTable 3-3 ).Wecan,however,drawcertainconclusionsbasedontheseresults,therstofwhichisfairlyintuitive. 52

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3.5 isobvious;sincethestandarddeviationincreaseswiththemarketdemand,thermincreasesthepricetobalancetheincreaseinuncertainty.Thefactthatamarket'spricemaydecreaseinresponsetoanincreaseinanothermarket'suncertainty,however,issurprising,andcanbeexplainedasfollows.BecauseM2becomesrelativelylessuncertain,thesupplierdecreasesthepriceinthismarkettobalancethedecreaseintotaldemandduetothehigherpricerequiredinM1.Hence,wemayconcludethatthebuyersinM2facealowerpriceasaresultofanincreaseinuncertaintyinM1,althoughthecharacteristicsofM2areunchanged.Insummary,withalineardemandmodel,ourcomputationalresultsindicatethatpotentialmarketsizeandpricesensitivityarecriticalfactorsindrivingmarketselectiondecisions,althoughcoecientofvariationandshortagecostsmayalsoplayasignicantroleincertainsituations.(MSP-MS)alwaysoutperforms(MSP-S)asexpected.Yet,themagnitudeofthedierencedependsontherelativecostparameters,thesimilaritiesbetweenmarketsintermsofresultingpricingdecisions,andthecoecientofvariationofthedemands.Wealsoobservethatamarketmaybenegativelyaected(evendropped)becauseofthechangesintheothermarketunder(MSP-S),whichmakesthefairnessassertionof(MSP-S)questionable. 53

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A.5 ,undermildassumptions 54

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MarketselectiondecisionsfortheEOQmodel. 221/11/11/01/11/11/1231/*1/01/01/11/01/1241/01/01/01/01/01/032*/11/11/11/11/11/1331/11/*1/01/11/11/1341/01/01/01/11/01/1420/11/11/10/11/10/1430/11/11/*1/11/11/1440/01/01/01/11/11/1 (inparticular,ifqi(p)andi(p)approachzeroatthesamepointoratthesamerate)theobservationswehavediscussedcontinuetohold. 3-6 providesthemarketselectiondecisionsfor(MSP-S)underdierentpotentialmarketsizesandpricesensitivities.ThemarketselectiondecisionsarealmostidenticaltotheNewsvendormodel,whichisquiteintuitivesincebothmodelscanberepresentedwithalmostthesamemathematicalmodel.Hence,ourobservationsinSection 3.3.2.1 regardingtherelationbetweenmarketselectionandpotentialmarketsizeandpricesensitivityarealsovalidfortheEOQmodel(SeeObservation 3.1 ).Thedierencesintheexpectedprotsresultingfromthe(MSP-MS)and(MSP-S)strategiesalsofollowasimilarpatterntotheNewsvendormodelforthesamereason(SeeTable 3-7 andObservation 3.4 ). 55

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AveragepercentagedierencesinprotsfortheEOQmodel. 220.018.7214.650.003.420.002323.466.833.2511.7819.118.67240.920.330.1814.756.9823.693223.460.143.4111.781.108.67330.1218.056.990.015.290.00344.650.810.3611.4115.056.45420.9214.920.2914.7513.4023.69434.651.6314.5911.410.046.4544n/a2.210.710.049.360.01 Wenextfocusonhowxedorderingcostandholdingcostsaectthemarketselectionandpricingdecisions. 56

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3.3.2 regardingtherelationbetween 57

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Whileittakesthreetofourmonthstoprintsilicon,ourcustomerswanttopromisetosupplytheircustomersintwotothreeweeks.Inordertocopewiththisproblem,supplychainpartnersareworkingmorecloselyandcompaniesarenowapproachingtheirsuppliersas`long-term'partners.Forinstance,intheautomobilesector,mostofthemanufacturershavelaunchedprogramstoreducethetotalnumberofdirectsuppliersandestablishstrategicpartnershipswiththem(Bensaou(1999)).Theselong-termrelationshipscanprovidemajorbenetsforbothparties.Whiletheupstreamsupplierprovidestheinvestmentandtechnologyforthedownstreampartner,thesupplierinturnreceivesamorestabledemandstream.Inthischapter,weinvestigatesupplier-buyerrelationsintermsofthetimingofordercommitmentsanditsimplicationsforboththeupstreamanddownstreamtierinasingle-periodcontext.Specically,weconsideratwo-echelonsupplychainconsistingofasinglemanufacturer(supplier)andmultipleretailers(buyers),wherebuyersfacestochasticdemands.Sincethemanufacturingleadtimeislongcomparedtothesellingseason,thesuppliermustdecideontheproductionquantityinadvanceofthesellingseason,andhencebeforedemanduncertaintyisresolved.Animportantquestioninsupplier-buyerrelationsistheallocationofdemandandsupplyrisksamongtheparties.Ifbuyersorderafterobservingdemand,thesupplierisforcedtodecideontheproductionquantityunderuncertaintyandhencebearsthe`demandrisk'.Inthiscase,thebuyers'orderquantities 60

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MattelwashurtlastyearbyinventorycutbacksatToys\R"Us,andocialsareeagertoavoidarepeatofthe1998Thanksgivingweekend.Mattelhadexpectedtoshipalotofmerchandiseaftertheweekend,butretailers,waryofexcessinventory,stoppedorderingfromMattel.Thatledthecompanytoreporta$500millionsalesshortfallinthelastweeksoftheyear....Forthecrucialholidayseasonofthisyear,MattelsailitwillrequireretailerstoplacetheirfullordersbeforeThanksgiving.And,forthersttime,thecompanywillnolongertakere-ordersinDecember,Ms.Baradsaid.ThiswillenableMatteltotailorproductionmorecloselytodemandandavoidbuildinginventoryforordersthatdon'tcome.Althoughsomesupplierscantailortheordertimingoftheirretailerstotheirbenet,somemaylackthechannelpowertodoso.Forinstance,QuantumCorporationstatesinits2005annualreportthatmorethanhalfofitssalescomefromafewcustomerswhohavenominimumorlong-termpurchasecommitments.Similarly,JabilCircuit,whichisthesolesuppliertoQuantumforcertainproducts,notesintheir2005annualreportthattheircustomersdonotcommittormproductionschedulesinadvance,whichmakesitdicultforJabiltoscheduleproductionandmaximizetheutilizationoftheproductionfacility.Hence,weconsiderasysteminwhichthestatusquoinvolvesdelayedcommitmentfortheprimaryretailer,i.e.,thesupplierproducesinadvanceofthesellingseason,and 62

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4.1 ,weformallydescribetheproblemenvironment,ourassumptions,andthebasicsofthedelayedandearlycommitmentmodels.InSection 4.2 ,weanalyzetheexpectedprotsofthesupplierandprimarymanufacturerunderbothcommitmentschemes.Section 4.3 identiesthedierencesbetweenasingle-retailersystemandamulti-retailersystem.AfternumericallycomparingthecommitmentschemesundervariousparametersettingsinSection 4.4 ,weanalyzethestrategicinteractionbetweentheretailersintermsofordertiminginSection 64

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.Section 4.6 investigatesacapacitatedsettinganditsimplicationsregardingtheordertiming.WeconcludethischapterinSection 4.7 byhighlightingourndings. 65

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4{1 ): wheredenotesthepercentageofthesupplier'sproductionquantityreservedforR1.Theessenceofgeneralizeduniformallocationcanbeexplainedasfollows:Thesupplierreservesacertainfractionoftheproductionquantityforeachretailer;thatis,eachretailerisassuredtohaveatleastafractionoftheproductionnomatterwhattheordersizeoftheotherretaileris.Furthermore,ifoneoftheretailersorderslessthanherreservedamount,theremainingpartofhersharecanbeutilizedtoservetheotherretailerwhenrequired.Proposition 4.1 characterizestheretailers'orderquantitieswhenthesupplierallocatesinventoryaccordingtoageneralizeduniformallocationmechanism. 66

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Proof. 4{1 ).Hence,givenQSandQ2,R1willorderherdemandrealization.SimilarargumentsarealsovalidforR2.Thatis,givenQSandQ1,wehaveQ2=x2.Hence,attheequilibrium,bothretailersordertheirtrueneeds. Thegeneralizeduniformallocationrulepreventstheretailersfrommanipulatingtheirorderquantitiesinordertogetagreatersharefromthesuppliersincetheallocationruledoesnotrelyontheorderquantities.Hence,theorderquantitiesoftheretailersmatchtheirdemandsunderthedelayedcommitmentschemeregardlessoftheproductionquantityofthesupplier,i.e.,Q1=x1andQ2=x2.ThereadermayrefertoCachonandLariviere(1999)foradetaileddiscussionoftheallocationmechanisms.Sincethesupplierdoesnotdierentiatebetweentheretailersunderthedelayedcommitmentscheme,hefacesanaggregatedemandofZ=X1+X2,andhisprotisgivenbyEquation( 4{2 ). DS(QS)=8><>:w(x1+x2)cQSx1+x2QS(wc)QSx1+x2>QS Hence,thesupplier'sproblemismaxQSE[DS(QS)].E[DS(QS)]ischaracterizedinEquation( 4{3 )whereh(z)andH(z)denotethepdfandcdfofZ=X1+X2respectively.E[DS(QS)]=(wc)(1+2)24(wc)1ZQS(zQS)h(z)dz+cQSZ0(QSz)h(z)dz35 67

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w).NotethatthesecondterminEquation( 4{3 )quantiesthedemandriskofthesuppliersincethesupplierwouldgenerateaprotof(wc)(1+2)iftheretailers'demandsweredeterministic.Wecannowderivetheexpectedprotfunctionoftheprimaryretailer,R1.Recallthatherorderquantityisequaltothedemandrealization.Hence,herprotcanbewrittenas(rw)q1.UtilizingEquation( 4{1 ),Equation( 4{4 )presentstheexpectedprotofR1: Ifthesupplier'sproductionquantitywasinnite,theprimaryretailer'sexpectedprotwouldbe(rw)1.Hence,wecandeducethatthesecondterminEquation( 4{4 )quantiesthesupplyriskofR1underadelayedcommitmentscheme. 4{5 ). TheoptimalorderquantityofR1isQ1=F11(rw r).ThesecondterminEquation( 4{5 )characterizesthedecreaseinR1'sexpectedprotduetothedemandrisk.However,thesupplyriskinthedelayedcommitmentscheme(seeEquation 4{4 )diminishesunderanearlycommitmentscheme.Sinceallinformationissymmetric,thesupplierinferstheorderquantityofR1andproducesexactlyQ1unitsforR1.TheproductionquantityforR2mustbedetermined 68

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4{6 ). w).Thesupplierstillfacessomedemandrisk.However,thedegreeofthisriskisreducedbecauseonlyR2'sdemandisuncertainasopposedtobothretailer'sdemandunderdelayedcommitment.Hence,wecandeducethatthedemandriskofthesupplierdecreasesunderanearlycommitmentscheme. 69

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w),and(:)and(:)denotethepdfandcdfofthestandardnormaldistribution,respectively.Underanearlycommitmentscheme,thecorrespondingproductionquantityandtheexpectedprotofthesupplierare r).Let[QS]denotetheincreaseinthesupplier'sproductionquantityduetoearlycommitment.Accordingly,let[SS]and[S]denotetheincreaseinexpectedsalesandexpectedprot,respectively.Then, [QS]=QESQDS=1k1(T2)km[SS]=E[SES]E[SDS]=1k1+(T2)L(km) 70

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(4{7)If[S]isgreaterthanzero,thesupplierbenetsfromtheearlycommitmentofR1.RecallthatifR1ordersafterdemandrealization,theburdenofdemandriskwillbeonthesupplier.WhenR1commitsearly,thesupplierdoesnotfullybearthedemandriskwithrespecttoR1'sdemandandhencefacesadecreasedlevelofuncertainty.AnotherfactorthataectstheprotabilityofthesupplieristheorderquantityofR1.IfQE1isgreaterthantheexpectedorderquantityunderadelayedcommitment(whichisequaltotheexpecteddemand,1),thesuppliercertainlybenetsfromearlycommitment,sincebothfactorsworkinhisfavor.Otherwise,thetradeobetweenuncertaintyandexpectedsalesdetermineswhichcommitmentschemeismoreprotable.Proposition 4.2 formallydenesthisrelationship. 71

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wc(km),thesupplierbenetsfromtheearlycommit-mentscheme.Otherwise,delayedcommitmentismoreprotable. Proof. 4{7 ). TherelationbetweentheorderquantitiesofR1underdelayedandearlycommitmentschemesisdeterminedbyk1,i.e.,ifk1>0,theorderquantityunderearlycommitmentisgreater,resultinginhigherexpectedprotforthesupplier,whichisapparentfromProposition 4.2 .WhentheorderquantityofR1decreaseswiththeearlycommitment,thedierencebetweenprotsdependsonthetradeowediscussedearlier;earlycommitmentmaystillprovidehigherprotwhenthedecreaseismoderate.Wenowanalyzetheeectsofuncertaintyonthebenetsofearlycommitmenttothesupplier,startingwiththeuncertaintyinR1'sdemand.Proposition 4.3 illustrateshowtheimprovementsinthesupplier'sexpectedprotfromearlycommitmentchangewithrespecttotheuncertaintyinR1'sdemand. wc(km). Proof. TheconditioninProposition 4.3 holdsautomaticallywhenk10,whichisamildassumptionifweconsideritsimplicationsontheservicelevelofR1.Hence,wecandeduce,ingeneral,thatearlycommitmentbecomesmorefavorabletothesupplierasthelevelofuncertaintyinR1'sdemandincreases.Letkp11andkp21denotethethresholdvaluesfork1inPropositions 4.2 and 4.3 ,respectively.Then,wehavekp11>kp21,whichleadstoCorollary 4.1

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Ifk10,increasinguncertaintyincreasestheorderquantityofR1andhencethesupplier'sprot.However,whenk1<0,theorderquantitydecreasesin1,whichisnotfavorableforearlycommitmentfromthesupplier'sperspective.Notethatd[S] Proof. TheonlyuncertaintyinvolvedintheearlycommitmentschemeforthesupplieristhedemanduncertaintyforR2.Indelayedcommitmentontheotherhand,bothdemandsarerandomandhenceaggregatedemandcanbepooledtoreducetheriskofuncertainty.Sincethisdoesnotholdfortheearlycommitmentcase,thesupplierismorevulnerabletotheuncertaintyinR2'sdemand,i.e.,anincreasein2resultsinalargerdecreaseinthe 73

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4.2 furtheranalyzesthedierencesbetweendelayedandearlycommitmentschemeswithrespectto2. a)Ifk10,earlycommitmentalwaysoutperformsdelayedcommitmentforany2. b)Ifw(km) 2u;u=(wc)k1 c)Ifk10.[S]decreasesin2andbecomeszerowhen2=02. Corollary 4.2 providesanintervalfork1wherethedegreeofuncertaintyinR2'sdemandeectsthecommitmentpreferenceofthesupplier.Specically,ifk12(w(km) 74

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4{4 )and( 4{5 ),respectively. Proof. Thereasoningbehindtheprecedingpropositioncanbeintuitivelyexplainedasfollows:WhenQE1QS,i.e.,k11kmT,R1canalreadygetQE1unitsfromthesupplierwithcertaintyevenunderthedelayedcommitmentscheme.Hence,thereisnoneedforhertopre-committhesamequantitythatshecanalreadyreceive,andbeartherisksassociatedwiththedemanduncertainty.WhenQE1>QS,ontheotherhand,thepreferenceofR1dependsonthetradeobetweencapacityallocationanddemanduncertainty. 75

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(4{8) whichleadstoProposition 4.6

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r.Recallthat(km)=wc w<(k)and(k1)=rw r<(k).Since()isamonotonenondecreasingfunctionandg()isconvex,wecandeducethatk1;kmk1,theng(km)
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4{9 )issatised. Proof. 4{9 )willnotbesatisedforsucientlylargevaluesof2.Let02denotethesmallest2valuethatviolatescondition( 4{9 ).Since1 Proposition 4.7 impliesthatifthesupplier'sservicelevelissucientlylow,itisoptimalforR1toorderearlyinordertoensureasucientamountofsalesforthesystem.WecanalsoconcludefromProposition 4.8 thatintermsofoverallsystemperformance,thesuppliershouldnottrytoinduceearlycommitmentwhentheuncertaintyinthesecondaryretailer'sdemandisaboveacertainlevel. [S]=(wc)k11+1w(km): 78

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[T]=1(rL(km)+ckm(rL(k1)+ck1)) (4{11) Proposition 4.9 characterizesthepreferenceofthecommitmentschemefromthesystem-wideprotperspective: Proof. 4.7 ),g(k1)km. ComparingEquation( 4{10 )withEquation( 4{7 ),weobservethattheincreaseinthesupplier'sexpectedprotduetoearlycommitmentisgreaterinasingle-retailersystem.Thatis,thesupplier'svaluationoftheearlycommitmentschemedecreaseswhenthereisanothercustomer(oracollectionofcustomers)whoutilizesthesupplierasanoverowsupplierpostdemandrealization.Weperformasimilaranalysisfromtheprimaryretailer'sperspective.Proposition 4.10 providesconditionsunderwhichtheprimaryretailerbenetsfromtheexistenceofR2underdelayedcommitment.

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TheexistenceofR2aectsthecommitmentpreferencefromthesystem'sperspective,whichischaracterizedbyProposition 4.11 Proof. 4{8 )andEquation( 4{11 ),andthefactthat1>T2. Summingupwhatwehaveobservedinthissection,theexistenceofanothercustomermakesdelayedcommitmentmorefavorablefromthesystem'sperspective.Theprimaryretailer(undercertainsettings,seeProposition 4.10 )andthesuppliervaluedelayedcommitmentmorewhencomparedtothesingle-retailersystem. 4.3 .Forthispurpose,weusetheparameterspresentedinTable 4-1 Table4-1: Parameters. Notethatk1>0forourparameterset.Hence,weknowthatthesupplierisalwaysbetterowiththeearlycommitmentofR1.Therefore,wheneverR1benetsfromthe 80

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4.5 ),underwhichwehaveshownanalyticallythatR1prefersthedelayedcommitmentscheme.TheparametersetinTable 4-1 correspondsto243testcases.Theprimaryretailerprefersearlycommitmentin141ofthesecases,whichisequalto58%(141outof243)ofthetime.Hence,wecanconcludethatalthoughdelayedcommitmentisfrequentlyobservedinindustry,retailerscanbenetfromorderingearlyincertaincontexts,evenwithouttheincentiveofadiscountedwholesaleprice.Furthermore,theaggregateprotofthesupplierandtheprimaryretailerishigherunderanearlycommitmentcasein89.3%(217outof243)ofthecases,whichimpliesthatinsomesettings,thesuppliergenerateshighenoughprotstocompensateforthelossoftheprimaryretailer.Whentheretailmarkupisrelativelylow(r=40),thereisnocasewhereR1benetsfromanearlycommitmentscheme.Inthiscase,theunitrevenueisnothighenoughforR1tojustifythedemandriskofearlycommitment.Whenr=100,ontheotherhand,R1isbetterowiththeearlycommitmentschemeinallcases.ThisisintuitivesinceR1doesnotwanttobearthehighsupplyriskasunsatiseddemandduetoinsucientsupplywouldresultinconsiderablelossofprot.Whenr=70,R1'spreferencedependsonthedemandparametersofbothretailers.WhenhermeandemandexceedsR2's,sheusuallyprefersearlycommitment.Thatis,whenherdemandissucientlylarge,R1preferstosecureinventorybycommittingearly.WhenR2'sdemandislargerthanR1's,thesupplyriskforR1isnotseverebecauseherinventoryallocationisrelativelylargeduetothelargedemandofR2.Hence,insuchcases,R1usuallyprefersdelayedcommitment.Figure 4-1 depictstheaveragepercentageincreaseinR1'sexpectedprotduetoearlycommitment(comparedtodelayedcommitment)asafunctionofunitrevenueandhermeandemand.R1valuesanearlycommitmentschememoreasherexpecteddemandincreasesforallretailmarkuplevelsconsidered.Similarly,Figure 4-2 depictstheaveragepercentageincreaseinR1'sexpectedprotduetoearlycommitment(comparedtodelayed 81

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PercentincreaseinR1'sexpectedprotduetoearlycommitment:eectsof1. Figure4-2 PercentincreaseinR1'sexpectedprotduetoearlycommitment:eectsof2. commitment)asafunctionofunitrevenueandR2'smeandemand.R1valuesanearlycommitmentschememoreastheexpecteddemandofthesecondaryretailerdecreasesforallretailmarkuplevelsconsidered,sincehigherexpecteddemandforR2increasestheprimaryretailer'schancetoreceivesucientinventoryunderadelayedcommitmentscheme.WenextanalyzetheeectsofuncertaintyontheexpectedprotofR1andhercommitmentpreference.Figures 4-3 and 4-4 illustratetherelationbetweentheaverageincreaseinR1'sexpectedprotduetoearlycommitmentandthecoecientsofvariationofdemandforR1andR2,respectively,fordierentvaluesofunitrevenue.Increasing 82

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PercentincreaseinR1'sexpectedProtduetoearlycommitment:eectsofcv1. uncertaintyinR1'sdemand(Figure 4-3 )worksinfavorofthecommitmentschemethatprovidesmoreprots.Whenr=40,R1prefersdelayedcommitment,whichbecomesmorefavorableas1=1increases.Whenr=70or100,earlycommitmentismorefavorableandtheimprovementinR1'sexpectedprotincreasesin1=1.Thisisduetotherelativeimportanceofsupplyanddemandrisks.Whendemandriskismorecritical(r=40),increasinguncertaintyindemandampliesthisrisk,andhencedelayedcommitmentbecomesmorefavorable.IfR1'sprimaryconcernisthesupplyrisk,thenincreasinguncertaintymakesitmorecrucialtocommitearly.IncreasinguncertaintyinR2'sdemand(Figure 4-4 )decreasesthepercentageimprovementinR1'sexpectedprotduetoearlycommitment,whichcanbeexplainedasfollows:underdelayedcommitment,R1isreservedacertainamountofinventorynomatterwhatR2orders.WhenuncertaintyinR2'sdemandincreases,herorderquantitybecomesmoreuncertain.HigherorderquantitieswillnothurtR1becauseofthereservedsupply,whereaslowerorderquantitiesresultinlargersupplyforher.Hence,increasinguncertaintyofR2'sdemandmakesdelayedcommitmentmorefavorableforR1.Fortherestofthissection,wefocusonthesingle-retailersystemanditscomparisontothemulti-retailersetting.Weusethesameparameterset(seeTable 4-1 )removing 83

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PercentincreaseinR1'sexpectedProtduetoearlycommitment:eectsofcv2. theparametersrelatedtothesecondaryretailer,whichresultsin27testcases.Eachofthesecasesiscompatiblewith9casesinthemulti-retailersettingthathavethesameparameters.Forthecomparisonofthesupplier'schoice,weusethe`increaseinexpectedprotsduetoearlycommitment'ratherthanthe`percentageincrease'sincethelattermaybemisleadinginevaluatingthesupplier'sprot(inthemulti-retailersetting,thesupplierservesthesecondaryretailerinbothcommitmentschemes.Hence,usingthepercentageincreaseandcomparingitwiththesingle-retailersystemunderestimatesthevalueofearlycommitmentinamulti-retailersystem).Inasingle-retailersystem,thesupplierprefersearlycommitmentinallthecasesasinthemulti-retailersetting.Theaverageincreaseinhisexpectedprotduetoearlycommitmentis$203.5whenprimaryretaileristheonlydemandsourcewhereasitis$155.2whenthereisanotherretailer(orcollectionofretailers).Furthermore,ineachofthe27casesconsidered,theincreaseinprotsinasingle-retailersystemexceedsthemaximumincreaseinthe9correspondingcasesinthemulti-retailersetting.Hence,weconcludethatthesuppliervaluesdelayedcommitmentmoreinthepresenceofanotherretailerregardlessofitsdemandparameters. 84

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4-2 andFigure 4-4 ).Thatis,R1valuestheexistenceofanothercustomermorewhenthatcustomer'sexpecteddemandandcoecientofvariationarehigher,whichcanbeexplainedasfollows:notethatkm=0forourparameterset,whichindicatesthattheprimaryretailer'sreservedsupplyunderadelayedcommitmentschemeisequalto1inboththesingle-retailerandmulti-retailersystems.However,inthemulti-retailersystem,R1hastheopportunitytousetheexcesssupplythatresultswhenR2'sdemandislessthanherreservedsupply.When2and/or2increase,theamountofpossibleexcesssupplyincreases.Notethattheparametersusedinourcomputationaltestarebiasedinthattheyexcludeinstanceswherewecouldproveanalyticallythroughsucientconditionsthattheprimaryretailerwouldselectadelayedcommitmentscheme.Hence,thedatasetpresentedinTable 4-1 mayleadtotheoverestimationoftheinstanceswheretheprimary 85

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4.1 86

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Strategicinteractionbetweentheretailers. Early(E)Delayed(D) Early(E)(E1jR2=E;E2jR1=E)(E1jR2=D;D2jR1=E)Delayed(D)(D1jR2=E;E2jR1=D)(D1jR2=D;D2jR1=D) 4-2 ,whererowscorrespondtotheactionsofR1,andcolumnscorrespondtotheactionsofR2.Eachentryinthematrixconsistsoftheexpectedprotsoftheretailerscorrespondingtotheselectedactionpair.Forinstance,thetop-rightcellcorrespondstoR1andR2selectingearlyanddelayedcommitmentschemes,respectively.TheexpectedprotsofR1andR2inthiscasearerepresentedbyE1jR2=DandD2jR1=E,respectively.Anequilibriumofastrategicgamecanbecharacterizedasfollows: 4.2 canbeillustratedforoursettingasfollows:considertheactionprole(E,D),thatis,R1andR2selectearlyanddelayedcommitment,respectively.ThisactionproleisaNashequilibriumifandonlyifR1hasnoincentivetoselectdelayedcommitmentgivenR2selectsdelayedcommitment(i.e.,E1jR2=DD1jR2=D),andR2hasnoincentivetoselectearlycommitmentgivenR1selectsearlycommitment(i.e.,D2jR1=EE2jR1=E).Beforefurtheranalyzingthestrategicinteractionbetweentheretailers,theoptimalproductionquantityofthesuppliercorrespondingtoeverypossibleoutcomeofthis 87

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4{12 ): w)=T+kmT(R1;R2)=(D;D)F11(rw e)+F12(wc w)=T+k11+km2(R1;R2)=(E;D)F11(wc w)+F12(rw r)=T+km1+k12(R1;R2)=(D;E)F11(rw r)+F12(rw r)=T+k11+k12(R1;R2)=(E;E) (4{12) Theresultingexpectedprotsofthesuppliercanalsobeinferredfromthepreviousanalysis,exceptthecase(R1;R2)=(E;E).Inthiscase,bothretailersorderearly,hencethesupplier'sprotisdeterministicandgivenbyS(E;E)=(wc)(T+k11+k12).Likewise,wecanalsocomputetheexpectedprotsoftheretailersasintheprevioussections,exceptfortheprotofaretailerwhoselectsdelayedcommitmentwhentheotherretailerselectsearlycommitment,whichisgivenbyE[Di]=(rw)[iiL(km)].WenextanalyzetheoutcomeofthissettingusingtherandomdatasetdiscussedinSection 4.4 ,consideringonlypurestrategyNashequilibria(seeOsborne2004forthediscussionofpurestrategyandmixedstrategyNashequilibria).NotethatthedenitionofNashEquilibriumforthestrategicgamebetweentheretailers(seeDenition 4.2 )doesnotguaranteeauniqueequilibrium.Thatis,morethanoneactionprolecansatisfytherequirementsofaNashEquilibrium.Inthosecases,werstusetheequilibriumrenementtechniquecalledpareto-dominance(payodominance),whichisdenedbelow:

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Illustrationofrisk-dominance. Whenpareto-dominancedoesnotprovideauniqueequilibrium,weutilizetheconceptofrisk-dominance,whichisdescribedbelow: 4-3 .AssumethatV=(V1;V2)andU=(U1;U2)aretwoequilibriumpoints.Then,thedeviationlossesfromequilibriumVaregivenbyv1=a22a12>0andv2=b22b21>0.Similarly,thedeviationallossesfromequilibriumUareu1=a11a21andu2=b11b12>0.ThedeviationallossesaregreaterthanzerosinceUandVareequilibriumpoints.In1,000testinstancesconsidered,thereare268caseswithmultipleequilibria.In232ofthese268cases,(E,E)and(D,D)aretheequilibria.Pareto-dominanceeliminatestheequilibrium(E,E)inall232cases.Intheremaining36cases,themultipleequilibriaare(E,D)and(D,E),andequilibriumrenementusingpareto-optimalitydoesnotrenderauniqueequilibrium.Inthesecases,risk-dominanceisutilizedtoproduceauniqueequilibrium,whichprovides15of36caseswith(E,D),andtheremainingwith(D,E).Asaresult,wehaveauniqueequilibriumforeachofthetestinstances.Bothretailerspreferearly(delayed)commitment7.8%(87.8%)ofthecases.R1andR2preferearlyanddelayed(delayedandearly)commitmentschemes1.8%(2.6%)ofthetestinstances,respectively.Delayedcommitmentisstillthedominantstrategyfortheretailers.Ontheotherhand,thesupplierprefersthe(D,D)outcomeonlyin53cases.Intheremaining947instances,thesupplierprefersbothretailerstocommitearly.However,onlyin78ofsuchinstancestheretailersprefertodoso.Hence,wecandeducethattheoutcomeofthe 89

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5.1.2 ). w)=T+kmT.Ifthisquantityisgreaterthanthecapacity,thenthesupplierwillproduceatthecapacitylevel.Thatis,we 90

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Thenthecorrespondingexpectedprotofthesupplierisgivenby Similarly,whentheprimaryretailerselectsearlycommitment,theunconstrainedproductionquantityis[QES]U=F11(rw r)+F12(wc w)=T+k11+km2.Ifthisquantityisgreaterthanthecapacity,thenthesupplierwillproduceatthecapacitylevel,Thatis,wehave TheexpectedprotofthesupplierunderanearlycommitmentschemeisthencharacterizedbyE[ES]=8><>:(wc)(T+k11)2(wL(KTk11

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B.1.1 ,whichleadstoProposition 4.12 : Proof. B.1 andLemma B.2 Whetherthesystemiscapacitatedornot,thesupplierbenetsfromanearlycommitmentschemeif[QDS]U[QES]U.However,thesebenetsdecreaseasthecapacityleveldecreases,whichcanbeexplainedasfollows:theinequality[QDS]U[QES]Uindicatesthatcapacityismorecrucialforthesupplierunderanearlycommitmentscheme.Hence,theexpectedprotunderearlycommitmentismorevulnerabletothelimitationsoncapacity.Thatis,theadvantagesofanearlycommitmentschemewillgraduallydisappearasthecapacityleveldecreases.Figure 4-5 illustrates[S]asafunctionofKforanexamplesetting(1=2=100,1=2=30,c=10,w=15,r=100).Forthisexample,wehave[QDS]U=181:73<[QES]U=218:17.Notethat[S]islessthan$1untilK>90sincethesupplierutilizesthecapacityat%100almostwithcertaintyunderbothcommitmentschemeswhencapacityistoolow.Fromthatpointonward,thebenetsofearlycommitmentincreasesgradually.Wenowanalyzethesettingwhere[QDS]U>[QES]U.Asforthecasewith[QDS]U[QES]U,weneedtoanalyzetheincreaseinexpectedprotofthesupplierduetoearlycommitmentinthreeintervals:(i)KT+k11+km2,(ii)T+k11+km2
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[S]asafunctionofKwhen[QDS]U<[QES]U. B.1.2 ,whichleadstothecharacterizationof[S]withrespecttoK: Proof. B.3 andthediscussionsthereafterinAppendix B.1.2 Thereisanimportantdierenceinthebehaviorof[S]when[QDS]U>[QES]U,comparedtothecase[QDS]U[QES]U:itrstincreases,andthendecreasesinK,whichcanbeexplainedasfollows:whencapacityistight,theprobabilitythatanincrementalincreaseincapacitywillbeutilizedunderanearlycommitmentschemeisgreaterthanthecorrespondingprobabilityunderdelayedcommitmentschemebecauseoftheearlysalestotheprimaryretailer.Hence,[S]increasesinK.Whencapacityexceedsacriticalvalue,itbecomesmorecrucialunderadelayedcommitmentschemesincethesupplierwillusecapacityonlyforthesecondaryretailer'sdemandunderanearlycommitmentschemeoncetheprimaryretailer'sorderissatised,whereasunderdelayedcommitment,thereisstillapossibilitythatbothretailerwillneedcapacity.Hence,[S]decreasesinK.Notethat 93

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[S]asafunctionofKwhen[QDS]U>[QES]U:sampledata(i). wecannotreadilycommentonthesignof[S]whenK2(K0s;T+kmT).However,utilizingProposition 4.2 ,wehavethefollowingcorollary: wc(km),thatis,ifthesupplierprefersdelayedcommit-mentinanuncapacitatedsetting,thereexistsacapacitylevel,K00s,suchthat[S]>0forallK2(0;K00s),and[S]<0forallK2(K00s;1).Otherwise,ifthesupplierprefersearlycommitmentinanuncapacitatedsetting,[S]isgreaterthanzeroforallK.Figures 4-6 and 4-7 illustrateProposition 4.13 andCorollary 4.3 fortwoexamplesettings:(i)1=2=100,1=2=30,c=10,w=30,r=55(Figure 4-6 ),and(ii)1=2=100,1=2=30,c=10,w=25,r=40(Figure 4-7 ).Insetting(i),thesupplieralwaysprefersdelayedcommitment.ThethresholdcapacitylevelcharacterizedinProposition 4.13 isequalto188.30.Thatis,whenK<188:3,[S]isincreasinginK,andwhenK>188:3,itisnonincreasing.Insetting(ii),thesupplierprefersdelayedcommitmentifK>K00s=196:43,andearlycommitmentotherwise(seeCorollary 4.3 ).Havinganalyzedbothcases,thatis,[QDS]U[QDS]Uand[QDS]U>[QES]U,wemaynowcombineourobservationsandcommentonthesupplier'schoiceofcommitmentscheme.

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[S]asafunctionofKwhen[QDS]U>[QES]U:sampledata(ii). 4.12 for[QDS]U[QDS]U,andCorollary 4.3 for[QDS]U>[QDS]U. Proof. 4.3 Contrarytotheinitialintuition,capacitydoesnotnecessarilyfavorearlycommitmentfromthesupplier'sperspective.Thatis,[S]ofacapacitatedsystemisnotnecessarilygreaterthanthatofanuncapacitatedsystem.Thisisduetohowacommitmentschemeaectscapacityutilization.Underanearlycommitmentscheme,partofthecapacityisutilizedwithcertainty.However,theremainingcapacityismorevulnerable.Underdelayedcommitment,alowrealizationofoneretailer'sdemandcanbecompensatedbyalargerealizationoftheotherretailer'sdemand. 95

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96

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4{13 ).Whenretailerdemandsarenormallydistributed,Equation( 4{17 )reducesto Similarly,theexpectedprotunderearlycommitmentisgivenby whereQE1=F11(rw r),QSEisgivenbyEquation( 4{15 ),and^Qe=QESQE1.Whendemandsarenormallydistributed,Equation( 4{19 )reducesto

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B.2.1 Wenextexaminetheincreaseintotalexpectedprotduetoearlycommitmentaswedidforthesupplier.Thatis,weconsidertwocasesseparately:(i)[QDS]U[QES]U,and(ii)[QDS]U>[QES]U.When[QDS]U[QES]U,thebreakpointsofE[ET]andE[DT]resultinfourintervalstoconsiderinordertoevaluatetheincreaseinthesupplier'sexpectedprotduetoearlycommitment:i)K1+k11,(ii)1+k11T+k11+km2.EachoftheseintervalsisanalyzedindetailinAppendix B.2.2 ,whichrevealsthefollowing: Proof. B.4 ,Lemma B.5 ,andLemma B.6 ,wecanconcludethat[T]<0forallKif[T]<0fortheuncapacitatedsystem.Forpart(b),weknowthat[T]<0whenK=T+kmT(Lemma 4.1 ),anditisnondecreasingthereafter(Lemma B.6 ).Hence,if[T]>0inanuncapacitatedsetting,part(b)follows. Figures 4-8 and 4-9 illustratetheanalysisinAppendix B.2.2 andProposition 4.16 fortwoexamplesettings:(i)1=2=100,1=2=30,c=10,w=25,r=100(Figure 4-8 ),and(ii)1=2=100,1=2=30,c=10,w=15andr=100(Figure 4-9 ),respectively.Forsetting(i),thebreakpointsof[T]areQE1=120:23,[QDS]U=210:75,and[QES]U=227:83.[T]isdecreasingin(0;120:23),increasingin(120:23;227:83),andconstantthereafter,ascharacterizedinAppendix B.2.2 .DelayedcommitmentoutperformsearlycommitmentregardlessofKasindicatedinProposition 4.16 .Forsetting(ii),thecorrespondingbreakpointsareQE1=131:09,[QDS]U=181:73,and[QES]U=218:17. 98

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[T]asafunctionofKwhen[QDS]U<[QES]U:sampledata(i) Figure4-9 [T]asafunctionofKwhen[QDS]U<[QES]U:sampledata(ii) Althoughthebehaviorof[T]withrespecttoKissimilartosetting(i),thereexistsathresholdcapacitylevel(K0T=214:58)inthissettingsuchthat[T]>0ifK>214:58,and[T]<0otherwise(seeProposition 4.16 ).When[QDS]U>[QES]U,weagainhavefourintervalstoconsiderinordertoevaluatetheincreaseinthetotalsystemprotduetoearlycommitment:(i)K1+k11,(ii)1+k11T+kmT.ThedetailedanalysisoftheexpectedtotalprotineachoftheseintervalsispresentedinAppendix B.2.3 ,whichleadstoProposition 4.17 : 99

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[T]asafunctionofKwhen[QDS]U>[QES]U. Proof. B.4 B.7 ,and B.8 .Part(b)followsfromLemma 4.1 andLemma B.8 Figure 4-10 illustratesProposition 4.17 foranexamplesettingwhere1=2=100,1=2=30,c=10,w=30,andr=50.Inthissetting,wehaveQE1=92:4,[QES]U=205:32,[QDS]U=218.27,andK00T=174:05.Nowthatwehaveanalyzedbothcases,thatis,[QDS]U[QES]Uand[QDS]U>[QES]U,wecanevaluatethecommitmentschemesfromthesystem'sperspectiveinacapacitatedsetting. Proof. 4.16 andProposition 4.17 100

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Proof. 4.16 andProposition 4.17 101

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102

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103

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104

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5.1 ,wecharacterizethestrategicinteractionbetweenthesupplierandtheprimaryretailerandintroducethesupplier'sproductioncapacityasademandmanagementtool.Section 5.2 presentsthestrategicuseofleftoversunderdierentpowerstructuresofthesupplychain.AfteranalyzingwholesalepricinginSection 5.3 ,weconcludethischapterbyhighlightingourmainndingsinSection 5.4 105

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5-1 ,wheretheprimaryretaileractsrstatthetoplevel,andthesupplieractsafterthatatthesecondlevel.Thesupplier'sdecisiondependsontheactionoftheprimaryretailer,i.e.,thenodeheisataftertheprimaryretailer'smove.NotethattheactionsetofthesupplierinFigure 5-1 islimitedtotwoforsimplicity,whereasheactuallyhasaninnitenumberofpossibleactionssinceweassumehisproductionquantityiscontinuous.Theoptimalsolutionofthedecisiontreecanbecharacterizedusingbackwardinduction.Beingtheleader,thatis,havingtherstmove,theprimaryretailertakesinto 106

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Sequentialgameundertheprimaryretailer'slead. accounttheoptimalresponseofthesupplier.RecallthatwehavealreadyanalyzedtheoptimalproductionquantityofthesuppliergiventheselectionoftheprimaryretailerinChapter 4 .ThesupplierproducesQDS=H1(wc w)=T+kmTiftheprimaryretailerselectsdelayedcommitment,andQES=F11(rw r)+F12(wc w)=T+k11+km2ifsheselectsearlycommitment.Hence,inthissetting,theprimaryretailerisabletodictateherchoiceofcommitmentschemetothesupplychain.Thatis,theprimaryretailercandeterminewhatwillhappen,andpicksherbestoutcomebasedondemand,costandrevenueparameters.Next,werevisitourcomputationalstudyinChapter 4 toillustratetheequilibriumsolutionfordierentparametervalues.Inordertoanalyzetheeectsofdierentparametersontheoutcomeoftheinteractionbetweenthesupplierandtheprimaryretailer,westartwiththedatasetdesignedinChapter 4 ,whichisduplicatedinTable 5-1 .Ofthe243testcases,theprimaryretailerwillselectdelayedcommitmentin102instances(42%).However,recallthatthecomputationaltestwasbiasedinthatweexcludedinstanceswherewecouldproveanalyticallythroughsucientconditionsthattheprimaryretailerwouldselectadelayedcommitmentscheme.Inordertohaveanunbiasedanalysis,weusethe1,000randomtestinstancesthatweregeneratedinChapter 4 withthefollowingparameters:iU[0;150],cviU[1=10;1=3],c=5,wU[c;10c],rU[w;10w].Tables C-1 107

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Parameters. C-2 ,and C-3 inAppendix C depictthedistributionoftheseinstancesoverdierentrangesofservicelevels,coecientofvariations,andmeandemandlevels,respectively.In105oftheseinstances,theequilibriumisearlycommitment.Thatis,only10.5%ofthetimedoestheprimaryretailerselectearlycommitment.Ineachofthese105instances,thesupplier'spreferenceisalsoearlycommitment.Hence,earlycommitmentispareto-optimal.However,thesupplierprefersearlycommitmentin933instances,i.e.,93.3%ofthetime.Intheremaining67instances,heprefersadelayedcommitmentscheme,whichisalsotheequilibriumsolution.Hence,inonly17.2%ofthetestinstances,thesupplier'spreferenceandtheequilibriumoutcomematch.Therefore,wemaydeducethatthesupplierhasasignicantincentivetoworkwiththeprimaryretailerinordertogethertoselectanearlycommitmentscheme.Fortherestofthischapter,weanalyzeanumberofdemandmanagementtoolsthatthesuppliercanutilizetoinducehisdesiredoutcome,andassesstheireectiveness. 5.1.1 revealedthattheoriginaloutcomeofthestrategicinteractionbetweenthesupplierandtheprimaryretailerisnotinalignmentwiththesupplier'spreferenceinmostcases,whichisearlycommitment.Hence,thesupplierhasasignicantmotivetogettheprimaryretailertocommitearly.Onesuchtoolthatthesuppliercanutilizeistoactbeforethecommitmentdecisionoftheprimaryretailerbybuildingupcapacityorinventory.Inthissection,wearguethatthesuppliercaninuencethedecisionoftheprimaryretailerifhecancrediblysharetheproductionquantityinformationwiththeprimaryretailer,oriftheprimaryretailerisabletoobservetheactiontakenbythesupplier.Note 108

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r)+F12(wc w)(QDS=H1(wc w)).Furthermore,thesuppliercannotgettheprimaryretailertoselectdelayedcommitmentbyimposingacapacitylimitifheroriginalchoiceisearlycommitment(seeSection 4.6 foradetaileddiscussionofthecapacitatedsetting).Hence,thesupplierisgoingtousetheleadershippositioneectively 109

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4.6 ,iftheprimaryretailerprefersearlycommitmentintheoriginalsetting,shewillcontinuetodosointhepresenceoflimitedcapacity.Ontheotherhand,ifsheprefersdelayedcommitment,wehavethefollowing: Proof. 00,suchthatE[E1(K)]>E[D1(K)]8K2(0;),whichprovesthepropositiontogetherwiththeconditionthat[E[D1]]U>[E[E1]]U,where[E[D1]]Uand[E[E1]]Udenotetheprimaryretailer'sexpectedprotundertheoriginaldelayedandearlycommitmentschemes,respectively. Animmediatecorollaryoftheabovepropositionis: 110

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5.1 doesnotnecessarilyguaranteethatthesupplierisgoingtosetcapacityequaltoalevelwheretheprimaryretailerselectsearlycommitment.WhetherthesuppliershouldsetcapacityequaltothisleveldependsonhisexpectedprotsundertheoriginaldelayedcommitmentschemeandunderacapacitatedearlycommitmentschemewherethecapacitylevelischaracterizedbyProposition 5.1 .Ourapproachtoresolvethisissueisasfollows:wenextshowthatifthesupplierprefersearlycommitmentintheoriginalsetting,thereexistsathresholdcapacitylevelabovewhichthesupplierwillgeneratemoreprotunderanearlycommitmentschemecomparedtotheoriginaldelayedcommitmentscheme.Then,bycomparingthisthresholdwiththeonethatmakestheprimaryretailerindierent,wecanconcludewhetherthesuppliershouldeectivelyassumetheleadershippositionandbuildupcapacitytohavetheprimaryretailerorderearly. Proof. [S(K)]=wk11+TL(km)2LKTk11 111

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c. Proof. 5.1 andLemma 5.1 Proposition 5.2 characterizestheoutcomeofthestrategicinteractionbetweenthesupplierandtheprimaryretailerwhenthesupplieristheleader.Whenbothpartiespreferthesamecommitmentscheme,thesuppliersetsthecapacityequaltohisproductionquantityintheoriginalsetting.Whenthesupplierandtheprimaryretailerpreferdelayedandearlycommitmentschemes,respectively,thesuppliercannotincreasehisprotbysettingadierentcapacitylevelthantheproductionquantityunderprimaryretailer'slead.Hence,theoutcomeisagainthesame,independentoftheleader.Ontheotherhand,whenthesupplierandprimaryretailerpreferearlyanddelayedcommitmentschemes,theoutcomedependsonthethresholdcapacitylevels.IfKr>Ks,thesupplierincreaseshisprotsbysettingcapacitytoKrandtheprimaryretailerselectsearlycommitment.Otherwise,theoutcomeisthesameasthecasewheretheprimaryretaileristheleader,whichistheoriginaldelayedcommitment(equivalently,K=[QDS]U).Table 5-2 summarizestheoutcomeofthegamebetweenthesupplierandtheprimaryretailerundertheleadofbothparties.Wenextprovidetheresultsofourcomputationalanalysistoseehoweectivecapacityisasatoolforthesuppliertomanipulatetheprimaryretailer'sdemand,thatis, 112

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Outcomeofstrategicordertimingundersupplier'sandprimaryretailer'slead. Supplier'sChoiceR1'sChoiceR1'sLeadSupplier'sLead EarlyEarlyEarlyEarly,K=[QES]UDelayedDelayedDelayedDelayed,K=[QDS]UEarlyDelayedDelayedEarly,K=KrifKr>KsDelayed,K=[QDS]UifKr0.Similarargumentsarealsovalidfortheinstancewherethesupplierhastheminimumgain(2.70%),whichoccurswhen1=80,2=120,cv1=cv2=1=7,andr=40.Theaveragegainforthesupplierfromusingcapacityasastrategictoolis7.41%.Inall102cases,theprimaryretailergetsherorderquantityinfullwhensheswitchestoearlycommitmentduetothecapacitylimit.Furthermore,weknowthatsheprefersdelayedcommitmentinalltheseinstancesintheoriginalsetting.Hence,wecanconcludethattheprimaryretailer'sexpectedprotdecreasesduetothelimitedcapacity.The 113

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5-2 forthesupplier,andFigure 5-3 fortheprimaryretailer.Whentheprimaryretailer'sdemandincreases,herorderquantityunderanearlycommitmentschemewillalsoincreasewithcertaintysincek1>0.Hence,thepercentageincreaseinthesupplier'sprotalsoincreases(seeFigure 5-2 ).Fromtheprimaryretailer'sperspective,thebenetsofdelayedcommitmentdecreaseas1increases(seeChapter 4 ,Figure 4-1 ).Hence,beingforcedtocommitearlyresultsinlowerlosseswhen1increases(seeFigure 5-3 ).When2increases,theportionofthesupplier'sprotduetoR2increases.Hence,weexpectthattheincreaseinprotsduetotheearlycommitmentofR1willdecrease.Furthermore,increasing2makesR1morereluctanttocommitearly(seeChapter 4 ,Figure 4-2 ),whichforcesthesuppliertosetatightercapacity.Hence,thepercentageincreaseinthesupplier'sprotdecreasesin2(seeFigure 5-2 ).Duetothesameconsiderations,thepercentagelossoftheprimaryretailerincreasesin2(seeFigure 5-3 ).Theeectsofcoecientsofvariationoftheretailerdemands(cv1andcv2)areillustratedinFigure 5-4 forthesupplier,andFigure 5-5 fortheprimaryretailer.Whencv1increases,theorderquantityoftheprimaryretailerunderanearlycommitmentschemeincreases,whichworksinfavorofcapacitatedearlycommitment.However,theretailerbecomesmorereluctanttocommitearlysincethebenetsofdelayedcommitmentincreaseincv1(seeChapter 4 ,Figure 4-3 ),resultinginatightercapacitythatdecreasesthesupplier'sprotsduetoR2'sdemand.Figure 5-4 showsthattheformerdominatesthelatter,resultinginhigherpercentageincreasesinthesupplier'sprotascv1increases.As 114

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Percentincreaseinthesupplier'sexpectedprot:eectsof1and2. Figure5-3 PercentdecreaseinR1'sexpectedprot:eectsof1and2. thebenetsofdelayedcommitmentincreaseincv1,thepercentagedecreaseintheprimaryretailer'sprotincreasesincv1(seeFigure 5-5 ).Theincreaseinthesupplier'sprotdoesnotshowasignicantchangeincv2.However,theprimaryretailer'spercentagelossincreasesincv2(seeFigure 5-5 ).Thisismainlyduetotheincreaseinthebenetsofadelayedcommitment(seeChapter 4 ,Figure 4-4 ).Recallthatthesupplierisabletousecapacityasastrategictoolandincreasehisprotsinall102instancesinourdatasetinwhichtheprimaryretailerselectsdelayedcommitmentoriginally.However,recallalsothatwehavek1>km,i.e.,theretailmarkupisgreaterthanthewholesalemarkup,inallinstances.WenextconsidertherandomdatasetdiscussedinSection 5.1.1 toseehoweectivecapacityiswhenwedonotassumeany 115

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Percentincreaseinthesupplier'sexpectedprot:eectsofcv1andcv2. Figure5-5 PercentdecreaseinR1'sexpectedprot:eectsofcv1andcv2. orderonk1andkm.In1000instancesexamined,thesupplierhastheopportunitytousecapacityin828instances.Thatis,thesupplierandtheprimaryretailerpreferearlyanddelayedcommitmentintheoriginalsetting,respectively.In505oftheseinstances(61%),thesupplierbenetsfrominducingtheprimaryretailertocommitearly.Thepercentageincreaseinthesupplier'sprotcanbeashighas63%,andtheaverageis6.68%.Theaveragedecreaseintheprimaryretailer'sprotis4.55%,themaximumbeing42%.Next,weanalyzehowtheseinstancesaredistributedoverdemandandrevenueparametersintermsofeectiveness,whichwemeasureasthepercentageofinstancesinwhichcapacitycanbeusedsuccessfullytoinduceearlycommitment. 116

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Strategicuseofcapacityoverk1andk2. Table 5-3 depictsthenumberofinstancesthatcapacitycanbesuccessfullyusedasastrategictoolfordierentrangesofservicelevelsfortheprimaryretailer(k1)andthesupplier(km).Recallingthattheunitcostwaskeptconstantinallinstances,kmalsoimpliesthewholesaleprice.Thenumbersinparenthesesarethenumbersofinstanceswherethesupplierhasachancetoinduceearlycommitment.Thestrategicuseofcapacityismoresuccessfulwhenk1islarger,whichisreasonablesincetheprimaryretailerwillbemoreinclinedtoswitchtoearlycommitmentwhencapacityisrestricted.Theoppositeholdstrueforkm:whentheservicelevelofthesupplierislarger,theprimaryretailerismorelikelytosticktodelayedcommitment,whichwouldcausethesuppliertoreducecapacitysignicantlytoinduceearlycommitment.Hence,thesuccessrateofusingcapacityasastrategicdemandmanagementtooldecreases.Notethatthereisnocasewithk11orkm1wherecapacitycanbeusedasastrategictool.Intheformercase,thesupplierprefersdelayedcommitmentaswell;hencehedoesnothaveanincentivetoinduceearlycommitment.Inthelattercase,theprimaryretaileralreadyprefersearlycommitment;hencethereisnoneedtouseanextrameasuretoinduceearlycommitment.Table 5-4 depictsthenumberofinstancesthatthesupplierbenetsfromtheuseofcapacityfordierentrangesofthecoecientofvariations,cv1andcv2.Thereisnotanevidentpatterninthenumberofinstancesascv1andcv2changes.Thisisbecausetheeectsofuncertaintyonthecommitmentschemearecloselyrelatedtotheservicelevelsofthesupplierandtheprimaryretailer.Forinstance,ifk1>0,theprimaryretailer'searlyorderquantityincreasesasuncertaintyincreases,whichmayallowthesuppliertoinduceearlycommitmentmoreeciently.Ontheotherhand,ifk1<0,increasinguncertaintyworksintheoppositedirection. 117

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Strategicuseofcapacityovercv1andcv2. 0:1cv20:1780:178
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Strategicuseofcapacityover1and2. 0<25050<2100100<2150 0<15046(74)30(87)24(82)50<110080(95)62(91)45(92)100<115073(100)83(108)62(99) Thereareanumberofstudiesinthesupplychainmanagementliteraturethatfocusoninformationasymmetrybetweensupplychainmembers.Thesestudiesdealwithsupplychainswhereonermpossessessuperiorinformationregardingitsowncosts(CorbettanddeGroote(2000),Ha(2001),Corbett(2001)),operatingcharacteristics(Gavirnenietal.(1999),Lim(2001)),anddemandforecasts(LauandLau(2001),Leeetal.(2000),Li(2002)).Whenthermwiththesuperiorinformationistherstpartytoact,theresultinginteractionischaracterizedasa`signalinggame'(seeCachonandNetessine(2004)).Notethatoursettinginthissectionisquitesimilartoa`signalinggame'.However,thereisanimportantdistinctionwhichpreventsusfromasimilaranalysis:intheliterature,theinformationasymmetrybetweenmembersisaboutanexogenousparameter,forwhichthermwithoutperfectinformationholdsapriorprobabilitydistribution,andthermwithsuperiorinformation`signals'theinformationbychoosingaspecicaction.Inoursetting,ontheotherhand,theinformationasymmetryisabouttheactionchosenbytherstpartytoact.TheresultinginteractionbetweenthesupplierandtheprimaryretailerisdepictedinFigure 5-6 asanextensivegamewithimperfectinformation(seeOsborne(2004)foradetaileddiscussion).Forsimplicity,itisassumedthatthesupplierhasonlytwostrategiestochoosefrom,K1andK2,whereasheactuallyhasinnitelymany.However,wewilllateronshowthatheactuallyhastwocandidatestrategiesforoptimality.Theotherstrategieswillbedominatedbythese.Thedashedlinebetweenthenodescorrespondingtotheprimaryretailer'sdecisionindicatesthattheprimaryretailerdoesnotknowthesupplier'scapacitydecisionwhenshechoosesthecommitmentscheme.Inotherwords,shedoesnotknowwhatnodesheisatwhensheistochooseherstrategy. 119

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Sequentialgamebetweenthesupplierandtheprimaryretailerwithimperfectinformation. Sinceeachplayermovesonlyonce,andnoplayer,whenmoving,isinformedofanyotherplayer'saction,thisgamemayalternativelybemodelledasastrategicgamewherethesupplier'sactionsetcontainsinnitelymanystrategies(capacitylevel),andtheprimaryretailer'sactionsetconsistsoftwostrategies(earlyanddelayedcommitment).ThereadermayrefertoChapter 4 ,Section 4.5 forthediscussionofstrategicgames.DuetoDenition 4.2 inSection 4.5 ,wecanlimittheactionsetofthesupplierasfollows:assumethattheprimaryretailerselectsdelayedcommitment.Then,theonlyviableactionforthesupplieristosetcapacityequalto[QDS]Usinceanyothercapacitylevelisdominatedby[QDS]U.Similarly,iftheprimaryretailerselectsearlycommitment,theonlyactionforthesupplieristosetthecapacityequalto[QES]U.Hence,thereareonlytwopossibleactionsforthesupplierthatwemayobserveatequilibrium:[QDS]Uand[QES]U.Forexpositionalclarity,letKD=[QDS]UandKE=[QES]U.TheresultinggameisdepictedinTable 5-6 ,wherethesupplieristhecolumnplayer.Eachentryinthematrixconsistsoftheexpectedprotsoftheprimaryretailerandthesupplier,correspondingtotheselectedactionpair.Forinstance,thetop-rightcellcorrespondstoR1andthesupplierselectingdelayedcommitmentandKE,respectively.TheexpectedprotsofR1andthesupplierinthiscasearerepresentedbyD1(KE)andDS(KE),respectively. 120

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Strategicgamebetweenthesupplierandtheprimaryretailer Theequilibriumconceptofastrategicgame(seeDenition 4.2 inSection 4.5 )canbeillustratedforoursettingasfollows:considertheactionprole(E;KE),thatis,R1andthesupplierselectearlycommitmentandKE,respectively.ThisactionproleisaNashequilibriumifandonlyifR1hasnoincentivetoselectdelayedcommitmentgiventhesupplierselectsKE(i.e.,E1(KE)D1(KE)),andthesupplierhasnoincentivetoselectKDgivenR1selectsearlycommitment(i.e.,ES(KE)ES(KD)).ThesupplierwillselectKD(KE)iftheprimaryretailerselectsdelayedcommitment(earlycommitment).Hence,only(D;KD)and(E;KE)arecandidateequilibriumsolutions.Proposition 5.3 characterizestheoutcomeofthegame. Proof. 121

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Example(i)forthestrategicgamewhenR1prefersearlycommitment. Delayed(8,126.93,843.12)(8,268.97,831.49)Early(8,343.42,859.75)(8,343.42,862.29) ToillustrateProposition 5.3 ,weconsidertwoexamplesettingsforthecasewheretheprimaryretailerprefersearlycommitment,i.e.,E1(KE)>D1(KD),withthefollowingparameters:(i)1=2=100,1=5,2=30,(c;w;r)=(10;15;100),and(ii)1=2=100,1=2=30,(c;w;r)=(5;10;100).Inexample(i)(Table 5-7 ),(E;KE)istheuniqueequilibrium,thatis,neitherthesuppliernortheprimaryretailerhasanincentivetodeviatefromthisactionprole,giventheotherplayerwilladheretoitsactioninthisprole.However,oneshouldnotethattheprimaryretailer'spreferenceofearlycommitmentdoesnotnecessarilymeanthatshewillcontinuetopreferearlycommitmentgiventhatthesupplierselectsKE.ConsiderthestrategicgameinTable 5-8 ,whichcorrespondstoexample(ii).Althoughtheprimaryretailerprefersanearlycommitmentscheme,shewillswitchtodelayedcommitmentifthesupplierdecidestoproduceKEunits.Inthiscase,thegameadmitsnoequilibrium,i.e.,wearenotabletopredicttheoutcomeofthegame.Proposition 5.3 doesnotprovethat(D;KD)isauniqueequilibriumwhentheprimaryretailerprefersdelayedcommitment,thatis,itdoesnotexclude(E;KE)asanequilibrium.IfKE>KD,thenD1(KE)>D1(KD)>E1(KE).Hence,(E;KE)isnotanequilibrium.IfKE
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Example(ii)forthestrategicgamewhenR1prefersearlycommitment. Delayed(8,238.34,830.74)(8,810.43,765.64)Early(8,473.51,985.80)(8,473.51,1,072.55) observedbytheretailer.Hence,thesuppliercannotinducetheprimaryretailertocommitearly.Thatis,thesupplierhasachancetousecapacityasastrategictooltomanipulatetheprimaryretailer'sdemandonlyifhecancrediblysharethecapacityinformation. 4 ,theprimaryretailerhasasingleorderopportunity,whichisutilizedbeforethedemandrealizations.Duetotheuncertaintyinvolvedinthedemandprocess,theprimaryretailerusuallyoptstoselectdelayedcommitmentandshareacommonpoolofinventorywiththesecondaryretailerafterdemandrealizationsinordertoeliminatecostsassociatedwithdemanduncertainty,ratherthanorderingearlyandfacingtheriskofuncertainty.Inthissection,weanalyzeamoreexibleversionoftheoriginalearlycommitmentmodel,`earlycommitmentwithrecourse',wherethesupplieroerstheleftovers,ifany,totheprimaryretailerafterdemandsarerealized.Assumingthatthesupplierproducesapositivequantityforthesecondaryretailer,thereisalwaysapositiveprobabilitythatthesupplierwillhaveleftoversundertheoriginalearlycommitmentschemebecausetheproductionforthesecondaryretailerisdeliveredafterherdemandrealization.Ifherdemandturnsouttobelessthantheproductionquantity,thesupplierwillhaveleftovers.Underearlycommitmentwithrecourse,theseleftoverswillbeallocatedtotheprimaryretailerifherdemandrealizationexceedsherinitialorder.Inthissetting,weaimtoanalyzewhethersuchanarrangementprovidesenoughincentivefortheprimaryretailertochooseearlycommitmentoveradelayedcommitmentscheme,andwhetherthesupplieriswillingtooerit.Theimplicationsofsuchanarrangementseemtrivialatrst.Boththesupplierandprimaryretailerappeartobenet;thesuppliergeneratesmoreprotfrompossible 123

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S(Q1;QS)=w8>>>>>>><>>>>>>>:(Q1+x2)x1Q1;x2QS(Q1+QS)x2>QS(x1+x2)Q1Q1+QSx2;x2QS9>>>>>>>=>>>>>>>;c(Q1+QS): TakingexpectationwithrespecttoX1andX2,wegetE[S(Q1;QS)]=w24QSZ0(Q1+x2)F1(Q1)+(Q1+QS)[1F1(Q1+QSx2)]!dF2(x2)+QSZ0Q1+QSx2ZQ1(x1+x2)dF1(x1)!dF2(x2)+(Q1+QS)[1F2(QS)]35c(Q1+QS) (5{3)Therstandsecondderivativesofthesupplier'sexpectedprotwithrespecttoQSare 125

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Proof. w)=w24QSZ0[1F1(Q1+QSx2)]f2(x2)dx235>0ThepropositionfollowssinceE[S(QSjQ1)]isconcave. Proposition 5.4 isintuitiveforthefollowingreason:intheoriginalearlycommitmentscheme,thereisnoleftoverreallocationandthesupplierproducesF2((wc)=w)unitsinadditiontotheorderoftheprimaryretailer,Q1.SincetheexpectedstockoutforR1isgreaterthanzeroregardlessofQ1,thesupplierproducesmoreforthesellingseasoninthepresenceofreallocationtoaccountforthisextrademand.Proposition 5.5 furthercharacterizestheoptimalresponseofthesupplier,QS,totheearlyorderoftheprimaryretailer,Q1. Proof. 5{4 )tozero.Thatis,wehave 5{5 )withrespecttoQ1,weget

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5{5 )yields(QSR0QSx2R0f1(x1)f2(x2)dx1dx2=wc w).Hence,wehaveQS=H1(wc w).WhenQ1!1,Equation( 5{5 )yields(F2(QS)=wc w).Hence,QS=F12(wc w). SincetheexpectedstockoutfortheprimaryretailerisdecreasinginQ1,thesupplierdecreasesQSasQ1increases.WhenQ1=0,thesystemisequivalenttoadelayedcommitmentschemefromthesupplier'sperspective,andheproducesH1(wc w)units.However,fromtheprimaryretailer'sperspective,thereisacrucialdierencefromadelayedcommitmentscheme:theallocationmechanismusedisnolongerproportionalsincethesuppliersatisesthesecondaryretailer'sdemandrst.Wenextanalyzetheoptimalorderquantityfortheprimaryretailer.GivenX1=x1andX2=x2,R1'sprotisgivenby 1(Q1;QS)=8>>>>>>><>>>>>>>:rx1wQ1x1Q1(rw)Q1x1>Q1;x2>QS(rw)x1x2QS;Q1x1Q1+QSx2(rw)(Q1+QSx2)x2QS;Q1+QSx2
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Proof. r)istheorderquantityunderanearlycommitmentschemewithoutrecourse. r+(rw)264w r(1F2(QS))+QSZ0(1+dQS rf2(x2)dx2375 r+(rw)264w r(1F2(QS))+QSZ0w rf2(x2)dx2375 =0Inequality( 5{9 )followssincewehaveF1(Q1)rw rforallQ1QE1.Inequality( 5{10 )followssince0<(1+dQS Itisintuitivethattheprimaryretailerorderslessthanherorderquantityunderanearlycommitmentschemewithoutrecourse,sincethereisapossibilitythattherewillbeinventoryleftatthesupplierafterdemandrealizations.Wenowfocusontheoutcomeoftheinteractionbetweentheprimaryretailerandthesupplier.Inordertoinvestigatewhethertheprimaryretaileroptstoselectearly 128

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5-1 .Aswepreviouslyprovedanalytically,theorderquantityoftheprimaryretailerunderearlycommitmentwithrecourseislessthanherorderundertheoriginalearlycommitmentscheme.AsthecoecientofvariationofR2'sdemand(cv2)increases,Q1decreasessincetheexpectedquantityofleftoversincreaseincv2,whichenablesR1torelymoreontheleftovers.ThisalsopromptsthesuppliertoproducemoreinexcessofQ1,i.e.,QSincreasesincv2.Asrincreases,theprimaryretailerincreasesQ1tosecuremoreinventory,whichleadsthesuppliertodecreaseQS.However,recallthatwehaveprovedanalyticallythatQSunderearlycommitmentwithrecourseisgreaterthanQSunderearlycommitmentwithoutrecourse.Inourcomputationaltests,earlycommitmentwithrecourseincreasestheprimaryretailer'sprotsby1%onaveragewhencomparedtothecasewithoutrecourse.Theincreasegetsmoresignicantasrdecreases.However,recallthatthebenetsofdelayedcommitmentoverearlycommitmentalsoincreaseasrdecreases.Asaresult,thesupplierisnotabletogettheprimaryretailertoselectearlycommitmentwithrecoursewhenr=40.Recallthattheprimaryretailerselectsdelayedcommitmentin102ofthe243cases.Theseincludeallcaseswithr=40;theremainingarewithr=70.Whenr=70,thesuppliercansuccessfullyemploytheleftover-reallocationschemetoinducetheprimaryretailertoorderearlywhenevertheprimaryretailer'soriginalpreferenceisdelayedcommitment,whichcorrespondsto21cases.Theaverageincreaseinthesupplier'sprotinthesecasesis9:85%,whereasitis0:5%forR1.Hence,wecanconcludethatalthoughthereallocationofleftoversisnotpowerfulenoughtoinducetheearlycommitmentoftheprimaryretailerinallcases,itprovidesasignicantincreaseinthesupplier'sprotwhenitispossibletoimplementit.Forthisdataset,whentheoriginaloutcomeisearlycommitment,thesupplierisneverbetterowithearlycommitmentwith 129

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5.1.1 .Mostofourndingsaboveremainvalid.In7:7%ofthesetestcases,thesupplieroersearlycommitmentwithrecourseandtheprimaryretaileracceptsit.In25ofthesecases,theoriginaloutcomewasearlycommitment,whichwedidnotobserveinthepreviousdataset.Intheseinstances,theprotmarginofthesupplierissolowthatheisbetterooeringleftoverstotheprimaryretailerinordertoreducethecostofuncertaintyduetothesecondaryretailer'sdemand.Sincetheprimaryretailerdoesnotdecreaseherinitialordersignicantly,thesupplierisabletoincreasehisprots.Intheremaining52instances,theleftoverreallocationintroducedbythesuppliersuccessfullychangestheordertimingoftheprimaryretailer.Hence,ourinitialobservationthatreallocationoftheleftoversisnotpowerfulenoughtoinduceanearlycommitmentoutcomeundertheprimaryretailer'sleadisstillvalid.However,againinalignmentwithourinitialndings,itcanincreasethesupplier'sprotasmuchas30%whenitissuccessfullyimplemented. 5.1.1 ),andevaluatetheeectivenessoftheearlycommitmentschemewithrecourseunderthesupplier'slead.Werstsolvetheprimaryretailer'sproblem,giventheproductionquantityofthesupplierforthesellingseason,QS.TheexpectedprotoftheprimaryretailerisgivenbyEquation( 5{7 ).GivenQS,therstandsecondderivativesoftheprimaryretailer's 130

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5{11 )and( 5{12 ),respectively. SinceE[1(Q1;QS)]isconcave(@2E[1(Q1;QS)] Proof. r)(rw)=rforallx22(0;QS).ThepropositionfollowssinceE[1(Q1;QS)]isconcaveinQ1. ConsideringPropositions 5.6 and 5.7 together,wecandeducethattheprimaryretailer'sorderquantityislessthanorequaltoherorderunderearlycommitmentwithoutrecourseregardlessoftheinventoryavailableafterdemandrealizationsandtheleadershipofthesupplychain.Proposition 5.8 furtheranalyzestheoptimalresponseoftheprimaryretailer(Q1)tothesupplier'saction,QS. 131

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Proof. 5{11 )tozero.Thatis,wehave 5{14 )withrespecttoQS,weget 5{14 )yieldsF1(Q1)=(rw)=r,i.e.,Q1=F11((rw)=r).WhenQS!1,wehavewF1(Q1)=0,i.e.,Q1=0. WhenQSincreases,theexpectedquantityoftheleftoversthatwillbeavailabletotheprimaryretaileralsoincreases.Becauseoftheriskofuncertaintyassociatedwiththeearlyorder,theprimaryretailerdecreasesherorderquantity,andreliesmoreontheleftoversasQSincreases.FromProposition 5.8 ,wecandeducethatthesupplierisabletoinducetheprimaryretailertoorderanyquantityin(0;QE1)bymanipulatingQS.Inotherwords,thesuppliercanincreaseQ1bydecreasingQS,whichcoststhesuppliertheexpectedrevenuethatwillbegeneratedafterthedemandrealizations.Thatis,thesupplierfacesthetradeobetweentherevenuefromR1'sorderandtherevenuesfromR2'sdemandandpossiblereallocationtoR1.Thistradeoiscapturedbyincorporatingtheoptimalresponseoftheprimaryretailer(Q1asafunctionofQS)intothesupplier'sexpectedprot(Equation 5{3 ).Wenowpresentourndingsviacomputationalanalysisregardingtheearlycommitmentwithrecourseundersupplier'slead,andinvestigatewhetherthesuppliercanusethisalternativetoinducetheearlycommitmentoftheprimaryretailer.Westart 132

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5-1 .Asinthesettingwiththeprimaryretailer'slead,Q1decreasesincv2.However,contrarytothatsetting,QSdecreasesincv2underthesupplier'sleadsincethesuppliertriestopreventafurtherdecreaseinQ1bylimitingtheexpectedquantityofleftovers.Thesupplier's(primaryretailer's)expectedprotisgreaterunderhis(her)lead.Furthermore,Q1(QS)isgreaterunderthesupplier's(primaryretailer's)lead.Underthesupplier'slead,thesupplierisstillunabletoinduceearlycommitmentwhenr=40,whereashemanagestodosowhenr=70.Hence,onemightconcludethatearlycommitmentwithrecoursedoesnotprovideenoughincentivefortheprimaryretailertocommitearly,evenunderthesupplier'slead.However,notethatouranalysisofthisschemedoesnottaketheprimaryretailer'schoiceofcommitmentintoaccount.Inotherwords,thesupplieroptimizeshisexpectedprotunderearlycommitmentwithrecourseandtheresultingoutcomemaynotprovidesucientincentivefortheprimaryretailertoswitchfromanexistingdelayedcommitmentscheme.Insuchcases,thesuppliermayinduceearlycommitmentbyincreasingtheproductionquantityforthesellingseason(andhencetheexpectedquantityofleftovers),whichincreasesprimaryretailer'sexpectedprotsaswell.IfthesuppliercanincreaseQStoacertainlevelwheretheprimaryretailerisindierentbetweenthedelayedcommitmentandearlycommitmentwithrecourse,andifhecanmaintainagreaterprotlevelthanthedelayedcommitmentcaseatthesametime,thenhecansuccessfullyinducetheprimaryretailertocommitearlybyusingthereallocationofleftoversstrategically.Forexample,considerthefollowinginstance:(c;w;r)=(5;10;40)and(1;1)=(2;2)=(80;80=3).Intheoriginalsetting(norecourse,primaryretaileristheleader),theprimaryretailerprefersdelayedcommitment,andthecorrespondingexpectedprotsoftheprimaryretailerandthesupplierare2,173.9and649.5,respectively.Ifthesupplieroersearlycommitmentwithrecoursebyoptimizinghisownexpectedprots,theresultingorderquantitiesareQ1=93:2and 133

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Strategicuseofleftoversoverk1andk2. 5-9 depictsthenumberofinstancesthatleftoverscanbesuccessfullyusedasastrategictoolfordierentrangesofservicelevelsfortheprimaryretailer(k1)andthesupplier(km).Asforthestrategicuseofcapacity,thestrategicuseofleftoversismore 134

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Strategicuseofleftoversovercv1andcv2. 0:1cv20:1780:178
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Strategicuseofleftoversover1and2: 0<15059(83)19(98)14(99)50<110094(105)75(105)44(103)100<1150109(112)98(117)69(106) Notethatin828ofthe928instanceswherehemayincreasehisprots,thesupplier'soriginalchoiceisnotinalignmentwiththeprimaryretailer(seeSection 5.1.1 ).Hence,in100instances,thesupplierhastheopportunitytoimprovehisprots,eveniftheoriginaloutcomewasinhisfavor,whichisanadvantageoftheearlycommitmentwithrecourseoverthestrategicuseofcapacity.Althoughearlycommitmentwithrecoursemayperformbetterthanthestrategicuseofcapacityincertaincases,onemayarguethatthecredibilityofthesupplier'scommitmenttoaproductionquantityislowerthanthecredibilityofcommittingtoacapacitylevel,ascapacitymaybeobservedbytheprimaryretailer,whereasthesuppliermaychangetheproductionquantityunderearlycommitmentwithrecourseoncetheprimaryretailerplacesherorder.Eventhoughthisargumentmaybecorrectinasingleperiodcontext,theprimaryretailercanpredictthesupplier'sactioninthelongtermsincetheinformationaboutthequantityofleftoversandsecondaryretailer'sdemandisavailabletotheprimaryretaileraswell.Hence,thesupplierisnotgoingtomanipulatetheproductionquantityinalong-termrelationship.Weinvestigatethesettingwheretheprimaryretailerdoesnotrelyonthesupplier'scommitmenttoaproductionquantityinthenextsection. 5.1.3 .Thatis,thesequentialdecisionmakingunderthesupplier'sleadwithanunreliablequotefromthesupplierabouttheproductionquantityisequivalenttosimultaneousdecision 136

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5{3 )and( 5{7 ),respectively. Proof. Whenthenumberofstrategiesavailabletoeachparty(inourmodel,Q1andQSfortheprimaryretailerandthesupplier,respectively)isacontinuum,andthepayos(inourmodel,expectedprots)arecontinuous,apure-strategyNashequilibriumsolutioncanbeobtainedasthecommonintersectionpointsofthereactioncurvesoftheparties.Thereactioncurveofapartyisconstructedbyitsoptimalresponsesetforeverypossiblestrategysetoftheremainingparties,providedthattheoptimalresponsesetisasingletonforagivenstrategysetoftheremainingparties(seeBasarandOldser(1995)foraformaldenitionofthereactioncurve).ThereactioncurvesofthesupplierandtheprimaryretailerarecharacterizedbyEquations( 5{15 )and( 5{16 ),respectively,sinceE[S(Q1;QS)]andE[1(Q1;QS)]arestrictlyconcaveinQSandQ1,respectively.RCS(Q1)=(QS:w"QSZ0[1F1(Q1+QSx2)]f2(x2)dx2+1F2(QS)#c=0) 137

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Proof. 5.9 ,theyintersectatleastonce.Hence,theequilibriumisunique.Figure 5-7 illustratesthereactionfunctionsandtheuniqueequilibrium. Theprecedinganalysischaracterizestheoutcomeoftheinteractionbetweenthesupplierandtheprimaryretailerassumingthatbothpartiesagreetooperateunderanearlycommitmentschemewithrecourse.Thatis,wehavenotyetaddressedwhethertheprimaryretaileriswillingtoswitchtoanearlycommitmentschemewithrecoursefromtheoriginalsettingornot.Iftheoriginaloutcomeisearlycommitment,andthesupplier 138

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Responsefunctionsforthestrategicgame oerstheleftovers,theprimaryretailerwillaccept.However,iftheprimaryretailer'soriginalchoiceisdelayedcommitment,andtheNashequilibriumprovideslowerprots,theprimaryretailerwillsticktodelayedcommitment.Insuchacase,onemayarguethatthesupplier,beingawareoftheprimaryretailer'sprotunderadelayedcommitmentscheme,maytrytouseQSstrategicallytogettoearlycommitment.Thatis,thesuppliermayincreaseQSuptoacertainlevelsuchthattheprimaryretailerisindierentbetweenearlycommitmentwithrecourseanddelayedcommitment.Wenowelaborateontheseissues,andarguethatthestrategicuseofQSisnotpossiblewithoutcredibleinformationsharing.Let'sassumefornowthatthereisnostrategicuseofQS,andlet(Q1;QS)betheuniqueequilibriumtotheearlycommitmentwithrecourse.Iftheprimaryretailer'sexpectedprotunderdelayedcommitmentislargercomparedtothatofearlycommitmentwithrecourse,theoutcomeoftheinteractionbetweenthesupplierandtheprimaryretailerwillbedelayedcommitment,wherethesupplierproducesQDSunitsandallocatestotheretailerssinceneitherofthepartieshasanincentivetodeviatefromthissolution.Otherwise,theoutcomeisearlycommitmentwithrecoursewheretheprimaryretailer 139

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5-1 .Withrespecttoexpectedprotsandorderquantities,theearlycommitmentmodelwithrecourseundertheleadofanunreliablesupplierispositionedbetweenthemodelwiththeretailer'slead,andtheonewithareliablesupplier'slead,whichisreasonablesincethissettingreducestoastrategicgamewhereneitherofthepartieshasthebenetoftheleadership.In21ofthe102caseswheredelayedcommitmentistheoriginaloutcome,thesuppliermanagestoinduceearlycommitmentwithrecourse,asundertheretailer'slead.Hisexpectedprotsincreasebyanaverageof10:1%,whichisslightlylargerthantheincreaseundertheretailer'slead.Consideringtherandomdatasetof1000instances,earlycommitmentwithrecourseisimplementedinonly45instances.Earlycommitmentwithoutrecoursewastheoriginaloutcomein30oftheseinstances.Hence,wecandeducethatthesupplierneedstoconcentrateoncredibleinformationsharinginordertouseearlycommitmentwithrecourseeectively. 140

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5.1.1 .Thesupplierdoesnothaveanyincentivetooeradiscountunderadelayedcommitmentschemesincehisexpectedprotincreasesinthewholesalepriceinsuchacase.Therefore,weanalyzetheexpectedprotofthesupplierunderanearlycommitmentschemewithadiscountedwholesaleprice,andexaminewhetherhecanusewholesalepricetoimprovehisexpectedprot.Notethattheprimaryretailerwillnotagreetocommitearlywiththediscountedpriceunlessherexpectedprotisatleastaslargeasherprotundertheoriginalsetting.Notealsothattheprimaryretailerwillcertainlybebetterowiththediscountedwholesalepriceifshealreadyprefersearlycommitmentundertheoriginalsetting. 141

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wd(5{19)FromEquation( 5{19 ),weget 5{20 )intoEquation( 5{18 ),weget [1F1(Q1)][1F2(QS)]=c r:(5{21)Thatis,Q1canalsobeexpressedasafunctionofQS,whichischaracterizedbyEquation( 5{21 ).Hence,theexpectedprotofthesupplierunderanearlycommitmentschemewithadiscountedwholesalepricecanbeexpressedas 1F2(QS)0@Q1+21ZQS(xQS)dF2(x)1AQSZ0(QSx)dF2(x)35(5{22)whereQ1isafunctionofQS,and0QSF12(wc w).Iftheoriginaloutcomeoftheinteractionbetweentheprimaryretailerandthesupplierisearlycommitment,thenmaximizingE[ES(QS)]issucienttosolvethesupplier'sproblem.Otherwise,theoptimalexpectedprotlevelunderanearlycommitmentschememaynotberealizablesincetheprimaryretailermayrefusetoswitchtoanearly 142

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5{22 )doesnotprovideusefulinsightsabouttheprecedingdiscussions.Hence,wenextpresentourndingsthroughcomputationalanalysisabouttheeectivenessofahomogenouswholesalepricediscountinincreasingsupplier'sprots.WerstevaluatethedatasetpresentedinTable 5-1 .Recallthatthereare102instances(outof243)wheretheoriginaloutcomeisdelayedcommitment,andthesupplieralwaysprefersearlycommitment.In40instancesofthese,thesupplierisbetterobyoeringadiscountedwholesalepriceandinducingtheprimaryretailercommitearly.Theseincludethe21caseswheretheunitrevenuefortheretailersis70.Recallthatintheremaininginstanceswherer=70,theoriginaloutcomeisalreadyearlycommitment.Whenr=40,thediscountedwholesalepriceschemeworksinjust19ofthe81instances,whichisreasonablesinceitwouldtakeadeeperdiscounttohavetheprimaryretailercommitearlywhenherunitrevenueislower,whichthesupplierisreluctanttooer.Hence,oeringahomogenousdiscountworksbetterwhenrislarger.Theaverageincreaseinthesupplier'sexpectedprotduetoadiscountinthewholesalepriceis6:87%.Itis4:71%whenr=40and8:83%whenr=70.Theaveragewholesalepriceis9.28whenr=40,and9.87whenr=70,whichisinalignmentwiththeabovediscussion(theoriginalwholesalepriceis10).Theincreaseinthesupplier'sexpectedprotincreasesas1and/orcv1increase,whichisreasonablesincetheearlyorderquantityoftheprimaryretailerincreasesin1andcv1.Thedemandparametersofthesecondaryretaileraecttheeciencyofthediscountingschemeaswell.Theincreaseinthesupplier'sprotincreasesas2and/orcv2decrease.Thisisalsointuitivelyreasonableforthefollowingreason:highermeandemandandcoecientofvariationforofthesecondaryretailerincreasetheavailabilityofbuerstockfortheprimaryretailerunderadelayed 143

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5.1.1 ,thesuppliermanagestoinduceearlycommitmentoftheprimaryretailerbyoeringadiscountedwholesalepricein84instances.Theaverageincreaseinhisexpectedprotintheseinstancesis8:57%.Themaximumincreaseisover28%.Hence,wecanconcludethatoeringahomogenousdiscountedwholesalepricecanprovideasignicantincreaseinthesupplier'sprotalthoughitcannotbeutilizedasfrequentlyasthepreviousmethodsthatwehavediscussedinSections 5.1 and 5.2 5.3.1 ,theoptimalwholesalepricethatmaximizesthesupplier'sprotmaybetoohighfortheprimaryretailertoselectearlycommitment.Insuchcases,thesuppliermayhavetodecreasethewholesalepricefortheearlycommitmenttoacriticallevelsuchthattheprimaryretailerisindierentbetweenearlyanddelayedcommitment,andthiscriticalwholesalepricelevelisalsoaectedbythedemandparametersofthesecondaryretailer.Werstcharacterizethewholesalepricethatmaximizesthesupplier'sexpectedprotgiventhattheprimaryretailercommitsearly.Then,throughacomputationalanalysis,weexplorehoweectivethediscountingschemeisintermsofgettingtheprimaryretailertocommitearly,andcompareitwiththehomogenouswholesalepricecasethatwasdiscussedinSection 5.3.1 144

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5{23 )aswd=r[1F1(Q1)],andsubstitutingintoEquation( 5{24 ),weget 5-1 .Thereare82outof102instanceswherethesuppliermanagestogettheprimaryretailertocommitearlybyoeringadiscountedwholesaleprice.Recallthattherewereonly40suchcaseswhensecondaryretaileralsobenetsfromthediscount.Theaverageincreaseinthesupplier'sprotis6:95%,whichatrstseemscounterintuitivesinceitislessthantheaverageincreasewiththehomogenouswholesalepricecase.However,ifweonlyconsiderthe40instanceswherethehomogenouswholesalepricewaseectivelyused,theaverageincreaseinthesupplier'sprotis11%whenthediscountisonlyoeredtotheprimaryretailer.Hence,itissuperiortothehomogenouswholesalepricecasebecauseitnotonlyworksmorefrequently,butprovideshigherreturnsaswell.Forinstance,whenr=70,bothschemesmanagetogettheprimaryretailercommitearly.However,theincreaseinthesupplier'sexpectedprotis10:16%withdierentwholesale 145

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Strategicuseofwholesalepriceoverk1andk2. Table5-13: Strategicuseofwholesalepriceovercv1andcv2. 0:1cv20:1780:178
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Strategicuseofwholesalepriceover1and2. 0<25050<2100100<2150 0<15021(74)11(87)6(82)50<110039(95)24(91)17(92)100<115047(100)44(108)23(99) Notethatthewholesalepriceisnotaspowerfulasthecapacityandleftoversintermsoftheeectivenessininducingearlycommitment.However,thisobservationaloneisnotsucienttoconcludethatitisinferior.Theremaybeinstanceswhereitprovideslargerincreasesinthesupplier'sexpectedprot. 5.1.1 servesasabenchmarktoassessthesuccessofthesetools.Weconsidertheuseof(i)capacityunderthesupplier'sleadwithcredibleinformation,(ii)earlycommitmentwithrecourse(leftoverreallocation)underthesupplier'sleadwithcredibleinformation,and(iii)dierentwholesaleprices.TherandomdatasetcharacterizedinSection 5.1.1 formsthebasisforourobservations.Sincetheuseofcapacityisonlyeectivewhenthepreferencesofthesupplierandtheprimaryretailerarenotinalignment,weevaluatetheeectivenessofthedemandmanagementtoolsinthese828instancesinordertoperformafaircomparison.However,itisworthwhiletonotethatearlycommitmentwithrecourseworksin40instanceswherethepreferenceofthesupplierandtheprimaryretaileristhesameintheoriginalsetting,whichcanbeconsideredasanadvantageofthismechanismovertheothers.Recallthatthestrategicuseofcapacityworksin505of828instanceswherethesupplierandtheprimaryretailerhaveconictingpreferences.Earlycommitmentwithrecourseanddierentwholesalepricesaresuccessfulin541and232instances,respectively.Thereare613(74%)instanceswhereatleastoneofthetoolsissuccessful.Thatis,the 147

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152

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Pni=1[qi(p)]22i.DenotingthepdfofXybyfy(x;p),theexpectedprotofthermis 153

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P`i=1[Ci(p)]2isconvex.Hence,wecanconcludethatG(p)isconcave.ThesecondderivativeofC(p)withrespecttopisgivenby 224`Xi=1Ci(p)C0i(p)!2+`Xi=1[Ci(p)]2!`Xi=1([C0i(p)]2+Ci(p)C00i(p))!#whereC0i(p)andC00i(p)denotetherstandsecondderivativesofCi(p),respectively.LetT=`Pi=1[Ci(p)]23 2.AlsodenoteCi(p),C0i(p),andC00i(p)byAi,BiandCi.Then,d2C(p) Pni=1[Ci(pi)]2isconvexinp.Let!C(p)=[C1(p1);;Cn(pn)]T.ThenC(p)isthe`2normof!C(p),i.e.,C(p)=jj!C(p)jj.Forany2(0;1),thefollowingholdsforanyp1andp2: 154

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3.1 Letp0idenotetheminimumpriceatwhichRi(pi)andhenceCi(pi)areequaltozero.Atp0i,Pnj=1[Cj(pj)]21=2iswelldenedsinceatleastonemarketisselectedandtherstderivativeevaluatedatp0iis (A{2) indicatingthattheoptimalpishouldbelessthanp0i,i.e.,marketishouldbeselected.Assumemarketjisnotselectedintheoptimalsolution.Wenextcharacterizethedecisionregardingmarketi.Ifmarketiistobeselected,thenduetotherstpartoftheproof,marketjshouldalsobeselected,whichviolatestheassumptionthatmarketjisnotselected.Hence,theoptimaldecisionformarketishouldbe`notselecting'. 3.3 ,forthenewsvendormodel,weonlyconsideredi(p)=cviqi(p)andvariedthevalueofcvitoexaminetheeectsofuncertainty.Wenowextendourcomputationstoincludegenerali(p)functions.Figures A-1 and A-2 illustratethenonlinearfunctionsusedinthelinearandiso-elasticdemandcases,respectively.Inbothcases,the(p)functionsaregeneratedsuchthatthecoecientofvariationis 155

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Standarddeviationfunctionforlineardemand. lessthan1=3.Inthelinearcase,thestandarddeviationfunctionisgivenby(p)=sp 3.3 .Replicatingourcomputationalstudywiththesestandarddeviationfunctions,weobservedthatallofourobservationsarestillvalidforboththelinearandtheiso-elasticdemandcases.Forthelineardemandwealsoconsideralinearstandarddeviationfunction.NotethattheconstantcoecientcasecorrespondstothelinearstandarddeviationfunctionasdepictedinFigure A-3 .Wenowconsideralinearfunctionthatapproachestozerobeforethedemanddoes.Insuchacase,thedemandbecomesdeterministicwhenpriceexceedsacertainthreshold.Wegeneratedierentlevelsofuncertaintybychangingthisthresholdpricelevel(SeeFigure A-4 ). 156

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Standarddeviationfunctionforiso-elasticdemand. Replicatingourcomputationalanalysiswiththisformofthestandarddeviationfunction,wefoundthatourobservationsregardingthemarketselectiondecisionsstillhold.However,ourobservationsabouttheeectsofuncertaintyonthepricingdecisionsdonotgeneralizetothiscase.Thisisbecausewehaveastandarddeviationfunctionthatallowsdeterministicdemandwhenpriceexceedsacertainthresholdvalue.Insomeinstances,thesuppliersetsthepricesuchthattheresultingdemandinthemarket(s)isdeterministic.Whenuncertaintyincreases,thatis,whenthethresholdpricelevelincreases,thesuppliermayalsoincreasethepricetoremaininthe(smaller)regionwheredemandiseectivelydeterministic.Furtherincreasinguncertaintyincreasesthethresholdpriceatwhichdemandiseectivelydeterministic,andmayforcethesuppliertodecreasethepriceinordertoincreasedemandattheexpenseofincurringsomedegreeofuncertainty.Hence,thereactionofoptimalpricestochangesinuncertaintyisnotnecessarilyconsistentinthisspecialcase. 157

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Linearstandarddeviationfunctionforlineardemand:I. FigureA-4 Linearstandarddeviationfunctionforlineardemand:II 158

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B.1.1[QDS]U[QES]U:kmTk11+km2 (i): 4{14 )andEquation( 4{16 ),respectively.Then,wehave [S]=w1k1+TLKT and whichleadstothefollowinglemma: Proof. NotethatbothE[ES]andE[DS]decreaseasthecapacitybecomesmorerestricted.ThedecreaseinE[ES]islargerthanthedecreaseinE[DS].Thiscanintuitivelyexplainedasfollows:theadvantageofanearlycommitmentschemeoveradelayedcommitmentschemefromthesupplier'sperspectiveistheguaranteedsalestotheprimaryretailer, 159

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Interval (ii): Proof. B.1 Interval (iii): B.2 ),wecanconcludethat[S]isgreaterthanzerointhisintervalaswell.Moreover,[S]isconstantsinceuncapacitatedoptimalsolutionsareachievable. (i): B{1 )andEquation( B{2 ),respectively. 160

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Proof. Interval (ii): Interval (iii): B.2.1ProofofLemma 4.1 161

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[S]=x1+21ZKx1(x1+x2K)f2(x2)dx2QE12+1ZKQE1(QE1+x2K)f2(x2)dx2=(x1QE1)+1ZKQE1(QE1+x2K)f2(x2)dx21ZKx1(x1+x2K)f2(x2)dx2>(x1QE1)+1ZKx1(QE1+x2K)f2(x2)dx21ZKx1(x1+x2K)f2(x2)dx2=(x1QE1)1ZKx1(x1QE1)f2(x2)dx2=(x1QE1)F2(Kx1)>0WhenX1=x1>K,theexpectedsalesunderdelayedcommitmentisequaltoK,whichisobviouslygreaterthantheexpectedsalesunderearlycommitment.Hence,theproofiscomplete. (i): 4.1 ,wecanconcludethatdelayedcommitment 162

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B.4 characterizeshowthedierencebetweenprotschangeswithrespecttoKinthisinterval. Proof. [T]=TKT Interval (ii): 4.1 isstillvalidinthisinterval.Thatis,delayedcommitmentalwaysoutperformsearlycommitment.ThefollowinglemmaillustratesthechangeinthedierencebetweenexpectedprotswithrespecttoK. Proof. [T]=TKT

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B.1 ,wededucethat[S]isincreasinginK. Interval (iii): 4.6 .Hence,wehavethefollowing: (iv): (i): B.2.2 isstillvalid.Thatis,delayedcommitmentalwaysoutperformsearlycommitment(seeLemma 4.1 ),andthebenetsofdelayedcommitmentincreaseinK(Lemma B.4 ). Interval (ii): 4.1 ,delayedcommitmentalwaysoutperformsearlycommitment. 164

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Proof. B.3 Interval (iii): (iv): 165

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TableC-1: Distributionofinstancesoverk1andkm. TableC-2: Distributionofinstancesovercv1andcv2. 0:1cv20:1780:178
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_IsmailSerdarBakalwasborntoparentsOmerandFatmaBakalinGenc,BingolinTurkey,onOctober1,1978.HeholdsBSandMSdegreesinindustrialengineeringfromtheMiddleEastTechnicalUniversityinAnkara,earnedin2001and2003,respectively.HehaspursuedhisPhDdegreeasadoctoralfellowintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridasince2003.Hismainareasofresearchareproductionandinventorytheory,operationsmanagementandclosedloopsupplychains. 174