Planning Models for Price Promotions in Multi-Level Supply Chains

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Title:
Planning Models for Price Promotions in Multi-Level Supply Chains
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1 online resource (135 p.)
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english
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Su, Yiqiang
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
GEUNES,JOSEPH PATRICK
Committee Co-Chair:
SMITH,JONATHAN COLE
Committee Members:
GUAN,YONGPEI
CARRILLO,JANICE ELLEN

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Subjects / Keywords:
inventory -- promotions
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Industrial and Systems Engineering thesis, Ph.D.
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Abstract:
Price promotions are playing a significant role in retail industry; however, inefficiency of price promotions has been demonstrated both in theoretical works and industrial practices. As a result, other price strategies, like every day low price (EDLP) strategies, are recommended to replace price promotions. Since there is plenty of evidence that price promotions are not replaceable from consumer's perspective, price promotions are certain to continue existing in the near future. This dissertation is going to demonstrate a number of profitable price promotion scenarios, and develop tools to help supply chain participants to increase price promotion performance and resolve potential conflicts that may occur in price promotion activities.
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In the series University of Florida Digital Collections.
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Includes vita.
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by Yiqiang Su.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: GEUNES,JOSEPH PATRICK.
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Co-adviser: SMITH,JONATHAN COLE.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-06-30

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PLANNINGMODELSFORPRICEPROMOTIONSINMULTI-LEVELSUPPLYCHAINSByYIQIANGSUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013YiqiangSu 2

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Idedicatethistomygirlfriend,Zhuofeiforbeingthereformethroughouttheentiredoctorateprogram. 3

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ACKNOWLEDGMENTS Firstandforemost,tomyparents,yourlove,supportandunderstandinghavebeeninvaluable.ThenIwouldliketoexpressthedeepestappreciationtomycommitteechair,ProfessorJosephGeunes,whohasgreatimpactonmyacademiclife.Ithankyouforyourenlighteningideas,extensiveknowledgeaboutsupplychainmanagementandremarkablepatience.Withoutyourguidanceandhelpthisdissertationwouldnothavebeenpossible.Ialsowishtothankmycommitteemembers,ProfessorJaniceCarrillo,ProfessorColeSmith,andProfessorYongpeiGuanwhoweremorethangenerouswiththeirexpertiseandprecioustime.Ithankyouallforsuggestionsonmydissertation.Finally,ItakethisopportunitytorecordmysincerethankstoallthefacultymembersoftheIndustrialandSystemsEngineeringprogramattheUniversityofFlorida,andmygraduatestudentcolleagues.SpecialthankstoRuiwei,Qianfan,Siqian,Zhili,ChinHon,Kelly,Hongsheng,Andrew,Soheil,Cinthia,Prince,andClay,yourfriendshipmademydoctoratestudyamemorableexperienceofmylife. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2PRICEPROMOTIONS,OPERATIONSCOST,ANDPROFITINATWO-STAGESUPPLYCHAIN ................................... 14 2.1ProblemDescriptionandLiteratureSurvey ................. 14 2.2ProblemStatementandFormulation ..................... 19 2.2.1ImpactofPromotionsonDemand ................... 21 2.2.2RetailerProtunderPromotions ................... 25 2.2.3SupplierProt .............................. 30 2.2.3.1Suppliertransportationcost ................. 30 2.2.3.2Supplierinventorycostandprot .............. 33 2.2.4QuantifyingtheBullwhipEffect .................... 35 2.3NumericalExperiments ............................ 38 3MULTI-PERIODPRICEPROMOTIONSINASINGLE-SUPPLIER,MULTI-RETAILERSUPPLYCHAINUNDERASYMMETRICDEMANDINFORMATION ...... 50 3.1ProblemDescriptionandLiteratureSurvey ................. 50 3.2ProblemStatementandFormulation ..................... 54 3.3SolutionProceduresforSTP ......................... 59 3.3.1DeterministicEquivalent ........................ 59 3.3.2Single-levelProblem .......................... 60 3.3.3Linearization .............................. 62 3.3.3.1Successivelinearprogrammingapproach ......... 62 3.3.3.2Al-Khayyal'sapproach .................... 63 3.3.3.3Penalty-basedmethod .................... 65 3.4NumericalExperiments ............................ 69 3.4.1ComparisonofSolutionMethods ................... 69 3.4.2ParameterAnalysis ........................... 74 4SALESPROMOTIONSANDASSORTMENTPLANNINGINMULTIPLESELLINGCHANNELS ...................................... 78 4.1ProblemDescriptionandLiteratureSurvey ................. 78 5

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4.2ProblemStatementandFormulation ..................... 83 4.2.1ProductRelatedAssumptions ..................... 83 4.2.2FacilitiesRelatedAssumptions .................... 84 4.2.3DemandRelatedAssumptions .................... 85 4.2.4Sequenceofevents .......................... 87 4.2.5DemandModels ............................ 89 4.2.6Chance-ConstrainedFormulation ................... 91 4.3SolutionProcedure ............................... 95 4.3.1MNLModelLinearization ....................... 95 4.3.2SampleAverageApproximation .................... 96 4.3.2.1Solutionvalidation ...................... 97 4.3.2.2SAAformulation ....................... 99 4.4GreedyandLocalSearchAlgorithm ..................... 100 4.5NumericalExperiments ............................ 102 4.5.1ConvergenceTests ........................... 102 4.5.2ComparisonofSolutionMethods ................... 104 4.5.3ParameterAnalysis ........................... 105 4.5.4ConsumerBehaviorAnalysis ..................... 109 5CONCLUSIONS ................................... 114 APPENDIX APROOFTHATRETAILER'SPROBLEMISACONVEXPROGRAMFORFIXEDlAND ........................................ 117 BPROOFOFSUFFICIENTOPTIMALITYCONDITIONFORCONVEXMIXEDINTEGERPROGRAMWITHONEINTEGERVARIABLE ............ 118 CDECOMPOSITIONOFUNCAPACITATEDMINIMUMCOSTFLOWPROBLEM 120 DDETAILEDAL-KHAYYAL'SAPPROACH ...................... 123 ENUMERICALRESULTSFORCHAPTER4 .................... 125 REFERENCES ....................................... 127 BIOGRAPHICALSKETCH ................................ 135 6

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LISTOFTABLES Table page 2-1Parametersettingsforcomputationaltests. .................... 40 2-2ComputationalresultsforcategoryAwithstandardparametersettings. .... 42 2-3ComputationalresultsforcategoryBwithstandardparametersettings. .... 43 2-4ComputationalresultsforcategoryCwithstandardparametersettings. .... 44 2-5Resultsofcomputationaltests. ........................... 44 2-6Parametersettingsforfactorialtests. ........................ 45 2-7Hypothesistestsforbullwhipeffectwithkeyparametersatdifferentlevels. ... 45 3-1Parameterdistributionsusedincomputationaltests. ............... 70 3-2ComputationalresultsI. ............................... 72 3-3ComputationalresultsII. ............................... 73 4-1Parameterdistributionsusedincomputationaltests. ............... 103 4-2ComputationalresultsI-instance1. ........................ 104 4-3ComputationalresultsII. ............................... 106 4-4Parameterdistributionsusedinparameteranalysis. ............... 108 E-1ComputationalresultsI-instance2. ........................ 125 E-2ComputationalresultsI-instance3. ........................ 125 E-3ComputationalresultsI-instance4. ........................ 126 E-4ComputationalresultsI-instance4. ........................ 126 7

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LISTOFFIGURES Figure page 2-1Consumerdemandovertime. ............................ 26 2-2Retailer'sinventory. ................................. 27 2-3Supplier'sinventory. ................................. 34 2-4Forwardbuyinginsupplierdiscountwithatdifferentlevels. .......... 47 2-5Retailerprotgaininsupplierdiscountwithkeyparametersatdifferentlevels. 48 2-6Supplierprotgaininsupplierdiscountwithkeyparametersatdifferentlevels. 48 2-7Systemprotgaininsupplierdiscountwithkeyparametersatdifferentlevels. 49 2-8Bullwhipeffectinsupplierdiscountwithkeyparametersatdifferentlevels. ... 49 3-1Supplierprotgaininwithatdifferentlevels. ................. 76 3-2Retailerprotgaininwithatdifferentlevels. .................. 76 3-3Totalsystemprotgaininwithatdifferentlevels. ............... 77 3-4Bullwhipeffectinwithatdifferentlevels. .................... 77 4-1Illustrationofshelfspaceallocation. ........................ 85 4-2Differentcriteriawithhiatdifferentlevels. ..................... 109 4-3Differentcriteriawithcdiatdifferentlevels. ..................... 110 4-4Differentcriteriawithatdifferentlevels. ...................... 111 4-5Effectsofconsumer'sconsiderationsetonassortment. ............. 112 4-6Effectsofconsumer'spreferenceweightsonassortment. ............ 113 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyPLANNINGMODELSFORPRICEPROMOTIONSINMULTI-LEVELSUPPLYCHAINSByYiqiangSuDecember2013Chair:JosephGeunesMajor:IndustrialandSystemsEngineering Pricepromotionsareplayingasignicantroleintheretailindustry;however,inefciencyofpricepromotionshasbeendemonstratedbothintheoreticalworksandindustrialpractice.Asaresult,otherpricestrategies,likeeverydaylowprice(EDLP)strategies,arerecommendedtoreplacepricepromotions.Sincethereisplentyofevidencethatpricepromotionsarenotreplaceablefromtheconsumer'sperspective,pricepromotionsarecertaintocontinueexistinginthenearfuture.Thisdissertationisgoingtodemonstrateanumberofprotablepricepromotionscenarios,anddeveloptoolstohelpsupplychainparticipantstoincreasepricepromotionperformanceandresolvepotentialconictsthatmayoccurinpricepromotionactivities. Werstconsideradeterministic,two-stagesupplychainmodelcomposedofasinglesupplierandasingleretailer.Inthismodel,thesupplierperiodicallyofferstheretailerapricediscountatasingletimepointwithineachplanningcycle.Theretailerneedstodetermineitsorderandpricingpolicyinordertomaximizeitsprot.Inresponsetotheretailer'spolicy,thesuppliermustdetermineitstransportationandinventorypolicyinordertominimizeitstotaloperationscost.Wedemonstratethateventhoughtheuseoftradepromotionscanindeedincreasearetailer'sandsupplier'soperationscosts,thesecostsmaybemorethanoffsetbyincreasedrevenues,evenintheabsenceofexplicitcoordination.Weprovideabroadsetofcomputationalresultsthatvalidatethisconclusionanddiscusstheresultingmanagerialinsights. 9

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Thenweconsiderastochastic,two-stagesupplychainmodelinwhichasupplierservesasetofstoresinaretailchain.Inthismodel,thesupplieroffersperiodicoff-invoicepricediscountstotheretailchainoveraniteplanninghorizonunderuncertaintyindemand.Basedonthepricediscountsofferedbythesupplier,andafterstoredemanduncertaintyisresolved,theretailchaindeterminesindividualstoreorderquantitiesineachperiod.Becausethesupplieroffersstore-specicprices,theretailermayshipinventorybetweenstores,apracticeknownasdiverting.Wedemonstratethat,despitetheresultingbullwhipeffectandassociatedcosts,acarefullydesignedpricepromotionschemecanimprovethesupplier'sprotwhencomparedtothecaseofeverydaylowpricing(EDLP).Wemodelthisproblemasastochasticbileveloptimizationproblemwithabilinearobjectiveateachlevelandwithlinearconstraints.WeprovideanexactsolutionmethodbasedonaReformulation-LinearizationTechnique(RLT).Inaddition,wecompareoursolutionapproachwithawidelyusedheuristicandanotherexactsolutionmethoddevelopedbyAl-Khayyal[ 7 ]inordertobenchmarkitsquality. Finally,weconsideranoptimizationproblemthathelpsaclicks-and-mortarretailercoordinateproductassortments,retailpricesandinventoriesbetweenaphysicalchannelandanonlinechannel.Inthisproblem,weconsideraclicks-and-mortarretailerwhosellsproductstoend-consumersthroughbothphysicalandvirtualchannels.Weuseamultinomiallogit(MNL)modeltodescribetheconsumerpurchaseprocessandachanceconstrainttolimittheprobabilityofshortage.Wemodelthisproblemasachance-constrainedandtwo-stagestochasticproblem.WeadoptaneffectivecombinedsampleaverageapproximationalgorithmdevelopedbyWangandGuan[ 97 ]tosolvethisproblem.Inaddition,wealsoproposeagreedyandlocalsearchheuristicasanalternativesolutionapproachforlargesizeproblemswhencomputingpowerislimitedorsolutiontimeiscritical. 10

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CHAPTER1INTRODUCTION Nowadays,inanincreasinglycompetitiveretailingenvironment,pricepromotionsarebecomingimportantmarketingactivitieswhichaffecteverystageinasupplychain.AnA.C.Nielsonsurveyshowsthatabout41%ofconsumersactivelylookdealsatretailstores[ 72 ].AnotherA.C.Nielsonsurveyillustratesthatalargepercentage(12%-25%)ofEuropeanretailersalesaremadeviasalespromotions[ 51 ].Moreover,theuseoftradepromotionsbysuppliersofconsumerpackagedgoodstodistributorsrosefrom$8billionin1990toapproximately$80billionin2004[ 52 ]. Inthisdissertation,wewilltoconsidertwodistincttypesofpricepromotionsinsupplychains:retailsalespromotionsandtradepromotions.Retailsalespromotionstargetconsumers,whiletradepromotionsareofferedbyanupstreamsupplychainplayertoadownstreamplayer.Westressthedifferencebetweenthesetwokindsofsalespromotionsbecausetheyaffectthesupplychainplayersandconsumersindifferentways. Despitebeingheavilyusedintheretailindustry,pricepromotionsarenotalwayseffectiveforeithersuppliersorretailers.Withthelion'sshare(52%)[ 6 ]ofthemoneyspentonadvertisingandpromotionsgoingtotraderatherthansalespromotions,85%ofsuppliersbelievethattheirtradepromotiondollarsarenotbeingspenteffectively[ 1 ],andonly19%thinktheygetagoodvaluefortheirmoney[ 19 ].Oneofthemajorreasonsfortheineffectivenessofthetradepromotionistheadoptionofoff-invoicedeals.Inanoff-invoicetradepromotion,asuppliersimplydeductssomepercentagefromtheinvoicepriceforpurchasesoverasetofperiods.Thus,suppliershavenocontroloverwhetherretailerswillactuallypasstheirpromotionalpricesontoendconsumers.Infact,asurveyshowsthat,onaverage,retailersonlypass60%oftradefundstoconsumers[ 23 ].Besidesthislowpass-throughrate,therearetworetailerstrategiesthatreducetheeffectivenessoftradepromotions.First,retailersmayrespondto 11

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anoff-invoicetradepromotionbyengaginginforward-buying,i.e.,theretailertakesadvantageofthesupplier'stemporarydiscountbypurchasingaquantitythatfarexceedsitscurrentneeds,andsubsequentlyonlyappliesadiscounttopartofthisorderwhensellingtoconsumers.Whenthepromotionperiodexpires,theretailersellstheremaininginventorythatitpurchasedatadiscounttoitsconsumersattheregularprice.Second,retailersmaytakeadvantageofoff-invoicetradepromotionsbyengagingindiverting,i.e.,orderingfromonestoreandredirectingtheshipmenttoanotherstore.Bronnenberg,Dhar,andDube[ 29 ]showthatgeographicvariationinbrandsharesforleadingconsumerbrandsisbothsubstantialandpersistent.Consequently,themarginaleffectivenessofpromotionsandpricesvariesconsiderablyamongretailstoresfromaroundthecountry[ 28 ].Toaccommodategeographicdifferencesinsalesofsimilarproducts,suppliersmayoffercertainoff-invoicetradepromotionsonlyincertainregionsofthecountry.Largenationalretailchainsmaycircumventtheserestrictionsbypurchasingaproductatalowerwholesalepricewherethedealisoffered,andsubsequentlyshippingtheitemtostoresinotherregions(wherethedealisnotoffered)forsaleattheregularprice.Conventionalwisdomholdsthat5%to10%ofgroceryproductsonpromotionarediverted[ 22 ]. Theseforward-buyinganddivertingstrategiescanbothresultinthebullwhipeffect,whichisviewedinextremelynegativetermsbecauseofitsnegativeimpactsonsupplychainoperationscosts.Leeetal.[ 64 65 ]alsocharacterizedpricevariationsasoneofthemajorcausesofthebullwhipeffect. Foraretailer,althoughsalespromotionscanincreasesalesvolumesignicantlyintheshortterm,someresearchersshowedthatalargeportionofthisimmediateincreaseinsalescomesfrombrandswitching[ 26 ].Itispossiblethatwhenaretaileroffersadiscountforaparticularproduct,someconsumerschoosetobuythediscountedproductinsteadoftheirrst-choiceproductwithahigherprice.Thisso-calledcannibalizationeffectcanpotentiallymakethesalespromotionsunprotableevenintheshortterm. 12

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Moreover,someresearchers[ 26 ]arguethatsalespromotionshaveeithernolongtermeffectornegativelongtermeffect.Thisoccursbecausefrequentsalespromotionsincreasepricesensitivityandunderminebrandloyalty[ 34 ]. Becausepricepromotionsarenotextremelyefcient,Leeetal.[ 64 65 ]suggestedthatsuppliersanddistributorsadoptcorrespondingmanagementpractices(anEDLPstrategy,forexample)tostabilizeprices.Meanwhile,somesuppliers,likeZaraRandP&GR,havealreadybegunadoptingEDLPstrategiesandachievedbusinesssuccesses.However,webelieveinthenearfuturepricepromotionswillcontinueplayinganimportantroleintheretailindustry.First,tradepromotionsallowsuppliertopasstemporaryreducedrawmaterialcoststoretailersandnalconsumers,whichmayincreaseasupplier'ssalesandenhanceasupplier'scapacityutilization.Second,forsomeconsumers,pricepromotionscannotbereplacedbyEDLPstrategies.EDLPstrategiescanminimizebothpurchasingcostfortheconsumersandoperationscostsfortheretailersandsuppliers;however,salespromotionscanprovideconsumerswithmorebenetsbeyondmerelysavingmoney[ 34 ].Forexample,salespromotionscanenhanceaconsumer'sself-perceptionasasavvyshopper.Aslongasconsumerscontinuerespondingtopricepromotions,retailersandsuppliersshouldkeepspendingonpricepromotions.A2012AMGStrategicAdvisorsstudybacksupthisidea,anditshowsthatoutofmorethan768suppliers,56%ofthemplacehighorhigherpriorityonincreasingspendingontradepromotionfundsinthenextthreeyears.A2012IEGreportalsoshowsthatinthelastthreeyearssalespromotionspendinggrew2%peryearonaverage,anditpredictsthatthisspendingwillincreaseby3%in2013.Sincepricepromotionsarecurrentlynotreplaceable,modelsthatimprovetheeffectivenessandefciencyofpricepromotionsshouldbedeveloped. 13

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CHAPTER2PRICEPROMOTIONS,OPERATIONSCOST,ANDPROFITINATWO-STAGESUPPLYCHAIN 2.1ProblemDescriptionandLiteratureSurvey Thephenomenoninwhichdemandvariabilityincreasesasonemovesupstreaminthesupplychainhasbeenoftenobservedinpractice.Thissocalledbullwhipeffecthasdrawnasurgeofresearchinterestforhalfacentury.Forrester[ 47 ]rstidentiedthephenomenonofincreasingdemandvariabilityasonemovesupasupplychainviaaseriesofcasestudies.Hesuggestedthatthiseffectwascausedbytimevaryingbehaviorswithinindustrialorganizations.Sterman[ 90 ]usedthewell-knownBeerDistributionGametoshowthattheamplitudeandvarianceofordersincreasesteadilyfromconsumertoretailertofactory,attributingthisphenomenontoplayers'misperceptionsoffeedback. Previousresearchhasmainlyfocusedonidentifyingcausesofthebullwhipeffect,quantifyingthecostimpactsofthebullwhipeffect,anddevelopingstrategiestoreducethebullwhipeffect.ArguablythemostinuentialworkinrecentyearswasprovidedbyLee,Padmanabhan,andWhang[ 64 65 ].Theyshowedthatevenundertheassumptionthatthesupplychainmembersarerational,optimizingagents,theso-calledbullwhipeffectmaystillexist,andtheyidentifyfourmaincausesofthiseffect:theuseofcertaindemandforecastingmethods,supplyshortagegames,orderbatching,andpriceuctuations.Inaddition,theyalsoproposedmechanismstocounterthenegativeimpactsofthebullwhipeffectonoperationscosts,oneofwhichwastheuseofanEDLPstrategytocounterthecomponentofthebullwhipeffectthatresultsfrompriceuctuations.Chen,DreznerandSimchi-Levi[ 35 36 ]quantiedtheimpactofdemandforecastingonthebullwhipeffectforatwo-echelonsupplychainconsistingofasinglesupplierandasingleretailer.Intheirstudy,thedemanddistributionparametersarenotknownwithcertainty,andtheretailerthereforeusesmovingaverageorexponentialsmoothingforecaststoestimatethedemandmeanandvariance.Theseestimatesare 14

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thenusedtoimplementanorder-up-toinventoryreplenishmentpolicy.Inmeasuringthebullwhipeffect,FransooandWouters[ 48 ]discussedpotentialmeasurementproblemsasaresultofdataaggregation,incompletenessofdata,andtheisolationofdemanddata.Theyusedtheratioofthecoefcientofvariationofordersplacedtothecoefcientofvariationofordersreceivedasameasureofthebullwhipeffect.Cachon[ 31 ]alsousedthisapproachtomeasurethebullwhipeffect.HeinvestigatedasupplychainwithonesupplierandNretailersfacingstochasticdemand,whoimplementscheduledorderingpolicies,andfoundthatthesupplier'sdemandvariabilitydeclinesastheretailer'sorderintervalsincreaseorthebatchsizeincreases.PotterandDisney[ 81 ]focusedontheimpactofbatchsizeonthebullwhipeffect,andtheyusedtheratioofvarianceofordersplacedtothevarianceofordersreceivedtomeasurethebullwhipeffect. However,littlemodel-basedresearchhasaddressedtheextenttowhichthebullwhipeffectmightbeanecessaryevilinaprot-maximizingsupplychain.Thatis,ifanactivitysimultaneouslyincreasesrevenueandcost(asaresultofthebullwhipeffect),howdoesadecisionmakerchoosethelevelofthisactivitythatoptimizessystemperformance?Inourstudy,thisactivitycorrespondstoapricepromotion.AswementionedinChapter 1 ,pricepromotionsstillprevailinindustrialpracticesnowadays,andwebelievetherearemanyreasonsthatpricepromotionswillnotbecompletelyreplacedbyEDLPstrategiesinthenearfuture.Researchinmarketingsciencealsoprovidesevidencethatwell-designedpricevariationscanbenettheentiresupplychainsystem. Blattberg,Briesch,andFox[ 24 ]summarizedthisviewbycontendingthat:1)temporaryretailerpricereductionssubstantiallyincreasesales;2)thefrequencyofdealschangesaconsumer'sreferenceprice;and3)thegreaterthefrequencyofdeals,thelowertheheightofthedealspike.Blattberg,Eppen,andLieberman[ 25 ]showedthatfoodretailerspredominantlypreferofferingtemporarypricediscountstoadopting 15

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anEDLPstrategy,attributingthistoadesiretoincreasemarketshare,andtoattractnewcomerstotheirproductsbyofferingalowerrisklevel.Hoch,Dreze,andPurk[ 57 ]thoroughlyinvestigatedtheuseofanEDLPstrategy,andfoundthatthisstrategytendstoimproveasupplier'sprotverylittle,whilepotentiallyleadingtobiglossesforaretailer.Ailawadi,FarrisandShames[ 6 ]usedanumericalexampletodemonstratethatawell-designedtradepromotioncanincreasethetotalsupplychainsystemprotsandthattheupstreamplayergainsalargershareofthetotalprotthanthecaseinwhichthesupplierxesasinglepriceanddoesnotusepromotions.Intheirsingle-supplier,single-retailerstudy,thesupplier'sunitcostsarexed,whiletheretailer'sunitcostequalsaxedvalueplusthesupplier'swholesaleprice.Moreover,theirmodeldoesnotaccountforxedordercostsorinventoryholdingcosts,andthereforethebullwhipeffectisnotafactor.Chandon,Wansink,andLaurent[ 34 ]builtaframeworkaddressingthemultipleconsumerbenetsofsalespromotions,andtheyclassiedthesebenetsintosixcategories:monetarysavings,qualityincreases,convenience,valueexpression,exploration,andentertainment.Theyalsorecommendedagainstaretailer'sblinduseofEDLP,arguingthatconsumersrespondtosalespromotionsforreasonsthatextendbeyondmonetarybenets. Thepresenceofapparentinconsistenciesbetweenthemarketingandoperationsviewsofpromotionsleadsustowonderhowpriceuctuationsaffecttotalsystemprotinasupplychain.Sincetheconclusionsfromthemarketingscienceliteraturetypicallydonotexplicitlyconsidersupplychainoperationscosts,amathematicalmodelthataccountsfortheseoperationscostsinadditiontothedemand-sideeffectsofpricepromotionsmayservetopartiallyreconciletheseconictingviews. Somepastresearchhaspartiallyaddressedthisquestion,takingaretailer'sview.Onegroupofstudiesisundertheassumptionofdeterministicdemand.Ardalan[ 14 15 ]developedamodelthatdeterminesboththeretailer'soptimalpriceandorderingpoliciesinresponsetoaone-timeonlypricediscountusingageneralprice-demandrelationship. 16

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ArcelusandSrinivasan[ 12 ]generalizedthisworkbyaccountingforpotentialforwardbuyingattheretailer.Forthemoregeneralcaseinwhichpricediscountscanbeofferedonmorethanoneoccasion,SogomonianandTang[ 88 ]developedamodelthatdeterminesoptimalpromotionandproductiondecisionsforasingleprot-maximizingrmusingamixed-integerprogram.Basedontheassumptionsthatpromotionlevelsbelongtoanitesetandtheconsumerresponsetopromotionsdependsonthetimeelapsedsincethelastpromotion,theyproposedanapproachtosolvetheproblemgloballybyreformulatingtheirproblemasalongestpathproblemonanetwork. Anothergroupofstudiesrelatedtosalespromotionsisundertheassumptionofstochasticdemand.PetruzziandDada[ 80 ]consideredaprice-settingrmthatstocksasingleproductsubjecttorandom,price-dependentdemand,withtheobjectiveofdeterminingstocklevelsovermultipleperiods.Theydevelopedaconditionunderwhichthesolutiontothisproblemisstationaryandmyopic.Onelimitationoftheirresearchistheimpositionofaconstantpriceassumption.Forajointpricingandinventorymanagementproblemwithstochasticdemand(inwhichpricesmaychangedynamicallyovertime),Thowsen[ 92 ]showedthattheoptimalpricing/inventorypolicyisabase-stocklist-pricepolicyundertheassumptionsthat:i)theexpecteddemandcurveandstockoutcostsarelinearfunctions;ii)holdingcostsareconvex;andiii)thedensityfunctionoftherandomcomponentofconsumerdemandisPF2(aPolyafrequencyfunctionoforder2).ChengandSethi[ 38 ]usedaMarkovdecisionprocess(MDP)modelforajointinventoryandpromotiondecisionprobleminwhichthepromotionhasonlytwostates(onandoff).Undercertainconditions,theyshowedthattheoptimalpromotionandorderingpolicyisan(S0,S1,P)policy.Thatis,iftheinitialinventorylevelisatleastP,theproductispromoted;iftheinitialinventoryislessthanS1andtheproductispromoted,anorderisplacedtoincreasetheinventorypositiontoS1;iftheinitialinventoryislessthanS0andiftheproductisnotpromoted,anorderisplacedtoincreasetheinventorypositiontoS0. 17

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Beyondtheretailer'sinventoryandpricingpolicies,wearealsointerestedinhowthesystemwide(supplierandretailer)policiesinteracttodriveperformance.Tothisend,Goyal[ 53 ]proposedanintegratedinventorymodelforatwo-echelonsystem;hisworkcomparedindependentandjointapproachesforasysteminventorycontrolproblem.ErnstandPyke[ 46 ]examinedinventorypoliciesatbothawarehouseandtheretaileritserves,andtheyalsoconsiderthewarehouse'soptimalshippingpolicy.Theycharacterizedtheimpactofconsumerdemandvariabilityonoptimalwarehousetruckingcapacity.Ineachofthesesettings,pricewastreatedasaxed(andstationary)parameterovertheplanninghorizon. Inthischapter,weprovideamathematicalmodelofatwo-echelonsupplychaininwhichthesupplier'sandretailer'spricesmaybeperiodicallyreducedinconjunctionwithapromotion.Sincewewishtofocusonpricingeffects,weassumethatconsumerdemandattheretailechelonisdeterministicandprice-dependent.Thisassumptionpermitsustogaininsightintosuchproblemsbyisolatingthewayinwhichpricingpoliciesinuenceoperationscostsintheabsenceofrandomdemand.Ourgoalistocapturethewayinwhichbotharetaileranditsconsumersreacttopromotions,aswellashowthis,inturn,translatestothebullwhipeffectandimpactsoperationscostsandprotlevels. Gavirneni[ 50 ]consideredamodelthatissimilarinspirittoours,althoughitusesadifferentsetofassumptionsandfactorsthatdriveperformance.Inparticular,hestudiedacapacitatedtwo-echelonsupplychainwithstochasticdemand.Heshowedthatifinformationsharingoccursinthesupplychain,priceuctuationscanimprovetotalsystemperformance.Hisstudyconsideredatradepromotiononlyanddidnotallowfortheretailertopassanyofthediscounttoconsumers.Incontrast,inourmodel,theimpactofretailerpricediscountsondemandservesasthemaindriveroftheincreasedsystemprotresultingfrompromotions. 18

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Theremainderofthischapterisorganizedasfollows.InSection 2.2 wedescribeoursupplychainmodel.Subsection 2.2.1 statesourassumptionsaboutconsumerdemand,followedbySubsections 2.2.2 and 2.2.3 ,whichderivemathematicalmodelstooptimizeretailerandsupplierprots,respectively.InSubsection 2.2.4 ,weelaborateonhowwecomputethevarianceofretailerordersandthevarianceofconsumerdemand,whichtogetherpermitcharacterizingthebullwhipeffect.Section 2.3 illustrateshowpromotionsaffectprotsinourmodelviaasetofnumericaltests;wecomparetheretailer'sprot,thesupplier'sprot,andthetotalsystemprotinboththepromotionscenarioandtheconstant-pricescenario. 2.2ProblemStatementandFormulation Webeginbyconsideringasimpledistributionsystemstructure,namely,asinglesupplierwhosellsaproducttoaretailer,whointurnsellsittoconsumers.Weassumethatconsumerdemandoccursataconstant,deterministicratethatisknownbytheretailerandisalsoprice-dependent. Thereisonemajorreasonwhysupplierpriceuctuationscanleadtothebullwhipeffect:theretailermayrespondtoatemporarypricediscountbyengaginginforwardbuying,i.e.,theretailertakesadvantageofthesupplier'stemporarydiscountbypurchasingaquantitythatfarexceedsitscurrentneeds,andsubsequentlyonlyappliesadiscounttopartofthisorderwhensellingtoconsumers.Whenthepromotionperiodexpires,theretailersellstheremaininginventorythatitpurchasedatadiscountfromthesuppliertoitsconsumersattheregularprice.Thismakestheretailer'sorderingpatterndifferfromtheconsumerdemandpattern,andconsequently,thevariationinaretailer'sorderingquantitymaybemuchlargerthanthevariationinend-consumerdemand. Clearly,ifthesupplierandtheretailerworkcooperatively,thiscanenablethesuppliertoaccesstheretailer'spoint-of-saledata,whichmakesitpossibleforthesuppliertoofferdiscountsonlyontheretailer'ssell-throughunits(i.e.,unitssoldbytheretaileratareducedpricetoconsumers).Asaresult,theretailercannolongeradopt 19

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aforwardbuyingpolicy,andthedriverdisappears.However,counteringthebullwhipeffectisnottheemphasisofourwork,asweconsidersystemsinwhichsuchexplicitcoordinationisabsent,andinvestigatetherelationshipbetweenpricediscountsandthebullwhipeffectinsuchcases.Thatis,inthischapterweseektodevelopandexploreamodelthatvalidatesthecoexistenceofthebullwhipeffectandincreasedprotswhencomparedtoanEDLPstrategy. Weconsiderasituationinwhichthesupplierperiodicallyofferstheretailerapricediscountatasingletimepointwithineachplanningcycle(wheretheplanningcycledurationisdenedbythetimebetweenpromotionsorpricediscounts).Theretailerisnotrequiredtopassthispricereductionontoitsconsumers,sincetherevenuefromtheextrademandgeneratedbythelowerpricemaynotoffsetthecorrespondingcost.Theretailerthusneedstoconsiderthreefactors:(i)thequantitytobeorderedatadiscountfromthesupplier;(ii)thediscountedpricetobeofferedtoitsconsumers;and(iii)theforwardbuyquantity,i.e.,thequantityitwillpurchaseatadiscountinexcessofitsnormalorderquantitythatitwillnotsell-throughatadiscount.Aftertheretailersetsitsorderingpolicy,theretailer'sorderandpricinginformationispassedtothesupplierwithoutdelay.Inresponsetotheretailer'sorderingandpricingpolicy,thesuppliermustdetermineitstransportationandinventorypolicyinordertominimizeitstotaloperationscost. Wecomparetwoscenarios,denotedbyS0andS1.Therstscenario,S0,isthenormalscenario,underwhichthesupplier'spriceisconstant,andtheretailersellstheproductatapriceP0perunit,withacorrespondingconsumerdemandrateofD0.Underthisscenario,theretailerrepeatedlyordersinlotsofsizeq0whenitsinventoryisdepleted,asinthenormaleconomicorderquantity(EOQ)model.Thisso-callednormalscenariowillcorrespondtotheEDLPcase.Thesecondscenario,S1,representsthesituationinwhichthesupplieroffersadiscounttotheretaileratthebeginningofeachplanningcycle.Basedonpreviousresearch[ 76 91 ],iftheretailerisarationaland 20

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optimizingagent,theretailerwillacceptthisofferandplaceaspecialorderofsizeQ1.Theretailerwillthenoffera(possibly)discountedpriceofP1P0toitsconsumers.Itisacommonpracticeinretailingindustrytohighlightthepromotionaloffersincertainwaystodrawconsumerattention,and,inthischapter,wealsoassumewhenevertheretaileroffersapricediscount,theretailerdisplaysthispromotionaloffereffectively,sotheconsumercandistinguishthedifferencebetweenpromotionalpriceandregularpricewithoutdifculty.Thecostassociatedwiththehighlightordisplayactivityisusuallynegligiblecomparedtoothercosts,forexample,acommonpracticetofeatureapricediscountistochangethecolorofpricetag.Aretailpricediscountwithin-storeretaildisplayinducesaconsumerdemandrateofD1D0.WeassumetheretailersubsequentlyreturnstoitsnormalpriceP0andorderquantityq0afterdepletingallofthespecialorderofsizeQ1.Inbothscenarios,weassumethatnoshortagesarepermittedattheretailer. Weareinterestedincapturingthetwoprimarymetricsoftotalsystemprotandthebullwhipeffect.Thetotalsystemprotisequaltothesumoftheretailer'sandthesupplier'sprot,wheretheretailer'sprotisdenedasthedifferencebetweenitsrevenueandallcosts,includingvariablepurchasingcost,inventoryholdingcost,andxedordercost;similarly,thesupplier'sprotequalsitsrevenuelessitsproduction,inventoryholding,setup,andtransportationcosts.WediscussourcomputationofthebullwhipeffectlaterinSection 2.2.4 2.2.1ImpactofPromotionsonDemand Thenotationweuseissummarizedbelow: d:supplier'sunitdiscount(indollars)totheretailer; :retailer'sunitdiscount(indollars)toitsownconsumers; i:retailer'sinventoryholdingcostrateperunitperyear; c:retailer'snon-discountedpurchaseprice; 21

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A:retailer'sxedordercost; S0:scenarioinwhichd=0; S1:scenarioinwhichd>0; Ps:retailer'ssellingpriceforalternatives=0,1; Ds:retailer'sannualdemandatsellingpricePs,s=0,1; PR0:retailer'smaximumaverageannualprotwhennodiscountisoffered; PR1:retailer'saverageannualprotwhenadiscountisofferedandtheEOQ(q0)isusedforordersplacedwhenthediscountisnotineffect; q0:retailer'sorderquantitywhennodiscountisoffered; Q1:retailer'sorderquantitywhenadiscountisoffered; q1:portionofQ1soldatthenon-discountedpriceofP0(equivalently,theforwardbuyquantity); t0:lengthofregulartimeintervalbetweensuccessiveretailersetups; t1:lengthoftimeintervalcoveredbyretailer'sspecialorderofsizeQ1; L:lengthoftheplanningcycle,i.e.,timebetweensupplierpricepromotions. Asnotedpreviously,theretailer'sdemandratedependsonthepriceitoffersconsumers.Wemakethreeassumptionswithrespecttotheretailer'sdemandfunction.First,thepriceelasticityofconsumerdemandunderthenormalpriceisdifferentfromthatunderapromotional(reduced)price.BellandLattin[ 20 ]andBlattberg,BrieschandFox[ 24 ]provideempiricalevidenceinsupportofthisassumption.Theintuitiveideabehindthisassumptionisthatbecauseapromotionaldiscountisonlyavailableforashorttime,consumersknowthattheyhavetorespondquickly.Therefore,ahigherdegreeofpromotionalpriceelasticityisexpectedascomparedtothenormalpriceelasticity.Incontrast,underthenormalprice,consumersmayanticipatethatapricedecreasewilllastforaconsiderabletime,potentiallyleadingtoweakerpriceelasticity. 22

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UnderthenormalscenarioS0,thepriceisstableovertime,andthedemand-pricerelationshipismodeledusingthefollowinglinearfunction: D0=rc)]TJ /F3 11.955 Tf 11.96 0 Td[(r0P0, (2) withrc0,r0>0, wherercisaconstantandr0istheregularpriceelasticityofconsumerdemand.UnderthescenarioS0,clearlyitisoptimalfortheretailertoorderitscorrespondingEOQ.UsingEquation( 2 ),theretailer'sproblemformaximizingprotunderthenormalscenariocanbewrittenas: PR0(P0,q0)=max(P0)]TJ /F3 11.955 Tf 11.96 0 Td[(c)D0)]TJ /F3 11.955 Tf 13.15 8.09 Td[(icq0 2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(AD0 q0(2) subjectto:D0=rc)]TJ /F3 11.955 Tf 11.95 0 Td[(r0P0withrc0,r0>0, whereP0andq0aretheoptimalpriceandtheoptimalorderquantity,respectively.ArcelusandSrinivasan[ 12 ]showedthattheoptimalsolutionsto( 2 )canbewrittenas: q0=q 2AD0 ic (2) P0=1 2hc+A q0+rc r0i, whereD0isthedemandratecorrespondingtotheoptimalretailerprice.Thus,underthenormalscenarioS0withoutanysupplierpricediscounts,theretailerreplenishesitsinventoryeveryt0=q0=D0timeunits. Second,weassumethattheconsumerpopulationconsistsoftwogroups:brandloyalsandimpulsiveconsumers.Thebrandloyalsalwaysbuytheproduct,whereasimpulsiveconsumersdonot.Moreover,impulsiveconsumerstendtomakebuyingdecisionswithoutthoughtfulanddeliberateconsiderationwhichmakestheirbuying 23

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behaviormoreeasilyalteredbythefeaturedpoint-of-purchasepromotions.Wetakeimpulsiveconsumersintoconsideration,becauseimpulsivebuyingisaprevalentphenomenonintheretailingindustry.Reportsshowbetween27%and62%ofalldepartmentstorepurchasesareidentiedasunplannedpurchases[ 17 ],and88%impulsepurchasesarebecauseshoppersndproductsareofferedatalowerprice(oronsale)[ 45 ].Whenatemporarypricediscountisoffered,theimpactsofapricediscountonbrandloyalscanbedecomposedintotwoaspects:purchaseaccelerationandforwardbuying.Inotherwords,thedemandrateofbrandloyalsisgoingtoincreaseattherater0unitsperdollarofpricereductionasinnormalscenarioS0;moreover,theseconsumersarewillingtocarrysomeadditionalinventoryinreturnforareductioninprice.Inadditiontothepurchaseaccelerationandforwardbuyingofbrandloyals,impulsiveconsumersalsotemporarilybuytheproductatthediscountedpriceforseveralreasons:someimpulsiveconsumersmayenjoytheprocessoflookingfordeals,somemaywanttotrytheproductatalowerrisklevel,andsomemaysimplybedrivenbyasuddenandspontaneousdesireorurgetobuythingsonsale. Third,weassumethattherelativefrequencyofpricepromotionsimpactstheperformanceofthepromotions.Forexample,supposetheretailerprovidesapricepromotioneveryLtimeunits,andthepromotionlastslunitsoftime.IflL,thenthisisnolongerahigh-lowpricingstrategy,anditdegeneratesintoanEDLPstrategy,sothatthepriceelasticityisequivalenttothatunderthenormalprice.Ontheotherhand,ifl
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where0
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Figure2-1. Consumerdemandovertime. providedinSection 2.2.1 ,andtheoptimalvalueofPR0canbedeterminedbyndingthecorrespondingstationarypointoftheobjectivefunction.Intherestofthissection,wefocusonconstructingamathematicalmodelfortheretailer'saverageprotperunittimeunderthepromotionscenarioS1,denotedasPR1. Asnotedpreviously,weassumethatthesupplierperiodicallyoffersadiscounttotheretailer.Formodeltractability,weassumethatthetimebetweenpromotionsislongwhencomparedtotheretailer'snormalreplenishmentcyclelengtht0andthatthelengthoftheplanningcyclemustbeanintegermultipleoft0,i.e.,L=nt0forsomen2N,wherethevalueofnispredetermined.Thatis,weassumeanexogenouslydeterminedtimebetweenpromotions(e.g.,yearlyorquarterly). UnderthescenarioS1,atthebeginningoftheplanningcycle,thesupplieroffersapricecut,d,totheretailer,andtheretailerthenrequestsalargeorderofsizeQ1,whichwillcoverdemandoveratimeinterval,t1,whichisatleastaslongastheregularreplenishmenttimeinterval,t0.Furthermore,thelengthofthisspecialtimeintervalisrestrictedtoanintegermultipleofoft0,i.e.,t1=mt0;m2N,wheremisadecision 26

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variableinourmodel.ThisassumptionismadestrictlyinordertofacilitatecomputationofthebullwhipeffectaswelaterdiscussinSection 2.2.3 .Clearlymshouldbesmallerthann(otherwisethesystemwouldnoteverreturntothenormaloperatingscenario).WeassumethattheretailerwillsellQ1)]TJ /F3 11.955 Tf 12.83 0 Td[(q1oftheunitspurchasedatadiscountedpricetoitsconsumers.Thatis,afterpurchasingQ1unitsatadiscountfromitssupplier,theretailerthendiscountsitspricetoitsconsumersforQ1)]TJ /F3 11.955 Tf 12.41 0 Td[(q1oftheseunits.Clearlythecaseofq1=Q1impliesthattheretailerdoesnotpassanyofthediscountontoitsconsumers.Notethatq1thusdeterminestheamountoftheretailer'sforwardbuy. Theretailer'sprotperpromotioncycleunderthescenarioS1iscomposedofthreeparts(asshowninFigure 2-2 ).Therstcomponentconsidersthetimeduringwhichtheretaileroffersadiscounttoitsconsumers,whichhasadurationofl=(Q1)]TJ /F3 11.955 Tf 12.02 0 Td[(q1)=D1timeunits.Includedhereare: Figure2-2. Retailer'sinventory. 27

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therevenuefromsellingtheitematthediscountedpriceP1; thecostofpurchasing(Q1)]TJ /F3 11.955 Tf 11.72 0 Td[(q1)unitsoftheitematthediscountedcost(c)]TJ /F3 11.955 Tf 11.72 0 Td[(d)perunit; thesetupcostforplacingtheorder,A;and thecostofholdinganaverageof(Q1+q1)=2unitsoveraperiodoflength(Q1)]TJ /F3 11.955 Tf -414.97 -14.45 Td[(q1)=D1atarateofi(c)]TJ /F3 11.955 Tf 11.95 0 Td[(d)perunit. Thesecondpart,oflength~rf(Q1)]TJ /F3 11.955 Tf 12.62 0 Td[(q1)=(D0D1)+q1=D0,accountsfortheprotassociatedwithanynon-discountedpartoftheretailer'sspecialorder.Thatis,duringthistime,theretailerhaseliminateditspricediscount,buthasremaininginventoryfromitsspecialpurchaseofsizeQ1.Includedhereare: therevenuefromsellingq1unitsatanon-discountedpriceP0; thecostofpurchasingq1unitsatthediscountedcostof(c)]TJ /F3 11.955 Tf 11.96 0 Td[(d)perunit; theholdingcostforq1unitsforaperiodoflength~rf(Q1)]TJ /F3 11.955 Tf 12.22 0 Td[(q1)=(D0D1)atarateofi(c)]TJ /F3 11.955 Tf 11.96 0 Td[(d)perunit;and theholdingcostforanaverageofq1=2unitsoveraperiodoflengthq1=D0atarateofi(c)]TJ /F3 11.955 Tf 11.95 0 Td[(d)perunit. Thethirdandnalpart,oflength(nt0)]TJ /F6 11.955 Tf 12.23 0 Td[((Q1)]TJ /F3 11.955 Tf 12.24 0 Td[(q1)=D1)]TJ /F4 11.955 Tf 12.24 0 Td[(~rf(Q1)]TJ /F3 11.955 Tf 12.23 0 Td[(q1)=(D0D1))]TJ /F3 11.955 Tf 12.23 0 Td[(q1=D0)coverstheprotresultingfromsellingatthenormalpriceP0whileorderinginlotsofsizeq0fortheremainderoftheplanninghorizon.WeemploythetimeproportionalcostassumptionofNaddor[ 76 ],i.e.,weassumethattheprotassociatedwiththisthirdcomponentisproportionaltothetimeremaininginthepromotioncycleandaccumulatesatanannualrategivenbyPR0(Equation( 2 )). Combiningalloftheaboveyieldsthegeneralizedmodelbelow:PR1(Q1,q1,,m)=max1 Lf(P0)]TJ /F4 11.955 Tf 11.96 0 Td[()(Q1)]TJ /F3 11.955 Tf 11.95 0 Td[(q1)+P0q1)]TJ /F6 11.955 Tf 11.95 0 Td[((c)]TJ /F3 11.955 Tf 11.96 0 Td[(d)Q1)]TJ /F3 11.955 Tf 11.95 0 Td[(A)]TJ /F12 11.955 Tf 11.29 16.86 Td[(Q21)]TJ /F3 11.955 Tf 11.95 0 Td[(q21 D1+q1(Q1)]TJ /F3 11.955 Tf 11.96 0 Td[(q1)~rf D0D1+q21 D0i(c)]TJ /F3 11.955 Tf 11.96 0 Td[(d) 2+1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(m nPR0 (2) 28

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subjectto: Q1)]TJ /F3 11.955 Tf 11.96 0 Td[(q1 D1+~rf(Q1)]TJ /F3 11.955 Tf 11.96 0 Td[(q1) D0D1+q1 D0=mq0 D0(2) D1=D0+(r0+~rf+~ri)(2) l=Q1)]TJ /F3 11.955 Tf 11.96 0 Td[(q1 D1(2) L=nq0 D0(2) 0q1Q1,0,mn,m2N.(2) Theaboveoptimizationproblemisamixedintegernonlinearprogrammingproblemwithoutaclosed-formsolution.Evenifwerelaxtheintegralityconstraintonm,becausetheobjectivefunctionisneitherconvexorconcave,wecanonlyusethenecessaryKKTconditionstoyieldcandidateKKTpointsforanoptimalsolution.Thus,theretailer'sproblemaswehaveformulateditisanonconvexmixedintegeroptimizationproblem,andwecannot,therefore,guaranteendinganoptimalsolution. Inpractice,however,itistypicallynotnecessaryfortheretailertosolvethisnonconvexmixedintegeroptimizationproblemexactlyfortworeasons.First,aretailerrarelyoffers,say,an18.79%discount,forexample,becauseconsumersaregenerallyinsensitivetoverysmallchangesinprice;itisnotuncommonforaretailertoroundtheoptimaldiscounttothenearestmultipleof5%or1%.Second,thelengthofthepromotionperiodisoftenstatedasamultipleofsomebaseplanningperiodlength,e.g.,onedayoroneweek.Consequently,theretaileronlyneedstoconsideranitenumberofdiscretecombinationsofland,thepromotiondurationandthediscountlevel,foranygivenvalueofthemanufacturer'sdiscountd.Thus,itisnotimpracticalfortheretailertoenumeratecandidatevaluesoflandinordertosolvetheresultingoptimizationproblem. Afterxingland,wecanshowthatunderverymildconditionson~rfand~ri,thecontinuousrelaxationoftheretailer'sproblemisaconcaveprogram(Appendix A forproofoftheconcavityoftheobjective,assuming~ri+r0~rfand~rf4D0). 29

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Moreover,thisoptimizationproblemcontainsonlyasingleintegervariable,m.Thus,forgivenvaluesofland,wecaneasilysolvetheretailer'smixedintegeroptimizationproblembyrstsolvingtherelaxation,thenroundingtheresultingvalueofmupanddown,andsolvingtheconvexprogramthatresultsateachofthesexedvaluesofm(thesubproblemwiththehigherobjectivefunctionvalueprovidestheretailer'soptimalsolution).Thatis,werstsolvetherelaxedconvexprogramand,ifmisinteger,westopwithanoptimalsolution.Otherwise,wesolvetwoadditionalconvexprogramswithm=bmcandm=dme,respectively,andselecttheonewiththehigherobjectivefunctionvalue.TheAppendix B demonstratesthatthisapproachprovidesanoptimalsolutionforxedlandvalues. 2.2.3SupplierProt Thissectiondiscussesthecostsincurredandrevenuesreceivedbythesupplierinthetwo-stagesystemwehavedescribed.Weassumethesupplierincurstransportationcostsinensuringdeliveryofitemstotheretailer,andthatthesupplieralsostocksinventory(andincursassociatedholdingcosts)inordertomeetretailerdemand.Thefollowingsubsectiondiscussesthestructureofsuppliertransportation-relatedcosts.Followingthis,weconsiderthesupplier'sinventory-relatedcostsandtheproblemofsettingawholesalediscountlevelassociatedwithapromotion.Thesefactorscombinetodeterminethesupplier'soverallprotfunction. 2.2.3.1Suppliertransportationcost Thebullwhipeffecthampersasupplier'sabilitytoefcientlyutilizecapacity.Ourmodelthereforeconsidershowthisaffectsbothasupplier'sinventorycostsanditstransportationcapacity.Althoughweexplicitlydiscusstransportationcapacity,thiscapacitymightcorrespondtoanytypeofgenericcapacity,includingproductioncapacity.Tocomputethetransportationcost(whichweassumeisbornebythesupplier),weusethefollowingnotation: W=regularin-housetruckcapacityinunits; 30

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SFR(W)=xedcostpershipmentasafunctionofregularcapacity; SF0=xedcostperoutsideoremergencyshipment; SPR=regularunitshippingcost; SP0=outsidecarrierunitshippingcost; TQ(W)=pershipmentcostifthesizeofdeliveryisW; T(W)=totaltransportationcostperpromotioncycle. ErnstandPyke[ 46 ]exploredoptimalinventoryandshippingpoliciesforatwo-leveldistributionsystemcomposedofawarehouseandaretailerwithrandomconsumerdemand.Weuseasimilartransportationcoststructureinourmodel,whichconsidersasupplier'sin-housetruckingcostaswellascostsassociatedwithanoutsidesource(commoncarrier).In-housetruckingcorrespondstoaeetoperatedbythesupplier.Weassumethatthesupplieralsocontractswithanoutsidetruckingrmsothatifthein-housetruckingcapacityisnotsufcienttodelivertheretailer'sorder,thesuppliermustusethisoutsidetruckingcapacity,atahighercost,toshipthedifference(alternatively,thesuppliermayarrangeforanunplannedshipment,whichtypicallycomesatanadditionalcost).Thisimpliesthatweassumethatthesuppliermustmeetalloftheretailer'sorderswithoutbackorderinganydeliveries. Shippingcostscontainseveralcomponents.First,weassumeaxedcostpershipmentusingin-housecapacity,andthatthisxedcostisalinearfunctionofin-housetruckcapacity,whichimpliesthatin-housetruckcapacityisoneofsupplier'sdecisionvariables.Second,weconsideracostperunitshippedusingin-housetruckcapacity,andassumethatthiscostisindependentofthexedcost.Moreover,weassumethattheoutsidetruckingrmchargesbothaxedcostpershipmentandacostperunitshipped.Thus,thesupplier'sper-shipmenttransportationcostis TQ(W)=SFR(W)+SPRmin(Q,W)+SFOI(Q)]TJ /F3 11.955 Tf 11.96 0 Td[(W>0) (2) +SPOmin(Q)]TJ /F3 11.955 Tf 11.96 0 Td[(W,0), 31

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whereQistheretailer'sorderquantity,andI(Q)]TJ /F3 11.955 Tf 10.9 0 Td[(W>0)istheidentityfunction,denedasI(Q)]TJ /F3 11.955 Tf 11.95 0 Td[(W>0)=8>><>>:0,Q)]TJ /F3 11.955 Tf 11.95 0 Td[(W0,1,Q)]TJ /F3 11.955 Tf 11.95 0 Td[(W>0. Asnotedbefore,weapproximatethexedcostofin-houseshippingusingalinearfunction SFR(W)=aW,(2) whereaisascalarthatdenotesthexedcostperunitoftruckcapacity,chargedonaper-shipmentbasis. Sincethesystemdemandisdeterministic,aftertheretailer'sinventorypolicyhasbeenset,thesuppliercanevaluatetheoptimalin-housetruckingcapacityindependentlyofitsinternalinventoryreplenishmentdecisions(becausethesupplier'sinventorycostisindependentofitsoutboundtransportationcapacity,whichdependsonlyontheretailer'sorderquantities). ForthenormalscenarioS0,itisstraightforwardforthesuppliertominimizetransportationcost.Sincethereisonlyasingleordersizeequaltoq0,thesupplier'sin-housetruckingcapacitycanbeeither0orq0,andarationalsupplierchoosestheonethatresultsinlowercost.ForscenarioS1,therearetwokindsofdeliveriesperpromotioncycle:alargesizedelivery,Q1and(n)]TJ /F3 11.955 Tf 12.87 0 Td[(m)regulardeliveriesofsizeq0.Wethereforeneedsolvethefollowingoptimizationproblemtodeterminetheoptimalin-housecapacity: T(W)=minfTQ1(W)+(n)]TJ /F3 11.955 Tf 11.96 0 Td[(m)t0Tq0(W)g, (2) whereTQ1(W)andTq0(W)arethesupplier'spershipmentcostsifthedeliverysizesequalQ1andq0,respectively,andT(W)isthetotaltransportationcostperpromotion.Theaboveobjectivefunctionisapiecewiselinearfunctionwithbreakpointsatq0andQ1withlimW!1T(W)=1;thus,anoptimalsolutioncanonlyoccuratoneofthethree 32

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criticalpoints0,q0,andQ1,andweneedonlycomparethevaluesofthefunctionatthesepointsandchoosetheminimum. 2.2.3.2Supplierinventorycostandprot Thenotationweusetoconstructthesupplier'sinventorycostmodelissummarizedasfollows: AS=supplier'ssetupcost; iS=supplier'sinventoryholdingcostrateperunitperyear; T0=lengthofsupplier'sregularreplenishmentperiod; T1=lengthofsupplier'sspecialreplenishmentperiod; cS=supplier'sregularproductioncost; dS=discountonrawmaterial; K0=T0=t0; K1=(T1)]TJ /F3 11.955 Tf 11.96 0 Td[(t1)=t0+1; PS0=supplier'soptimalyearlyprotwhennodiscountisoffered; PS1=supplier'sprotwhendiscountisofferedandanoptimalpolicyforPS0isused. Inadditiontotransportationcost,wealsoconsiderinventory-relatedcostsincomputingthesupplier'sprot.UnderthescenarioS0,weassumethatthesupplierusesadiscreteEOQquantitydenotedbyQS.ItiseasytoshowthatQSmaybeconstrainedtoanintegermultipleofq0(Goyal[ 53 ],whousedasimilarassumptiontostudyatwo-stageinventorymodel).Wecandeterminethesupplier'soptimalprotunderthenormalscenariobysolvingthefollowingmaximizationproblem: PS0(K0)=max(c)]TJ /F3 11.955 Tf 11.96 0 Td[(cS)D0)]TJ /F12 11.955 Tf 11.95 16.86 Td[(AS K0t0+(K0)]TJ /F6 11.955 Tf 11.96 0 Td[(1)D0t0iScS 2)]TJ /F3 11.955 Tf 13.15 9.07 Td[(Tq0(W) t0,(2) whereK0isthesupplier'smultipleoftheretailer'snormalorderquantityinanormalproductionrun,andTq0(W)istheminimumpershipmenttransportationcostwhenthe 33

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retailer'sordersizeisq0,whichisdeterminedindependentlybytheretailer.Goyal[ 53 ]showedthatthesupplier'soptimalproductionmultipleK0satises: K0(K0+1)2AS D0iS(t0)2K0(K0)]TJ /F6 11.955 Tf 11.95 0 Td[(1).(2) WeextendthistothecaseinwhichtheretailerplacesaspecialorderofsizeQ1,whichimpliesthatthesupplierneedstoplaceaspecialorderwithaquantityequaltooneofthevaluesQ1,Q1+q0,Q1+2q0,...Afterexhaustingthisspecialproductionbatch,thesupplier'sproductionisassumedtoreturntoitsnormalsize,K0q0,whichistheoptimalsolutionderivedfrom( 2 ). Inadditiontotransportationcost,thesupplier'sprotperpromotioncyclecontainstwoparts(asshowninFigure 2-3 ).Therstpart,oflengthT1,consistsoftheprotassociatedwiththespecialorderatadiscountedrawmaterialprice.Includedhereare: Figure2-3. Supplier'sinventory. therevenuefromsellingthespecialorderofsizeQ1at(c)]TJ /F3 11.955 Tf 13.11 0 Td[(d)perunitand(K1)]TJ /F6 11.955 Tf 11.96 0 Td[(1)q0unitsatcperunit; thecostofproducingQ1+(K1)]TJ /F6 11.955 Tf 11.96 0 Td[(1)q0unitsatacostof(cS)]TJ /F3 11.955 Tf 11.95 0 Td[(dS); thexedsetupcost,AS; 34

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thecostofholding(K1)]TJ /F6 11.955 Tf 12.39 0 Td[(1)q0unitsforaperiodoflengtht1atarate(cS)]TJ /F3 11.955 Tf 12.4 0 Td[(dS)iS;and ifK12,thecostofholdinganaverageof(K1)]TJ /F6 11.955 Tf 12.86 0 Td[(2)q0=2unitsforaperiodof(K1)]TJ /F6 11.955 Tf 11.96 0 Td[(1)t0atarate(cS)]TJ /F3 11.955 Tf 11.95 0 Td[(dS)iSperunit. Thesecondpart,oflength(nt0)]TJ /F3 11.955 Tf 12.09 0 Td[(T1),coverstheprotarisingfromthereturntoregularproductionfortheremainderofthepromotioncycleatarateofPS0peryear(Equation( 2 )). Combiningeachoftheabovecomponentsyieldsthesupplier'sprot-maximizingmodelbelow: PS1(K1)=1 nt0maxf(c)]TJ /F3 11.955 Tf 11.95 0 Td[(d)Q1+(K1)]TJ /F6 11.955 Tf 11.95 0 Td[(1)q0c)]TJ /F6 11.955 Tf 11.96 0 Td[((cS)]TJ /F3 11.955 Tf 11.96 0 Td[(dS)[Q1+(K1)]TJ /F6 11.955 Tf 11.95 0 Td[(1)q0])]TJ /F3 11.955 Tf 11.96 0 Td[(AS)]TJ /F12 11.955 Tf 11.3 13.28 Td[(h(K1)]TJ /F6 11.955 Tf 11.95 0 Td[(1)t1+(K1)]TJ /F10 7.97 Tf 6.59 0 Td[(1)(K1)]TJ /F10 7.97 Tf 6.58 0 Td[(2)t0 2iq0iS(cS)]TJ /F3 11.955 Tf 11.95 0 Td[(dS) (2) +[nt0)]TJ /F6 11.955 Tf 11.96 0 Td[((t1+(K1)]TJ /F6 11.955 Tf 11.95 0 Td[(1)t0)]PS0(K0)+Tq0(W) t0)]TJ /F3 11.955 Tf 11.96 0 Td[(T(W?)osubjectto:K12Z. TheaboveobjectiveisconcaveinK1,andsowecaneasilyshowthattheoptimalspecialproductionsizeK1satisesthefollowinginequalities maxn1,q0(c)]TJ /F5 7.97 Tf 6.59 0 Td[(cS+dS))]TJ /F5 7.97 Tf 6.59 0 Td[(t0PS0 q0iS(cS)]TJ /F5 7.97 Tf 6.59 0 Td[(dS)t0)]TJ /F5 7.97 Tf 13.7 4.88 Td[(t1 t0+1oK1maxn1,q0(c)]TJ /F5 7.97 Tf 6.58 0 Td[(cS+dS))]TJ /F5 7.97 Tf 6.59 0 Td[(t0PS0 q0iS(cS)]TJ /F5 7.97 Tf 6.59 0 Td[(dS)t0)]TJ /F5 7.97 Tf 13.7 4.88 Td[(t1 t0+2osuchthatK12Z. (2) 2.2.4QuantifyingtheBullwhipEffect Inthisstudy,weusetheratioofstandarddeviationoforderquantitytostandarddeviationofdemandstd[Q] std[D]BWE,asameasureofthebullwhipeffect,whereQistheretailer'sorderquantityatthebeginningofeachregularreplenishmentperiodoflengtht0andDistheconsumerdemandthatoccursduringthissameintervallength.(Becausetheaverageconsumerdemandperunittimemustequaltheaverageretailer-to-supplierorderquantityperunittimeinthelongrun,thisisequivalenttoconsideringtheratioofcoefcientofvariationvaluesatthestagesinourmodel.) 35

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UnderthenormalscenarioS0,theretailerrepeatedlyordersq0atthebeginningofeachregularreplenishmentperiod,andsounderthisscenario,theretailer'sorderquantityhaszerovariance.Moreover,theconsumerdemandoccursataconstantrate,sotheconsumerdemandratealsohaszerovariance.Consequently,thebullwhipeffectisnon-existentunderthenormalscenarioS0. UnderthepromotionscenarioS1,whenpresentedwiththesupplier'spricediscount,theretailerplacesaspecialorderofsizeQ1atthebeginningofthepromotioncycle,whichcoversmt0regularreplenishmentperiods,whichimpliesthattheretailerplacesnoorderatthebeginningofthefollowingm)]TJ /F6 11.955 Tf 12.6 0 Td[(1regularreplenishmentperiods.AfterdepletingthespecialorderquantityQ1,theretailerreturnstothenormalstrategyandordersq0atthebeginningoftheremainingn)]TJ /F3 11.955 Tf 12.24 0 Td[(mreplenishmentregularreplenishmentperiods.Combiningtheseobservations,themeanandstandarddeviationoftheretailer'sorderquantitycanbecalculatedas: E[Q]=q0(n)]TJ /F3 11.955 Tf 11.95 0 Td[(m)+Q1 n,(2) std[Q]=r (q0)]TJ /F3 11.955 Tf 11.96 0 Td[(E[Q])2(n)]TJ /F3 11.955 Tf 11.95 0 Td[(m)+(Q1)]TJ /F3 11.955 Tf 11.95 0 Td[(E[Q])2+(0)]TJ /F3 11.955 Tf 11.96 0 Td[(E[Q])2(m)]TJ /F6 11.955 Tf 11.95 0 Td[(1) n.(2) UnderthepromotionscenarioS1,theconsumerdemandiscomposedofthreeparts(asshowninFigure 2-1 ).Therstcomponenthasadurationoft=(Q1)]TJ /F3 11.955 Tf 12.78 0 Td[(q1)=D1timeunitsandademandrateofD1;thesecondcomponenthasadurationoft=~rf(Q1)]TJ /F3 11.955 Tf 12.07 0 Td[(q1)=(D0D1)timeunitsandademandrateofzero;andthethirdcomponenthasadurationoft=q1=D0+(n)]TJ /F3 11.955 Tf 12.15 0 Td[(m)t0timeunitsandademandrateofD0.Becauset,t,andtmaynotbeintegermultiplesoft0,weneedtodeneadditionalparametersbeforecomputingthedemandvariance.Todothis,wedenethefollowingquantities: n1=jt t0k; n2=maxfjt t0+t t0)]TJ /F3 11.955 Tf 11.95 0 Td[(n1)]TJ /F6 11.955 Tf 11.95 0 Td[(1k,0g; 36

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n3=jt t0k; I1=(1t t0+t t0)]TJ /F3 11.955 Tf 11.96 0 Td[(n1<1,0otherwise; I2=lt t0)]TJ /F3 11.955 Tf 11.96 0 Td[(n3m. Apromotioncyclebeginswithn1regularreplenishmentintervals,eachwithdemandD1t0,andendswithn3regularreplenishmentperiods,eachwithdemandD0t0.Fortherestofthepromotioncycle,therearetwopossiblecases:(i)I1=0,whichimpliesthedurationoftimewithazerodemandrateislessthantheregularreplenishmentintervallengtht0,andasingleintervalexiststhateitherbeginswithademandrateofD1andendsatzero,beginswithazerodemandrateandendswithademandrateofD0,orbeginswithademandrateofD1andendswithademandrateofD0.Thedemandduringthisintervalequalsd3=(t t0)]TJ /F3 11.955 Tf 12.45 0 Td[(n1)D1+(1+n1)]TJ /F5 7.97 Tf 13.64 4.71 Td[(t t0)]TJ /F5 7.97 Tf 13.63 4.71 Td[(t t0)D0;(ii)I1=1,whichimpliesthattherearen)]TJ /F3 11.955 Tf 10.76 0 Td[(n1)]TJ /F3 11.955 Tf 10.76 0 Td[(n3periodsoflengtht0containingintervalsofzerodemand.Includedhereare: aregularreplenishmentperiodstartingwithademandrateD1andendingwithazerodemandrate,whichhastotaldemandofd4=(t t0)]TJ /F3 11.955 Tf 11.96 0 Td[(n1)D1; n2regularreplenishmentperiodscontainingzerodemand; aregularreplenishmentperiodstartingwithademandrate0andendingwithademandrateD0,whichhasatotaldemandd5=t t0)]TJ /F3 11.955 Tf 11.95 0 Td[(n3D0. Combiningeachoftheabovecomponents,themeanandvarianceofconsumerdemandarecomputedas E[D]=n1D1t0+n3D0t0+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(I1)d3+I1d4+I2I1d5 n,(2) Var[D]=1 nfn1(D1t0)]TJ /F3 11.955 Tf 11.95 0 Td[(E[D])2+n2(0)]TJ /F3 11.955 Tf 11.96 0 Td[(E[D])2+n3(D0t0)]TJ /F3 11.955 Tf 11.96 0 Td[(E[D])2+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(I1)(d3)]TJ /F3 11.955 Tf 11.95 0 Td[(E[D])2+I1(d4)]TJ /F3 11.955 Tf 11.95 0 Td[(E[D])2+I2I1(d5)]TJ /F3 11.955 Tf 11.96 0 Td[(E[D])2g, (2) 37

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andwehavestd[D]=p Var[D]. Insummary,ourmodelconsiderstwoscenarios,anormalscenarioandapromotionscenario,andweassumethattheretailerandsupplieroptimizetheirprotsindependently.Werstcomputetheretailer'soptimalprot,PR0,andthenthesupplier'soptimalprot,PS0,underthenormalscenario.Forthepromotionscenario,givenatemporarysupplierpricediscount,theretaileroptimizesitsprot,PR1,bychoosingappropriatevaluesofQ1,q1and.Giventheretailer'sorderingpolicy,thesupplierchoosesitsin-housetruckingcapacityWandarrangesaspecialproductionorder(determinedbyK1)tomaximizeitsprot,PS1.Finally,wecomputethebullwhipeffect,BWE=std[Q] std[D],theretailer'sprotgain,R=PR1)]TJ /F5 7.97 Tf 6.58 0 Td[(PR0 PR0100%,thesupplier'sprotgain,S=PS1)]TJ /F5 7.97 Tf 6.59 0 Td[(PS0 PS0100%,andthetotalsystemprotgain,=PS1+PR1)]TJ /F5 7.97 Tf 6.59 0 Td[(PS0)]TJ /F5 7.97 Tf 6.58 0 Td[(PR0 PS0+PR0100%. 2.3NumericalExperiments Thegoalofournumericalexperimentsinthischapteristocharacterizehowpricediscountsinuencethesystemprot.Weareparticularlyinterestedintheconditionsunderwhichincreasedsystemprotscancoexistwiththebullwhipeffect.Togainsomeinsightintothisquestion,weperformednumerouscomputationalteststhatparameterizeonthediscountgivenbythesupplier,d. Inordertocoverabroadrangeofpotentialproductcharacteristics,weconsiderthreecategoriesofinventorybasedonunitvalues.Weapplyan80=20Paretoprincipleandclassifyitemsbasedonannualdollarmovement(asmeasuredbycD)asfollows: CategoryAitemscomprise20%oftheSKUsandcontributeto80%ofannualdollarmovement. CategoryBitemscomprise30%oftheSKUsandcontributeto15%ofannualdollarmovement. CategoryCitemscomprise50%oftheSKUsandcontributeto5%ofannualdollarmovement. Theretailer'sstandardparametervaluesforCategoryBitems(exceptfor,andrf)correspondtothoseusedbyArdalan[ 13 ].Thevaluesof,andrffor 38

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CategoryBitemswerechosenbasedontheworkofGupta[ 54 ],whichestimatesthatonly16%ofthesalesincreasefrompromotionsresultsfrombrandloyals'forwardbuying.Wenotethatitispossibletostackthedeckinfavorofpromotionsbycreatingnumericalexampleswithanextremelyhighrateofdemandfromimpulsiveconsumers(bychoosingahighvalueof).Aswelaterdiscuss,however,theparameterswehavechosenarereasonablyconservativeintermsoftheimpactofimpulsiveconsumersontheprotfrompromotions.AccordingtothestudybyBlattbergandNeslin[ 26 ],theincreaseinsalesduetopricecutscanbeasmuchas10timesthenormallevel(ora1000%increase).However,theparameterswehavechosenleadtoprobleminstancesinwhichthedemandfromimpulsiveconsumersdoesnotexceed21%oftotaldemand.Thuswecancreateinstancesinwhichtheprotfrompromotionsismuchhigherbyincreasingthedemandfromimpulsiveconsumers(althoughwehavechosennottodothis).Thesupplier'sstandardparametervaluesforcategoryBitemswerechosentoreectthefactthatthesupplierusuallyhaslowerholdingcostsandhighersetupcoststhantheretailer.AsshowninTable 2-1 ,allproductcategoriesusethesamevaluesofi=0.25,is=0.15,=ln(2)and=ln(1.33). BasedontheparametervaluesofCategoryBitems,thoseforCategoryAandCitemswerecomputedaccordingtothe80=20principle(asshowninTable 2-1 ).Forexample,wesettheparametersforCategoryAitemstoensurethat,underthenormalscenario,theannualdemandforCategoryAitems,DA,equalstwothirdsofDB.WethensettheperunitcostofCategoryAitemsusingcA=cBDB DA80 15.Theothercost-relatedparametersforCategoryAitemswerethencomputedbyscalingtheCategoryBparametersinproportiontotherelativevaluesofDAandcA.Finally,wesettheregularpriceelasticityofconsumerdemandforCategoryAitemsusingr0A=r0BDA DB15 80;thevalueoftheonlyremainingparameter,rc,canbefoundbytrialanderrortoensurethatthe80=20principleissatised.Foreachitemcategory,wesolvedthemodelforvariousvaluesofsupplierdiscountpercentage,i.e.,d c100%,rangingfrom0.01to 39

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Table2-1. Parametersettingsforcomputationaltests. CategoryccSiiSAASdS A80480.250.15804804.84B1060.250.1510600.60.5C21.20.250.152120.121CategorySPRSF0SP0rcr0rfri A8120012ln(2)ln(1.33)42800375133375B11501.5ln(2)ln(1.33)49000300010003000C0.2300.3ln(2)ln(1.33)6160015000670015000 0.3.WeapplytheapproachdiscussedinSection 2.2.2 tosolvetheretailer'sproblemforadiscretesetofcandidatevaluesofandlunderthepromotionscenario,S1.Weexaminethirtypossiblevaluesof(fromc1%toc30%)andonehundredpotentialvaluesofl(from1to100).Tables 2-2 2-4 providedetailedresultsofthemodelsforCategoriesA,BandC,andforgivenvaluesofthesupplier'sdiscountd. Theresultsinthetablessuggestthefollowing.Forallinventorycategories,asthesupplierdiscount,d,increases,theretailer'sdiscount,,alsoincreases,and,somewhatsurprisingly,theretailer'soptimaldiscount,,isnotalwayssmallerthanthesupplier'sdiscount,d.Notethatforallinventorycategories,theretailer'sspecialordersize,Q1,andcorrespondingprotgain,R,bothincreaseasdincreases,whichindicatesthatwhenasufcientnumberofimpulsiveconsumersexist(determinedby~ri),ourresultsfortheretailerareconsistentwithintuitionandwiththeresultsofArcelusandSrinivasan[ 12 ].Thus,theretailer'sincentivetopressthesupplierforperiodicdiscountsisclear.Moreover,fromasystemperspective,apositivelevelofsupplierdiscountisdesirableforallitemcategories,whichimpliesthatthebullwhipeffectdoesnotalwaysimplyasystemlossandcancoexistwithhigherprotlevelsascomparedwiththecaseofEDLP.However,whenthesupplierandretaileroperateindependently,itwilllikelybethecasethatthesupplier'soptimalactionisnotoptimalforthesystem,i.e.,thediscountofferedbythesupplier(ifany)islessthanthelevelofsupplierdiscountthatmaximizesthesystemprot.Inaddition,itispossiblethatthesupplierandthetotalsystemmaybe 40

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worseoffasaresultofprovidingadiscounttotheretailer.Boththesupplier'sprotgain,S,andthetotalsystemprotgain,,rstincreaseind,andthen,afterreachingapeakatsomepositivevalueofd,decreaseind.ForCategoryCitems,Sisalwaysnegative,whichindicatesthatthesuppliershouldnotdiscountCitemsunderourmodelassumptions.Thus,thesupplier'schoiceofwhetherornottoofferadiscountdependsontheitemcategory.ThelastcolumnofTables 2-2 2-4 providesthepercentageofimpulsiveconsumers'demandtoillustratethatourexamplesdonotarticiallyinatetheextrademandfromimpulsiveconsumerswhencomparedtopracticalcases. Wealsoobservethatforallinventorycategories,theratioofthestandarddeviationofretailerorderstothatofconsumerdemand,theBWE,isalwaysgreaterthanone,whichindicatesthatbullwhipeffectexistsforourprobleminstances.Notethatforallcategories,theBWEtendstoincreaseasthediscountdincreases,whichisconsistentwiththendingsofLee,Padmanabhan,andWhang[ 64 65 ]thatpricevariationsserveasoneofthecausesofbullwhipeffect. Thus,forthemodelwehavedeveloped,itisnotonlyintheretailer'sbestinteresttoseekperiodicdiscounts,butitisalsoofteninthesupplier'sbestinteresttoprovidethem,despitetheassociatedincreaseincostsduetothebullwhipeffect.Wenote,however,thattheoptimalindependentdecisionforthesupplierdoesnotleadtotheoptimalsystemprot,i.e.,thesupplierandretailercantypicallyreaphighergainsthroughcoordinatingtheiractions(althoughweleavethespecicsofsuchcoordinationmechanismsforfurtherresearch). Table 2-5 summarizesaverage,minimum,andmaximumprotchangesandtheBWEforeachcategory.ThenumbersinparenthesesinTable 2-5 representthevalueofthesupplier'sdiscountatwhichthecorrespondingmaximum(orminimum)protincreaseforthesupplieroccurs.Althoughthereappearstobenodirectrelationshipbetweenprotchanges(R,Sand)andtheBWE,itemcategorieswithgenerallysmallervaluesoftheBWEtendtoproducehighergainsasaresultofasupplier 41

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Table2-2. ComputationalresultsforcategoryAwithstandardparametersettings. CategoryA d c100% c100%Q1q1RSBWESW% 1%2%731.68210.380.10%0.18%0.15%11.490.25%2%3%1026.91192.460.28%0.37%0.33%9.220.56%3%3%1293.7182.20.53%0.32%0.42%10.290.75%4%3%1560.26172.230.84%0.19%0.49%11.230.93%5%3%1826.58162.571.23%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.03%0.54%12.081.11%6%4%2469.39117.141.72%0.19%0.89%10.681.97%7%4%2748.5105.422.29%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.19%0.93%11.222.21%8%4%3027.3194.092.94%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.67%0.97%11.742.45%9%4%3583.9972.553.68%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.26%0.99%12.702.92%10%5%3973.53259.514.53%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.53%1.23%11.303.65%11%5%4264.6246.755.48%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.29%1.24%11.663.95%12%5%4845.61222.586.54%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.22%1.22%12.414.53%13%6%5499.6627.257.70%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.07%1.28%12.395.43%14%6%5802.5613.569.00%)]TJ /F21 9.963 Tf 7.75 0 Td[(5.17%1.28%12.825.78%15%6%6330.04825.510.41%)]TJ /F21 9.963 Tf 7.75 0 Td[(6.72%1.07%13.746.12%16%7%6784.81000.611.95%)]TJ /F21 9.963 Tf 7.75 0 Td[(7.63%1.28%12.887.14%17%7%7413.9971.7313.64%)]TJ /F21 9.963 Tf 7.75 0 Td[(9.23%1.18%13.527.94%18%7%7952.761183.315.46%)]TJ /F21 9.963 Tf 7.75 0 Td[(11.23%0.92%14.258.33%19%8%8670.841577.4417.42%)]TJ /F21 9.963 Tf 7.75 0 Td[(13.04%0.82%13.659.52%20%8%9096.841323.4619.55%)]TJ /F21 9.963 Tf 7.75 0 Td[(14.71%0.88%14.0010.41%21%8%9646.591535.1121.83%)]TJ /F21 9.963 Tf 7.75 0 Td[(17.29%0.51%14.7010.86%22%9%10402.231923.9524.28%)]TJ /F21 9.963 Tf 7.75 0 Td[(19.73%0.30%14.3612.21%23%9%11074.471895.9226.91%)]TJ /F21 9.963 Tf 7.75 0 Td[(22.56%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.05%14.9513.20%24%9%11634.72108.0929.71%)]TJ /F21 9.963 Tf 7.75 0 Td[(25.77%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.52%15.5513.69%25%10%12548.792253.7332.71%)]TJ /F21 9.963 Tf 7.75 0 Td[(28.80%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.81%14.9915.75%26%10%13119.812466.0535.92%)]TJ /F21 9.963 Tf 7.75 0 Td[(32.47%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.35%15.5716.29%27%10%14034.62667.9239.31%)]TJ /F21 9.963 Tf 7.75 0 Td[(37.06%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.31%16.4817.35%28%11%14875.323047.2642.94%)]TJ /F21 9.963 Tf 7.75 0 Td[(41.05%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.83%16.0919.09%29%11%15456.073260.8446.79%)]TJ /F21 9.963 Tf 7.75 0 Td[(45.44%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.47%16.6319.67%30%11%16165.033240.3150.86%)]TJ /F21 9.963 Tf 7.75 0 Td[(50.01%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.11%17.2020.82% discount(forboththeretailerandsupplier),whichimpliesthatactionsotherthanstabilizingpriceshouldperhapsbetakentomitigatethenegativeimpactsofthebullwhipeffect. Usingourbaseparametersettings,wenextexaminehowchangesinspecicparametervaluesaffectthenumericalresults.Foraparameterofinterest,e.g.,theretailer'ssetupcostA,wexthevaluesofallotherparametersandconsiderhowchangingthevalueoftheparameterofinterestaffectstheresults.Thatis,forthesetupcostA,weconsiderthevaluesAHandAL,whereAHishigherthanthestandardvalueofA,andALislowerthanthestandardvalueofA;wethencomputethecorresponding 42

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Table2-3. ComputationalresultsforcategoryBwithstandardparametersettings. CategoryB d c100% c100%Q1q1RSBWESW% 1%3%893.94257.910.09%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.01%0.05%11.620.25%2%4%1558.13230.870.22%0.25%0.24%11.570.67%3%4%1878.28220.650.42%0.14%0.29%12.640.83%4%4%2198.18210.730.67%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.06%0.34%13.611.00%5%4%2837.2191.810.99%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.30%0.41%15.341.32%6%5%3258.18159.771.36%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.34%0.60%13.321.85%7%5%3917.59138.321.80%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.79%0.64%14.542.25%8%5%4575.74118.322.31%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.40%0.65%15.672.65%9%5%4904.35108.852.88%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.12%0.65%16.212.84%10%5%5560.6290.953.53%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.01%0.61%17.273.23%11%6%6287.9603.934.25%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.85%0.64%16.333.87%12%6%6689.11309.25.06%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.71%0.70%16.444.33%13%6%7301.49574.775.95%)]TJ /F21 9.963 Tf 7.75 0 Td[(6.17%0.54%17.534.56%14%7%8060.251082.96.93%)]TJ /F21 9.963 Tf 7.75 0 Td[(7.41%0.53%16.975.32%15%7%8754.831062.537.99%)]TJ /F21 9.963 Tf 7.75 0 Td[(8.98%0.42%18.055.85%16%7%9376.371327.929.15%)]TJ /F21 9.963 Tf 7.75 0 Td[(10.87%0.21%19.146.11%17%8%10166.071831.7810.41%)]TJ /F21 9.963 Tf 7.75 0 Td[(12.56%0.16%18.736.98%18%8%10878.431811.611.77%)]TJ /F21 9.963 Tf 7.75 0 Td[(14.61%0.00%19.597.57%19%8%11508.782077.1513.23%)]TJ /F21 9.963 Tf 7.75 0 Td[(17.00%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.27%20.527.87%20%9%124192292.4914.81%)]TJ /F21 9.963 Tf 7.75 0 Td[(19.14%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.34%19.719.18%21%9%13333.242832.8716.50%)]TJ /F21 9.963 Tf 7.75 0 Td[(22.49%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.90%20.969.50%22%9%14060.582815.7118.31%)]TJ /F21 9.963 Tf 7.75 0 Td[(25.39%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.20%21.6910.15%23%10%14920.383311.8320.25%)]TJ /F21 9.963 Tf 7.75 0 Td[(28.35%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.45%20.7811.28%24%10%159403570.0322.31%)]TJ /F21 9.963 Tf 7.75 0 Td[(32.36%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.09%21.8611.99%25%11%16819.294064.4724.50%)]TJ /F21 9.963 Tf 7.75 0 Td[(35.82%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.42%21.4713.19%26%11%17855.864323.5626.85%)]TJ /F21 9.963 Tf 7.75 0 Td[(40.38%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.16%22.6013.96%27%11%18510.644591.4929.33%)]TJ /F21 9.963 Tf 7.75 0 Td[(44.51%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.63%23.3414.34%28%12%19693.625358.7531.97%)]TJ /F21 9.963 Tf 7.75 0 Td[(49.55%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.42%23.0515.64%29%12%20745.335620.4734.77%)]TJ /F21 9.963 Tf 7.75 0 Td[(54.98%)]TJ /F21 9.963 Tf 7.75 0 Td[(5.30%24.1016.47%30%12%21794.315884.8337.72%)]TJ /F21 9.963 Tf 7.75 0 Td[(60.71%)]TJ /F21 9.963 Tf 7.75 0 Td[(6.22%25.1817.28% valuesofR,S,,andBWE.Table 2-6 showsthehighandlowparametersettingsforeachcategoryofitems.Figures 2-5 2-8 illustratetheresultsoftheseexperiments,inwhichXH(XL)correspondstotheresultsfromanexperimentwithparameterXatahigh(low)level(thelackofsmoothnessofmanyofthecurvesarisesfromtheintegralityrestrictiononmandtheselectionofl,,anddfrompredeneddiscretesets).ThetrendsforCategoryBandCitemsareconsistentwiththoseforCategoryAitems,andtheyarethereforeomitted.Inordertovalidatewhetherparametervalueshaveasignicantimpactonthebullwhipeffect,weperformedaseriesofhypothesistests(usingtheMinitabstatisticalsoftwarepackage)onthedifferenceinbullwhipeffectvalues 43

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Table2-4. ComputationalresultsforcategoryCwithstandardparametersettings. CategoryC d c100% c100%Q1q1RSBWESW% 1%4%1527.17320.310.08%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.59%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.14%14.900.30%2%5%2351.89284.710.19%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.48%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.03%14.510.62%3%5%3148.86259.390.33%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.61%0.02%16.380.87%4%6%4023.69211.730.53%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.66%0.14%15.281.33%5%6%4835.66185.040.76%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.06%0.16%16.481.62%6%6%5241.2172.361.04%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.60%0.17%17.041.76%7%6%6051.4148.341.37%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.27%0.17%18.112.04%8%6%6860.45126.061.74%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.09%0.16%19.132.32%9%7%7822.0863.872.17%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.69%0.24%17.593.03%10%7%8644.9541.792.64%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.80%0.20%18.443.35%11%7%9821.55376.663.17%)]TJ /F21 9.963 Tf 7.75 0 Td[(6.33%0.05%20.023.66%12%7%10641.8358.343.76%)]TJ /F21 9.963 Tf 7.75 0 Td[(7.77%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.03%20.833.97%13%8%11553.41018.194.40%)]TJ /F21 9.963 Tf 7.75 0 Td[(9.18%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.06%20.914.54%14%8%12742.951354.385.10%)]TJ /F21 9.963 Tf 7.75 0 Td[(11.25%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.26%22.664.89%15%8%13514.71700.775.87%)]TJ /F21 9.963 Tf 7.75 0 Td[(13.27%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.41%23.835.06%16%9%14804.62712.246.70%)]TJ /F21 9.963 Tf 7.75 0 Td[(15.54%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.60%23.975.70%17%9%15652.182694.447.60%)]TJ /F21 9.963 Tf 7.75 0 Td[(17.77%)]TJ /F21 9.963 Tf 7.75 0 Td[(0.73%24.846.08%18%9%16785.473396.478.57%)]TJ /F21 9.963 Tf 7.75 0 Td[(20.72%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.05%26.406.28%19%10%18173.464041.669.61%)]TJ /F21 9.963 Tf 7.75 0 Td[(23.59%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.29%25.877.19%20%10%18958.364388.9610.73%)]TJ /F21 9.963 Tf 7.75 0 Td[(26.54%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.50%26.747.40%21%10%20171.774730.311.93%)]TJ /F21 9.963 Tf 7.75 0 Td[(30.15%)]TJ /F21 9.963 Tf 7.75 0 Td[(1.89%27.967.81%22%11%21514.355735.913.21%)]TJ /F21 9.963 Tf 7.75 0 Td[(33.89%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.25%27.278.59%23%11%22660.316439.4314.57%)]TJ /F21 9.963 Tf 7.75 0 Td[(38.11%)]TJ /F21 9.963 Tf 7.75 0 Td[(2.72%28.498.82%24%12%24097.867082.916.02%)]TJ /F21 9.963 Tf 7.75 0 Td[(42.35%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.14%27.829.87%25%12%25605.178142.2617.56%)]TJ /F21 9.963 Tf 7.75 0 Td[(47.81%)]TJ /F21 9.963 Tf 7.75 0 Td[(3.90%29.4210.12%26%12%26842.958487.7419.20%)]TJ /F21 9.963 Tf 7.75 0 Td[(52.81%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.44%30.5610.60%27%13%28221.829490.9320.94%)]TJ /F21 9.963 Tf 7.75 0 Td[(58.01%)]TJ /F21 9.963 Tf 7.75 0 Td[(4.98%30.2811.49%28%13%29826.7910193.5422.78%)]TJ /F21 9.963 Tf 7.75 0 Td[(64.40%)]TJ /F21 9.963 Tf 7.75 0 Td[(5.84%31.7112.01%29%13%30983.4810900.9524.72%)]TJ /F21 9.963 Tf 7.75 0 Td[(70.29%)]TJ /F21 9.963 Tf 7.75 0 Td[(6.47%32.8012.26%30%14%32836.6411901.2126.78%)]TJ /F21 9.963 Tf 7.75 0 Td[(77.30%)]TJ /F21 9.963 Tf 7.75 0 Td[(7.39%32.5413.48% Table2-5. Resultsofcomputationaltests. CategoryRSBWE AverageA16.22%)]TJ /F6 11.955 Tf 9.3 0 Td[(13.33%0.12%13.32B12.21%)]TJ /F6 11.955 Tf 9.3 0 Td[(16.92%)]TJ /F6 11.955 Tf 9.3 0 Td[(0.79%18.33C8.80%)]TJ /F6 11.955 Tf 9.3 0 Td[(22.86%)]TJ /F6 11.955 Tf 9.3 0 Td[(1.59%23.43MaximumA50.86%(30%)0.37%(2%)1.28%(13%)17.20(30%)B37.72%(30%)0.25%(2%)0.70%(12%)25.18(30%)C26.78%(30%))]TJ /F6 11.955 Tf 9.3 0 Td[(0.48%(2%)0.24%(9%)32.54(30%)MinimumA0.10%(1%))]TJ /F6 11.955 Tf 9.3 0 Td[(50.01%(30%))]TJ /F6 11.955 Tf 9.3 0 Td[(4.11%(30%)9.22(2%)B0.09%(1%))]TJ /F6 11.955 Tf 9.3 0 Td[(60.71%(30%))]TJ /F6 11.955 Tf 9.3 0 Td[(6.22%(30%)11.57%(2%)C0.08%(1%))]TJ /F6 11.955 Tf 9.3 0 Td[(77.30%(30%))]TJ /F6 11.955 Tf 9.3 0 Td[(7.39%(30%)14.90(1%) 44

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Table2-6. Parametersettingsforfactorialtests. CategoryLevelAirf Ahigh8000.5500ln(1.66)low400.12500Bhigh1000.54000ln(1.66)low50.12500Chigh200.52000ln(1.66)low10.12500 Table2-7. Hypothesistestsforbullwhipeffectwithkeyparametersatdifferentlevels. ParameterAlternativehypothesisP-value H0:BWEHBWELAH1:BWEH
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placinglargeorders.Thiseffectreducesthebullwhipeffectandhinderstheretailer'spotentialtogainadditionalrevenuefromalongerpromotionperiodlength. Forallinventorycategories,ahighervalueof~ri(thenumberofunitspurchasedperdollarofpricereductionforimpulsiveconsumers)impliesahighervalueandalowervalueoftheBWE.Thisisbecausethepresenceofimpulsiveconsumersmakesitmorelikelyfortheretailertopassalargerpercentageofthesupplier'sdiscounttoconsumersinsteadofforwardbuying.Consequently,theretailer'sorderpatternismoreconsistentwiththeendconsumers'purchasepattern,whichleadstoasmallerdegreeofbullwhipeffect. Forallinventorycategories,ahighervalueof~rf(thenumberofunitspurchasedperdollarofpricereductionforbrandloyals)leadstoalowervalueof,whichindicatesthatifaretailerpricediscountinducesalargedegreeofconsumerforwardbuying,thesuppliershouldadoptaconservativepromotionstrategy. Forallinventorycategories,thebullwhipeffectexistsand,exceptfortheresultswhenisatalowlevel,theBWEtendstoincreaseasdincreases;whenisatalowlevel,theBWEdecreases.Tobetterunderstandthisphenomenon,Figure 2-4 illustrateshowtheratiooftheretailer'sforwardbuyquantitytotheretailer'sspecialorderquantity,q1 Q1,changeswithd.Weobservethatwhenisatalowlevel,theBWEisconsistentwiththevalueofq1 Q1.Thisindicatesthatthesupplierdiscount,d,inducesthebullwhipeffectviatheinteractionoftwoquantities:Q1andq1.Aswementionedbefore,whenpresentedwithapricediscount,theretailerwillplaceaspecialorderofsizeQ1>q0,andalargervalueofdclearlyleadstoalargervalueofQ1.AlargervalueofQ1,inturnleadstoalargervalueofstd(Q).Aspartofitslargeorder,theretailermayalsoadoptaforwardbuyingstrategy,i.e.,q1>0.Aforwardbuyquantityincreasesthevarianceofordersplacedwiththesupplier,butdoesnotstimulateconsumerdemand.Thus,thelargertheratioofq1 Q1,thelargerthebullwhipeffect;thebullwhipeffectisthenmaximalwhenq1 Q1=1(whichimpliesstd(D)=0andstd(Q) std(D)=1),andminimalwhenq1=0. 46

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Consequently,whenisatalowlevel,q1issignicantlylarge,sothattheimpactofq1overwhelmstheimpactofQ1;asaresult,thebullwhipeffectchangesinresponsetothechangeinq1 Q1.Nosystematicrelationshipcanbeobservedbetweenthebullwhipeffect Figure2-4. Forwardbuyinginsupplierdiscountwithatdifferentlevels. andR,Sand.Inotherwords,ahigherlevelofbullwhipeffectdoesnotuniversallyimplylowerlevelsofprot. Insummary,wedemonstratedthateventhoughtheuseoftradepromotionscanindeedincreaseasupplier'soperationscosts,thesecostsmaybemorethanoffsetbyincreasedrevenues,evenintheabsenceofexplicitcoordination.Thatis,thesupplychainprotcanexceedthatunderanEDLPstrategy,if(i)thesupplierjudiciouslysetsthepricediscount;(ii)thereisasufcientnumberofimpulsiveconsumerswhobuytheproductatthediscountedprice;(iii)thepricediscountdoesnotinduceahighdegreeofend-consumerforwardbuying. 47

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Figure2-5. Retailerprotgaininsupplierdiscountwithkeyparametersatdifferentlevels. Figure2-6. Supplierprotgaininsupplierdiscountwithkeyparametersatdifferentlevels. 48

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Figure2-7. Systemprotgaininsupplierdiscountwithkeyparametersatdifferentlevels. Figure2-8. Bullwhipeffectinsupplierdiscountwithkeyparametersatdifferentlevels. 49

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CHAPTER3MULTI-PERIODPRICEPROMOTIONSINASINGLE-SUPPLIER,MULTI-RETAILERSUPPLYCHAINUNDERASYMMETRICDEMANDINFORMATION 3.1ProblemDescriptionandLiteratureSurvey AstudybyAndersenConsulting[ 42 ]foundthattradepromotionisthebiggest,mostcomplexandcontroversialdilemmafacingtheretailindustrytoday.AswementionedinChapter 1 ,tradepromotionshavelongplayedanimportantroleinU.S.retailsupplychains,however,tradepromotionsdonoteffectivelyimprovesuppliers'performance.Adoptionofoff-invoicedealsisoneofthemajorreasonsforthepoorperformanceoftradepromotions,becauseitisprotableforretailerstorespondtoanoff-invoicetradepromotionbyengaginginforward-buyinganddiverting,andtheseforward-buyinganddivertingstrategiesresultinthebullwhipeffect,whichisviewedinextremelynegativetermsbecauseofitsnegativeimpactsonsupplychainoperationscosts.Lee,PadmanabhanandWang[ 64 ]characterizedpricevariationsasoneofthemajorcausesofthebullwhipeffectandsuggestedusingcorrespondingmanagementpractices(anEDLPstrategy,forexample)toreducetheoperationscostsassociatedwiththebullwhipeffect. Despitethemanyundesirableconsequencesdiscussedintheoperationsliterature,therehasbeennosignofdeclineintheuseoftradepromotionsinindustry.Somemarketingscienceliteratureshowsthatacarefullydesignedpricepromotionschemecanimprovethesupplier'sprotwhencomparedtothecaseofeverydaylowpricing(EDLP).Blattberg,EppenandLieberman[ 25 ]suggestedthateffectivetimingofoff-invoicetradepromotionscanreduceasupplier'sinventorycost.DrezeandBell[ 42 ]showedthatscan-backtradepromotionsmayimproveboththeretailer'sandthesupplier'sperformance.Scan-backtradepromotionsaresonamedbecausetheyarebasedonstores'scannerdataand,therefore,theamountsoldtoendconsumersfromeachstore. 50

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Althoughthemarketingscienceliteratureprovidesimportantinsightsabouttradepromotions,theytypicallydonotexplicitlyconsidersupplychainoperationscosts.Mathematicalmodelsthataccountforoperationscostsinadditiontothedemand-sideeffectsofpricepromotionshavebeendevelopedintheoperationsresearchliterature.Neslin,PowellandStone[ 77 ]developedamulti-echelonmodelthatconsiderstheactionsofasupplier,asetofretailers,andconsumersfromthepointofviewofthesupplier,whoattemptstomaximizeprotbytargetingadvertisingtoconsumersandtradepromotionstoretailers.Duetothecomplexityoftheirmodel,theycouldnotprovideaprocedureforobtainingagloballyoptimalsolution.Moreover,theirmodelingapproachparameterizedonalloftheretailer'sdecisions.However,tradepromotionsaresupplierdecisions,whichaffecttheretailer'sactions,andtherearenumerousinteractionstakingplacebetweenthesupplierandtheretailer.Usuallyintheseinteractions,thesupplieristheleaderwhosetsthewholesalepricesanddiscountsrst,andtheretaileristhefollowerwhosetstheretailpricesandplacespurchaseordersgiventhesupplier'sdecisions.Asaresult,aStackelberggameinwhichthesuppliermovesrstisabetterapproximationofthisprocess.Kopalle,MelaandMarsh[ 62 ]wereamongtherstwhodevelopedanormativemodelusingaStackelberggametodetermineoptimalsupplierandretailerpricesovertime.Intheirstudy,theyindicatedthattradepromotionshaveeffectsnotonlyintheperiodstheyareoffered,butalsoaffectfutureperiods.Asaresult,theydevelopedadynamic,descriptivebrandsalesmodelwhichaccountsfordynamiceffectsofdiscounts,suchasconsumerforwardbuyingandcompetitionbetweenbrands,andtheyintegratedthisdescriptivemodelwiththenormativemodeltostudythecontemporaneouseffectsandfutureeffectsofpromotions.However,theirmodelallowednoretailerforwardbuyinganddiverting.KoganandHerbon[ 60 ]consideredasimilartwo-echelonsupplychainmodelwithasupplierandaretailerfacingstochasticdemands,andtheyusedadynamiccontinuous-timeStackelberggametosimulatetheinteractionsbetweenthesupplierandheretailer. 51

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IncontrasttothestudyofKopalleetal.[ 62 ],intheirmodel,KoganandHerbon[ 60 ]incorporatedretailerforwardbuyingandconsideredinstantaneousandexogenouschangeschangeintheconsumerpricesensitivitywhichwilltemporarilyincreasetheconsumerdemandpotential.Inadditiontothesetwostudies,therearemanyotherstudiesaboutchannelcoordinationdealingwithpricingstrategiesforasupplier-retailerchannelrelationshipintheliterature(e.g.,JeulandandShugan[ 59 ];EliashbergandSteinberg[ 44 ];XieandWei[ 100 ]). Inthispaper,weprovideamathematicalmodelofadecentralizedtwo-echelonsupplychaininwhichthesupplier'spricingdecisions(tradepromotionlevels)andtheretailer'soperationsdecisions(orderquantities,inventorylevels,andtransshipmentquantities)aredeterminedsimultaneously.Tobroadentheapplicabilityofourmodel,weassumethatthepricecanbedynamicallychangedovertimeandthatdemandisstochasticandprice-dependent.Tothebestofourknowledge,noexistingworkconsidersanexactsolutionmethodforajointpromotionandoperationsprobleminamulti-echelonsupplychainunderuncertaindemand,aswedointhischapter.Intheeldofmathematicalprogramming,thisproblemfallsintheclassoflinearlyconstrained,bilevel,nonconvexoptimizationproblems.Mostoftheavailablealgorithmsintheeldofbilevelprogrammingapplytobilevellinearproblems(where,forxedvaluesofonesetofdecisionvariables,theremainingproblembecomesalinearprogram).Ben-Ayed[ 21 ]andWenandHsu[ 98 ]provideddetailedreviewsofbilevellinearprogrammingproblems.Theypresentedabasicmodelalongwithcharacterizationsofoptimalsolutionpropertiesfortheproblemclassandsomeexistingsolutionapproaches.Tosolvelinearlyconstrainedbilevelconvexquadraticproblems,MuuandVanQuy[ 74 ]developedabranch-and-boundalgorithmforndingaglobaloptimalsolution.Aswewillsee,noneofthepreviouslymentionedsolutionmethodsforbilinearoptimizationproblemscanbedirectlyappliedtothemodelwedevelop,whichfallswithinamoregeneralclassofnonconvexbileveloptimizationproblems.Thesemethodshave,however,provided 52

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substantialguidanceandinspirationforthesolutionprocedurewehavedevelopedforsolvingtheproblemwedene. Aswelaterdiscuss,ourmodelcanbecastasageneralizedbilinearprogram(GBP).Al-Khayyal[ 7 ]providedagenericapproachforsolvingGBPsglobally;thisapproachwasessentiallyanextensiontotheAl-KhayyalandFalk[ 9 ]algorithmwhichwasproposedtosolvejointlyconstrainedbilinearprograms.Al-Khayyal,LarsenandVanVoorhis[ 8 ]furtherextendedtheapplicabilityofthismethodtononconvexquadraticallyconstrainedquadraticprograms.Oursolutionapproachadaptsthebranch-and-boundalgorithmofAl-Khayyal[ 7 ]andalsodrawsonpreviouslydevelopedReformulation-Linearization(RLT)techniques.SheraliandAlameddine[ 84 ]developedabranch-and-boundalgorithmbasedonanRLTforjointlyconstrainedbilinearprograms.AlthoughthelinearrelaxationobtainedfromtheRLTapproachistheoreticallytighterthanthatderivedbyAI-Khayyal'smethod,thisRLTbranch-and-boundapproachcannotbeappliedtoGBPsdirectly.Inourstudy,wedevelopamethodwhichisabletosolveourproblembyintegratingtheRLTbranch-and-boundalgorithmwithinapenalty-function-basedapproach,whichpenalizesviolationsofarelaxedconstraintsetintheobjectivefunction. Theremainderofthischapterisorganizedasfollows.Section 3.2 describesoursupplychainmodel.WepresentthreeapproachesforsolvingtheresultingprobleminSection 3.3 .WersttransformouroriginalmodeltoageneralizedbilinearprogrammingprobleminSections 3.3.1 and 3.3.2 .Section 3.3.3.1 presentsawidelyusedheuristicsolutionmethodpreviouslydevelopedforsolvingbilinearproblems,followedbySections 3.3.3.2 and 3.3.3.3 ,whichpresenttwoexactsolutionmethods.TherstoftheseisamodicationofAl-Khayyal'sapproach[ 7 ]forsolvinggeneralizedbilinearprogramsandthesecondisanewapproachwehavedevelopedforthisproblemclass.InSection 3.4 ,wediscusstheresultsofacomputationalstudyusedtovalidateoursolutionmethodandcompareitwiththeresultsoftheheuristicmethodandAl-Khayyal'sapproach. 53

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3.2ProblemStatementandFormulation Toformalizeourmodel,wedenethefollowingnotation:InputsandParameters i,j:retailstoreindices,i,j=1,...,S.l:periodindex,l=1,...,L.LP:lengthofpromotionperiod,1
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inperiodl,denotedbyxli;(2)theinventorylevelatstoreiattheendofperiodl,denotedbyIli;and(3)thedivertingquantityfromstoreitostorejinperiodl,denotedbyylij,forallperiodsl=1,...,Landallstoresi,j=1,...,S,i6=j.Thesupplierexplicitlyincorporatestheanticipatedreactionsoftheretailerinitsoptimizationprocess,andweassumefullinformationisavailabletothesupplierregardingtheretailer'sinventoryandtransportationcosts.ThiststheclassicalStackelberggameparadigm,wheretheleader,whoisawareofthefollower'sbestresponse,choosesamovethatmaximizesitsownexpectedpayoff. Weassumerandomnessindemandisprice-independentandcanbemodeledinanadditivefashion.Specically,demandisdenedas~dli(!)=dli+lilizli+"li(!),where!isanoutcomebelongingtosomeprobabilityspace,and!inuencesallrandomvariables"li.Notethatthequantitylizliisthepricediscountseenbyconsumersasitistheproductoftheretailer'sdiscountpass-throughrateandthesupplier'stradediscount.Thus,thequantitylilizlicorrespondstotheincreaseindemandatretailstoreiinperiodlasaresultofthediscountpassedthroughtoconsumers. AsnotedinSection 3.1 ,weassumethattheretailermakesitsdecisions(xli,ILi,andylij)afterdemanduncertaintyisresolvedbutbeforedemandoccurs.Oneinterpretationofthisassumptionisthat,fromamarketingperspective,retailersare,bydenition,closertoconsumersthanmanufacturingcompanies,andsoretailerscanmoreeasilyengageinpersonalcontactwithconsumers,gatherinformationonconsumerbehavior,andanticipateconsumerpurchasepatternsintheshortrun.And,becauseretailer-to-supplierleadtimesareoftenveryshort,theretailerhastheluxuryofplacingordersimmediatelybeforedemandoccurs(oreveninresponsetodemandinsomecases,asinamake-to-ordersetting).Moreover,effectivepromotionsareusuallythoseofferedonlyoverashorttime-span,becausefrequentusageofpromotionsdiminishestheireffectiveness[ 26 ].Asaresult,wecanoftenassumethataretailercanaccuratelyforecastconsumerdemandduringpromotionperiodsinthenearfuture. 55

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However,weassumethattheretailerdoesnotshareconsumerinformationwiththesupplierinthisdecentralizedsupplychain.Becausethesupplierdoesnotpossesstheretailer'sspecializedknowledgeoflocalmarkets,thesuppliercannotaccuratelyforecastconsumersdemandatthelocallevel;asaresult,thesuppliermustdetermineitspromotionaldiscountlevels(zlivalues)intheabsenceofpreciselocaldemandinformation.Thus,theasymmetryinthedegreeofdemanduncertaintybetweenthesupplierandretailerisattributabletotheretailer'sabilitytoobtainmorepreciselocaldemandinformation.Fromthesupplier'sperspectiveitiseithertoocostlytoobtainthislocalinformationortheretailerisunwillingtosharethisinformationforcompetitivereasons. Ourmodelcapturestheinteractionbetweenthesupplierandretailchain.Thepurposeofthemodelistodeterminetheoptimalpromotionplanforthesupplierwhenanticipatingtheretailchain'susageofforwardbuyinganddivertingstrategies.Tosimplifytheexpositionofthemodelandformodeltractability,weassumethat:1)replenishmentdelaysarenegligible;2)xedcostsarezero(orconstant);3)transhipmentbetweenanytwostorescanbedonewithinoneperiod;and4)theretailer'sdiscountpass-throughratesarexedorpre-determined1.Basedontheabovenotationandmodeldescription,thetradepromotionproblemcanbeformulatedasastochasticbilevelprogramwithbilinearobjectivesatbothdecisionlevels,andwithlinearconstraints.Foragivenrealization!,theorderquantityxli(!),forwardbuyingquantityIli(!),anddivertingquantityylij(!)correspondtoanoptimalsolutiontothelowerlevellinearprogramforgivenupperleveltradepromotionlevelszli,theoptimalvaluesof 1Whilewerecognizethatthepass-throughratesaredecisionvariablessetbytheretailer,weassumexedvaluesinourmodelformodeltractability.Thus,whenusingthismodelfordecisionmaking,thesuppliermustparameterizeonthepass-throughrateinordertodetermine(atleastapproximately)thevaluesthattheretailerislikelytoapply. 56

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whicharedeterminedbymaximizingtheexpectedvalueofnetsupplierprotacrossallpossiblerealizations. Wecannowformulateourstochasticmulti-period,two-stagetradepromotionproblemas:(STP)maxzE"LXl=1SXi=1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli)]TJ /F3 11.955 Tf 11.96 0 Td[(zlixli(!)# (3)s.t.0zlicli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli8i=1,...,S,l=1,...,LP (3)zli=08i=1,...,S,l=LP+1,...,L (3) where(x(!),I(!),y(!))isanoptimalsolutionofthefollowingproblem:minx,y,ILXl=1SXi=1")]TJ /F3 11.955 Tf 5.48 -9.68 Td[(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(zlixli(!)+hliIli(!)+Xj6=itlijylij(!)# (3)s.t.Ili(!)=Il)]TJ /F10 7.97 Tf 6.59 0 Td[(1i(!)+xli(!)+Xk6=iyl)]TJ /F10 7.97 Tf 6.58 0 Td[(1ki(!))]TJ /F6 11.955 Tf 12.48 2.66 Td[(~dli(!))]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xj6=iylij(!)8i=1,...,S,l=1,...,L (3)~dli(!)=dli+lilizli+"li(!)8i=1,...,S,l=1,...,L (3)xli(!)0,Ili(!)08i=1,...,S,l=1,...,Lylij(!)08i=1,...,S,i6=j,l=1,...,LI0i(!)=08i=1,...,S, where(!)isarandomvectorconsistingofallrandomcomponents"li(!).Theupperlevelobjective( 3 )istomaximizeexpectednetprotandisexpressedasthedifferencebetweenthesumofrevenuesarisingfromwholesalepricing(cli)]TJ /F3 11.955 Tf 12.68 0 Td[(zli)andthesumofvariablecosts.Theupperlevelconstraints( 3 )statethatthewholesalepricediscountszlishouldbenonnegativeandlessthanorequaltotheperunitprotmargin,cli)]TJ /F3 11.955 Tf 12.13 0 Td[(mli.Theupperlevelconstraints( 3 )statethat,afterthepromotionperiod,thesupplierstopsofferingatradepromotiontoallretailstores,andweaddthissetofconstraintstoourmodelsothatitisabletocapturethecarry-overeffect(forward 57

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buying,forexample)ofdecisionsinpromotionperiods.Theobjectiveofthelowerlevelproblem( 3 )istominimizetheretailer'stotalcostofordering,holdinginventoryanddiverting.Inourstudy,sincediscountpass-throughratesarexed,itfollowsthattheretailpricesareindirectlydeterminedbythesupplier.Asaresult,forthelowerlevelproblem,minimizingcostisequivalenttomaximizingprot.Therstsetoflowerlevelconstraints( 3 )requiresthatendinginventoryatstoreiinperiodlequalstheendinginventoryatstoreiinperiodl)]TJ /F6 11.955 Tf 12.3 0 Td[(1,plustheamountorderedfromthesupplieratstoreiinperiodl,plustheamountshippedfromotherstorestostoreiinperiodl)]TJ /F6 11.955 Tf 12.44 0 Td[(1,minusthedemandatstoreiinperiodl,minustheamountshippedfromstoreitootherstoresinperiodl.Thesecondsetofconstraints( 3 )modelsconsumerdemandasalinearfunctionofthewholesalepricediscount.Theremainingconstraintsindicatethatallvariablesarereal-valuedandnonnegative. Notethatforarealization!andspecicpromotionlevels,theobjectiveofthelowerlevelproblemisconvex(butnotstrictlyconvex),whichimpliesthesolutiontothelowerlevelproblemmaynotnecessarilybeunique.Asaresult,weassumethatgiventhechoicebetweensolutionstothelowerlevelproblemwithequalcost,thesolutionselectedistheoneyieldingthehighestexpectedprotforthesupplier.Onemayalternativelyapplyaworst-caseapproachfromthesupplier'sperspective,assumingthattheretailerchoosesthesolutionyieldingthelowestexpectedprotforthesupplier. TherearetwodifcultiesinherentinsolvingSTP.First,foreachoutcomeofthedemandrealization,theresultingproblemisabilevelproblemwithbilinearobjectivesatbothlevelsandwithlinearconstraints,whichfallsintheclassofNP-hardbilinearlyconstrainedbilinearprograms(orgeneralizedbilinearprograms).Theseconddifcultyliesinthecomputationalburdenofcomputingtheexpectedprotfunction. Inthenextsection,weproposeathreestepmethodologytosolvetheSTPproblemglobally.First,thestochastictradepromotionproblemisconvertedtoanequivalentdeterministicproblem.Then,wetransformtheresultingdeterministicbilevelproblem 58

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intoasingle-levelproblem.Finally,weadaptabranch-and-boundalgorithmbasedonaReformulation-LinearizationTechnique(RLT)forsolvingtheresultingsinglelevelgeneralizedbilinearprogram. 3.3SolutionProceduresforSTP 3.3.1DeterministicEquivalent InordertoanalyzetheSTP,werstconstructasupplychaintime-expandednetworkG=fN,Agrepresentingthetwo-echelon,multi-periodproblem,whereN=f(i,l):i=1,...,S,l=1,...,Lg[f(0,0)gisthesetofnodesandAisthesetofarcs.Node(0,0)representsthesupplier,whilenode(i,l)correspondstoretailstoreiinperiodl.ThesetNl=f(1,l),(2,l),...,(S,l)gNcorrespondstothesetofnodesassociatedwithretailstoresinperiodl.ThreetypesofarcsarecontainedinthenetworkG:(1)((0,0),(i,l)),8(i,l)2N)-272(f(0,0)g,witharccostcli)]TJ /F3 11.955 Tf 12.56 0 Td[(zliperunitofow(orderquantity)fromthesuppliertonode(i,l);(2)((i,l),(i,l+1)),8(i,l)2N)]TJ /F3 11.955 Tf 12.19 0 Td[(NL)-241(f(0,0)gwithaunitcostofhlicorrespondingtotheinventorycarriedfromnode(i,l)to(i,l+1);and(3)((i,l),(j,l+1)),8i,j=1,...,S,l=1,...,L)]TJ /F6 11.955 Tf 12.45 0 Td[(1,i6=jwithaunitcostoftlijfortransshipmentfromnode(i,l)tonode(j,l+1). FromtheconstructionofgraphG,wemightviewthelowerlevelproblemofSTPasaminimumcostowproblem,whichrequiressendingdliunitsofowascheaplyaspossiblefromnode(0,0)toeachnode(i,l)inthesetN)-323(f(0,0)ginanuncapacitatednetwork.ObservethataminimumcostowproblemwithnoarccapacitiescanbedecomposedintoasetofSLshortestpathproblemsthatareindependentofthedemandlevels(Appendix C forademonstrationofthevalidityofthisdecomposition).ThisobservationimpliesthatthelowerlevelproblemofSTPisequivalenttoasetofdeterministicproblemsthatareindependentoftherandomterms.Basedonthisobservation,itisnothardtoseethatfortheentireproblem,thestochasticcomponentsonlyappearintheobjectivefunctionofthetop-levelproblem.Aftertakingtheexpectationoftheobjectivefunction,theSTPisthusadeterministicproblem,and 59

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thedistributionsofthestochasticcomponentsdonotimpacttheoptimizationproblemformulation.Wenextdiscusshowtotransformthisdeterministicequivalentforthetwo-levelproblemintoasingle-leveloptimizationproblem. 3.3.2Single-levelProblem Thedeterministictradepromotionproblem(DTP)isderiveddirectlyfromtheSTPproblembyreplacingtherandomtermwithitsexpectation.TheresultsofSection 3.3.1 implythatDTPisequivalenttoSTP.NotethatthelowerlevelproblemcanbewrittenwithoutholdingcostsbymakingsubstitutionIli=Pl=1[xi+Pk6=iy)]TJ /F10 7.97 Tf 6.59 0 Td[(1ki)]TJ /F12 11.955 Tf 11.9 8.97 Td[(Pj6=iyij)]TJ /F6 11.955 Tf 11.89 0 Td[((di+iizi)];thentheDTPproblemcanbeformulatedas:(DTP)maxx,y,zLXl=1SXi=1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli)]TJ /F3 11.955 Tf 11.96 0 Td[(zlixlis.t.0zlicli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli8i=1,...,S,l=1,...,LPzli=08i=1,...,S,l=LP+1,...,L, where(x,y)isanoptimalsolutionofminx,yLXl=1SXi=1")]TJ /F6 11.955 Tf 5.66 -9.69 Td[(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(zlixli+Xj6=itlijylij)]TJ /F6 11.955 Tf 12.06 2.66 Td[(hli)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(dli+lilizli#s.t.lX=1(xi+Xk6=iy)]TJ /F10 7.97 Tf 6.58 0 Td[(1ki)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xj6=iyij)lX=1(di+iizi)8i=1,...,S,l=1,...,L (3)xli08i=1,...,S,l=1,...,Lylij08i=1,...,S,i6=j,l=1,...,L, wherehli=PL=lhi,cli=cli+hliandtlij=tlij+hl+1j)]TJ /F6 11.955 Tf 12.78 2.66 Td[(hli.Foranyxedvalueofzli,thelowerlevelproblemisalinearprogram,soanyoptimalsolutionofthelowerlevelproblemsatisesthestrongdualityproperty.Asaresult,wecanreplacethelowerlevelprogrambyitsprimal-dualoptimalityconditions,whereisthevectorofdualvariables 60

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associatedwiththesetofconstraints( 3 ).Letusrstdene:F8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:(x,y,z,):lP=1(xi+Pk6=iy)]TJ /F10 7.97 Tf 6.58 0 Td[(1ki)]TJ /F12 11.955 Tf 11.96 8.97 Td[(Pj6=iyij)]TJ /F4 11.955 Tf 11.96 0 Td[(iizi)lP=1di,8i=1,...,S,l=1,...,LLP=li+zlicli,8i=1,...,S,l=1,...,LPL=l+1j)]TJ /F12 11.955 Tf 11.96 8.96 Td[(PL=litlij,8i=1,...,S,i6=j,l=1,...,L9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>; and8>>>>>>>>>><>>>>>>>>>>:(x,y,z,):zli0andcli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)]TJ /F3 11.955 Tf 11.95 0 Td[(zli0,8i=1,...,S,l=1,...,LPzli=0,8i=1,...,S,l=LP,...,Lxli0andUl)]TJ /F3 11.955 Tf 11.96 0 Td[(xli0,8i=1,...,S,l=1,...,Lylij0andUl)]TJ /F3 11.955 Tf 11.96 0 Td[(ylij,0,8i=1,...,S,i6=j,l=1,...,Lli0andcli)]TJ /F4 11.955 Tf 11.96 0 Td[(li0,8i=1,...,S,l=1,...,L9>>>>>>>>>>=>>>>>>>>>>;, whereUl=PL=lPSi=1(di+iimi)andisahyper-rectangle.UsingtheabovedenitionswecanformulatetheDTPasfollows:(SDTP())maxx,y,z,(x,y,z,)=LXl=1SXi=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)]TJ /F3 11.955 Tf 11.95 0 Td[(zlixli (3)s.t.LXl=1SXi=1")]TJ /F6 11.955 Tf 5.66 -9.69 Td[(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(zlixli+Xj6=itlijylij#=LXl=1SXi=1LX=li)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(dli+lilizli (3)(x,y,z,)2F\. Theaboveproblemmaximizesabilinearobjectivefunction( 3 )overafeasibleregiondenedbyabilinearconstraint( 3 )andasetoflinearconstraints.Thisproblemthusfallswithintheclassofgeneralizedbilinearprograms(GBPs),andsotheSDTPreducestoalinearprogramwhenevereitherzor(x,y,)arexed.However,theobjectivefunction( 3 )andthebilinearconstraint( 3 )arenonconvexfunctions.SolvingaGBP 61

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isNP-hard[ 79 ],and,tothebestofourknowledge,nomethodsexistthatguaranteeconvergencetoanexactsolutioninanitenumberofsteps.Inthefollowingsection,wewillpresentsomeclassicaltechniquesforsolvingtheSDTPproblem,aswellasapenalty-basedmethodthatwehavedevelopedforsolvingtheproblem. 3.3.3Linearization 3.3.3.1Successivelinearprogrammingapproach Successivelinearprogrammingisacommonlyusedheuristicmethodforsolvingbilinearprogrammingproblems.Thisprocedureiteratesbetweenxingthesupplier'spricediscountszandtheretailer'sprimalanddualvariables(x,y,)forsolvingSDTP.Atagiveniterationk,werstndthevaluesof(xk,yk,k)thatoptimizetheobjectivefunctionforaxedzk)]TJ /F10 7.97 Tf 6.58 0 Td[(1,andthenndthevectorzkthatoptimizestheobjectivefunctionforxedvaluesof(xk,yk,k).Werepeatthisprocedureuntiltheobjectivedoesnotimprovebetweentwosuccessiveiterations. Theclassicalbilinearprogramisaclassofquadraticprogramswiththefollowingstructure:maxx,ycTx+xTAy+dTys.t.x2X:=fx:B1xb1,x0gy2Y:=fy:B2yb2,y0g. Iftheabovebilinearprogramhasaniteoptimalsolution,thenthereexistsanextremepointx2Xandanextremepointy2Ysuchthat(x,y)isanoptimalsolutionoftheclassicalbilinearprogram[ 96 ].Sincethefeasibleregionisdenedbytwoseparablepolyhedralsets,theclassicalbilinearprogramisalsocalledabilinearprogramwithseparableconstraints.SheraliandShetty[ 85 ]showedthat,foraclassicalbilinearprogram,thelimitpointofthesuccessivelinearprogrammingapproachisalocallyoptimalsolution.However,SDTPisaGBPproblem,whichdoesnothaveseparable 62

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constraintsinthebilinearterms.Therefore,wehavenoguaranteesonsolutionqualityforasolutionobtainedusingsuccessivelinearprogrammingapproach. 3.3.3.2Al-Khayyal'sapproach Al-KhayyalandFalk[ 9 ]developedaninnitelyconvergentbranchandboundalgorithmforjointlyconstrainedbilinearprograms(JCBPs)usinglowerboundsderivedfromconvexenvelopesofthebilinearterms.Al-Khayyal[ 7 ]foundthatthesameapproachcanalsobeappliedforGBPs. Therststepoftheapproachistoobtaintheconcaveoverestimateoftheobjectivefunction( 3 )over.Piecingtogetherallthevariables,weobtainavector(x,y,z,,,,),withuptoN=(5+S+L)(SL)components,andtheconcaveoverestimateof( 3 )overcanberepresentedas: ()=LXl=1SXi=1(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)xli)]TJ /F4 11.955 Tf 11.95 0 Td[(lis.t.2F1(), (3) whereF1()8><>::li08i=1,...,S,l=1,...,Lli(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli)xli+Ulzli)]TJ /F6 11.955 Tf 11.96 0 Td[((cli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli)Ul8i=1,...,S,l=1,...,L9>=>;. Thesecondstepoftheapproachistoobtainapolyhedralapproximationoftheconvexhulloftheregiondenedbyconstraint( 3 ). Bytheweakdualitytheorem,whenever(x,y)andarefeasiblefortheprimalanddualproblems,respectively,theleft-hand-sideofequation( 3 )isalwaysgreaterthanorequaltoitsright-hand-side.Asaresult,replacingconstraint( 3 )withthefollowingconstraintdoesnotchangethefeasibleregionoftheSDTP: LXl=1SXi=1(clixli+Xj6=itlijylij)]TJ /F5 7.97 Tf 19.26 14.94 Td[(lX=1dili)]TJ /F3 11.955 Tf 11.95 0 Td[(zlixli)]TJ /F5 7.97 Tf 19.26 14.94 Td[(lX=1iizili)0.(3) 63

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Next,denethepolyhedralsetF2()8>>>>>>>>>><>>>>>>>>>>::PSi=1PLl=1(li+Pl=1li)0(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(cli+mli)xli+Pj6=itlijylijli8i=1,...,S,l=1,...,Lclixli+Pj6=itlijylij)]TJ /F3 11.955 Tf 11.96 0 Td[(Ulzlili8i=1,...,S,l=1,...,L)]TJ /F6 11.955 Tf 9.3 0 Td[([di+ii(ci)]TJ /F3 11.955 Tf 11.96 0 Td[(mi)]lili8i=1,...,S,l=1,...,L,l)]TJ /F3 11.955 Tf 9.3 0 Td[(dili)]TJ /F4 11.955 Tf 11.95 0 Td[(iiclizili8i=1,...,S,l=1,...,L,l9>>>>>>>>>>=>>>>>>>>>>;. Al-Khayyal[ 7 ]showedthatforany(x,y,z,)satisfyingconstraint( 3 ),thereexists2F2(). Fromtheresultsabove,wehavethefollowingtwoobservations:i)8(x,y,z,)2F\and2F\\F1(),wehave(x,y,z,) ();andii)foranyfeasiblesolution(x,y,z,)toSDTP,thereexists2F\\F1()\F2().Consequently,wehavethefollowingconvexprogramforapproximatingtheSDTP:(LP())max ()=LXl=1SXi=1[(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)xli)]TJ /F4 11.955 Tf 11.96 0 Td[(li]s.t.2F\\F1()\F2(). TosolveSDTPglobally,wecanimplementabranch-and-boundalgorithmwhichisproventoconvergetoaglobaloptimalsolutionbasedontheaboveapproximationscheme,wherepartitioningisperformedbydecomposingintosub-hyper-rectangles.Anoutlineofthealgorithmisasfollows: InitializationStep:TheinitialproblemistheproblemLP().Initialize(1,1)=andletT1=f(1,1)gbetheindexsetofasinglenodeatiterationoneofthebranch-and-boundtree.LetUB(1,1)=1betheupperboundassociatedwithnode(1,1).LetLB=andUB=1betheinitiallowerandupperboundsoftheproblem.Setk=1,andgototheMainStep. MainStep:Atiterationk,selectanode(u,v)fromTkandremovethisnodefromTk.SolveLP((u,v))toobtainthepartialsolution(x,y,z,).If(x,y,z,)satisestheconstraint( 3 )and(x,y,z,)= (),thenthealgorithmterminateswith(x,y,z,)asanoptimalsolutiontoSDTP().Otherwise,therearetwopossiblecases:i)if(x,y,z,)satisesconstraint( 3 )but(x,y,z,)< (),thenlet 64

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LBk=(x,y,z,)bethecurrentiterationlowerbound,andsetthepartitioningindex(p,q)asfollows:(p,q)=argmax(i,l)fzlixli)]TJ /F6 11.955 Tf 12.57 0 Td[(lig; elseii)if(x,y,z,)doesnotsatisfyconstraint( 3 ),thenletLBk=bethecurrentiterationlowerbound,andsetthepartitioningindex(p,q)asfollows:(p,q)=argmax(i,l)(max"clixli+Xj6=itlijylij)]TJ /F6 11.955 Tf 12.1 0 Td[(zlixli)]TJ /F6 11.955 Tf 12.81 2.65 Td[(li,)]TJ /F5 7.97 Tf 17.52 14.94 Td[(LX=l(dlii+lilizlii+li)#). Afternding(p,q),partitiontheregion(u,v)intotwomutuallyexclusiveandexhaustivesubregions.First,noticethat(u,v)isahyper-rectanglewhichcanbeexpressedasfollows:(u,v)(x,y,z,)2:ZLlizliZUli8i=1,...,S,l=1,...,LP, whereZLliandZUliarethelowerandupperboundsofthecomponentzli.Usingtheabovenotation,thetwonewsubregionscanberepresentedasfollows:(k+1,1)=(u,v)\fZLqpzqpzqpg(k+1,2)=(u,v)\fzqpzqpZUqpg. SetUB(k+1,1)=UB(k+1,2)= ().ThenaddthesetwonodestothesetTkandupdatethetree,ifnecessary.Setk=k+1,andgotothenextiteration. Thedetailedbranch-and-boundalgorithmanditsupdatingoperationsaredescribedintheAppendix D Al-KhayyalandFalk[ 9 ]showedthatthisalgorithmconvergestoagloballyoptimalsolutionforaGBP;however,forourproblem,theconvergencerateforthisalgorithmcanbedisappointing,evenforsmall-sizedinstances.Inthenextsection,wethereforedevelopaspecicpenalty-basedmethodforsolvingtheSDTP. 3.3.3.3Penalty-basedmethod Wenextdiscussacustomizedpenalty-basedmethodforsolvingtheSDTPbyexploitingapropertyofconstraint( 3 ).ObservethattheSDTPisajointlyconstrainedbilinearprogramwithoutconstraint( 3 ),andjointlyconstrainedbilinearprogramscanbesolvedusingasuitableReformulation-LinearizationTechnique(RLT). 65

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Asaresultoftheaboveobservation,therststepofourmethodistoobtainarelaxationoftheSDTPbyeliminatingconstraint( 3 )fromtheconstraintsetandpenalizingviolationsofthisconstraintintheobjectivefunction( 3 ).Sincetheleft-handsideof( 3 )isalwaysgreaterthanorequaltotheright-handside,penalizingviolationsofconstraint( 3 )yieldsthefollowingrelaxedproblem:maxx,y,z,LXl=1SXi=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli)]TJ /F3 11.955 Tf 11.96 0 Td[(zlixli)]TJ /F3 11.955 Tf 11.96 0 Td[(M(LXl=1SXi=1")]TJ /F6 11.955 Tf 5.66 -9.68 Td[(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(zlixli+Xj6=itlijylij)]TJ /F5 7.97 Tf 18.18 14.94 Td[(LX=li)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(dli+lilizli#)s.t.(x,y,z,)2F\. whereMisasufcientlylargepositivenumber(whichcorrespondstoapenaltyperunitofviolationoftheconstraint).Byrewritingtheobjective,weobtainthefollowingequivalentformulation:(PEN)maxx,y,z,LXl=1SXi=1((cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)]TJ /F3 11.955 Tf 11.95 0 Td[(Mcli)xli)]TJ /F3 11.955 Tf 11.96 0 Td[(MXj6=itlijylij+MlX=1dili+(M)]TJ /F6 11.955 Tf 11.95 0 Td[(1)zlixli+MLX=lliliizli)s.t.(x,y,z,)2F\. Theaboveformulationisajointlyconstrainedbilinearprogram.SheraliandAlameddine[ 84 ]developedanRLTforthisproblemclassandembeddeditwithinaprovablyconvergentbranch-and-boundalgorithm.SheraliandAlameddine'sRLTreformulatesthebilinearprogrambyrstconstructingvalidnonlinearinequalitiesfromtheoriginalconstraintsdeningF\.Thefollowingaretwogeneralmethodsforgeneratingtheseadditionalnonlinearinequalities: Multiplyinganytwoconstraintsinpairwise,e.g.,(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)]TJ /F3 11.955 Tf 11.95 0 Td[(zli)(Uk)]TJ /F3 11.955 Tf 11.95 0 Td[(xkj)0;and MultiplyingaboundingconstraintinwithaconstraintinF,e.g.,(cli)]TJ /F3 11.955 Tf 12.95 0 Td[(mli)]TJ /F3 11.955 Tf -394.36 -15.45 Td[(zli)(PL=kj+zkj)]TJ /F6 11.955 Tf 12.13 0 Td[(ckj)0. DeningthesetofconstraintsgeneratedusingtheabovepairwiseproductoperationsasF(),thentheoriginalPENproblemconstraintstogetherwiththeconstraintsinF() 66

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yieldanewequivalentformulationoftheproblemPEN,whichwedenoteasPEN':(PEN')maxx,y,z,LXl=1SXi=1()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(cli)]TJ /F3 11.955 Tf 11.95 0 Td[(mli)]TJ /F3 11.955 Tf 11.96 0 Td[(Mclixli)]TJ /F3 11.955 Tf 11.95 0 Td[(MXj6=itlijylij+MlX=1dili+(M)]TJ /F6 11.955 Tf 11.96 0 Td[(1)zlixli+MLX=lliliizli)s.t.(x,y,z,)2F\\F(). AllofthenonlineartermsofthePEN'formulationarebilinearand,asaresult,PEN'canbelinearizedthroughanappropriatevariablesubstitutionstrategy,whichtransformsthenonlinearconstraintsofthesetF()toasetoflinearconstraints.Forexample,wesubstitute:kli=xkizli8i=1,...,S,k,l=1,...,L,kli=kizli8i=1,...,S,k,l=1,...,L. Letrepresentthevectorcontainingallsuchnewvariablescreatedotherthanand,andletFl()representthelinearizedsetofconstraintsfromF().ThisleadstothefollowingreformulationofPEN':maxLXl=1SXi=1()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)]TJ /F3 11.955 Tf 11.95 0 Td[(Mclixli)]TJ /F3 11.955 Tf 11.96 0 Td[(MXj6=itlijylij+MlX=1dili+(M)]TJ /F6 11.955 Tf 11.95 0 Td[(1)lli+MLX=llilili)s.t.(x,y,z,,,,)2F\Fl()\. Notethatafterlinearization,theresultingproblemisarelaxationofthePEN'problem,whichcorrespondstoanupperboundinglinearprogramfortheoriginalbilinearprogram;SheraliandAlameddine[ 84 ]showedthattheresultingupperboundisatleastasgoodasthatobtainedusingAl-Khayyal'sapproach.Theresultingbranch-and-boundalgorithmweuseisverysimilartoAlgorithm 2 ofAl-Khayyal'sapproach.Thereisonlyonedifference:thebranch-and-boundalgorithminourmethoddoesnotneedtocheckwhetherthepartialsolution(x,y,z,)isfeasible,sothepartitioningindex(p,q)canbe 67

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foundasfollows:(p,q)=argmax(i,l)(max"lli)]TJ /F3 11.955 Tf 11.96 0 Td[(zlixlk,LX=llili(li)]TJ /F4 11.955 Tf 11.95 0 Td[(izli)#). Theremainderofourbranch-and-boundalgorithmusesthesameupdatingoperationsandstoppingcriteriaasAlgorithm 2 fromAl-Khayyal[ 7 ]. Fromtheabovedescription,wederiveaprocedureforsolvingtheSDTPusingthepenalty-basedmethodandRLTasshowninAlgorithm 1 Algorithm1Penalty-basedapproach 1: M 0,UB 1andLB ; 2: whileUB)]TJ /F3 11.955 Tf 11.95 0 Td[(LB>do 3: (x',y',z',') anoptimalsolutioncorrespondingtoPEN'; 4: UB theoptimalobjectivevalueofPEN'; 5: (x,y,) anoptimalsolutioncorrespondingtoSDTPproblemforxedvectorsz'; 6: LBk PLl=1PSi=1cli)]TJ /F3 11.955 Tf 11.96 0 Td[(mli)]TJ /F6 11.955 Tf 11.95 0 Td[((z')lixli; 7: ifLB
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3.4NumericalExperiments Thissectionpresentscomputationaltestresultsforourpenalty-basedapproach,theiterativeLPheuristicapproach,andAl-Khayyal'sapproachforsolvingtheSDTP.Wewilldemonstratethatourpenalty-basedapproachoutperformstheothertwosolutionmethodsfromtheliterature.Inadditiontoevaluatingtheperformanceofdifferentsolutionapproaches,wealsoanalyzetheimpactsofdifferentparametersonthebullwhipeffectanditsassociatedcosts,aswellasonthesupplier'snetprotfromwholesalediscounts.WeimplementedallthreeofthesolutionapproachesintheC#programminglanguage,withtherelaxedlinearprogramssolvedusingILOGR'sCPLEXR12.5solverwithConcertTechnology.WeperformedalltestsonacomputerwithanIntelRDualCore1.70GHzand6GBmemory. 3.4.1ComparisonofSolutionMethods Tobenchmarktheperformanceofourpenalty-basedapproachwiththeiterativeLPheuristicandAl-Khayyal'sapproach,wetestedoursolutionmethodusingnineproblemsets.Eachproblemsetcorrespondstoaxednumberofretailstoresandnumberoftimeperiods,(S,L),whereS2f2,3,4gandL2f2,3,4g.Wetestedtwentyrandomlygeneratedinstancesforeachcombinationof(S,L)values,foratotalof180probleminstances.Forbothexactsolutionapproaches,wesettherelativeoptimalitytoleranceto10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,andthetimelimitto180seconds. Table 3-1 summarizesthedistributionsusedingeneratingparametersinourcomputationalstudy.Foreachprobleminstance,thesupplier'sunitwholesalepricescli,deterministiccomponentsofconsumerdemanddli,andretailer'spassthroughratesliweregeneratedfromuniformdistributionsrst.Then,basedonthegeneratedvaluesofclianddli,thesupplier'sunitcostsmli,theretailer'sunitinventoryholdingcostshli,theretailer'sunittransportationcoststlijandtheconsumerpromotionalpriceelasticityliweregeneratedbasedoncontinuousuniformdistributions.WeletU(l,u)denotethecontinuousuniformdistributionwithlowerboundlandupperboundu. 69

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Table3-1. Parameterdistributionsusedincomputationaltests. Supplier'sunitwholesaleprice,cliSupplier'sunitcost,mliU(10.00,20.00)cliU(0.50,0.80)Retailer'sunitinventoryholdingcosthliRetailer'sunittransportationcost,tlijcliU(0.01,0.20)jcli)]TJ /F3 11.955 Tf 11.95 0 Td[(cljjU(1.01,1.20)Deterministiccomponentofconsumerdemand,dliRetailer'spass-throughrate,liU(100,500)U(0.30,1.00)Consumerpromotionalpriceelasticity,lidlicli 20U(0.10,0.50) Observethatthelargestsizeinstanceinourcomputationalstudyonlyconsidersfourstores,andinpractice,itisnotcommonthatacompanyoperatesasmallnumberofretailstoresbutstillservesgeographicallydispersedareas.However,retailerslikeWal-MartRusuallyuseaspoke-and-hubstrategy:theyrstmovetheproductsfromsupplierstodistributioncenters,andthenfromdistributioncenterstolocalstores.Byadoptingthisstrategy,retailersonlyneedtooperateasmallnumberofdistributioncenters.Infact,atthestartof2010,40outofthetop75foodretailersinNorthAmericahadnomorethanfourdistributioncentersinU.S.andCanada[ 75 ].Thedistributioncentersareusuallygeographicallydispersedacrossthecountry,andthelocalstoreorderswillbeaggregatedatthedistributioncenter.Moreover,retailstoresservedbythesamedistributioncenterwillhavethesamewholesaleprice,andasaresult,transshipmentwillonlyoccurbetweenthedistributioncenters.Ourchoiceofparametersmatchwellwiththistypeofspoke-and-hubstrategy,andtheonlychangethatneedstobemadeisinusingdistributioncentersinsteadofstoresinourmodel.Inaddition,thereasonweselectedthenumberofplanningperiodstobenomorethanfourisbecauseweassumetheretailercanmakeaccuratedemandforecastsovertheplanninghorizon;asaresultthenumberofperiodsintheplanninghorizoncannotbetoolarge,assumingthateachtimeperiodrepresentsoneortwoweeks,forexample. 70

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Foreachprobleminstance,inadditiontothepricepromotiongame,wealsoconsiderano-discountcaseandabasicpromotioncase.Intheno-discountcase,thesupplierdoesnotofferanypricediscounttotheretailerovertheentireplanninghorizon;inthiscasetheretailer'soptimalprotisPR0andthesupplier'soptimalprotisPM0.Inthebasicpromotioncase,thesuppliersetsitsdiscountpolicybyassumingtheretailerwillpasstheentirediscountontoitsconsumersandwillneitherforwardbuynordivert(when,infact,theretailerwillminimizeitscostbyapplyingbothforwardbuyinganddivertingstrategies).Forbothpromotioncases,wedenetheretailer'soptimalprotandthesupplier'sprotasPR1andPM1,respectively.Theperformancemeasureswewilluseforcomparativepurposesincludethebullwhipeffect,BWE=q Var[x] Var[d],theretailer'sprotgain,r=PR1)]TJ /F5 7.97 Tf 6.59 0 Td[(PR0 PR0100%,thesupplier'sprotgain,m=PM1)]TJ /F5 7.97 Tf 6.58 0 Td[(PM0 PS0100%,andthetotalsystemprotgain,=PS1+PR1)]TJ /F5 7.97 Tf 6.58 0 Td[(PS0)]TJ /F5 7.97 Tf 6.59 0 Td[(PR0 PS0+PR0100%. TocompareouralgorithmwithAl-Khayyal'sapproach,weconsidertherunningtimeandrelativeoptimalityperformance.Theresultsofourtests,averagedoverthetwentyrandomprobleminstancesforeachcombinationof(S,L)values,arepresentedinTable 3-2 .Thetableshowsthatforsmaller-sizeproblemswith(S,L)2f(2,2),(2,3),(2,4),(3,2),(3,3),(4,2)g,ourapproachonaveragetakeslessthan150secondstoreducetherelativeoptimalitygapunder3%.Forthesamesetoftheproblems,exceptforproblemswith(S,L)=(2,2),theAl-Khayyal'sapproachcouldnotsolvetheproblemsgloballywithinthe180secondtimelimit,andtheaveragerelativeoptimalitygapisunacceptablylarge(72.55%).Forlarge-sizeproblemswith(S,L)2f(3,4),(4,3),(4,4)g,neitherapproachwasabletoreducetherelativeoptimalitytoleranceunder5%within180seconds.However,thesolutionsgivenbyourapproachhavemuchsmallerrelativeoptimalitygapswhencomparedwiththesolutionsobtainedusingAl-Khayyal'sapproach. WealsotestedeachprobleminstanceusingthecommercialnonlinearprogrammingsolversGAMS/LINDOGlobalandGAMS/BARON.Bothofthesesolversonlyguarantee 71

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Table3-2. ComputationalresultsI. ProblemsetAl-KhayyalPenaltymethod SLRelativeTimeRelativetimegap(%)gap(%) 224.46147.540.9128.52349.53180.001.5759.85478.36180.002.33123.743217.02180.001.3454.333114.41180.002.58141.484126.61180.005.29180.0042103.45180.002.38125.053126.89180.005.35180.004134.02180.008.78180.0052119.21180.002.94130.91 localoptimalityoftheirsolutions,andtheyfailedtoobtainmeaningfulupperboundsfortheproblemwith(S,L)2f(3,4),(4,3),(4,4)gforallinstances.Fortheotherproblemswithameaningfulupperbound,theaveragerelativegapismorethan4%andthemaximumrelativegapis9.3%. Inadditiontorelativeoptimalitygaps,wealsoconsiderthebestfeasiblesolutionobtainedasanotherperformancecriterion.Asdenedabove,PM0isthesupplier'snetprotforthecaseofnodiscount,andPM1isthesupplier'snetprotforthepromotioncases.Thequantitym=PM1)]TJ /F5 7.97 Tf 6.59 0 Td[(PM0 PS0100%measurestheperformanceofthepromotioncasecomparedwiththenodiscountcase.Ifm>0,thismeansthecorrespondingpromotioncaseismoreprotablethanthenodiscountcase;otherwise,thenodiscountcaseisabetteroptionforthesupplier.Moreover,alargervalueofmmeansagreaterlevelofprotforthecorrespondingpromotionplan.FromTable 3-3 ,weobservethatourapproachfoundbetterfeasiblesolutionsthanAl-Khayyal'sapproachonaverage,andthegeneralpromotiongamecaseoutperformsthebasicpromotioncaseandthecaseofnodiscountonaverage.However,thebasicpromotioncaseisnotnecessarilymoreprotablethanthenodiscountcase.AnotherinterestingobservationisthatthebestfeasiblesolutionobtainedfromourapproachhasasmallervalueofBWEthaninthebasicpromotioncaseonaverage,whichisconsistentwith 72

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Table3-3. ComputationalresultsII. ProbleminstanceBaseAl-KhayyalPenaltymethod SLm(%)BWEm(%)m(%)BWE 22-11.018.013.1210.104.353-5.586.85-7.945.925.164-8.266.32-14.254.233.4032-8.105.482.045.054.923-8.127.87-19.694.295.914-8.997.33-15.263.984.8342-8.824.61-20.633.123.993-5.836.46-20.153.304.244-6.976.97-19.013.035.25 thendingsofLee,Padmanabhan,andWhang[ 64 ]thatthebullwhipeffectimpairsupstreamperformance.Ontheotherhand,theBWEonaverageisgreaterthanoneforthebestsolutionobtainedfromourapproach,whichindicatesthattherevenuegainfrompricepromotionscancompensatefortheextracostinducedbythebullwhipeffectifthesuppliertakestheretailer'sreactionsintoconsiderationandjudiciouslyappliesatradepromotionstrategy. Totesttheperformanceofthesuccessivelinearprogrammingheuristic,wesettheinitialvalueofthepromotionvectorzto0.Wecontinuedtouse180secondsasthetimelimit.Iftheimprovementbetweentwosuccessiveiterationsislessthan10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,westoptheheuristicevenifitisbeforereachingthetimelimit.Theresultsderivedfromtheheuristicareverydisappointing:evenwhenthereisanobvioussolutionbetterthanthenodiscountcase(forexample,thebasicpromotioncaseusuallygivesabettersolutionthannodiscountcase),theheuristicalwaysstopsatz=0aftertwoiterations.Thereasonforthisisasfollows:assumingthat(x0,y0,0)optimizestheobjectivefunctionfortheinitialvalueofz,thenthevalueofzthatoptimizestheobjectivefunctionforthexed(x0,y0,0)isstillavectorofzeroesbecause,aswecanobservefromtheobjectivefunction,foraxedvalueofx,itisalwaysoptimaltosetzto0whenthisisfeasible. 73

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3.4.2ParameterAnalysis Thegoalofthissectionistostudyhowthepass-throughrate,,andconsumerpromotionalpriceelasticity,,inuencetheprotperformance(,m,r)andtheBWEforthecaseof(S,L)=(3,3).Tothisend,weconsideredtenlevelsofandthreelevelsofwhenappliedtothreerandomlygeneratedprobleminstances,foratotalof90additionaltestcases.Inthissection,weconsidertheresultsobtainedbysolvingeachofthesetestprobleminstancesusingourpenalty-basedapproach,becauseofitsabilitytoconsistentlyobtainasuperiorfeasiblesolutionforSDTPforthepreviouslytestedinstances. Figures 3-1 3-4 illustratetheresultsoftheseexperiments.Theresultsshownintheguresleadtothefollowingobservations. Figures 3-1 3-3 showthat,forallprobleminstances,ahighervalueofimplieshighervaluesofm,rand.Thiseffectisquiteintuitive,becausewhenconsumersaremoreresponsivetopricereductions,weexpectthattheperformanceofboththesupplierandtheretailerwillimproveasaresult. Figure 3-1 showsthat,forallprobleminstances,alargervalueofthepass-throughrate,implieslargervaluesofm.Thisobservationisconsistentwiththefactthatalowpass-throughratefortradepromotionsisamajorcauseofinefciencyintradepromotions. InFigure 3-2 ,forprobleminstances1and2,theretailer'sprotgaindecreasesasthepass-throughrateincreaseswhenthelevelofislow.So,whentheretailpricecutdoesnotattractasufcientnumberofadditionalconsumers,theretailerhasincentivetolowerthepass-throughratetogainmoreprot.However,whenthelevelofishigh,theretailer'soptimalpass-throughratesareusuallynon-zero,andsometimesthebestdecisionfortheretaileristopassthroughmorethan100%ofthesupplier'stradepromotiontoconsumers,asshownforinstance3. 74

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Figure 3-3 showsthatwhenthevalueofissufcientlylarge,mayinitiallydecreasein(orremainunaffected),andthen,atsomepositivevalueof,beginincreasingin.However,whenthelevelofislow,alwaysdecreasesinforinstances1and2.Thus,fromasystemperspective,thesupplier'schoiceofwhetherornottoofferadiscountdependsonthevalueof. Figure 3-4 showsaninterestingpatternwithrespecttothebullwhipeffect.Forallinstances,thebullwhipeffectmayinitiallyincreaseorremainunchangedin;itthenjumpstoamuchhighervalueatsomepositivevalueof,afterwhichitultimatelydecreasesin.Weobservethatthejumpinthebullwhipeffectoccursatthesamepointatwhichtheincreaseinrsuddenlybecomeslarge.Wealsondthatthelargerthevalueof,theearlierthisjumpoccurs.Fromournumericaltests,wefoundthattherearetypicallytwotypesofpromotionplanpatternsforeachprobleminstance,whichwewillcallPlanLandPlanH,respectively.PricediscountsforPlanHaremuchdeeperthanpricediscountsforPlanL.However,whenthepass-throughrateislow,itisoptimalforthesuppliertoadoptPlanLinsteadofPlanH.Asthevalueofincreases,afterreachingacertainpositivevalueof,theoptimalpromotionplanforthesupplierchangesfromtheformofPlanLtoPlanH,andwecallthisthethresholdvalueof.Whenthevalueofislarge,thismeanstheconsumersaremoresensitivetoretailpricediscounts,so,forthesamelevelofretaildiscount,ahighervalueofmeansmorenewconsumers,whichresultsinasmallerthresholdvalueof.BecausethenumberofnewconsumersattractedbyPlanHisusuallymuchlargerthanthatofPlanL,PlanHtypicallybringsalargersystemprotandahighervalueofthebullwhipeffect. 75

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Figure3-1. Supplierprotgaininwithatdifferentlevels. Figure3-2. Retailerprotgaininwithatdifferentlevels. 76

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Figure3-3. Totalsystemprotgaininwithatdifferentlevels. Figure3-4. Bullwhipeffectinwithatdifferentlevels. 77

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CHAPTER4SALESPROMOTIONSANDASSORTMENTPLANNINGINMULTIPLESELLINGCHANNELS 4.1ProblemDescriptionandLiteratureSurvey U.S.B2Ce-commercehasbeenboominginrecentyears,andthetotalsalesgrewfrom72billionU.S.dollarsin2002to256billionU.S.dollarsin2011[ 63 ].Inthisprosperoustimeofonlineshopping,asocalledclicks-and-mortarbusinessmodelhasemerged.Inthisbusinessmodel,theonlineretailchannelisintegratedwiththetraditionalretailchannel.Bothtraditionalretailersandpreviouspure-playInternetretailershavecontributedtothedevelopmentoftheclicks-and-mortarbusinessmodel.Ontheonehand,traditionalretailerslikeWal-MartRandTargetRhavetolimittheirassortmentonlytopopularitemsatthestoresbecauseofthehighnumberofproductscompetingforthelimitedstorespace[ 43 ].However,MittelstaedtandStassen[ 73 ]showedthatsuchreductionsinassortmentencouragevariety-seekingshopperstovisitotherstores,whichleadstopotentialretailernancialloss.Toovercomespacelimitations,manyofthelargestretailers(suchasWal-MartR,TargetRandBestBuyR)haveadoptedtheclicks-and-mortarmodeltomakeabroaderassortmentavailableattheironlinestores.Ontheotherhand,pure-Internetretailersarealsomovingintoofineshoppingbyopeningphysicalstoresorcollaboratingwithtraditionalretailers[ 67 ],asinthecaseofWarbyParkerR,asuccessfulonlineeyeglassretailer,whichjustopeneditsrstphysicalstoreinNewYorkCityin2013.Oneofthereasonsforthemovebytheseonlineretailersisthatcertaincategoriesofproductsarelessamendabletoonlineshopping,soanonline-onlyretailchannelisnotenoughtoensurelong-termsuccess.Forinstance,itishardforapure-Internetfashionretailertosucceedinthelongterm,becauseconsumersneedtotryandfeeltheproductbeforehand.Lackofphysicalexperienceinrealtimecausesahighreturnrate(upto45%)associatedwiththeonlineapparelsales[ 2 ],andcostsofreturnhandlingusuallyarehighforanonline-onlyretailer.Inaddition,thecompetitionfromtraditionalretailerslikeWal-MartRwhoareactively 78

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expandingtheirvisualchannelalsourgesthepure-Internetretailerstomoveintotheofine-shoppingworld. Inthisstudy,adecisiontoolthathelpsretailerscoordinateproductassortments,retailpricesandinventorybetweenthephysicalchannelandonlinechannelisdevelopedinthehopetoreducechannelconictsandimprovetheretailer'sprotability.TheresearchincoordinatingtraditionalandInternetchannelsisvoluminous.Agatzetal.[ 2 ]providedanexcellentreviewwhichaddressesvarioussupplychainmanagementissuesspecictoe-fulllmentinaclicks-and-mortarmodel.Intheirreview,Agatzetal.discussedbothpracticalmanagerialplanningissuesandcorrespondingoperationsresearchmodels.Accordingtotheirreview,managementissuesthatarerelatedtoourstudyincludepricing,distributionnetworkdesignandinventoryandcapacitymanagement. First,pricingdecisionsplayanimportantinroleinanybusinessmodel,andasignicantstreamofresearchisavailableformulti-productpricingoptimization,whichappearstobeapplicabletoaclicks-and-mortarbusinessmodel.TheMultinomialLogit(MNL)modelandtheNestedLogit(NL)modelaremainlyusedasmodelsofconsumerchoice.StudiesbyHansonandMartin[ 56 ],Dongetal.[ 41 ],SongandXue[ 89 ]andChenandHausman[ 37 ]areexamplesofrecentworkswhichusedtheMNLmodeltosimultaneouslydeterminepricesforsubstitutableproducts.AlthoughtheMNLmodeliswidelyusedinthemulti-productpricingliterature,theindependenceofirrelevantalternatives(IIA)propertyoftheMNLmodelrestrictsitsapplication.TheIIApropertygenerallydoesnotholdinthecasewheretheconsiderationsetcouldbedividedintosubsetssuchthatproductswithinonesubsetaremoresimilartoeachotherthanacrosssubsets.Toaddressthisdrawback,LiandHuh[ 66 ]andGallegoandWang[ 49 ]consideredthemulti-productpricingproblemwiththeNLmodel. 79

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Distributionnetworkdesignisanotherkeystrategicdecisioninaclicks-and-mortarsupplychain.Withthedevelopmentofthetechniquecalleddrop-shipping1,clicks-and-mortarretailers(e.g.,StaplesRInc.)havestartedadoptingadualsupplychainstrategy.Theystillholdtheirowninventoryforhighsalesvolumeproducts,buttheydeliverotherlesspopularproductsusingdrop-shipping[ 86 ].Inthisway,theycanachievebenetsintermsofbothwiderproductselectionandlowerinventorycosts.Randalletal.[ 82 83 ]discussedthecostsandbenetsofusingthedrop-shippingtechniqueandthetrade-offsbetweentraditionalandvirtualsupplychainstructures.BaileyandRabinovich[ 18 ]developedananalyticalmodelwhichexaminesanonlinebookretailer'sdecisionsunderin-stockanddrop-shippinginventorystrategies,andtheyshowedthattheretailershouldadoptbothinventorymanagementstrategiesasproductsbecomemorepopularandtheretailerincreasesitsmarketshare.NetessineandRudi[ 78 ]developedanoncooperativegametomodelthedualinventorystrategy(i.e.,in-stockanddrop-shippingstrategies)ofatwo-levelsupplychainwithasinglewholesalerandmultipleretailers,andtheirstudyprovidespracticingmanagerswithguidelinesforchoosingappropriatedistributionchannels. Thethirdmanagementissuerelatedtoourstudyisinventoryandcapacitymanagement.Inventorymanagementintheclicks-and-mortarbusinessmodelcanbeveryexible.Forexample,theclicks-and-mortarretailercouldmakein-storeinventoriesavailabletoonlinebuyersordeliverout-of-stockitemsdirectlytolocalconsumersbyusingvirtualchannelinventory.Theclicks-and-mortarinventorymanagementissuesusuallyarisefromtheinteractionsbetweendifferentconsumersegmentswhichcanbeaddressedbyinventoryrationingmodels.CattaniandSouza[ 33 ],Ayansoetal.[ 16 ]and 1Thedrop-shoppingtechniqueisanoutsourcingprocessinwhichtheretailertransferstheconsumerorderandshipmentdetailstothemanufacturerortoawholesaler,whothenshipstheorderdirectlytotheconsumeratthecostofahigherwholesaleprice. 80

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Dingetal.[ 40 ]allmadecontributionstotheliteratureintheareaofinventoryrationingintheonlinechannel. Inadditiontothethreeissuesmentionedin[ 2 ],anothermanagementissuerelatedtoourstudyisassortmentplaning.Academicstudyonassortmentplanningisrelativelynew,butithasexperiencedquickgrowthrecently.Koketal.[ 61 ]didanextensivereviewoftheassortmentplanningliterature.AnearlymodelforassortmentplanningisproposedbyvanRyzinandMahajan[ 95 ].Inthismodel,vanRyzinandMahajanusedaMNLtodescribetheconsumerchoiceprocessandanewsvendormodeltodescribethesupplyprocess.Theyassumedthatthesetofpossiblevariantsishomogeneous,whichmeanseachvariantissoldatanidenticalpriceandchargedanidenticalcost.Theyshowedthat,undertheseassumptions,theoptimalassortmentcanberestrictedtooneofnpossibletypes,wherenisthenumberofnumberofvariantsunderconsideration.Theresultofthisstudyiselegant,butfewpracticalproblemstthestrictassumptionsofthismodel.Intheirsecondpaper,MahajanandvanRyzin[ 70 ]stillstudiedanewsvendor-likemodelwithconsumerchoicedecisionsbasedonMNL.Butinthisstudyproductscanhavenonidenticalpriceandcost,andconsumersdynamicallysubstituteamongproductswhenastock-outoccurs.Theirresultsfromthisstudyshowedthatthestocklevelofapopularvariantshouldbehigherthanthestocklevelsuggestedbyatraditionalnewsvendoranalysisunderdynamicsubstitution.Cachonetal.[ 32 ]extendedthevanRyzinandMahajan[ 95 ]modelbyincorporatingconsumersearch,andtheyshowedthatinthepresenceofconsumersearchitmayevenbeoptimaltoaddanonprotableproducttoanassortmentsoastopreventconsumersearch.AnotherimportantassortmentplanningstudybasedontheMNLmodelwasdevelopedbyMilleretal.[ 71 ].Theirstudyusedconsiderationsetstocapturetheheterogeneityofconsumerpreferenceandusedintegerprogrammingtodeterminetheoptimalretailassortments. 81

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InadditiontotheMNLmodel,theexogenousdemand(EXD)modelisanothercommonlyusedmodelintheassortmentplanningliterature.TheEXDmodelovercomestheshortcomingsofMNLmodelandhasmoredegreesoffreedom.TheEXDmodeldirectlyspeciesthedemandforeachproductandthesubstitutionrateiftheproductisnotavailable,butthereisnounderlyingutilitymodeltoexplaintheconsumerbehavior.SmithandAgrawal[ 87 ]consideredtheproblemofoptimizingassortmentswiththeEXDmodel.Intheirmodel,theyassumedthattheconsumerpurchasebehaviorisdynamic,whichmeansaconsumer'snalchoicedependsontheconsumer'spreferences,theavailabilityofallproductsuponthearrivaloftheconsumer,thechoiceofpreviousconsumers,andthenumberofsubstitutionattemptsmadebytheconsumers.Theiroptimizationmodeliscomputationallyinfeasibleinmostcases,sotheydevelopedanapproximationmethodtosolveit. Onegroupofproblemsthatisstronglyrelevanttotheassortmentplanningproblemisshelfspaceallocationplanningproblems.Undertheassumptionthatallocatingzerospacetoaproductisequivalenttoeliminatingitfromtheassortment,shelfspaceallocationandassortmentselectioncanbedeterminedsimultaneously.Incontrasttotheassortmentplanningliteraturewhichassumesshelfspacehasnoeffectonconsumerpreferences,nearlyalltheshelfspaceallocationliteratureisbasedontheassumptionthatspaceallocatedtoproductshaseffectsonsales,andtheseeffectsaremeasuredbyspaceelasticity.TheshelfspaceallocationproblemhasbeenconsideredbyAndersonandAmato[ 10 ],HansenandHeinsbroek[ 55 ],CorstjensandDoyle[ 39 ],BorinandFarris[ 27 ],Urban[ 94 ],Hwangetal.[ 58 ]andmanyotherresearchers.Fortheliteratureonshelfspaceallocation,anoften-statedcriticismisthattherelationshipbetweensalesandspaceallocationisweak,andtheimpactofshelfspaceallocationonsalesisverysmallrelativetoothermarketingvariables(e.g.,advertisingandpromotion)[ 101 ]. 82

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4.2ProblemStatementandFormulation Weconsideraclicks-and-mortarretailerwhosellsgoodstoend-consumersthroughbothphysicalandvirtual(e-commerce)channels.Thisretaileroperatestwophysicalfacilities(awarehouseandaretailstore)andawebsiteforonlineshopping.Goodscanbestoredinthewarehouseandthestore,andmaybetransferredfrombothfacilitiestoconsumersindirectshipments.Theretailerhaslimitedstoragecapacity,andwouldliketodeterminethesetofproductstooffer,bothinthestoreandonline,inordertomaximizeitsprotfromsalesduringasinglesellingseason. 4.2.1ProductRelatedAssumptions LetN1=f1,...,ngandN2=fn+1,...,2ng,respectively,denotetheproductselectionsetsatthelocalandonlinestores.Weassumeproducti2N1andproducti+n2N2areidentical,exceptthatiissoldatthelocalstorebuti+nissoldattheonlinestore.Wemakethisassumptionforthesakeofnotationalsimplicity,andbecauseitalsoreectsthefactthatproductiisdifferentfromproducti+nfromtheconsumer'sperspectiveandshouldbetreateddifferentlywhenkeptinstockduringthesellingseason.Wethusmaketheassumptionthat,duringthesellingseason,theinventoryofproducticanonlybeusedtosatisfydemandsforproducti;attheendoftheseason,however,anyleftoverinventoryofproducti2N1maybeusedtosatisfyoutstandingordersforproducti+n2N2.Wewilldiscussthespecicsoforderfulllmentlateringreaterdetail. Inaddition,eachoftheproductsischaracterizedbyPpossiblesellingprices.Theupperlimitforthepriceofproducti,whichisalsothehighestpossiblesellingprice,ri1,istheoriginalretailpriceofproductibeforeanydiscounting.Thelowerlimitofthepriceofproductiisthewholesalepriceofproducti,whichisdenotedasci.Wedenethepthhighestpossiblesellingpriceasrip=ri1)]TJ /F6 11.955 Tf 13.15 8.09 Td[((ri1)]TJ /F3 11.955 Tf 11.96 0 Td[(ci)(p)]TJ /F6 11.955 Tf 11.96 0 Td[(1) P. 83

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ThevalueofPdependsonthedesiredresolution.Moreover,foreachpossiblesellingpriceoftheproducti,wecreateapseudo-product(i,p). 4.2.2FacilitiesRelatedAssumptions Withoutlossofgenerality,weassumethatthelocalstorehasbothdisplayedinventoryandbackroominventory.Partofanyreceivedorderisdeliveredtotheshelvesdirectly,withtheremainderplacedinbackroomstoragebeforebeingbroughttothedisplayarea.Theshelfspaceatthelocalstorehasthreedimensions,namelywidth,depthandheightasshowninFigure 4-1 .Inthisstudy,weonlyconsiderthewidthdimensionwhenallocatingshelfspace.Theshelfwidthisdividedintofacings,andthesizeofproducti'sfacingisequaltothewidthiofitsfrontface,inotherwords,eachindividualunitofaproductisonefacingwide.Capacityofproducti'sfacingdependsontheheightanddepthoftheshelfandthephysicalsizeofaunitofproducti.Thenumberoffacingsallocatedtoeachproductdeterminesthemaximumlevelofitsdisplayedinventory.Forexample,therearetwotypesofproducts(i.e.,productsiandj)inFigure 4-1 .Twofacingsareallocatedtoproducti,whilefourfacingsareallocatedtoproductj.Byobservation,sixunitscanbeheldinonefacingofproducti,sothetotalspaceallocatedtoproducticanstore12units.Similarly,thefourfacingsallocatedtoproductjcanhold36units. Wefurtherassumethattheorderquantityforeachproductislargeenoughtollitsinitiallyallocateddisplayedinventoryspace(fullstocked),anditemsarerestockedfromthebackroomontoshelvesasdisplayeditemsaredepletedbyconsumerdemands. Therearemanypossiblewaysfortheclicks-and-mortarretailertomanageinventoryatthelocalstorebackroomandthewarehouse,andourmodelissufcientlyexibletoincorporateseveraltypesofinventorystrategies.Inordertoconciselypresentourmodelingmethods,weassumethatthedistancebetweenthewarehouseandlocalstoreistoolongtotransshipanyproductfromonefacilitytotheotherduringtheselling 84

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season.WealsoassumethatalloftheproductsinthesetN1canonlybestoredatthelocalstore,whilethevariantsinN2canonlybestoredinthewarehouse. Figure4-1. Illustrationofshelfspaceallocation. 4.2.3DemandRelatedAssumptions Betweenthelocalandonlinestores,theretailerservesKconsumersegments,andeveryconsumerhasachoiceofpurchasingeitheratthestoreorplacinganorderonthewebsite.Thesegmentationofconsumersintodistinctgroupsmaybebasedonconsumerpurchasebehavior,consumerneeds,channelpreferences,and/orinterestincertainproductfeatures.Inourstudy,thereareatleastthreeconsumersegments,namelywalk-in,onlineandhybridconsumers.Whilewalk-inconsumersareassumedtoonlyshopatthephysicalstore,onlineconsumersplaceordersonthewebsiteandhavetheordershippeddirectlytotheirresidence.Moreover,hybridconsumershaveaprimarychannelpreferenceandarelikelytoswitchtotheotherchanneliftheirpreferredproductisnotavailableinthepreferredchannel.Thesethreebasicsegmentscanbe 85

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furtherdividedintosub-segmentstoconveymoreusefulinformationaboutconsumerattributes. LetNN1[N2denotetheentiresetof2nproducts.Oneoftheretailer'sdecisionsistheset(s)ofproductstomakeavailabletoconsumersduringthesellingseason.WecallthesubsetSNofavailableproducts,theretailer'sofferset.Theotherretailerdecisionistodeterminethesellingpriceofeachproductselectedintheofferset.Noticethatifproductiisselectedintheofferset,oneandonlyoneofthePpossiblesellingpricesshouldbeselected.Thenaloutcomeofthesetworetailerdecisionsisasetofpseudo-products,whichisdenotedasSPNf1,...,Pg.GivenapseudooffersetSP,anarrivingconsumerfromsegmentkchoosesproduct(i,p)2SPwithprobabilityPkip(SP),wherePkip(SP)=0,if(i,p)=2SP.Wedenotetheno-purchaseprobabilitybyPk0(S),andbytotalprobability,wehavethatP(i,p)2SPPkip(SP)+Pk0=1.Ifthenumberofsegment-kconsumersarrivingduringtheplanninghorizonfollowsaPoissondistributionwithmeank,andthechoiceofeachconsumerisindependentoftheothers,thenthedemandofsegment-kconsumersforpseudo-product(i,p)whentheretailerofferssetSPfollowsaPoissondistributionwithmeankPkip(SP). Beforeproceedingintodetailsaboutthesequenceofevents,wemakethefollowingthreeimportantassumptionsconcerningtheconsumerchoiceprocess,whichareverysimilartothestaticsubstitutionassumptionsusedbySmithandAgrawal[ 87 ],vanRyzinandMahajan[ 95 ],Topaloglu[ 93 ]. Assumption1. ConsumerschoosebasedonlyonknowledgeofthesetSP,andtheirchoiceswillbenotbeaffectedbytheinventorylevelsandshelfspaceallocationofthepseudo-productsinSP. Assumption2. Ifanin-storeconsumerselectsaproductinS\N1andthestoredoesnothaveitinstock,theconsumerdoesnotundertakeasecondchoice,andthesaleislost. 86

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Assumption3. IfanonlineconsumerselectsaproductinS\N2andtheonlinestoredoesnothaveitinstock,theconsumerdoesnotknowthis,andthesaleisnotlost.Theretailersatisessuchademandthrougheitheraproducttransferbetweenfacilitiesordrop-shipping. Underourassumptions,forthetraditionalchannel,aconsumer'sinitialchoiceisonlyinuencedbythesetofalternativesoffered,andthereisnodynamicsubstitutioniftheirrstchoiceisoutofstock.VanRyzinandMahajan[ 95 ]showedseveralexampleswheretheseassumptionsserveasareasonableapproximationofconsumerbehaviorintraditionalsellingchannels.However,theconsequencesofstockoutsintraditionalchannelscouldbedire:localstoreconsumersmayswitchstoresandpurchasetheitemelsewhere,andthisstoreswitchingcouldbepermanent.Asaresult,itisboththeoreticallyandpracticallydesirabletodampenthenegativeeffectsofstockouts.Toachievethisgoal,weapplyachanceconstrainttolimittheprobabilityofstockingoutatthelocalstore.Foronlineretailchannels,ourassumptionsapproximatethepracticeinwhichthestocklevelofproductisnotshownonthewebsite,buttheconsumerorderisalwayssatised. 4.2.4Sequenceofevents Thesinglesellingseasoninourmodelcanbedividedintothreestages:beforethesellingseason,duringthesellingseason,andafterthesellingseason.Wewilldiscussthesequenceofeventsoccurringineachstageintherestofthissection. Asequenceofdecisionsismadebeforethesellingseason.Atrst,theretailerchoosesanoffersetSN.GiventheoffersetS,theretailerdeterminesthesellingpricepforeachselectedproductandallocatesshelfspacetoeachselectedproductatthelocalstore.Aftersettingpricesandallocatingshelfspace,theretailerplacesanorderandpaysaregularwholesalepriceforeachproduct.Deliveryofordersalsooccursbeforethesellingseason.ThevariantsinsetN1aresenttothelocalstore:part 87

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oftheorderisusedtolltheshelfspace;andtheremainderisstoredinthebackroom.ThevariantsinthesetN2aresentdirectlytothewarehouseandstoredthere. Duringthesellingseason,thestockonhandofeachproductisdepletedbythedemandduringthesellingseason,whichisarealizationfromadistributionwhosedensityfunctionisdeterminedbySP.Atthelocalstore,ifdemandforproductiisgreaterthanthesumofitsdisplayinventoryandbackroominventory,thentheunsatiseddemandislost;otherwise,therewillbeleftoverinventoryofproductiattheendofthesellingseason.Attheonlinestore,iftheproducti+nisinstockatthewarehouse,theretailerguaranteestodelivertheorderwithinacertainnumberofdays,andthisshippingoptionisnamedregularshipping.Whentheinventoryofproducti+ninthewarehouseisdepleted,anonlineconsumerisofferedtwodeliveryoptions:regularshippingandsuper-savershipping.Iftheconsumerchoosesregularshipping,theretailerchargesthisconsumertheregularpriceandusesadrop-shoppingstrategytoshiptheorderimmediately.Ontheotherhand,ifsuper-savershippingischosen,theconsumerischargedadiscountedprice,buttheretailerdoesnotshiptheorderuntiltheendofthesellingseason. Afterthesellingseason,theretailerfulllsthesuper-savershippingordersatthelowestcost.Theretailerchecksthestocklevelofproductirst:ifanyinventoryofproductiisleftoveratthelocalstore,thenthisleftoverinventoryofproductiisusedtofullltheorderofproducti+n;otherwise,theretaileragainusesthedrop-shippingstrategytosatisfytheoutstandingorders.Thisstrategyachievesthelowestcostbecauseweassumethatmanufacturerschargetheretailerhigherwholesalepricesfordrop-shippingitems,butthereisnoextracostassociatedwithshippingfromitsowninventory.Afterdeliveringsupersavingshippingitems,theretailerdisposesofanyremaininginventoryatasalvagecost. 88

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4.2.5DemandModels Asmentioned,consumersarecategorizedwithindifferentmarketsegments,andwefurtherassumethateachconsumersegmentisonlyinterestedinasubsetoftheentireproductset.LiuandvanRyzin[ 68 ]calledthissettheconsumer'sconsiderationset.WeassumethateachconsumerbelongstooneofKmarketsegments,andeachsegmentkischaracterizedbyoneconsiderationsetCkNf1,...,Pg.Forexample,segment-lconsumersarelowfarewalk-inconsumers,whichmeanstheyonlyshopatthelocalstoreandonlypurchaseproductswithapricelessthan.Asaresult,theirconsiderationsetClcanbedescribedasCl=f(i,p)ji2N1,pg.AsinBront,Mendez-DazandVulcano[ 30 ],weallowforoverlappingsegments,i.e.,wepermitCk\Cl6=;fork6=l. Inthispaper,weuseamultinomiallogitmodel(MNL)asthedemandmodel.TheMNLmodelisautilitymodelthatiscommonlyusedintheeconomicsandmarketingliterature.ForabriefdescriptionoftheMNL,seeAndersonetal.[ 11 ]orvanRyzinandMahajan[ 95 ].UndertheMNLmodel,theprobabilitythatasegment-kconsumerchoosespseudo-product(i,p)fromSP\Ckisdenedbyapreferencematrixvk,wherethecomponent(i,p)ofvk,vkip,isasegment-kconsumer'spreferenceweightforpseudo-product(i,p).Thismatrix,togetherwiththeno-purchasepreferencevk0,determinestheprobabilityPkip(SP)asfollows: Pkip(SP)=vkip P(j,q)2Ck\SPvkjq+vk08i,p,k.(4) If(i,p)=2SPor(i,p)=2Ck,thenvkip=0.Moreover,vkipisincreasinginconsumerutility,soahighvalueofvkipcorrespondstoapseudo-productwithahigherexpectedutility. OneadvantageoftheMNLmodelisthatitallowsonetoeasilyincorporatemarketingvariablessuchaspricesandpromotionsintothechoicemodel.Inourstudy,pseudo-products(i,p1)and(i,p2)areidenticalproducts,andtheonlydifferenceisthatp16=p2.Ifp1
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k,wewillhavevkip1>vkip2.Inaddition,wecanusesimpleconstraintsinourmodeltoreectthefactthatpseudo-products(i,p1)and(i,p2)maynotbeincludedinasetSPatthesametime. Despiteitsadvantages,therearetwomajorshortcomingsoftheMNLmodel.OneoftheseshortcomingsstemsfromitsIndependenceofIrrelevantAlternatives(IIA)property.Toillustratethisproperty,considertwodistinctconsiderationsetsS1andS2,andtwodistinctproductsiandj.TheMNLformulaimpliesthat Pi(S1) Pj(S1)=Pi(S2) Pj(S2)=vi vj.(4) Thus,therelativepreferencesofproductsiandjareindependentofthecompositionoftheoffersetsS1andS2.TheIIApropertywouldnotholdinthecasewheretheconsiderationsetcouldbedividedintosubsetssuchthatproductswithinonesubsetaremoresimilartoeachotherthanacrosssubsets.Forexample,supposethereisasmartphonemaniacwhohasthesameprobabilityofpurchasingaSamsungGalaxyIIIoraniPhone4S,i.e.,PfSamsungg=PfiPhoneg=1 2.TherearetwomodelsofiPhone4Sthatareidenticalexceptfortheircolors,blackorwhite.Assumethatthemaniacisindifferentaboutthecolorofthephonehepurchases.IftheoffersetisfSamsungGalaxyIII,blackiPhone4S,whiteiPhone4Sg,thenonewouldintuitivelyexpectthatPfSamsungg=1 2andPfblackiPhoneg=PfwhiteiPhoneg=1 4.However,theMNLmodelimpliesthatthatPfSamsungg=PfblackiPhoneg=PfwhiteiPhoneg=1 3. ThesecondshortcomingoftheMNLmodelisthatitisunabletofullycapturesubstitutionbehavior.Koketal.[ 61 ]showedthatitisnotpossibleundertheMNLmodeltohavetwocategorieswiththesamepreferenceweightbutdifferentsubstitutionrates. Weuseabinaryvectorx2f0,1g2nPtocharacterizethesetSP,i.e.,xip=1if(i,p)2SP,andxip=0otherwise.Wecanthenexpress( 4 )intermsofthebinary 90

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variablesxipfortheMNLchoicemodel: Pkip(SP)=Pkip(x)=vkipxip P(j,q)2Ckvkjqxjq+vk08i,p,k.(4) 4.2.6Chance-ConstrainedFormulation Werstdenethenotationwewilluseinformulatingachance-constrainedformulationoftheproductselectionandinventoryallocationproblem.Asdiscussedearlier,weuseiandjtoindexproducts,ptoindexpricesandktoindexconsumersegments.Weassumethatshelfspacehasthreedimensions,butweonlyusewidthtomeasureshelfspace.Onefacingofproducticonsumesiunitsofshelfwidth,wherethetotalwidthoftheshelfspaceisAunits.Theheightanddepthofshelfdeterminethecapacityofonefacing,andeachfacingofproducticanholdiunitsofproducti.Lettingwipdenotethenumberoffacingsallocatedtopseudo-product(i,p),theniwipdenotestheshelfwidthconsumedbypseudo-product(i,p),andiwipdenotesthemaximumnumberofunits(i,p)thatcanbeheldontheshelf. Forconvenience,wemeasurebackroomandwarehousecapacityusingasingledimension,whereBdenotesthestore'sbackroominventorycapacity,Bwdenotesthewarehousecapacity,andidenotestheamountofthiscapacityconsumedperunitofproducti.Ifyipunitsofpseudo-product(i,p)areordered,wherei2N1,thenthenumberofunitsthatmustbeallocatedtothebackroomequalsyip)]TJ /F4 11.955 Tf 12.33 0 Td[(iwip.Asaresult,thetotalamountofbackroomcapacityconsumedbypseudo-product(i,p)equalsi(yip)]TJ /F4 11.955 Tf 11.95 0 Td[(iwip). WeletIipdenotethenumberofunitsofpseudo-product(i,p)inventoryremainingattheendofthesellingseason,whilesipdenotestheamountofdemandthatexceedstheorderquantityyip.Ifi2N1,thensipisthenumberoflostsalesatthestoreforpseudo-product(i,p);ifi2N2,thensipcorrespondstothetotalnumberofunitseitherdropshippedfromthemanufacturerortransferredfromthelocalstore.Letusfurtherassumethatifconsumersareofferedregularshippingandsuper-savershippingfor 91

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pseudo-product(i,p)atthesametime,thenonaverageipofthemchooseregularshippingand1)]TJ /F4 11.955 Tf 11.96 0 Td[(ipofthemchoosesuper-savingshipping,whereip2[0,1]. Wealsoletydipdenotethenumberofunitsofpseudo-product(i,p)dropshippedfromthemanufacturerdirectlytotheconsumer,whileytipdenotesthenumberofunitsofpseudo-product(i,p)transferredfromleftoverinventoryofproducti)]TJ /F3 11.955 Tf 12.5 0 Td[(n.Notethatregularshippingofout-of-stockpseudo-product(i,p)requiresusingdrop-shipping,butdrop-shippingmaybealsousedattheendofsellingseasoniftheremaininginventoryisinsufcienttosatisfyordersforsuper-savershipping.Thisimpliesydipisatleastaslargeasipsip.NotethatIip,sip,ydipandytiparerandomvariablesthatdependontheproductorderquantitiesandconsumerdemands. Letripdenotetheunitretailpriceofpseudo-product(i,p).Letciandfidenote,respectively,theunitwholesalepriceandxedcostassociatedwithorderingproducti,andletgiandhidenote,respectively,theunitshortagecost(lostsalesorcompensationforsuper-savershipping)andunitsalvagevalueforproducti.Letcdidenotetheunitdrop-shippingwholesalepriceassociatedwithproducti,andnotethatitisintuitivetohavecdi>ci.Weassumethateachtimethatshelfinventoryforproductiisreplenishedfromthebackroominventory,acostofmiperunitreplenishedisincurred. Inordertoformulatetheretailer'sdecisionproblem,weneedtocharacterizeafunctionthatdetermineswhetheranyshortageshaveoccurred.Letkdenotearandomnumberofconsumerswithinsegmentkthatarriveduringthesellingseason.Then,becausePkip(x)givestheproportionofsegmentkconsumersthatdemandpseudo-product(i,p)whenthesetxisoffered,PKk=1kPkip(x)givesthetotalnumberofdemandsforpseudo-product(i,p)duringthesellingseason.WedeneG(x,y,)=max1in,1pPnPKk=1kPkip(x))]TJ /F3 11.955 Tf 11.96 0 Td[(yipo.Then,ifG(x,y,)islessthanorequaltozero,thisimpliesthatnoshortageshaveoccurred,andwecanwritetheprobabilityofnoshortagesasP(G(x,y,)0). 92

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Inordertocomputetheexpectedprot,werstdenethefunctionQ(w,x,y,)=Xi2N1PXp=1(rip"KXk=1kPkip(x))]TJ /F3 11.955 Tf 11.95 0 Td[(sip#+hi)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Iip)]TJ /F3 11.955 Tf 11.96 0 Td[(yt(i+n)p)]TJ /F3 11.955 Tf 11.96 0 Td[(gisip)]TJ /F3 11.955 Tf 11.95 0 Td[(mi"KXk=1kPkip(x))]TJ /F3 11.955 Tf 11.96 0 Td[(sip)]TJ /F4 11.955 Tf 11.95 0 Td[(iwip#+)+Xi2N2PXp=1(ripKXk=1kPkip(x)+hiIip)]TJ /F3 11.955 Tf 11.95 0 Td[(gi(1)]TJ /F4 11.955 Tf 11.95 0 Td[(ip)sip)]TJ /F3 11.955 Tf 11.95 0 Td[(cdiydip). (4) whereIip=hyip)]TJ /F12 11.955 Tf 11.95 8.96 Td[(PKk=1kPkip(x)i+,sip=hPKk=1kPkip(x))]TJ /F3 11.955 Tf 11.96 0 Td[(yipi+andytip=minfPPp=1I(i)]TJ /F5 7.97 Tf 6.58 0 Td[(n)p,(1)]TJ /F4 11.955 Tf 12.59 0 Td[(ip)sipgandydip=maxfsip)]TJ /F3 11.955 Tf 12.59 0 Td[(ytip,ipsipg.Next,lettingQ(w,x,y)=E[Q(w,x,y,)],weformulatetheretailer'sinventoryplanningproblemasfollows:=maxQ(w,x,y))]TJ /F12 11.955 Tf 11.95 11.36 Td[(Xi2NPXp=1(fixip+ciyip) (4)s.t.Xi2N1PXp=1iwipA (4)Xi2N2PXp=1iyipBw (4)Xi2N1PXp=1i(yip)]TJ /F4 11.955 Tf 11.95 0 Td[(iwip)B (4)PXp=1xip18i2N (4)iwipyip8i2N1,p=1,...,P (4)yipMxip8i2N,p=1,...,P (4)P(G(x,y,)0)1)]TJ /F4 11.955 Tf 11.95 0 Td[( (4)xip2f0,1g,wip0,yip08i2N,p=1,...,P. (4) Theobjectivefunction( 4 )iscomposedoftheorderingcostandtheexpectedprotfromsales.Therstpartistheprotatthelocalstore,whichisexpressedasthedifferencebetweenthesumofrevenuesandthesumofsalvagecost,stockoutcost 93

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andshelfreplenishmentcost.Thereareseveralthingstobeawareofwhencomputingtheprotforproducti2N1:rst,alostsaleispossibleforaproductsoldatthelocalstore,whichimpliestherealizedsalesforpseudo-product(i,p)isKPk=1kPkip(x))]TJ /F3 11.955 Tf 12.67 0 Td[(sip;second,partoftheleftoverinventoryofproductimaybeusedtofulllonlineconsumerdemandforproducti+n,thereforeonlyIip)]TJ /F3 11.955 Tf 12.41 0 Td[(ytipunitswillrealizethesalvagevaluehi;andnally,theshelfreplenishmentcostwillonlybechargediftherealizeddemandisgreaterthantheon-shelfinventory,andthereplenishmentcostislinearinthenumberofunitsofproductreplenished.Theotherpartoftheprotequationisfromtheonlinestore,anditisthedifferencebetweenthesumofrevenuesandthesumofsalvagecost,compensationforsuper-savershipping,anddrop-shippingcost.Notethattherearenolostsalesforonlineproductsandcompensationisonlypaidtoconsumerswhochoosesuper-savershipping. Constraint( 4 )ensuresthatshelfcapacityAatthelocalstoreisnotviolated.Constraint( 4 )statesthatthecapacityallocatedtoonlineproductsisboundedabovebyBw.Constraint( 4 )statesthattheorderquantitiesofproductssoldatthelocalstorebeyondthedisplayinventoryarelimitedbythebackroomcapacityBatthelocalstore.Constraint( 4 )guaranteesthatifpseudo-product(i,p)isnotorderedatthelocalstore,thennoshelffacingcapacityisallocatedtopseudo-product(i,p);italsoensuresthattheorderquantityislargeenoughtolltheallocatedshelfspace.Ifpseudo-product(i,p)isnotincludedinthelocal/onlinestoreassortment,thennopseudo-product(i,p)inventoryisorderedatthelocal/onlinestore,andconstraint( 4 )capturesthisassumption.Finally,thenonnegativityandintegralityconstraintsareprovidedin( 4 ).Constraint( 4 )indicatesthatG(w,x,y,)shouldbenon-positivewithaprobabilityofatleast1)]TJ /F4 11.955 Tf 12.04 0 Td[(,whichimpliesthatthereisatmostanprobabilityofashortageforanyproductatthelocalstore. 94

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4.3SolutionProcedure TherstdifcultyinsolvingourproblemisthattheMNLmodelincludesnonlineartermswhichneedtobelinearizedinordertoconvertourproblemtoaformthatiswellbehavedintermsofchance-constrainedstochasticprograms.StandardlinearizationtechniquesareusedtoconverttheexpressionsundertheMNLdemandmodelstolinearexpressions. 4.3.1MNLModelLinearization Inthissection,wefollowtheprocedureusedbyMendez-DazandVulcano[ 30 ]tolinearizeourMNLmodel.Bydeningthevariables k=1 P(j,q)2Ckvkjqxjq+vk0,8k=1...,K, theprobabilityPkip(x)intheMNLmodelcanberewrittenasPkip(x)=vkipkxip8i2N,p=1,...,P,k=1,...,KX(j,q)2Ckvkjqkxjq+vk0k=18k=1,...,K. Wu[ 99 ]provedthatthepolynomialmixed0)]TJ /F6 11.955 Tf 13.21 0 Td[(1termukip=kxip,wherexipisabinaryvariableandkisanonnegativecontinuousvariable,canberepresentedbythefollowinglinearinequalities:(i)k)]TJ /F3 11.955 Tf 12.04 0 Td[(ukipM1)]TJ /F3 11.955 Tf 12.03 0 Td[(M1xip;(ii)ukipk;(iii)ukipM1xip;and(iv)ukip0,whereM1isalargenumbergreaterthan1 mink,i,pfvkipg.Byapplyingthisresult, 95

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weobtainthefollowingreformulationoftheMNLmodel:Pkip(x)=vkipukip8i2N,p=1,...,P,k=1,...,K (4)X(j,q)2Ckvkjqukjq+vk0k=18k=1,...,K (4)k)]TJ /F3 11.955 Tf 11.96 0 Td[(ukipM1)]TJ /F3 11.955 Tf 11.95 0 Td[(M1xip8(i,p)2Ck,k=1,...,K (4)ukipk8(i,p)2Ck,k=1,...,K (4)ukipM1xip8(i,p)2Ck,k=1,...,K (4)ukip08(i,p)2Ck,k=1,...,K. (4) 4.3.2SampleAverageApproximation Theseconddifcultyinourproblemisthatourmodelcontainsbothchance-constrainedandtwo-stagestochasticprogramfeatures.Ingeneral,evaluatingtheobjectivefunctionandcheckingsolutionfeasibilityofthistypeofproblemisnoteasy,andthefeasibleregiondenedbythechanceconstraintgenerallyisnotconvex.WangandGuan[ 97 ]callthistypeofprogramaCCTSprogram,andtheydevelopedacombinedsampleaverageapproximation(SAA)algorithmtosolvetheCCTSprogrameffectively. Asampleaverageapproximationofourtrueproblemisobtainedbyreplacingthetruedistributionofdemandsbyanempiricaldistributioncorrespondingtoarandomsample.Inthisstudy,weassumethatconsumersinanysegmentkarriveaccordingtoaPoissondistributionwithratek.Intheprevioussection,wedenedkasarandomnumberofconsumersinsegmentkthatarriveduringthesellingseason,andnowwefurtherdenef1,...,Kgastherandomdemandvector.Nowlet1,...,Lbeanindependentidenticallydistributed(i.i.d)sampleofLrealizationsofgeneratedby 96

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MonteCarlosimulation,andconsiderthefollowingproblemassociatedwiththissample:max^L(u,w,x,y)=)]TJ /F10 7.97 Tf 16.09 14.94 Td[(2nXi=1PXp=1(fixip+ciyip)+1 LLXl=1Q(u,w,x,y,l)s.t.( 4 )-( 4 ),( 4 ),( 4 )-( 4 )1 LLXl=11(0,1)(G(u,w,x,y,l)). (4) Theaboveoptimizationproblemisusuallyreferredasasampleaverageapproximate(SAA)problemofthetrueproblem.Incontrasttothetrueproblem,asampleaveragefunction1 LPLl=1Q(u,w,x,y,l)isusedinplaceoftheexpectedvaluefunctionE[Q(u,w,x,y,)],andafunction1 LPLl=11(0,1)(G(u,w,x,y,l))isusedtoestimatetheprobabilityofnoshortage2.NoticethattheSAAproblemisalsoaCCTSprogram,asistheoriginalproblem,butwithadifferentdemanddistribution. Dene(u,w,x,y)asthevectorofdecisionvariables.Assumethatisanoptimalsolutionforthetrueproblemand^LisanoptimalsolutionfortheSAAproblem.Letrepresenttheoptimalobjectivevalueofthetrueproblemand^LrepresenttheoptimalobjectivevalueoftheSAAproblem.WangandGuan[ 97 ]showedthat^L!andD(^L,)!0withprobabilityoneasL!1,whereD(^L,)representsthedistancebetween^Land. 4.3.2.1Solutionvalidation Acandidatesolution^LoftheSAAproblemisnotnecessaryanoptimalsolutionforthetrueproblem.Actually,^Lmaynotevenbefeasibleforthetrueproblem.Asaresult, 2Inthisstudy,wedenetheindicatorfunction1(0,1)(t):R!Rasfollows1(0,1)(t)=1ift>00ift0. Thatis,1 LPLl=11(0,1)(G(u,w,x,y,l))isequaltotheproportionofrealizationsinwhichastockoutoccurs. 97

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afterobtainingacandidatesolution^LoftheSAAproblem,itisimportanttovalidateitsqualityasasolutionofthetrueproblem. Thesolutionvalidationstartswiththevericationoffeasibilityof^L.Given^L,letq(^L)=PG(^L,)0. Becauseitisdifculttocomputeq(^L)exactly,inordertocheckthefeasibilityof^L,wefollowedthemethoddevelopedbyAhmedandShapiro[ 3 4 ]toconstructanapproximate(1)]TJ /F4 11.955 Tf 12.38 0 Td[()condenceupperboundonq(^L).First,generateani.i.dsample1,...,Lwhichshouldbeindependentofthesampleproducingthecandidatesolution^L.Giventhisnewlygeneratedsampleand^L,let^qL(^L)=1 LLXl=11(0,1)(G(^L,l)). NoticeherethesizeLofthissamplecouldbeverylarge,sincewedonotneedtosolveanyoptimizationproblemusingthissample.Moreover,fornottoosmallq(^L)andlargeL3,^qL(^L)isnotonlyanunbiasedestimatorofq(^L),butthedistributionofq(^L)canalsobeapproximatedreasonablywellbyanormaldistributionwithmean^qL(^L)andvariance^qL(^L)(1)]TJ /F6 11.955 Tf 12.69 0 Td[(^qL(^L))=L.Asaresult,wecanconstructthefollowing(1)]TJ /F4 11.955 Tf 11.96 0 Td[()-condenceupperboundonq(^L):U,L(^L)=^qL(^L)+)]TJ /F10 7.97 Tf 6.59 0 Td[(1(1)]TJ /F4 11.955 Tf 11.96 0 Td[()q ^qL(^L)(1)]TJ /F6 11.955 Tf 12.2 0 Td[(^qL(^L))=L, where)]TJ /F10 7.97 Tf 6.59 -.01 Td[(1istheinverseofthecumulativedistributionofstandardnormaldistribution.ItiseasytoseethatifU,L(^L)islessthan,thenweare(1)]TJ /F4 11.955 Tf 12.52 0 Td[()condentthat^Lisfeasibleforthetrueproblemand^L(^L)isalowerboundfor. 3Tobeonthesafeside,oneshouldrequireLq(^L)tobegreaterthanorequalto5. 98

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Aftercheckingthefeasibilityofsolution^L,thenextstepofthesolutionvalidationistoestimatetheoptimalitygap)]TJ /F6 11.955 Tf 13.07 2.65 Td[(^L(^L),andweagainfollowedthemethodprovidedbyAhmedandShapiro[ 3 4 ]toobtainaupperboundfortheobjectivevalue.Inthismethod,wegeneratedMindependentsamples1,m,...,L,m,m=1,...,M,eachofsizeL.FortheseMsamples,wepickedtheTthlargestoptimalvalueastheapproximateupperboundforwithcondencelevel(1)]TJ /F4 11.955 Tf 12.21 0 Td[(),whereTiscalculatedasdescribedin[ 4 ].Thentheoptimalitygapcanbeestimatedbythedifferencebetweenthisapproximatedupperboundand^L(^L). 4.3.2.2SAAformulation TosolvetheSAAproblemusingacommercialsolver,thechanceconstraint( 4 )shouldbereformulatedasfollows:KXk=1lkPkip(x))]TJ /F3 11.955 Tf 11.96 0 Td[(yipM2zl8i2N1,p=1,...,L,l=1,...,LLXl=1zlLzl2f0,1g8l=1,...,L, whereM2isalargenumber.Afterreformulation,theSAAproblembecomesamixedintegerprogramwithalargenumberofbinaryvariables.IntheSAAliterature,inordertospeedupthealgorithm,star-inequalitiesareoftenusedtostrengththeformulationoftheSAAproblem.However,forourproblem,star-inequalitiesarenotapplicable. Whenthereisonlyoneconsumersegmentunderconsideration(i.e.,K=1),aftertakingasampleofLrealizations,thedemandrealizationscanbesortedinnonincreasingorder,andweassumethatlareindexedsuchthat1...L.TheconstraintPLl=1zlLimpliesthatwecannothavezl=1foralll=bLc+1,...,L.WecanthereforetightentheSAAformulationbyxingzl=0,8l=bLc+1,...,L. 99

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Itcanbeobservedthatthissortingmethoddoesnotworkforthegeneralcaseinwhichthereismorethanoneconsumersegment,becauseeachdemandrealizationlisavectorwhichiscomposedofmorethanonecomponent.Atthisstage,wecouldnotndstronginequalitiesforthegeneralcaseoftheSAAformulation.Asaresult,inthenextsection,weproposeaheuristicforsolvingourproblemingeneral. 4.4GreedyandLocalSearchAlgorithm Inthegreedyandlocalsearchalgorithm,thesetSPisconstructedfromscratch(anemptyset),choosingateachiterationthepseudo-productbringingthehighestadditionalprot. GivenasetSP,theoriginalCCTSproblemwithbothlinearandbinaryvariablescanbereducedtoanotherCCTSproblemwithonlylinearvariables,whichcanbewrittenasfollows(SP)=max)]TJ /F12 11.955 Tf 20.05 11.36 Td[(X(i,p)2SP(fi+ciyip)+E[Q(SP,w,y)] (4)s.t.X(i,p)2SP1iwipA (4)X(i,p)2SP2iyipBw (4)X(i,p)2SP1i(yip)]TJ /F4 11.955 Tf 11.96 0 Td[(iwip)B (4)iwipyip8(i,p)2SP1 (4)P(G(SP,y,)0)1)]TJ /F4 11.955 Tf 11.96 0 Td[( (4)wip0,yip08(i,p)2SP, (4) 100

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whereQ(SP,w,y)=X(i,p)2SP1(rip"KXk=1kPkip(SP))]TJ /F3 11.955 Tf 11.96 0 Td[(sip#+hi)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Iip)]TJ /F3 11.955 Tf 11.96 0 Td[(yt(i+n)p)]TJ /F3 11.955 Tf 11.96 0 Td[(gisip)]TJ /F3 11.955 Tf 11.96 0 Td[(mi"KXk=1kPkip(SP))]TJ /F3 11.955 Tf 11.95 0 Td[(sip)]TJ /F4 11.955 Tf 11.96 0 Td[(iwip#+)+X(i,p)2SP2(ripKXk=1kPkip(SP)+hiIip)]TJ /F3 11.955 Tf 11.96 0 Td[(gi(1)]TJ /F4 11.955 Tf 11.95 0 Td[(ip)sip)]TJ /F3 11.955 Tf 11.96 0 Td[(cdiydip), andG(SP,y,)=max(i,p)2SP1nPKk=1kPkip(SP))]TJ /F3 11.955 Tf 11.96 0 Td[(yipo,andSPjf(i,p)j(i,p)2SP,i2Njg,8j=1,2.Theaboveproblemcanbeconsideredasarestrictedversionoftheoriginalproblem.Itiseasiertosolvethisrestrictedproblemthantheoriginalone,becausetherestrictedproblemdoesnotincludeanybinaryvariables,andtheprobabilitytermsPkip(SP)areconstant.WestillusetheSAAalgorithmtosolvethisrestrictedproblembutwithtwomodications.First,thestar-inequalitiesareaddedtotheSAAproblemtospeedupthecomputation.Todothis,weintroduceanewsetofbinaryvariablesfzipl2f0,1g:i2N;p=1,...,P;l=1,...,bLcganddenedipl=PKk=1lkPkip(SP).Wesortthediplindecreasingorderofmagnitude,i.e.,dip1dip2...dipbLc.Thenthestar-inequalitiescanbewrittenasfollows:zipl)]TJ /F3 11.955 Tf 11.95 0 Td[(zip(l+1)08(i,p)2SP,l=1,...,bLczl)]TJ /F3 11.955 Tf 11.96 0 Td[(zipl08(i,p)2SP,l=1,...,bLcyip+bLcXl=1(dipl)]TJ /F3 11.955 Tf 11.95 0 Td[(dip(l+1))ziplzip18(i,p)2SPLXl=1zlLzl,zipl2f0,1g. Luedtkeetal.[ 69 ]provedthatthestar-inequalitiescanstrengthentheformulationoftheSAAproblem.Second,thegoalofthisheuristicistogenerateagoodfeasible 101

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solutionandnoupperboundneedstobegenerated,sothesolutionvalidationstepthatgeneratesanupperboundisskippedaftergettinga(1)]TJ /F4 11.955 Tf 12.6 0 Td[()-condencelowerbound^(SP)of(SP).Thegreedyandlocalsearchalgorithmcanbesummarizedasfollows. 1. SetSP=;. 2. Set(i,p)=argmaxf^(SP[(j,q)),(j,q)=2SPg 3. If^(SP[(i,p))^(SP),thenstop. 4. Otherwise,setSP=SP[(i,p),andreturnto2. 4.5NumericalExperiments ThissectionpresentscomputationalresultsforthecombinedSAAalgorithmandtheheuristicapproachforsolvingourCCTSproblem.First,wedemonstratetheconvergencepropertyofthecombinedSAAalgorithmforourproblem.Second,wecomparetheperformanceofthesetwodifferentsolutionapproaches.Inadditiontothese,wealsoanalyzedtheimpactsofdifferentparametersandconsumerbehaviorsontheassortmentandpricingdecisions.WeimplementedbothapproachesintheC#programminglanguage,withthemixedintegerSAAproblemsolvedusingILOGR'sCPLEXR12.5solverwithConcertTechnology.WeperformedalltestsonacomputerwithanIntelRDualCore1.7GHzand6GBmemory. 4.5.1ConvergenceTests TodemonstratetheconvergencepropertyofthecombinedSAAalgorithm,weusedoneproblemsetwithtwoconsumersegments(i.e.,K=2),tenproducts(i.e.,jN1j=jN2j=10)andthreepricelevels(i.e.,P=3).Wetestedverandomlygeneratedinstancesforthisproblemset.Foreachprobleminstance,wesetdifferentvaluesforL,LandMtoshowthatoptimalitygapdecreasesasthesamplesizeincreases.Weused(1)]TJ /F4 11.955 Tf 11.96 0 Td[()=99%asourestimationcondencelevel. Table 4-1 summarizesthedistributionsusedingeneratingparametersinourcomputationalstudy.Foreachprobleminstance,theproductwidthsi,volumesi,unitwholesalecostsci,consumer'saveragearrivalrateskandconsumer'spreferences 102

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Table4-1. Parameterdistributionsusedincomputationaltests. Producti'swidthiProducti'svolumeiU(1,3)U(1,9)Retailer'sunitwholesalecost,ciConsumer'saveragearrivalrateskU(0.5,5)U(1000,5000)Retailer'sxedcost,fiConsumer'spreferencesatregularpricevki1ciU(10,100)U(0,15)Retailer'sunitpriceriConsumer'spreferencesatpricepvkipciU(1.3,1.7)vki11.3p)]TJ /F10 7.97 Tf 6.58 0 Td[(1Retailer'sunitinventorysalvagevaluehiRetailer'sunitshortagecost,giciU(0.5,0.9)ciU(0.1,0.5)Retailer'sunitdrop-shippingvaluecdiRetailer'sunitreplenishedcost,miciU(1.1,1.5)ciU(0.01,0.1)ShelfspacelengthABackroominventorycapacityBU(0.01,0.5)U(0.2,0.5)WarehouseinventorycapacityBwProducti'sfacingcapacityiU(0.2,0.5)b9i icLocalstoreservicelevel1)]TJ /F4 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(U(0.01,0.2) atregularpricevki1wererstgeneratedfromsomeuniformdistributions.Second,wecalculatedtheaveragewidthandtheaveragevolumeoverallproducts,andtotalexpecteddemandoverallconsumersegments.Finally,basedonthesegeneratedvaluesandcalculatedvalues,thevaluesofotherparametersweregeneratedbasedonadditionalindependentcontinuousuniformdistributions.AsinChapter 3 ,weletU(l,u)denotethecontinuousdistributionwithlowerboundlandupperboundu. Table 4-2 summarizestheresultsforinstance1,andtablesforotherinstancesarepresentedinAppendix E .AlltablesshowthatalargervalueofLresultsinasmallerrelativeoptimalitygap,whichshowsempiricallythattheoptimalsolution 103

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Table4-2. ComputationalresultsI-instance1. (M,L)LLBUBGap=LB UB100(%)Time(s)(25,1000)2032810331050.901145032813328670.1622410032807328520.13448 (25,2000)2032810328770.512205032814328600.1430710032821328580.11564 (50,1000)2032812329480.412295032809328610.1547610032830328500.06828 (50,2000)2032820329340.344345032821328660.1359410032831368480.051083 indeedconvergesasLincreases.However,thebestfeasiblesolution(i.e.,LB)maynotnecessarilyincreaseasLincreases.Thisisbecause,asLincreases,thefeasibleregionoftheapproximationbecomessmallerwhichwouldleadtomoreconservativesolutions.ThisobservationisconsistentwiththendingsofLuedtkeandAhmed[ 69 ].Moreover,foraxedvalueL,therelativeoptimalitygapdecreasesasthevalueofMLincreases. 4.5.2ComparisonofSolutionMethods TobenchmarktheperformanceoftheSAAalgorithmwiththegreedyandlocalsearchheuristic,wetestedoursolutionmethodsusingtwelveproblemsets.Eachproblemsetcorrespondstoaxednumberofconsumersegments,numberofproductsandnumberofpricelevels,(K,jN1j,jN2j,P),whereK2f1,3g,jN1j,jN2j2f5,10gandP2f2,3,4g.Wetestedtwentyrandomlygeneratedinstancesforeachcombinationof(K,jN1j,jN2j,P)values,foratotalof240probleminstances.FortheSAAalgorithm,wesetthescenarioparametersL=100,M=25,L=1000and=1%.Forthegreedyandlocalsearchheuristic,wesetL=1000. TocomparetheSAAalgorithmandthegreedyandlocalsearchheuristic,weconsidertherunningtimeandrelativeoptimalityperformance.Theresultsofourtests,averagedoverthetwentyrandomprobleminstancesforeachcombinationof 104

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(K,jN1j,jN2j,P)values,arepresentedinTable 4-3 .Inthetable,SAArepresentstheSAAalgorithmandGreedyrepresentsthegreedyandlocalsearchheuristic.Thetableshowsthatforbothmethods,therunningtimeincreasesinL,LandM.Forthemajorityofproblems,theheuristicrunsfasterthantheSAAalgorithm,exceptforproblemswith(jN1j,jN2j,P)=(10,10,2).ThisisbecausetherunningtimeoftheSAAalgorithmincreasesexponentiallyasthesizeofoftheproblemincreases,buttherunningtimeoftheheuristicincreasespolynomiallyasthesizeoftheproblemincreases4.Asaresult,forsmallprobleminstances,itispossiblethattheSAAalgorithmmayrunfasterthantheheuristic,butweexpectthatthespeeddominanceoftheheuristicwillbecomemoreandmoreobviousasthenumberofproductsorthenumberofpricelevelsincreases. Forallproblemsets,therelativeoptimalityperformanceofbothmethodsdecreasesastheproblemsizeincreases.Asamyopicmethod,theheuristicstillhasanaveragerelativeoptimalitygapunder2%.Inconclusion,forlargesizeprobleminstances,theheuristichasmuchshorterrunningtimeandacceptableoptimalityperformance.Asaresult,thegreedyandlocalsearchheuristiccanbeusedasanalternativesolutionapproachforlargesizeproblemswhencomputingpowerislimitedorsolutiontimeiscritical. 4.5.3ParameterAnalysis Thegoalofthissectionistostudyhowdifferentparametersinuencetheclicks-and-mortarretailerdecisions.Forcomparisonpurposes,weusedabaseparametersettinginthissection,assummarizedinTable 4-4 .Foreachrandomprobleminstanceused,i,i,ci,kandvkp1arestillgeneratedfromcontinuousuniform 4Bothmethodssolveaseriesofmixedintegerprograms.FortheSAAalgorithm,thenumberofmixedintegerprogramssolvedisMandthenumberofbinaryvariablesineachmixedintegerprogramisequalto(jN1j+jN2j)P+L.Atthesametime,thenumberofmixedintegerprogramssolvedintheheuristicisatmost(jN1j+jN2j+1)(jN1j+jN2j) 2PandthenumberofbinaryvariablesineachprogramisequaltoL. 105

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Table4-3. ComputationalresultsII. ProblemsetSAAGreedy KjN1j=jN2jPRelativeTimeRelativeTimegap(%)gap(%) 1520.12740.662930.17950.914240.201621.11541020.151050.8513830.202250.9415340.383481.101933520.092040.697330.122830.919440.374001.301151020.312611.2128430.694721.3231740.888781.70364 distributions.However,incontrasttoTable 4-1 usedinSubsections 4.5.1 and 4.5.2 ,thevaluesofotherparametersaredeterminedbydeterministicfunctionsofi,i,ci,korvkp1.Usingthisbaseparametersetting,weexaminedhowchangesinspecicparametervaluesaffectthenumericalresults.Foraparameterofinterest,e.g.,theretailer'ssalvagevaluehi,wegeneratedthreerandomprobleminstancesforthecaseof(K,jN1j,jN2j,P)=(2,10,10,3)usingthebaseparametersetting,thenwexedthevaluesofallotherparametersforeachprobleminstance,andconsideredhowchangingthevalueoftheparameterofinterestaffectsthenumericalresults.Thatis,forthesalvagevaluehi,weconsidervecases: 1. hi=0.1ci,i=1,...,jNj; 2. hi=0.3ci,i=1,...,jNj; 3. hi=0.5ci,i=1,...,jNj; 4. hi=0.7ci,i=1,...,jNj;and 5. hi=0.9ci,i=1,...,jNj. Wethencomputedthefollowingcriteriaforeachcase: maximumprot; 106

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shelfspaceutilization(SU)Pi2N1wi A100%; backroomcapacityutilization(BU)Pi2N1(yi)]TJ /F16 7.97 Tf 6.59 0 Td[(iwi) B100%; warehousecapacityutilization(WU)Pi2N2yi Bw100%;and adjustedservicelevel(AS)Pi2N[PKk=1lkPkip(x))]TJ /F5 7.97 Tf 6.58 0 Td[(slip] PKk=1PLl=1lk. Thecriterionadjustedservicelevelisdifferentfromtheservicelevelfortworeasons:rst,itaccountsforboththelocalstoreandonlinestore,becausetheonlineout-of-stockitemstillinducesextracost,eventhoughtheonlinestorealwayssatisesdemands;second,theservicelevelonlyconsidersconsumerswhodecidetomakeapurchase,whereastheadjustedservicelevelconsidersallcustomerarrivals,eveniftheconsumerpurchasesnothing. Theguresthatfollowillustratetheresultsoftheseexperiments,whichleadtothefollowingobservations. InFigure 4-2 ,wevariedthevalueoftheretailer'ssalvagevaluehiandxedthevaluesofallotherparameters.Thegureshowsthatahighervalueofhiimplieslargervaluesoftheretailer'soptimalprot,theretailer'swarehousecapacityutilizationWUandtheadjustedservicelevelAS.Thisisbecauseahighervalueofhiimpliesalowercostofremaininginventoryattheendofthesellingseason.Asaresult,ahighervalueofhiusuallyleadstoalargeronlineorderquantityyi,whichutilizesmorewarehousecapacityandcoversmoreconsumerdemands.Butnorelationshipcanbeobservedbetweenhiandtheretailer'slocalinventoryspaceutilization(i.e.,BUandSU).Thisisbecausethelocalinventorylevelisrestrictedbytheservicelevel1)]TJ /F4 11.955 Tf 12.65 0 Td[(,andthe0.95servicelevelinthebaseparametersettingleaveslittleroomforincreasinglocalorderquantitywhenvalueofhiishigh. InFigure 4-3 ,weshowhowperformancecriteriachangewiththeretailer'sunitdrop-shippingcostcdi.Forallprobleminstances,ahighervalueofcdiimpliesalowervalueof.Thiseffectisquiteintuitive,becausetheincreasedcostofon-linechannel 107

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Table4-4. Parameterdistributionsusedinparameteranalysis. Producti'swidthiProducti'svolumeiU(1,3)U(1,9)Retailer'sunitwholesalecost,ciConsumer'saveragearrivalrateskU(0.5,5)U(1000,5000)Retailer'sxedcost,fiConsumer'spreferencesatregularpricevki1ci50U(0,15)Retailer'sunitpriceriConsumer'spreferencesatpricepvkipci1.5vki11.3p)]TJ /F10 7.97 Tf 6.58 0 Td[(1Retailer'sunitinventorysalvagevaluehiRetailer'sunitshortagecost,gici0.7ci0.3Retailer'sunitdrop-shippingvaluecdiRetailer'sunitreplenishedcost,mici1.3ci0.05ShelfspacelengthABackroominventorycapacityB0.20.5WarehouseinventorycapacityBwProducti'sfacingcapacityi0.5b9i icLocalstoreservicelevel1)]TJ /F3 11.955 Tf 11.96 0 Td[(epsilon0.95 stockoutsreducestheretailer'sprot.Inaddition,forallprobleminstances,WUandASincreaseasthecdiincreases.Thisisbecauseahighervalueofcdimakesitlessprotabletodrop-shipunsatisedonlineconsumerorders.Asaresult,theretailerwillincreasetheinventorylevelatthewarehousetoreducetheprobabilityofstockoutattheonlinestore.. Figure 4-4 showsthatwhenthevalueofincreases,alsoincreases.Becauseahighervalueofimpliesalargerfeasibleregionforthethechance-constrainedtwostageprogram,weexpecttheretailerwillgainahigherprotasaresult.WealsoobservedthatBUandSUdecreaseasincreases.Thisobservationisconsistentwith 108

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thefactthatanincreaseinservicelevelrequiresanincreaseintheinventorylevel,whichresultsinhighutilizationoflocalstorespace. Figure4-2. Differentcriteriawithhiatdifferentlevels. 4.5.4ConsumerBehaviorAnalysis Inadditiontotheaboveanalysis,wearealsointerestedinhowconsumerpurchasingbehaviorsaffecttheretailerassortmentdecisions.Tothisend,westillusedthebasicsettingsummarizedinTable 4-4 .Wegeneratedthreerandomprobleminstancesforthecaseof(K,jN1j,jN2j,P)=(5,5,5,3)usingthebaseparametersetting. 109

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Figure4-3. Differentcriteriawithcdiatdifferentlevels. ThereisonedifferencefromSubsection 4.5.3 :hereweassumethatallproductsareidentical,i.e.,eachproducthassamecosts,priceandsize. Werststudiedtheeffectsoftheconsumer'sconsiderationset,andweconsideredthefollowingthreecases: 1. completelyoverlappingcase:Ck=f(i,p)ji2N,p=1,...,Pg; 2. partiallyoverlappingcase:Ck=f(i,p)ji2N)-222(fk,k+5g,p=1,...,Pg;and 3. completelydifferentiatedcase:Ck=f(i,p)ji2fk,k+5g,p=1,...,Pg. 110

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Figure4-4. Differentcriteriawithatdifferentlevels. Inthecompletelyoverlappingcase,theconsiderationsetsofanytwoconsumersegmentsareidentical.Incontrast,inthecompletelydifferentiatedcase,theconsiderationsetsofanytwodifferentconsumersegmentsaremutuallyexclusive.Thepartiallyoverlappingcaseisinbetweenthesetwocases.TheeffectscanbeseeninFigure 4-5 inwhichLi(Oi)correspondstotheproductiatlocal(online)channel.Beforeinterpretingtheresults,werstdenetheretailerstockingrateastheratioofthenumberofproductsincludedintheoffersetStothetotalnumberofproductsintheentiresetN.Accordingtothisdenition,ahigherretailerstockingrateimpliesawider 111

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Figure4-5. Effectsofconsumer'sconsiderationsetonassortment. productassortment.Itshouldbenotedthatforallprobleminstances,thecompletelyoverlappingcasehasthelowestretailerstockingrateandthecompletelydifferentiatedcasehasthehighestretailerstockingrate.Thisisbecause,undertheMNLmodel,increasingtheretailerstockingrateincreasesconsumersopportunitiestondproductsthatbestsatisfytheirneedsanddecreasestheprobabilityofnopurchase.Atthesametime,awiderassortmentmayalsoincreasethewithin-retailercannibalization.Inthecompletelyoverlappingcase,addingaproducttoanonemptyoffersetwillincreasesalescannibalizationforsure.Ontheotherhand,inthecompletelydifferentiatedcase,addingaproducti2N1willnotinducecannibalizationunlessproducti+5isalreadyintheofferset.Insummary,amarketwithhigherconsumerdiversitytendstorequireawiderproductassortment. Inadditiontotheconsumer'sconsiderationsets,wealsoconsidertheeffectsofconsumerpreferenceweightsontheretailerassortmentdecisions.Tothisend,wegeneratedanotherthreerandomprobleminstancesforthecaseof(K,jN1j,jN2j,P)=(5,5,5,3)usingthebaseparametersetting,andwestillassumedthatalltheproductsareidentical.Moreover,weassumedalltheconsumersegmentsareidenticalandhavetheentireproductsetasaconsiderationset.Thesetwoassumptionsimplythat,foranyvalueofp,theconsumer'spreferenceweightsvkiparethesameforallkandi.Foreachprobleminstance,weconsiderfourcases: 112

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1. vki1=vk0; 2. vki1=3vk0; 3. vki1=5vk0;and 4. vki1=7vk0. TheresultsarepresentedinFigure 4-6 .Wenoticedthat,astheconsumerpreferenceweightvkipincreases,theretailerstockingratedeclines,andtheretailer'soptimalprotincreases.Thisisbecausethatalowconsumerpreferenceweightimpliesahighnopurchaseutility.vanRyzinandMahajan[ 95 ]pointedoutthatahighnopurchaseutilityrepresentstheexistenceofmanyattractiveexternaloptions,anditisintheretailer'sinteresttohaveawiderassortment.Ontheotherhand,alownopurchaseutilitymeanslessexternalthreat,andmorethreatscomefromwithin-retailercannibalization,sotheretailershouldlimitthebreadthoftheassortment.Lastbutnotleast,ahighervalueofnopurchaseutilityimpliesamorecompetitiveretailenvironment,anditisintuitivethattheretailerwillhaveloweroptimalprotunderamorecrowdedmarket. Figure4-6. Effectsofconsumer'spreferenceweightsonassortment. 113

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CHAPTER5CONCLUSIONS Inthischapter,wesummarizetheresearchwediscussedinChapter 2 throughChapter 4 .Wereviewtheemployedmethodologiesandcorrespondingnumericalresults,andwealsosuggestfutureresearchdirections. Chapter 2 examinesthedegreetowhichthebullwhipeffectresultsfrompriceuctuationsinatwo-echelonsupplychainwithdeterministicandprice-sensitivedemand.Weprovidenumericalevidencethatincreasedsystemprotcancoexistwiththebullwhipeffectasaresultofpricepromotionsif:(i)thesupplierjudiciouslysetsthepricediscount;(ii)thereisasufcientnumberofimpulsivecustomerswhobuytheproductatthediscountedprice;(iii)thepricediscountdoesnotinduceahighdegreeofend-customerforwardbuying.However,evenwhenthetotalsystemprotincreases,theretailertakesadisproportionatelylargershareofthisprotgain,whilethesupplierincursgreateroperationscostsandtendstoobserveamarginalprotgain.Thenumericalexperimentsalsoillustratetwomechanismsthatcausethebullwhipeffect.Inaddition,thischapterbuildsabasicstructureforfutureresearch.Forexample,wemayconsiderageneralizedmodelwithstochasticdemand,treatingthesupplier'spricediscountdasadecisionvariableinsteadofparameterizingonit.Wearealsointerestedinissuesrelatedtohandlingmultipleproductsanddesigningsupplier-retailercontractsfordiscountpolicies,bothofwhichserveasinterestingdirectionsforfurtherrelatedresearch. Chapter 3 considersastochasticbilevelmodelthatsimultaneouslydeterminesasupplier'stradepromotionpolicyandaretailer'soperationsdecisions.Wepresentedaprocedurewhichtransformsthestochasticbilevelmodeltoadeterministicsingle-levelproblemintheformofageneralizedbilinearprogrammingproblem.WeprovidedanexactsolutionmethodforsolvingthisGBPproblem,andcomparedthisapproachwithAl-Khayyal'sapproachandawidely-usedheuristicmethodforsolvingGBPs.Basedon 114

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ournumericalstudy,ourexactalgorithmhasproventobequiteefcient.InadditiontothendingsinChapter 2 ,weagainprovidednumericalevidencethatincreasedsupplierprotandincreasedsystemprotcancoexistwiththebullwhipeffectasaresultofpricepromotionsif:(i)thesupplieraccountsfortheretailer'sreactionswhenmakingpromotiondecisions;(ii)thereisasufcientnumberofadditionalconsumerswhoareattractedbythediscountedprice;and(iii)thepass-throughrateissetjudiciouslybytheretailer.Thischapterbuildsafoundationforfutureresearch.Forexample,wemayconsidertheretailer'spass-throughrateasadecisionvariable.Wearealsointerestedinthepromotiondesignproblemwhentheeffectivenessofapromotiondependsonthesupplier'sandretailer'spreviousdecisions.Finally,solvinglarge-scaleprobleminstanceswouldlikelynecessitatetheapplicationofheuristicsolutionmethods,whereinthemethodswehaveproposedmaybeveryusefulforprovidingboundsonoptimalsolutions. Chapter 4 considersachance-constrainedtwostagemodelthatdeterminesaclicks-and-mortarretailer'sassortmentandpricingdecisionsintwosellingchannels.Weadoptedawell-developedcombinedsampleaverageapproximationalgorithmforsolvingthisCCTSmodelapproximately,andournumericalresultsshowtheapproximatesolutionobtainedfromtheSAAalgorithmwillconvergetotherealoptimalsolutionasthesamplesizeincreases.InadditiontotheSAAalogrithm,wedevelopedourowngreedyandlocalsearchheuristic,whichcanreducetheproblemsolvingtimesignicantlyforlargesizeprobleminstanceswithoutcompromisingtheoptimalityperformancesignicantly.Thischapterbuiltafoundationforfutureresearchonassortmentandpromotionplanninginmultiplesellingchannels.Forexample,itishardtousetheMNLmodelinthecasewherecustomersubstitutionbehaviorisdynamic,buttheexogenousdemand(EXD)modelcanbeeasilyextendedtoincludedynamicstockoutsubstitutions.Asaresult,futurestudymayconsiderusingtheEXDmodelinplaceoftheMNLmodeltodesignproductassortmentandpromotionsinasimilar 115

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problemsetting.Wemayalsoconsiderthedistributionstructureinwhichthevirtualchannelinventoryisavailabletothelocalcustomerswhichallowsalocalstockouttobefullledfromtheonlinewarehouse. 116

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APPENDIXAPROOFTHATRETAILER'SPROBLEMISACONVEXPROGRAMFORFIXEDLAND Werstassumethattheimpulsivebuyingrateisatleastasgreatastheforwardbuyingrate,andthattheincreaseindemandrateduetoforwardbuyingisnomorethanfourtimesthenormaldemandrate.Notethatforxedvaluesofland,D1isxedandtheconstraintsarelinearintheremainingvariables.TheHessianmatrixoftheobjectivefunctionisasfollows: 52PR1(Q1,q1,m)=i(c)]TJ /F3 11.955 Tf 11.95 0 Td[(d) 2LD1)]TJ /F6 11.955 Tf 9.3 0 Td[(2)]TJ /F16 7.97 Tf 6.59 0 Td[(~rf D00)]TJ /F16 7.97 Tf 6.59 0 Td[(~rf D0)]TJ /F10 7.97 Tf 6.58 0 Td[(2(~ri+r0) D00000(A) ThisHessianmatrixisnegativesemi-deniteif4D0(~ri+r0)2~r2f,andthisconditioncanbewrittenas4D0~r2f ~ri+r0.Nextnotethatif~ri+r0~rf,thenwehave~r2f ~ri+r0~rf.Theterm~rfcorrespondstothetotalforwardbuyfrombrandloyalsduringthepromotionperiod.Thus,wehavethattheobjectivefunctionisconcave,providedthatthecondition4D0~rfholds.Thatis,aslongastheamountoftheforwardbuyisnomorethanfourtimesthenon-discounteddemandrate,ourobjectivefunctionisconcave.Theconcavityoftheobjectiveandthelinearityoftheconstraintswhenlandarexedimplythatundertheconditionsstated,theretailer'sproblemisaconvexprogram. 117

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APPENDIXBPROOFOFSUFFICIENTOPTIMALITYCONDITIONFORCONVEXMIXEDINTEGERPROGRAMWITHONEINTEGERVARIABLE TheoremB.1. Letf(x1,...,xn,y)beaconcaveandcontinuouslydifferentiablefunc-tionoverSR,whereSisanopenconvexsubsetofRn.Let(x1,...,xn,y)beaglobaloptimalsolutionoftheproblemmaxff(x1,...,xn,y)j(x1,...,xn,y)2SRg.Thenweclaimeither(x1,...,xn,byc)=argmax(x1,...,xn)2Sff(x1,...,xn,bycgor(x1,...,xn,dye)=argmax(x1,...,xn)2Sff(x1,...,xn,dyegistheoptimalsolutionoftheproblemmaxff(x1,...,xn,y)j(x1,...,xn,y)2SZg. Proof. Firstwewanttoshowf(x1,...,xn,byc)istheoptimalsolutionoftheproblem maxff(x1,...,xn,y)jybyc,(x1,...,xn,y)2SRg.(B) Sincef(x1,...,xn,y)isaconcaveandcontinuouslydifferentiablefunctionoverSR,andybycisalsoconcaveandcontinuouslydifferentiableoverSR,itfollowsthatapoint(x1',...,xn',y')isanoptimalsolutionof( B )ifandonlyif(x1',...,xn',y')isanKKTpoint,i.e., rf(x1',...,xn',y')+0BBBBBBB@0...011CCCCCCCA=0 (B)(y')-222(byc)=0 (B)y'byc (B)(x1',...,xn',y')2SR. (B) Ify'=byc,wearedone;otherwise,wehave=0,whichimpliesrf(x1',...,xn',y')=0andf(x1',...,xn',y')=f(x1,...,xn,y).Alsoy'byc,sothereexists2[0,1]such 118

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thatbyc=y'+(1)]TJ /F4 11.955 Tf 12.14 0 Td[()y.Bythepropertiesofconcavefunctionsandthedenitionof(x1,...,xn,byc),wehavef(x1,...,xn,byc)f(x1'+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()x1,....,xn'+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()xn,byc)f(x1',...,xn',y')+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()f(x1,...,xn,y)=maxff(x1,...,xn,y)j(x1,...,xn,y)2SRgmaxff(x1,...,xn,y)jybyc),(x1,...,xn,y)2SRg. Fromabove,wecanconcludethat(x1,...,xn,byc)isanoptimalsolutionof( B ),andwecanfurtherconcludethat f(x1,...,xn,byc)=maxff(x1,...,xn,y)jybyc),(x1,...,xn,y)2SZg(B) Similarly,wecanshowthat f(x1,...,xn,dye)=maxff(x1,...,xn,y)jydye),(x1,...,xn,y)2SZg(B) Combining( B )and( B ),theproofiscomplete. 119

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APPENDIXCDECOMPOSITIONOFUNCAPACITATEDMINIMUMCOSTFLOWPROBLEM Thissectionshowsaproofthatanuncapacitatedminimumcostowproblemcanbedecomposedintoasetofshortestpathproblems. TheoremC.1. Asingle-origin,N-destinationuncapacitatedminimumcostowproblemcanbedecomposedintoNshortestpathproblemsthatdonotdependonthedemandlevel. Proof. AssumewithoutlossgeneralitythatwehaveadirectednetworkG=(V,E),whereV=f0,1,...,NgisthesetofnodesandEisthesetofarcs.Eacharc(i,j)2Ehasanassociatedcostcijthatdenotesthecostperunitowonthatarc.Wealsoassumethateacharchasaninnitecapacity,i.e.,thereisnoupperboundonthemaximumamountthatcanowonanyarc.Thenetworkhasauniquenode0,calledthesource,withsupplyPNi=1di.Foreachnonsourcenodei2V,weassociatewithitademandleveldi(0).Thesingle-originN-destinationuncapacitatedminimumcostowproblemistodetermineaminimumcostowthroughtheuncapacitatednetworkinordertosatisfydemandsattheNnonsourcenodesfromavailablesuppliesatthesourcenode0.Thedecisionvariablesinthisproblemarearcows,andwerepresenttheowonanarc(i,j)2Ebyxij.Thesingle-originN-destinationuncapacitatedminimumcostowproblemcanbeformulatedasfollows:(UMCF)minX(i,j)2Ecijxijs.t.Xj:(0,j)2Ex0j)]TJ /F12 11.955 Tf 20.31 11.36 Td[(Xj:(j,0)2Exj0=NXi=1diXj:(i,j)2Exij)]TJ /F12 11.955 Tf 19.43 11.36 Td[(Xj:(j,i)2Exji=)]TJ /F3 11.955 Tf 9.3 0 Td[(di8i2f1,...,Ngxij08(i,j)2E. 120

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Theowdecompositiontheorem[ 5 ]impliesthattheowxijoneacharc(i,j)2Ecanbedecomposedbydestination,withxlijrepresentingofowonarc(i,j)usedtosatisfydemandatnodel.Withthisnotation,wecanreformulatethesingle-originN-destinationuncapacitatedminimumcostowproblemasfollows:(UMCF')minNXl=1X(i,j)2Ecijxlijs.t.Xj:(0,j)2Exl0j)]TJ /F12 11.955 Tf 20.3 11.36 Td[(Xj:(j,0)2Exlj0=dl8l2f1,...,NgXj:(i,j)2Exlij)]TJ /F12 11.955 Tf 19.43 11.36 Td[(Xj:(j,i)2Exlji=8>><>>:0ifi6=l)]TJ /F3 11.955 Tf 9.3 0 Td[(dlifi=l8i,l2f1,...,Ngxlij08(i,j)2E,l2f1,...,Ng. Bydenition,wehavexij=PLl=1xlij.Supposewedenevariablesylij=xlij dl.Usingthese,thesingle-originN-destinationuncapacitatedminimumcostowproblemcanbeexpressedasfollows:(UMCF)minNXl=1dlX(i,j)2Ecijylijs.t.Xj:(0,j)2Eyl0j)]TJ /F12 11.955 Tf 20.31 11.36 Td[(Xj:(j,0)2Eylj0=18l2f1,...,NgXj:(i,j)2Eylij)]TJ /F12 11.955 Tf 19.43 11.36 Td[(Xj:(j,i)2Eylji=8>><>>:0ifi6=l)]TJ /F6 11.955 Tf 9.3 0 Td[(1ifi=l8i,l2f1,...,Ngylij08(i,j)2E,l2f1,...,Ng. 121

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Inthisaboveformulation,thesetoffeasiblesolutionsYcanbedecomposedintoNsetsY1,Y2,...,YN,whereYl8>>>>>><>>>>>>:yl:Xj:(i,j)2Eylij)]TJ /F12 11.955 Tf 19.43 11.36 Td[(Xj:(j,i)2Eylji=8>>>>>><>>>>>>:1ifi=00ifi6=l)]TJ /F6 11.955 Tf 9.3 0 Td[(1ifi=l8(i,j)2E9>>>>>>=>>>>>>;. Byobservation,thevariablesforeachsubsetdonotappearinanyotherset.Consequently,wecandecomposeUMCPintoNshortestpathproblemswhichdonotdependentonthedemandlevelinthefollowingformulation:(SPl)mindlX(i,j)2Ecijylij=dlminX(i,j)2Ecijylijs.t.Xj:(0,j)2Eyl0j)]TJ /F12 11.955 Tf 20.31 11.35 Td[(Xj:(j,0)2Eylj0=1Xj:(i,j)2Eylij)]TJ /F12 11.955 Tf 19.43 11.36 Td[(Xj:(j,i)2Eylji=8>><>>:0ifi6=l)]TJ /F6 11.955 Tf 9.3 0 Td[(1ifi=l8i2f1,...,Ngylij08(i,j)2E. AftersolvingalloftheseNshortestpathproblems,wecanobtainthesolutionstotheoriginalUMCPusingthefollowingrelationship:xij=LXl=1xlij=LXl=1dlylij8(i,j)2E. 122

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APPENDIXDDETAILEDAL-KHAYYAL'SAPPROACH ThedetailedAl-Khayyal'sbranch-and-boundalgorithmisshownbelow. Algorithm2Al-Khayyal'sBranch-and-BoundAlgorithm 1: (1,1) ,T1 f(1,1)g,k 1,UB(1,1) 1,UB 1andLB ; 2: whileUB)]TJ /F3 11.955 Tf 11.95 0 Td[(LB>do 3: chooseanactivenode(u,v)2Tkandremovenode(u,v)fromactivenodeset; 4: anoptimalsolutiontoLP((u,v)); 5: UBk (); 6: if(x,y,z,)satisfyingconstraint( 3 )then 7: LBk (x,y,z,); 8: ifLBkLBthen 23: LB LBkand(x,y,z,) (x,y,z,); 24: Tk Tk)-222(f(m,n)2Tk:UB(m,n)LBg; 25: Tk+1 Tk[f(k+1,1),(k+1,2)g; 26: UB=maxfUB(m,n),8(m,n)2Tk+1g; 27: endif 28: k k+1; 29: endwhile Afterinitializingthebranch-and-boundtreeinline 1 ,thealgorithmrepeatstheMainStepinthewhileloopinlines 2 29 untiltheoptimalitygapislessthanapredeterminedvalue.Atthestartofeachiterationkofthewhileloop,anode(u,v)isselectedandremovedfromtreeTkinline 3 .Inlines 4 and 5 ,thelinearprogramLP((u,v))issolvedtoobtainthesolutionandthecorrespondingoptimalvalue()isassignedtoUBk. 123

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Inlines 6 16 ,thecurrentiterationlowerboundLBkisobtainedandthepartitioningindex(p,q)isfoundbycheckingtheoptimalityandthefeasibilityofthesolution.Afternding(p,q),partitiontheregion(u,v)intotwomutuallyexclusiveandexhaustivesubregionsinlines 18 21 .Attheendofthewhileloop,thebranch-and-boundtreeisupdatedbythreetypesofoperationsinlines 22 27 .Includedhereare 1. updatethelowerboundoftheproblemifLB
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APPENDIXENUMERICALRESULTSFORCHAPTER4 TableE-1. ComputationalresultsI-instance2. (M,L)LLBUBGap=LB UB100(%)Time(s)(25,1000)2029664297220.191475029674297030.0920910029674296890.05436 (25,2000)2029673297080.112915029674296970.0743210029674296740.00949 (50,1000)2029669297410.242545029670296780.0231910029674296760.01828 (50,2000)2029666297210.184935029669296910.0763010029674296740.001106 TableE-2. ComputationalresultsI-instance3. (M,L)LLBUBGap=LB UB100(%)Time(s)(25,1000)2032678327110.101385032682327020.0520610032681327010.05406 (25,2000)2029667327270.182645032670327040.1038610032678296840.02745 (50,1000)2032673327330.182545032671327010.0931910032670326830.03520 (50,2000)2032666327220.174875029676326900.0462910032674326830.021004 125

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TableE-3. ComputationalresultsI-instance4. (M,L)LLBUBGap=LB UB100(%)Time(s)(25,1000)2032492325910.301325032491325290.1118310032502325130.03325 (25,2000)2032497325570.182685032501325210.0636210032498325010.01742 (50,1000)2032494325550.182395032493325160.0729010032497325020.02479 (50,2000)2032492325680.234735032498325290.0957110032496324990.01857 TableE-4. ComputationalresultsI-instance4. (M,L)LLBUBGap=LB UB100(%)Time(s)(25,1000)2037866394120.301325032491325290.1118310032502325130.03325 (25,2000)2032497325570.182685032501325210.0636210032498325010.01742 (50,1000)2032494325550.182395032493325160.0729010032497325020.02479 (50,2000)2032492325680.234735032498325290.0957110032496324990.01857 126

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BIOGRAPHICALSKETCH YiqiangSuwasborninBeijingin1987.YiqiangistheoldestsonofChangchengSuandBingyiJi.Yiqiangearnedhisbachelor'sdegreeinindustrialengineeringandengineeringmanagementfromtheHongKongUniversityofScienceandTechnology(HKUST)in2009,andhewasalsotherecipientoftheAcademicAchievementAwardwhichisthehighestacademichonorbestowedbytheUniversityonundergraduatesupongraduation.YiqiangjoinedtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida(UF)inAugust2009,startinghisdoctoralstudyundertheguidanceofDr.JosephGeunes.HereceivedhisPh.D.fromtheUniversityofFloridainthefallof2013.Followinggraduation,hejoinstheBNSFRailwayCompanyasasenioroperationsresearchspecialist. 135