Citation
Optimization Models for Inventory Systems with Price-Dependent Supply

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Title:
Optimization Models for Inventory Systems with Price-Dependent Supply
Creator:
Teksan, Zehra Melis
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
GEUNES,JOSEPH PATRICK
Committee Co-Chair:
RICHARD,JEAN-PHILIPPE P
Committee Members:
GUAN,YONGPEI
PAUL,ANAND ABRAHAM

Subjects

Subjects / Keywords:
convex
cost
eoq
inventory
lot-sizing
newsvendor
optimization
planning
price-dependent
production
supply

Notes

General Note:
In this dissertation we consider a class of production planning problems for a producer who procures an input component for production by offering a price to suppliers. The available supply quantity for the production input depends on the price the producer offers, and this supply level constrains production output. We consider this problem in several problem settings. First, we study the case where the producer seeks a time-phased production and supply-pricing plan that minimizes the cost incurred while meeting a set of demands over a finite horizon consisting of a discrete number of time periods. We model the problem as a finite-horizon, discrete-time production and component-supply-pricing planning problem with non-stationary costs, demands, and component supply levels. This leads to a class of two-level lot-sizing problems with objective functions that are neither concave nor convex. Although the most general version of the problem is NP-Hard, we provide polynomial-time algorithms for practical special cases. Motivated by this problem we also provide a polynomial-time algorithm for economic lot-sizing problems with convex costs in the production and inventory quantities. The resulting algorithm is based on a primal-dual approach that takes advantage of the problem's special structure. Second, we consider the problem in an infinite-horizon setting with stationary production and inventory holding costs as well as stationary demand and supply rates. We analyze the behavior of the optimal replenishment and pricing policy, which depends on the economics of production and procurement costs and the prices associated with input components and end-items. This analysis sheds light on how to deal with price-dependent supply in production planning, as well as on the value of supplier heterogeneity and information on the relationship between component price and supply. Third, we consider the problem in a single planning period where the demand is random, which provides a generalization of the newsvendor problem. We characterize the optimal order quantity for cases in which the supply versus price relationship is either linear or nonlinear. We also examine cases where demand is also price sensitive, and analyze the behavior of the optimal profit margin.

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UFRGP
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All applicable rights reserved by the source institution and holding location.
Embargo Date:
5/31/2018

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OPTIMIZATIONMODELSFORINVENTORYSYSTEMSWITHPRICE-DEPENDENTSUPPLYByZEHRAMELISTEKSANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2016

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c2016ZehraMelisTeksan

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Tomymother,Fatma,myfather,Rza,andmybrother,Onur

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ACKNOWLEDGMENTSFirstandforemostIwouldliketothankmyadvisorJosephGeunes.DuringmyrstyearinISEPh.D.programIaskedhim,whetherIcouldbehisPh.D.student.Hedidnothesitatetoacceptme.Fromtheverybeginninghetrustedmycapabilitiesandencouragedmetodevelopasagoodresearcher.IamverygratefulforhisunlimitedsupportthroughoutmyyearsatUF.Iamdeterminedtoworkpersistentlynottodisappointhimwithmyfutureacademiccareer.IwouldalsoliketothankmydissertationcommitteemembersJean-PhilippeRichard,YongpeiGuan,andAnandPaulfortheirvaluablecontributioninthedevelopmentofmydissertation.IwouldalsoliketothankmyformeradvisorAliTamerUnalandmyformerco-advisorandcolleagueZ.CanerTasknatBogaziciUniversityfortheirendlesssupportandencouragementthatleadmetopursueaPh.D.degreeandanacademiccareer.Theircontributioninmyprofessionallifeisunquestionable.IamveryhappytobeapartoftheGatorNation.IwouldliketothankeverybodywhotookapartinmylifeduringmyyearsinGainesville,Florida.Iamverygratefultoallmyfriendsandcolleaguesformakingthisexperienceahappilymemorableone.LastbutnotleastIwouldliketothankmyparentsFatmaandRzaandmybrotherOnurfortheirendlessloveandsupportduringalltheseyearswehavebeenapart.Itwasverytoughformeandespeciallyforthemtobethousandsofmilesawayforsuchalongperiodoftime.Therearenowordstodescribemyappreciationfortheirpatience.IwouldalsoliketothankmyanceMehmetforhisconsistentlyrisingloveandmakingmehappy. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 2PRODUCTIONPLANNINGWITHPRICE-DEPENDENTSUPPLY ...... 16 2.1Motivation .................................... 16 2.2RelatedWorkandContributions ....................... 16 2.2.1DynamicLotSizinginSerialProductionSystems .......... 17 2.2.2Price-DependentProductionPlanning ................. 18 2.2.3DynamicLotSizingwithConvexCosts ................ 18 2.2.4PricingandDynamicLotSizinginRemanufacturing ......... 19 2.2.5ContributionsofthisChapter ..................... 21 2.3ProblemDenition,Formulation,andComplexity .............. 21 2.4AnalysisofP(FC)andPolynomiallySolvableCases ............. 25 2.4.1P(FC)withUniformPrice-SupplyFunctionsandVariableProcurementCosts ................................... 31 2.4.2P(FC)withNondecreasingProcurement-RelatedCosts ....... 36 2.5ComputationalTestResults .......................... 39 2.5.1PerformanceoftheDynamicProgram ................. 40 2.5.2PerformanceasaHeuristicSolutionMethod ............. 44 3APOLYNOMIALTIMEALGORITHMFORCONVEXCOSTLOTSIZINGPROBLEMS ..................................... 47 3.1ModelandRelatedWork ............................ 47 3.2ProblemFormulationandSolutionMethod .................. 51 3.2.1GeneralizedKKTConditions ...................... 52 3.2.2SolutionApproach ............................ 52 3.2.3AlternativeImplementation ...................... 58 3.2.4RelationshiptoVeinott'sApproach .................. 59 3.3ProductionPlanningwithPrice-DependentSupply ............. 60 4ANEOQMODELWITHPRICE-DEPENDENTSUPPLYANDDEMAND .. 63 4.1Motivation .................................... 63 4.2RelatedLiterature ............................... 64 5

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4.2.1Price-DependentEOQModels ..................... 64 4.2.2EOQModelswithDiscounts ...................... 65 4.2.3EOQModelsforRemanufacturing ................... 66 4.3ProductionPlanningandPricingModel ................... 67 4.4OptimalSupplyandSellingPrices ....................... 70 4.4.1OptimalDecisionsunderaFixedPrice-to-PriceResponse ...... 73 4.4.2EectsoftheSupply-PriceRelationship ................ 77 4.4.3DecisionMakingwithoutAccountingforSupply-PriceRelationship 83 5ANEWSVENDORPROBLEMWITHPRICE-DEPENDENTSUPPLYANDDEMAND ....................................... 90 5.1MotivationandRelatedLiterature ...................... 90 5.1.1NewsvendorProblemwithPrice-DependentDemand ......... 90 5.1.2NewsvendorProblemwithSupplierDiscounts ............ 91 5.1.3RemanufacturingProblemsinNewsvendorSetting .......... 92 5.2Price-DependentSupply ............................ 93 5.2.1LinearSupply-PriceFunctionCase .................. 96 5.2.2IsoelasticSupply-PriceFunctionCase ................. 97 5.2.3ComparisonwiththeStandardNewsvendorProblem ........ 99 5.2.4RandomSupplyQuantity ........................ 108 5.3Price-DependentSupplyandDemand ..................... 111 5.3.1AdditiveDemandModel ........................ 112 5.3.2MultiplicativeDemandModel ..................... 116 6SUMMARYANDCONCLUSIONS ......................... 121 APPENDIX APROOFOFPROPOSITION2.1 .......................... 125 BPROOFOFPROPOSITION2.2 .......................... 129 CPROOFOFPROPOSITION4.2 .......................... 131 DPROOFOFPROPOSITION4.3 .......................... 133 EPROOFOFPROPOSITION4.4 .......................... 134 FPROOFOFPROPOSITION4.5 .......................... 135 GPROOFOFPROPOSITION4.6 .......................... 136 HPROOFOFPROPOSITION4.7 .......................... 137 IPROOFOFPROPOSITION4.8 .......................... 139 JFIRSTORDERNECESSARYOPTIMALITYCONDITIONSFOR(c;p) ... 140 REFERENCES ....................................... 142 6

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BIOGRAPHICALSKETCH ................................ 149 7

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LISTOFTABLES Table page 2-1Uniformdistributionparameters. .......................... 40 2-2Scenarios. ....................................... 40 2-3Runtimecomparison:Uniformprice-supplyfunctioncase. ............ 42 2-4Sizeofthesetofvertices,Vandarcs,A. ...................... 43 2-5Runtimecomparison:Nondecreasingprocurementrelatedcostcase. ...... 44 2-6Averagefractionofperiodsinwhichasetupoccurs. ................ 44 2-7Performanceofthedynamicprogramasaheuristic. ................ 46 5-1Supply-pricerelationshipexamples. ......................... 101 5-2Marginalcostofprocuringnextunitofsupplyatc. ............... 102 5-3OptimalsolutionstothescenariospresentedinTable5-1comparedwithstandardnewsvendorsolutions. ................................. 105 5-4^cvaluesandcorrespondingsolutionsforthescenariospresentedinTable5-1. .. 108 8

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LISTOFFIGURES Figure page 2-1Supplyvs.pricefunction. .............................. 24 2-2PartialrepresentationofthedirectedgraphG(V;A). ............... 33 4-1Supplyanddemandcurves. ............................. 67 4-2Componentandend-iteminventorylevelswithtimewithabatchsizeofQ. ... 69 4-3Supplyanddemandcurvesforlinearsupply-price-selling-pricerelationship(k=1:9;^p=6). ....................................... 71 4-4Protfunctionfordierentvaluesofb,where^p=4. ................ 74 4-5ProtfunctionfordierentvaluesofF,where^p=4. ............... 75 4-6pcvsFfordierentvaluesofa. ........................... 76 4-7Optimalsellingpricewithrespecttok. ....................... 82 4-8Optimalannualprotwithrespecttok. ...................... 82 4-9Optimalnetrevenueperunitwithrespecttok. .................. 83 4-10Comparisonoftheoptimalchoiceofpcwithrespecttopredeterminedsupplypriceps. ........................................ 86 4-11Comparisonoftheoptimalchoiceofpcwithrespecttopredeterminedsupplypricepsviatheeectsonprot. ........................... 87 4-12Demandandsupplycurveswithrespecttops. ................... 87 5-1Expectedprotfunction(c)withrespecttoc,whereQ(c)islinearwith=500and=1000. .................................. 97 5-2Expectedprotfunction(c)withrespecttoc,whereQ(c)isisoelasticwith=2and=50. .................................. 100 5-3Optimalexpectedprotfunctions(c)andoptimalexpectedprotvaluesforprice-dependentsupplycase. ............................. 102 5-4Optimalexpectedprotfunctions(c)andexpectedprotfunctions(c)withrespecttocwhenQ(c)islinear. ........................... 103 5-5Optimalexpectedprotfunctions(c)andexpectedprotfunctions(c)withrespecttocwhenQ(c)isisoelastic. ......................... 104 5-6OptimalorderquantityfunctionQ(c)andsupply-pricefunctionQ(c). ..... 106 9

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5-7OptimalservicelevelfunctionSL(c)andoptimalservicelevelvaluesforprice-dependentsupplycase. ...................................... 107 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATIONMODELSFORINVENTORYSYSTEMSWITHPRICE-DEPENDENTSUPPLYByZehraMelisTeksanMay2016Chair:JosephP.GeunesMajor:IndustrialandSystemsEngineeringInthisdissertationweconsideraclassofproductionplanningproblemsforaproducerwhoprocuresaninputcomponentforproductionbyoeringapricetosuppliers.Theavailablesupplyquantityfortheproductioninputdependsonthepricetheproduceroers,andthissupplylevelconstrainsproductionoutput.Weconsiderthisprobleminseveralproblemsettings.First,westudythecasewheretheproducerseeksatime-phasedproductionandsupply-pricingplanthatminimizesthecostincurredwhilemeetingasetofdemandsoveranitehorizonconsistingofadiscretenumberoftimeperiods.Wemodeltheproblemasanite-horizon,discrete-timeproductionandcomponent-supply-pricingplanningproblemwithnon-stationarycosts,demands,andcomponentsupplylevels.Thisleadstoaclassoftwo-levellot-sizingproblemswithobjectivefunctionsthatareneitherconcavenorconvex.AlthoughthemostgeneralversionoftheproblemisNP-Hard,weprovidepolynomial-timealgorithmsforpracticalspecialcases.Motivatedbythisproblemwealsoprovideapolynomial-timealgorithmforeconomiclot-sizingproblemswithconvexcostsintheproductionandinventoryquantities.Theresultingalgorithmisbasedonaprimal-dualapproachthattakesadvantageoftheproblem'sspecialstructure.Second,weconsidertheprobleminaninnite-horizonsettingwithstationaryproductionandinventoryholdingcostsaswellasstationarydemandandsupplyrates.Weanalyzethebehavioroftheoptimalreplenishmentandpricingpolicy,whichdependsontheeconomicsofproductionandprocurementcostsandthepricesassociatedwithinput 11

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componentsandend-items.Thisanalysisshedslightonhowtodealwithprice-dependentsupplyinproductionplanning,aswellasonthevalueofsupplierheterogeneityandinformationontherelationshipbetweencomponentpriceandsupply.Third,weconsidertheprobleminasingleplanningperiodwherethedemandisrandom,whichprovidesageneralizationofthenewsvendorproblem.Wecharacterizetheoptimalorderquantityforcasesinwhichthesupplyversuspricerelationshipiseitherlinearornonlinear.Wealsoexaminecaseswheredemandisalsopricesensitive,andanalyzethebehavioroftheoptimalprotmargin. 12

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CHAPTER1INTRODUCTIONInthecenturyfollowingtheseminalpaperbyHarris[ 32 ],theeconomicorderquantity(EOQ)modelhasbeenwidelyusedtosolveinventoryplanningproblemsunderdeterministicandstationarycostanddemandassumptionstominimizelongtermaverageproductionandholdingcosts.TheEOQmodelhasdrawntheattentionofnumerousresearchersbecauseofitswell-established,simpleandeasy-to-modifynature(seeNahmias[ 55 ]andSilver,Pyke,andPeterson[ 75 ]).Anothersegmentofproductionplanningandinventorymanagementresearchfocusesondynamiclot-sizingmodels,whichareusedtosolveproductionplanningproblemswhendynamiceectsexistinmarketdemandandtheeconomiesofproduction(e.g.,WagnerandWhitin[ 88 ]).Thisresearchstreamconsidersproductionplanningproblemswithaniteplanninghorizondividedintodiscretetimeperiods,wheretheproductionandinventoryholdingcostsaswellasthemarketdemandchangefromperiodtoperiod.Anotherresearchstreamfocusesonplanningforitemsforwhichdemandonlyarisesinasinglesellingperiodorseason(e.g.,fashiongoods,newspapers).Thedecisionmakerfacesrandomdemandandneedstodeterminehowmuchtoorder/producetominimizetheexpectedcostofinventoryholding(incaseofoverstocking)andtheexpectedcostoflosingcustomers(incaseofunderstocking).Thisproblem,whichiscalledthenewsvendorproblem,wasrstintroducedbyArrow,HarrisandMarshak[ 5 ],andnumerousvariationshavebeenstudiedbymanyresearcherssincethen.Foreachoftheliteraturestreamsmentionedabove,asegmentofresearchexistswhichhasrecognizedtherelationshipbetweenpricingdecisionsandthedemandlevelsthatdriveproductionrequirements.Pricingdecisionsandprice-dependentdemandrateshavebeenconsideredintheEOQliteraturebymanyresearchersoverthepastseveraldecadesstartingwithWhitin[ 89 ],whopresentedtherstmodelinwhichprice-dependentdemandwasincorporatedwithinthetraditionalEOQmodel(alsoseee.g.,ArcelusandSirinivasan 13

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[ 4 ]andRay,Gerchak,andJewkes[ 62 ]).Inthedynamiclot-sizingcontextthedecisionmaker'sgoalistodeterminetheoptimalpricingscheduleinadditiontoproductionandinventorylevelsusingaprot-maximizingapproach(see,e.g.,Thomas[ 80 ],Gilbert[ 26 ],DengandYano[ 14 ],Geunes,Romeijn,andTaae[ 25 ],vandenHeuvelandWagelmans[ 82 ],andGeunes,Merzifonluoglu,andRomeijn[ 24 ]).Thenewsvendorproblemwithprice-dependentdemandhasalsobeenstudiedbymanyresearchers.ExamplesincludeWhitin[ 89 ],Mills[ 53 ],KarlinandCarr[ 40 ],andPetruzziandDada[ 59 ].Modelsforproductionplanningandinventorymanagementhavetypicallytakenthepricesofproductioninputsasxed.Thatis,thecostofproducinganumberofunitstypicallyequalssomepredened(usuallyconcave)andexogenouslydeterminedfunctionofthequantityproduced.Inacquiringaproductioninputcomponentthatisnotacommonlyavailablemarketcommodity,aproducermayinsomecaseshavediscretionoverthecostoftheinputcomponentviatheirwillingnessorabilitytopaysuppliersoftheinput.Thatis,forcertainclassesofproductioninputs,thepricetheproduceroerstosuppliersforthecomponentmaydeterminethequantitythatsuppliersarewillingtoprovide.Ahigherpriceoeredtosupplierswilllikelyattractagreaterquantityofthecomponent.Thisphenomenonisparticularlyrelevantforaremanufacturerwhorequiresinputsfromindividualconsumerswhoownaproductthattheremanufacturerwishestoacquireasaproductioninput.Oeringahigherpricetoconsumersincreasesthenumberofconsumerswhowillsellbacktheproducttotheproducer.Beyondtheremanufacturingcontext,however,producersroutinelyseeksuppliersforcustomizedproductioninputs;thepriceoeredbytheproducerdirectlyimpactsthequantitytheyareabletoacquirefrompotentialsuppliers.Motivatedbytherelationshipbetweensupplypriceandquantity,weconsidervariousinventorymodelswheretheavailablesupplyservingasinputforproductionisprice-dependent.Eachnalproductrequiresaninputcomponentfromanexternalsupplier.Forexample,intheremanufacturingcontext,eachnalproductisremanufactured 14

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usinganolditemreturnedbyanend-user.Toensureavailabilityofproductioninputs,theproduceroersapricetosuppliers,wherethetotalsupplyofproductioninputsdependsonthepriceoeredbytheproducer/remanufacturer.Intheproblemsettingsweconsider,theproducerneedstodecideonthepriceofproductioninputaswellastheproductionandinventorylevels.Chapter 2 presentstheproductionplanningproblemwhereweconsideraproducerwhowishestomeetasetofdemandsforanalproductoveraniteplanninghorizondividedintodiscretetimeperiods.Theproducerseekstomeetallnalproductdemandswhileminimizingitsproduction,inventory,andsupply-input-relatedcosts.Theresultingmodeltakestheformofatwo-leveldynamiclotsizingproblemwithdiseconomiesofscaleinprocurementcostsandeconomiesofscaleinproduction.Thespecialcaseofthisproblemclasswithzeroxedchargesinproductionandprocurementlevelsleadustothestudyofdynamiclot-sizingproblemswithconvexcosts,forwhichwedevelopedapolynomialtimesolutionalgorithm(seeChapter 3 ).Chapter 4 introducesaproductionplanningprobleminwhichaninputcomponentisrequiredforproductionandforwhichallcost,demandandpricingparametersareassumedtobestationary.Chapter 5 introducesanewsvendorproblemwhereagaintheavailabilityofinputcomponentsdependsonthepriceoeredtosuppliers.InChapter 6 wesummarizetheresultsofourstudy. 15

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CHAPTER2PRODUCTIONPLANNINGWITHPRICE-DEPENDENTSUPPLY 2.1MotivationInthischapter,weconsideraproductionplanningproblemwheretheproducerseeksatime-phasedproductionandsupply-pricingplanthatminimizesthecostincurredwhilemeetingasetofdemandsoveranitehorizonconsistingofadiscretenumberoftimeperiods.Wemodeltheproblemasanite-horizon,discrete-timeproductionandcomponent-supply-pricingplanningproblemwithnon-stationarycosts,demands,andcomponentsupplylevels.Thisleadstoaclassoftwo-levellot-sizingproblemswithobjectivefunctionsthatareneitherconcavenorconvex.AlthoughthemostgeneralversionoftheproblemisNP-Hard,weprovidepolynomial-timealgorithmsforpracticalspecialcases.Wethenapplytheresultingalgorithmsheuristicallytothemoregeneralproblemversion,andprovidecomputationalresultsthatdemonstratethehighperformancequalityoftheresultingheuristicsolutionmethods.Therestofthischapterisorganizedasfollows.Section 2.2 summarizesliteraturerelatedtoourworkanddiscussesthecontributionsofthischapter.Section 2.3 introducesthenotationweusethroughoutthechapter,statesthemodelingassumptions,providesageneralmodelformulation,anddemonstratestheNP-Hardnessofthegeneralmodel.Section 2.4 considerspracticallyrelevantspecialcasesofthegeneralmodelanddemonstratesthepolynomialsolvabilityofthesespecialcases.Section 2.5 appliestheresultingalgorithmstodemonstratetheirperformancecharacteristics,bothasanexactapproachforthespecialcasesforwhichtheyweredeveloped,andasaheuristicapproachformoregeneralproblemcategories. 2.2RelatedWorkandContributionsThissectionprovidesareviewoftheliteraturerelatedtoourwork,whichfallswithinthreemainresearchstreams:dynamiclotsizinginserialproductionsystems,pricingindynamiclotsizing,anddynamiclotsizingwithconvexcosts.Anaturalapplicationarea 16

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ofourmodeliswithinaremanufacturingsettingwherearemanufacturerwishestocollectusedproductsfromthemarkettoserveasinputtoitsproductionprocess.Therefore,wewillreviewasegmentoftheremanufacturing/reverselogisticsliterature,focusingonthoseworkswhicharemostcloselyrelatedtoours. 2.2.1DynamicLotSizinginSerialProductionSystemsAswewilllatersee,themodelweconsiderresultsinadynamiclotsizingproblemwithtwostagesinseries,i.e.,theinputcomponentprocurementstagefollowedbytheproductionstage.Zangwill[ 92 93 ]providedseminalworkonmultistageproductionsystems,characterizingextremepointsolutionpropertiesandtheiroptimalityunderconcavecosts.Love[ 50 ]subsequentlyfocusedonserialproductionsystemsanddemonstratedtheoptimalityofnestedproductionschedulesundernonincreasingproductioncostsatastageasafunctionoftimeandnondecreasingholdingcostsaswemovedownstreamintheproductionsystem.Anestedscheduleimpliesthatifproductionatastageinaperiodequalszero,thenproductionattheimmediatepredecessoralsoequalszero.KaminskyandSimchi-Levi[ 39 ]studiedadynamiclotsizingprobleminatwo-stageserialcapacitatedproductionsettingwithxedcostsfortransportationbetweenstages.Understationarycapacitylevels,theydevelopedapolynomialtimealgorithmtosolvetheproblemwithso-called\non-speculative"costs(thenatureofwhichwewilldiscusslater)andtime-invariantcapacitylevels.VanHoeseletal.[ 85 ]andSargutandRomeijn[ 72 ]developedpolynomialtimealgorithmsfortwo-levelseriallotsizingproblemswithconcavecostsandstationaryproductioncapacities.Incontrasttopreviousstudies,SargutandRomeijn[ 72 ]considerbacklogging,outsourcingandovertimeproductionoptions.MeloandWolsey[ 52 ]consideredtheuncapacitated,two-levelseriesdynamiclotsizingproblem,andprovidedanO(T2logT)dynamicprogrammingalgorithm,whereTcorrespondstothenumberofperiodsintheplanninghorizon.Solyalietal.[ 76 ]studytheuncapacitatedandcapacitatedversionsofthisproblem,wherebackloggingisallowed,andtheydevelopecientshortestpathformulations.Foreachoftheseproblemclasses,allcostsareconcave 17

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intheproductionandinventorylevels.Fortheproblemweconsider,incontrast,thetotalcostconsistsofasumofconvexandconcavecostfunctions. 2.2.2Price-DependentProductionPlanningIntheproblemweconsider,theavailablequantityofproductioninputsdependsonthepricetheproduceroerstosuppliersforeachunitinputorcomponent,i.e.,supplyisprice-dependent.Whileasignicantbodyofliteratureexiststhatconsidersproductionplanningproblemswithpricingeects,toourknowledge,thisliteraturefocusesexclusivelyonthepricingimpactonend-itemdemand(see,forexample,Thomas[ 80 ],KunreutherandSchrage[ 44 ],Pekelman[ 58 ],KimandLee[ 42 ],ZhaoandWang[ 95 ],ElmaghrabyandKeskinocak[ 17 ],DengandYano[ 14 ],Ahnetal.[ 1 ],Haugenetal.[ 33 ],OnalandRomeijn[ 57 ],vandenHeuveletal.[ 81 ],Gilbert[ 26 ],Geunes,Romeijn,andTaae[ 25 ],vandenHeuvelandWagelmans[ 82 ],andGeunes,Merzifonluoglu,andRomeijn[ 24 ]).Fortheseproblemclasses,agivenprice,orpricevector,determinesthesetofdemandsthatmustbesatisedinaproductionplanningproblemwithconcavecosts. 2.2.3DynamicLotSizingwithConvexCostsAswewilllatersee,theprice-sensitivityofsupplyavailabilityweconsiderleadstoaclassoftwo-levelseriallotsizingproblemswhosecostisthesumofconcaveandconvexfunctions,andwherethesupplyavailabilitylimitsproductionquantities.Theliteratureonlot-sizingproblemscontainingconvexproductioncostsisreasonablysparse,withafewnotableexceptions.Veinott[ 86 ]consideredasingle-stage,dynamiclotsizingprobleminwhichproductionandinventorycostsarepiecewise-linearandconvex.Hedesignedaparametric-programming-basedprocedureinwhichthesolutionforaproblemwithaxedparametersetisbuiltuponthesolutionofanotherproblemwithasimilarparameterset.Veinott[ 86 ]assumedallparameterswereintegervalued,andhisprocedureresultedinapseudo-polynomialsolutionalgorithm.Florianetal.[ 22 ]mentionedVeinott'sprocedureasthemostattractiveapproachtosolvelotsizingproblemswithconvexproductionandinventorycostsandwithoutxedsetupcosts,eventhoughtheproblemisdemonstrated 18

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tobenoharderthanlinearprogramming,whichispolynomiallysolvable.ErengucandAksoy[ 18 ]consideredasingle-item,capacitateddynamiclotsizingproblemwithxedproductionsetupcostsandlinearinventorycosts,whilevariableproductioncostswerepiecewiselinearandconvexintheproductionquantityinaperiod.Theyusedabranch-and-boundalgorithmforthisproblem,whichcontainsneitheraconvexnorconcaveobjectivefunction.ShawandWagelmans[ 74 ]developedapseudo-polynomialdynamicprogramtosolveacapacitatedsingle-itemlot-sizingproblemwithpiecewiselinearproductioncosts.Theiralgorithmcanbeutilizedtosolveproblemswithpiecewise-linearandconvexproductioncosts,althoughitdoesnotrequireanyspecialstructureforthepiecewise-linearcostfunction.Fengetal.[ 19 ]developedanO(TlogT)algorithmforthesingle-itemlot-sizingproblemwithconstantcapacity,convexinventorycosts,andnon-increasingxedordercosts.TheworkbyKianetal.[ 41 ]iscloselyrelatedtoours,astheyanalyzeasingle-stage,uncapacitatedeconomiclotsizingproblemwithxedsetupcostsandvariablecostsineachperiodthatareconvexintheproductionquantity(takingtheformofapolynomialfunction).Theyderiveseveralkeyoptimalityconditionsforthisproblemclass,aswellasadynamicprogrammingsolutionalgorithmthatisexponentialinthelengthofthetimehorizon.Theyalsoprovideseveralheuristicsolutionapproaches,alongwithacomprehensivenumericaltestsetthatdemonstratestheeectivenessoftheirproposedheuristics.Theproblemweconsider,incontrast,isatwo-levelseriallotsizingproblem,andwefocusonreasonablecostassumptionsthatleadtopolynomialsolvability,aswellastheuseoftheresultingalgorithmsasheuristicsolutionmethodswhenthesecostassumptionsareviolated. 2.2.4PricingandDynamicLotSizinginRemanufacturingNumerousfactors,includingpotentialcostsavingsandenvironmentalconcerns,havemotivatedmanufacturerstoemphasizereverselogisticsinthepastdecade.Asaresult,reverselogisticsandremanufacturinghavereceivedsignicantinterestbyresearchersinrecentyears,andtheliteratureisvoluminous.Ourdiscussionoftherelevantliterature 19

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focusesonthoseworksinwhichdynamiclotsizingproblemsarisewithinthereverselogisticsliterature.Fleischmannetal.[ 20 ]provideacomprehensiveoverviewofreverselogisticsproblems,classifyingvariousresearchthrustswithintheeld.Golanyetal.[ 27 ]studyaversionofthesingle-itemdynamiclotsizingprobleminwhichdisposalofremanufacturableitemsisanoption.TheyshowthattheproblemisgenerallyNP-Completeunderconcavecostsanddevelopapolynomial-timealgorithmforthecasewithlinearcosts.Teunteretal.[ 79 ]studyadynamiclotsizingprobleminwhichremanufacturedreturnscanbeusedfordemandsatisfactionalongwithnewlymanufacturedproducts.Theyconsidertwoversionsoftheproblem;therstversionassumesjointsetupcostsforbothmanufacturingandremanufacturing,andthesecondconsidersseparatesetupcostsforeachproductionmode.Therstversionturnsouttobepolynomiallysolvable,andtheauthorsprovideanexactalgorithm,whereasthesecondversionisNP-Hardandissolvedusingheuristicmethods.Schulz[ 73 ]focusesonimprovingtheSilver-MealheuristicforsolvingthecasewithseparatesetupcostsformanufacturingandremanufacturingproposedbyTeunteretal.[ 79 ].Helmrichetal.[ 34 ]studytheproblemwithseparateandjointsetupcostsformanufacturingandremanufacturingoptions.Theyshowthatbothversionsare,ingeneral,NP-Hard,andprovidetightmixedintegerprogrammingformulationsthat,onaverage,performbetterthanthewell-knownsingleitemlotsizingformulation.RichterandSombrutzki[ 65 ]considerthecasewhereremanufacturingistriggeredbythereturnofproductsfromcustomers,whichtheycallthepurelyreverseWagner/Whitinmodel.RichterandWeber[ 66 ]extendthissettingtoamodelinwhichmultipleproductionmodes(manufacturingandremanufacturing)areavailable.Ineachofthenotedstudies,thequantityofavailableproductioninputsforremanufacturingisdeterminedexogenouslyandisknowninadvance.Guideetal.[ 29 ]studyasomewhatsimilarsettingtotheoneweconsider,inwhichpricesaredeterminedforproductioninputswithvaryingqualitylevels.Theysolvea 20

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price-matchingproblemfordemandandsupplyinasingle-periodsetting.Sunetal.[ 77 ]consideracaseinwhichreturnstoaremanufacturerarealsoprice-sensitive,butrandom.Theyformulatetheproblemasamultiperiodstochasticdynamicprogramwherethedecisionvariablesincludethepriceandremanufacturingquantityineachperiod.Thesestudiesarerelatedtoourwork,astheyconsiderpricedecisionsforsupplyinputs.Incontrast,weincorporatetheideaofprice-sensitivesupplyintoclassicalproductionplanningproblems,wherethegoalistominimizeproductionandinventoryrelatedcostswhilesatisfyingdeterministicdemandduringamultiperiodplanninghorizon. 2.2.5ContributionsofthisChapterTothebestofourknowledge,priorworkdoesnotexistthatconsiderstwo-levellotsizingproblemswithconvexcomponentsupplycostfunctions,alongwiththeusualxedproductionsetupcostsandstandardlinearholdingcosts.WeshowthatthisproblemisNP-Hardingeneral,evenintheabsenceofexplicitproductioncapacityrestrictions,andconsiderseveralpracticallyrelevantspecialcasesthatresultinpolynomialsolvability.Theresultingalgorithmsforthesespecialcasesservetwopurposes.First,theydemonstratethepolynomialsolvabilityofpracticallyrelevantspecialcasesofthisclassofnonconvex,nonlinearmixedintegeroptimizationproblems.Second,thealgorithmicdevelopmentprovidesinsightsonthestructureofoptimalsolutionsandsuggeststheuseoftheresultingalgorithmsasheuristicsolutionapproachesformoregeneralclassesofprobleminstances.Ourworkalsocontributestothereverselogisticsliterature,asitencompassesproblemclassesinwhichprice-sensitivereturnsfromcustomersserveasthesupplysourceforproduction. 2.3ProblemDenition,Formulation,andComplexityWeconsideraproductionprocessthatrequiresthetransformationofaninputcomponentintoanalgoodforwhichcustomerdemandexists.Werefertotheownerofthisproductionprocessastheproducer.Theavailabilityoftherequiredcomponentdependsonaunitpricethattheproduceroersforthecomponent.Thatis,thoseagents 21

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withtheabilitytosupplythecomponentarewillingtosupplyanamountthatdependsontheunitpriceoeredbytheproducer.WeconsideranitesetTofplanningperiods,whereT=f1;2;:::;jTjg,suchthattheproducermayoeradierentunitpricefortheinputcomponentineachperiod.Letptandtdenote,respectively,theunitpricetheproduceroerstosuppliersinperiodt,andthetotalsupplyobtainedbytheproduceratthisprice.Weassumethattisdeterminedbyanonnegative,nondecreasingfunctiont=Kt(pt),suchthatKt(pt)isinvertibleandaone-to-onemappingexistsbetweenpriceptandcomponentsupplylevelt,i.e.,pt=K)]TJ /F7 7.97 Tf 6.59 0 Td[(1t(t).TheamounttheproducertransferstosuppliersinperiodtthenequalsptKt(pt)=tK)]TJ /F7 7.97 Tf 6.59 0 Td[(1t(t),andweassumelimpt!1t=1,i.e.,noniteboundexistsonavailablesupply.Inadditiontothecostincurredforobtainingcomponentsfromsuppliers,theproducermayincuradditionalprocurementrelatedcosts;therefore,weletfC(t)denoteafunctionequaltothetotalprocurement-relatedcostincurredbytheproducerinperiodt,wherefC(t)tK)]TJ /F7 7.97 Tf 6.59 0 Td[(1t(t),t2T.Weassumethatproductionandcomponentprocurementleadtimesarenegligible,althoughthemodelcanaccommodateniteandconstantleadtimesaswell.Foranal-goodsproductionlevelofxtinperiodt,theproducerincursaproductioncostofft(xt).Weassumeforconveniencethatoneunitoftheinputcomponentisrequiredforeachunitofthenalgoodproduced.TheproducermayholdcomponentsandnalgoodsininventoryatacostofhtperunitofthenalgoodheldattheendofperiodtandacostofhCtperunitofinputcomponentheldattheendofperiodt,t2T.LettingItandICtdenote,respectively,thenal-goodsandcomponentinventorylevelsattheendofperiodt,theholdingcostincurredattheendofperiodtequalshtIt+hCtICt.Weassumezeroinitialinventoriesforbothnalgoodandcomponentlevels.Theproducerwishestomeetdemandforthenalgoodwithoutshortagesineachperiodoftheplanninghorizon,wheredtdenotesthenumberofunitsofthenalgooddemandedinperiodt.Theproducerseekstominimizethecostsincurredinmeeting 22

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thisdemand,wheretotalcostiscomprisedofcomponentprocurementcosts,nal-goodsproductioncosts,andinventoryholdingcosts.Weformulatethegeneralversionofthisproblem,denotedasP(G),asfollows.P(G):MinimizeXt2Tft(xt)+fCt(t)+htIt+hCtICt (2{1)Subjectto:It)]TJ /F7 7.97 Tf 6.58 0 Td[(1+xt=dt+It;8t2T; (2{2)ICt)]TJ /F7 7.97 Tf 6.58 0 Td[(1+t=xt+ICt;8t2T; (2{3)xt;It;ICt;t0;8t2T: (2{4)Theobjectivefunctionminimizestotalcost,whileconstraints( 2{2 )and( 2{3 )correspondtoinventorybalanceconstraintsforthenalgoodsandcomponents,respectively.Constraintset( 2{4 )requiresallvariablestotakenonnegativevalues.Observethatifft(xt)andfCt(t)areconcavefunctionsforallt2T,thenthisproblemfallsintotheclassofuncapacitatedtwo-levelseriallot-sizingproblems,whichhavebeenwidelystudiedandarewellsolvedinpolynomialtimeasafunctionofjTj(see,e.g.,Zangwill[ 93 ],vanHoesel,Romeijn,MoralesandWagelmans[ 85 ],SargutandRomeijn[ 72 ],andMeloandWolsey[ 52 ]).Forpracticalcontexts,however,whilewemightexpectconcavitytoholdfortheproductioncostfunctions(ft(xt),t2T),itisunlikelytoholdfortheprice-dependentcomponentsupplycostfunctionfCt(t),t2T,whensupplyquantityisincreasinginprice.Incontrast,theconvexityofthisfunctionappearstobeamorereasonableassumption,ashigheracquisitionpricesarelikelytoincreasesupplyatanincreasingrate.Ifft(xt)isconcaveforallt2TandfCt(t)isnondecreasingandconvexforallt2T,thenproblemP(G)becomesNP-Hard,whichwenextdemonstrateusingaspecialcaseofP(G).Inordertoreectpracticalproductionplanningcontexts,wewillgenerallyassumethatft(xt)takestheformofaxed-chargepluslinearcostfunctionforeacht2T,i.e.,ft(xt)=Ftyt+ctxt,whereytdenotesabinaryvariableequaltooneifxt>0andequal 23

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tozerootherwise.Thisisacommonlyusedconcavefunctionalformforrepresentingproductioncoststhatariseintheformofaxedsetupcostplusavariablecostforeachunitofproduction.Wefurtherassume,exceptwherespecicallynoted,thatfCt(t)takesaparticularfunctionalformdescribedasfollows.Supposethataminimumreservationpriceexistsineachperiodt,denotedbyp0t,suchthatforanypricenotexceedingthisvalue,nosupplyisavailable.Forpricesexceedingthisthreshold,weassumethatsupplyincreaseslinearlyinpriceatarateoft>0perunitpricechange.Asaresultoftheseassumptions,thesupplyinperiodt,t(pt)0isisanonnegativeandnondecreasingpiecewiselinearfunctionofprice,pt,withonebreakpointatp0t,denedby t(pt)=8><>:tpt)]TJ /F3 11.955 Tf 11.96 0 Td[(t;ifptp0t;0;otherwise,(2{5)forpt0,witht0forallt2T(seeFigure 2-1 ). Figure2-1. Supplyvs.pricefunction. Foreachunitofthecomponentprocured,theproducerpayspt,whichimpliesthatthetotalamountpaidtosuppliersinperiodtequalsptt=tp2t)]TJ /F3 11.955 Tf 12.12 0 Td[(tpt=2t=t+tt=t(wherewesuppressthedependenceoftonptforconvenience).Inaddition,weassumetheproducerincursacostofFCtyCt+ttforaprocurementleveloftinperiodt,whereFCtdenotesaxedcostincurredforanypositivelevelofprocurement,andyCtdenotesabinaryvariableequaltooneift>0andequaltozerootherwise.Thus,atotalprocurementrelatedcostofFCtyCt+(pt+t)t=FCtyCt+ttpt+tp2t)]TJ /F3 11.955 Tf 12.47 0 Td[(tpt= 24

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FCtyCt+2t=t+(t+t=t)tisincurredineachperiodt2T.WewillrefertothisversionoftheproblemwithxedchargesforbothproductionandprocurementasproblemP(FC). Proposition2.1. Theproductionplanningproblemwithprice-dependentsupplycapacityisNP-Hard. Proof. PleaseseeAppendix A IntheproofofProposition 2.1 inAppendixA,allxedproductionsetupcostsequaloneandallxedprocurementcostsequalzero.ThisimpliesthateventhespecialcaseinwhichnoxedprocurementcostsexistandallproductionsetupcostsareequalforallperiodsisNP-Hard.Wecanalsoshowthatthespecialcaseinwhicht=,t=,andt+1=t+hCtforallt2T,i.e.,whentheprice-supplyfunctionsareidenticalineveryperiodandtheunitcomponentprocurementcostincreasefromoneperiodtothenextequalsthecomponentholdingcostinthepriorperiod,isNP-Hardaswell.(Notethatthislatterconditionimpliesthatthevariableprocurementcostassociatedwithacomponentininventoryinanyperiodisindependentoftheperiodinwhichthecomponentwasprocured.) Proposition2.2. Forthespecialcaseinwhicht=,t=,andt+1=t+hCtforallt2T,theproductionplanningproblemwithprice-dependentsupplycapacityremainsNP-Hard. Proof. PleaseseeAppendix B ThefollowingsectionanalyzeskeypropertiesofoptimalsolutionsforproblemP(FC)andidentiespracticallyrelevantspecialcasesthatmaybesolvedinpolynomialtime. 2.4AnalysisofP(FC)andPolynomiallySolvableCasesInordertoformallyanalyzepropertiesofoptimalsolutionsforproblemP(FC),wedeneaproduction(component)regenerationintervalRIp(t;t0)(RIc(t;t0))asapartialsolutionforperiodstthrought0)]TJ /F1 11.955 Tf 12.51 0 Td[(1,suchthatIt)]TJ /F7 7.97 Tf 6.58 0 Td[(1=It0)]TJ /F7 7.97 Tf 6.58 0 Td[(1=0(ICt)]TJ /F7 7.97 Tf 6.58 0 Td[(1=ICt0)]TJ /F7 7.97 Tf 6.58 0 Td[(1=0)andIj>0(ICj>0)forallj=t;:::;t0)]TJ /F1 11.955 Tf 12.06 0 Td[(2.NotethatanyfeasiblesolutionforproblemP(FC) 25

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canbecharacterizedasasequenceofcomponentregenerationintervalsthatprovideinputtoproductionregenerationintervals.Moreover,anoptimalsolutionforproblemP(FC)existsconsistingofsequencesoftheseregenerationintervals.ThefollowingpropositioncharacterizesanimportantpropertyofanoptimalcomponentregenerationintervalRIcforproblemP(FC). Proposition2.3. IfRIc(t;t0)ispartofanoptimalsolution,thenforanypairofperiodsi1;i2suchthatti10andsucientlysmall,andlet~ICj=ICj+forj=i1;:::;i2)]TJ /F1 11.955 Tf 12.1 0 Td[(1(thevaluesofallothervariablesremainunchanged).Thevalueoftheoriginalsolutionminusthenewsolutioncanbewrittenas= 2i2 i2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2i1 i1)]TJ /F5 7.97 Tf 11.96 14.99 Td[(i2)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xj=i1hCj+i2+i2 i2)]TJ /F3 11.955 Tf 11.95 0 Td[(i1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(i1 i1)]TJ /F3 11.955 Tf 11.96 0 Td[(i1+i2 i1i2!:Forpositivesuchthat0,whichcontradictstheoptimalityoftheoriginalsolution(notethattheright-handsideoftheaboveisstrictlypositivebyassumption).Asimilarargumentshowsthatasolutioninwhich2i1=i1+i1+i1=i1+Pi2)]TJ /F7 7.97 Tf 6.58 0 Td[(1j=i1hCj>2i2=i2+i2+i2=i2cannotbeoptimal. ObservethatforaperiodiwithinaRIc(t;t0)(i.e.,tit0)]TJ /F1 11.955 Tf 12.75 0 Td[(1),themarginalcostforsupplyingacomponenttoperiodithatisprocuredinperiod
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iwithintheregenerationinterval(i.e.,suchthattit0)]TJ /F1 11.955 Tf 12.74 0 Td[(1),anoptimalsolutionexistssuchthatthemarginalcostforsupplyingacomponenttoperiodiisthesameforallpriorperiodswithinRIc(t;t0).AsimilarpropertyarisesinDengandYano[ 14 ]andGeunes,andMerzifonluoglu,andRomeijn[ 24 ]inthecontextofproductionplanningwithprice-sensitivedemand,aswellasinKian,Gurler,andBerk[ 41 ]insingle-stagelotsizingwithconvexproductioncosts.Remark.SupposeprocurementcoststaketheformofageneralconvexfunctionfCt(t),andlet@fCt(t)=@tdenotethesetofsubgradientsoffCtattforallt2T.Forthismoregeneralcase,Proposition 2.3 canbegeneralizedtostatethatifRIc(t;t0)ispartofanoptimalsolution,thenforperiodsi1andi2suchthatti10andi2>0,wehave@fCt(i1)=@i1+Pi2)]TJ /F7 7.97 Tf 6.59 0 Td[(1j=i1hCj\@fCt(i2)=@i26=;.IfeachfCtiseverywheredierentiable,thenthisconditionbecomesdfCt(i1)=di1+Pi2)]TJ /F7 7.97 Tf 6.59 0 Td[(1j=i1hCj=dfCt(i2)=di2.Despitethesestructuralresults,theproblemP(FC)is,ingeneral,NP-Hardintheabsenceofanyadditionalconditionsimposedontheproblem'sparameters.Wenextimposeafairlymildassumptiononthestructureofproductionandholdingcoststhatleadstotheoptimalityofthewell-knownZero-InventoryProduction(ZIP)property,underwhichanoptimalsolutionexistssuchthatproductioninaperiodoccursifandonlyifnoend-iteminventoryisheldoverfromthepriorperiod. Assumption1. Foreveryperiodt=1;:::;jTj)]TJ /F1 11.955 Tf 17.93 0 Td[(1,weassumect+htct+1+hCt.Assumption 1 statesthatitisalwaysatleastascostly(intermsofvariableproductionandholdingcosts)toproduceandholdaproductasitistoholdacomponentforlaterproduction(thisassumptionisconsistentwiththeassumptionsonproductionandholdingcostsinLove[ 50 ];KaminskyandSimchi-Levi[ 39 ]imposedasimilarassumptionontransportationandholdingcostsintheiranalysisofatwo-levelcapacitatedlotsizingproblemwithconcaveproductionandtransportationcosts).Itisnotuncommoninpracticeforholdingcoststoincreasedownstreaminamulti-stagesystem,asvalue 27

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isaddedtoaproduct;thus,wetypicallyexpecttheconditionhthCttoholdineachperiod.ObservethatAssumption 1 doesnotrequirenonincreasingproductioncosts;whenevertheincreaseinproductioncostfromperiod-to-perioddoesnotexceedthedierencebetweendownstreamandupstreamholdingcost,thisstructuralcostassumptionwillapply.(Wenotethatthecaseinwhichct+ht0andIi)]TJ /F7 7.97 Tf 6.58 0 Td[(1>0,andlet(;x;IC;I)denotethevectorsofcomponentprocurement,production,componentinventory,andnalproductinventoryvariablevaluesinthecorrespondingsolution.Consideraunitofinventoryinperiodi)]TJ /F1 11.955 Tf 12.3 0 Td[(1,andlet
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Substitutingthisinequalityforsuccessivevaluesoft=+2;:::;igivescci+i)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xj=hCj)]TJ /F5 7.97 Tf 13.74 14.95 Td[(i)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xj=hj;whichgivesc+i)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xj=hjci+i)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xj=hCj:Thus,thenewsolutioncostsnomorethantheoriginalsolution.Repeatingthisargumentforeachunitofinventoryinperiodi)]TJ /F1 11.955 Tf 12.55 0 Td[(1impliesthatwecaneitherndanewoptimalsolutionsatisfyingthedesiredpropertyorwehaveacontradictiontotheoptimalityoftheoriginalsolution.Ineithercase,anoptimalsolutionexistssatisfyingtheZIPproperty. TheZIPpropertyimpliesthateveryproductionsetupproducesaquantityequaltoaconsecutivesetofperioddemands,i.e.,lettingDi;j=Pjt=idt,intheworstcaseweonlyneedtoconsidersolutionssuchthatxt2f0;Dt;t;Dt;t+1;:::;Dt;jTjfort=1;:::;jTj.Nextobservethatanyproductionrequirescomponentprocurement,andtheproductioninanyperiodmustdrawitscomponentsupplyfromsomeRIc(t;t0).Moreover,theRIc(t;t0)canserveproductioninanyperiodgreaterthanorequaltot,andiftheRIc(t;t0)iscontainedinanoptimalsolution,thenthenalperiodofproductionthisRIc(t;t0)servesmustoccurinperiodt0)]TJ /F1 11.955 Tf 12.24 0 Td[(1(otherwisewehaveheldcomponentinventoryattheendofperiodt0)]TJ /F1 11.955 Tf 12.01 0 Td[(1thatisnotusedtosatisfyproduction,violatingthedenitionofRIc(t;t0)). Proposition2.5. UnderAssumption 1 ,iftheRIc(t;t0)iscontainedinanoptimalsolution,thenthetotalprocurementamountinperiodst;t+1;:::;t0)]TJ /F1 11.955 Tf 12.41 0 Td[(1equalsDu;vforsomeuandv,whereutandvt0)]TJ /F1 11.955 Tf 11.96 0 Td[(1. Proof. SupposeRIc(t;t0)servesproductioninperiodst1;t2;:::;tnwherett1
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inRIc(t;t0)mustbeequaltothesumofproductionsinperiodst1;:::;tn.BytheZIPproperty,weknowthattherespectiveproductionamountsinperiodst1;:::;tnareDu;t2)]TJ /F7 7.97 Tf 6.59 0 Td[(1;Dt2;t3)]TJ /F7 7.97 Tf 6.58 0 Td[(1;:::;Dt0)]TJ /F7 7.97 Tf 6.58 0 Td[(1;vwherevt0)]TJ /F1 11.955 Tf 12.41 0 Td[(1.ThesumoftheseproductionamountsareequaltoDu;v.ThusthecomponentprocurementamountinRIc(t;t0)equalsDu;v. WhentheperiodswithpositiveprocurementwithinacomponentRI,sayRIc(t;t0),areknown,wecancomputetheprocurementlevelsforeachoftheseperiodsusingtheresultsofProposition 2.3 .Wedenet=t+t=t+PjTjj=thCj,andsupposetherearemperiodswithpositiveprocurementt1;t2;:::;tmwithinRIc(t;t0),andthesecomponentsservedemandinperiodsuthroughv.ByProposition 2.3 ,wehavet1+2t1 t1=t2+2t2 t2=:::=tm+2tm tm:Anyprocurementleveltjforj=2;:::;mcanbewrittenintermsoft1asfollows:tj=t1)]TJ /F1 11.955 Tf 12.39 0 Td[(tj+2t1 t1tj 2: (2{6)BecauseDu;v=Pmj=1tj,wecanthuscomputet1asfollows:t1=Du;v)]TJ /F9 11.955 Tf 11.95 8.97 Td[(Pmj=2(tj=2)(t1)]TJ /F1 11.955 Tf 12.39 0 Td[(tj) 1+Pmj=2tj=t1: (2{7)Wecannowusethevalueoft1andEquation( 2{6 )toobtaineachofthemvaluesoftj.Unfortunately,foragivenvalueofmt0)]TJ /F3 11.955 Tf 12.31 0 Td[(t+1,wehave2m)]TJ /F1 11.955 Tf 12.32 0 Td[(1uniquechoicesofasubsetofmperiodsinwhichpositiveprocurementmayoccurwithinRIc(t;t0).Thus,determiningthebestsetofmpositiveprocurementperiodscorrespondstoadicultcombinatorialoptimizationproblemingeneral.However,underadditionalassumptionsonthebehaviorofprocurement-relatedcosts,itispossibletodetermineanoptimalsetofmpositiveprocurementperiodsinpolynomialtimeforanymt0)]TJ /F3 11.955 Tf 11.95 0 Td[(t+1.Theremainderofthissectionconsistsoftwosubsections,eachofwhichconsidersasetofassumptionsonthebehaviorofprocurementcostsovertime.Section 2.4.1 rstconsiderscaseswithstationaryprice-supplyfunctionsandvariableprocurementcostsbut 30

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permitsarbitraryxedchargevaluesforbothprocurementandproduction.Section 2.4.2 thenallowsforslightlymoregeneral,nondecreasingprice-supplyfunctionsandvariableprocurementcostsovertime,butrequiresnondecreasingxedprocurementcostsaswell. 2.4.1P(FC)withUniformPrice-SupplyFunctionsandVariableProcurementCostsInmanypracticalcontexts,theresponseofsupplierstothepriceoeredisnotlikelytobevariableintheshortrun.RecallfromProposition 2.2 thatevenwhentheprice-supplycurveparametersarestationaryandvariableprocurementcostsareindependentoftheprocurementperiod,i.e.,t=,t=,andt+1=t+hCt,t=1;:::;jTj,theproblemremainsNP-hard.UnderthenonspeculativecoststructureofAssumption 1 ,however,theproblembecomessolvableinpolynomialtime,aswenextdemonstrate. Assumption2. Foreveryperiodt2T,wehavet=,t=,andforeveryt2TnfjTjg,t+1=t+hCt. Corollary1. UnderAssumption 2 ,Proposition 2.3 impliesthattheprocurementlevelsinperiodswithpositiveprocurementwithinanRIcmustbeequal.UnderAssumptions 1 and 2 ,wecansolvetheprobleminpolynomialtimeunderarbitraryvaluesofthexedchargesatboththeprocurementproductionlevels.Werstdeneasubplanbythequadruple(s;u;t;v),whichindicatesthatthecomponentregenerationintervalRIc(s;t+1)servesproductioninperiodsuthroughv,whichisusedtosatisfyalldemandinperiodsuthroughv,wheresutv.Subplan(s;u;t;v)impliesthatweprocureDu;vunitsofthecomponentinRIc(s;t+1)andproducethisamountofend-productinperiodsuthroughv,usingasetofconsecutiveproductionregenerationintervalsRIp(u;i1);RIp(i1;i2);:::;RIp(in;v+1).Givenaxednumbermofperiodswithpositiveprocurement(form=1;:::;t)]TJ /F3 11.955 Tf 12.49 0 Td[(s+1),wecaneasilycomputetheprocurementamountineachperiodwithpositiveprocurement,asthismustequal 31

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Du;v=m.However,theexactperiodswithpositiveprocurementareunknownandmustbedetermined.LetSPC(s;u;t;v)betheminimumcostofsubplan(s;u;t;v),includingthesumofthexedandvariableproductioncostsandnalproductholdingcostsinperiodsuthroughv,plusthexedandvariablecomponentprocurementandholdingcostsinperiodssthrought.InordertocomputeSPC(s;u;t;v),weneedtodeterminethesubsetofperiodsfromuthroughvinwhichproductionwilloccur.However,wealsoneedtodeterminetheperiodswithpositiveprocurementfromsthrought.Bythedenitionofsubplan(s;u;t;v),weknowthatthelastproductionsetupperiodwithinthesubplanmustoccurinperiodt,withaproductionquantityequaltoDt;v,asaresultoftheZIPpropertyfromProposition 2.4 .Giventhenumberofperiodswithpositiveprocurement,m,anddemandrequirementsineachperiod,wecandeterminethesubsetofperiodsuthroughvinwhichproductionoccursandthesubsetofperiodssthroughtinwhichprocurementoccursbysolvingashortestpathproblemonagraphcreatedasfollows:LetG(V;A)beadirectedgraphwithvertexsetVandarcsetA(seeFigure 2-2 ).ThevertexsetVconsistsofanoriginvertexsandadestinationvertextm.Theintermediateverticesaredenotedbyil,wherei2fu;u+1;:::t)]TJ /F1 11.955 Tf 12.67 0 Td[(1g,1lminfm;i)]TJ /F3 11.955 Tf 12.67 0 Td[(s+1g,andm)]TJ /F3 11.955 Tf 12.67 0 Td[(lt)]TJ /F3 11.955 Tf 12.67 0 Td[(i.Vertexilcorrespondstoproductionperiodiwhenthenumberofprocurementsetupsuptoperiodiequalsl.Thedestinationvertextmimpliesallmprocurementsetupsoccurreduptoperiodt,whichisalsoimpliedbythedenitionofthesubplan(s;u;t;v).ThearcsetAconsistsoftwosubsetsofarcs:A1,andA2,i.e.,A=A1[A2.ThearcsetA1includesarcs(s;ul)where1lminfm;u)]TJ /F3 11.955 Tf 12.93 0 Td[(s+1g.Anarcfromstoulimpliesthattherearelprocurementsetupsfromsthroughu.ArcsetA2isdenedasfollows:A2=f(il;jk)jil;jk2V;i
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Figure2-2. PartialrepresentationofthedirectedgraphG(V;A). NextweshowhowtocomputethecostsofarcsinA1,andA2,giventhatthenumberofprocurementsetupperiodsequalsm.Notethattheprocurementamountsinallperiodswithpositiveprocurementareequal,andletdenotethisprocurementlevel,i.e.,=Du;v=m.Letcmadenotethecostofarca2A.Fora2A1,cmaiscomputedasfollows,wherea=(s;ul)and1lminfm;u)]TJ /F3 11.955 Tf 10.13 0 Td[(s+1g.Weneedtodeterminethelcheapestprocurementperiodsfromsthroughu)]TJ /F1 11.955 Tf 12.29 0 Td[(1.LetFC[j]denotethejthcheapestxedprocurementcostfromsthroughu)]TJ /F1 11.955 Tf 12.14 0 Td[(1.Here[j]returnstheindexoftheperiodwiththejthcheapestxedprocurementcost.Fromsthroughu)]TJ /F1 11.955 Tf 12.34 0 Td[(1,theprocurementlevels,i,areequaltoforalli=[1];:::;[l],andzerootherwise.UsingEquation( 2{8 ),wecancomputethecostofarcaforalla2A1.WedeneKs;itodenotethecumulativeprocurementamountfromperiodstoperiodi,i.e.,Ks;i=Pik=sk. cma=lXj=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(FC[j]+[j]+u)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xk=shCkKs;k+l 2+l :(2{8)Fora2A2,cmaincludesbothproductionandprocurementrelatedcosts,aswellasinventoryholdingcostsatbothlevels.Hereadenotesanarcoftheform(il;jk)wherei
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setupsfromi+1throughjcannotexceedthenumberperiodswithinthetimewindow.Whenk)]TJ /F3 11.955 Tf 12.67 0 Td[(l<>:1;ifDu;j)]TJ /F7 7.97 Tf 6.58 0 Td[(1>Ks;i;C(il;jk)+P(il;jk);otherwise,(2{9)where C(il;jk)=k)]TJ /F5 7.97 Tf 6.59 0 Td[(lXt=1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(FC[t]+[t]+j)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xt=ihCt(Ks;t)]TJ /F3 11.955 Tf 11.95 0 Td[(Di;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+k)]TJ /F3 11.955 Tf 11.96 0 Td[(l 2+(k)]TJ /F3 11.955 Tf 11.95 0 Td[(l) ;(2{10)and P(il;jk)=Fi+ciDi;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1+j)]TJ /F7 7.97 Tf 6.58 0 Td[(2Xk=ihkDk+1;j)]TJ /F7 7.97 Tf 6.58 0 Td[(1:(2{11)HereC(il;jk)denotesthetotalxedandvariableprocurementcostsandcomponentinventoryholdingcostsfromperiodithroughj.P(il;jk)isthesumofxedandvariableproductioncostsandend-iteminventoryholdingcostsfromithroughj)]TJ /F1 11.955 Tf 11.96 0 Td[(1.Letm(s;u;t)denotethecostoftheshortestpathinthegraphG(V;A)fromstot.Thecostofsubplan(s;u;t;v)iscomputedasfollows: SPC(s;u;t;v)=min1mt)]TJ /F5 7.97 Tf 6.58 0 Td[(s+1(m(s;u;t)+Ft+ctDtv+v)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=thiDi+1;v):(2{12)TosolvetheproblemunderAssumption 2 ,wecannowapplyadynamicprogrammingapproach.LetCs;ubetheminimumcostofsatisfyingdemandfromperioduuntiltheendofthehorizon,giventhatallcomponentprocurementoccursinperiodsandlater,where 34

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su.Cs;ucanbecomputedusingthefollowingdynamicprogrammingrecursion: Cs;u=minv;t:utvfSPC(s;u;t;v)+Ct+1;v+1g:(2{13)Theassociatedboundaryconditionsareasfollows: CjTj+1;i=Ci;jTj+1=0foralli=1;:::;jTj+1:(2{14)HereC1;1givesanoptimalproductionandprocurementplanfortheentireplanninghorizon.TocomputeC1;1,weneedtocomputeO(jTj4)valuesofSPC(s;u;t;v).ComputingSPC(s;u;t;v)foreveryquadruple(s;u;t;v)requiressolvingmshortestpathproblemstodeterminem(s;u;t)foreachm2f1;:::;t)]TJ /F3 11.955 Tf 12.31 0 Td[(s+1g.Foreachoftheseshortestpathproblemsweneedtoconstructagraph,whichrequiresO(jTj5)operations.Therefore,thetimecomplexitytocomputem(s;u;t)foragivenmequalsO(jTj5).Thus,theoveralltimecomplexityofthedynamicprogramisO(jTj9).Remark.Supposethatforeveryperiodt2T,wehaveFCt=0.NotethatunderthisassumptionandAssumption 2 ,therearenoxedchargesattheprocurementlevel,andtheconvexprocurementcostfunctionsareidenticalineveryperiod.Inthiscase,wecanshowthatthereexistsanoptimalsolutionwheretheprocurementquantitiesareequalinallperiodswithpositiveprocurementwithinanRIc.WecanalsoshowthatifanRIc(t;t0)existsinanoptimalsolutionwithpositiveprocurementinperiodi1,thenanoptimalsolutionexistswithpositiveprocurementineachoftheperiodsi1+1;i1+2;:::;t0)]TJ /F1 11.955 Tf 12.4 0 Td[(1.BecausepositiveprocurementmustoccurintherstperiodofanRIc,thepreviousobservationsimplythatpositiveprocurementmustoccurineveryperiodoftheRIcandthattheprocurementlevelmustbethesameineachperiodoftheRIc.Followingtheseresults,thespecialcaseofproblemP(FC)underthesemorerestrictivecostassumptionscanbesolvedusingadynamicprogramthatisdesignedinasimilarmannertothedynamicprogramdiscussedinthissection.BecausepositiveprocurementexistsineachperiodofanRIcwithequalprocurementquantitiesineachperiodoftheRIc,weneednot 35

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considerdierentvectorsofpositiveprocurementperiods,i.e.,weonlyneedtoconsideroneprocurementscenarioforanysubplan(s;u;t;v),andthetimecomplexityofthisdynamicprogramwouldthereforeequalO(jTj6). 2.4.2P(FC)withNondecreasingProcurement-RelatedCostsThissectionconsidersamoregeneralsetofassumptionsonthenatureprice-supplyfunctionsandvariableprocurementcoststhaninthepriorsubsection,attheexpenseoflessgeneralityinthenatureofxedprocurementcosts.Inparticular,inthissectionweassume Assumption3. Foreveryperiodt=f1;:::;jTj)]TJ /F1 11.955 Tf 18.43 0 Td[(1g,weassumeFCtFCt+1,tt+1,andtt+1. Proposition2.6. UnderAssumption 3 ,ifanRIc(t;t0)existsinanoptimalsolutionwithmpositiveprocurementperiods,thentheseperiodsaretherstmperiodsofRIc(t;t0). Proof. SupposeoptimalprocurementperiodswithinRIc(t;t0)aret=t1t2:::tmwheretm>t+m)]TJ /F1 11.955 Tf 12.05 0 Td[(1.Sincetm>t+m)]TJ /F1 11.955 Tf 12.05 0 Td[(1,thereexistsaperiodiwithinrstmperiodsoftheRIwithnoprocurement.Wecreateanewsolutionbymovingtheprocurementinperiodtm,tm,toperiodiwhereeverythingelseremainsthesame.Letbethedierencebetweencostsoftheoriginalsolutionandthenewsolution.=FCtm+tmtm+2tm tm)]TJ /F3 11.955 Tf 11.96 0 Td[(FCi)]TJ /F1 11.955 Tf 12.39 0 Td[(itm)]TJ /F3 11.955 Tf 13.15 8.62 Td[(2tm i;=FCtm)]TJ /F3 11.955 Tf 11.95 0 Td[(FCi+(tm)]TJ /F1 11.955 Tf 12.39 0 Td[(i)tm+1 tm)]TJ /F1 11.955 Tf 15.22 8.09 Td[(1 i2tm:ByAssumption 3 ,FCtmFCi,tmi,andtmi,thus0.If>0thenthiscontradictstheoptimalityoftheoriginalsolution.Andif=0thenalternativeoptimalsolutionsexist.Bothcasesleadtothedesiredresult. Proposition2.7. UnderAssumption 3 ,ifthereexisttwoconsecutivecomponentregener-ationintervalsRIc(s;t)andRIc(s0;t0)inanoptimalsolution,wherests0t0,thent=s0. 36

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Proof. SupposeanoptimalsolutioncontainstwoconsecutiveRIcs,RIc(i1;i2)andRIc(i3;i4)wherei1i2
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withvertexsetV=fu;u+1;:::;t)]TJ /F1 11.955 Tf 11.99 0 Td[(1;tgandarcsetA=f(i;j)ji;j2Vandj>ig.Anarcfromitojimpliesthataproductionsetupoccursinperiodithatsatisesalldemandinperiodsithroughj)]TJ /F1 11.955 Tf 11.96 0 Td[(1.Aproductionsetupinperiodiwhichwillsatisfydemandsfromperioditoperiodj)]TJ /F1 11.955 Tf -456.75 -23.91 Td[(1isonlyfeasibleifthecomponentprocurementlevelissucienttocovertheproductioninthisperiod.NotethattheproductionamountinperiodiequalsDi;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1,andthesumofallproductionsuptoandincludingperiodiequalsDu;j)]TJ /F7 7.97 Tf 6.58 0 Td[(1,bytheZIPproperty.Ifthetotalprocurementamountuptoandincludingperiodiisgreaterthanthetotalproductionamountuptoandincludingperiodi,i.e.,Ks;iDu;j)]TJ /F7 7.97 Tf 6.58 0 Td[(1,thenitisfeasibletohaveaproductionsetupinperioditosatisfydemandinperiodsithroughj)]TJ /F1 11.955 Tf 12.39 0 Td[(1.Giventhatprocurementoccursintherstmperiodsofthesubplanforagivenvalueofm,thearccosts,cmij,forall(i;j)2A,arecomputedasfollows: cmij=8><>:Fi+ciDi;j)]TJ /F7 7.97 Tf 6.58 0 Td[(1+Pj)]TJ /F7 7.97 Tf 6.58 0 Td[(2k=ihkDk+1;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1+Pj)]TJ /F7 7.97 Tf 6.59 0 Td[(1k=ihCk(Ks;k)]TJ /F3 11.955 Tf 11.95 0 Td[(Du;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1);ifKs;iDu;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1;1;otherwise.(2{15)Let0m(s;u;t)equalthelengthoftheshortestpathfromutotonthedirectedgraph;0m(s;u;t)correspondstotheminimumcostproductionplanfromperiodutoperiodt)]TJ /F1 11.955 Tf 12.15 0 Td[(1,includingcomponentandend-itemholdingcostsintheseperiods,giventhatcomponentregenerationintervalstartsatperiodsandthenumberofperiodswithpositiveprocurementequalsm.Bytheconstructionoftheshortestpathnetwork,thevalueof0m(s;u;t)canbedeterminedinO(jTj2)timeforany(s;u;t;m)quadruple.Thecostofsubplan(s;u;t;v),SPC0(s;u;t;v),isthencomputedasfollows: SPC0(s;u;t;v)=min1mt)]TJ /F5 7.97 Tf 6.58 0 Td[(s+18<:s+m)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=s(FCi+2i i+ii i+ii)+u(s;m)Xi=shCiKs;i+Ft+ctDtv+v)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xi=thiDi+1;v+0m(s;u;t)); (2{16) whereu(s;m)=minfu)]TJ /F1 11.955 Tf 11.96 0 Td[(1;s+m)]TJ /F1 11.955 Tf 11.95 0 Td[(1g. 38

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LetC0s;udenotetheminimumcostofsatisfyingdemandfromperioduuntiltheendofthehorizon,giventhatallcomponentprocurementoccursinperiodsandlater,wheresu.WecancomputeC0s;uusingthefollowingdynamicprogrammingrecursion: C0s;u=mint;v:utvfSPC0(s;u;t;v)+C0t+1;v+1g(2{17)Theboundaryconditionsforthisdynamicprogramarewrittenas C0jTj+1;i=C0i;jTj+1=0foralli=1;:::;jTj+1:(2{18)NotethatC01;1givesthecostofanoptimalplanfortheentireplanninghorizon.ThebottleneckoperationforcomputingC01;1arisesincomputingeachoftheO(jTj4)valuesofSPC0(s;u;t;v),wherecomputingthisvaluerequiresexplicitlyconsideringO(jTj)valuesofmin( 2{16 ),aswellasO(jTj2)timetocompute0m(s;u;t)foreachm.Thus,theworst-casecomplexityofthisdynamicprogramequalsO(jTj7). 2.5ComputationalTestResultsThissectionpresentstheresultsofasetofcomputationaltestsintendedtoassesstheperformanceofthedynamicprogramspresentedinSection 2.4 .InSection 2.5.1 wecomparetheperformanceoftheproposeddynamicprograms(codedinC++)withthegeneralpurposenonconvexmixedintegernonlinearprogramingsolverCouenne1.Forthisperformanceanalysis,weconsideredproblemswithvariousplanninghorizonlengths.Thedataforeachprobleminstancewererandomlygeneratedusingasetofuniformdistributions;weletU(a;b)denoteagivenuniformdistributionon[a;b],wherea
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whereallothersremainasshowninTable 2-1 .Table 2-2 providesthedierentscenariosconsideredfortheFCtanddtvalues.Ourgoalsingeneratingtherandomdatausedintheexperimentsweretwofold:(1)thedatashouldleadtoprobleminstancesinwhichoptimalsolutionsarenotextremeortrivial,meaningthatwedonothaveasinglesetupatthebeginningofthehorizonateitherlevel;and(2)thedatashouldreectrelativecostswemightexpecttondinpracticalsettings.Achieving(1)requiredhighersettingsforprocurementxedcoststhanweinitiallyexpected,astheconvexityofprocurement-relatedcostsleadstodiseconomiesofscaleinprocurementand,thus,morefrequentprocurementsetups. Table2-1. Uniformdistributionparameters. Problemparameterdescriptionab FtFixedproductioncost5001000ctVariableproductioncost15tPrice-vs-procurementamountfunctionslope15tComponenthandlingcost15htEnditeminventoryholdingcost0.0010.005hCtComponentinventoryholdingcost0.00050.002p0tThresholdprice(t=t)15 Table2-2. Scenarios. ScenarioNo.FCt(a,b)dt(a,b) 1(500,1000)(10,100)2(500,1000)(100,500)3(1000,5000)(100,500)4(1000,5000)(10,100) Section 2.5.2 laterdiscussestheperformanceofourdynamicprogramwhenusedasaheuristicmethodforsolvingP(FC)withoutanyspecicrestrictionsoncoststructures.Allnumericalstudieswereperformedonacomputerwith64-bitWindowsoperatingsystem,a2.67GHzIntelrCoreTM2CPU,and4GBRAM. 2.5.1PerformanceoftheDynamicProgramInthissectionwebenchmarktheperformanceofthedynamicprogramsproposedinSection 2.4 versusthenonconvexmixedintegernonlinearprogramming(MINLP)solver 40

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Couenne.Firstweanalyzethecasewithuniformprice-supplyfunctionsandgeneralxedprocurementcharges,asdiscussedinSection 2.4.1 (forthesecases,werandomlygenerateasinglevalueofeachoftheparameterst=,p0t=p0,andt=).ThetimecomplexityofthedynamicprogramtosolvethiscaseisO(jTj9)wherejTjistheplanninghorizonlength.Recallthatthebottleneckoperationrequiresdeterminingthecostofasubplan(s;u;t;v)whereashortestpathproblemissolvedforeachpossiblenumberofprocurementsetups,m,for1mt)]TJ /F3 11.955 Tf 12.15 0 Td[(s+1.Foreachsubplan(s;u;t;v),weperformapreprocessingstepinwhichwecomputealowerboundandanupperboundonSPC(s;u;t;v)foreachmandeliminatethosethatcannotbepartofanoptimalsolution.Fortheupperboundcomputation,werequirethemprocurementsetupstooccurintherstmperiodsoftheRIc(s;t+1),andthensolvethelot-sizingproblemattheproductionlevel.Notethatifthissolutionisnotfeasible,thennofeasiblesolutionexistsforsubplan(s;u;t;v)withmsetups.Tocomputealowerbound,weassumethatthemprocurementsetupswilltakeplaceinthemperiodswiththecheapestprocurementsetupcosts(withoutconsideringinfeasibility),andsolvetheuncapacitatedlot-sizingproblemattheproductionlevel,assuminganinnitesupplyofcomponents.Wetheneliminateanymvaluefromconsiderationthatproducesahigherlowerboundthantheminimumupperboundobtainedinthisprocess.Clearly,theseeliminatedcasescannotbepartofanoptimalsolution.ThispreprocessingstepsignicantlyreducesthenumberofshortestpathproblemssolvedforeachSPC(s;u;t;v).Table 2-3 comparestheperformanceofthenonconvexMINLPsolverCouenneversusthedynamicprogramproposedinSection 2.4.1 .Thevalues(inseconds)aretheaveragesover40probleminstances(10instancesforeachscenarioshowninTable 2-2 ).Herewesetthetimelimitparameterforthesolverto3,600seconds.Weobservedthatforsomeoftheprobleminstances,thesolverstoppedbeforendinganoptimalsolutionbecauseitreachedthetimelimit(thenumberofinstancesforwhichthisoccurredisnotedinthesuperscriptinthesecondcolumnofthetable). 41

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Table2-3. Runtimecomparison:Uniformprice-supplyfunctioncase. AverageruntimeStandarddev.ofruntimejTjCouenneDPCouenneDP 40.326<0.0010.139<0.00150.534<0.0010.210<0.00160.9750.0010.4060.00471.5240.0020.7180.00682.2310.0021.4850.00693.7230.0052.4210.008106.9880.0125.3810.0101111.5330.02011.0670.0121227.3910.03226.8850.0201351.9130.06164.9000.02714151.6080.082178.7990.03815329.9190.135413.8060.062161057.167(5)0.2241386.4040.114171567.080(10)0.3251580.3800.156181955.395(12)0.5271603.5750.261191983.928(13)0.6401660.7190.349202198.771(9)0.8951594.9850.495 ()SuperscriptsshowthenumberofinstancesinwhichCouennereachedthetimelimit. Clearly,thedynamicprogramoutperformsthegeneralpurposeMINLPsolver,becauseitutilizesthestructuralcharacteristicsofanoptimalsolutionforthisproblemclass.Astheproblemsizegrows,thecomputationtimeincreasesdrasticallyfortheMINLPsolver,asdoesasthevariationintotalsolutiontime.Theaverageandstandarddeviationofruntimesremainmuchlowerwhenusingthedynamicprogrammingmethod.Herewenotethatanotherbottleneckoperationinthedynamicprogrammingalgorithmisthegenerationoftheshortestpathgraphtocomputethecostofasubplan(s;u;t;v)foragivenvalueofm(numberofprocurementsetupsinthesubplan).ThenumberofarcsofthisgraphisboundedbyO(jTj4),whereweneedO(jTj)operationstocomputeeacharc'scost.Ournumericalanalysisshowsthattheaveragesizeismuchsmaller(seeTable 2-4 ),andthattheworst-caseboundsonthenumberofverticesandarcsarequiteloose.This,infact,impliesthatourdynamicprogramhasanaverage-caseperformancethatismuchbetterthantheworst-caseperformancebound. 42

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Table2-4. Sizeofthesetofvertices,Vandarcs,A. jTjAveragejVjMaximumjVjAveragejAjMaximumjAj 43.6972.75954.06103.151264.32133.632274.56174.073684.83214.554395.11265.2474105.33315.8191115.61376.31111125.78437.02177136.17507.89179146.33578.82214156.836510.15338166.827310.72342177.178211.68393187.379112.21571197.8410114.11471207.7811114.56678 Table 2-5 showssimilarresultsforcaseswithnondecreasingprocurementrelatedcosts,asdiscussedinSection 2.4.2 (togeneratetheseprobleminstances,wesortthejTjrandomlygeneratedvaluesofFCt,t,t,andt;ift>t+1,weincreaset+1untilt=t+1).Hereagainthevalues(inseconds)aretheaveragesover40probleminstances(10instancesforeachscenarioshowninTable 2-2 ;wedidnotgobeyondjTj=18periodsforthiscase,astheresultsandtrendswereverysimilartotheuniformprocurementcostcase,andtheruntimestendtobeveryhighforCouenne).Asnotedatthebeginningofthissection,thenumberofsetupsthatoccurinanoptimalsolutionistypicallyunusuallyhighwhencomparedtoproblemsintheabsenceofconvexprocurementcosts.ForeachofthescenariosshowninTable 2-2 ,Table 2-6 showstheaveragefractionofperiodsinwhichprocurementandproductionsetupsoccurredinanoptimalsolution,forboththeuniformprocurementcostandnondecreasingprocurementcostcases.RecallthatScenarios3and4correspondtohigherxedprocurementcosts,whileScenarios2and3correspondtohigherdemandlevels.Asasimplebenchmark,understationarydemand,evenwithourlowestsettingofxedcost(500),highestlevel 43

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Table2-5. Runtimecomparison:Nondecreasingprocurementrelatedcostcase. AverageruntimeStandarddev.ofruntimejTjCouenneDPCouenneDP 40.293<0.0010.146<0.00150.499<0.0010.2130.00260.837<0.0010.346<0.00171.3510.0010.5130.00381.8760.0010.7450.00492.6880.0021.0460.005104.5220.0032.3600.006117.0170.0064.2290.0081211.1550.0086.5960.0081321.6970.01115.6680.0071445.0900.01741.7350.0041596.2530.02490.8410.00816177.1100.035190.2590.00717454.3530.048501.6300.005181059.7580.0671373.0650.007 ofdemandperperiod(500)andhighestlevelofholdingcostperperiod(0:005),wewouldnotsetupmorefrequentlythanonceevery20periods(usingtheEOQformula,T=p 2F=hD).Table 2-6 showsthatanoptimalsolutionsetsupmuchmorefrequentlythanthis(insomecases,weobserveprocurementineveryperiod).Thesediseconomiesofscaleinprocurementalsoleadtomorefrequentproductionsetups(asaresultoflimitingtheamountofprocurementinanyperiod),despitetheabsenceofcapacitylimits. Table2-6. Averagefractionofperiodsinwhichasetupoccurs. UniformcostsNondecreasingcostsScenarioNo.ProductionProcurementProductionProcurement 10.320.620.260.5220.861.000.741.0030.860.990.710.9840.300.390.210.34 2.5.2PerformanceasaHeuristicSolutionMethodInthissectionwetesttheperformanceofthedynamicprogrampresentedinSection 2.4.1 whenusedasaheuristicmethodforprobleminstancesthatviolatethecostassumptionsinSection 2.4.1 .Theideaofthisheuristicapproachistoimpose 44

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arequirementonthesolutiontoprocureequalamountsinallperiodswithpositiveprocurementwithinasubplan.Inaddition,theheuristicwillrequireproductionplanstosatisfytheZIPproperty,evenwhenthecoststructuredoesnotimplythatthispropertymustholdinanoptimalsolution.Theheuristicsolutionmethodworksasfollows.Wecomputethecostofsubplan(s;u;t;v),i.e.,SPC(s;u;t;v),asdescribedinSection 2.4.1 withafewmodications.Giventhenumberofperiodswithpositiveprocurement,m,wecreatetheshortestpathgraphG(V;A),wherethesetofverticesVandarcsAaregeneratedthesamewayasdescribedbefore.ThecostofarcsinthesetsA1andA2iscomputeddierently,inordertoaccountforthetime-varyingprocurementcostparameters.Hereagain,weletbetheamounttobeprocuredgivenm;hence,=Du;v=m.Equations( 2{19 )and( 2{20 )presentthearccostcomputationsforthearcsinsetsA1andA2,respectively. cma=lXj=1FC[j]+[j]+2 [j]+[j] [j]+u)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xk=shCkKs;k;8a2A1:(2{19) cma=8><>:1;ifDu;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1>Ks;i;C0(il;jk)+P(il;jk);otherwise,;8a2A2;(2{20)where C0(il;jk)=k)]TJ /F5 7.97 Tf 6.59 0 Td[(lXt=1FC[t]+[t]+2 [t]+[t] [t]+j)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xt=ihCt(Ks;t)]TJ /F3 11.955 Tf 11.96 0 Td[(Di;j)]TJ /F7 7.97 Tf 6.59 0 Td[(1);(2{21)andP(il;jk)iscomputedasinequation( 2{11 ).Here[j]returnstheindexoftheperiodwiththejthcheapestxedprocurementcost.ThecomputationofSPC(s;u;t;v)andformulationofthedynamicprogramfollowasinSection 2.4.1 .Weperformedtwosetsofteststobenchmarktheperformanceofthedynamicprogrammingalgorithmasaheuristicmethod.Intherstset,wecreatedinstancesforwhichAssumption 1 and,therefore,theZIPproperty,holdswithageneralcoststructureforremainingproblemparameters.Inthesecondset,wecreatedinstanceswithgeneral 45

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costsstructures,i.e.,noneoftheproblemparametersarerequiredtofollowanyofthespecializedcostassumptionsstatedinSection 2.4 .Table 2-7 demonstratestheperformanceofthedynamicprogramasaheuristic,wheretheoptimalsolutionwasobtainedineachcasebyallowingCouennetorununtilanoptimalsolutionwasobtained.Theaverageoptimalitygapachievedbytheheuristicwasreasonablysmall,evenfortheprobleminstanceswherethecoststructuredoesnotguaranteetheoptimalityofaZIPsolution.Themaximumoptimalitygapdidnotexceed5.5%forinstanceswhereaZIPsolutionisnotguaranteedtobeamongtheoptimalsolutions,anditwassmallerthan3.9%forinstanceswhereaZIPsolutionisoptimal.ThisclearlyshowsthatthedynamicprogramcanbeusedasaneectiveheuristicapproachtosolvegeneralinstancesofP(FC).Moreover,thisanalysisindicatesthat,onaverage,thecostpenaltyforimposingaZIPrequirementonproductionandequalprocurementbatchsizeswithcomponentregenerationintervalsisgenerallysmall. Table2-7. Performanceofthedynamicprogramasaheuristic. withZIPwithoutZIPjTjAverage%GapStandarddev.Average%GapStandarddev. 40.6230.8590.9301.28761.0261.0421.1921.15081.0230.8771.1590.695100.8710.6101.4251.050121.1110.7601.2680.711141.2860.7441.3570.538161.3890.8151.4020.737 46

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CHAPTER3APOLYNOMIALTIMEALGORITHMFORCONVEXCOSTLOTSIZINGPROBLEMS 3.1ModelandRelatedWorkWeconsidertheclassicdiscrete-time,nite-horizoneconomiclot-sizingproblemwithnondecreasingandconvexcostsintheproductionquantitiesandinventorylevelsineachperiod.Thisproblemconsidersasetofconsecutivedemandperiodsandseekstomeetdeterministicdemandforaproductineachperiodt=1;2;:::;Twithoutlostsalesorbacklogging,ataminimumtotalcostovertheplanninghorizonoflengthT.Thetotalcostincurredoverthehorizonconsistsofthosecostsassociatedwithproducingtheproductinanyperiodaswellasthecostofholdinginventoryoftheproductbetweenperiods.Atthebeginningofeachperiod,theproductionquantityintheperiodisaddedtoremaininginventoryfromthepriorperiodwithzeroleadtime,andthesumofthesetwoquantitiesmustbeatleastasgreatasthedemandintheperiod(withthedierencecomprisingtheamountofinventorythatwillremainattheendoftheperiod).Becausealldemandsandcostsareassumeddeterministic,allproductiondecisionsmaybemadeinadvanceoftherstplanningperiod,andthemodelcaninprincipleaccommodateanynite,deterministicproductionplanningleadtime.Weassumethroughoutthatnocapacitylimitexistsontheproductionquantityortheinventoryheldinanyperiod.Ofparticularimportanceinapproachingthesolutionofproblemsinthisclassarethenatureandstructureoftheproductionandinventoryholdingcosts.Becauseawiderangeofpracticalsettingsexistwitheconomiesofscaleinproduction,thecostofproductioninaperiodisoftenmodeledasanondecreasingandconcavefunctionoftheproductionquantity,whileconventionalapproachestypicallytreatthecostofholdinginventoryasalinearfunctionoftheinventoryamountattheendoftheperiod.Thisleadstoatotalcostfunctionthatmaybeexpressedasthesumofconcavefunctionsoftheproductionquantities,implyingaconcavetotalcostfunction.Conversely,productionprocessesandinventorycostswithdiseconomiesofscalemayoftenbemodeledusingconvexfunctionsof 47

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theproductionquantitiesandinventorylevels.Thischapterconsidersthislatterclassofproblemswithnondecreasingconvexcostfunctions.Ourmaincontributionliesinprovidingapolynomial-timealgorithmforthelot-sizingproblemwehavedescribedwithgeneralnondecreasingconvexproductionandholdingcostfunctions.Wediscusstwowaystoimplementourapproach.TherstisaniterativenumericalapproachthatrunsinO(T2maxflogT;logSlogM=g)time.HereTdenotesthenumberoftimeperiodsandSprovidesanupperboundonthenumberofnon-dierentiablepointsofthecosttosupplydemandinperiodsusingproductioninperiodtforany(t;s)pairwith1tsT.ThevalueofMisanupperboundonthemarginalproductioncostinanyperiod,anddenotesastoppingcriterionforabisectionsearchroutineintheproposedalgorithm.AswewillseeinSection 3.2 ,forthecaseinwhichallofthecostfunctionsarepiecewiselinearandconvex,thisiterativeapproachrequiresO(T2maxflogT;(logS)2g)time.ThesecondimplementationweproviderequiresrepeatedsolutionofasystemofequationswithatmostT+1equationsandT+1variables.AssumingthissystemofequationscanbesolvedinO((T))time,where(T)isapolynomialfunctionofT,thissolutionapproachrequiresO(TmaxfTlogT;(T)g)time.Whenallcostsaredierentiableandquadratic,thenO((T))=O(T),andthisexpressionbecomesO(T2logT).Tothebestofourknowledge,theliteraturedoesnotcontainaspecial-purposealgorithmforthisgeneralproblemclassthatrunsinpolynomialtimeintheworstcase.Althoughtheresultingproblemisaconvexoptimizationproblem,theapplicationofageneralpurposenonlinearprogrammingsolvertoproblemswithspecialstructuremayresultinunnecessarilylongsolutiontimes,andtheassociatedrunningtimewouldbeweaklypolynomial,regardlessofthestructureofthecostfunctions.Thealgorithmweprovideisbasedonaprimal-dualsolutionapproachderivedfromanalyzingthespecialstructureoftheproblem'sgeneralizedKarushKuhn-Tucker(KKT)conditions,whicharenecessaryandsucientforoptimality.Wespecializetheresultingalgorithmforapplicationtoaproductionplanningcontextinvolvingprice-dependent 48

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supplycomponents,wherecomponentsupplyislinearlyincreasinginprice.Theresultingproblemisaspecialcaseofthegenerallot-sizingproblemwithnondecreasingandconvexproductioncosts,wheretheproductioncoststakeaquadraticformandtheassociatedcomplexityisO(T2logT).Veinott[ 86 ]consideredasingle-stage,dynamiclotsizingprobleminwhichproductionandinventorycostsarepiecewise-linearandconvex.Hedesignedaparametric-programming-basedprocedureinwhichthesolutionforaproblemwithaxedparametersetisbuiltuponthesolutionofanotherproblemwithasimilarparameterset.Veinott[ 86 ]assumedallparameterswereintegervalued,andhisprocedureresultedinapseudo-polynomialsolutionalgorithm.ThetimecomplexitythealgorithmisO(TD),whereD=PTt=1dtdenotesthesumofalldemandsovertheplanninghorizon.Florianetal.[ 22 ]mentionedVeinott'sprocedureasthemostattractiveapproachtosolvelotsizingproblemswithconvexproductionandinventorycostsandwithoutxedsetupcosts,eventhoughtheproblemisdemonstratedtobenoharderthanlinearprogramming,whichispolynomiallysolvable.TheworkbyKianetal.[ 41 ]isalsocloselyrelatedtoours,astheyanalyzedasingle-stage,uncapacitatedeconomiclot-sizingproblemwithxedsetupcostsandvariablecostsineachperiodthatareconvexintheproductionquantity(takingtheformofapolynomialfunctionoftheproductionquantity).Theyderivedseveralkeyoptimalityconditionsforthisproblemclass,aswellasadynamicprogrammingsolutionalgorithmthatisexponentialinthelengthofthetimehorizon.Theypresentedanexactsolutionalgorithmforthelot-sizingproblemwithzerosetupcostsandproductioncoststakingtheformofpolynomialconvexfunctions,andstatedthattheworst-casetimecomplexityofsuchanalgorithmwouldbeO(T2).InSection 3.3 ,wewillexplainwhythisboundappliestothenumberofsubplansthatmustbeconsidered,andnottotheproblem'soverallworst-casecomplexity. 49

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Theproblemweconsiderfallsintotheclassofthesingle-stage,dynamiclot-sizingproblems,whichhasbeenstudiedbymanyresearchers,startingwiththeseminalworkofWagnerandWhitin[ 88 ],whorstconsideredtheproblemwithconcaveproductioncosts.Brahimietal.[ 9 ]providedanextensivereviewofuncapacitatedandcapacitatedversionsofthesingle-stage,dynamiclot-sizingproblem.Theuncapacitatedversionoftheclassicalsingle-stage,dynamiclot-sizingproblemwithconcavecostsispolynomiallysolvable,whereasthegeneralcapacitatedversionoftheproblemwithconcavecostsisNP-Hard,althoughtheuniform-capacityversioncanbesolvedinpolynomialtimeviadynamicprogramming(seeFlorianandKlein[ 21 ]).Manystudieshaveconsideredvariationsofthesingle-stage,dynamiclot-sizingproblem(seee.g.VanHoeselandWagelmans[ 84 ],andVandenHeuvelandWagelmans[ 83 ]).Inthesestudies,allcostsareassumedtobeconcaveintheproductionandinventorylevels.Theliteratureonlot-sizingproblemscontainingconvexproductioncostsisreasonablysparse,withafewnotableexceptions.ErengucandAksoy[ 18 ]consideredasingle-item,capacitateddynamiclotsizingproblemwithxedproductionsetupcostsandlinearinventorycosts,whilevariableproductioncostswerepiecewiselinearandconvexintheproductionquantityinaperiod.Theyusedabranch-and-boundalgorithmforthisproblem,whichcontainsneitheraconvexnorconcaveobjectivefunction.ShawandWagelmans[ 74 ]developedapseudo-polynomialdynamicprogramtosolveacapacitatedsingle-itemlot-sizingproblemwithpiecewiselinearproductioncosts.Theiralgorithmcanbeutilizedtosolveproblemswithpiecewise-linearandconvexproductioncosts,althoughitdoesnotrequireanyspecialstructureforthepiecewise-linearcostfunction.Fengetal.[ 19 ]developedanO(TlogT)algorithmforthesingle-itemlot-sizingproblemwithconstantcapacity,convexinventorycosts,andnon-increasingxedordercosts.Therestofthischapterisorganizedasfollows.Section 3.2 providesageneralmodelformulation,descriptionoftheKarushKuhn-Tucker(KKT)conditionsfortheproblem,anddevelopmentofapolynomial-timealgorithm,alongwithacomparisonof 50

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thisapproachtotheoneprovidedbyVeinottin[ 86 ].Section 3.3 discussestheproductionplanningproblemwithprice-dependentsupply,andillustratestheapplicationofthemodeltoapracticalspecialcasethatfallswithinthegeneralproblemclassweconsider.Section??containsbriefconcludingremarks. 3.2ProblemFormulationandSolutionMethodThelot-sizingproblemrequiresmeetingasetofdemandsdt,fort=1;:::;TwithoutshortagesataminimumtotalproductionandinventoryholdingcostoverthehorizonoflengthT.Lettingxtdenotetheproductionquantityinperiodt,itwillbeconvenientandusefultokeeptrackoftheamountproducedinperiodtinordertomeetdemandinperiod,xt,wherext=PT=txtfort=1;:::;T.Weassumethatthecosttoproducextunitsinperiodtisanondecreasingconvexfunctionft(xt).Inaddition,ht(it)denotesanondecreasingandconvexinventoryholdingcostfunction,whichdependsontheinventoryattheendofperiodt,denotedbyit.Theinventoryremainingattheendofperiodtcanbeequivalentlywrittenasit=Pt=1PTi=xi)]TJ /F9 11.955 Tf 12.16 8.96 Td[(Pt=1d;thus,wecanalternativelywritehtasafunctionoftheproductionvariablesusingtheexpressionhtPt=1PTi=xi,wherewehavesuppressedthedependenceofhtoncumulativedemanduptoperiodtfornotationalconvenience.WeassumethateachofthefunctionsftandhtiseverywherelocallyLipschitzcontinuous.Theconvexcostlot-sizingproblemmaythenbeformulatedasfollows.P:MinimizeXTt=1nftXT=txt+htXt=1XTi=xio (3{1)Subjectto:Xt=1xt=d;=1;:::;T; (3{2)xt0;=1;:::;T;andt: (3{3)Asthesumofconvexfunctions,theobjectivefunctionofproblemPisconvex;thiscombinedwiththelinearconstraintsetimpliesthatthegeneralizedKKTconditionsarenecessaryandsucientforoptimality(seeHiriart-Urruty[ 36 ]). 51

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3.2.1GeneralizedKKTConditionsTocharacterizethegeneralizedKKTconditions,let@ft(Xt)denotethegeneralizedgradientofftatXt,whereXt=(xtt;xtt+1;:::;xtT).IfftisdierentiableatXt,then@ft(Xt)consistsofasingletonequaltothepartialderivativewithrespecttoanyelementofXtatXt;otherwise,@ft(Xt)correspondstothesetofsubgradientsatXt.Similarly,letXt=(X1;X2;:::;Xt),andlet@ht(Xt)denotethegeneralizedgradientofhtatXt.NotethatthesumoverallvariablesinthesetXtgivesthecumulativeproductionthroughtheendofperiodt.Wealsonotethatagivenvariablext(=1;:::;Tandt)appearsintheargumentofthefunctionftandintheargumentofeachfunctionhiforit.LetandvdenotevectorsofKKTmultipliersassociatedwithconstraints( 3{2 )and( 3{3 ),respectively.ThefollowingprovidethegeneralizedKKTconditionsforproblemP,separatedintodualfeasibilityandcomplementaryslacknessconditions. Dualfeasibilityconditions:02@ft(Xt)+TXi=t@hi(Xi))]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(vt;=1;:::;Tandt; (3{4)vt0=1;:::;Tandt: (3{5) Complementaryslacknessconditions:vtxt=0=1;:::;Tandt: (3{6) 3.2.2SolutionApproachLetPs)]TJ /F7 7.97 Tf 6.59 0 Td[(1denotethesubproblemobtainedbyeliminatingperiodss;:::;TfromproblemP(forsomepositivesT),andsupposewehaveanoptimalsolutionforPs)]TJ /F7 7.97 Tf 6.59 0 Td[(1.Letxs)]TJ /F7 7.97 Tf 6.59 0 Td[(1tdenotethevalueofxtinthisoptimalsolutionfor=1;:::;s)]TJ /F1 11.955 Tf 12.44 0 Td[(1andt,andlets)]TJ /F7 7.97 Tf 6.59 0 Td[(1andvs)]TJ /F7 7.97 Tf 6.58 0 Td[(1tdenotethecorrespondingoptimalKKTmultipliers.BecausethegeneralizedKKTconditionsarenecessaryandsucient,( 3{2 )through( 3{6 )mustholdforthissolution.Next,considerproblemPs,andletxst=xs)]TJ /F7 7.97 Tf 6.59 0 Td[(1t,vst=vs)]TJ /F7 7.97 Tf 6.59 0 Td[(1t,ands=s)]TJ /F7 7.97 Tf 6.59 0 Td[(1for=1;:::;s)]TJ /F1 11.955 Tf 12.02 0 Td[(1andt.WewillcreateasolutionforproblemPsbydeterminingvalues 52

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ofxstsandvstsfort=1;:::;s,aswellasss,whileleavingallremainingprimalvariablesunchanged.Wedothisbyrstinitializingeachofthesenewvariablesatzero,andthendeterminingtheoptimalvaluesofthesenewvariables.Observethatbycondition( 3{6 ),ifxsts>0forsomets,thenwemusthavevsts=0.IfweletTdenotethesetofperiodstfrom1throughssuchthatxsts>0,thenwemusthave ss2@ft(Xt)+Psi=t@hi(Xi);8t2T; (3{7) atageneralizedKKTpointforproblemPs.OurgoalistondavalueofssandacorrespondingsetT,withassociatedxstsvalues,suchthat( 3{7 )holdsandPt2Txsts=ds.Werstcharacterizeakeypropertyoftherelationshipbetweenssandxstsin( 3{7 ).Tocharacterizethisproperty,letusdeneXt)]TJ /F5 7.97 Tf 6.59 0 Td[(s=(xstt;xst;t+1;:::;xst;s)]TJ /F7 7.97 Tf 6.59 0 Td[(1),andXi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s)=Xinfxstsg.Property3.1.Foranyt2TandforanyxedXt)]TJ /F5 7.97 Tf 6.58 0 Td[(sandaxedXi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s),i=t;:::;s,theminimumvalueofxststhatsatises( 3{7 )ismonotonicallyincreasinginss.Property3.1followsfromtheconvexityofftandeachht.Wecanreplacetheword\minimum"inthispropertywith\maximum"andthepropertycontinuestohold(ifftandallhtfunctionsarestrictlyconvex,thentheword\minimum"mayberemovedfromthestatement;the\minimum"or\maximum"qualieraccountsforthepossibilityoflinearsegmentsofthefunction,whereinagivenvalueofssdoesnotmaptoauniquevalueofxstsin( 3{7 )).Forclarity,wewrite@ft(Xt)foraxedXt)]TJ /F5 7.97 Tf 6.59 0 Td[(s=(xstt;xst;t+1;:::;xst;s)]TJ /F7 7.97 Tf 6.59 0 Td[(1)as@ft(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;xsts),where@ft(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;0)correspondstothegeneralizedgradientatxsts=0.Similarly,foraxedXi)]TJ /F7 7.97 Tf 6.58 0 Td[((t;s),wewrite@ht(Xi)]TJ /F7 7.97 Tf 6.58 0 Td[((t;s);xsts).Wealsousethenotation@lftand@lht(@uftand@uht)todenotetheminimum(maximum)subgradientvalueoftheproductionandholdingcostfunctioninperiodtatapoint. 53

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Property3.2.SupposesscorrespondstoanoptimalmultiplieratageneralizedKKTsolution.Ifss<@lft(Xt)]TJ /F5 7.97 Tf 6.58 0 Td[(s;0)+Psi=t@lhi(Xi)]TJ /F7 7.97 Tf 6.58 0 Td[((t;s);0)forsomets,thenthecorrespondinggeneralizedKKTpointmusthavevsts>0andxsts=0.Property3.2followsfrom( 3{4 )through( 3{6 ),themonotonicityproperty3.1,andthenonnegativityofxsts.Notealsothatifss2@lft(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;0)+Psi=t@lhi(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);0)forsomets,thenwemaysetxsts=vsts=0toobtainacorrespondingKKTpoint.Ifss>@uft(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;0)+Psi=t@uhi(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);0)forsomets,however,thenwecansatisfycondition( 3{7 )byincreasingxsts.Assumethatforagivenss,wecandetermineanintervalforxstssuchthatss2@ft(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;xsts)+Psi=t@hi(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);xsts),i.e.,suchthat( 3{7 )holds.Let[lts(ss);uts(ss)]denotethecorrespondinginterval.Next,deneT(ss)asthesetofperiodstsuchthatss>@fut(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;0)+Psi=t@hui(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);0).InordertondageneralizedKKTpoint,weseekafeasiblesolutiontothefollowingsystem: xsts2[lts(ss);uts(ss)];t2T(ss); (3{8) Pt2T(ss)xsts=ds: (3{9) Theabovesystemdenesapolyhedron,andwecandeterminewhetherafeasiblesolutionexistsbyevaluatingtheleft-handsideof( 3{9 )atxsts=lts(ss)foreacht2T(ss)andatxsts=uts(ss)foreacht2T(ss).Iftheformervalueislessthanorequaltodsandthelattervalueisgreaterthanorequaltods,thenafeasiblesolutionexistsandcanbeconstructedinO(T)time.Iftheleft-handsideof( 3{9 )evaluatedatxsts=uts(ss)isstrictlylessthands,thenthegivenvalueofsscannotcorrespondtoanoptimalKKTmultiplier;inthiscase,byProperty3.1,theoptimalvalueofsforproblemPsmustexceedthechosenvalueofss.Ontheotherhand,iftheleft-handsideof( 3{9 )evaluatedatxsts=lts(ss)isstrictlygreaterthands,thenthegivenvalueofsscannotcorrespondtoanoptimalKKTmultiplier;inthiscase,byProperty3.1,theoptimalvalueofsforproblemPsmustbelessthanthechosenvalueofss.Asaresultofthemonotonicityproperty(3.1)andthecontinuityofeachftandht,wecanperformabisectionsearchon 54

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sinordertondavalueofsssuchthatafeasiblesolutionexistsfor( 3{8 )and( 3{9 )towithinachosentolerance>0(notethateachssevaluatedinthebisectionsearchrequiresexplicitlydeningthesetT(ss)).Wecallsuchasolutionan-approximategeneralizedKKTpoint.Theresultingsolutiondeterminesssandxstsfort2T(ss).Afterndingasolutionto( 3{8 )and( 3{9 )viabisectionsearchons,foreacht=2T(ss)wesetvstsequaltoanelementof@ft(Xt)+Psi=t@hi(Xi). Proposition3.1. Thesolutionobtainedafterperformingabisectionsearchonsasdescribedprovidesan-approximategeneralizedKKTpointforconditions( 3{2 )through( 3{6 )for=sandts. Proof. Thesolutionisprimalfeasiblewithatoleranceofasaresultofsatisfying( 3{9 )towithinfor=sandallts(andthefactthatnoprimalvariablesxtwerechangedfor0.Weconsidertwocases.CaseI:xst=0:Inthiscase,anincreaseinvstcanabsorbanyincreasein@ft(Xt)+Psi=t@hi(Xi).CaseII:xst>0:Ifxst>0,thensupposeweincreasesto@flt(Xt)+Psi=t@hli(Xi);notethatsss2@ft(Xt)+Psi=t@hi(Xi).Supposethisincreaseinscausesaviolationinaconstraintoftheform( 3{4 )forsomeperiodr,whichmayonlyoccurifxsrs=0(otherwise,( 3{7 )wouldhavetoholdatt=r).Thisconstraintviolationimpliesthats+vsr>@fur(Xr)+Psi=r@hui(Xi).Ifxsr>0thenthisimpliesvsr=0ands>@fur(Xr)+Psi=r@hui(Xi);however,becausesss,thisimpliesss> 55

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@fur(Xr)+Psi=r@hui(Xi),whichwouldhaveledtor2T(ss),whichinturnwouldhaveledtoasolutioninwhichss2@fr(Xr)+Psi=r@hi(Xi),creatingacontradiction.Therefore,itmustbethecasethatvsr>0andxsr=0.Inthiscase,wecanreducevruntiltheconstraintissatisedoruntilithitszero.Ifthelatteroccursrst,thenwearebackatthecaseinwhichsss>@fur(Xr)+Psi=r@hui(Xi),leadingtoacontradiction.Asaresult,wehaveconstructedasolutionthatsatisesthegeneralizedKKTconditionsforproblemPstowithin,andtherefore,an-approximategeneralizedKKTsolution. ItisasimplemattertosolveP1,bysettingx111=d1,11=@f1(d1),andv111=0.Algorithm 3.1 thensolvesPsfors=2;:::;T. Algorithm3.1. GivenoptimalsolutionforPs)]TJ /F7 7.97 Tf 6.58 0 Td[(1,solvePs. 1: Input:OptimalsolutionforPs)]TJ /F7 7.97 Tf 6.59 0 Td[(1:xs)]TJ /F7 7.97 Tf 6.59 0 Td[(1tfor=1;:::;s)]TJ /F1 11.955 Tf 11.95 0 Td[(1andt. 2: Output:SolutionforPs:xstfor=1;:::;sandt. 3: Initialization:i=0,xst=xs)]TJ /F7 7.97 Tf 6.58 0 Td[(1tfor=1;:::;s)]TJ /F1 11.955 Tf 11.13 0 Td[(1andt;xsts=0fort=1;:::;s. 4: Sort@fut(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;0)+Psi=t@hui(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);0)valuesinnondecreasingorder,t=1;:::;s. 5: whilexsts=0forallt=1;:::;sdoBisectionsearchonss 6: Givenss,letT(ss)=ftsjss>@fut(Xt)]TJ /F5 7.97 Tf 6.58 0 Td[(s;0)+Psi=t@hui(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);0)g. 7: Fort2T(ss)determineinterval[lts(ss);uts(ss)]usingbinarysearch. 8: ComputePt2T(ss)lts(ss)=LandPt2T(ss)uts(ss)=U. 9: ifLdsUthen 10: Find-feasiblesolutiontoxsts2[lts(ss);uts(ss)];t2T(ss);Pst=1xsts=ds. 11: else 12: Adjustssvalueinbisectionsearchandcontinue. 13: endif 14: endwhileThefollowingpropositioncharacterizestheworst-casecomplexityofAlgorithm 3.1 .Instatingthisproposition,weletSdenoteaboundonthemaximumnumberof 56

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pointsofnon-dierentiabilityofanyofthefunctionsoftheformft+Psi=thi,wherewehavesuppressedthedependenceofthesefunctionsonthextvariablesfornotationalconvenience. Proposition3.2. Algorithm 3.1 requiresO(T2maxflogT;logSlogM=g)time. Proof. Characterizingtheworst-casecomplexityofAlgorithm 3.1 requiressomeassumptionsonthecharacteristicsofthefunctionscontainedintheobjectivefunction.WerequirethecharacterizationofO(T2)functionsoftheformFtft+Psi=thi.Tothisend,supposethefunctionFtiscomposedofStsegments,suchthatthefunctionisdierentiableontheinteriorofeachsegment,theunionofthesegmentsequalsR+,andnotwosegmentsintersect(theendpointsofeachsegmentcorrespondtothefunction's\breakpoints,"whereljtandujtdenotethelowerandupperbreakpointsofsegmentj).LetS=maxt=1;:::;TfStg.ForeachfunctionFtandeachintervalofthefunction,weneedtocharacterizethegeneralizedgradientfunction@Ft.ForthejthsegmentofFt,denotedassjtforj=1;:::;St,letm jtand mjtdenotetheminimumandmaximumslopes(derivatives)forthesegment.Ifasegmentislinear,thenm jt= mjt;otherwise mjt>m jt.Thesetofsubgradientsatthelower(upper)breakpointofsegmentjisgivenby[ mj)]TJ /F7 7.97 Tf 6.59 0 Td[(1t;m jt]([ mjt;m j+1t]).Withintheinteriorofeachsegment,thefunction'sderivativeprovidesthesubgradientvalue.AlthoughStep 3 ofAlgorithm 3.1 requirestheinitializationofO(T2)variables,thisstepisnotstrictlynecessary,asallxtvariablesfor
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andmaximumslopevaluesassociatedwitheachsegmentofthefunction;afteridentifyingtheappropriatesegment,weassumethatthecorrespondingintervalmaybedeterminedinconstanttime.BecausethisisrequiredforO(T)functionsFt,thisstepofthealgorithmrequiresO(TlogS)operations.Asdiscussedpreviously,Steps 8 through 10 requireO(T)operations.Thewhileloop(steps5-14)performsabisectionsearchonss.IfMdenotesanupperboundontheslope(derivative)valuethatneedstobeevaluatedforallofourconvexfunctionsanddenotesthestoppingcriterionofthebisectionmethod,thewhilelooprequiresO(logM=)iterationsforeachofthefunctionsFt;thus,thewhilelooprequiresatotalofO(TlogSlogM=)iterations.AnalgorithmfordetermininganoptimalsolutiontoproblemPs,givenanoptimalsolutiontoproblemPs)]TJ /F7 7.97 Tf 6.59 0 Td[(1,requirestwostepsandasubroutine.TheoverallcomplexityofsolvingPsisthenO(TmaxflogT;logSlogM=g)givenasolutionforproblemPs)]TJ /F7 7.97 Tf 6.59 0 Td[(1.InordertosolveproblemP=PT,weapplyAlgorithm 3.1 fors=2;:::;T.ThecomplexityassociatedwithsolvingPthereforeequalsO(T2maxflogT;logSlogM=g). Observethatforthepiecewise-linearcostcase,wecanperformabinarysearchonssinsteadofabisectionsearch.BecauseSdenotesanupperboundonthenumberofsegmentsofthepiecewise-linearcostfunctioninanyperiod,thewhilelooprequiresatotalofO(T(logS)2)iterations,whichleadstoanoverallcomplexityofO(T2maxflogT;(logS)2g). 3.2.3AlternativeImplementationWenextbrieydescribeanalternativeimplementationofAlgorithm 3.1 ,whicheliminatestheneedforbisectionsearchonssandprovidesanexactsolution.InStep4ofthealgorithm,wesortthevaluesof@fut(Xt)]TJ /F5 7.97 Tf 6.59 0 Td[(s;0)+Psi=t@hui(Xi)]TJ /F7 7.97 Tf 6.59 0 Td[((t;s);0)innondecreasingorderfort=1;:::;s.Letgst=@fut(Xt)]TJ /F5 7.97 Tf 6.58 0 Td[(s;0)+Psi=t@hui(Xi)]TJ /F7 7.97 Tf 6.58 0 Td[((t;s);0),andletgs[]correspondtothethsuchvalueaftersortinginnondecreasingorder,e.g.,gs[1]gs[2]gs[s].NextnotethatthedenitionofthesetT(ss)isthesameforanyssvalueintheinterval(gs[)]TJ /F7 7.97 Tf 6.58 0 Td[(1];gs[]],for=1;:::;s+1(wheregs[0]=0andg[s+1]=1);letTdenotethe 58

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correspondingsetdenitionfortheintervalendingatgs[].Ifwecanndsomevalueofthefreevariableonthisintervalsuchthat 2@ft(Xt)+Psi=t@hi(Xi);8t2T; (3{10) Pt2Txsts=ds; (3{11) wewillhavefoundageneralizedKKTpoint.Asaresult,insteadofperformingthebinarysearchonss,wemayattempttodeterminewhetherasolutiontotheabovesystemexistsdirectlyforeachintervalofssonwhichthedenitionofTisinvariant.Thecomplexityassociatedwithndingsuchasolutionwilldependonthestructureoftheftandhtfunctionsandtheirsubgradients.Let(T)denoteafunctionthatcharacterizesthenumberofoperationsassociatedwithndingtheO(T)solutionsto( 3{10 ){( 3{11 )foreachdenitionofT,=1;:::;s.Thus,wereplacethewhileloopinAlgorithm 3.1 withaprocedurethatrunsinO((T))time,witharesultingoverallcomplexityofO(TmaxfTlogT;(T)g)time.If,forexample,thesefunctionsareeverywheredierentiableandquadratic,thenthegradientfunctionswillbelinearfunctions,andEquation( 3{10 )takestheformofalinearequation.Thispermitsexpressingeachxstsasalinearfunctionofforeacht2T,substitutingtheresultingexpressionforxstsintoEquation( 3{11 ),andsolvingdirectlyforforeachsetT,=1;:::;s.Inthiscase,(T)=O(T),andtheworst-casecomplexityforsolvingproblemPbecomesO(T2logT).WediscussapracticalapplicationwithquadraticanddierentiablecostfunctionsinSection 3.3 3.2.4RelationshiptoVeinott'sApproachWenextdescribehowthealgorithmweproposeforlotsizingwithconvexcostscomparestoVeinott'sapproach([ 86 ])forlotsizingwithpiecewise-linearandconvexproductionandholdingcostsandintegerparametervalues.Veinott'smethodisaparametricapproachthatproceedsforwardintime,onedemandunitatatime.Beginninginperiod1,fortherstunitofdemand,hedeterminestheleastcostoptionforproduction 59

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ofthisunit(inthecasewithoutbacklogging,thereisonlyoneoptionforproducingthisunit,aswellasallunitsofperiod1demand).Aftersatisfyingalldemandinperiod1,hemovestoperiod2.Foreachunitofdemandinperiod2,hedetermineswhetheritischeapertoproducetheunitinperiod1or2,basedontheincrementalcostofproducingtheunit,whichinturndependsonthepartialsolutiondeterminedatpreviousiterations.Ifitischeapertoallocatetheunittoperiod1,thenheincreasesperiod1productionbyoneunit;otherwise,heincreasesperiod2productionbyoneunit.Foraunitofdemandinperiodt,giventhepartialsolutioncreatedatthepreviousiteration,hedeterminestheleastcostsolutionamongsolutionsthatproduceanadditionalunitinperiod1,2,:::;t)]TJ /F1 11.955 Tf 12.63 0 Td[(1,ort.Intheworst-case,therefore,therunningtimeofVeinott'salgorithmisafunctionofthetotaldemand.Infact,onecanshowthatthetimecomplexityofthisalgorithmisO(TD),whereDequalsthesumofalldemandsovertheplanninghorizon.Ouralgorithm,ontheotherhand,permitshandlingmultipledemandsateachiteration,insteadofconsideringonedemandunitatatime.Thisisenabledbysearching(viasubgradientsearch)forasubgradientvaluesuchthatthesumoftheproductionlevelswiththecorrespondingsubgradientvalue(forasubsetofperiods)equalsdemand.Forthecaseofpiecewise-linearandconvexproductioncostfunctions,thisrequireskeepingtrackoftheslopesandbreakpointsassociatedwiththepiecewise-linearcostfunctionforeachproduction/demandperiodpair.Basedonourprioranalysis,theworst-casecomplexityofthisapproachisO(T2maxflogT;(logS)2g),whereScorrespondstothemaximumnumberoflinearsegmentsassociatedwiththecosttosatisfydemandinperiodsusingproductioninperiodt,1tsT. 3.3ProductionPlanningwithPrice-DependentSupplyThissectiondescribesanapplicationofthelot-sizingproblemwithconvexproductioncosts,whichresultsinareductionintheworst-caserunningtimeofAlgorithm 3.1 ingeneral.Consideraproductionplanningsettingforaproductsuchthattheproduct'smanufacturingprocessrequiresaninputcomponentforwhichtheproduceroersaunit 60

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price,pt,ineachperiodttosuppliersofthatproductioninput.Onesuchexamplewouldcorrespondtoaresellerofarefurbishedproductwhoneedstoacquiretheuseditemfromconsumers.(Assumeforconveniencethatoneunitoftheinputcomponentisrequiredforeachunitoftheendproduct.)Theamountthatsuppliersarewillingtoprovidedependsonthepricetheproduceroers.Supposethattheproducerrefurbisheseachitemintheperiodinwhichitisreceived,whichimpliesthattheproductionamountinanyperiodt,xt,isafunctionofthepriceoeredtosuppliers.Tocharacterizethisfunction,supposethataminimumreservationpriceexistsineachperiodt,denotedbyp0t,suchthatforanypricenotexceedingthisvalue,nosupplyisavailable.Forpricesexceedingthisthreshold,weassumethatsupplyincreaseslinearlyinpriceatarateoft>0perunitpricechange.Asaresultoftheseassumptions,thesupplyinperiodtisanonnegativeandnondecreasingpiecewiselinearfunctionofprice,pt,withonebreakpointatp0t,denedby xt=8><>:tpt)]TJ /F3 11.955 Tf 11.96 0 Td[(t;ifptp0t;0;otherwise,(3{12)forpt0,witht0forallt=1;:::;T.Theparametert>0correspondstothepriceelasticityofsupplyinperiodt,fort=1;:::;T.Foreachunitofthecomponentprocured,theproducerpayspt,whichimpliesthatthetotalamountpaidtosuppliersinperiodtequalsptxt=tp2t)]TJ /F3 11.955 Tf 12.94 0 Td[(tpt=(x2t+txt)=t.Inaddition,weassumetheproducerincursacostoftxtforaprocurementlevelofxtinperiodt,wheretdenotesamaterialhandlingcostperunitcomponent.Thus,atotalprocurementrelatedcostof(pt+t)xt=tp2t)]TJ /F3 11.955 Tf 12.75 0 Td[(tpt+t(tpt)]TJ /F3 11.955 Tf 12.75 0 Td[(t)=(x2t+txt)=t+txtisincurredineachperiodt=1;:::;T.Anadditionalinventoryholdingcostofhtisassessedagainsteachunitremainingininventoryattheendofperiodt.Lettingct=ct+t+(t=t)+Pjthj, 61

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theresultingformulation,P0,iswrittenasfollows.P0:MinimizeTXt=1ctTX=txt+TXt=1PT=txt2 t (3{13)Subjectto:Xt=1xt=d;=1;:::;T; (3{14)xt0;=1;:::;Tandt: (3{15)Notethatwehaveomittedtheconstantterm)]TJ /F9 11.955 Tf 11.29 8.96 Td[(PTt=1dtPT=thfromtheobjectiveinthisformulation.Constraintset( 3{14 )ensuressatisfyingdemandineachperiod,whereasconstraints( 3{15 )enforcethenonnegativityrequirementsonthextvariables.ProblemP0isaspecialcaseofproblemPinwhicheachhtfunctionequalszeroandeachftfunctioniseverywheredierentiableandquadratic.Asaresult,problemP0canbesolvedinO(T2logT)timeusingAlgorithm 3.1 .Kianetal.[ 41 ]studytheuncapacitateddynamiclot-sizingproblemwithconvexcoststhattaketheformofapolynomialfunctionoftheproductionquantities.Theyprovideaforwarddynamicprogramtosolvetheversionoftheproblemwithxedcharges,whichtheymodifyinordertosolvefortheproblemwithzeroxedcharges,wheretheystatethecomplexityofthisalgorithmasO(T2).Similartoourmethod,theirsolutionapproachusesthemarginalcoststodeterminetheproductionamountsforeachperiod.TheirapproachsolvesasetofO(T2)subproblemsinadynamicprogrammingrecursion.ThisconstructivealgorithmrequiressolvingasystemofpolynomialequationsforeachsubprobleminordertocomputetheproductionquantityforeachofO(T)periods(aswellasthecostoftheassociatedsubproblem,whichcontainsO(T)periods).Thus,aworst-casecomplexityofO(T2)isonlyachievablewheneachofthesesubproblemsisassumedtobeperformedinconstanttime. 62

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CHAPTER4ANEOQMODELWITHPRICE-DEPENDENTSUPPLYANDDEMAND 4.1MotivationInthecenturyfollowingtheseminalpaperbyHarris[ 32 ],theeconomicorderquantity(EOQ)modelhasbeenwidelyusedtosolveinventoryplanningproblemsunderdeterministicandstationarycostanddemandassumptions.TheEOQmodelhasdrawntheattentionofnumerousresearchersbecauseofitswell-established,simpleandeasy-to-modifynature(seeNahmias[ 55 ]andSilver,Pyke,andPeterson[ 75 ]).Thischapterconsidersthisclassicprobleminthepresenceofabroadersetoffactorsthatmayinuencetheeconomicsofproductioninpractice.Morespecically,westudyaproductionplanningproblemwhereaninputcomponentisrequiredtoproduceaparticulartypeofend-item.Thecostoftheinputcomponentdependsonthepriceoeredtosuppliersbytheproduceroftheend-item.Asthepriceoeredbytheproducerincreases,theaggregatenumberofunitsofthecomponentavailabletotheproducerincreasesaswell.Thisphenomenonfollowsafundamentalprincipleineconomicsthattheavailablesupplyofagoodinamarketincreasesinprice.AsintheclassicalEOQmodel,aproductionprocessisrequiredtotransformtheinputcomponenttoanend-item.Thisproductionprocessinvolvesaxedcostforproducingabatchofend-items,aswellasavariable(per-unit)productioncostandinventoryholdingcostsforcomponentsandend-items.Demandfortheend-itemisanonincreasingfunctionofitssellingprice.Thus,theproducerwishestodeterminethepriceoeredtosuppliersfortheinputcomponent,thesellingpriceoftheend-item,anditsperiodicproductionlotsize,inordertomaximizeaverageprotperunittime.Wemodelthisproductionplanningandpricingproblemandcharacterizethecorrespondingoptimaldecisions.Inaddition,weanalyzethemodelinordertobetterunderstandthewayinwhichprice-dependentsupplyaectspricingandinventoryplanningdecisions. 63

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Thisworkcontributestotheliteraturebyconsideringhowthepricingofinputs(components)andoutputs(end-items)inuencestheeconomicsandprotabilityofproduction.Intheprocess,weidentifysomeinterestingrelationshipsbetweenthemodel'sparametersandthecharacteristicsofoptimalpricingandproductionplanningdecisions,aswellastheoptimalprotperunittimeandproductionprotmargins.Therestofthischapterisorganizedasfollows.Section 4.2 summarizestherelatedworkintheliterature.Section 4.3 denestheproblemunderconsideration,andformalizestheEOQmodelthatwillserveasthebasisforouranalysis.Section 4.4 providesthebulkofouranalysis.Werstproposeafunctionalrelationshipbetweensupplyandsellingpricesinequilibrium.Section 4.4.1 characterizesoptimalsupplyandsellingpricesforourmodel,aswellasthewayinwhichkeyparametervaluesaectthesedecisionvariablesatoptimality.Section 4.4.2 nextexploresthewayinwhichthenatureofthecomponentsupply-pricecurveinuencesprotabilityandpricingdecisions.Section 4.4.3 thendemonstrateshowalackofaccountingforthecomponentsupply-pricerelationshipcanreduceprotability. 4.2RelatedLiteratureThissectionprovidesareviewoftheliteraturecloselyrelatedtotheworkpresentedinthischapter.OurworkismostcloselyrelatedtopriorworkonEOQmodelswithprice-dependentdemandandEOQmodelswithprocurementquantitydiscounts.Inaddition,anaturalapplicationareaofourmodellieswithinaremanufacturingsetting,wherearemanufacturerwishestocollectuseditemsfromamarketinordertoprovideinputtoitsproductionprocess.Therefore,wewilldiscusstherelationofourworktoasegmentoftheremanufacturing/reverselogisticsliterature,focusingonthoseworkswhicharemostcloselyrelatedtoours. 4.2.1Price-DependentEOQModelsPricingdecisionsandprice-dependentdemandrateshavebeenconsideredintheinventorycontrolliteraturebymanyresearchersoverthepastseveraldecades.Whitin[ 89 ]presentedtherstmodelinwhichprice-dependentdemandwasincorporatedwithinthe 64

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traditionalEOQmodel.ArcelusandSirinivasan[ 4 ]subsequentlystudiedadeterministicEOQmodelinwhichdemandwaspricedependent,withagoalofcharacterizinganoptimalpricingandreplenishmentpolicy.Intheirmodel,thesellingpricewassetaccordingtoamarkupontheitem'sunitcost,wherethemarkuprateactedasadecisionvariable,andtheyconsideredtheobjectivesofmaximizingprot,returnoninvestment,andresidualincome.Rayetal.[ 62 ]characterizedprot-maximizingsolutionsunderaprice-sensitivedemandratefortwoclassesofpolicies.Intherstpolicyclass,pricewasanindependentdecisionvariable,whileinthesecond,pricewassetaccordingtoamarkuponunitcost.Inourwork,boththeinputtotheproductionprocess(component)andtheoutput(end-item)areprice-dependent.AdditionalworkongeneralizationsoftheEOQmodelwithprice-dependentdemandaccountsforperishabilityandbacklogging(Sana[ 68 ]),ademandratethatdependsonpriceandpromotionaleort(DeandSana[ 13 ]),anddemandthatdependsonthesefactorsaswellason-handinventory(SanaandPanda[ 71 ]). 4.2.2EOQModelswithDiscountsAnumberofstudiesexistintheliteraturethatdealwithEOQmodelsinwhichthepurchasecostvariesdependingonthequantitythatthebuyer/produceriswillingtopurchase.Thispricevariationtypicallycomesintheformofdiscountsasthepurchasequantityincreases.Monahan[ 54 ]devisedanoptimalquantitydiscountingschemeforasupplierinasingle-supplier,single-retailersystem,assumingthesupplierfollowsalot-for-lotpolicyinresponsetotheretailer'sorders.LeeandRosenblatt[ 47 ]generalizedMonahan's[ 54 ]modeltoallowthesuppliertodeviatefromalot-for-lotpolicy.Matsuyama[ 51 ]studiedcasesinwhichthepurchasepriceisanonincreasingstepfunctionoftheorderquantity.Arcelusetal.[ 3 ]determinedaprot-maximizingstrategyforaretailer/buyerwhenthevendorproposeseitheradiscountonthepurchasepriceoradelayinpayments.ViswanathanandWang[ 87 ]consideredtheeectivenessofquantitydiscountsandvolumediscountsinasingle-vendor,single-buyersystemwithprice-sensitivedemandfacedbythebuyer.Theycharacterizedanequilibriumpointfor 65

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determiningoptimalpricinganddiscountoptionsinaStackelberggamewherebothplayersaimtomaximizeprot.Qinetal.[ 60 ]consideredpricediscountsandfranchisefeesascoordinationmechanismsinasystemconsistingofavendorandabuyer.InamannersimilartotheapproachofViswanathanandWang[ 87 ],theyformulatedtheproblemasaStackelberggamewherethevendoristheleader(rstsettingitspricetothebuyer)andthebuyeristhefollower,whosubsequentlysetsthemarketpriceand,hence,determinesthemarketdemandforthecommodity.SanaandChaudhuri[ 70 ]consideredtheimpactofdelayedpaymentstothesupplier,togetherwithquantitydiscounts.Inthesecontexts,wholesalepricingisusedtofacilitatecoordinationbetweenasupplierandbuyer,ortoprovideincentivestothebuyertopurchasegoodsfromthesupplier.Intheproblemweconsider,theproducerattractssuppliersbyestablishingthepriceitiswillingtopayforeachunitofsupply.Insuchasetting,whenhigherpricesareoeredtosuppliers,thenumberofsuppliersandthenumberofunitsthatthesuppliersarewillingtoprovideincrease. 4.2.3EOQModelsforRemanufacturingPrice-dependentsupplyisaparticularlyrelevantphenomenoninreverselogisticssettings,wheretheinputsrequiredbyaremanufacturerareownedbyindividualconsumerswhomaybewillingtoselltheirproductsbacktotheremanufacturer,dependingonthepriceoered.TheexistingoperationsliteraturecontainsasignicantnumberofstudiesinwhichremanufacturingisconsideredasanoptionfordemandsatisfactionwithinanEOQframework.ExamplesincludeRichter[ 63 ],RichterandDobos[ 64 ],Teunter[ 78 ],DobosandRichter[ 15 ],andElSaadanyandJaber[ 16 ].Guideetal.[ 29 ]considerasettingwherereturnanddemandratesarepricedependent.Theysolveasingle-periodproblemandseektheprotmaximizingacquisitionandsellingpricesfortheremanufacturedproduct.Theirmodel,however,doesnotincludeoperationscostssuchasinventoryholdingandsetupcosts.Tothebestofourknowledge,noexistingstudyintheliteratureincorporatessupplyanddemandpricingdecisions,andtheimplicationsof 66

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thesedecisionsontheeconomicsofproduction,withagoalofcharacterizinganoptimalreplenishmentandpricingpolicy. 4.3ProductionPlanningandPricingModelWeconsideraproductionplanningprobleminwhichaninputcomponentisrequiredforproductionofanend-item,andforwhichallcost,demandandpricingparametersareassumedtobestationary.Thecostoftheinputcomponentdependsonthepricetheproduceroerstosuppliers.Letps0denotetheunitpurchasepricetheproduceroerstoitssuppliers,andsupposethatthesupplyrateofcomponentsforproductionisastationary,nondecreasing,nonnegative,anddierentiablefunctionofpsdenotedbyK(ps).ThecostperunittimeforprocuringcomponentsisthereforepsK(ps).Weassumethattheproducermustmeetend-itemdemand(withoutshortages),wherethedemandrateispricesensitiveanddeterministic,andequalsD(pc)unitsperunittimeatthepricelevelpc,whichcorrespondstothesellingpriceoeredtocustomers.Wealsoassumethatthetimerequiredtoconvertabatchofinputcomponentstoend-itemsisnegligible. Figure4-1. Supplyanddemandcurves. ConsiderFigure 4-1 ,whichshowsthebehaviorofthesupplyratefunction,K(p),andthedemandratefunction,D(p),asafunctionofprice.Thesupplyratefunctionisanondecreasingfunctionofthepriceoeredtosupplierswhere,foranypriceoeredthatis 67

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belowaspeciedpricelimit,p0s,thesupplyrateiszero,i.e.,nosupplierinthemarkethasalowerreservationpricethanp0s.Weletp0cdenotethesellingpriceatthecorrespondingdemandlevel.Thedemandratefunctionisanonincreasingfunctionofthesellingpriceoeredtocustomersforaunitoftheend-item.Asthesellingpriceapproachesinnity,thedemandrateapproacheszero.Theintersectionofthecurves,attheprice^p,providesalowerboundonthepricetocustomers,pc,andanupperboundonthepriceoeredtosuppliers,ps(ifpcisbelowthispriceorifpsisabovethispricethenwewillrequiresettingps>pc,whichimpliesanegativeprotlevel).Givenavalueofpc^p,weassumethatauniquepricepsexistssuchthatthesupplyrateequalsthedemandrate.Thus,somefunctionexistssuchthatpsisadecreasingfunctionofpc(or,equivalently,wecouldexpresspcasadecreasingfunctionofps).SupposethatK(ps)isthesupplyrate,expressedasafunctionofthesupplyprice,andD(pc)isthedemandrate,expressedasafunctionoftheend-itempriceoeredtocustomers.Weconsideraproductionprocessthatconvertsthesuppliedcomponentstoenditems,whereweassumewithoutlossofgeneralitythateachend-itemrequiresoneunitofthecomponent(alternatively,ifeachend-itemrequiresmultiplecomponentsperunitofend-item,wecanscalethedemandrateandassociatedcostsaccordingly,sothattheend-itemdemandrateisexpressedinunitsofthecomponent).Weassumethatoperatingthisprocessrequiresincurringsomexedcost,denotedbyF,whichleadstoproductionbatching.ComponentsarrivetotheproductionstageatarateofK(ps),andareheldininventoryuntilbeingconvertedintoend-itemsviathebatchproductionprocess.End-itemsaredepletedfromend-iteminventoryaccordingtothedemandrateD(pc).Becausetheproducermustmeetend-itemdemandswithoutshortages,thisimpliesthatwerequireK(ps)D(pc).Moreover,becausethesupplyrateisnondecreasingin 68

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price,thisimpliesthat,atoptimality,K(ps)=D(pc).1Thus,ifQdenotestheproductionbatchsize,theninsteadystate,whenevercomponentinventoryreachesQunits,thesecomponentsareconvertedtoend-itemsviaproductionofabatchofsizeQ.Letcdenotethecostofconvertingoneunitofsupplytodemand,andlethsdenotetheunitcostperunittimeforholdingsuppliedcomponents,whileheistheholdingcostperunitperunittimeforenditems.Ifbothsupplyanddemandoccurataxedrate,thenforgivenpricespcandpsandagivenbatchsizeQ,Figure 4-2 illustratesthebehaviorofcomponentandend-iteminventoryovertime. Figure4-2. Componentandend-iteminventorylevelswithtimewithabatchsizeofQ. Giventheinventorylevelsofcomponentsandend-itemsillustratedinFigure 4-2 ,theaverageinventorylevelforbothcomponentsandend-itemsequalsQ=2,leadingtoanaverageinventoryholdingcostperunittimeof(hs+he)Q=2.AsinthestandardEOQmodel,becauseaxedcostofFisincurredforeveryproductionbatch,andwehaveanaveragenumberofD(pc)=Qproductionbatchesperunittime,theaveragexedcostperunittimeequalsFD(pc)=Q.Asaresult,forQ0,p0sps^ppcp0c,theaverage 1Givenanysolutionwithpricesp0sandp0csuchthatK(p0s)>D(p0c),notethatasolutionexistswithpricesp00sandp0csuchthatK(p00s)=D(p0c)andp00s
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costperunittimecanbeexpressedas AC(Q;ps;pc)=FD(pc) Q+(hs+he)Q 2+psK(ps)+cD(pc): (4{1) Theaverageannualcostisthesumoftheaveragexedcost,componentandend-iteminventoryholdingcost,andpurchasingandproductioncostperunittime.Lettingh=hs+he,wenotethatforanypricepcwehaveanaverageannualcostminimizingsolutionforQ:Q(pc)=p 2FD(pc)=h2.Wecancharacterizetheaverageannualprotequationforp0sps^ppcp0cas (ps;pc)=(pc)]TJ /F3 11.955 Tf 11.95 0 Td[(ps)]TJ /F3 11.955 Tf 11.96 0 Td[(c)D(pc))]TJ /F9 11.955 Tf 11.96 10.81 Td[(p 2FhD(pc): (4{2) Asmentionedpreviously,thesupplypricepscanbecharacterizedasadecreasingfunctionofthesellingpricepc,e.g.,ps(pc).Thus,theaverageannualprotcanbewrittenasafunctionofpconlyfor^ppcp0c,i.e., (pc)=(pc)]TJ /F3 11.955 Tf 11.96 0 Td[(ps(pc))]TJ /F3 11.955 Tf 11.95 0 Td[(c)D(pc))]TJ /F9 11.955 Tf 11.95 10.81 Td[(p 2FhD(pc): (4{3) Inthefollowingsection,wecharacterizethesupplyandsellingpricesthatmaximizetheaverageannualprot,whereweassumespecicfunctionalformsforD(pc)andps(pc).Wealsoanalyzethebehavioroftheoptimalprices,prot,andbatchsizewithrespecttokeyproblemparameters. 4.4OptimalSupplyandSellingPricesInthissection,wecharacterizetheoptimalsolutionwhenthedemandratefunctiontakesaniso-elasticform.Thatis,weassumethatD(pc)=ap)]TJ /F5 7.97 Tf 6.59 0 Td[(bc,wherea;b>0andpc^p.Asnotedintheprevioussection,weassumethesupplypricepscanbecharacterizedasadecreasingfunctionofpc.Inparticular,weassumethatthisrelationshipmay 2NotethatAC(Q;ps;pc)isconvexinQ.Thus,solvingtherstorderconditionyieldstheoptimalsolutionforQ,whichisQ(pc)=p 2FD(pc)=h. 70

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becharacterizedbytheequationps(pc)=^p)]TJ /F1 11.955 Tf 13.2 0 Td[((k)]TJ /F1 11.955 Tf 13.2 0 Td[(1)(pc)]TJ /F1 11.955 Tf 14.18 0 Td[(^p),wherek>1andp0sps^ppcp0c.Theparameterk)]TJ /F1 11.955 Tf 12.12 0 Td[(1characterizestheresponseoftheequilibriumsupplyprice3tochangesintheend-customersellingprice.Forthisreason,wewillsometimesrefertok)]TJ /F1 11.955 Tf 12.47 0 Td[(1astheprice-to-priceresponse.Whilethisrelationshipbetweenpricesischaracterizedbyasimplelinearfunction,observethattheimpliedsupplyratefunctioninpricebecomesK(ps)=a(k)]TJ /F1 11.955 Tf 12.82 0 Td[(1)b(k^p)]TJ /F3 11.955 Tf 12.81 0 Td[(ps))]TJ /F5 7.97 Tf 6.58 0 Td[(b,whichpermitsconsiderableexibilityincharacterizingarangeofnonlinearsupplycurvesthatareincreasinginprice(e.g.,seeFigure 4-3 ). Figure4-3. Supplyanddemandcurvesforlinearsupply-price-selling-pricerelationship(k=1:9;^p=6). 3Theequilibriumsupplypricecorrespondstothesupplypricesuchthat,foragivensellingprice,thecorrespondingsupplyrateequalsthedemandrate. 71

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Usingtherelationshipbetweensupplyandsellingprices,wecanwritetheaverageannualprotasafunctionofpconly,for^ppcp0c: (pc)=(pc)]TJ /F1 11.955 Tf 11.95 0 Td[((^p)]TJ /F1 11.955 Tf 11.95 0 Td[((k)]TJ /F1 11.955 Tf 11.95 0 Td[(1)(pc)]TJ /F1 11.955 Tf 12.94 0 Td[(^p)))]TJ /F3 11.955 Tf 11.96 0 Td[(c)ap)]TJ /F5 7.97 Tf 6.58 0 Td[(bc)]TJ /F9 11.955 Tf 11.96 10.84 Td[(p 2Fhap)]TJ /F5 7.97 Tf 6.58 0 Td[(bc;=(k(pc)]TJ /F1 11.955 Tf 12.94 0 Td[(^p))]TJ /F3 11.955 Tf 11.95 0 Td[(c)ap)]TJ /F5 7.97 Tf 6.59 0 Td[(bc)]TJ /F9 11.955 Tf 11.95 10.83 Td[(p 2Fhap)]TJ /F5 7.97 Tf 6.59 0 Td[(bc: (4{4) Observethatmaximizing(pc)isthesameasmaximizing^(pc)whichisequalto(pc)=k.Thus, ^(pc)=(pc)]TJ /F3 11.955 Tf 11.96 0 Td[(c0)ap)]TJ /F5 7.97 Tf 6.59 0 Td[(bc)]TJ /F9 11.955 Tf 11.95 10.84 Td[(p 2F0hap)]TJ /F5 7.97 Tf 6.58 0 Td[(bc; (4{5) wherec0=c=k+^pandF0=F=k2.ThefunctionalformofthisaverageannualprotfunctionisequivalenttotheonepresentedinRayetal.[ 62 ],whoconsideredtheEOQmodelwithprice-dependentdemand.Thus,foraxedvalueofk,theresultsofRayetal.[ 62 ]permitcharacterizingtheprotfunctionandtheoptimalsellingprice.Thelimitingspecialcase4oftheprotequation( 4{4 )inwhichtheprice-to-priceresponse(k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)equalszero,andthevariableproduction/procurementcostequalsc+^p,ispreciselythemodelprovidedbyRayetal.[ 62 ].Ourinterestthusliesprimarilyinexploringandunderstandinghowgeneralizingthemodeltopermitk>1aectsoptimalpricingdecisionsandtheeconomicsofsupply,demand,andproductiondecisions.Inordertodothis,thefollowingsectionrstcharacterizesoptimalpricingandproductiondecisionsforourmodelforaxedvalueofkwhenk>1,largelydrawingontheresultsofRayetal.[ 62 ].Followingthis,inSection 4.4.2 ,weexplorethewayinwhichtheprice-to-priceresponseaectsthemodel'sresults.Section 4.4.3 thenconsidershowalackofconsiderationofthesupply-pricerelationshipcanimpactdecisionsandprotperformance. 4Strictlyspeaking,ourmodelassumesk>1,whichprovidesafunctionalrelationshipbetweenthesup-plypriceandthesellingprice.Werefertothecaseofk=1asthelimitingspecialcaseask!1fromabove.Atk=1,thefunctionalrelationshipbetweenthesupplyandsellingpricesbreaksdown,andsup-plypricebecomesindependentofsellingprice. 72

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4.4.1OptimalDecisionsunderaFixedPrice-to-PriceResponseAsnotedearlier,foraxedprice-to-priceresponse(k)]TJ /F1 11.955 Tf 12.2 0 Td[(1),ouraverageannualprotequation( 4{5 )ismathematicallyequivalenttothatconsideredbyRayetal.[ 62 ].Wewishtomaximizethisprotequationoverpcsuchthat^ppcp0c,where^pcorrespondstotheminimumpricesuchthatnonnegativeprotispossible,andp0ccorrespondstothemaximumpricelevelsuchthatthedemandlevelensuresthatsomepositivesupplyisavailableatthecorrespondingequilibriumsupplylevel.WecanthususetheresultsfromRayetal.[ 62 ]tocharacterizetheoptimalsellingprice,poptc,aswellashowthispriceandthecorrespondingoptimalbatchsizeandprotmarginareinuencedbyvariousproblemparameters.AsinRayetal.[ 62 ],wehavethatfor01.Proposition 4.1 showsthat,forb>1,astationarypointsolutionexistswhichistheunconstrainedmaximizerof^(pc),andthattheoptimalsellingpriceistheminimumbetweenthisstationarypointandp0c. Proposition4.1. Forb>1,theoptimalsellingpriceistheminimumofthesmallestpositivestationarypointsolutionfor^(pc)andp0c. Proof. SeeAppendix C Forb=2,weobtainthefollowingclosed-formstationarypointsolution,pc: pc=2a(c+k^p) ak)]TJ 11.95 10.19 Td[(p 2Fha:(4{6)NotethattheoptimalpriceRayetal.[ 62 ]obtainforthecaseofb=2equalstheaboveequationwithk=1.Theoptimalsellingprice,poptc,canbecharacterizedas: poptc=minpc;p0c;(4{7) 73

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wherep0c=(k^p)]TJ /F3 11.955 Tf 12.4 0 Td[(p0s)=(k)]TJ /F1 11.955 Tf 12.4 0 Td[(1).Thecorrespondingoptimalsupplyprice,popts,isgivenby^p)]TJ /F1 11.955 Tf 11.91 0 Td[((k)]TJ /F1 11.955 Tf 11.91 0 Td[(1)(poptc)]TJ /F1 11.955 Tf 12.89 0 Td[(^p).Forb=2,wecanobtainthefollowingclosed-formsolutionforpoptsaswell: popts=maxfp0s;psg;(4{8)where ps=k^p)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2a(k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)(c+k^p) ak)]TJ 11.95 10.19 Td[(p 2Fha:(4{9)Figures 4-4 and 4-5 illustrateexamplecurvesfortheaverageannualprotasafunctionofprice((pc))fordierentparametersets. Figure4-4. Protfunctionfordierentvaluesofb,where^p=4. Wenextinvestigatethebehavioroftheoptimalsellingprice,optimalsupplyprice,andoptimalbatchsizewithrespecttokeyproblemparameters,wherekisassumedtobexedandb>1. Proposition4.2. Forb>1andxedk,theoptimalsellingprice,poptc"asa#,F",c",h",^p",whenpoptc=pc.Whenpoptc=p0c,poptc"as^p"andp0s#. Proof. SeeAppendix D 74

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Figure4-5. ProtfunctionfordierentvaluesofF,where^p=4. Whenpoptc=pc,theoptimalsellingpricecorrespondstoastationarypointsolution,whichisaectedbytheproblemparametersinthesamewayasintheprice-dependentEOQmodelofRayetal.[ 62 ].Note,however,thatwhenpoptchitstheboundaryvalueofp0c,itsvalueisnolongeraectedbytheoperationscostordemandparametervalues,onlybytheequilibriumsupplyanddemandparameters.Figure 4-6 showssomeexamplesofpoptcasafunctionofFfordierentvaluesofa.NotethatasFincreases,poptcincreasesuntilitreachesitsmaximumpossiblevalue,i.e.,p0c. Proposition4.3. Forb>1andxedk,theoptimalsupplyprice,popts"asa",F#,c#,h#,whenpoptc=pc(otherwisepopts=p0swhichisconstant).As^p",poptsrstdecreasesandthenincreases. Proof. SeeAppendix E Asthepropositionshows,theeectsofoperationscostanddemandparametervaluesontheoptimalsupplypriceactintheoppositedirectionofthesellingprice,whichfollowsbecauseahighersellingpriceimpliesalowersupplypriceinequilibrium(wherethedemandrateequalsthesupplyrate). 75

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Figure4-6. pcvsFfordierentvaluesofa. Proposition4.4. Forb>1andxedk,theoptimalbatchsize,Qopt"asa",c#,h#,^p#whenpoptc=pc.Whenpoptc=pc,asF",Qoptrstincreasesandthendecreases.Whenpoptc=p0c,Qopt"asa",F",h#,^p#,andp0s". Proof. SeeAppendix F Whenpoptc=pc,thewayinwhichoperationsanddemandparametersaecttheoptimalbatchsizeisthesameasintheEOQmodelwithprice-dependentdemandofRayetal.[ 62 ].Observethatinthiscase,theoptimalbatchsizestartsoincreasinginthexedcostFandlaterdecreasesinF,asanincreasingFleadstoanincreasedsellingprice(andthusadecreaseddemandrate).RecallthatQopt=p 2FD(pc)=h;atlowervaluesofF,theimpactofanincreaseinFontheproductionbatchsizeoutweighsthecorrespondingdecreaseindemand,whileathighervaluesofF,thereductionindemandleveloutweighstheimpactofacorrespondingincreaseinFonthebatchsize.Whenpoptc=p0c,theimpactofoperationsanddemandparametersontheoptimalbatchsizeisthesameasinthestandardEOQmodelwithoutprice-dependentdemand,becausethepricep0cisindependentoftheoperationscostanddemandparameters. 76

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Netrevenueperunit.Wedene=pc)]TJ /F3 11.955 Tf 12.55 0 Td[(ps)]TJ /F3 11.955 Tf 12.55 0 Td[(casthenetrevenueearnedperunit.Whenbothsellingandsupplypricesaredecisionvariables,thebehaviorofthenetrevenueperunitisnotasstraightforwardasinthecasewherethesellingpriceistheonlydecisionvariable.Thereforeitisinterestingforourcasetoinvestigatethenetrevenueperunitatoptimalityasaperformancemeasureofthesystem.Atoptimality,thenetrevenueperunit,opt,equalspoptc)]TJ /F3 11.955 Tf 13.23 0 Td[(popts)]TJ /F3 11.955 Tf 13.23 0 Td[(c=kpoptc)]TJ /F3 11.955 Tf 13.23 0 Td[(k^p)]TJ /F3 11.955 Tf 13.24 0 Td[(c.Notethatopt=kpc)]TJ /F3 11.955 Tf 12.17 0 Td[(k^p)]TJ /F3 11.955 Tf 12.17 0 Td[(cwhenthestationarypointsolutionisfeasible(andthereforeoptimal).Otherwise,opt=k(^p)]TJ /F3 11.955 Tf 11.95 0 Td[(p0s)=(k)]TJ /F1 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(c.Forb=2,wecanexpressoptasfollows. opt=8><>:ak+p 2Fha ak)]TJ 6.59 6.8 Td[(p 2Fha(c+k^p);ifpc1andxedk,theoptimalnetrevenueperunit,opt"asa#,F",c",h",and^p",whenpoptc=pc.Whenp0cistheoptimalsellingprice,opt"as^p",c#,andp0s#. Proof. SeeAppendix G Whenpoptc=pc,asanyoftheoperationscostparametersincreases(orasthedemandparameteradecreases),theoptimalsellingpricepoptcincreasesaswell,whiletheoptimalsupplypricepoptsdecreases,leadingtoahigher-margin,lowervolumesolution.Incontrast,whenpoptc=p0c,reducingtheunitproductioncost,c,servestoonlyincreasethenetrevenueperunit,astheoperationscostanddemandparametersdonotaectp0c. 4.4.2EectsoftheSupply-PriceRelationshipThissectionexplorestheeectsofprice-sensitivesupplyonoptimalpricingdecisions,aswellasontheoptimalprotandtheoptimalnetrevenueperunit.Weareparticularlyinterestedinhowtheprice-to-priceresponse(viatheparameterk)aectsthebehavioroftheoptimalsellingprice,poptc,theoptimalprot,(poptc),andtheoptimalnetrevenue 77

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perunit,opt(seeProposition 4.6 )5.Proposition 4.6 indicatesthatpoptcisdecreasingink.Thus,astheprice-to-priceresponseincreases,weoeralowerpricetocustomersinordertocapturegreaterdemandvolume. Proposition4.6. Forb>1ask",poptc#andQopt".Whenpoptc=p0c,poptsisconstant(p0s),andopt#ask".Whenpcistheoptimalsellingpriceandb=2,popts#,andathresholdvalueofkexists,~k,suchthatoptisdecreasinginkfork<~kandincreasinginkfork>~k(thevalueof~kisprovidedintheappendix). Proof. SeeAppendix H Thefollowingpropositionshowsthat,despitethisreducedsellingprice,theaverageannualprotisincreasingintheprice-to-priceresponsewhentheoptimalpriceoccursatthestationarypointpc. Proposition4.7. Forb>1,theoptimalaverageannualprot,(poptc),isincreasingintheprice-to-priceresponseparameterk,whenpoptc=pc. Proof. SeeAppendix I Becausep0cisdecreasingink,itisnotpossibletoshowthat(p0c)ismonotonicallyincreasingordecreasinginkingeneral.However,whenb=2,wecanestablishconditionsontheproblemparametersthatguaranteethatwhenpoptc=p0c,athresholdvalueforkexists,k0,suchthat(p0c)isincreasinginkforkk0(theseconditionsareprovidedinAppendixG).WecangainsomeinsightontheimplicationsofProposition 4.7 byexaminingextremevaluesofkandwhattheseextremevaluesimplyintermsofthecharacteristicsofthesupplierbase.Considertheextremecasesfork,i.e.,thecaseswherek=1andk=1.Asdiscussedearlier,whenk=1,ps(pc)becomesequalto^p,whichisindependentofpc.That 5Notethatasweincreasek,weholdp0sand^pxedinordertoensureanonnegativesupplypricecorrespondingtoeveryfeasiblesellingprice. 78

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is,whenk=1,themodelreducestotheEOQmodelwhereonlythedemandrateisprice-dependent,whichwaswellcharacterizedbyRayetal.in[ 62 ].Althoughthesupplyrateisindependentofps,wecanenvisionaprice-supplycurvethatequalszeroforallps<^pandequals1forallps^p.Undersuchaprice-supplycurve,arationalproducerwillthussetps=^p,butwillpurchasefromsuppliersonlytheamountnecessarytomeetdemandattheoptimalsellingprice,whichisexactlywhatoccursinthemodelofRayetal.[ 62 ].RecallthatourmodelassumesthatK(ps)isanondecreasingfunctionofps.Theeconomicreasoningunderlyingthisrelationshipassumesthataheterogeneouscollectionofsuppliersexists,eachwithaminimumreservationpricerequiredforsellingtheirsupplytotheproducer.Inotherwords,wecanthinkofthiscurveasrepresenting,foranygivenps,thenumberofsuppliers,eachpossessinganindividualunit,withareservationpricelessthanorequaltops.Aspsincreases,moreandmoresuppliersexistwithareservationpricelessthanorequaltops.Thecurvewedescribedcorrespondingtothespecialcaseofk=1thenrepresentszerosupplierheterogeneity,orauniformlyhomogeneoussupplierbase,whereallsupplyhasareservationpriceof^p.Asweincreasetheprice-to-priceresponse(k)]TJ /F1 11.955 Tf 12.16 0 Td[(1)fromzero,thesupplybasebecomesincreasinglyheterogeneous,andwewouldliketounderstandhowthisheterogeneityinthesupplybaseinuencesdecisionmakingandprotability.Attheoppositeextreme,ask!1,observethatthesupplyrateK(ps)approachesthexedvalueofa^p)]TJ /F5 7.97 Tf 6.59 0 Td[(bforanypricevalueps,i.e.,theprice-supplycurvebecomesat.Thus,foranypossiblesupplypricepssuchthatp0sps^p,theresultingsupplyrateequalsa^p)]TJ /F5 7.97 Tf 6.58 0 Td[(b,whichisthedemandratewhenpc=^p.So,astheprice-to-priceresponseapproachesinnity,foranysupplypricetheproducerobtainsaxedsupply.Thus,^pbecomestheonlyrationalchoiceforthesellingprice,whiletheoptimalsupplypriceequalsp0s. 79

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Givenanypcthen,whichmustbegreaterthanorequalto^p,foranynitek>1,thesupply-pricecurvepermitsmatchingthesupplyleveltothedemandlevelusingapricelessthanorequalto^p.Thisispossiblebecauseoftheheterogeneityofsupplierreservationprices.Inotherwords,thereisincreasingvalueintheheterogeneityofthesupplierbase,whichisconsistentwithProposition 4.7 .Wecangainsomeadditionalinsightintotheinuenceoftheprice-to-priceresponseparameterkbyconsideringthecaseinwhichb=2.Whenb=2,thefunctionalformoftheaverageannualprotequationpermitsobtainingclosed-formexpressionsfortheoptimalsellingandsupplyprices.Wecanshowthatpoptcisstrictlydecreasinginkwhetheritequalsthestationarypointsolutionpcortheboundaryvaluep0c.However,therateofdecreaseinpoptcisdierentinthesetwocases,andavalueofkexists,denotedbyk,atwhichtheoptimalpoptcswitchesfrompctop0c.InFigure 4-7 ,whichillustratespoptcasafunctionofkforanexampleparameterset,thispointcanbeclearlyseenasabreakpointbetweentwosmoothfunctions.Thevalueofthisbreakpoint,k,correspondstothevalueofkatwhichpcequalsp0c.Hence,forb=2,wehave k=)]TJ /F3 11.955 Tf 9.3 0 Td[(+q 2+4a^p(2ac+p0sp 2Fha) 2a^p;(4{11)where=ap0s+^pp 2Fha+2a(c+^p).6IntheexampleshowninFigures 4-7 4-8 ,and 4-9 ,thevalueofkequals1.686.Theaverageannualprottakesthefollowingformforb=2: (poptc)=8><>:(ak)]TJ 6.58 6.79 Td[(p 2Fha)2 4a(c+k^p);ifpc
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theoptimalprottakestheformofthetopequationin( 4{12 ).Whenkk,theoptimalsellingpriceisp0c,andtheoptimalprottakestheformofthebottomequationin( 4{12 ).Askapproachesinnity,theoptimalsellingpriceapproaches^p.Hence,wehave limk!1(p0c)=a(^p)]TJ /F3 11.955 Tf 11.95 0 Td[(p0s)]TJ /F3 11.955 Tf 11.95 0 Td[(c) ^p2)]TJ 13.15 18.27 Td[(p 2Fha ^p:(4{13)FortheexampledepictedinFigure 4-8 ,theabovelimitingvalueequalsapproximately1103.75.Recallthatwedenedoptasequaltopoptc)]TJ /F3 11.955 Tf 13 0 Td[(popts)]TJ /F3 11.955 Tf 13 0 Td[(c(optisnot,therefore,themaximumvalueof=pc)]TJ /F3 11.955 Tf 12.88 0 Td[(ps)]TJ /F3 11.955 Tf 12.88 0 Td[(c,asthesolutionsthatgivethemaximumaverageannualprotandthemaximumnetrevenuevaluesarenotlikelytocoincide).AsnotedinProposition 4.6 ,wecanshowthatoptisdecreasinginkwhentheoptimalsellingpriceequalsp0c.InFigure 4-9 ,observethatoptisdecreasingforkk(totherightofthebreakpoint),wherepoptcequalsp0c.Inthisexample,optisincreasingwhenpoptc=pc,i.e.,when1
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Figure4-7. Optimalsellingpricewithrespecttok. Figure4-8. Optimalannualprotwithrespecttok. 82

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Figure4-9. Optimalnetrevenueperunitwithrespecttok. 4.4.3DecisionMakingwithoutAccountingforSupply-PriceRelationshipThissectionanalyzesthecaseinwhichthesupply-pricerelationshipisnotaccountedforbytheproducer(e.g,thisrelationshipiseitherunknownorignored).Becauseexistingmodelsdonotaccountforprice-dependentsupply,wewouldliketoexplorehowthislackofaformalizedmethodologicalapproachcanimpactprotperformance.Thus,weassumethattheproducerchoosesasupplypricewithoutknowledgeofhowitaectstheavailablesupplyquantity.Theproducerthendeterminestheend-customerprice,givenitschosensupplyprice,usinganexistingmodelthataccountsforprice-dependentdemand(Rayetal.[ 62 ]).Asaresult,wecanexpectamismatchbetweentheresultingsupplyanddemandrates.Thiswouldleadtounanticipatedcoststhatwerenotaccountedforindeterminingthesellingprice.Wewishtoquantifytheseadditionalcostsinordertocharacterizethevalueofinformationregardingthesupply-pricerelationship.Supposethesupplyprice,ps,ischosenheuristicallybytheproducerwithoutknowledgeofthesupply-pricerelationship.Thispre-determinedsupplypricepsimpliesasupplyrateofK(ps),thatisinitiallyunknowntotheproducer.Theproducersubsequentlydeterminesthesellingpricepcthatmaximizestheaverageannualprot 83

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givenin( 4{2 )forthegivensupplypriceps,i.e.,viatheEOQmodelwithprice-dependentdemandfromRayetal.[ 62 ],assumingavariablecostofps+c.Oneofthreepossiblescenariosresults:(1)K(ps)>D(pc),i.e.,theresultingsupplyrateexceedsthedemandrate,inwhichcasewecanexpectthattheproducerwouldcollectonlyanamountofsuppliedcomponentssucienttosatisfydemand(and,perhaps,turnawayasubsetofsupplierswithareservationpricelowerthanthechosensupplyprice);(2)K(ps)D(pc),presumablytheproducerwouldonlybuyasucientquantityofinputcomponentsfromsupplierstomeetitsend-customerdemand.Hence,theaverageamountpaidbytheproducertosupplierswouldequalpsD(pc).However,asupplypricep0sexiststhatprovidesasupplyrateK(p0s)suchthatK(p0s)=D(pc).Notethatp0spc.Atasellingpriceofp0c=D)]TJ /F7 7.97 Tf 6.58 0 Td[(1(K(ps)),however,theproducercouldachieveahigherprotmargin.Allelsebeingequal,theresultingaverageannualprotwouldincreaseby(p0c)]TJ /F3 11.955 Tf 12.84 0 Td[(pc)K(ps),whichprovidesalowerboundonthecostpenaltyfornotaccountingfortheprice-supplyrelationship. 84

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SupposeK(ps)=D(pc),andnotethatthiscanonlyoccurifD(pc)=ap)]TJ /F5 7.97 Tf 6.58 0 Td[(bc=K(ps)=a(k)]TJ /F1 11.955 Tf 11.26 0 Td[(1)b(k^p)]TJ /F3 11.955 Tf 11.25 0 Td[(ps))]TJ /F5 7.97 Tf 6.59 0 Td[(b.Thisonlyresultsifthevalueofpcthatmaximizes( 4{2 )atthechosenvalueofpshappenstoequal(k^p)]TJ /F3 11.955 Tf 12.62 0 Td[(ps)=(k)]TJ /F1 11.955 Tf 12.62 0 Td[(1),whichishighlyunlikelyintheabsenceofinformationaboutthesupply-pricerelationship(and,therefore,withoutknowingk).Whenb=2,forexample,theclosed-formexpressionforthevalueofpcthatmaximizes( 4{2 )foragivenpsisequalto pc=2a(c+ps) a)]TJ 11.96 10.19 Td[(p 2Fha:(4{16)Inthiscase,theonlyvalueofpsthatleadstoK(ps)=D(pc)istheuniquevalue ps:=k^p(a)]TJ 11.96 10.19 Td[(p 2Fha))]TJ /F1 11.955 Tf 11.95 0 Td[(2a(k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)c a(2k)]TJ /F1 11.955 Tf 11.96 0 Td[(1))]TJ 11.95 10.19 Td[(p 2Fha:(4{17)AlthoughK(ps)=D(pc),thepair(ps;pc)doesnotmaximizethetrueaverageannualprotEquation( 4{4 ),aspcdoesnotequaltheoptimalvaluefromEquation( 4{6 ).Thus,usingtheEOQmodelwithprice-sensitivedemandonly,itisimpossiblefortheproducertochooseapairofprices(ps;pc)suchthatthesupplyrateequalsthedemandrateandthetrueaverageannualprotEquation( 4{4 )ismaximized.Thefollowingexampleillustratesthecostpenaltyincurredwhenthesupply-pricerelationshipisnotproperlytakenintoaccount.Inthisexample,weassumethatD(pc)=ap)]TJ /F5 7.97 Tf 6.59 0 Td[(bc,withb=2,a=10000,F=5000,h=0:0077,c=0:5andp0s=1.Theprice-to-priceresponse(k)]TJ /F1 11.955 Tf 12 0 Td[(1)andlowerboundonsellingprice(^p)equal0:6and6,respectively,andareunknowntotheproducer.Hereagainweassumethattherelationshipbetweenthesupplyandsellingpricesisgivenbyps(pc)=^p)]TJ /F1 11.955 Tf 12.33 0 Td[((k)]TJ /F1 11.955 Tf 12.33 0 Td[(1)(pc)]TJ /F1 11.955 Tf 13.32 0 Td[(^p),whichresultsinasupplyratefunctionequaltoK(ps)=a(k)]TJ /F1 11.955 Tf 11.95 0 Td[(1)b(k^p)]TJ /F3 11.955 Tf 11.96 0 Td[(ps))]TJ /F5 7.97 Tf 6.59 0 Td[(b.Supposetheproducerrstdeterminesasupplypriceps(whichmustbegreaterthanorequaltop0s),andthendeterminesthesellingprice,pcusing( 4{16 ).ThebluecurveinFigure 4-10 showspcasafunctionofps.(Notethatpcisanondecreasinglinearfunctionofps.)Theanticipatedaverageannualprot,whichwedenoteas(pc),computedusingEquation( 4{2 ),isshownbythebluecurveinFigure 4-11 .Thisprotlevel,however,is 85

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notalwaysachievable,becausethesupplylevelimpliedbytheinitialchoiceofpsmaybetoolowtosatisfythedemandrateassociatedwiththesellingpricepc.Figure 4-12 showsthesupplycurveasafunctionofpsandtheresultingdemandratewhenpsischosenasthesupplyprice.Thisgureshowsthatuptoaparticularsupplypricevalue(ps)thedemandrateishigherthanthesupplyrate,anditislowerforanypsvaluethatexceedsps.WhenpsK(ps),andthebesttheproducercandoistosatisfyademandrateequaltothesupplyrateK(ps).Therefore,theactualprotwillbelowerthantheinitiallyanticipatedprotlevel.TheredcurveinFigure 4-11 showstheactualprotlevel.Wecanclearlyseethattheactualprotislowerthantheanticipatedlevelwhenps
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Figure4-11. Comparisonoftheoptimalchoiceofpcwithrespecttopredeterminedsupplypricepsviatheeectsonprot. Figure4-12. Demandandsupplycurveswithrespecttops. 87

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Whenpsps,thesupplypriceresultsinasupplyratethatexceedsthedemandrate.TheproducercouldreducethesupplypricetoalevelwhichwouldensureD(pc)=K(ps).TheadjustedsupplypricesareshowninFigure 4-10 bytheyellowcurve.Thesupplypricewouldthusdecreaseinordertomatchthedemandratewhenps>ps.ThecorrespondingprotlevelisshowninFigure 4-11 bytheyellowcurveforps>ps.ThedierencebetweentheyellowandredcurvesinFigure 4-11 correspondstothepotentialincreaseinprotifeitherthesupplypriceorsellingpriceissubsequentlyadjustedtoensurethatsupplyequalsdemand,asafunctionoftheinitiallyselectedsupplypriceps.Thedierenceinthevalueoftheyellowcurveattheoptimalpoint(denotedbytheasterisk,)andthevalueoftheyellowcurveatanyotherpsgivesthereductioninprotperformanceassociatedwithusingasuboptimalsupplyprice,whileadjustingeitherthesellingpriceorthesupplypricetoensurethatsupplyequalsdemand.Thus,thepenaltyforusingasuboptimalsupplypriceisgenerallyquitesmallrelativetothepenaltyfornotensuringequalsupplyanddemandrates.Whenthesupplypriceischosensuchthatps=ps,theresultingdemandandsupplyrateshappentoequaloneanother;hence,nosupplyimbalanceexistsinthiscase(thiscorrespondstothepointatwhichthethreecurvescoincideinFigure 4-11 ).Eventhoughthesupplyanddemandratesturnouttomatchnicely,andthereisnotapenaltyatthispointfornotensuringequalsupplyanddemandrates,thispointdoesnotcorrespondtotheoptimaldecisionthattheproducerwouldmakewhenproperlyaccountingforthesupply-pricerelationship.Moreover,thissolutioncorrespondstothemaximumpenaltyforusingasuboptimalsupplyprice.Theoptimalsellingandsupplypricesaredenoted 88

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inFigure 4-10 by;ps=1:5855andpc=13:3576.Theoptimalprotforthissystemis(ps;pc)=566:0646(seeFigure 4-11 denotedby).Itisinterestingtoconsiderwhatwouldhappeninthisexampleproblemiftheproducerhadchosentheoptimalsupplyprice(ps=1:5855)andthensolvedtheEOQmodelwithprice-dependentdemandonlyofRayetal.[ 62 ].Thismodelprescribesasellingpriceequalto2a(c+ps)=(a)]TJ 12.71 10.18 Td[(p 2Fha)=4:5722andacorrespondingdemandrateof478.3542unitsperunittime,withanaverageannualprotvalueof997.6033.However,thisvalueofsupplypriceimpliesasupplyrateofonly56.0466unitsperunittime,whichisinsucienttomeetthedemandrate.Asaresult,usingthispairofprices(ps=1:5855;pc=4:5722)whilemeetingademandrateof56.0466unitsperunittimewillleadtoanannualprotofonly73.678.Iftheproducerweretoadjustitssellingpricetotheoptimalvalueprescribedbyourmodel(poptc=pc=13:3576),itcouldthenextracttheoptimalprotlevelof566:0646.Thisexampleillustrateshowaccountingfortheimpactsofprice-dependentsupplycandrasticallyaectprotability. 89

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CHAPTER5ANEWSVENDORPROBLEMWITHPRICE-DEPENDENTSUPPLYANDDEMAND 5.1MotivationandRelatedLiteratureThenewsvendorproblemconsidersplanningforitemsforwhichthedemandarisesonlyinonesellingseason.Theproducerfacesarandomdemandwithaknowndistribution,andaimstodeterminetheorderquantitysuchthattheexpectedprotismaximizedforthecorrespondingplanningperiod.Inthestandardversionoftheproblem,allunitcostandrevenueparametersareknownwithcertainty,andtheoptimalorderingpolicyisdeterminedbythebalancebetweenunitcostofunderstockingandoverstocking.Inthischapter,weconsidertheplanningproblemofaproducerforasingleperiodwheretheend-itemdemandisrandomandthesupplyquantitydependsontheunitpriceoeredbytheproducertosuppliers.Werststudythecasewhereonlysupplyisprice-dependent.Inparticular,wecharacterizetheoptimalsupply-pricingdecisionsforcasesinwhichthesupplyquantityisdenedbyalinearoranisoelasticfunctionofthesupplyprice.Wetheninvestigatethedierencesintheoptimalpoliciesbetweenthestandardnewsvendorproblemandourmodel.Wealsopresentsomeresultsforthecasewherethesupplyisprice-dependentandrandom.LaterinSection 5.3 westudythecasewherethedemandisalsoprice-dependent.Ourworkinthischapterisrelatedtoseveralresearchstreamsconsistingofvariousextensionsofthenewsvendorproblem.Themostrelevantworktoourstudyincludesnewswendorproblemswithprice-dependentdemandandproblemswithsupplydiscounts.Asapracticalapplicationareaweconsidertheremanufacturingsetting,andwereviewtheexistingworksinthisareawhicharemostrelatedtoourstudyinthischapter. 5.1.1NewsvendorProblemwithPrice-DependentDemandTheeectofpricingdecisionsondemandvalueshasreceivedasignicantinterestwithintheproductionplanningandinventorymanagementresearch.Theconceptofprice-dependentdemandwasrstintroducedwithinthenewsvendorsettingbyWhitin 90

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in[ 89 ].Innewsvendorproblemswithprice-dependentdemandthesellingpriceisalsoadecisionvariableinadditiontotheorderquantity.SomeexamplesincludeMills[ 53 ],KarlinandCarr[ 40 ],Zabel[ 91 ],LauandLau[ 46 ],andPetruzziandDada[ 59 ].PetruzziandDada[ 59 ]recapitulatetheexistingresultswhileintroducinginsightfulformulationsoftheproblemforvariousdemandmodels.Incontrasttothesestudies,Sana[ 69 ]considersthenewsvendorproblemwithprice-dependentdemandwherethesellingpriceisarandomvariable.Inalloftheaforementionedstudies,theunitcostofthesupplyisexogenouslydetermined,andtheproducercanacquireaninniteamountofsupplybypayingthesameamountforeachunitofsupply.Inourstudy,theunitcostofsupplyisgivenbytheunitpriceoeredbytheproducertosuppliers,anditisadecisionvariablefortheproducer. 5.1.2NewsvendorProblemwithSupplierDiscountsTheliteratureonnewsvendorproblemscontainsasignicantnumberofresearcharticlesaddressingtherelationshipsbetweentheorderquantityanddiscountoersinitiatedbythesupplier.Therearetwotypesofdiscountschemesstudiedwidelyintheliterature;all-unitdiscountsandincrementaldiscounts.Inthecaseofall-unitdiscounts,theproducerpaysahigherunitpricetothesuppliersunlesstheorderquantityexceedsapredeterminedthreshold.Iftheproducerordersaquantitygreaterthanthethreshold,theunitcostdrops.Somestudiesconsideringanall-unitdiscountschemeincludeJuckerandRosenblatt[ 38 ],Burnetas,Gilbert,andSmith[ 10 ],Haji,Haji,andDarabi[ 31 ],Altintas,Erhun,andTayur[ 2 ],andZhang[ 94 ].Incontrast,whentheproducerpaysalowerunitpriceonlyforthoseunitsorderedinexcessofthethresholdquantity,thisisreferredtoasanincrementaldiscountscheme.ExamplesfromtheliteratureinanewsvendorsettingincludeLinandKroll[ 49 ],Guder,Zydiak,andChaudhry[ 28 ],andJiandShoa[ 37 ].LinandKroll[ 49 ]studybothtypesofquantitydiscountsinaneorttodeterminetheorderingpolicythatmaximizesexpectedprotandalsoguaranteesthattheprobabilityofreachingacertainprotlevelexceedsatargetlevel. 91

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Inallofthesestudiesthesupplierprovidesanincentivetotheproducertobuymorebyreducingthepriceastheorderquantityincreases.Inourmodel,theproducerwishestoattractsuppliersbyoeringaunitpurchaseprice.Clearlyahigherpriceresultsinahighersuppliedquantity.Therelationshipbetweenthesupplyquantityandsupplycostweconsiderrunscountertothatconsideredintheexistingworksinthisparticularresearchstream. 5.1.3RemanufacturingProblemsinNewsvendorSettingTheliteratureonproductionplanningwitharemanufacturingoptioncontainsasignicantnumberofstudieswherethisplanningproblemisconsideredwithinanewsvendorsetting.SomeexamplesincludeRobotis,Bhattacharya,andVanWassenhove[ 67 ],Xanthopoulos,Vlachos,andIakovou[ 90 ],andLi,Li,andCai[ 48 ].Inallofthesestudiestheunitcostofinputsforremanufacturingprocessesisknowninadvance.Aswepointedoutintheearlierchapters,theconceptofprice-dependentsupplyisparticularlyrelevantinreverselogisticssettings.Insuchsettings,theinputsrequiredbyaremanufacturerareownedbyindividualconsumerswhomaybewillingtoselltheirproductstotheremanufacturer,dependingonthepriceoered.Somestudiesintheliteraturecharacterizetherelationshipbetweenthequantityoftheused-itemsavailableforremanufacturingandtheacquisitionpriceoeredbytheremanufacturertoused-itemsuppliers,andanalyzetheeectsonremanufacturingdecisions(seeforexampleGuideandvanWassenhove[ 30 ],KlausnerandHendrickson[ 43 ],andRay,Boyaci,andAras[ 61 ]).ThestudyofBakalandAkcali[ 6 ]iscloselyrelatedtoours,wheretheyconsiderprice-dependentacquisitionofusedvehiclesasasourceofpartremanufacturing.Theirmodelincludesarandomyieldfortheremanufacturablepartsandaprice-dependentdemand.IncontrasttothestudyofBakalandAkcali[ 6 ],werstconsidercaseswhereend-itemdemandisrandombutnotprice-dependent,andlaterwestudyoptimalsupply-pricingdecisionswheretheend-itemdemandisbothprice-dependentandrandom. 92

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5.2Price-DependentSupplyWeconsiderasingleperiodplanningproblemforaproducerwhofacesrandomdemandforasingleitem.Wedenotetheend-itemdemandasarandomvariableDwithaknowndistribution.Theprobabilitydensityfunctionandthecumulativedistributionfunctionofthedemanddistributionaredenotedbyf()andF(),respectively,whereasDandDare,respectively,themeanandthestandarddeviationofthisdistribution.Theproducerobtainscomponentsfromthesuppliersandconvertsthemtoend-itemsviaaproductionprocess.Thenumberofcomponentsobtainedfromsuppliers,equivalentlytheorderquantity,dependsonthepricethattheproduceroerstoitssuppliers.Letcdenotethepriceoeredtosuppliersbytheproducer.Weassumethattheorderquantityisanonnegative,nondecreasingandconvexfunctionofc,whichwedenotebyQ(c).Thesuppliedcomponentsmustbeprocessedataunitcostvtoconvertcomponentsintosellableend-items.Eachend-itemcanbesoldforaunitpricedenotedbyp.Attheendoftheplanningperiodtheend-itemsininventorycanbesalvagedforanamountofsperunit.Whentheend-itemisout-of-stock,i.e.,theproducerisunabletosatisfyacustomerdemand,apenaltycost,g,isincurredperunitunsatiseddemandtoaccountforthelossofgoodwill.Theproducerseekstodeterminethesupplypricethatmaximizesitsexpectedprot.Let(c;D)denotethetotalprotoftheproducerwhenthesupplypriceequalscandthedemandisequaltoD.Wecanwrite(c;D)as (c;D)=8><>:pQ(c))]TJ /F3 11.955 Tf 11.96 0 Td[(g[D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c)])]TJ /F1 11.955 Tf 11.96 0 Td[((c+v)Q(c);ifDQ(c);pD+s[Q(c))]TJ /F3 11.955 Tf 11.96 0 Td[(D])]TJ /F1 11.955 Tf 11.95 0 Td[((c+v)Q(c);ifD
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Let(c)denotetheexpectedprot,i.e.,(c)=E[(c;D)].Wecanwritethisexpectedprotas (c)=ZQ(c)0(px+s[Q(c))]TJ /F3 11.955 Tf 11.96 0 Td[(x])f(x)dx+Z1Q(c)(pQ(c))]TJ /F3 11.955 Tf 11.96 0 Td[(g[x)]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c)])f(x)dx)]TJ /F1 11.955 Tf 9.3 0 Td[((c+v)Q(c): (5{2) Theproducer'sgoalistodetermineaunitpurchaseprice,c,suchthatitsexpectedprotismaximized.Todeterminetheoptimalsupplyprice,c,weneedtoconsidercvaluesthatsatisfytherstandsecondorderoptimalityconditions,i.e.,werequirethat0(c)=0and00(c)<0.Therstandsecondderivativesof(c)areasfollows. 0(c)=(p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)Q0(c)[1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(Q(c))])]TJ /F1 11.955 Tf 11.96 0 Td[((c)]TJ /F3 11.955 Tf 11.96 0 Td[(s)Q0(c))]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c): (5{3) 00(c)=(p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)fQ00(c)[1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(Q(c))])]TJ /F3 11.955 Tf 11.95 0 Td[(f(Q(c))(Q0(c))2g)]TJ /F1 11.955 Tf 9.3 0 Td[((c+v)]TJ /F3 11.955 Tf 11.96 0 Td[(s)Q00(c))]TJ /F1 11.955 Tf 11.96 0 Td[(2Q0(c): (5{4) Solvingtherstorderconditionweobtaintheequality( 5{5 ).Anycsatisfying( 5{5 )isastationarypointoftheexpectedprotfunction(c).Notethat(c)isstrictlyconcavewhen00(c)<0forallc.Notealsothatthisdependsonboththecharacteristicsofthedemanddistributionandtheformofthesupply-pricefunctionQ(c).When(c)isconcave,apositivecvaluesatisfying( 5{5 )isamaximizerof(c). F(Q(c))=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F5 7.97 Tf 14.49 5.7 Td[(Q(c) Q0(c) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s: (5{5) HereF(Q(c))givestheprobabilitythatthecustomerdemandwillbefullysatisedwhentheorderquantityequalsQ(c)or,equivalently,whenthepriceoeredtothesuppliersbytheproducerequalsc.Thisprobabilityofsatisfyingthedemandisalsoknownastheserviceleveloftheproducer.Thevalueofcthatmakestheservicelevelequaltotheright-handsideof( 5{5 ),whichcanbereferredasthecriticalfractile,isastationarypointof(c).Notethatifthisproblemwerethestandardnewsvendorproblem,wheretheunitpurchasingprice,c,isxedandaninniteamountofsupplyisavailableatthatprice,the 94

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criticalfractilewouldbeequalto(p+g)]TJ /F3 11.955 Tf 12.13 0 Td[(c)]TJ /F3 11.955 Tf 12.13 0 Td[(v)=(p+g)]TJ /F3 11.955 Tf 12.13 0 Td[(s),wheretheunitoverstockingcost,co,isc+v)]TJ /F3 11.955 Tf 12.02 0 Td[(sandtheunderstockingcost,cu,isp+g)]TJ /F3 11.955 Tf 12.02 0 Td[(c)]TJ /F3 11.955 Tf 12.02 0 Td[(v.Hencethesolutiontothestandardnewsvendorproblemisgivenby F(Q)=cu cu+co:(5{6)Notethatinthestandardnewsvendorproblem,themarginalcostofthenextunittobesuppliedisequaltothexedunitsupplycost,c.AsshowninProposition 5.1 ,thismarginalcostisequaltoc+Q(c)=Q0(c)whenthesupplyquantitydependsontheunitpriceoeredbytheproducer.Hence,inourmodel,themarginalunderstockingcost,cu,andthemarginaloverstockingcost,co,arep+g)]TJ /F3 11.955 Tf 9.33 0 Td[(v)]TJ /F3 11.955 Tf 9.33 0 Td[(c)]TJ /F3 11.955 Tf 9.34 0 Td[(Q(c)=Q0(c)andc+Q(c)=Q0(c)+v)]TJ /F3 11.955 Tf 9.33 0 Td[(s,respectively,whenthepriceoeredtosuppliersequalsc.Consequently,thisshowsthattheinterpretationofthecriticalfractileremainsthesameasinthestandardnewsvendorproblemwheretheorderquantityischosenbasedonthebalancebetweenoverstockingandunderstockingcosts.Thus,( 5{6 )stillholdswheretheoptimalorderquantityQisgivenbyQ(c)inourmodel,andwherecdenotestheoptimalsupplyprice. Proposition5.1. Giventhatthecurrentchoiceofsupplypriceisc,themarginalcostofsupplyingthenextunitequalsc+Q(c)=Q0(c). Proof. Letc(q)denotetheunitsupplypricethattheproducerneedstooertosupplierstoreceiveanamountofsupplyofq.Thatisc(q)denotestheinverseofthefunctionQ(c)givenasupplyquantitylevelq,i.e.,c(q)=Q)]TJ /F7 7.97 Tf 6.58 0 Td[(1(q).Foratotalsupplyquantityofqtheproducerspendsc(q)q.Themarginalcostofthenextunitisgivenbyd(c(q)q) dq=c(q)+qc0(q):Bytheinversefunctiontheorem,wecanwritec0(q)=1 Q0(c(q)): 95

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Selectingqtobethequantityreceivedbysettingthesupplypriceatc,weobtaind(c(q)q) dqq=Q(c)=c+Q(c)=Q0(c): Notethattheexpectedprotfunction(c)isconcaveincwhen00(c)0.Thisconcavityconditionof(c)issatisedwhen Q00(c) Q0(c)2)]TJ /F9 11.955 Tf 11.96 16.86 Td[(1 Q0(c)2F(Q(c))(c+v)]TJ /F3 11.955 Tf 11.95 0 Td[(s)Q00(c)+2Q0(c) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(sf(Q(c)) F(Q(c));(5{7)whereF()=1)]TJ /F3 11.955 Tf 12.16 0 Td[(F().Notethattheright-handsideoftheinequality( 5{7 )isthefailurerateofthedemanddistributionandtherefore,alwayspositive.Nextweconsidertwospecialcasesforthesupply-pricefunction.Firstweassumethattherelationshipbetweenthesupplyquantityandpriceislinear(seeSection 5.2.1 ).TheninSection 5.2.2 weassumewehaveanisoelasticsupply-pricefunction.LaterinSection 5.2.4 weconsiderthecasewherethesupplyquantityisbothprice-dependentandrandom. 5.2.1LinearSupply-PriceFunctionCaseInthissectionweanalyzeaspecialcasewheretheprice-supplyfunctionislinear.HereweassumethatQ(c)=c)]TJ /F3 11.955 Tf 12.49 0 Td[(,where>0and0.Letc0bethemaximumpricewhereQ(c0)=0.Werefertoc0asthethresholdprice,wheretheproducercannotpurchaseanysupplyifthepriceoeredtosuppliersisbelowthisthresholdpricec0.Clearly,c0==forthisparticularsupply-pricerelationshipcase.Notethatforalinearsupply-pricefunction,Q00(c)=0andtherighthandsideoftheinequality( 5{7 )isnegative,implyingthattheexpectedprotfunctionisstrictlyconcaveforallcforthecaseofalinearsupply-pricerelationship.Thus,csatisfying( 5{5 )istheuniquemaximizerof(c).Notethatthisresultisindependentofthedemanddistribution.Thus,wehave F(Q(c))=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F5 7.97 Tf 13.15 5.25 Td[(c)]TJ /F5 7.97 Tf 6.59 0 Td[( p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s;=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F1 11.955 Tf 11.96 0 Td[((c)]TJ /F3 11.955 Tf 11.95 0 Td[(c0) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s: (5{8) 96

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Figure 5-1 illustratesanexampleoftheexpectedprotfunction(c).Heretheproblemparametersarechosenasfollows:p=10,g=5,s=3,andv=1.ThedemandisassumedtofollowanormaldistributionwithD=2000andD=100.TheparametersandofQ(c)are500and1000,respectively.Herethethresholdprice,c0equals2,andtheoptimalsupplypricecis5:921. Figure5-1. Expectedprotfunction(c)withrespecttoc,whereQ(c)islinearwith=500and=1000. 5.2.2IsoelasticSupply-PriceFunctionCaseInthissection,weassumethatQ(c)takesthefollowingisoelasticform:Q(c)=cwhere>0and>1.InthiscasewehaveQ0(c)=c)]TJ /F7 7.97 Tf 6.58 0 Td[(1andQ00(c)=()]TJ /F1 11.955 Tf 12.11 0 Td[(1)c)]TJ /F7 7.97 Tf 6.58 0 Td[(2.Forthissupply-pricerelationship,thestationarypointcondition( 5{5 )becomesthefollowing. F(Q(c))=p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(v)]TJ /F5 7.97 Tf 13.15 5.25 Td[(+1 c p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s:(5{9)Nextweexaminetheconcavityconditionoftheexpectedprot(c),i.e.00(c)0.Foranalyticalpurposes,wemultiplybothsidesoftheinequality( 5{7 )withQ(c).Ontheleft-handsideoftheinequalityweobtainthegeneralizedfailurerateofthedemanddistributionasafunctionofc.Hence,inequality( 5{7 )canbewrittenasfollows. Q(c)Q00(c) Q0(c)2)]TJ /F9 11.955 Tf 11.96 16.86 Td[(Q(c) Q0(c)2F(Q(c))(c+v)]TJ /F3 11.955 Tf 11.95 0 Td[(s)Q00(c)+2Q0(c) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(sG(Q(c));(5{10) 97

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where G(Q(c))=Q(c)f(Q(c)) F(Q(c)):(5{11)NotethatG(Q(c))isthegeneralizedfailurerateofthedemanddistribution.Forthisparticularsupply-pricefunctioncase,wehaveQ(c)Q00(c) Q0(c)2=()]TJ /F1 11.955 Tf 12.44 0 Td[(1)=.Therefore( 5{10 )reducesto )]TJ /F1 11.955 Tf 11.96 0 Td[(1 )]TJ /F9 11.955 Tf 11.96 16.86 Td[(1 F(Q(c))+1 c+)]TJ /F7 7.97 Tf 6.58 0 Td[(1 (v)]TJ /F3 11.955 Tf 11.95 0 Td[(s) p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(sG(Q(c)):(5{12)Proposition 5.2 showsthatwhenthedemanddistributionhasanincreasinggeneralizedfailurerate,thereexistsavalueofc,denotedbyc ,suchthat(c)isconcaveforanycc .Furthermore,thestationarypointsatisfying( 5{9 )istheonlystationarypointanditisgreaterthanc ,whichimpliesthatitisthemaximumof(c). Proposition5.2. Giventhatthedemanddistributionhasanincreasinggeneralizedfailurerate(IGFR),thenthereexistsa c ,suchthatforanyc c ,(c)isconcave.Furthermore,thestationarypointsatisfying( 5{9 )istheuniquemaximizerof(c). Proof. Byourinitialassumption,Q(c)isincreasingincandequalszerowhenc=0.SincethedemanddistributionhasanIGFR,1theright-handsideofinequality( 5{12 )increaseswithcandisalwayspositive.Wecanrewritetheleft-handsideof( 5{12 )asfollows: )]TJ /F1 11.955 Tf 11.95 0 Td[(1 1)]TJ /F3 11.955 Tf 48.69 8.09 Td[(v)]TJ /F3 11.955 Tf 11.96 0 Td[(s F(Q(c))(p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s))]TJ /F5 7.97 Tf 51.28 15.59 Td[(+1 c F(Q(c))(p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s):(5{13)Theterm()]TJ /F1 11.955 Tf 13.02 0 Td[(1)=ontheleft-handsideisapositiveconstantlessthan1anditismultipliedwithatermthatdecreasesascincreases.Thisisbecauselimc!1F(Q(c))=0: 1TheclassofdistributionswithIGFRcoversawiderangeofdistributionsincludingsomeofthedistri-butionswhichdonothaveanincreasingfailurerate[ 45 ].Examplesincludesomewidelyuseddistributionssuchasthenormal,uniform,gamma,andWeibulldistributions. 98

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Thesecondtermalsoincreaseswithc,whichimpliesthatthetermin( 5{13 )decreasesascincreases.Thatimpliesthatascincreases,theleft-handsideof( 5{12 )decreasesandtheright-handsideof( 5{12 )increases.Ifatc=0theinequality( 5{12 )issatised,thenitissatisedforallpositivecvaluesand(c)isconcaveforallc0.Ifatc=0theinequality( 5{12 )isnotsatised,thereexistsapositivecvalue,c ,where( 5{12 )issatised,i.e.,theprotfunctionisconcaveintheinterval[c ;1).Wehavelimc!0(c)=)]TJ /F3 11.955 Tf 9.3 0 Td[(g;andlimc!00(c)=0:Wecanalsoshowthat,ascapproachestoinnity,(c)and0(c)gotonegativeinnity.Andsincethereisonlyoneinectionpointc ,(c)isunimodal.Wenextshowthatthestationarypointisthemaximizerof(c)usingtheidentity( 5{9 ).From( 5{9 ),weobtainF(Q(c))=+1 c+v)]TJ /F3 11.955 Tf 11.96 0 Td[(s p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s:AftersubstitutingF(Q(c))in( 5{12 )weobtain )]TJ /F1 11.955 Tf 11.2 8.09 Td[(1 +1 c0,sincetheleft-handsideoftheinequalityisalwaysnegative,whereastheright-handsideoftheinequalityispositive.Thusthestationarypointsatisesthesecondorderoptimalityconditionandthereforeitisthemaximizerof(c). Figure 5-2 illustratesanexampleoftheexpectedprotfunction(c)whenQ(c)=50c2.Heretheproblemparametersarechosenasfollows:p=10,g=5,s=3,andv=1.ThedemandisassumedtofollowanormaldistributionwithD=2000andD=100.Forthiscase,theoptimalsupplypricecisequalto6.277. 5.2.3ComparisonwiththeStandardNewsvendorProblemNextwecompareourmodel'soutcomeswiththestandardnewsvendorproblem.Inthestandardversionoftheproblem,thenewsvendorfacesrandomdemandfollowingsome 99

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Figure5-2. Expectedprotfunction(c)withrespecttoc,whereQ(c)isisoelasticwith=2and=50. distributionF()inasingleperiod.Giventhecostparameters(unitcostc,salvagevalues,lossofgoodwillcostg,andunitproductioncostv)andtheunitsellingpricep,theoptimalorderquantityQequatestheserviceleveltothewell-knowncriticalratiogivenbythebalancebetweenaverageoverstockingandunderstockingcosts.Morespecically,Qisthequantitythatsatisesthefollowingcondition. F(Q)=p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(v)]TJ /F3 11.955 Tf 11.95 0 Td[(c p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s:(5{15)Notethatleavingallotherparametersconstant,theoptimalorderquantityQcanbeexpressedasafunctionofthesupplypricec.Hence,withaslightabuseofnotation,wecanwrite Q(c)=F)]TJ /F7 7.97 Tf 6.58 0 Td[(1p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(v)]TJ /F3 11.955 Tf 11.95 0 Td[(c p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s:(5{16)Similarly,theoptimalexpectedprotforthiscase,(Q),canbereformulatedasafunctionofthesupplypricec.Wedenotethisoptimalexpectedprotfunctionass(c).Nextweillustrateacomparisonbetweenthestandardnewsvendorproblemsolutionandourmodel'ssolutionusingsomeexamples,wheretheproblemparametersarechosenasfollows:p=10,g=5,s=3,andv=1.Thedemandisassumedtofollowanormal 100

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distributionwithD=2000andD=100.Table 5-1 showsourselectionofvarioussupply-pricefunctionsusedinthiscomparison.Hererstvescenariosincludelinearsupply-pricefunctionsandthelastveincludeisoelasticsupply-pricefunctions. Table5-1. Supply-pricerelationshipexamples. ScenarioNo.Supply-pricefunction 1Q(c)=c)]TJ /F3 11.955 Tf 11.96 0 Td[(10005002Q(c)=c)]TJ /F3 11.955 Tf 11.96 0 Td[(15007503Q(c)=c)]TJ /F3 11.955 Tf 11.96 0 Td[(10007504Q(c)=c)]TJ /F3 11.955 Tf 11.96 0 Td[(5007505Q(c)=c)]TJ /F3 11.955 Tf 11.96 0 Td[(100010006Q(c)=c1001.57Q(c)=c5028Q(c)=c10029Q(c)=c150210Q(c)=c1003 Figure 5-3 showstheoptimalsolutionsforthescenariosinTable 5-1 .Theoptimalexpectedprotvaluesareallstrictlybelowthes(c)curve.Thisimpliesthatgiventheoptimalsupplypricecofourmodel,theexpectedprotobtainedinthestandardnewsvendorsolutionisalwayshigherthantheexpectedprotobtainedinourmodel,wherethesupplyquantitydependsonthesupplyprice(alsoseeTable 5-3 ).Thisdierenceintheoptimalexpectedprotofthetwomodelsarisesfromthefactthat,inourmodel,theavailablesupplyquantitydependsonthepriceoeredtothesuppliers,andtheproducercanreceivemoresupplyonlyifsheoersahigherpriceforeachsupplyunittobepurchased.However,inthestandardnewsvendorsetting,itisassumedthatthereisaninniteamountofsupplyatthegivenunitcost,c.Forthesameunitsupplyprice,thestandardnewsvendorcanobtainmoresupplyandsatisfyahigherproportionofthedemand,andhence,achieveahigherexpectedprot.Therefore,theexpectedprotlevelofthestandardnewsvendorisnotachievablebytheproducerinourproblemsetting.Table 5-2 showtheoptimalsupplypricesandthemarginalcostsofsupplyinganadditionalunitforthescenariospresentedinTable 5-1 .Ahighermarginalsupplycostactuallyindicatesthatthecorrespondingscenarioleadstoalowservicelevelandtherefore 101

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Figure5-3. Optimalexpectedprotfunctions(c)andoptimalexpectedprotvaluesforprice-dependentsupplycase. alowvolumeoperation.Theresultingoptimalexpectedprotisalsolowerforthecaseswithhighermarginalsupplycosts. Table5-2. Marginalcostofprocuringnextunitofsupplyatc. Sc.#cc+Q(c) Q0(c) 15.9219.84224.6847.36934.0366.73843.3886.10853.0645.12867.13721.41176.27712.55484.5029.00393.7007.400102.7624.144 Figures 5-4 and 5-5 illustratethes(c)functionand(c)functionsforthevariousscenariospresentedinTable 5-1 .Inthosegures,wecanobservethatforanycvalues(c)isalwaysgreaterthanorequaltothecorresponding(c)forallscenarios.Heres(c)formsanupperenvelopeforallcasesof(c).Figures 5-4 and 5-5 alsodemonstratehowthesupplymarketconditionsaecttheexpectedprotoftheproducer.Figure 5-4 illustratesthescenarioswithlinear 102

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Figure5-4. Optimalexpectedprotfunctions(c)andexpectedprotfunctions(c)withrespecttocwhenQ(c)islinear. supply-pricefunctions.Figure 5-4 ashowstheexpectedprotfunctionsforscenarios1,3,and5,where,theslopeofthefunctionQ(c),takesvalues500,750and1000,respectively,andstaysconstantatlevelof1000.Herescenario1hasthelowestrateofincreaseinsupplywhenthesupplypriceincreases,whereasscenario5hasthehighestincreaserateinsupply.Hence,forthesameamountofsupply,weneedtooerahigherpriceinscenario1comparedtotheothers.Scenario1isalsothecasewiththehighestthresholdprice.Thisresultsinanoptimalsupplypriceforscenario1thatishigherthanintheothertwoscenarios.Notethatthehighestdierencebetweenthestandardnewsvendor'soptimalexpectedprotandtheexpectedprotinourmodeloccursinscenario1.Astheslopeofthesupplypriceincreases,theoptimalexpectedprotofourmodelconvergestothestandardnewsvendor'soptimalexpectedprot.(Assumingthatthereexistsanitepositivethresholdprice,as!1,theproducercouldaccessaninniteamountofsupplyforaunitpriceequaltothethresholdprice.Thisispracticallythecaseofastandardnewsvendorproblem,wheretheunitcostofsupplyequalsthethresholdprice.)Notealsothatasincreasesthefunction(c)getssteeper,whichmeansanysmallchangeinthesupplypricecresultsinbiggerchangesintheexpected 103

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prot.Whentheproduceroersaslightlyhigherpricethantheoptimalsupplyprice,thedecreaseintheexpectedprotishighestinscenario5.Herethetotalcostofsupplyincreasesfastercomparedwithotherscenarios,sincethesameamountofincreaseinthesupplypriceresultsinahigherincreaseinthesupplyquantity.Hence,astheresponsivenessofthesupplymarkettotheincreaseintheoeredsupplypriceincreases,thesensitivityoftheexpectedprottothechangesinthesupplypriceincreasesaswell.Figure 5-4 bshowstheexpectedprotfunctionsforscenarios2,3,and4,where,theslopeofthefunctionQ(c),staysconstantatlevelof750,andtakesvalues1500,1000and500,respectively.Whiletheslopeofthesupply-pricefunctionstaysconstant,theshapeofthefunction(c)staysthesameamongthesethreescenarios.Theoptimalsupplypricesforscenarios2and3andforscenarios3and4dieralmostbythesameamountwhichisequaltothedierenceinthethresholdprices(thresholdpricesdierby2/3).Herethemarginalresponseofthesupplymarkettothechangesinsupplypriceisthesameforthethreescenarios.Weobtainaloweroptimalsupplypriceasthethresholdpricedecreases. Figure5-5. Optimalexpectedprotfunctions(c)andexpectedprotfunctions(c)withrespecttocwhenQ(c)isisoelastic. 104

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Figure 5-5 illustratesthescenarioswithisoelasticsupply-pricefunctions.InFigure 5-5 a,weillustratetheexpectedprotfunctionsforscenarios6,8,and10,wheretakesvalues1.5,2and3,respectively,andstaysconstantatalevelof100.Scenario6,withthelowestvalue,hasthehighestoptimalsupplyprice.Asincreases,toobtainthesameamountofsupplyweneedpaythelowestunitpriceinscenario8comparedtoscenarios6and7.Figure 5-4 bshowstheexpectedprotfunctionsforscenarios7,8,and9,wherestaysconstantatlevelof2,andtakesthevalues50,100and150,respectively.Unlikethecaseofthelinearsupply-pricefunction,theparameteraectstherateofincreaseinthesupplyquantityasthesupplypriceincreases.Therefore,changesinalsochangetheshapeof(c),andasitincreases(c)becomessteeper.Wecanobservethat,asincreases,theoptimalsupplypricecdecreases,andtheoptimalexpectedprotincreases.Notealsothatasand/orincrease,theresponsivenessofthesupplymarkettochangesinthesupplypriceincreasesaswell.Therefore,incaseswithhigherorhighervaluesweobtainexpectedprotfunctionsthataremoresensitivetochangesintheunitsupplyprice. Table5-3. OptimalsolutionstothescenariospresentedinTable 5-1 comparedwithstandardnewsvendorsolutions. StandardNVSc.#cQ(c)(c)SL(c)Q(Q)SL 15.9211960.505560.280.3472044.895725.160.67324.6842013.238192.340.5532075.988272.690.77634.0362026.709538.960.6052095.569625.560.83043.3882040.6310894.680.6582119.7210991.180.88453.0642064.1011614.340.7392134.9011679.070.91167.1371906.662972.100.1752018.133255.070.57276.2771969.974894.310.3822036.814998.820.64484.5022026.358593.930.6042081.188652.660.79293.7002053.6110284.400.7042107.2810330.540.858102.7622107.9512307.160.8602152.5812325.720.937 Figure 5-6 illustratestheoptimalorderquantityQofthestandardnewsvendorproblemasafunctionofc,i.e.Q(c)(seebluecurve),andtheoptimalsupplyquantityQ(c)withitscorrespondingoptimalsupplypricecforeveryscenariopresentedinTable 105

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5-1 .Hereagainwecanobservethatthe(c;Q(c))pointscorrespondingtotheoptimalsolutionsofourmodelliebelowtheQ(c)curve.InTable 5-3 wecanseethatgiventheoptimalchoiceofthesupplyprice,theoptimalorderquantityinthestandardnewsvendorsolutionisalwaysgreater.Thisresultcanalsobeobservedinthecriticalfractiletermsofbothmodels.Thenumeratorofthecriticalfractileoftheprice-dependentsupplycasehasanadditionalterm,)]TJ /F3 11.955 Tf 9.3 0 Td[(Q(c)=Q0(c).Thistermisalwaysnegative,andtherefore,forthesamecvalue,thecriticalfractileofournewsvendormodelisalwayslessthanthestandardnewsvendor'scriticalfractile.Thisleadsournewsvendortooperatewithalowerservicelevel(alsoseeFigure 5-7 )andhenceatalowervolume.ThiscanbeclearlyobservedinTable 5-3 ,wherewecomputethestandardnewsvendor'soptimalorderquantity,QandoptimalservicelevelSLgiventheoptimalsupplypricecofthemodelswithprice-dependentsupply. Figure5-6. OptimalorderquantityfunctionQ(c)andsupply-pricefunctionQ(c). Thedierencebetweenourmodelandthestandardnewsvendormodelarisesfromthefactthatinourmodelagreatersupplyquantitycanbeonlyattractedbyoeringahigherpricetothesuppliers.Hence,toreachtotheoptimalorderquantitylevelofthestandardnewsvendor,ourproducerhastopayahigherunitprice,whichisclearlynotthe 106

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Figure5-7. OptimalservicelevelfunctionSL(c)andoptimalservicelevelvaluesforprice-dependentsupplycase. optimalbehavior.Thusthegapbetweenouroptimalexpectedprotandthestandardnewsvendor'siscausedbytheincreasingmarginalcostofthesupply.Remark.InFigures 5-4 and 5-5 ,wecanobservethatthereexistsasupplypricevaluewheres(c)=(c).Thissupplypricevaluecorrespondstothepointwheretheoptimalorderquantityofthestandardnewsvendorequalstheprice-dependentsupplyofourmodelQ(c).Wedenotethissupplypricevalueby^c.Notethat^cisthevaluesatisfyingthefollowingcondition: F(Q(c))=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F3 11.955 Tf 11.96 0 Td[(c p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s:(5{17)Foragivenc,theright-handsideof( 5{17 )isthestandardnewsvendor'scriticalfractile,anditishigherthanthecriticalfractileobtainedinournewsvendormodelwithprice-dependentsupply.Therefore,solvingforthevalueofcsatisfying( 5{17 )wouldleadtoahigherorderquantityandhence,toahighersupplypricevaluec.Clearly^c>c,wherecistheoptimalchoiceofthesupplyprice.Theresultswhensolving( 5{17 )arepresentedinTable 5-4 .Theresultspresentedinthistableclearlyshowthatwhentheincreasingmarginalcostisignoredwhilechoosingtheoptimalsupplyprice,thevolumeofoperationwillbehigher;howevertheoptimalexpectedprotandtheoptimalservicelevel 107

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cannotbeachieved.Furthermore,dependingonthesupplymarket,thesolutionobtainedcanleadtoanexpectedprotandaservicelevelwhichhavearbitrarilylargeoptimalitygaps. Table5-4. ^cvaluesandcorrespondingsolutionsforthescenariospresentedinTable 5-1 %DierenceSc.#^cQ(^c)(^c)SL(^c)cQ(c)(c)SL(c) 16.0822041.205395.410.3202.7264.116-2.965-7.76324.7652073.758105.250.5391.7233.006-1.063-2.42534.1242092.709441.270.5912.1813.257-1.024-2.42943.4872115.5510779.630.6412.9493.672-1.056-2.52453.1322131.5011535.290.7282.2003.265-0.681-1.51567.3992012.572727.050.1393.6705.555-8.245-20.75376.3792034.524791.190.3691.6253.276-2.107-3.35084.5602079.458530.750.5971.3022.621-0.735-1.20993.7472105.5510232.600.6981.2572.529-0.504-0.838102.7812151.2812285.270.8580.6812.056-0.178-0.244 5.2.4RandomSupplyQuantityThestandardnewsvendorproblemassumesthat,whentheproducerplacesanorder,shereceivestheexactsameamountthatsheorders.Inthischapter,wegenerallyassumethatthesupplyquantityobtainedbytheproducerdependsonthepricethattheproduceroertoitssuppliers,andtheresponseofsupplierstoagivenpriceisknownwithcertainty.Wealsoassumethattheproducerreceivestheexactamountgivenbythisdeterministicsupply-pricerelationship.Inthissection,westudyaparticularcasewherethesupplyquantityisprice-dependentandalsorandom.Theliteratureonthenewsvendorproblemcontainssomestudieswheretheorderquantityisassumedtoberandom.Therandomnessisincorporatedinvariousways.Somestudiesassumethatthesuppliedamountfollowsadistributionwithameanequaltotheorderedquantity(examplesincludeCiarallo,Akella,andMorton[ 11 ],andOkyay,Karaesmen,andOzekici[ 56 ]).Therearesomeotherswhichassumerandomyield,whichmeansthatonlyarandomfractionoftheorderedquantityarrives,orisusefulinproduction(examplesincludeHenigandGerchak[ 35 ],Bassokand 108

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Akella[ 7 ],andDada,Petruzzi,andSchwarz[ 12 ]).Toourknowledge,nostudyexistsintheliteraturewhichassumesthatthesupplyquantityisprice-dependentandalsorandom.Weconsiderthecasewherethesupplyquantityfollowsadistributionwhosemeandependsonthesupplypricec.Weassumethatthetotalquantitysuppliedgivenanoeredpricecisanonnegativerandomvariablewithmean(c)andstandarddeviation.Here,(c)isanonnegativenondecreasingfunctionofc.Thestandarddeviationofthesupplyisconstantandknown,whileh()andH()aretheprobabilitydensityfunctionandthecumulativedistributionfunctionofthesupplydistribution,respectively.Givenasupplypricec,Q(c)givestherandomsupplyquantity.Theprotfunctionisformulatedin( 5{18 ),whereQ(c)andDarebothrandomvariables. (c;D)=pminfQ(c);Dg)]TJ /F3 11.955 Tf 20.58 0 Td[(g[D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c)]++s[Q(c))]TJ /F3 11.955 Tf 11.96 0 Td[(D]+)]TJ /F3 11.955 Tf 11.95 0 Td[(cQ(c)(5{18)UsingtheidentityminfQ(c);Dg=Q(c))]TJ /F1 11.955 Tf 11.96 0 Td[([Q(c))]TJ /F3 11.955 Tf 11.96 0 Td[(D]+,weobtain (c;D)=(p)]TJ /F3 11.955 Tf 11.96 0 Td[(c)Q(c))]TJ /F3 11.955 Tf 11.95 0 Td[(g[D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q(c)]+)]TJ /F1 11.955 Tf 11.96 0 Td[((p)]TJ /F3 11.955 Tf 11.95 0 Td[(s)[Q(c))]TJ /F3 11.955 Tf 11.96 0 Td[(D]+:(5{19)Ourgoalistodeterminethesupplypricecsuchthattheexpectedprotismaximized.Theexpectedprot(c)forthiscaseisasfollows: (c)=(p)]TJ /F3 11.955 Tf 11.96 0 Td[(s)D)]TJ /F1 11.955 Tf 11.95 0 Td[((c)]TJ /F3 11.955 Tf 11.95 0 Td[(s)(c))]TJ /F1 11.955 Tf 11.96 0 Td[((p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)E[(D)]TJ /F3 11.955 Tf 11.95 0 Td[(Q(c))+]:(5{20)WenextassumethatDandQ(c)arenormallydistributed.WedeneanewrandomvariableW,whereW=D)]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c).Theexpectedprotfunctionbecomes (c)=(p)]TJ /F3 11.955 Tf 11.95 0 Td[(s)D)]TJ /F1 11.955 Tf 11.95 0 Td[((c+v)]TJ /F3 11.955 Tf 11.95 0 Td[(s)(c))]TJ /F1 11.955 Tf 11.96 0 Td[((p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)Z10wfW(w)dw:(5{21)HerefW()isthenormaldensityfunctionwiththemeanD)]TJ /F3 11.955 Tf 12.11 0 Td[((c)andvariance2+2D.LetLdenotethestandardnormallossfunctionL(z)=R1z(x)]TJ /F3 11.955 Tf 12.27 0 Td[(z)(x)dx,whereisthestandardnormaldensityfunction.Wedenotethecorrespondingcumulativedistributionfunctionby,and(x)=1)]TJ /F1 11.955 Tf 12.29 0 Td[((x).WehaveL(z)=(z))]TJ /F3 11.955 Tf 12.29 0 Td[(z(z)andL0(z)=)]TJ /F1 11.955 Tf 10.6 3.02 Td[((z). 109

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Hence,wecanreformulatetheexpectedprot(c)usingthestandardnormallossfunction. (c)=(p)]TJ /F3 11.955 Tf 11.95 0 Td[(s)D)]TJ /F1 11.955 Tf 11.95 0 Td[((c)]TJ /F3 11.955 Tf 11.96 0 Td[(s)(c))]TJ /F1 11.955 Tf 11.96 0 Td[((p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)L(z(c))q 2+2D:(5{22)wherez(c)=((c))]TJ /F3 11.955 Tf 12.78 0 Td[(D)=p 2+2D.NotingthatdL(z(c))=dc=)]TJ /F3 11.955 Tf 9.3 0 Td[(z0(c)(z(c))andz0(c)=0(c)=p 2+2D,wenextpresenttherstandsecondderivativesof(c). 0(c)=)]TJ /F3 11.955 Tf 9.3 0 Td[((c))]TJ /F1 11.955 Tf 11.96 0 Td[((c)]TJ /F3 11.955 Tf 11.96 0 Td[(s)0(c)+(p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)q 2+2Dz0(c)(z(c)):(5{23) 00(c)=)]TJ /F1 11.955 Tf 9.3 0 Td[(20(c))]TJ /F1 11.955 Tf 11.96 0 Td[((c)]TJ /F3 11.955 Tf 11.95 0 Td[(s)00(c)+(p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)"00(c)(z(c)))]TJ /F1 11.955 Tf 20.51 8.09 Td[((0(c))2 p 2+2D(z(c))#:(5{24)Solvingfortherstorderoptimalityconditionyieldsequation( 5{25 ).Thevalueofcsatisfying( 5{25 )isastationarypointof(c).Here(z(c))givestheprobabilitythatthesupplyisgreaterthanthedemand.Notethattheright-handsideofthisequationisverysimilartotheratioweobtainedinEquation( 5{5 )atthebeginningofthissection.TheratioQ(c)=Q0(c)forthedeterministiccaseisreplacedby(c)=0(c)intherandomsupplycase.Inthiscase,themarginalunderstockingcost,cu,isgivenbyp+g)]TJ /F3 11.955 Tf 10.7 0 Td[(c)]TJ /F3 11.955 Tf 10.7 0 Td[(v)]TJ /F3 11.955 Tf 10.7 0 Td[((c)=0(c),andthemarginaloverstockingcost,coequalsc+(c)=0(c)+v)]TJ /F3 11.955 Tf 11.96 0 Td[(s. (z(c))=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F5 7.97 Tf 14.49 5.7 Td[((c) 0(c) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s:(5{25)(c)isconcaveincwhen00(c)0forallc>0.Thesecondorderoptimalityconditionissatisedwhenthefollowinginequalityholds. 00(c) 0(c)2)]TJ /F9 11.955 Tf 11.95 16.85 Td[(1 0(c)2(z(c))(c)]TJ /F3 11.955 Tf 11.96 0 Td[(s)00(c)+20(c) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s(z(c)) (z(c))p 2+2D:(5{26)Notethatthecondition( 5{26 )dependsontheformofthefunction(c)andthecharacteristicsofthedistributionoftherandomvariableW.Whenthefunction(c)takesalinearformasdescribedinSection 5.2.1 ,(c)isconcaveincindependentofthedistributionofW.Whenthefunction(c)takesanisoelasticformasdescribedinSection 5.2.2 ,concavityof(c)alsodependsonthedistributionofW.ByProposition 5.2 in 110

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Section 5.2.2 ,weknowthatwhenthedemanddistributionhasanincreasinggeneralizedfailurerate,wecanobtainauniquemaximizeroftheexpectedprot.SinceWfollowsanormaldistributionandthenormaldistributionhasanincreasinggeneralizedfailurerate,thereexistsauniquestationarypointcthatsatisestheinequality( 5{26 ).Theresultsinthissectionshowthat,inasingleperiodsetting,therandomsupplyquantitydoesnotaecttheoptimalpolicythatischaracterizedbythebalancebetweenmarginaloverstockingandunderstockingcosts.OurndingsherecoincidewiththeonesinthestudyofCiarallo,Akella,andMorton[ 11 ].Thatis,theproducerhastodecideonasupplypriceassumingthatshewillreceivetheexpectedsupplyquantity. 5.3Price-DependentSupplyandDemandWenextstudythecasewherethedemandisalsoprice-sensitive.Inthismodel,thesellingpricepisalsoadecisionvariableinadditiontothesupplypricec.Hereweconsidertwodemandmodels:anadditivemodel(D(p)=y(p)+wherey(p)=a)]TJ /F3 11.955 Tf 12.45 0 Td[(bpfora;b>0andistherandomcomponent)andamultiplicativemodel(D(p)=y(p)wherey(p)=ap)]TJ /F5 7.97 Tf 6.59 0 Td[(bfora>0,b>1andistherandomcomponent).WeassumethatisarandomvariablebetweenAandB,whereB>A.Fortheadditivedemandcase,werequireA>)]TJ /F3 11.955 Tf 9.29 0 Td[(a,andforthemultiplicativedemandcase,werequireA>0.Theprobabilitydensityfunctionandthecumulativedistributionfunctionofaredenotedbyf()andF(),respectively.Wealsoletandrepresentthemeanandthevarianceof,respectively.ThedemandmodelsconsideredinthissectionarethesameasthosedescribedinthestudyofPetruzziandDada[ 59 ],whoconsiderthenewsvendorproblemwhereonlydemandisprice-dependent.Ourgoalistoinvestigatethedierencescausedbytheprice-dependenceofthesupplyontheoptimalpricingdecisionscomparedtothecasewhereonlydemandisprice-sensitive. 111

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5.3.1AdditiveDemandModelInthissectionweconsiderthefollowingdemandmodel.Theprice-dependentandrandomdemanddenotedbyD(p;)takestheformy(p)+wherey(p)=a)]TJ /F3 11.955 Tf 12.01 0 Td[(bp(a;b>0).Foragivenrealizationofandgivensupplyandsellingpricescandp,wecanwritetheprotfunctionasfollows: (c;p;)=8><>:pQ(c))]TJ /F3 11.955 Tf 11.95 0 Td[(g(D(p;))]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c)))]TJ /F1 11.955 Tf 11.96 0 Td[((c+v)Q(c);ifD(p;)>Q(c);pD(p;)+s(Q(c))]TJ /F3 11.955 Tf 11.95 0 Td[(D(p;)))]TJ /F1 11.955 Tf 11.95 0 Td[((c+v)Q(c);ifD(p;)<>:p(y(p)+z))]TJ /F3 11.955 Tf 11.96 0 Td[(g()]TJ /F3 11.955 Tf 11.95 0 Td[(z))]TJ /F1 11.955 Tf 11.96 0 Td[((c(z;p)+v)(y(p)+z);if>z;p(y(p)+)+s(z)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F1 11.955 Tf 11.95 0 Td[((c(z;p)+v)(y(p)+z);if
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expectedleftovers,i.e., (z)=ZBz(x)]TJ /F3 11.955 Tf 11.95 0 Td[(z)f(x)dx;(5{32) (z)=ZzA(z)]TJ /F3 11.955 Tf 11.96 0 Td[(x)f(x)dx:(5{33)Nextconsidertherstandsecondderivativesof(z;p)withrespecttoz. @(z;p) @z=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(@c(z;p) @z(y(p)+z))]TJ /F1 11.955 Tf 11.95 0 Td[((c(z;p)+v)]TJ /F3 11.955 Tf 11.96 0 Td[(s)+(p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(z)):(5{34) @2(z;p) @z2=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(@2c(z;p) @z2(y(p)+z))]TJ /F1 11.955 Tf 11.95 0 Td[(2@c(z;p) @z)]TJ /F1 11.955 Tf 11.95 0 Td[((p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)f(z):(5{35)Notethatforagivenpvalue,theconcavityoftheexpectedprotfunction(z;p)withrespecttozdependsonthesupply-pricerelationship.Inthissectionwewillassumethatthesupply-pricefunctionQ(c)takesthelinearformdescribedinSection 5.2.1 ,i.e.,Q(c)=c)]TJ /F3 11.955 Tf 12.69 0 Td[(where;>0.Whenthesupply-pricerelationshipislinear,wehavec(z;p)=(1=)(z++a)]TJ /F3 11.955 Tf 12.28 0 Td[(bp),@c(z;p)=@z=1=,and@2c(z;p)=@z2=0.Thereforeforthiscase,(z;p)isconcaveinzforagivenp.Notealsothatwhen(z;p)isconcaveinz,thezvaluesatisfyingthefollowingconditionmaximizes(z;p)foragivenvalueofp. F(z)=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c(z;p))]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F5 7.97 Tf 13.16 5.7 Td[(@c(z;p) @z(y(p)+z) p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s:(5{36)SimilartoEquation( 5{5 )inSection 5.2 ,Equation( 5{36 )showsthat,atoptimality,theprobabilitythattheuncertainportionofthedemandissatisedequalsthewell-knowncriticalfractile,cu=(cu+co).Notethaty(p)+z=Q(c(z;p))andbytheinversefunctiontheorem,@c(z;p)=@z=1=Q0(c(z;p)),whichshowsthatweobtainthesamecriticalfractilebehaviorasdiscussedinSection 5.2 .Thatis,themarginalunderstockingcost,cu,equalsp+g)]TJ /F3 11.955 Tf 12.83 0 Td[(c(z;p))]TJ /F3 11.955 Tf 12.84 0 Td[(v)]TJ /F5 7.97 Tf 14.03 5.69 Td[(@c(z;p) @z(y(p)+z),andthemarginaloverstockingcost,co,equalsc(z;p)+@c(z;p) @z(y(p)+z)+v)]TJ /F3 11.955 Tf 11.96 0 Td[(s. 113

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Wenextconsidertherstandsecondderivativesof(z;p)withrespecttothesellingpricep. @(z;p) @p=1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@c(z;p) @py(p)+)]TJ /F3 11.955 Tf 11.95 0 Td[(b(p)]TJ /F3 11.955 Tf 11.96 0 Td[(c(z;p))]TJ /F3 11.955 Tf 11.95 0 Td[(v))]TJ /F1 11.955 Tf 11.95 0 Td[((z))]TJ /F3 11.955 Tf 11.96 0 Td[(z@c(z;p) @p:(5{37) @2(z;p) @p2=)]TJ /F3 11.955 Tf 9.3 0 Td[(b2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@c(z;p) @p)]TJ /F3 11.955 Tf 13.15 8.09 Td[(@2c(z;p) @p2(z+y(p)):(5{38)IntheadditivedemandcaseconsideredinPetruzziandDada[ 59 ],wheretheunitpriceofsupplyisxedandthesupplyquantitydoesnotdependonthisprice,theexpectedprotfunctionisconcaveinthesellingpricepforaxedz.Equation( 5{38 )showsthattheconcavityof(z;p)withrespecttopdependsontheformofthefunctionc(z;p).Thisimpliesthatinourproblemsetting,therelationshipbetweenthepriceoeredtosuppliersandthesupplyquantitythattheproducerreceivesinresponsetotheoeredpriceaectsthebehavioroftheexpectedprotfunction(z;p)withrespecttothesellingpricep.Whenthesupply-pricerelationshipislinearasdiscussedabove,wehave@c(z;p)=@p=)]TJ /F3 11.955 Tf 9.3 0 Td[(b=and@2c(z;p)=@p2=0.Hence,whenthesupply-pricerelationshipislinear,theexpectedprot(z;p)isstrictlyconcaveinpforagivenz,andforagivenzthesellingpricepthatmaximizes(z;p)canbewrittenasafunctionofz. p(z)=p0(z))]TJ /F1 11.955 Tf 13.15 10.33 Td[((b )(+z)+(z) 2b(1+b );(5{39)where p0(z)=(1+b )(+a)+bv+(b )(a++z) 2b(1+b ):(5{40)Herep0(z)isthevalueofpthatmaximizestherisklessprot(z;p)foragivenz.If(b )(+z)+(z)ispositive,wehavep(z)p0(z).Thisrelationshipwasalsoshownin[ 59 ]byPetruzziandDadaforthecasewhereonlydemandisprice-dependentandtheadditivedemandmodelisassumed.Theydidnot,however,needanyfurtherassumptionsontheproblemparameterstoshowthisresult,whichwasrstdemonstratedbyMillsin[ 53 ].Notethatifweassumethatisarandomvariablewithapositivemean(i.e., 114

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if>0),thenthesameresultalsoholdsforourcase.Notealsothatbothp0(z)andp(z)areincreasinginz,andaszgoestoinnityp0(z)goestoinnityaswell.However,p(z)approachesanitevalueasshownin( 5{41 ).Inthecasewhereonlydemandisprice-dependent,thepricep0thatmaximizestherisklessprotisindependentofz,andtheoptimalsellingprice,p(z)approachesp0aszgoestoinnity. limz!1p(z)=(1+b )a++b(v+(+a)=) 2b(1+b ):(5{41)Foranygivenzvalueweknowthatp(z)in( 5{39 )maximizestheexpectedprot.Using( 5{39 ),wecanreducethenumberofdecisionvariablesofourmaximizationproblemtoone.Inthereformulatedproblem,wewanttodeterminethevalueofzsuchthat(z;p(z))ismaximized.ThisapproachisalsopresentedbyPetruzziandDada[ 59 ]forthecasewhereonlydemandisprice-dependent.Nextweconsidertherstandsecondderivativesof(z;p(z)). d(z;p(z)) dz=)]TJ /F1 11.955 Tf 9.3 0 Td[((y(p(z))+z) )]TJ /F1 11.955 Tf 10.97 0 Td[((c(z;p(z))+v)]TJ /F3 11.955 Tf 10.97 0 Td[(s)+(p(z)+g)]TJ /F3 11.955 Tf 10.97 0 Td[(s+z +b)F(z):(5{42) d2(z;p(z)) dz2=F(z) 2b(1+b )4b +F(z))]TJ /F1 11.955 Tf 13.86 8.08 Td[(2 )]TJ /F3 11.955 Tf 11.95 0 Td[(f(z)p(z)+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s+z +b:(5{43)Solvingfortherstorderconditionyieldsthefollowing.Anyzvaluesatisfyingthecondition( 5{44 )isastationarypointofthefunction(z;p(z)). F(z)=p(z)+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c(z;p(z)))]TJ /F3 11.955 Tf 11.95 0 Td[(v+z +b)]TJ /F7 7.97 Tf 13.61 4.7 Td[(1 (y(p(z))+z) p(z)+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s+z +b:(5{44) Proposition5.3. Giventhattheprice-dependentdemandfollowsanadditivemodelandthesupply-pricerelationshipislinear,andassumingthatthedistributionofhasanincreasingfailurerate,thereexistsa z suchthatthefunction(z;p(z))isconcaveforallz z ,andthevalueofzsatisfying( 5{44 )isthemaximizerof(z;p(z)). 115

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Proof. Forconcavity,werequired2(z;p(z))=dz2<0,andthisconditionissatisedwhencondition( 5{45 )issatised.(Weobtaincondition( 5{45 )usingEquation( 5{43 ).) 4b=+F(z) 2b(1+b=))]TJ /F7 7.97 Tf 22.31 4.71 Td[(2 F(z) p(z)+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s+z +b0,hencetheleft-handsideof( 5{45 )decreasesinz.Thisprovesthatazvalue,z ,mustexistsuchthatforallzz condition( 5{45 )issatised,i.e.,(z;p(z))isconcaveforallzz .Aszapproaches)]TJ /F3 11.955 Tf 9.3 0 Td[(a(thisistheminimumvaluethatzcantake),(z;p(z))approachesavalue,i.e.,itdoesnottendtoinnity.Andaszapproachestoinnity,(z;p(z))goestonegativeinnity.Andsincethereisonlyoneinectionpointz ,(z;p(z))isunimodal,andthezvaluesatisfying( 5{44 )isitsmaximizer. Remark.Appendix J showsthederivationofrstordernecessaryoptimalityconditionsforasolutionthatmaximizestheexpectedprot.Hereweformulatetheprotfunctionintermsofthesellingpricepandthesupplypricec.Ouranalysisshowsthatanyoptimalsolution(c;p)mustsatisfythefollowingequality(seeAppendix J forthederivation). p)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 11.95 0 Td[(v)]TJ /F3 11.955 Tf 14.55 8.09 Td[(Q(c) Q0(c)=y(p)+)]TJ /F1 11.955 Tf 11.96 0 Td[((c;p) b:(5{46)Ontheright-handsideof( 5{46 ),thenumeratorequalstheexpecteddemandminustheexpectedshortages,whichgivestheexpectedsalesatoptimality.Ontheleft-handsideof( 5{46 )wehavethenetrevenuethatcanbegeneratedbythenextunittobepurchasedatoptimality.Thisequalityshowsthat,atoptimality,themarginalnetrevenueequalstheexpectedsalesdividedbythepriceelasticityofdemand. 5.3.2MultiplicativeDemandModelInthissectionweconsiderthecasewherethedemandD(p;)equalsy(p),wherey(p)=ap)]TJ /F5 7.97 Tf 6.58 0 Td[(b(a>0andb>1).Givenasupplypriceandasellingprice,andarealization 116

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of,wecanformulatetheprotasfollows. (c;p;)=8><>:pQ(c))]TJ /F3 11.955 Tf 11.95 0 Td[(g(D(p;))]TJ /F3 11.955 Tf 11.96 0 Td[(Q(c)))]TJ /F1 11.955 Tf 11.96 0 Td[((c+v)Q(c);ifD(p;)>Q(c);pD(p;)+s(Q(c))]TJ /F3 11.955 Tf 11.95 0 Td[(D(p;)))]TJ /F1 11.955 Tf 11.95 0 Td[((c+v)Q(c);ifD(p;)<>:py(p)z)]TJ /F3 11.955 Tf 11.95 0 Td[(gy(p)()]TJ /F3 11.955 Tf 11.95 0 Td[(z))]TJ /F1 11.955 Tf 11.96 0 Td[((c(z;p)+v)y(p)z;if>z;py(p)+sy(p)(z)]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F1 11.955 Tf 11.96 0 Td[((c(z;p)+v)y(p)z;if
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@2(z;p) @z2=y(p))]TJ /F1 11.955 Tf 10.49 8.09 Td[(2y(p)z )]TJ /F1 11.955 Tf 11.96 0 Td[((p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)f(z):(5{53)Notethatforagivenpvalue,(z;p)isconcaveinz.Notealsothatwhen(z;p)isconcaveinz,thezvaluesatisfyingthefollowingconditionmaximizes(z;p)foragivenvalueofp. F(z)=p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(c(z;p))]TJ /F3 11.955 Tf 11.95 0 Td[(v)]TJ /F5 7.97 Tf 13.15 5.7 Td[(y(p)z p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s:(5{54)Notethaty(p)z=Q(c(z;p))andy(p)z==Q(c(z;p))=Q0(c(z;p)),whichshowsthatweobtainthesamecriticalfractilebehaviorasdiscussedinSection 5.2 .Similartotheadditivedemandmodel,Equation( 5{54 )showsthat,atoptimality,theprobabilitythattheuncertainportionofthedemandissatisedequalsthewell-knowncriticalfractile,cu=(cu+co).Themarginalunderstockingcost,cu,equalsp+g)]TJ /F3 11.955 Tf 11.51 0 Td[(c(z;p))]TJ /F3 11.955 Tf 11.5 0 Td[(v)]TJ /F5 7.97 Tf 12.7 5.7 Td[(y(p)z ,andthemarginaloverstockingcost,co,equalsc(z;p)+y(p)z +v)]TJ /F3 11.955 Tf 11.95 0 Td[(s.Nowweconsidertherstandsecondderivativesof(z;p)withrespecttop. @(z;p) @p=y0(p)(b)]TJ /F1 11.955 Tf 11.95 0 Td[(1)()]TJ /F1 11.955 Tf 11.96 0 Td[((z))p)]TJ /F3 11.955 Tf 11.96 0 Td[(z2zap)]TJ /F5 7.97 Tf 6.59 0 Td[(b + +v)]TJ /F3 11.955 Tf 11.95 0 Td[(g(z)+s(z):(5{55) @2(z;p) @p2=y00(p)(b)]TJ /F1 11.955 Tf 11.95 0 Td[(1)()]TJ /F1 11.955 Tf 11.95 0 Td[((z))p)]TJ /F3 11.955 Tf 11.95 0 Td[(z2zap)]TJ /F5 7.97 Tf 6.58 0 Td[(b + +v)]TJ /F3 11.955 Tf 11.96 0 Td[(g(z)+s(z)+y0(p)(b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)()]TJ /F1 11.955 Tf 11.95 0 Td[((z))+2 z2bap)]TJ /F5 7.97 Tf 6.59 0 Td[(b)]TJ /F7 7.97 Tf 6.59 0 Td[(1: (5{56) Foragivenzvalue,wecancomputethevaluesofpsatisfying( 5{57 ).Anypvalueobtainedbysolving( 5{57 )isastationarypointof(z;p)givenz. (b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)()]TJ /F1 11.955 Tf 11.95 0 Td[((z))p)]TJ /F3 11.955 Tf 11.95 0 Td[(z2zap)]TJ /F5 7.97 Tf 6.59 0 Td[(b + +v)]TJ /F3 11.955 Tf 11.96 0 Td[(g(z)+s(z)=0:(5{57)Notethatatapositivestationarypointp,wehave@2(z;p)=@p2<0.Therefore,atmostonesuchstationarypointcanexist.Assumingthatthereexistsapositivestationarypointpgivenz,itistheglobalmaximizerof(z;p)givenz.Wenextproposeasequentialprocedurewhichstopsatapointwherethegradientof(z;p)equalszero.Westartwithaninitialvalueofpandsolveforthecorresponding 118

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zvaluethatsatises( 5{54 ).Clearly,thisvalueofzmaximizestheexpectedprotgiventheinitialp.Then,usingthezvalue,wesolve( 5{57 )toobtainthispvaluewhichwillmaximizetheexpectedprotforthexedz.Ifthecorrespondingvalueofpequalstheinitiallyselectedvalue,westop.Otherwise,wecontinuetheprocedurebysolving( 5{54 )forthenextzvalue.ThisprocedureisoutlinedinAlgorithm 5.1 .Thealgorithmstopswhenthezandpvaluesobtainedbysolving( 5{54 )and( 5{57 )donotchangeanymore. Algorithm5.1. Findastationarypointof(z;p). 1: Input:Aninitialvalueofp,p0. 2: Output:Astationarypointsolution(z;p). 3: Sett=1. 4: Solve( 5{54 )forzwherep=pt)]TJ /F7 7.97 Tf 6.58 0 Td[(1.Thesolutioniszt. 5: Solve( 5{57 )forpwherez=zt.Thesolutionispt. 6: ifpt=pt)]TJ /F7 7.97 Tf 6.58 0 Td[(1then 7: Stop.(z;p)=(zt;pt). 8: else 9: Sett=t+1.Goto 4 10: endifBythedevelopmentofthealgorithm,weknowthatr(z;p)=0.WhenAlgorithm 5.1 stops,itclearlystopsatastationarypointsolutionof(z;p).Ifthesolution(z;p)obtainedbyAlgorithm 5.1 satisesthesecondordersucientoptimalityconditions,thenwecanconcludethat,(z;p)isalocalmaximum.Hence,(z;p)isalocalmaximumwhenr2(z;p)0.From( 5{53 )and( 5{56 ),weknowthatthetraceoftheHessianmatrixr2(z;p)isnegative.Wemusthavedet(r2(z;p))>0toconcludethat(z;p)isalocalmaximum.Notethatdet(r2(z;p))=r21;1(z;p)r22;2(z;p))]TJ /F1 11.955 Tf 11.96 0 Td[((r21;2(z;p))2; 119

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andr21;2(z;p)=r22;1(z;p)=@2(z;p) @p@z.Wehave @2(z;p) @p@z=y0(p)((b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)F(z))]TJ /F1 11.955 Tf 13.15 8.09 Td[(4zy(p) )]TJ /F3 11.955 Tf 13.15 8.09 Td[( )]TJ /F3 11.955 Tf 11.96 0 Td[(v+s:(5{58)Hence,thestationarypoint(z;p)isalocalmaximumof(z;p),ifitsatisescondition( 5{59 ). 2zy(p)p b+(p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)f(z)p b(b)]TJ /F1 11.955 Tf 11.95 0 Td[(1)()]TJ /F1 11.955 Tf 11.95 0 Td[((z)))]TJ /F1 11.955 Tf 13.15 8.09 Td[(2(z)2y0(p) >((b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)F(z))]TJ /F1 11.955 Tf 13.15 8.09 Td[(4zy(p) )]TJ /F9 11.955 Tf 11.95 16.86 Td[( +v)]TJ /F3 11.955 Tf 11.95 0 Td[(s: (5{59) 120

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CHAPTER6SUMMARYANDCONCLUSIONSInthisstudyweconsideredinventorymodelswherethequantityofinputavailableforproductiondependsonthepricethattheproduceroerstoitssuppliers.Thisisaparticularlyrelevantphenomenoninareverselogisticssettingwheretheinputsrequiredbyaremanufacturerareownedbyindividualconsumerswhomaybewillingtoselltheirproductsbacktotheremanufacturerdependingonthepriceoered.Supplypricingisalsorelevantbeyondthiscontext,wherethepriceoereddirectlyimpactsthequantitymadeavailablebypotentialsuppliers.Thismotivatedtheconsiderationofvariousmodelsforasingle-itemproductionplanningproblemwithprice-sensitivesupply.Werststudiedthenite-horizon,discrete-timeproductionandcomponent-supply-pricingplanningproblemwithnon-stationarycosts,demands,andcomponentsupplylevels.Theproducerseeksaproductionandsupply-pricingplanthatminimizesthecostincurredwhilemeetingasetofdemandsoveradiscrete-timeniteplanninghorizon.Theprice-sensitivityofsupplyavailabilityledtoaclassoftwo-levellotsizingproblemswhosecostisthesumofconcaveandconvexfunctions,andwherethesupplyavailabilitylimitsproductionquantities.WeshowedthattheresultingproblemisNP-Hardingeneral,andconsideredseveralpracticallyrelevantspecialcasesthatpermitpolynomialsolvability.Ourworkprovidescontributionstotheclassofconvex-costlotsizingproblemsandtothereverselogisticsliterature,aswestudiedproblemclassesinwhichprice-sensitivereturnsfromend-usersserveasinputforproduction.Directionsforfutureresearchinthiscontextincludedevelopingsolutionapproachesforhandlingproblemsrequiringmultiplecomponentsandmultipleproducts.Theimpactofpricingdecisionsoncustomerdemandlevels,whichdirectlyaectproductionrequirements,hasreceivedsignicantattentionintheliterature.Thus,anotherhigh-valuedirectionwouldincorporateprice-sensitivedemandstogetherwithprice-sensitivesupply. 121

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Motivatedbythezero-xed-chargedversionofthetwo-leveldynamiclotsizingproblem,inChapter 3 ,wedevelopedapolynomialtimealgorithmfordynamiclot-sizingproblemswithconvexcostsinproductionandinventoryquantities.Theclassofconvex-costdynamiclot-sizingproblemsweconsideredpermittedconvexfunctionsoftheproductionandinventoryquantitiesineachperiodoveranitehorizon,whichistheclassofproblemsVeinott[ 86 ]consideredinhisclassicpaperonthetopic.Hereapromisingdirectionforfutureresearchwouldbethedevelopmentofasolutionmethodforamodelthatpermitsconvexholdingcoststhatexplicitlydependontheproductionanddemandperiod(e.g.,ht(xt)fort).Anotherworthwhileresearchdirectionmayaccountforbothprice-sensitivesupplyandprice-dependentdemandinproductionplanning.Wenextstudiedtheinnite-horizonversionoftheproblemwheretheproductionandinventoryholdingcostsaswellasthepricingparametersarestationary.Chapter 4 providedanovelgeneralizationofthedeterministicEOQmodelthatpermitsusingpricingonboththesupplysideandthedemandsideinordertooptimallymatchsupplyanddemandratesinaninventorysystem.Hereweassumedthatthesupplyforproductionarriveswitharatedependingonthepriceoeredtosuppliers.Wealsoassumedthatthedemandratedependsonthepriceoeredtocustomersbytheproducer.Wecharacterizedoptimalsolutionsforpricingdecisionsandinvestigatedtheirbehaviorwithrespecttochangingproblemparameters.Whilepastworkhasaccountedforthewaysinwhichpricinginuencesdemandlevels,ourworkalsoaccountsfortheimportantimpactofprice-dependentsupplyofinputcomponents.Werstexaminedtherelationshipbetweentheprice-dependentdemandandsupplyrates,andshowedthattheseratesmustbeequalinequilibrium.Wethenformulatedtheprotmaximizationproblemintermsoftheend-itemsellingprice,whichimpliesanunderlyingequilibriumcomponentsupplyprice.Underaniso-elasticfunctionalformforthedemandrate,wecharacterizedtheprot-maximizinglotsizeandprices. 122

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Ourmodelandanalysispermitteddemonstratingthewayinwhichprice-dependentsupplyimpactsoptimaldecisionsandprotability.Inparticular,thisanalysisshowedthevalueofheterogeneityinthesupplybase,aswellastheimportanceofunderstandingandaccountingfortherelationshipbetweenthesupplypriceandthequantitysupplied,intermsofprotability.Futureresearchinthissettingmayconsidergeneralizingourmodeltoaccountforshortages(intheformoflostsalesorbackorders)atanassociatedcost.Additionally,whileourmodelconsideredaniso-elasticprice-demandfunctionandalinearprice-to-priceresponsefunction,extensionsofthismodelmayconsiderdierentclassesofprice-demandandprice-to-priceresponsefunctions.Inthelastchapter,weconsideredasingleperiodplanningproblemwheretheend-itemdemandfacedbytheproducerwasrandom,andthesupplyquantitydependedontheunitpriceoeredbytheproducertosuppliers.Thisplanningproblemcorrespondstoanewsvendorproblemwheretheavailablesupplyquantityisprice-dependent.Firstwestudiedthecasewhereonlythesupplyisprice-dependent.Wecharacterizedtheoptimalsupplypricingpolicieswherethesupplyversuspricerelationshipswerelinearandisoelastic.Weshowedthatthelinearsupply-pricerelationshipledtoanexpectedprotfunctionwhichisconcaveinthesupplyprice.Thisresultwasindependentofthedemanddistribution.Whenthesupply-pricerelationshipisisoelastic,weshowedthattheexpectedprotfunctionisunimodalandthestationarypointsolutionmaximizestheexpectedprotwhenthedemanddistributionhasanincreasinggeneralizedfailurerate.Inbothofthesecases,weshowedthatthewell-knowncriticalfractile(underagecostdividedbythesumofoverageandunderagecosts)givestheprobabilitythattheend-itemdemandissatised.Weinvestigatedthedierencesintheoptimalpoliciesbetweenthestandardnewsvendorproblemandourmodel.Weshowedthat,forthesamesupplyprice,thestandardnewsvendorsolutionresultsinahigherservicelevel,andconsequently,inahigheroptimalorderquantity.Wealsoshowedthattheoptimalexpectedprotobtained 123

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bythestandardnewsvendorisalwaysgreaterthantheoptimalexpectedprotofourmodel.Thisdierencearisesfromthefactthat,inthestandardnewsvendorsetting,itisassumedthatthereisaninniteamountofsupplyatthegivenunitcost.However,inourmodel,theavailablesupplyquantitydependsonthepriceoeredtothesuppliers,andtheproducercanpurchasemoresupplyonlyifsheoersahigherpricetosuppliers.Forthesameunitsupplyprice,thestandardnewsvendorcanobtainmoresupplyandsatisfyahigherproportionofthedemand,andhence,achieveahigherexpectedprot.Wealsopresentedtheoptimalsupply-pricingpolicyforthecasewherethesupplyisprice-dependentandalsorandom.Inparticular,weconsideredthecasewherethesupplyquantityfollowsanormaldistributionwithaprice-dependentmeanandaknownvariance.Ouranalyticalresultsshowedthattheproducer'soptimalsupply-pricingpolicyisnotdierentfromthedeterministicprice-dependentsupplycase.Wethenstudiedthecasewherethedemandisalsoprice-sensitiveandanalyzedtheoptimalpricingdecisions.Weconsideredtwodemandmodels,additiveandmultiplicative,whichwerealsopresentedinthestudyofPetruzziandDadain[ 59 ].Wealsoassumedthatthesupply-pricerelationshipislinear.Fortheadditivedemandcase,weobtainedclosed-formsolutionsfortheoptimalsellingandsupplypricesgiventhattheend-itemdemandfollowsadistributionwithanincreasingfailurerate.Unlikethecasewhereonlydemandisprice-dependent,wecouldnotobtainclosed-formsolutionsforthemultiplicativedemandmodel.Therefore,weproposedasequentialproceduretoobtainstationarypointsolutionsfortheexpectedprotfunctionandprovidedconditionsthatalocalmaximumhastosatisfy. 124

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APPENDIXAPROOFOFPROPOSITION2.1WebeginwithaspecialcaseofP(FC)inwhichFCt=0andhCt=1forallt2T,i.e.,noxedchargesexistforprocurementandnocomponentinventoryisheld(notethatforthisspecialcase,fC(t)isaconvexfunctionoftforeacht2T).LetSP1denotethisspecialcaseoftheproblemwithzeroxedcostsforcomponentprocurementandwherecomponentinventoryholdingisnotpermitted.ToformulateproblemSP1,letxtidenotetheamountproducedinperiodttosatisfydemandinperiodi,wheret;i2Tandti.Toensurethatproductionineveryperioddoesnotexceedavailablecomponentsupply,werequire jTjXi=txtit;8t2T: (A{1) Infact,becausenocomponentinventoryholdingispermitted,itisstraightforwardtoshowthatanoptimalsolutionexistssuchthatweprocureanumberofcomponentsinaperiodequaltotheproductionamountintheperiod,i.e.,suchthat( A{1 )issatisedatequality.Thus,wecanformulateproblemSP1asfollows:SP1:MinimizeXt2TFtyt+Xt2TctjTjXi=txti+Xt2T1 t0@jTjXi=txti1A2 (A{2)Subjectto:iXt=1xti=di;8i2T; (A{3)jTjXi=txtiMyt;8i2T; (A{4)xti0;8t;i2Tandwhereti; (A{5)yt2f0;1g;8t2T; (A{6)wherect=ct+PjTji=thi+t+t=tandMcorrespondstoabig-Mvalue(alargepositivenumberthatwecansettoDt;T=PTi=tdiwithoutlossofoptimality).Here,constraintset( A{3 )correspondstothedemandsatisfactionrequirements.Constraintset( A{4 ) 125

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forcesyttoequaloneifanyproductionoccursinperiodt,while( A{5 )and( A{6 )providenonnegativityandbinaryrestrictions.NotingthatPjTji=txti=xtforallt2T,wedenethefunctionPt(xt)asthesumoftheproductionandprocurementcostsinperiodt,wherePt(xt)=8><>:Ft+ctxt+x2t t;ifxt>0;0;otherwise.WeshowthatproblemSP1isNP-HardusinganapproachsimilartotheoneusedbyFlorian,etal.[ 22 ]forcapacitatedlotsizing.Clearlyonecanevaluatethecostofanyfeasiblesolutioninpolynomialtime,whichimpliesthattherecognitionversionofproblemSP1isinNP.WecanshowthattheSUBSETSUMproblem,whichisknowntobeNP-Complete[ 23 ],isreducibletoaspecialcaseofSP1.SUBSETSUM:Givenpositiveintegersa1;:::;at,A,doesthereexistasubsetST=f1;:::;tgsuchthatPi2Sai=A?GivenanyinstanceofSUBSETSUM,wedenethefollowinginstanceofSP1.LetT=f1;:::;tgdenotethesetofperiodsandletdi=0;i=1;:::;t)]TJ /F1 11.955 Tf 11.96 0 Td[(1;dt=A;i=a2i;i2T;ci=1)]TJ /F1 11.955 Tf 14.99 8.09 Td[(2 ai;i2T;Fi=1;i2T:WeclaimthatSUBSETSUMhasasolutionifandonlyifthereexistsafeasiblesolutiontothecorrespondinginstanceofSP1withtotalcostofatmostA.ObservethatPi(0)=0andPi(ai)=aiforalli2T.WecanshowthatPi(xi)>xiifxi>0andxi6=aiand,infact,Pi(xi))]TJ /F3 11.955 Tf 12.1 0 Td[(xiattainsitsminimumatxi=ai.Wedenegi(xi)=Pi(xi))]TJ /F3 11.955 Tf 12.1 0 Td[(xiforxi>0, 126

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wheregi(xi)=1 a2ix2i+1)]TJ /F1 11.955 Tf 14.99 8.09 Td[(2 aixi+1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi=1 a2ix2i)]TJ /F1 11.955 Tf 14.98 8.09 Td[(2 aixi+1;andnotethatg0i(xi)=2 a2ixi)]TJ /F1 11.955 Tf 14.99 8.09 Td[(2 ai:Observethatgi(xi)isastrictlyconvexfunctionofxi,andg0i(xi)=0forxi=ai,whichistheminimizerofgi(xi).Infact,gi(ai)=0,whichshowsthatgi(xi)>0,i.e.,Pi(xi)>xiforanyxi6=ai,andxi>0.Foralli2T,wehavePi(xi)=xiifxi=0orxi=ai;Pi(xi)>xiotherwise:Hence,anyfeasibleproductionplanhastotalcostofatleastA,i.e.,Xi2TPi(xi)Xi2Txi=A:ThecostoftheproductionplanwillbeexactlyequaltoAifandonlyifxi2f0;aigforalli2T,i.e.,ifandonlyifthereexistsasubsetSTsuchthatPi2Sai=A.Observethatthecapacitatedlotsizingproblemcanbeviewedasalimitingcaseoftheproblemwithxedsetupcostsandpiecewiselinearandconvexvariablecostswithasinglebreakpointineachtimeperiod(wherethebreakpointoccursatthecapacitylevelineachperiod,andastheslopeofthelinearpiecetotherightofthebreakpointtendstoinnity).TheproofofNP-CompletenessbyFlorian,etal.[ 22 ]isthusequivalenttousingaproductioncostfunctiondenedby 127

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Pt(xt)=8><>:Ft+c1txt;if0at;withFt=1,c1t=(1)]TJ /F1 11.955 Tf 12.43 0 Td[(1=at),andanyc2t>1forallt2T(thevalueofatbecomesthecapacitylimitinperiodtasc2t!1). 128

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APPENDIXBPROOFOFPROPOSITION2.2Hereagain,wecanshowthatSUBSETSUMproblemisreducibletoaspecialcaseofP(FC).GivenanyinstanceofSUBSETSUMwithpositiveintegersa1;:::;at;A,supposewithoutlossofgeneralitythattheaivaluesaresortedinnonincreasingorderfori=1;:::;t.WecreateaninstanceofaspecialcaseofP(FC)asfollows.LetT=f1;:::;tgdenotethesetofplanningperiods,with(asinthepreviousproof)di=0;i=1;:::;t)]TJ /F1 11.955 Tf 12.2 0 Td[(1anddt=A.WeconsiderthespecialcaseofP(FC)suchthatFt=ht=hCt=0forallt2T.Byassumptionwehavet=,t=,andt=forallt2T.Inaddition,wesetci=1)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2ai ;i2T;FCi=a2i ;i2T:Notethatci=1)]TJ /F1 11.955 Tf 12.62 0 Td[(2ai=impliesci=1)]TJ /F1 11.955 Tf 12.62 0 Td[(2ai=)]TJ /F3 11.955 Tf 12.61 0 Td[(=)]TJ /F3 11.955 Tf 12.61 0 Td[((andwecanselectvaluesof,,andthatensureallci0).WeclaimthatSUBSETSUMhasasolutionifandonlyifthereexistsafeasiblesolutiontothisspecialcaseoftheproductionplanningproblemwithtotalcostofatmostA.Werstnotethatforthisspecialcase,anoptimalsolutionexistssuchthatanyunitprocuredinperiodiisusedinproductioninperiodi(thisresultsfromthefactthatallholdingcostsandxedproductioncostsequalzero,variableproductioncostsarenondecreasingintimeasaresultoftheaivalueshavingbeensortedinnonincreasingorder,andtheabovecivalues).Letxidenotetheprocurement(andproduction)amountinperiodianddenetheassociatedcostfunction~Pt(xt)=8><>:FCt+ctxt+x2t ;ifxt>0;0;otherwise.Observethat~Pi(0)=0and~Pi(ai)=aiforalli2T.Additionally,~Pi(xi)>xiforanyxi>0suchthatxi6=ai.FollowingthesamestepsasintheproofofProposition 2.1 ,wecanshowthatthecostofanyfeasiblesolutionequalsatleastA,andtheproductionplan 129

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costwillexactlyequalAifandonlyifxi2f0;aigforalli2T,i.e.,ifandonlyifthereexistsasubsetSTsuchthatPi2Sai=A. 130

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APPENDIXCPROOFOFPROPOSITION4.2TherstorderconditionyieldsEquation( C{1 ).Anypcsatisfyingthisequationisastationarypointof^(pc). c0ab=(b)]TJ /F1 11.955 Tf 11.95 0 Td[(1)apc)]TJ /F3 11.955 Tf 13.59 8.09 Td[(b 2p 2F0hapb=2c:(C{1)Observethatfor1(b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)bapc)]TJ /F3 11.955 Tf 13.59 8.09 Td[(b 2b 2+1p 2F0ha(pc)b=2:(C{2)Equation( C{1 )impliesthatb 2p 2F0ha(pc)b=2=(b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)apc)]TJ /F3 11.955 Tf 11.96 0 Td[(c0ab:Thus,wecansubstituteb 2p 2F0ha(pc)b=2intheinequality( C{2 ).Wethenhaveb 2+1((b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)apc)]TJ /F3 11.955 Tf 11.96 0 Td[(c0ab)>(b)]TJ /F1 11.955 Tf 11.95 0 Td[(1)bapc)]TJ /F1 11.955 Tf 11.96 0 Td[((b+1)c0ab;whichimplies1)]TJ /F3 11.955 Tf 13.59 8.09 Td[(b 2(b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)apc>)]TJ /F3 11.955 Tf 10.93 8.09 Td[(b 2c0ab:For12,therighthandsideof( C{1 )increasesinpcuntilthepointpxcdenedin( C{3 )andthendecreases.Thisimpliesthattherearetwosolutionsto( C{1 ),oneofwhichisintheinterval[0;pxc] 131

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andtheotherofwhichisin[pxc;1]. pxc= (b)]TJ /F1 11.955 Tf 11.95 0 Td[(1)a b2 4p 2F0ha!2 b)]TJ /F12 5.978 Tf 5.75 0 Td[(2:(C{3)Therighthandsideofinequality( C{2 )increasesinpcuntilthepointpxxcdenedin( C{4 )andthendecreases.Thisimpliesthat^(pc)isrstconcave,thenconvex,andthenconcaveagain. pxxc= (b)]TJ /F1 11.955 Tf 11.96 0 Td[(1)ba b2 4)]TJ /F5 7.97 Tf 6.99 -4.98 Td[(b 2+1p 2F0ha!2 b)]TJ /F12 5.978 Tf 5.76 0 Td[(2:(C{4)Notethatpxc2.Letpcdenotethesmallerstationarypoint,thatispc2[0;pxc].Thus,pcpxc2.Whenthemaximumpointpcisgreaterthanp0c,thentheoptimalchoiceofsellingpriceisp0c. poptc=minfpc;p0cg:(C{5) 132

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APPENDIXDPROOFOFPROPOSITION4.3Forpoptc,whentakingthestationarypointsolutionpc,wewillshowtheresultforF.Therestfollowssimilarly.Takingthepartialderivativeof@^(pc)=@pc=0withrespecttoF,weobtain @(@^(pc)=@pc) @F+@(@^(pc)=@pc) @pc@pc @F=0:(D{1)Therefore @pc @Fpc=pc=)]TJ /F5 7.97 Tf 14.88 14.48 Td[(@(@^(pc)=@pc) @F @2^(pc) @p2cpc=pc:(D{2)Notethatthedenominatoroftherighthandsideof( D{2 )isnegative,since^(pc)isstrictlyconcaveatpc.Andwehave @(@^(pc)=@pc) @F=b 4r 2ha F1 k(pc))]TJ /F5 7.97 Tf 6.58 0 Td[(b=2)]TJ /F7 7.97 Tf 6.59 0 Td[(1;(D{3)whichisclearlypositive.Thus,@pc @Fispositive,whichshowstheresult.Whenpc>p0c,theoptimalsellingpricepoptcequalsp0c,whichonlydependson^pandp0sforaxedk.Notethatp0c=(k^p)]TJ /F3 11.955 Tf 11.35 0 Td[(p0s)=(k)]TJ /F1 11.955 Tf 11.34 0 Td[(1),@p0c=@^p=k=(k)]TJ /F1 11.955 Tf 11.35 0 Td[(1)and@p0c=@p0s=)]TJ /F1 11.955 Tf 9.3 0 Td[(1=(k)]TJ /F1 11.955 Tf 11.34 0 Td[(1).Thisshowsthedesiredresult. 133

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APPENDIXEPROOFOFPROPOSITION4.4Whenpoptc=p0c,theoptimalchoiceforthesupplypriceequalsp0swhichisaconstantdeterminedbythemarket.Wedenotetheoptimalsupplypricewhenpoptc=pcasps.Notethatps=k^p)]TJ /F1 11.955 Tf 11.95 0 Td[((k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)pc.Wehave @ps @^p=k)]TJ /F1 11.955 Tf 11.96 0 Td[((k)]TJ /F1 11.955 Tf 11.95 0 Td[(1)@pc @^p:(E{1)ByProposition 4.2 ,weknowthat@pc=@^p>0.Wecanalsoshowthat@2pc=@^p2<0,whichimpliesthat@2ps=@^p2>0,because@2ps=@^p2=)]TJ /F1 11.955 Tf 9.3 0 Td[((k)]TJ /F1 11.955 Tf 12.19 0 Td[(1)@2pc=@^p2.Thereforepsisconvexin^pandas^pincreases,itdecreasesuntilthe^pvalueisreachedwherek=(k)]TJ /F1 11.955 Tf 12.04 0 Td[(1)=@pc=@^p.Afterthispoint,psstartsincreasingagain.Fortheotherparametersa,F,candh,psbehavesintheoppositedirectionaspc.Thisisbecauseforanyoftheseparameters(e.g.,forx),wehave@ps=@x=)]TJ /F1 11.955 Tf 9.3 0 Td[((k)]TJ /F1 11.955 Tf -434.15 -23.91 Td[(1)(@pc=@x). 134

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APPENDIXFPROOFOFPROPOSITION4.5Theoptimalbatchsizecanbecomputedgivenpoptc,whereQopt=p 2Fa=h(poptc))]TJ /F5 7.97 Tf 6.59 0 Td[(b=2.Firstwewillshowtheresultwhenpoptc=pc.Toshowtheresultfora,wetakethederivativeofQoptwithrespecttoa:dQopt da=@Qopt @a+@Qopt @pc@pc @a:FromProposition 4.2 weknowthattheoptimalsellingpricepcisdecreasingina,hence,@pc=@aisnegative.Wehave@Qopt=@a=(1=2)q 2F ha(pc))]TJ /F5 7.97 Tf 6.58 0 Td[(b=2,whichispositive,and@Qopt=@pc=)]TJ /F5 7.97 Tf 10.8 4.71 Td[(b 2q 2Fa h(pc))]TJ /F5 7.97 Tf 6.58 0 Td[(b=2)]TJ /F7 7.97 Tf 6.59 0 Td[(1,whichisnegative.Therefore,dQopt=da>0.Theproofoftheresultsforotherparameters(h,c,and^p)followssimilarly,exceptforF.AvalueofF,F,existswheredQopt=dF=0,anddQopt=dF>0forFFanddQ=dF<0forF0.Theresultsfortheparametersa,F,h,and^pfollowsimilarly. 135

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APPENDIXGPROOFOFPROPOSITION4.6Whentheoptimalsellingpriceequalsp0c,thenetrevenueperunitonlydependsonlyon^p,p0s,andcforxedk.Inthiscase,wehavedopt d^p=k k)]TJ /F7 7.97 Tf 6.59 0 Td[(1,dopt dp0s=)]TJ /F5 7.97 Tf 15.9 4.71 Td[(k k)]TJ /F7 7.97 Tf 6.58 0 Td[(1,anddopt dc=)]TJ /F1 11.955 Tf 9.3 0 Td[(1,hencetheresultfollows.Whentheoptimalsellingpriceequalspc,optbehavesthesamewayaspcwithrespecttotheparametersa,F,andh.1Thederivativewithrespectto^pisasfollows: dopt d^p=k@pc @^p)]TJ /F3 11.955 Tf 11.96 0 Td[(k:(G{1)If@pc=@^p>1forany^p,thenwehavetheresult.Thisconditionissatisedwhen pc>b2 2(c=k+^p) b2 2)]TJ /F5 7.97 Tf 13.45 4.7 Td[(b 2+1:(G{2)pcmustbegreaterthanc=k+^pinordertoobtainpositiveprot.Themultipliertermofc=k+^p,0:5b2=(0:5b2)]TJ /F1 11.955 Tf 11.28 0 Td[(0:5b+1),islessthanorequalto1when12,thetermisslightlygreaterthan1.Ittakesitsmaximumvalue(8/7)whenb=4anditapproaches1asbincreases.Theinequality( G{1 )islikelytoholdforb>2,sincethechosensellingpricemustalsoaccountfortheoperationscosts.Similarly,if@pc=@c>1=kforanyc,wehavetheresultforc.Theconditionsdiscussedaboveholdforcaswell. 1Forexample,dopt=da=k(dpc=da).ByProposition 4.2 ,weknowthatdpc=da<0,thereforedopt=da<0,aswell. 136

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APPENDIXHPROOFOFPROPOSITION4.7Whenpoptcequalspc,wecandeterminethederivativewithrespecttokasfollows. @pc @kpc=pc=)]TJ /F5 7.97 Tf 14.88 14.48 Td[(@(@^(pc)=@pc) @k @2^(pc) @p2cpc=pc:(H{1)Notingthatthesecondderivativeof^(pc)atpcisnegativeand @(@^(pc)=@pc) @kpc=pc=)]TJ /F3 11.955 Tf 10.49 8.08 Td[(cab(pc))]TJ /F5 7.97 Tf 6.58 0 Td[(b)]TJ /F7 7.97 Tf 6.58 0 Td[(1 k2)]TJ /F1 11.955 Tf 15.83 8.08 Td[(1 k2b 2p 2Fha(pc))]TJ /F5 7.97 Tf 6.59 0 Td[(b=2)]TJ /F7 7.97 Tf 6.59 0 Td[(1<0;(H{2)andwecanconcludethatpcisdecreasingink.Whenpoptc=p0c,wehave dp0c dk=)]TJ /F1 11.955 Tf 16.33 8.09 Td[(^p)]TJ /F3 11.955 Tf 11.96 0 Td[(p0s (k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)2:(H{3)Since^p>p0s,dp0c dk<0,hencep0cisalsodecreasingink.Notealsothatp0capproaches^pask!1.Whenpoptc=p0c,popts=p0s,andp0sisaconstant.Forpoptc=pc,wehave dpopts dk=^p)]TJ /F3 11.955 Tf 11.96 0 Td[(pc)]TJ /F1 11.955 Tf 11.96 0 Td[((k)]TJ /F1 11.955 Tf 11.95 0 Td[(1)@pc @k:(H{4)Since@pc @k<0,itisnotclearwhetherdpopts dkispositiveornegative.Forb=2,wecanusetheclosedformexpressionforpctoshowthatpoptsisdecreasing1.ForQopt,wehave dQopt dk=r 2Fa h)]TJ /F3 11.955 Tf 10.93 8.09 Td[(b 2(poptc))]TJ /F5 7.97 Tf 6.58 0 Td[(b=2)]TJ /F7 7.97 Tf 6.58 0 Td[(1@poptc @k:(H{5)Wehaveseenthat@poptc @k<0,whichshowsdQopt dk>0. 1Theproofrequiresthatp 2Fh=a<1.p 2Fh=aistheunitoperationscostwhenthedemandrateequalsa.Thisconditionisalsorequiredforthelimitingcasewherek=1tohaveanonnegativeoptimalsellingprice. 137

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Whenpoptc=p0c,opt=p0c)]TJ /F3 11.955 Tf 12.45 0 Td[(p0s)]TJ /F3 11.955 Tf 12.45 0 Td[(c,andforthiscaseoptisclearlydecreasingink.Whenpoptc=pc,opt=kpc)]TJ /F3 11.955 Tf 11.95 0 Td[(k^p)]TJ /F3 11.955 Tf 11.96 0 Td[(c.Wehave dopt dk=pc)]TJ /F1 11.955 Tf 12.94 0 Td[(^p+k@pc @k:(H{6)Since@pc @k<0,itisnotclearwhetherdopt dkispositiveornegative.Wecanshowthatoptisincreasinginkforb=2whenthefollowingconditionholds: a^p(k2)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F1 11.955 Tf 11.95 0 Td[(2k)>2c;(H{7)where=p 2Fh=a,andisassumedtobelessthan1.Thisimpliesthatoptisincreasinginkfor k>~k:=2^pp 2Fha+p 4^p(2Fha)+4a^p(2c+^p2Fh) 2a^p:(H{8) 138

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APPENDIXIPROOFOFPROPOSITION4.8Weneedtoshowthatd(poptc)=dk>0.Wehave d(poptc) dk=@(poptc) @k+@(poptc) @poptc@poptc @k:(I{1)Notethatpoptc=minfpc;p0cg.Thepartialderivativeoftheoptimalprotwithrespecttokis @(poptc) @k=(poptc)]TJ /F1 11.955 Tf 12.94 0 Td[(^p)a(poptc))]TJ /F5 7.97 Tf 6.59 0 Td[(b:(I{2)Sincepoptc^p,@(poptc)=@kisalwaysnonnegative.Whenpoptc=pc,@(poptc)=@poptciszero.Therefore,theresultholdswhenthestationarypointsolutionpcisoptimal.Whenpoptc=p0c,wehave @p0c @k=)]TJ /F1 11.955 Tf 16.33 8.09 Td[(^p)]TJ /F3 11.955 Tf 11.95 0 Td[(p0s (k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)2;(I{3)andweknowthat@(p0c) @p0c>0,sincep0c@(p0c) @p0c.Forb=2,thisconditionissatisedforanykvaluewhen 1)]TJ /F9 11.955 Tf 11.96 19.09 Td[(r 2Fh a>2p0s+c ^p:(I{5)If( I{5 )isnotsatisedthenthereexistsakvalue,k0,suchthatd(p0c) dk=0,d(p0c) dk>0forkk0.k0iscomputedasfollows: k0=(1+p 2Fh=a)p0s+2c 2(p0s+c))]TJ /F1 11.955 Tf 11.96 0 Td[((1)]TJ /F9 11.955 Tf 11.96 10.22 Td[(p 2Fh=a)^p:(I{6) 139

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APPENDIXJFIRSTORDERNECESSARYOPTIMALITYCONDITIONSFOR(C;P)Inthisappendix,wepresentapartoftheanalysisforthecasewherethedemandisalsoprice-dependentanditfollowsanadditivemodelasdescribedinSection 5.3.1 .Herewepresenttheprotfunctionasafunctionofthesupplypricecandthesellingpricep.Formathematicalconvenience,wedenez(c;p)=Q(c))]TJ /F3 11.955 Tf 12.4 0 Td[(y(p).Afternecessarysubstitutions,theprotfunctioncanbereformulatedasfollows. (c;p;)=8><>:p(y(p)+z(c;p)))]TJ /F3 11.955 Tf 11.96 0 Td[(g()]TJ /F3 11.955 Tf 11.95 0 Td[(z(c;p)))]TJ /F1 11.955 Tf 11.96 0 Td[((c+v)(y(p)+z(c;p));if>z(c;p);p(y(p)+)+s(z(c;p))]TJ /F3 11.955 Tf 11.96 0 Td[())]TJ /F1 11.955 Tf 11.96 0 Td[((c+v)(y(p)+z(c;p));if
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@2(c;p) @c2=(p+g)]TJ /F3 11.955 Tf 9.46 0 Td[(s)Q00(c)F(z(c;p)))]TJ /F1 11.955 Tf 11.96 0 Td[((Q0(c))2f(z(c;p)))]TJ /F1 11.955 Tf 9.46 0 Td[((c+v)]TJ /F3 11.955 Tf 9.46 0 Td[(s)Q00(c))]TJ /F1 11.955 Tf 9.46 0 Td[(2Q0(c):(J{8)Notethattheconcavityof(c;p)dependsontherelationshipbetweenthesupplyquantityandthepriceoeredtosuppliersaswellasthedistributionoftherandomvariable.However,bytherstordernecessaryoptimalitycondition,weknowthatifcmaximizestheexpectedprot,itmustsatisfythefollowingconditionforanygivenp. F(z(c;p))=p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(c)]TJ /F3 11.955 Tf 11.96 0 Td[(v)]TJ /F5 7.97 Tf 14.49 5.7 Td[(Q(c) Q0(c) p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s:(J{9)Nextconsidertherstandsecondderivativesof(c;p)withrespecttop. @(c;p) @p=y(p)+)]TJ /F3 11.955 Tf 11.96 0 Td[(gy0(p)+(p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s)y0(p)F(z(c;p)))]TJ /F1 11.955 Tf 11.95 0 Td[((c;p):(J{10) @2(c;p) @p2=(Q0(c))]TJ /F3 11.955 Tf 11.95 0 Td[(b)F(z(c;p)))]TJ /F3 11.955 Tf 11.95 0 Td[(b2(p+g)]TJ /F3 11.955 Tf 11.96 0 Td[(s)f(z(c;p)):(J{11)From( J{11 )wecanobservethattheconcavityof(c;p)withrespecttopdependsonthedistributionoftherandomvariableandtothesupply-pricerelationship.However,bytherstordernecessaryoptimalitycondition,weknowthatifpmaximizestheexpectedprot,itmustsatisfythefollowingconditionforanygivenc. F(z(c;p))=y(p)+)]TJ /F7 7.97 Tf 6.59 0 Td[((c;p) b+g p+g)]TJ /F3 11.955 Tf 11.95 0 Td[(s:(J{12)Thus,ifweobtaintheoptimalsellingpricepandtheoptimalsupplypricec,bothconditions( J{9 )and( J{12 )mustbesatised.Thatimpliesthatatoptimality,wehave p)]TJ /F3 11.955 Tf 11.95 0 Td[(c)]TJ /F3 11.955 Tf 11.95 0 Td[(v)]TJ /F3 11.955 Tf 14.55 8.09 Td[(Q(c) Q0(c)=y(p)+)]TJ /F1 11.955 Tf 11.96 0 Td[((c;p) b:(J{13) 141

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BIOGRAPHICALSKETCHZehraMelisTeksanwasbornin1986inIstanbul,Turkey.Aftershereceivedherhighschooldegreein2005fromGermanHighSchoolIstanbul(DeutscheSchuleIstanbul),shestartedherundergraduatestudiesintheIndustrialEngineeringDepartmentatBogaziciUniversity.ShereceivedherB.S.andM.S.degreesinthesamedepartmentin2009and2011,respectively.Duringhermaster'sstudies,shealsoworkedinseveralpositionsforICRONTechnologies,includingseniorconsultant,projectmanager,andR&Dsoftwareproductdeveloper.InAugust2011,shejoinedthePh.D.programintheIndustrialandSystemsEngineeringDepartmentattheUniversityofFlorida.Hermainresearchfocusliesintheeldofproductionplanningandinventorytheory,and,ingeneral,sheisinterestedinresearchproblemsthatarerelevanttoreal-lifeindustrypractice.ShereceivedherPh.D.degreeinSpring2016. 149