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Operations Management and Marketing Interface

Permanent Link: http://ufdc.ufl.edu/UFE0024879/00001

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Title: Operations Management and Marketing Interface Making Supply Chain Decisions under Various Marketing Strategies
Physical Description: 1 online resource (115 p.)
Language: english
Creator: Seref, Michelle
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: generations, inventory, marketing, operations, pricing, timing
Information Systems and Operations Management -- Dissertations, Academic -- UF
Genre: Business Administration thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation focuses on supply chain management (SCM) decisions under various marketing strategies in the Operations Management (OM) / Marketing interface research area. It is composed of three primary research chapters. The first research chapter examines optimal inventory and pricing decisions under advance selling. Advance selling is a marketing strategy in which consumers have a chance to reserve a product in an advance sales period which occurs prior to the sales period. The retailer in this scenario must make an inventory order decision before the advance sales period begins to best meet demand in both the advance sales and consumption periods. I derive optimal inventory and pricing policies. The second research chapter focuses on optimal pricing and time-to-market decisions in a new product technology (NPD) environment. I consider two generations of a new technology product considering both price and diffusion effects on sales. I derive optimal pricing and time-to-market decisions for three different sales functions. The final research chapter considers the innovation speed of new technologies in a pricing and time-to-market model. I determine the optimal number of generations to offer of a new product in this scenario. All three research chapters contribute to the OM/Marketing research literature by solving business problems from a combined OM and Marketing perspective.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Michelle Seref.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Carrillo, Janice E.
Local: Co-adviser: Erenguc, Sahin S.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024879:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024879/00001

Material Information

Title: Operations Management and Marketing Interface Making Supply Chain Decisions under Various Marketing Strategies
Physical Description: 1 online resource (115 p.)
Language: english
Creator: Seref, Michelle
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: generations, inventory, marketing, operations, pricing, timing
Information Systems and Operations Management -- Dissertations, Academic -- UF
Genre: Business Administration thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation focuses on supply chain management (SCM) decisions under various marketing strategies in the Operations Management (OM) / Marketing interface research area. It is composed of three primary research chapters. The first research chapter examines optimal inventory and pricing decisions under advance selling. Advance selling is a marketing strategy in which consumers have a chance to reserve a product in an advance sales period which occurs prior to the sales period. The retailer in this scenario must make an inventory order decision before the advance sales period begins to best meet demand in both the advance sales and consumption periods. I derive optimal inventory and pricing policies. The second research chapter focuses on optimal pricing and time-to-market decisions in a new product technology (NPD) environment. I consider two generations of a new technology product considering both price and diffusion effects on sales. I derive optimal pricing and time-to-market decisions for three different sales functions. The final research chapter considers the innovation speed of new technologies in a pricing and time-to-market model. I determine the optimal number of generations to offer of a new product in this scenario. All three research chapters contribute to the OM/Marketing research literature by solving business problems from a combined OM and Marketing perspective.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Michelle Seref.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Carrillo, Janice E.
Local: Co-adviser: Erenguc, Sahin S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024879:00001


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OPERATIONS MANA GEMENTANDMARKETINGINTERFACE: MAKINGSUPPLYCHAINDECISIONSUNDERVARIOUSMARKETING STRATEGIES By MICHELLEM.H.SEREF ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c 2009 Mic helle M.H.Seref 2

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T o m yparents,Dr.MagdiandRoblynHanna,fortheircontinuoussupportand encouragementtoaimhighandworkhard,andto myhusband,Dr.OnurSeref,atruescientist,whoseloveismygreatestsupportandwhose exampleismygreatestmotivation 3

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ACKNO WLEDGMENTS I wouldliketothankmythesiscommitteefortheirhelpinthisresearch.Iwould especiallyliketothankDr.AydnAlptekinogluforhishelpintheresearchchapter InventoryManagementUnderAdvanceSelling:OptimalOrderandPricingPolicies ,Dr. SelcukErengucforhishelpinallofmyresearchandforhiscareerguidance,andDr. JaniceCarrilloforhercollaborationintheresearchchapters Multi-GenerationPricing andTimingDecisionsinNewProductDevelopment and OptimalNumberofGenerations foraMulti-GenerationPricingandTimingModelinNewProductDevelopment andfor allofheradviceineverysubject.IalsoappreciatethevaluablefeedbackofDrs.Anand PaulandJosephGeunesinthisdissertationwork.Finally,IwouldliketothanktheISOM DepartmentchairDr.AsooVakhariaforhissupportthroughoutmyworkinthePh.D. program. 4

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TABLE OF CONTENTS page A CKNO WLEDGMENTS ................................4 LISTOFTABLES .....................................7 LISTOFFIGURES ....................................8 ABSTRACT ........................................9 CHAPTER 1INTRODUCTION ..................................10 1.1OperationsManagement(OM)/MarketingInterfaceResearch .......10 1.2InventoryManagementunderAdvanceSelling:OptimalOrderandPricing Policies ......................................10 1.3Multi-GenerationPricingandTimingDecisionsinNewProductDevelopment 11 1.4OptimalNumberofGenerationsforaMulti-GenerationPricingandTiming ModelinNewProductDevelopment .....................11 1.5OverviewoftheDissertation ..........................12 2LITERATUREREVIEW ..............................13 2.1OM/MarketingInterface ............................13 2.2InventoryManagementunderAdvanceSelling:OptimalOrderandPricing Policies ......................................13 2.3Multi-GenerationPricingandTimingDecisionsinNewProductDevelopment 17 2.4OptimalNumberofGenerationsforaMulti-GenerationPricingandTiming ModelinNewProductDevelopment .....................19 3INVENTORYMANAGEMENTUNDERADVANCESELLING:OPTIMAL ORDERANDPRICINGPOLICIES ........................21 3.1Introduction ...................................21 3.2Model ......................................23 3.3Analysis .....................................30 3.3.1OptimalOrderQuantity Q (X a ) ....................30 3.3.2OptimalAdvanceSalesInventory X a (Q) ...............31 3.3.3OptimalOrderPolicy( Q ;X a ) .....................33 3.3.4OptimalPricingStrategy( p a ;p s ) ...................35 3.4NumericalExperiments .............................36 3.5AnExtension ..................................40 4MULTI-GENERATIONPRICINGANDTIMINGDECISIONSINNEWPRODUCT DEVELOPMENT ..................................46 4.1IntroductionandMotivation ..........................46 5

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4.2Mo del .....................................47 4.3Analysis .....................................51 4.3.1CASE1:PriceEectOnly,NoDiusionEect ............52 4.3.2CASE2:DiusionEectOnly,NoPriceEect ............56 4.3.2.1Timehorizonthreshold ....................57 4.3.2.2Abenchmarkscenario ....................59 4.3.3CASE3:PriceandDiusionEects ..................60 4.4NumericalAnalysis ...............................62 4.4.0.1Cumulativesalesandsalesrate ...............63 4.4.0.2Sensitivityanalysis ......................66 5OPTIMALNUMBEROFGENERATIONSFORAMULTI-GENERATION PRICINGANDTIMINGMODELINNEWPRODUCTDEVELOPMENT ..71 5.1Introduction ...................................71 5.2Model ......................................71 5.3Analysis .....................................74 5.4NumericalExperiments .............................77 6CONCLUSIONS ...................................85 6.1InventoryManagementunderAdvanceSelling:OptimalOrderandPricing Policies ......................................85 6.2Multi-GenerationPricingandTimingDecisionsinNewProductDevelopment 87 6.3OptimalNumberofGenerationsforaMulti-GenerationPricingandTiming ModelinNewProductDevelopment .....................88 6.4OM/MarketingInterface ............................89 APPENDIX APROOFSFORCHAPTER3:INVENTORYMANAGEMENTUNDERADVANCE SELLING:OPTIMALORDERANDPRICINGPOLICIES ...........91 BPROOFSFORCHAPTER4:MULTI-GENERATIONPRICINGANDTIMING DECISIONSINNEWPRODUCTDEVELOPMENT ...............100 CPROOFSFORCHAPTER5:OPTIMALNUMBEROFGENERATIONSFOR AMULTI-GENERATIONPRICINGANDTIMINGMODELINNEWPRODUCT DEVELOPMENT ..................................107 REFERENCES .......................................112 BIOGRAPHICALSKETCH ................................115 6

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LISTOF T ABLES T able page 3-1 Sensitivit y AnalysisforVarying Values ......................37 3-2SensitivityAnalysisforVarying H L SpreadValues ...............39 3-3SensitivityComparisonof vs. H L SpreadValues ...............40 3-4SensitivityAnalysisforVarying p s Values(V Uniform (20 ; 80)) ........43 3-5SensitivityAnalysisforVarying H L SpreadValuesforUniformandBernoulli CustomerValuations .................................45 4-1SensitivityanalysisresultsforCASE2. .......................67 4-2SensitivityanalysisresultsforCASE3. .......................67 5-1SummaryofSensitivityAnalysisResults. ......................81 6-1SummaryofSensitivityAnalysisTrials .......................86 7

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LISTOF FIGURES Figure page 3-1 Timeline ......................................24 3-2Graphsof X a vs. E []forVarying Values ....................38 3-3Graphsof X a vs. E []forVarying H L SpreadValues .............39 3-4Graphof X a vs. E []( V Uniform (20 ; 80)) ...................43 4-1Evaluating 3 (t m ) c t m when K 2 >K 1 andcostsareconstant. .........54 4-2Theswitchingbehaviorof 3 (t m )as T increases. ..................57 4-3CASE2:Behaviorofcumulativesales x 1 and x 2 overtime. ............64 4-4CASE2:Behaviorofsalesrates f (x 1 )and g (x 2 )overtime. ............64 4-5CASE3:Behaviorofcumulativesales x 1 and x 2 overtime. ............65 4-6CASE3:Behaviorofsalesrates f (x 1 ;p 1 )and g (x 2 ;p 2 )overtime. ........65 4-7CASE3:Behaviorofoptimalprice p 1 and p 2 overtime. ..............66 5-1Protasafunctionof n ...............................78 5-2OptimalPrice p (t),overtime. ...........................79 5-3PriceComponentandDiusionComponentofSalesRate,overtime. ......79 5-4SalesRate f (p;x ),overtime. ............................80 5-5IncreasingShiftinPriceCurve: p (t)vs. c t m ...................82 5-6IncreaseinPriceVariation: p (t )vs. a 2 ......................84 8

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AbstractofDissertationPresentedto the GraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy OPERATIONSMANAGEMENTANDMARKETINGINTERFACE: MAKINGSUPPLYCHAINDECISIONSUNDERVARIOUSMARKETING STRATEGIES By MichelleM.H.Seref August2009 Chair:JaniceCarrillo CoChair:S.SelcukErenguc Major: InformationSystemsandOperationsManagement Thisdissertationfocusesonsupplychainmanagement(SCM)decisionsundervarious marketingstrategiesintheOperationsManagement(OM)/Marketinginterfaceresearch area.Itiscomposedofthreeprimaryresearchchapters.Therstresearchchapter examinesoptimalinventoryandpricingdecisionsunderadvanceselling.Advancesellingis amarketingstrategyinwhichconsumershaveachancetoreserveaproductinanadvance salesperiodwhichoccurspriortothesalesperiod.Theretailerinthisscenariomustmake aninventoryorderdecisionbeforetheadvancesalesperiodbeginstobestmeetdemandin boththeadvancesalesandconsumptionperiods.Ideriveoptimalinventoryandpricing policies.Thesecondresearchchapterfocusesonoptimalpricingandtime-to-market decisionsinanewproducttechnology(NPD)environment.Iconsidertwogenerationsof anewtechnologyproductconsideringbothpriceanddiusioneectsonsales.Iderive optimalpricingandtime-to-marketdecisionsforthreedierentsalesfunctions.Thenal researchchapterconsiderstheinnovationspeedofnewtechnologiesinapricingand time-to-marketmodel.Ideterminetheoptimalnumberofgenerationstooerofanew productinthisscenario.AllthreeresearchchapterscontributetotheOM/Marketing researchliteraturebysolvingbusinessproblemsfromacombinedOMandMarketing perspective. 9

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CHAPTER1 INTR ODUCTION In thischapter,Idenethegeneralareaofresearchforthedissertationandintroduce thevariousresearchchapters. 1.1OperationsManagement(OM)/MarketingInterfaceResearch Thefocusofthisdissertationworkisonmakingjointoperationsandmarketing decisionsforstandardsupplychainproblemsinvariousmarketingsettings.The importanceofOM/Marketinginterfaceworkhasbeenhighlightedinrecentliteraturein boththeOMandMarketingelds.Thisnewacademicperspectivehasbeenmotivatedby problemsarisinginindustrywhenOMandMarketingdepartmentsdiscovertheirseparate goalsmaybeconictingandhavenegativeeectsonthecompanyasawhole.Thus OM/MarketinginterfaceresearchstrivestosimultaneouslyconsiderOMandMarketing decisionsandobjectivesinordertomaximizebenetsfortheentirerm. 1.2InventoryManagementunderAdvanceSelling:OptimalOrderand PricingPolicies Advancesellingisamarketingstrategyinwhichanadvancesalesperiodprecedes thespotperiod,andcustomersareuncertainabouttheirfutureproductvaluation.I consideraninventorymanagementdecision,inwhichIdecideaninventoryorderquantity aswellasaportionofthisinventorytoreserveforadvancesales.Iuseanexpectedprot maximizationmodeltondtheoptimalorderandpricingpolicies. Indtheoptimaladvancesalesinventoryleveltobeanextremepointsolution.This leadstoa"go/no-go"advancesalesdecisioninwhichIeitherspotsell(sellinthespot period)toallcustomersandadvanceselltonooneoradvancesell(sellintheadvance salesperiod)toalmostallcustomers.Ideriveseveralanalyticalresultsandperform numericalexperimentswhichexaminethebehavioroftheoptimalorderpolicyand providesensitivityanalysisonthemodelparameters. 10

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1.3Multi-Generation Pricing andTimingDecisionsinNewProduct Development Whenplanningfortheintroductionofastreamofnewproductsintothemarketplace, managersmustconsiderboththetiminganddynamicpricingdecisionstodetermine anappropriateentrystrategyintothemarketplace.Literatureinthenewproduct development(NPD)areahasaddressedoptimaltimingofmultiplegenerationsofproducts andthedynamicpricingdecisionsindependently.However,noanalyticresultshavebeen developedwhenthesedecisionsareconsideredsimultaneously. InChapter4,Idevelopananalyticalmodelofcoordinatedproductintroductionand pricingdecisionswhentherearetwogenerationsofanewproductunderconsideration. Factorsdrivingthedecisionsincludetheunitsalesandcostrelationshipsforeach generationaswellasNPDcostsforintroducingthenextgenerationofproducts.I deriveanalyticresultsthatcharacterizetheoptimaltimingandpricingstrategies.In addition,Iidentifyanoptimalthresholdvalueforthelengthoftheplanninghorizonwhich dictatesthenewproductintroductionstrategy.Furtherinsightsareobtainedforaspecial caseofthemodelwherethetwogenerationsofproductshavesimilarsalesandpricing characteristics. 1.4OptimalNumberofGenerationsforaMulti-GenerationPricingand TimingModelinNewProductDevelopment InChapter5,Iseektondtheoptimalnumberofgenerationstointroduceunder thepricingandtime-to-marketNPDmodeldevelopedinChapter4.InChapter5,I consideramulti-generationnewtechnologyproductwithsalesasafunctionofboth priceanddiusion.Inadditiontosolvingfortheoptimalpricingpolicyandoptimal timetomarketforeachgeneration,Indtheoptimalnumberofgenerationsthata rmshouldintroduce.Thenewproductdevelopment(NPD)literaturehasaddressed thetime-to-marketdecisionformulti-generationproductsandsimultaneousoptimal pricingpolicieshaverecentlybeenintroducedtotheOperationsManagement(OM)/ MarketingInterfaceliterature.However,verylittleworkhasbeendoneontheoptimal 11

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num b erofgenerationstointroduce.Incomparisonwithinnovationspeed(orclockspeed) papers,Iconsideraspecicadditivemodelofsaleswithbothpricinganddiusioneects andsimultaneouslysolveforoptimalpricing,timing,andnumberofgenerationsfora maximumprotobjective.Theanalyticsemployoptimalcontroltheory.Anextensive numericalanalysisisalsoperformed. 1.5OverviewoftheDissertation Theremainderofthedissertationisorganizedasfollows.InChapter2,Ipresenta detailedreviewoftheliterature.Idiscusspaperswhichpertaintotheimportanceand scopeofOM/MarketingInterfacework,aswellaspapersrelatedtotheresearchdonein eachofthedissertationchapters.Thenextthreechapterscomprisethemainresearch ofthedissertation.InChapter3,Idiscusstheworktitled\InventoryManagement UnderAdvanceSelling:OptimalOrderandPricingPolicies",inChapter4,Ipresent \Multi-GenerationPricingandTimingDecisionsinNewProductDevelopment", andinChapter5,Ipresenttheresearchfor\OptimalNumberofGenerationsfora Multi-GenerationPricingandTimingModelinNewProductDevelopment".Ineachof theseresearchchapters,Iintroducetheproblem,describethemodelandassumptions, performtheanalysis,andreviewnumericalexperiments.Thenalchapterofthe dissertation,Chapter6,providestheconclusionsforeachoftheseresearchchapters anddiscussesfutureresearchextensionsfortheseworksaswellasinthegeneral OM/MarketingInterfaceresearcharea. 12

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CHAPTER2 LITERA TURE REVIEW Inthischapter,Ipresentareviewoftheliteraturerelatedtotheresearchineachof themainchapters.IalsodiscusstheliteraturepertainingtothegeneralOM/Marketing Interfaceresearcharea. 2.1OM/MarketingInterface SeveralpapersintherecentliteratureofboththeOMandMarketingeldshave highlightedtheimportanceofinterfaceresearch.SomesurveypaperssuchasKarmarkar (1996)[ 1]andBalasubramanian,et.al.(2004)[ 2]aswellassomeempiricalworksuch asHausman,et.al.(2002)[ 3]examineproblemscausedbymiscommunicationbetween businessdepartmentsanddisjointobjectivesinthesupplychain.Theyshowhowa marketingpushforhigherpricesandquickerturnovercanhaveanegativeinuence onoperationscostandproductionrequirements.Likewiseoperationsprioritiessuchas lowercostsandotherinventoryandproductionmotivationscanhurtmarketinggoals. Theacademiccommunityhasnowbeenchallengedbyindustrytohelpcoordinatethese multi-disciplinarygoalsbyresearchingjointdecisionproblems.Intheextensivework ofEliashberg,et.al.(1993)[4],thesenewmarketing-productionjointdecisionproblems arediscussedasthisnewoptimizationareacontinuestoexpand.Below,Irefertomore specicworkscorrespondingtoeachoftheresearchchaptersinthisdissertation. 2.2InventoryManagementunderAdvanceSelling:OptimalOrderand PricingPolicies Theadvancesalesstrategyismostclearlydenedinthemarketingliterature.There ishoweveralsorelatedliteratureintherevenuemanagementandinventoryresearch areas.Inthissection,Idiscusstherelativeliteratureforadvancesellingandoperations managementdecisionmaking. XieandShugan(2001)[ 5]writeacomprehensivepaperdescribingtheideaand benetsbehindadvanceselling.Theyshowthatadvancedsellingisprotableinthe generalscenarioofunknownfutureconsumervaluation.Theirpaperassumesthatthe 13

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consumer'sv aluation functionisBernoulliwithhigh, H ,orlow, L,valuesoccurringwith probability and(1 ),respectively.Theyalsoassumethatadeterministic N customers arriveinboththeadvancesalesandspotsalesperiod.Theystatethatifthemarginalcost ofoeringadvancedsellingislowtomedium,thenadvancedsellingshouldbeoeredand theadvancesalespriceshouldbesettotheexpectedprice H +(1 )L andthespot priceshouldbesetto H .Theythendescribeoptimaladvancesellingstrategiesforvarious capacityscenarios.Theyshowthatifcapacityislimited,butlarge,thentheadvance salespricecanbesettoapremiumandthespotpriceshouldbesetto L.Inanother limitedcapacityscenariowithmediumcapacity,theadvancesalespriceshouldbesetto theexpectedprice H +(1 )L andthespotpriceshouldbesetto H .Inthethird limitedcapacityscenariowithsmallcapacity,thereshouldbelimitedadvancedsaleswith theadvancesalespricesettotheexpectedprice H +(1 )L andthespotpricesetto H .Themodelingtechniquesusedintheirpaperincludedynamicprogrammingtondthe optimalspotprice,followedbytheoptimaladvancesalesprice.Iadvancetheirmodeling assumptionsbyconsideringamorerealisticdescriptionofcustomerarrivalsanddynamic customervaluations. ShuganandXie(2004)[ 6]writeanotherpaperdescribingtheadvancesellingin serviceindustries.ThispaperisquitesimilartoShuganandXie(2001),butitusesan exponentialcustomervaluationfunctioninsteadoftheBernoullidistribution.Italso givesacomparisonofadvancedsalestoyieldmanagementsystems(YMS).Itexplains thatYMSarelimitedbecauseitrequiresbindingcapacityconstraints,verylowmarginal costofadditionalsales,andaninverserelationshipbetweenconsumerpricesensitivity andcustomerarrivaltime.Theirpaperalsoreviewsvariousscenariosinwhichadvanced salesisbenecial,againincludinglimitedcapacity(limitedadvancedsalesandpremium advancedsales).Theirmodelinginthispaperusesabuyerdecisiontreewhenadvanced salesarepricedatapremiumandasimple2-casemodelisusedtocompareadvancedsale andspotsaleprots. 14

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Asdiscussed in theShuganandXie(2004)paper,advancesellingissimilartoyield management,orrevenuemanagement.Thus,therearesomerelatedideasfoundinthe revenuemanagementliterature.TheDesirajuandShugan(1999)paper[ 7]clearlydenes yieldmanagementsystems(YMS).PricingstrategiesbasedonYMSmayindeedbe protable.Theyconsiderdiscounting,overbooking(whichisshowntobebenecialin somecases),andlimitedearlysalesforcapacity-constrainedservices.InYMS,there aretwodistinctmarketsegments:price-insensitive,whichhavehighvaluations,and price-sensitive,whichhavelowvaluations.Therearealsothreedistinctserviceclasses. ClassArepresentsearlyarrivalsfromtheprice-sensitivemarket,ClassBrepresentsearly arrivalsfromtheprice-insensitivemarket,andClassCrepresentsearlyarrivalsfromboth markets(thatis,thereisnodistinctionamongthemarketsegments).Theirresultsshow thatforClassA,limitedearlysalesatlower/increasingpricesisthebeststrategy,for ClassB,unlimitedearlysalesathigher/decreasingpricesisbest,andforClassC,YMS isnotprotablesothermshouldsellattheirbestprice.Theirmodelingtechniques includedynamicprogramming:foragivenoptimalspotprice,theyconsidertotalprots tondthebestadvancesalesprice.Theyleavesomeopenextensionstotheirwork,such asconsideringcompetition,additionalmarketsegments,channelsofdistribution,dierent qualitylevels,andsignalingonYMS. Thispapermakesthecleardistinctionbetweenrevenuemanagementandadvance sellingintheassumptionrequirements.Advancesellingonlyrequiresthatcustomers areuncertainabouttheirfuturevaluation,whereasrevenuemanagementassumesa certaintimingofdierentcustomersegmentsandusuallyacapacityconstraint.Revenue managementseeksmoretoallocateagivencapacityamongcustomersegmentsbycreating apricemenu.Inadvanceselling,customersarehomogeneousandmayarriveatanytime tothemarket. Ideassimilartoadvancesellingcanalsobefoundintheinventoryliterature. 15

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Tang, et. al.(2004)[ 8]describeaninventorystrategycalledAdvanceBooking Discounts(ABD).TheydiscussthebenetsofthisstrategyoverQuickResponse(QR), asimilarinventorystrategy.InQR,armproducesinarstperiod(basedonretailer estimates)andalsoproducesinasecondperiod(basedonretailerupdatedestimateafter someearlysales).InABD,however,armproducesinonlyoneperiod(afterpre-season discountsales).TheirresultsshowthatprotsfromtheABDstrategywithforecast updatingaregreaterthanprotsofABDwithoutforecastupdating.Theyalsoshow thatprotsofABDcanbegreaterthanabasecasescenario(wherenoABDstrategyis employed).Theyndthatdiscountpriceswithforecastupdatingaregreaterthanthe discountpriceswithoutupdating.Theirmodelseekstomaximizeprot.Theyachieve newsvendorresults.Someextensionstotheirworkincludendingtheoptimallength oftheABDadvancedperiod,consideringABDpremiumsinsteadofdiscounts,and consideringcapacityconstraints. Inanearlierpaper,IyerandBergen(1997)[ 9]introducethebenetsofQR.They discussconditionsunderwhichitisPareto-improving(bothretailerandmanufacturer benetorareaswello).WithQR,thereisaninitialdemandobservationperiod,then theretailerplacesorder,thentheleadtimeforproduction/shippingoccurs,thentheorder isreceivedandtheseasonbegins.TheirresultsshowthatQRisalwaysbenecialforthe retailer,butnotnecessarilyforthemanufacturer.InordertomakeQRPareto-improving, thermmustexaminetheirservicelevel,price,andvolumecommitments.QRalso dependsontheaccuracyofthedemandestimatefromtheinitialobservationperiod. Theyuseaprot-maximizationmodel.Theyndanewsvendorsolutionforanoptimal inventorylevelandservicelevel.Theydeterminethatinordertondexpectedprotsfor QR,thermshouldchooseanordersizewhichmaximizesprotforaposteriordemand distribution.Someextensionstotheirworkincludecompetition,multiplesalesperiods, andmultiplemanufacturers.TheycouldalsoconsiderifQRisbenecialinothermarkets? 16

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Also,what w ouldbetheeectofthedemandobservationperiodlengthontheQRbenet totheretailerandmanufacturer? Ingeneral,thereseemstobeanopenresearchdirectionforapplyingoperations managementdecisionstosalesstrategiessuchasadvanceselling.Myresearchcontributes totheliteraturebypresentingamorerealisticmodelofconsumer'svaluationwith uncertainty,ndinganoptimalpricingstrategyusingdynamicpricingfortheadvance salesperiod. 2.3Multi-GenerationPricingandTimingDecisionsinNewProduct Development TheliteraturerelatedtothemodelinChapter4canbedividedintothefollowing categories:SalesBehaviorforSingleandMultipleGenerations,PricingDecisions,and TimingDecisionsresearch.Whilemostofthisliteratureaddressesasubsetofthesetopics inisolation,Icombinethedynamicpricingandgenerationaltimingdecisionsintoasingle model. Alargebodyofliteratureaddressesthesalesand/ordiusionprocessforproductsin bothsingleandmultiplegenerationsettings.ManyofthesestemfromBass(1969)[ 10 ] whodescribesthediusionprocessforasinglegenerationofproductsasafunctionof bothinnovation(i.e.earlyadopters)andimitation(i.e.laterbuyers).Thisempirically basedmodelhasbeenshowntobearobustcharacterizationforthediusionprocessof durablegoodsincludinggrowth,maturityanddeclinephasesoftheproductlifecycle. NortonandBass(1987)[ 11]createamultiplegenerationversionoftheoriginalBass modelwhichincorporatessubstitutioneectsandincreasingsalesacrossthegenerations. Onekeyfacetoftheirmodelutilizedhereconcernstheassumptionthattheinnovation andimitationparametersremainthesameacrossmultiplegenerationsofthesame product. Anotherbodyofliteratureaddressesthedynamicpricingproblemassociatedwiththe introductionofasinglegenerationofanewproductintothemarketplace.Kalish(1983) 17

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[12]analyzes dieren tscenariosillustratingtheimpactofdiscounting,learningeectsand diusiononthedynamicpricingproblems.OneofthescenariosthatKalishexaminesis thecaseofadurablegoodwherepositivewordofmouthstimulatesdemandearlyinthe lifecycle,whilesaturationtakesoveranddemandincreaseslaterinthelifecycle.The optimaldynamicpriceundertheseconditionswillstartrelativelylow,increaseaslong astheword-of-moutheectovercomesthesaturationeect,andthendecreaseforthe remainderoftheplanninghorizon.Bassetal.(1994)[ 13]proposeageneralizedversionof theoriginalBassmodelwhichincludesthedynamiceectsofpricingand/oradvertising onproductdiusion.Krishnanetal.(1999)[ 14]usestheGeneralBassModel(GBM)to identifyanoptimalpricepathforanewgenerationofproducts.Incontrasttoprevious literature,theseauthorsndthattheoptimaldynamicpricedoesnotfollowatraditional salesgrowthpattern,but(inmanycases)isdecreasing.SethiandBass(2003)[ 15]also ndthatbothpriceandsalesratedeclineovertimeforaspecialcaseofGBM.Tengand Thompson(1996)[ 16]considertheimpactofbothqualityandpricesimultaneouslyon cumulativesalesandprot.BothBassetal.(1994)[ 13]andKrishnanetal.(1999)[ 14] oercomprehensiveoverviewsoftheliteraturewhichincorporatespriceand/oradvertising factorsintodiusionmodels. Thedynamicpricingproblemhasbeenextendedtoaddressoptimalpricingstrategies formultiplegenerationsofnewproducts.However,theentrytimeforthenewgenerations isconsideredexogenousorgiveninthesemodels.PadmanabhanandBass(1993)[ 17 ] analyzeamodelwhichcapturessubstitutionandcannibalizationeectsforarmthat introducestwogenerationsofnewproductsduringaniteplanninghorizon.Results fromthismodelshowthattheactualpriceschangedateachinstantoftimeforthe twoproductsaresignicantlydierentwiththeconsiderationofproductlineissuesin theprotmaximizationproblemascomparedtothesinglegenerationmodelssuchas Kalish(1983)[ 12].Intheirconclusionsection,theseauthorsalsocommentthat,\The demandmodelusedin(this)analysisassumesthatthetimeofentryofthesecondproduct 18

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isdetermined exogenously.. Endogenousconsiderationofthisissuewouldbeavery worthwhilecontributiontotheliterature."Kornish(2001)[ 18]alsoconsidersthepricing problemfortwogenerationsofaproductbasedontheconsumersvaluationsforeach generation. Severalauthorsaddresstheproblemofoptimalintroductorytimingformultiple generationsofnewproductsintoamarketplace.Basedonnumericalanalysis,Mahajan andMuller(1996)[ 19 ]developanoworatmaturityrulefortheintroductiontimingofa secondgenerationofproducts.Specically,theyndthatthermshouldeitherintroduce thesecondgenerationassoonasitisavailableordelayitsintroductiontothematurity stageoftheprecedinggeneration.Anothermodelwhichdeterminestheoptimaltiming oftheintroductionofasecondproductintothemarketplaceisdevelopedinCarrilloand Franza(2004)[ 20],whoalsoconsidertheimpactofbothprocessdevelopmentandproduct developmentactivitiesonthisdecision.Morganetal.(2001)[ 21]consideraqualityversus time-to-markettradeowhenmultiplegenerationsareintroduced.Carrillo(2004)[ 22 ]and Carrillo(2005)[ 23]addresstheoptimalnumberofgenerationstointroduceduringagiven planninghorizonandanalyzetheimpactofdynamicprotmarginsonthetimingdecision. However,noneofthesemodelsconsiderstheimpactoftimingsimultaneouslywithpricing asdecisionvariables. 2.4OptimalNumberofGenerationsforaMulti-GenerationPricingand TimingModelinNewProductDevelopment TheimportanceofconsideringclockspeedasacomponentofNPDdecisionmaking hasbeenhighlightedintherecentmarketingliterature.InCarbonellandRodriguez (2006)[ 24],theauthorsanalyzetheeectthatinnovationspeedhasontheperceptionof marketingadvantage.Theyrecognizethatinnovationspeedisanequivalent,ifnotmore important,marketingcharacteristicaectingNPDsales.Othermarketingliterature,such asNadlerandTushman(1999)[ 25 ],Pearce(2002)[ 26 ],andLambertandSlater(1999)[ 27] havediscussedthedirectimpactthatinnovationspeed,orclockspeed,mayhaveonthe 19

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salesrate. In fact,someempiricalworkhasshownthatsalesmayincreaseasthenumber ofgenerationsincreases. Intheoperationsmanagement(OM)literature,limitedresearchhasbeendoneonthe optimalnumberofgenerationstointroduceinanNPDscenario.InCarrillo(2005)[ 23], theauthorsolvesfortheoptimalclockspeedundervarioussalescurves.Conditionsare derivedtodeterminewhenrmsmayhaveanincentivetoincreasetheirclockspeed.In anotherpaper,bySouza,Bayus,andWagner(2004)[ 28],innovationspeediscompared withqualitydecisionsandtimetomarket. ComparativemodelscanbefoundinLukasandMenon(2004)[ 29 ],wholookatthe jointqualityandinnovationspeedproblem,andDahanandMendelson(2001)[ 30]who examineconcepttestinginNPD.AnotherOMworkbyXuandLi(2007)[ 31]addressthe jointtechnologyinvestmentandinnovationdecisioninassemble-to-ordersystems. InChapter4,optimalpricingandtime-to-marketdecisionsarederivedfora two-generationnewtechnologyproduct.Thischaptermakesasubstantialcontribution totheOM/MarketingInterfaceliteraturefortheNPDmarketingscenario.Chapter5 alsomakesasignicantcontributiontotheOM/MarketingInterfaceliteraturebysolving bothpricingandtime-to-marketdecisionsinadditiontosolvingfortheoptimalnumberof generationsofanNPDproduct. 20

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CHAPTER3 INVENTOR Y MANAGEMENTUNDERADVANCESELLING:OPTIMALORDER ANDPRICINGPOLICIES 3.1Introduction Advancesellingisamarketingstrategyinwhichanadvancesalesperiodprecedes thestandardconsumption,orspot,period.Theadvancesalesperiodisusedtoincrease salesbyoeringcustomersachancetocommitearlytopurchasingatwhatisusuallya discountedprice.Thisstrategytakesadvantageofcustomersbeinguncertainabouttheir futureproductvaluation.Advancesellinghasbecomeincreasinglymorepopularwith recenttechnologiessuchassmartcardsandonlinebooking(seetheEconomist(2005) [32]). Mostapplicationsofadvancesellingareintheserviceindustry.Forexample,consider ticketsalesforaconcert.Ticketpricesmaybe$50atthedoorbutonsalefor$30if boughtinadvance.Otherexamplesofadvancesellingintheserviceindustrymayinclude conferenceregistration,movietickets,andvacationpackages. Muchoftheliteratureinadvancesellinghastodowithndingtheoptimalpricing policyfortheadvanceandspotsalesperiods.Iaminterestedinextendingthismarketing analysistoincludeoperationsdecisions,specicallyaninventorymanagementdecision. Iassumethataone-timeinventoryordermustbeplacedatthebeginningofthe advancesalesperiod.Theinventorywillnotarriveuntilconsumption,whichoccursatthe endofthespotperiod.Imayconsiderasituationinwhichthereisalongleadtimeforan inventoryorderandnoopportunitytoplaceanotherorderbeforeconsumption. Anexampleofthisscenariomayoccurinthetoyindustry.Imayneedtodecide onanorderquantityoftoysforanupcomingsalesseason.Demandmaybeuncertain inthatcustomershavenotyetrealizedtheirfuturevaluationofaparticulartoy.Ifthe toysareproducedinadistantfacilitywithalongleadtime,Imayonlyhaveoneorder opportunity.Anadvancesellingstrategywouldbetooerreservationsofsomeportionof thesetoysandtoreservetheremainingportionforin-storesales. 21

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Anotherexample ma yoccurineventplanning.Imayneedtoreservealocation foranupcomingeventwithuncertaindemand.Again,customersmaybeuncertainof theirfuturevaluationofattendingthisevent.Theorderquantityinthiscasewouldbe equivalenttothecapacityofthelocationIreserve.Sincemanyeventlocationsmustbe reservedpriortotheeventwithsomekindofnon-refundabledeposit,Imayconsiderthis tobeaone-timeorderdecision.Advancesellinginthisscenariowouldincludereserving someofthiscapacityforadvancesalesticketsandleavingtheremainingforat-the-door sales. Athirdexamplemaybeinreal-estatemarketingofnewcondominiumdevelopment. Whendevelopinganewcondominiumcomplex,Imustdecideaheadoftimehowmany unitstobuild.SinceIcannotaddorsubtractunitsonceconstructionbegins,Imay considerthistobeaonetimeorderdecision.Demandisuncertaininthatcustomersare unsureoftheirfuturevaluationofpurchasingacondominiumunit.Theadvanceselling strategywouldinvolvereservingsomeunitstosellinadvanceandkeepingtheremaining portionavailableforsaleafterthecondominiumshavebeencompleted. Inadditiontodecidingtheinventoryorderquantity,Ialsoconsiderwhatportionof thisinventorytoreserveforadvancesales.Iuseanalyticalandnumericalresultstobetter understandwhenitisoptimalforthermtooeradvancesales,andifso,howmany advancesalestooer.Ialsoseektodeterminetheoptimalpricingpolicyfortheadvance andspotsalesperiods.Iuseanexpectedprotmaximizationmodeltondtheoptimal orderandpricingpolicies. Therestofthechapterisorganizedasfollows.InSection2,Idescribethemodel andassumptions.InSection3,Iperformtheanalysisandgivestructuralsolutionsto theoptimalinventoryandpricingpolicies.InSection4,Idescribeseveralnumerical experimentsanddiscusssensitivityanalysis.InSection5,Iconsideranextensionfora dierentcustomervaluationdistributionanddescriberelatednumericalexperiments. PleaserefertotheChapter2forareviewoftherelatedliterature. 22

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3.2Mo del There aretwoperiodsinwhicharmsellsaproductorservice:theadvancesales periodandthespotperiod.Theadvancesalesperiodprecedesthespotperiod,and consumptionoccursattheendofthespotperiod.Ihaveamarketofsize M ,withunits equaltothenumberofcustomers.Aportionofthismarketwillarrivetotheadvance salesperiod, N a ,andtheremainderwillarrivetothespotperiod, N s = M N a .Aswith themarketsize, N a and N s representanumberofcustomers. Thecustomersthatarrivetotheadvancesalesperiodwillbeoeredtheproduct(or service)ataprice p a .Iassumethatthespotprice p s isalsoannouncedtothecustomers intherstperiod. Customersdecidewhetherornottobuytheproductbasedontheirvaluationofthe product V ,measuredasadollarvalue.Iassumethatthetruevaluationoftheproduct isnotrealizeduntilthespotperiod.Thus,duringtheadvanceperiod,customersare uncertainabouttheirfuturevaluationbutknowthedistributionofthefuturevaluation andtheexpectedfuturevaluation, E [V ].Customersthatarrivetotheadvancesales periodmustthusdecidewhethertobuytheproductintheadvanceperiodorwaituntil thespotperiodbasedontheirexpectedfuturevaluation,theadvancesalesprice,andthe announcedspotprice.Duringthespotperiod,customersdecidewhetherornottobuythe productbasedonthespotpriceandtheirrealizedproductvaluation. Iconsideraninventorymanagementdecisionunderthisadvancesellingscenario. Thermmustplaceaninventoryorder Q atthebeginningoftheadvancesalesperiod. Withoutlossofgenerality,Iassumethatthisinventoryorderisdeliveredatconsumption, whichoccursattheendofthespotperiod.Someportionofthisinventory, X a ,isreserved fortheadvancesalesdemand,andtheremaininginventory, X s = Q X a ,isusedtosatisfy spotsalesdemand,whereIhave X a Q M .ConsiderthetimelineinFigure 3-1. 23

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Iseek to determinetheoptimalvaluesfortheorderquantity Q ,theadvancesales inventoryportion X a ,theadvancesalesprice p a andthespotprice p s suchthattotal expectedprotismaximized. Notations M Mark etsize N a Num b er ofcustomersarrivingtoadvancesalesperiod N s Num b er ofcustomersarrivingtospotsalesperiod p a Adv ance sales price p s Sp ot sales price V Customer v aluation ofproductatconsumption Q In v en toryorderquantity X a In v en toryallocatedtoadvancesales X s In v en toryallocatedtospotsales Letusnowderivetheexpressionsfortheadvancessalesandspotsalesdemand.The spotperioddemand D s isaBinomialrandomvariablewiththenumberofeventsequal tothenumberofspotperiodcustomers N s = M N a andtheprobabilityofsuccess,or probabilityofpurchase,dependentonthecustomervaluation, Pr fp s V g. D s Binomial (N s ;Pr fp s V g) (3{1) Intheadvancesalesperiod,acustomerwillonlybuytheproductiftheexpected utilityofanadvancepurchaseisgreaterthanorequaltotheexpectedutilityofwaitingto Figure 3-1. Timeline 24

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purchase in thespotperiod.Theutilityofanadvancepurchaseisthedierencebetweena customer'svaluationandtheadvancesalesprice. U a = V p a (3{2) ThusIhavetheexpectedutilityofanadvancepurchaseasfollows. E [U a ]= E [V ] p a (3{3) Iassumethatcustomersareawarethatthereislimitedinventoryavailableand thusthereisariskthattheremaynotbeenoughinventorytosatisfyalldemand.This informationwillaectthecustomer'sutilityofwaitingtopurchaseinthespotperiod. Let representtheprobabilitythatacustomerwillndavailableinventoryinthespot period.Icanthinkofthisprobabilityastherm'sdemandllrate.Idene astheratio ofsatiseddemandtototaldemand.Idenesatiseddemandastheminimumofthe availableinventoryandthedemand:min( Q X a ;D s ),where D s isthedemandinthespot period. = min(Q X a ;D s ) D s (3{4) The utilit y ofwaitingtopurchaseinthespotperiodisthenthepositivedierence betweentheirvaluationandthespotprice,( V p s ) + ifinventoryremains,or0ifthereis noinventory.Using asdenedaboveastheprobabilityofhavingavailableinventoryin thespotperiod,Idenetheutilityofwaitingtopurchaseinthespotperiodasfollows. U s = 8 > < > : (V p s ) + ; withprobability 0; withprobability1 (3{5) Although isafunctionofthespotdemand, D s ,whichisdependentonthecustomer valuation V ,Iassumethevaluationswhichdeterminethespotdemandareforcustomers whowillsportpurchase,whichdoesnotincludetheadvancesalescustomerIamcurrently 25

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considering.That is, thevaluationsconsideredinthespotdemandthroughtheBinomial probabilityofsuccess Pr fp s V g areindependentofthevaluationofthedecision-making advancesalescustomerwhosevaluationappearsin( V p s ) + .Thus,Icancalculatethe expectedutilityofwaitingtopurchaseinthespotperiodasfollows. E [U s ]= E [(V p s ) + ] E [ ] (3{6) Thus,acustomerwilldecidetopurchaseintheadvancesalesperiodifandonlyif E [U a ] E [U s ].Evaluatingthiscomparisontranslatesthisconditionintoamaximum advancesalesprice,^ p a forwhichacustomerwilldecidetoadvancepurchase. E [U a ] E [ U s ] (3{7) E [V ] p a E [( V p s ) + ]E [ ] (3{8) p a E [ V ] E [(V p s ) + ]E [ ]=^ p a (3{9) Thus,ifIsettheadvancesalesprice p a equaltothismaximumadvance-purchase-inducing price^ p a ,thenallcustomerswhoarrivetotheadvancesalesperiodwillchoosetoadvance purchase.SinceIwilldecidetheportionoftheinventoryreservedforadvancesales, X a Icanassumethatallofthisinventorywillbesoldintheadvanceperiodwhen p a ^ p a Thus,Icanconsiderthenumberofadvancesalescustomerstobeequivalenttothis advancesalesinventoryportion: N a = X a ,andthenumberofcustomerswhodecideto waitiszero.Thatis,Idecide X a andthenadvanceselltothatmanycustomers, N a = X a knowingthattheywillallagreetoadvancepurchaseif p a ^ p a Theadvancesalesdemand D a isthusequaltothenumberofcustomerswhoarriveto theadvancesalesperiod( D a = N a ),whichisthesameastheportionofinventorythatI reserveforadvancesales( N a = X a ). Icannowwritemyprot-maximizationobjectivefunction.Weassumeaunitadvance salescost c a ,unitordercost c ,andnosalvagevalue.Theprotexpressionconsidersprot earnedfromtheadvancesalespurchases X a ,revenuefromspotsales,andtheinventory 26

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ordercost. = ( p a c a ) X a + p s min(Q X a ;D s ) cQ (3{10) E []=(^ p a c a ) X a + p s ( D s D s ( Q X a )) cQ (3{11) Where D s isthelossfunction R 1 QX a (t Q + X a )f D s (t)dt and f D s (t )istheNormal pdf. Thus,myoptimizationproblemistodeterminetheoptimalvaluesfortheorderquantity Q,theadvancesalesinventoryportion X a ,theadvancesalesprice p a andthespotprice p s suchthattotalexpectedprotismaximized. MAX E []=(^ p a c a )X a + p s ( D s D s (Q X a )) cQ (3{12) subjectto 0 X a Q M (3{13) Letusassumethatacustomer'sproductvaluationisaBernoullirandomvariable whichcanhaveahighvalue H withprobability oralowvalue L withprobability1 aswasdonesimilarlyinXieandShugan(2001)[ 5].Acustomerwillonlybuytheproduct ifthepriceislessthanorequaltothisvaluation.Inthespotperiod,thisvaluationis realized,thusacustomerwillbuytheproductwithprobability Pr fp s V g.Withthe Bernoullidenitionofvaluation,thisprobabilityis: Pr fspotpurchase g = Pr fp s V g = 8 > > > > > > > < > > > > > > > : 0; if p s >H ; ; if p s = H ; ; if L

H ,sinceno customerswouldpurchase.Sincetheprobabilityofaspotpurchaseisthesamefor p s = H 27

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and L< p s < H ,itwouldbemoreprotabletooeraspotpriceof p s = H .Oeringa spotpriceof p s = L wouldbemoreprotablethananypricelowerthan L.Thus,from thispointforwardIwillassumethatthermconsidersoeringaspotprice p s ofeither H or L ,butnotanyvalueinbetween.Iwillthereforeperformmyanalysisforthesetwo cases. Icannowcalculatethespotdemand, D s ,tobeasfollows. D s 8 > < > : Binomial (N s ; ); if p s = H ; = N s ; if p s = L. (3{15) E [D s ]= 8 > < > : N s =(M X a ); if p s = H ; N s = M X a ; if p s = L. (3{16) Forthecasewhen p s = H ,IwillapproximatetheBinomialwithaNormal distributionwithmean D s and D s denedasfollows. D s Normal ( D s ; D s ) D s =(M X a ) (3{17) D s = p (M X a ) (1 ) (3{18) E [ ] can no wbecalculatedasfollows: E [ ]= E [min(Q X a ;D s )] E [D s ] (3{19) = 8 > < > : E [D s ( D s Q+X a ) + ] E [D s ] ; if p s = H ; min( QX a ;M X a ) M X a ; if p s = L. (3{20) = 8 > < > : 1 D s (QX a ) ( M X a ) ; if p s = H ; (QX a ) M X a ; if p s = L. (3{21) 28

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Notethat I evaluatemin( Q X a ;M X a )= Q X a basedonthefactthat X a Q M .NotealsothatIusetheapproximation E [ ]= E [ min ( Q X a ;D s )] E [ D s ] although it has beenshownintheliteraturethatthiswillyieldlargervaluesthantheexpectedvalue E [ min(QX a ;D s ) D s ]. I no w returntomydenitionofthemaximumadvancesalesprice,^ p a = E [V ] E [( V p s ) + ]E [ ].UsingtheBernoullidistributionforcustomervaluation,Icalculatethe following. E [V ]= H + L(1 ) (3{22) E [(V p s ) + ]= H X v = p s (v p s )Pr fV = v g (3{23) = 8 > < > : 0; if p s = H ; (H L); if p s = L. (3{24) NowIcanexpress^ p a usingtheabovedenitionsandtheexpressionsfor E [ ]as follows. ^ p a = 8 > < > : H + L(1 ); if p s = H ; H + L(1 ) (H L) ( QX a M X a ); if p s = L. (3{25) = 8 > < > : L + ( H L ) ; if p s = H ; L +(H L) ( M Q M X a ) ; if p s = L. (3{26) The exp ected prot E []forBernoullicustomervaluationsisthenasfollows. E []= 8 > > > > > > > < > > > > > > > : (L +(H L) c a )X a +H ((M X a ) D s (Q X a )) cQ; if p s = H ; (L +(H L) ( M Q M X a ) c a )X a +L(Q X a ) cQ; if p s = L. (3{27) 29

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3.3Analysis 3.3.1 Optimal OrderQuantity Q (X a ) Firstletusndtheoptimalorderquantity Q asafunctionof X a .Iwillsolvefor eachspotpricecaseseparately( p s = L and p s = H ). Forthecasewhen p s = L ,Indexpectedprot E [(L)]tobelinearin Q. E [(L)]is increasingin Q,underthecondition X a (H L) M X a + L c > 0, and decreasingin Q otherwise. Ihavethefollowingtheorem. Theorem1. For p s = L andBernoullicustomervaluations,theexpectedprot E [(L)] is linearincreasingin Q for X a (H L) M X a + L c > 0 Thustheoptimalorderquantity Q for agiven X a and p s = L is: Q (X a ;L )= 8 > < > : M; if (H L) Lc < M X a X a 0; otherwise. (3{28) (Se e App endix A fortheproof.) Thatis,foralowspotprice,whentheaboveconditionismet,theoptimalorder quantityisequaltotheentiremarket.OtherwiseIdonotorderanything.This"allor nothing"resultisduetotheBernoullicustomervaluationandlowspotprice.Whenthe spotpriceislow( p s = L),theadvancesalespriceisalsolow(^ p a = L ).Thus,Iorder enoughfortheentiremarketsinceeveryonewillbuy. Inthecasewhen p s = H ,Indexpectedprot E [(H )]tobeconcavein Q withthe optimalinventorysize Q (X a ;H )asfollows. Theorem2. For p s = H andBernoullicustomervaluations,theexpectedprot E [(H )] isconcavein Q andtheoptimalorderquantity Q foragiven X a and p s = H is: Q (X a ;H )= F 1 D s ( H c H ) + X a (3{29) (Se e Appendix A fortheproof.) 30

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Inthis case, sincethespotpriceishigh,Icannotbecertainhowmanycustomerswill buyinthespotperiod.Thatis,Imayhavecustomersinthespotperiodwitharealized valuationlowerthan H .Iseethat Q (X a ;H )resemblesthestandardnewsvendorsolution, whichcapturestheeectofthisdemanduncertainty,withanadditionalquantityforthe advancesalesinventory, X a ThusIhavefoundtheoptimalorderquantity Q asafunctionof X a tobe: Q (X a )= 8 > > > > < > > > > : F 1 D s ( H c H ) + X a ; if p s = H ; M ; if p s = L and ( H L) Lc < M X a X a 0; if p s = L, otherwise. (3{30) 3.3.2 Optimal AdvanceSalesInventory X a (Q) Nowletusndtheoptimaladvancesalesinventoryquantity X a asafunctionof Q.I willsolveforeachspotpricecaseseparately( p s = L and p s = H ). For p s = L,Indexpectedprot E [(L)]tobeconvexin X a .SinceIwantto maximizeexpectedprot,theoptimaladvancesalesinventory X a (Q;L)isthusan extremepointsolution,with0 X a (L) Q. Theorem3. For p s = L andBernoullicustomervaluations,theexpectedprot E [(L)] is convexin X a .Thustheoptimaladvancesalesinventorylevel X a foragiven Q and p s = L isanextremepointsolution. X a (Q;L)= 8 > < > : Q; if ( H L ) c a 0; otherwise. (3{31) (SeeAppendix A fortheproof.) SinceIhaveanextremepointsolution,Ieitherreserveallofmyorderquantityfor advancesalesorIdonotadvancesellatall.Theaboveconditionstatesthataslongas thecostofadvancesellingisrelativelylow,Iwilladvanceselltoeveryone.Thisseems 31

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reasonablesince the spotpriceislow.Thus,thereisnoadvantagetoreservinginventory forthespotperiodifIcanselleverythingforthesamepriceintheadvancesalesperiod. For p s = H ,Ialsondtheoptimaladvancesalesinventory X a (Q;H )tobean extremepointsolution,with0 X a (H ) Q. Theorem4. For p s = H andBernoullicustomervaluations,theoptimaladvancesales inventorylevel X a foragiven Q and p s = H isanextremepointsolution. X a (Q;H )= 8 > < > : Q; if Q[L(1 ) c a ] H [ D s D s (Q)], 0; otherwise. (3{32) Underthefollowingconditions. c 0:31H (3{33) D s 1 4 k 1 8 (3{34) LB U B (3{35) wher e LB = 8 > < > : 1 q 1 D s k 1= 2 2 for c H 0:00169 1 D s (2 D s +3k D s ) 2 k 2 D s otherwise (3{36) U B = 1 + q 1 D s k 1 =2 2 (3{37) (Se e App endix A fortheproof.) Again,sinceIhaveanextremepointsolution,Ieitherreserveallofmyorderquantity foradvancesalesoroernoadvancesalesatall.Theconditionforadvancesellingallof theorderquantityisdependentonhowuncertainthespotdemandis.Thatis,thelower thechanceofearningsalesinthespotperiod,theriskieritistoreservemorespotsales, despitethehighspotprice.Thusitismoreprotabletoadvancesellallofmyinventory. 32

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Notethat the conditionfor LB = 1 q 1 D s k 1=2 2 of c H 0:00169implies that D s 0 :61.Thatis,forhighercostvalues,orcostvaluescloserto H ,thevarianceofthespot demandmustbesmallforthislowerboundon tohold.This LB valueisincreasing intheratio D s D s creating a tighterbound.Theoppositeistruefortheothervalueof LB =1 D s (2 D s +3k D s ) 2 k 2 D s In this case,thecostisrelativelyinsignicant( c H 0:00169) and the v arianceofthespotdemandcanbemuchhigher( D s 0:61)forthelowerbound tohold.This LB valueisdecreasingintheratio D s D s creating a looserbound.The UB valueisdecreasingintheratio D s D s creating a tighterbound. ThusIhavefoundtheoptimaladvancesalesinventory X a asafunctionof Q tobe: X a (Q)= 8 > > > > > > > < > > > > > > > : Q; if p s = H and[ L(1 ) c a ]Q H [ D s (0) D s (Q)]; 0; if p s = H ,otherwise; Q; if p s = L and(H L) c a 0; if p s = L ,otherwise. (3{38) 3.3.3OptimalOrderPolicy (Q ;X a ) Tondtheoptimalorderpolicy( Q ;X a ),Iwillusethevariablesubstitutionmethod commonintheprice-dependentnewsvendorliterature(seePetruzziandDada(1999)[ 33 ]). Iwillreplace Q intheexpectedprotexpressionwiththesolutionfor Q (X a )foundin section 3.3.1.Iwillthenusetherstorderconditionof E [(Q )]tosolvefor X a Forthecasewhen p s = L,ifconditionsholdfor Q ( X a ;L )= M ,Indthat E [(Q (L))]islinearlydecreasingin X a .Thus X a (L)=0andIdonotadvancesell.If conditionsholdfor Q (X a ;L )=0,thenIknow X a (L)=0since X a Q M .Thus, Iconcludethatif p s = L itisneveroptimaltoadvancesell.Todeterminetheoptimal inventorypolicy,Icomparetheprotearnedfroma( Q;X a )=(M; 0)policywiththat froma(0 ; 0)policy.Clearly,forthe p s = L case,Ihavemaximumprotwiththe( M; 0) policy. 33

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Theorem5. F or p s = L and Bernoullicustomervaluations,theoptimalorderpolicyis: (Q (L);X a (L))=(M; 0) (3{39) (SeeAppendix A fortheproof.) Thus,whenthespotpriceislow,itisoptimaltoorderfortheentiremarketbutnot oeranyadvancesales. Forthecasewhen p s = H Iagainndanextremepointsolutionfor X a (H ). However,since Q ( X a ;H )isafunctionof X a (refertoequation 3{29 )and Q M ,I donothaveanupperboundof X a Q,butrather X a M k 2 (1 ) where I dene k = p 2er f 1 (2( H c H ) 1) as a constant(here, erf istheerrorfunction).Thus,Ihavetwo possibleorderpolicies( Q;X a )for p s = H :(M;M k 2 (1 ) ) and ( F 1 D s ( H c H ); 0). The optimal p olicyisdeterminedfromthefollowingcondition. Theorem6. For p s = H andBernoullicustomervaluations,theoptimalorderpolicyis: (Q (H );X a (H ))= 8 > < > : ( M;M k 2 (1) ); if L(1 ) + c c a ( F 1 D s ( H c H ); 0); otherwise. (3{40) (Se e App endix A fortheproof.) Therefore,whenthespotpriceishigh,iftheadvancesalescostisrelativelylow,it isoptimaltoorderfortheentiremarketreservealmostallofthisinventoryforadvance selling.Otherwise,Iorderthestandardnewsvendorquantityanddonotoeranyadvance sales. Icanwritetheoptimalorderpolicyasfollows. (Q ;X a )= 8 > > > > < > > > > : (M;M k 2 (1) ); if p s = H and L(1 ) + c c a ( F 1 D s ( H c H ); 0); if p s = H otherwise, ( M ; 0); if p s = L. (3{41) 34

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3.3.4Optimal Pricing Strategy ( p a ;p s ) Frommyutilityanalysisofthecustomer'sdecisionofwhetherornottoadvance purchase,Iconcludedthatacustomerwilladvancepurchase i p a ^ p a (refertoequation 3{9).Therefore,since^ p a isthemaximumpriceIcanoerintheadvancesalesperiodin ordertoearn X a sales,theoptimaladvancesalespriceis p a =^ p a Theorem7. Theoptimaladvancesalesprice p a isequivalenttothemaximumadvance salesprice ^ p a .ForBernoullicustomervaluations,thisoptimalpriceisasfollows. p a =^ p a = 8 > < > : L +(H L); if p s = H ; L +(H L) ( M Q M X a ); if p s = L (3{42) T o determine theoptimalspotprice p s ,Icomparetheexpectedprotfromeach pricecase( H or L)undertherespectiveoptimalorderpolicy: E [(Q (L);X a (L))]and E [(Q (H );X a (H ))].Ihavethefollowing. E [(Q (L);X a (L))]=(L c) M (3{43) E [(Q (H ) ;X a (H ))]=( L +(H L) c a ) X a (H ) +H ((M X a (H )) D s (F 1 D s ( H c H ))) c (F 1 D s ( H c H ) + X a ( H )) (3{44) I thenhaveoptimalspotprice p s = H when E [(Q (H );X a (H ))] >E [(Q (L );X a (L ))] and p s = L otherwise. IfIre-examinetheoptimaladvancesalesprice p a for p s = L giventheoptimalorder policy(Q (L);X a (L))=(M; 0),Ihave p a = L .Thatis,ifthespotpriceis p s = L thentheadvancesalespricewillalsobe p a = L.Thisexplainsthepreviousresultfor X a (L )=0(see 3{39 ).Thatis,ifthespotpriceis p s = L thenIdonotadvancesellsince theprobabilityofcustomersspotpurchasingis1(see 3{14 )andthereisnoextrarevenue tobeearnedfromadvancesales( p a = p s = L ).ThusIcanwritetheoptimalpricingpolicy 35

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asfollo ws. ( p a ;p s )= 8 > > > > < > > > > : (L +(H L);H ); if E [(Q (H );X a (H ))] >E [(Q (L );X a (L ))], (L;L ); otherwise. (3{45) Wheretheoptimaladvancesalesprice p a when p s = H issimplythecustomer's expectedfuturevaluation, E [V ],asdenedinequation 3{22.Thus,iftheexpectedprot fromoeringahighspotpriceishigherthantheexpectedprotfromoeringalowspot price,theoptimalpricingstrategyisahighspotpriceandanadvancesalespriceequal totheexpectedcustomervaluation.Iftheexpectedprotfromalowspotpriceishigher, thentheoptimalpricingstrategyistooeralowpriceinbothperiods. 3.4NumericalExperiments Iperformseveralnumericalexperimentstoanalyzethesensitivityofmyanalytical resultstothecustomervaluationparameters H ,and L,andtobetterunderstand thebehaviorofexpectedprotintheadvancesalesinventorydecision, X a .Inthese experiments,Iusethevariablesubstitutionfor Q (X a ;H )tomaximize E [(Q (X a ;H ))] bysetting X a asthedecisionvariable.Findingtheoptimaladvancesalesinventorythen determinestheoptimalvalueof Q (X a ;H )(whichisafunctionof X a (H )).Thus,Ifocus onvaluesfor X a Q ,andexpectedprot E []for p s = H Iperformasensitivityanalysisontheeectofthevaluationprobability andthe spreadbetweenthehighandlowvaluationlevels( H L spread)ontheoptimalvalues Q X a ,andtheexpectedprot.Ialsoexaminetheeectonthepercentofadvancesales inventory(% AdvInv = X a =Q ). Iassumethefollowingparameterstobeconstant:advancesalescost, c a =5,unit cost, c =3,andmarketsize, M =100.Ivarythevaluesofthevaluationprobability between0.1and0.9.Iinitiallyset H =80and L =20.Thisyieldsthepricingpolicy p s = H =80and p a =^ p a =62.Ithenvarythe H L spreadsuchthattheadvancesales price p a =^ p a remainsxedatthisvalue. 36

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T able 3-1. SensitivityAnalysisforVarying Values Asthevaluationprobability changes,IseetheresultsshowninTable 3-1 for Q X a X s = Q X a E [],and% AdvInv = X a =Q .Icanmakeseveralobservations. As increases, Q isconstantwhile X a > 0,then Q increasesafter X a =0.Also,as increases, X a decreasesasitbecomesmoreprotabletoreservespotsaleswhenthe probabilityofahighspotpricepurchaseincreases.Icanalsoobservethatas increases, theexpectedprotincreases.Thisisduetomorespotsales,whichhaveahigherpotential revenue( p s = H>p a ). Thebehaviorof E []in X a canbeseeninthegraphsinFigure 3-2 ,correspondingto the valuesbetween0.6and0.8fromthetableinTable 3-1.FromthesegraphsIconrm theextremepointsolutionfor X a .Thisbehaviorimpliesa"go/no-go"decisionforoering advancesales.Icanseethatthereissomethreshold valueabovewhichadvancesales arenolongerprotable.Thatis,oncecustomershaveahighenoughprobabilityofspot purchasing,itismoreprotabletofacetheriskofholdingallinventoryforspotsalessince againIhave p s = H>p a Asthe H L Spreadchanges,whilekeepingtheadvancesalesprice p a =^ p a thesame,IseetheresultsshowninTable 3-2 for Q X a X s = Q X a E [],and 37

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Figure 3-2. Graphs of X a vs. E []forVarying Values %AdvInv = X a =Q .Asthe H L Spreaddecreases,Iobserve Q tobedecreasingwhile X a =0andthenconstantfor X a =0.Asthe H L Spreaddecreases,Iobserve X a tobe increasing. X a increasesbecausethebenetofreservingspotsalesdecreasesasthe H L Spreaddecreases.Thatis,thespotpriceandadvancesalespricebecomecloseenoughto outweighthebenetofhigherrevenuefortheuncertainspotsales.Icanalsoobservethat asthe H L Spreaddecreases,theexpectedprotdecreases.Thisisduetosmallerprice values. Thebehaviorof E []in X a canbeseeninthegraphsinFigure 3-3 ,correspondingto the H L Spreadvaluescorrespondingtothe H valuesbetween80.4and80.8fromthe tableinTable 3-2.FromthesegraphsIcanagainconrmtheextremepointsolutionfor X a implyinga"go/no-go"decisionforoeringadvancesales.Icanseethatthereissome 38

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T able3-2. SensitivityAnalysisforVarying H L SpreadValues Figure 3-3. Graphs of X a vs. E []forVarying H L SpreadValues threshold H L Spreadvalueabovewhichadvancesalesarenolongerprotable.Thatis, oncethe H L spreadislargeenough,itismoreprotabletofacetheriskofholdingall inventoryforspotsalessinceIhave p s = H becomingincreasinglylargerthan p a Icancomparethechangeinexpectedprottothechangein valuesand H L Spreadvaluestodeterminewhichparameterhasthemoresensitiveeect.Fromthe resultsshowninTable 3-3,itisclearthatthemaximumexpectedprotismoresensitive tothevaluationprobability thantothe H L Spreadfortheparametervaluesinthe trialsperformedinTable 3-1 andTable 3-2. 39

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T able 3-3. SensitivityComparisonof vs. H L SpreadValues Thismaymotivateadiscussiononwhetherornotthermcansettheseparameter values.ForaBernoullidistributedcustomervaluation,canarmchoosetheprobability ofahighvaluation, ?Forthisdistribution,would actuallybesomefunctionofthe parameters H and L?Canthermsetthesevalues,andthusthe H L Spreadvalues, orarethesedeterminedbythemarket?InSection 3.5 ,Iexplorewhatwouldhappenif thecustomervaluationdistributionwasUniforminsteadofBernoulli.Inanycase,ifthe rmisnotabletoeectthevaluationdistributionparameters,itcanstilldeterminethe optimalorderandpricingpolicyfromtheresultsofmymodelanalysis. 3.5AnExtension Letusconsiderthesensitivityofmyresultstothecustomervaluationdistribution, specicallywhathappensifthecustomervaluationdistributionisUniforminstead ofBernoulli.Assumethatacustomer'sproductvaluationisdistributedaccordingto acontinuousUniformdistributionbetweenthelowandhighvalues( L;H ).Nowthe probabilitythatacustomerwillbuytheproductinthespotperiodis Pr fp s V g = H p s H L I will usetheexpression F V (p s )torepresentthisspotpurchaseprobability.Now Icanconsideranyspotprice L p s > > > < > > > > : 0; if p s = H ; H p s H L ; if L < p s < H ; 1; if p s = L. (3{46) 40

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Ino w havespotdemand D s distributedasfollows. D s Binomial (N s =(M X a ); F V (p s )) (3{47) E [D s ]=( M X a ) F V (p s ) (3{48) IwillagainapproximatetheBinomialspotdemandwithaNormaldistribution.I denethemeanandthestandarddeviationasfollows. D s =(M X a ) F V (p s ) (3{49) D s = q (M X a ) F V ( p s )F V (p s ) (3{50) I canthen calculatetheexpectedprobabilityofavailableinventory asfollows. E [ ]= E [min(Q X a ;D s ) E [D s ] (3{51) = 1 D s ( Q X a ) (M X a ) F V ( p s ) (3{52) Where D s is the lossfunction R 1 QX a (t Q + X a )f D s (t ) dt and f D s (t)istheNormal pdf with D s and D s asdenedin 3{49 and 3{50 Icannowcalculatethecomponentsofthecustomer'sutilityevaluationforan advancesalespurchase, E [U a ]and E [U s ]asfollows. E [V ]= H + L 2 (3{53) E [( V p s ) + ] = Z H p s (t p s ) f V (t)dt (3{54) = 1 H L Z H p s (t p s )dt (3{55) = (H p s ) 2 2(H L) (3{56) Where E [V ] and f V ( t ) arecalculatedfromthecontinuousUniformdistribution. 41

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Themaxim um advancesalesprice^ p a isthen. ^ p a = E [V ] E [(V p s ) + ]E [ ] (3{57) = H + L 2 (H p s ) 2 2( H L ) + D s (Q X a )(H p s ) 2( M X a ) (3{58) I can no wwritetheexpectedprotexpressionasfollows. E []=( H + L 2 ( H p s ) 2 2(H L) + D s (Q X a )(H p s ) 2(M X a ) c a )X a +p s ((M X a ) H p s H L D s (Q X a )) cQ (3{59) I then solv efortheoptimalinventoryquantity Q asafunctionof X a Theorem8. ForUniformcustomervaluations,theexpectedprot E [] isconcavein Q andtheoptimalorderquantity Q foragiven X a is: Q = F 1 D s ( p s c X a ( H p s ) 2( M X a ) p s X a ( H p s ) 2(M X a ) ) + X a (3{60) (Se e Appendix A fortheproof.) IndaresultthatresemblesthenewsvendormodelIfoundpreviously(see 3.3.1), exceptthistimewithasmallervalue.Thatis, F 1 D s ( p s c X a (H p s ) 2( M X a ) p s X a (H p s ) 2( M X a ) ) < F 1 D s ( H c H ). T o nd X a IperformsimilarnumericalexperimentsasdoneinSection 3.4 .Inthese experiments,Iagainusethevariablesubstitutionfor Q tomaximize E [(Q )]bysetting X a asthedecisionvariable.Forthesetrials,Ixtheadvancesalescost, c a =5,unit cost, c =3,andmarketsize, M =100.Then,sinceIdonotknowthespotprice p s ,Iset H =80and L =20andcomparetheresultsforthespotpricevalues L

0,then Q increasesafter X a =0.Also,as p s decreases, X a decreasesandtheexpectedprotdecreases.Thevaluesof X a and E []arehigherwith larger p s valuessincealargerspotpricecreatesalargeradvancesalesprice p a =^ p a 42

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T able 3-4. SensitivityAnalysisforVarying p s Values( V Uniform (20; 80)) Figure 3-4. Graph of X a vs. E []( V Uniform (20; 80)) Icanconcludethatthelargestprotisattainedwhen p s ismaximum.ThusIhave anoptimalpricingpolicyof(p s H;p a =^ p a (p s ))andanoptimalorderpolicyof (Q = F 1 D s ( H c H ) + X a ;X a ). Thatis,Iusethehigh p s valuetodriveupthevalueof p a .Sincetheprobabilityofspotpurchase, F V (p s ),becomesverylowwithsuchahigh p s value,Iadvanceselltoeveryoneat p a Thebehaviorof E []in X a canbeseeninthegraphsinFigure 3-4 .Fromthese graphsIobservetheextremepointsolutionfor X a .Icanseethatthereissomethreshold p s valuebelowwhichadvancesalesarenolongerprotable.Indsimilarresultsfor various H and L values. LetustrytocomparetheseresultstothecasewithBernoullicustomervaluations.In therstsetoftrialsinSection 3.4,showninTable 3-1 ,Iset H =80and L =20andvary the values(where istheprobabilityofaspotpurchasewhen p s = H ).Inthecaseof Uniformcustomervaluations,theprobabilityofaspotpurchase, F V (p s )isafunctionof 43

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p s .In order tocomparemyresultswiththeBernoullicustomervaluationcase,Imustset p s H ,whichyieldsaverylowvaluefor F V (p s ).Therefore,Icancomparetherstrow inTable 3-1 withtherstrowinTable 3-4 .Icanobservethefollowing.Theadvancesales price p a =^ p a ishigherintheUniformcasethanintheBernoullicase.Theinventorylevels Q and X a arebothhigh.Theexpectedprot E []ishigherintheUniformcasethanin theBernoullicase.ThusIcanobservethatfor p s = H and H =80, L =20,and or F V (p s )verylow,IhavesimilarinventorypoliciesbuthigherprotintheUniformcase. Tofurthercomparethesetwocustomervaluationdistributions,Irepeatthetrialsfor various H L Spreadvalues,thistimewiththesamespotpriceandspotpurchase probabilityvalues.SinceIexaminethecasewhen p s = H forBernoullicustomer valuations,IwanttoonlycomparetheUniformcustomervaluationtrialsfor p s H or p s = H 1.Then,sincetheprobabilityofaspotpurchaseforUniformcustomer valuations, F V (p s ),isaectedbytheprice p s ,IwillsettheBernoullispotpurchase probability = F V (p s )forafaircomparison.IthuscomparethetrialsshowninTable 3-5. Iobservethatasthe H L Spreaddecreases,theoptimaladvancesalesinventory X a decreasesinbothcases.Theoptimalorderquantity Q isdecreasingintheUniform case,butconstantintheBernoullicase.Thisdierencecanbeexplainedbythedierence inthe Q expressiondescribedearlier,whereIhave Q fortheUniformcasesmallerthan Q fortheBernoullicase(seeTheorem 8).Themaindierencebetweenthesevaluation casesisinthebehavioroftheexpectedprot.IntheUniformcase,Ihaveexpectedprot E []decreasingasthe H L Spreaddecreases,whereasintheBernoullicase, E [] isincreasing.Thisdierenceisattributedtothecalculationof p a =^ p a .Since X a is highinbothcases,mostoftheprotcomesfromadvancesales,andthusisaectedby theadvancesalesprice p a .IntheUniformcase,theadvancesalesprice,whenthespot priceishigh( p s H ),istheexpectedvaluation E [V ]= H +L 2 whic h is constant.Inthe Bernoullicase,theadvancesalespriceisalsotheexpectedvaluation,butinthiscase E [V ]= H + L(1 )isafunctionof .Therefore,Ihaveexpectedprotincreasing 44

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T able 3-5. SensitivityAnalysisforVarying H L SpreadValuesforUniformand BernoulliCustomerValuations asthe H L SpreaddecreasesfortheBernoullicasebecausetheadvancesalespriceis increasing. Thus,ifthecustomervaluationdistributionisdierent,thestructuralresultshold (extremepointsolutionfor X a Q hasanewsvendorcomponentplus X a ),butthe sensitivitytothevaluationdistributionparametersmayvary. PleaserefertoChapter6fortherelatedconclusionsandfutureresearchextensions. 45

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CHAPTER4 MUL TI-GENERA TIONPRICINGANDTIMINGDECISIONSINNEWPRODUCT DEVELOPMENT 4.1IntroductionandMotivation Theanalyticmodelintroducedhereutilizesatwogenerationframeworkand incorporateselementsfromdynamicpricingandtime-to-marketbodiesofliterature. mykeydecisionvariablesarethedynamicpricingstrategyforeachgenerationofproducts andtheoptimaltimetointroducethesecondgenerationofproducts.Amainfactor inuencingthesedecisionsistheanticipatedshapeofthedemand/salescurveforeach generationofnewproducts.Forexample,theproductlifecyclecurveisoftenassociated withtheintroduction,growthanddeclineofaproductinthemarketplaceviasomekind ofdiusionprocess.Conversely,acommonassumptionintheliteratureaddressingthe optimaltime-to-marketfornewproductintroductionsisthatprice(andconsequently sales)isstaticforbotholdandnewgenerationsofproducts.Otherfactorsincludedin themodelarethedynamicunitcostsforeachgenerationaswellasthedevelopmentcosts associatedwiththesecondgenerationofnewproducts. Iutilizeoptimalcontrolmethodologiestocharacterizetheoptimalpricingstrategyfor bothgenerationsandthetimingfortheintroductionofthesecondgeneration.Analytic resultsforspeciccasesreectingdierentassumptionsconcerningthedemandprocessare developedwhichdirectlylinkthepriceandtimingdecisions.Whensalesaredependent onpriceonly(i.e.nodiusioneects),theoptimalpolicyistointroduceonlythesingle mostprotablegenerationsofproductsinmostsituations.Specically,Ieitherintroduce thesecondgenerationatthestartoftheplanninghorizon,ornotatall(i.e.anowor neverpolicy).Whensalesaredependentondiusiononly(i.e.notadirectfunctionof price),thentheoptimaltimingofintroductionofthesecondgenerationofproductsis dependentonthelengthoftheplanninghorizon.Ideriveathresholdvaluesuchthatif thelengthoftheplanninghorizonissmallerthanthethreshold,thenasinglegeneration solutionisoptimal.Ifthelengthoftheplanninghorizonexceedsthethresholdvalue,then 46

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itis optimal tointroducebothgenerationstomarketinasequentialmanner.Inthiscase, theoptimaltimetomarketforthesecondgenerationisdependentontheprice,cost,and diusionparametersforbothgenerationsofproducts. PleaserefertotheChapter2forareviewoftherelatedliterature. 4.2Model Mymodelconsiderstwogenerationsofanewproduct:Generation1andGeneration 2.IassumethatsalesforGeneration1startattime0,whilethemarketentrytime forGeneration2isadecisionvariableinthemodel.Iwanttodeterminetheoptimal dynamicpriceofeachgeneration, p 1 (t)and p 2 (t),aswellastheoptimaltimetointroduce Generation2tothemarket, t m IconsiderasinglerolloverscenariowherethesalesforGeneration1willstoponce Generation2isintroducedtothemarket.Bothpriceandcostaredynamicvariables inmymodel.IalsoincludeaxedcostforintroducingGeneration2tothemarket, c t m whichisaone-timexedcostincurredifandwhenIintroduceGeneration2tothe market.Thiscostmaybeattributedtodevelopmentneedsormarketingexpenses.I assumethatthermhasaxedtimehorizon, T .AlthoughIconsiderthetimehorizonto beexogenous,Iwilldiscusslatertheeectofitsvalueontheoptimalintroductiontimeof Generation2. Thus,Ihavethefollowingdecisionvariables: p 1 (t) unit price at time t ofcurrentGeneration1 p 2 (t) unit price at time t ofcurrentGeneration2 t m time at whic hGeneration2isintroducedtothemarket And I dene thefollowingnotation: 47

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T lengthof the planninghorizon c 1 (t) unit cost at time t ofcurrentGeneration1 c 2 (t) unit cost at time t ofcurrentGeneration2 c t m xed cost of introducingGeneration2tothemarketattime t m x 1 (t) sales rate at time t ofGeneration1 x 2 (t) sales rate at time t ofGeneration2 x 1 (t) cum ulativ e salesinthetimeinterval(0 ;t)ofGeneration1 x 2 (t) cum ulativ e salesinthetimeinterval(0 ;t)ofGeneration2 1 (t) marginal v alue ofsellingoneunitofGeneration1 2 (t) marginal v alue ofsellingoneunitofGeneration2 3 (t) marginal v alue ofintroducingGeneration2tothemarket Y ( t ) binary indicator v ariablecorrespondingtotime-to-market Y (t) = 8 > < > : 0 ; t < > : 1 ; t = t m 0; otherwise. the dirac delta function My ob jectiv eistodeterminetheoptimalpricesforthetwogenerationsandthe optimalintroductiontimeforthesecondgenerationsuchthattotalprotismaximized overthetimehorizon.TheobjectivefunctionisstatedmathematicallyinEquation 4{1.Idenetotalprotasthenetrevenueforbothgenerationsearnedoverthetime horizon,minusthecostofintroductiontomarketforthesecondgeneration.Notethat Iignorediscountinginmymodel.Thisisdoneforclaritypurposes.Idesiretofocuson theoptimalpricingandtimingdecisionsderivedfromasimpleexpressionofprot.Ihave conductedmyanalysisforthediscountingcaseandndsimilarresults.Thus,forthe scopeofthischapterIwillassumethereisnodiscounting,althoughmyresultsholdfor 48

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thediscoun ting case. Max Z T 0 (_ x 1 (t)(p 1 (t ) c 1 (t))+_ x 2 (t )(p 2 (t) c 2 (t)))dt c t m v (t m )(4{1) s: t: 1 ( t ):_ x 1 ( t )= f (x 1 (t);p 1 (t ))(1 Y (t )) (4{2) 2 ( t ):_ x 2 ( t )= g (x 2 (t);p 2 ( t ))Y ( t ) (4{3) 3 ( t ): Y (t)= (t t m )v (t) (4{4) TheconstraintinEquation 4{2 denesthesalesrateforGeneration1.Sincemy binaryindicatorvariable Y (t)isinitially0,thesalesrateforGeneration1ispositivefor t
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H = ( p 1 c 1 + 1 ) f (x 1 ;p 1 )(1 Y )+(p 2 c 2 + 2 )g (x 2 ;p 2 )Y (4{5) H I =( 3 c t m )v (t m ) (4{6) Iderivethefollowingexpressionsutilizingthenecessaryconditionsforoptimality. Therstexpressionsbelowarefortheadjointvariables,whichformymodelaredened astheratesofchangeforthemarginalvaluesassociatedwiththecumulativesalesforeach generation x 1 ( t )and x 2 (t). 1 = H x 1 = ( p 1 c 1 + 1 ) f x 1 (1 Y ) (4{7) = 8 > < > : ( p 1 c 1 + 1 ) f x 1 ;t < > : 0 ; t
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Thenext expressions correspondtothecontinuouscontrolvariables,whichinmy modelaretheoptimalpriceforeachgeneration. p 1 : H p 1 = 0 (4{12) f ( x 1 ;p 1 )+( p 1 c 1 + 1 ) f p 1 = 0 (4{13) p 2 : H p 2 = 0 (4{14) g ( x 2 ; p 2 )+(p 2 c 2 + 2 ) g p 2 = 0 (4{15) Notice thattheoptimaldynamicpricesdependonthefunctionalformsofthesales ratesforeachgeneration. Thenextexpressionscorrespondtotheimpulsecontrolvariable,inmymodelthe optimaltimetomarketforGeneration2. v ( t m ): H I v ( t m ) = 0 (4{16) H I v = 3 c t m (4{17) v ( t m ) = 8 > < > : 1; 3 c t m 0; 3
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arelinear additiv efunctionsofpriceandsalesdiusion.(RefertoPadmanabhanandBass (1993)[ 17],TengandThompson(1996)[ 16],Mahajan,Muller,andBass(1990)[ 34],and Kalish(1983)[ 12]forsimilarfunctions.) f (x 1 ;p 1 )= a 0 a 1 p 1 + a 2 [(M 1 x 1 )+ M 1 (M 1 x 1 )x 1 ](4{20) g (x 2 ; p 2 ) = b 0 b 1 p 2 + b 2 [ (M 2 x 2 )+ M 2 (M 2 x 2 )x 2 ](4{21) F urthermore, I denethefollowingvariablesassociatedwiththesalesfunctions: a 0 ; a 1 ; a 2 p ositiv e constants b 0 ; b 1 ; b 2 p ositiv e constants M 1 mark et size ofGeneration1 M 2 mark et size ofGeneration2 co ecien t ofinnovation co ecien t ofimitation In thefollo wingsubsections,Ianalyzespecialcasesofthesefunctions. 4.3.1CASE1:PriceEectOnly,NoDiusionEect FirstIconsidertheisolatedeectofpriceonsales.Letusassumethat a 2 = b 2 =0 suchthattherearenodiusioneectsimpactingthesalesfunction.Thesalesrate functionscanthenbewrittenasfollows: f (x 1 ;p 1 )= a 0 a 1 p 1 (4{22) g (x 2 ;p 2 )= b 0 b 1 p 2 (4{23) Theorem1:OptimalPricesforCASE1 Whenthesalesfunctionincludespriceseect only,theoptimalprice p 1 forGeneration1and p 2 forGeneration2are: p 1 = 1 2 ( a 0 a 1 + c 1 ) (4{24) p 2 = 1 2 ( b 0 b 1 + c 2 ) (4{25) 52

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Notethat the optimalpricesarelinearadditivefunctionsoftheunitcosts.Therefore, thepricechangesatasamerateovertimeasthecorrespondingcostforthatgeneration.I canalsoobservethatpriceisdecreasingconvexinthepriceweightconstants;thatis p 1 is decreasingconvexin a 1 ,and p 2 isdecreasingconvexin b 1 .Astheweightofpriceincreases, thesamepricewillhavealarger(negative)eectonsales(refertoEqns4.3and4.5). Therefore,theoptimalpricedecreasesinordertomaximizethetradeobetweentheprot marginandsales.Imayalsonotethatwhenunitcostsareconstant,thentheoptimal pricesarealsoconstant. Theorem2:OptimalTimetoMarketforCASE1 Whenthesalesfunctionincludes priceeectonly,theoptimaltimetomarket t m is: t m : Z T t m ( K 1 K 2 )dt c t m (4{26) WhereIdene K 1 =(p 1 c 1 )( a 0 a 1 p 1 ) and K 2 =(p 2 c 2 )( b 0 b 1 p 2 ) astheprotrateforeach generation. Notethatif K 1 >K 2 8t ,then 3 (t m ) < 0andthereforenevergreaterthan c t m > 0,so IneverintroduceGeneration2.If,however, K 2 >K 1 8t,then 3 (t m )isdecreasingin t m Inthissituation,if 3 (0) >c t m ,IintroduceGeneration2immediately.Butif K 2 >K 1 8t and 3 (0) K 1 attheendoftheplanninghorizon.Theshapeof 3 (t )inthissituationissuchthatit startsoutfairlylow,increasestoapeakat t = t s ,andthendecreasestozeroatt=T. If 3 (t s ) >c t m ,thentheoptimaltimetomarketforthesecondgenerationissometime before t s .ThisresultconcurswithCarrilloandFranza(2004)whondthattheoptimal 53

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time-to-market can occurbeforethemarginalvalueofsalesofthenewproductexceed thoseoftheoldproduct. Ifunitcostsareconstantovertheentireplanninghorizon,thenIhave 3 (t m )= (K 1 K 2 )(T t m ).InFigure 4-1,Iillustratethisbehaviorof 3 (t m ).Icanseeclearly, thatwhen K 2 >K 1 ,because 3 (t m )isdecreasing,Ihaveanoworneveroptimaltimeto marketdependingonthevalueof 3 (0)relativeto c t m Figure 4-1. Ev aluating 3 (t m ) c t m when K 2 >K 1 andcostsareconstant. Corollary1:OptimalTimetoMarketforCASE1UnderConstantCosts When theunitcostsareconstant,theoptimaltimetomarketisasfollows: t m = 8 > < > : 0(now) ; (K 2 K 1 )T c t m never ; (K 2 K 1 )TK 2 (4{27) Keepinginmindthat K 1 and K 2 arefunctionsofpriceandcost,Icanobservethat t m isdecreasingin c 1 andincreasingin c 2 .Thisisintuitivesincealargerunitcostfor Generation1willmakeitlessprotableandtherefore t m willdecrease,orshift,from "never"to"now",meaningonlyGeneration2willbesold.Likewise,anincreasingunit costforGeneration2willmakeitlessprotableandtherefore t m willshiftfrom"now"to "never",meaningonlyGeneration1willbesold.Itisclearthat t m isincreasingin c t m Thatis, t m shiftsfrom"now"to"never"ifthecostofintroducingGeneration2istoo high. Ialsoobservethat t m decreasesin a 1 andincreasesin b 1 .Similartothesensitivity ofpricetotheseparameters,astheweightofpriceincreases,thesamepricewillhavea 54

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larger(negativ e) eectonsales.Therefore,anincreaseintheseparametersmakesselling thecorrespondinggenerationlessprotableduetoanegativeeectonsalesandalsoan implicitdecreaseinprice,andthereforethetimeeachgenerationissoldonthemarket decreases.Soas a 1 increases, t m shiftsfrom"never"to"now",andas b 1 increases, t m shiftsfrom"now"to"never". Corollary2:OptimalTimetoMarketforCASE1UnderConstantCostswith HorizonThreshold Givenahorizonthresholddenedas T = c t m ( K 2 K 1 ) the optimal t m c anbe writtenasfollows: t m = 8 > < > : 0(now) ;T T never ;T< T or K 1 >K 2 (4{28) Here, T representsthetradeobetweentheprotmarginandthecostofintroducing asecondgeneration.Icanobservethat t m isdecreasingin T .Thatis,astheplanning horizonincreases,itismoreprotabletosellGeneration2only(giventhat K 1 K 2 ). Note:If K 1 = K 2 ,thenIhave T = 1 and T< T willalwaysbetrue.Thus,ifIhave equalgenerations,thenIonlysellGeneration1andneverintroduceGeneration2.This seemsintuitive;whenIhaveequalgenerations,apositiveintroductiontomarketcost c t m and 3 (t m )decreasingin t m bothmakeGeneration1tobethemoreprotablesingle generationtobesold. Insummary,thekeydriverdeterminingtheoptimaltimetomarketforalinearprice demandmodelistheunitmarginsforeachofthegenerations.WhenIconsideraprice eectonlyonsaleswithconstantunitcosts,theoptimaltimetomarketforGeneration 2iseithernowornever.Theoptimalpriceinthissituationisconstantoverthetotal planninghorizon.Thisconcurswiththeliteratureforsingle-generationmodelswitha price-onlysaleseect(seeKalish[ 12]). 55

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4.3.2CASE 2: DiusionEectOnly,NoPriceEect Inowconsidertheisolatedeectofdiusiononsales.Letusassumethat a 0 = a 1 =0 and b 0 = b 1 =0,yieldingthefollowingsalesratefunctions: f (x 1 ;p 1 )= a 2 [(M 1 x 1 )+ M 1 (M 1 x 1 ) x 1 ] (4{29) g (x 2 ; p 2 ) = b 2 [ (M 2 x 2 )+ M 2 ( M 2 x 2 )x 2 ] (4{30) Theorem 3: Optimal PricesforCASE2 Whenthesalesfunctionincludesdiusion eectonly,theoptimalprices p 1 and p 2 areequaltothemaximummarketprices ^ p 1 and ^ p 2 respectively. Theorem4:OptimalTimetoMarketforCASE2 Whenthesalesfunctionincludes diusioneectonly,theoptimaltimetomarket t m is: t m :[( p 2 c 2 )g (x 2 (T t m )) (p 1 c 1 )f (x 1 (t m ))](T t m ) c t m (4{31) Theaboveequationshowsthattheoptimal t m issuchthattheprotearnedforthe remainingtimefromintroducingGeneration2isgreaterthanthecostofintroducing Generation2tothemarket. Iamnotabletondaclosedformsolutionfor t m .However,Idoinvestigateits sensitivitytootherparameters.Ind 3 ( t m )tobedecreasingin p 1 c 2 ,and M 1 ,which yieldsalarger t m value.Thisresultimpliesthatforlarger p 1 and/or M 1 values,Iwantto sellGeneration1foralongertime.Similarly,forlarger c 2 values,IwanttosellGeneration 2forashortertime,andthus t m islarger.Indoppositebehaviorsforthecomplimentary parameters: 3 (t m )isincreasingin p 2 c 1 ,and M 2 .Inthiscase,IwanttosellGeneration2 foralongertimeso t m issmaller.Icanobserve,thatalthoughIdonotconsidertheprice eectonthesalesrateinthissection,theoptimalprices p 1 and p 2 stillimpacttheoptimal timetomarket t m IntheNumericalAnalysissection,Iinvestigatethebehaviorof t m undervarious parameterassumptions. 56

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4.3.2.1Time horizon threshold Anothersignicantparameterdrivingtheoptimaltimetomarketisthelengthof theplanninghorizon.Instudyingthebehaviorof 3 (t m ),Inoticethatforsmall T values, 3 (0) > 0andisthengenerallydecreasing.Consequently,itisoptimaltosellonlyone generationif 3 (0) >c t m .However,forlarger T values, 3 (0) < 0andisthengenerally increasing.(RefertoFigure 4-2 foranillustrationofthisswitchingbehaviorin 3 ( t m ).) Figure 4-2.The switchingbehaviorof 3 (t m )as T increases. Iconcludethatthereexistsathresholdvalueof T ,whichIcall T ,afterwhichthere isashiftinthebehaviorand 3 (t m ).Theexplanationofthisthresholdvalueissomewhat intuitive.Ifthetimehorizon( T )istooshort,thencumulativesalesbenetscannotbe fullyrealized.Specically,thepeaksalesmaynotbereached.Thusitmaybemore protabletosellonlyonegeneration.Whereasforlarger T values,morecumulativesales canbeaccruedandthusitmaybemoreprotabletointroducebothgenerations.This thresholdbehaviorrevealsthatalthough T isconsideredtobeaxedparameter,itplays animportantroleinthedeterminationof t m Theorem5:HorizonThresholdforCASE2UnderNegligibleIntroductionCost Whenthesalesfunctionincludesdiusioneectonly,andtheintroductioncostisnegligible 57

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(c t m =0 ), ahorizonthresholdcanbederived: T = log 0 B B B B B B B B @ (2 a 2 M 1 ( c 1 + p 1 ) + b 2 M 2 (c 2 p 2 )( + ) 2 ( + ) p b 2 M 2 (c 2 p 2 )(4a 2 M 1 ( c 1 + p 1 ) + b 2 M 2 (c 2 p 2 )( + ) 2 )) 2a 2 M 1 (c 1 p 1 ) 2 1 C C C C C C C C A b 2 ( + ) (4{32) Corollary 3: Optimal TimetoMarketforCASE3withHorizonThreshold Given thehorizonthreshold T denedinTheorem5,theoptimaltimetomarket t m canbewrittenas follows: t m 8 > < > : =0(now)or= T (never) ;T T > 0(later) ; T> T (4{33) Where,forthecasewhere t m > 0,Ihave t m asdenedinTheorem4. Therefore,theoptimaltimetomarketisdenedrelativetotheplanninghorizon threshold.Forrelativelyshortplanninghorizons,asinglegenerationisoptimal.Forlonger planninghorizons,itisoptimaltointroducetwogenerationsofproductstothemarket. Forthesinglegenerationoptimalsolution,todeterminewhetherGeneration1or Generation2shouldbesold,Imustcomparetherelativeprotmarginstothecostof introductiontomarket c t m .Forexample,underashortplanninghorizon( T< T ),if c t m islowandGeneration2hasahighprotmargin( p 2 c 2 ),thenitisoptimaltosell Generation2only,thatisintroducenow( t m =0).If,againforashorthorizon,both generationshaveequalprotmargins( p 1 = p 2 and c 1 = c 2 )andapositiveintroduction cost(c t m > 0),thenIshouldneverintroduceGeneration2( t m = T )andsellGeneration1 only. Furthermore,thisthresholdisdependentonprice,cost,andmarketcharacteristics forbothofthegenerations.Ind T tobedecreasingin p 1 c 2 ,and M 1 .Asmaller T value impliesthattheoptimalsolutionismorelikelytocontaintwogenerationsofproducts. 58

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Ican also observethat 3 (0)isincreasingintheseparameters.Thus,asconditions forGeneration1becomemorefavorable,Iammorelikelytoeitherneverintroduce Generation2( t m = T )forasmallersetof T valuesorintroduceGeneration2later (t m > 0)foralargersetof T values. Theoppositebehaviorisfoundfor p 2 c 1 ,and M 2 .Thatis,Ihavealarger T value, whichimpliesthatmore T valueswillfallunderthethreshold.Iobservethat 3 (0) isdecreasingintheseparameters.Thus,asconditionsforGeneration2becomemore favorable,Iammorelikelytoeitherintroducelater( t m > 0)foralargersetof T valuesor never(t m = T )forasmallersetof T values. Noticealsothatalthoughpriceisnotapartofthesalesratefunction,itdoes inuencethethresholdvalueandthustheoptimaltimetomarket.IntheNumerical Analysissection,Ialsoexaminethebehaviorof T undervariousparameterassumptions. 4.3.2.2Abenchmarkscenario Forfurtherinsightsintothisproblem,Ianalyzeabenchmarkscenario.Considerthe situationwhenbothgenerationsareequalinprice,cost,andmarketsize,andIassumethe introductiontomarketcostisnegligible: p 1 = p 2 c 1 = c 2 M 1 = M 2 ,and c t m =0.The expressionforthethresholdvalue T thenreducestotheexpressioninCorollary4. Corollary4:HorizonThresholdforCASE2UnderaBenchmarkScenario Under abenchmarkscenariowithequalgenerationsandzerointroductiontomarketcost,thehorizon thresholdisderivedtobethefollowing: T = log( 2 2 ) + (4{34) Corollary 5: Optimal TimetoMarketforCASE2UnderaBenchmarkScenario Underabenchmarkscenariowithequalgenerationsandzerointroductiontomarketcost,witha horizonthreshold T = log( 2 2 ) + the optimal timetomarketcanbewrittenas: t m = 8 > < > : T (never) ;T T T 2 ; T > T (4{35) 59

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Thus, according toCorollary5,forequalgenerations,IeithersellGeneration1only (for T< T )orIbisectthetimehorizonamongthetwogenerations. Ifurtherobservethat T=2isactuallyequivalenttothepeaksalestime t p : f (x 1 (t)) t = 0. Then, forvaluesof T> T ,withequalgenerations,Ihave t m = T=2whichislargerthan T= 2= t p .Thisbringsustoaconclusioncontrarytothecurrentliterature. Corollary6:OptimalTimetoMarketvsPeakSalesUnderaBenchmark Scenario Underabenchmarkscenariowithequalgenerationsandzerointroductiontomarket cost,itisoptimaltointroducethesecondgenerationafterpeaksalesarerealized. t m = T=2 >t p (4{36) Wherepeaksalesaredenedas t p : f ( x 1 (t)) t = 0 MahajanandMuller(1996)utilizenumericalanalysistosolveasimilarproblem withaninniteplanninghorizon,andndthata"now"or"atmaturity"(i.e.whenpeak salesarereached)ruleisoptimal.Akeydierencebetweenmymodels,however,isthat Iconsideraniteplanninghorizon.Consequently,myanalyticresultsshowthatduring aniteplanninghorizon,itisoptimaltointroducethesecondgeneration"now"or"after maturity"suchthateachgenerationisinthemarketplaceforanequivalentamountof time. 4.3.3CASE3:PriceandDiusionEects Inowconsiderthemostgeneralcaseforthesalesfunctioninwhichbothpriceand diusioneectsales.Here,Iassumethat a 0 ;a 1 ;a 2 and b 0 ;b 1 ;b 2 arepositiveconstants (6=0).Ithenhavethefollowingfunctionalforms. f (x 1 ;p )= a 0 a 1 p + a 2 [ (M 1 x 1 )+ M 1 (M 1 x 1 )x 1 ](4{37) g (x 2 ; p 2 )= b 0 b 1 p 2 + b 2 [ (M 2 x 2 )+ M 2 (M 2 x 2 ) x 2 ](4{38) 60

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Theorem6: Optimal PricesforCASE3 Whenthesalesfunctionincludesbothprice anddiusioneects,theoptimalprices p 1 and p 2 are: p 1 = 0 B B B B B B B @ (2 a 2 1 a 2 c 1 t 2 t 2 m ( t 2 m + a 2 t ( )t 2 m + t 2 (2+ a 2 ( + )t m )) p a 2 1 M 1 (4a 2 (a 0 a 1 c 1 ) t 2 m + M 1 (4 + a 2 t m (4 4 + a 2 ( + ) 2 t m ))) + a 1 (2 a 0 a 2 t 2 m (t 2 + t 2 m )+ M 1 (a 2 t( )t 2 m (2+ a 2 ( )t m ) + t 2 m (2+ a 2 t m ( +2a 2 t m )) t 2 (4+ a 2 t m (4 4 + a 2 ( 2 + 2 )t m ))))) 1 C C C C C C C A 2a 2 1 a 2 t 4 m (4{39) p 2 = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ b 2 M 2 ( T t )( T t m ) 2 ( 2 t + 2 t +2T ( + ) 2 t m ) 2 t 2 + T 2 +4tt m t m (2T + t m ) b 1 b 2 [ 2b 1 M 2 + v u u u u u u u t b 2 1 M 2 (M 2 (4 + 4 b 2 ( )( T t m ) + b 2 2 ( + ) 2 (T t m ) 2 ) +4 b 2 ( b 0 b 1 c 2 ) (T t m ) 2 ) ] T t m b 1 [2b 2 1 c 2 (t t m ) 2 (T t m ) +( )( t + T ) v u u u u u u u t b 2 1 M 2 ( M 2 (4 + 4 b 2 ( )( T t m ) + b 2 2 ( + ) 2 (T t m ) 2 ) +4b 2 (b 0 b 1 c 2 ) (T t m ) 2 ) ( t + t m ) + b 1 ( 2b 0 (t T )(t + T 2t m )( T t m ) M 2 ( )(4 t 2 T 2 +4Tt m + t 2 m 2t(T +3t m )))] 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A 2b 1 (T t m ) 4 (4{40) 61

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Theorem7: Optimal TimetoMarketforCASE3 Whenthesalesfunctionincludes bothpriceanddiusioneects,theoptimaltimetomarket t m is: t m : 0 B B B B B B B B B B B B @ (T t m )(( 2b 1 M 2 + b 1 b 2 M 2 ( + )(T t m ) + v u u u u t b 2 1 M 2 (M 2 (4 + 4 b 2 ( )( T t m ) + b 2 2 ( + ) 2 (T t m ) 2 ) +4b 2 (b 0 b 1 c 2 ) (T t m ) 2 ) ) 2 1 a 3 1 a 2 2 t 4 m (a 1 M 1 (2 + a 2 ( + ) t m ) + p a 2 1 M 1 (4a 2 (a 0 a 1 c 1 ) t 2 m + M 1 (4 + 4 a 2 ( ) t m + a 2 2 ( + ) 2 t 2 m ))) 2 ) 1 C C C C C C C C C C C C A 4b 3 1 b 2 2 2 (T t m ) 4 c t m (4{41) I cannot nd aclosedformsolutionfor t m .Thus,Iperformseveralnumerical experimentstofurtherstudythebehaviorof 3 (t m ).Iconsiderabenchmarkscenario,as inCASE2,withequalgenerationsand c t m =0.AsinCASE2,Indthat t m = T=2. However,IdonothaveathresholdbehaviorforthetimehorizoninCASE3.Inthe NumericalAnalysissection,Ifurtheranalyzethesensitivityof t m tovariousparameters underthebenchmarkscenario. 4.4NumericalAnalysis IperformnumericalanalysisforCASE2andCASE3.Iobservethebehaviorof thecumulativesales( x 1 and x 2 ),andsalesrates( f and g ).Ialsostudythesensitivity of t m T p 1 ,and p 2 toseveralparameters.Iusethefollowingparametervaluesfromthe benchmarkscenariodiscussedinsection4.2.2inwhichIconsiderequalgenerations.I assume p>c and0 ; 1;valuesfor and arevaluescommonlyusedinempirical literature. 62

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c 1 = c 2 =3 c t m = 0 M 1 = M 2 = 100 = 0 : 05 = 0 : 5 a 0 = b 0 = 0 a 1 = b 1 = 8 > < > : 0, CASE 2 1, CASE3 a 2 = b 2 = 1 F or CASE 2,Ihave a 1 = b 1 =0(nopriceeect)and a 2 = b 2 =1.ForCASE3,Ihave a 1 = b 1 =1and a 2 = b 2 =1. 4.4.0.1Cumulativesalesandsalesrate Letusbeginbystudyingthebehaviorofthecumulativesales( x 1 and x 2 ),andsales rates(f and g )forCASE2andCASE3. InCASE2,sincetheoptimalpricevaluesarethemaximummarketprice,Iset p 1 = p 2 =5.Fortheparametervaluesdenedabove,Ind T =8:373.Fornow,Iwill set T =10(sothat T> T )and t m =5(t m = T=2).Iusetheexpressionsderivedin 4.21,4.23,4.29,and4.30tocreatetheCASE2graphsbelow.Icanobserveanincreasein cumulativesalesandapeakbehaviorinsalesrateforbothgenerationsinFigures 4-3 and 4-4 ,respectively. Icancalculatethepeaksalesvaluetobe t p = T=2=8:373=2=4 : 187.Thus Icanconrmmytheoreticalndingsfromthebenchmarkcasethat,for T> T ,the optimaltimetomarket t m = T=2willbegreaterthanpeaksales.(Forexample,for T =9 > T =8:373,Ihave t m = T=2=4:5 >t p =4 :187.Likewise,for T =10 > T =8 :373, Ihave t m = T=2=5 >t p =4: 187.) InCASE3,forthegivenparametervalues,Icalculate T =89:9.Thus,Iinitiallyset T =90( T> T )andndthecorrespondingtimetomarkettobe t m =45(t m = T= 2). 63

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Figure 4-3. CASE 2:Behaviorofcumulativesales x 1 and x 2 overtime. Figure 4-4. CASE 2:Behaviorofsalesrates f (x 1 )and g (x 2 )overtime. Iusetheexpressionsderivedin4.42,4.43,4.52,and4.59tocreatetheCASE3graphs below.Iobservelinearlyincreasingcumulativesalesandaconstantsalesrateforboth generationsinFigures 4-5 and 4-6,respectively. LetusdiscussfurtherthecumulativesalesandsalesratebehaviorsforCASE2 andCASE3.InCASE2,thecumulativesalesareincreasingconvexlyandsalesrateis concave.Thesearethestandardbehaviorsforsaleswithdiusioneectonlyandcanbe foundinthediusionliterature. 64

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Figure 4-5. CASE 3:Behaviorofcumulativesales x 1 and x 2 overtime. Figure 4-6. CASE 3:Behaviorofsalesrates f (x 1 ;p 1 )and g (x 2 ;p 2 )overtime. InCASE3,Ihavebothdiusionandpriceeect.Indthatthecumulativesales becomeslinearandthesalesrateconstant.Thisisaverydierentandinteresting behavior.ItseemsthatthediusioneectscapturedinCASE2,areosetbytheprice eect.Thatis,forcumulativesales,havingapositiveweightcoecientforpricedecreases sales,andthusattenstheconvexcurvefoundinCASE2toalinearcurveinCASE3. Forthesalesrate,thepricecoecienthasanegativeeectonsalesratewhichissmaller thanthepositiveeectofthediusioncoecient,thusyieldingapositiveconstantsales rate.Thisbehaviormaybeduetomymodelassumptions.Thatis,inmysalesfunctions, 65

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theprice and diusioneectsareadditive.Thesegraphsillustrate,thatevenforthe benchmarksituationofequalgenerations,includingtheeectofpriceonsalesmakesa majordierenceincumulativesalesandsalesratebehavior. Ialsolookatthebehaviorof p 1 and p 2 overtimeforthisbenchmarkscenario.In Figure 4-7,Iseethatboth p 1 and p 2 followaconcavecurve.Thatis,theoptimalpricing behaviorappearstobeinitiallylowsoastoincreasethesales,andthenastheproduct diusestothemarketplaceitincreases,andthendecreasesagaintowardstheendofthe horizon.SinceIconsiderequalgenerationsinthisbenchmarkscenario, p 1 and p 2 are identical. Figure 4-7. CASE 3:Behaviorofoptimalprice p 1 and p 2 overtime. 4.4.0.2Sensitivityanalysis Inowconductasensitivityanalysisofmydecisionvariables t m p 1 ,and p 2 tothecost, populationsize,andpriceand/ordiusioncoecientparametersforCASE2andCASE3. Ivarytheparametersincreasinglyaccordingtothefollowingvalues: c 1 c 2 c t m = (0,5) M 1 M 2 = (20, 110) a 1 b 1 = (1,3) a 2 b 2 = (1,1.8) 66

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T able 4-1. SensitivityanalysisresultsforCASE2. T able 4-2. SensitivityanalysisresultsforCASE3. ThesensitivityresultsforCASE2andCASE3aresummarizedinTables6and7. ForCASE2,sincethehorizonthreshold T issensitivetotheselectedparametersand t m dependson T ,Irstanalyzethesensitivityof T andthencalculate t m foreachparameter value.Similarly,forCASE3,since p 1 and p 2 arefunctionsof t m ,forallparameter changes,Irstnd t m .(Note:Fortheseexperiments,whenmultiplesolutionsfor t m were found,Ievaluatedandcomparedthetotalprotforeachsolutionandchosethe t m which yieldedthemaximumprotvalue.) LetusrstexaminetheresultsforCASE2.As c 1 increases,sellingGeneration1 becomeslessprotableandso t m decreasesfromalargevaluecloseto T (almost"never") to"now"(Generation2only).Likewise,as c 2 increases,sellingGeneration2becomesless 67

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protableand so t m increases from"now"(Generation2only)to"never"(Generation1 only).As c t m increases,itbecomesmorecostlytointroduceGeneration2tothemarket andso t m increasesfrom T=2to"never"(Generation1only).Thisincreaseoccursquickly, illustratinghowsensitive t m isto c t m .As M 1 increases,itbecomesmoreprotabletosell Generation1,thus t m increasesfrom"now"(Generation2only)toahighvaluecloseto T (almost"never").Likewise,as M 2 increases,itbecomesmoreprotabletosellGeneration 2,thus t m decreasesfrom"never"(Generation1only)toalowvaluecloseto0(almost "now"). Toexplaintherelationshipbetween a 2 and t m (andlikewise, b 2 and t m )Imust re-considerthedenitionof a 2 and b 2 .Thesecoecients,whichrepresenttheweightof theeectofdiusiononsales,canalsobedenedasthespeedofdiusion.Thatis,as a 2 increases,thespeedofdiusionofsalesforGeneration1increases,meaningthatit takeslesstimetoreachmaximumsales.Thus t m decreasessinceGeneration2canbe introducedearlierwithoutlosingprotfromGeneration1.Likewise,as b 2 increases,the diusionrateforGeneration2increases,meaningGeneration2doesnotneedasmuch timeonthemarketinordertoreachmaximumsales,thus t m increases.(Note: t m only increasesordecreasesslightlyfrom T= 2.Sincethechangein t m issmall,itshowsthat theseparametershaveaminimaleecton t m .). Ialsolookatthesensitivityofoverallprottotheseparameters.ForCASE2,prot isdecreasingincosts.Thisisclearfor c 1 and c 2 sinceanincreaseinthesevaluesdecreases theircorrespondinggeneration'sprotmargin.ForthecostofintroducingGeneration2 tothemarket,sinceanincreasein c t m increases t m to"never",thismeansGeneration2 willnotbeoered,andthereforetotalprotsdecrease.Protisincreasinginpopulation size.Thisisintuitivesincealargerpopulationsizewillincreasethesalesrateandthus thecorrespondinggeneration'sprotwillalsoincrease,increasingtotalprot.Anincrease inthediusioncoecients a 2 and b 2 alsoincreasesprot.Sincethesecoecientsdirectly eectthesalesrate,thecorrespondinggeneration'sprotcomponentalsoincreases. 68

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For CASE 3,Iconsidertheeectoftheparameterson t m andthenimplicitlyon p 1 and p 2 .Indresultsforcosts,populationsize,anddiusioncoecientstobesimilar toCASE2.Toexplaintherelationshipbetween a 1 and t m (andlikewisebetween b 1 and t m ),Imayreferbacktotheeectsoftheseparametersonthetimetomarketand pricingdecisionsdiscussedintheanalyticalsectionforCASE1.Indthat t m decreases in a 1 andincreasesin b 1 .Astheweightofpriceincreases,thesamepricewillhavea larger(negative)eectonsales.Thatis,anincreaseintheseparametersmakesselling thecorrespondinggenerationlessprotableduetoanegativeeectonsalesandalso animplicitdecreaseinprice.Thereforethetimeeachgenerationissoldonthemarket decreases.Soas a 1 increases, t m decreases,andas b 1 increases, t m increases. ForCASE3,Ialsostudythesensitivityoftheoptimalprices p 1 and p 2 tothese parameters.Forcosts,Indthatasageneration'sunitcostincreases,itscorresponding optimalpricealsoincreases.Thisseemslikeanaturalreactiondrivenbythemaximize protobjective.Thus,as c 1 increases, p 1 increases,andas c 2 increases, p 2 increases.When thesegenerationcostsincrease,thecomplimentarygenerationhasadecreaseinitsprice variation,oritismorestableovertime.Forexample,as c 1 increases,thevariationin p 2 overtimedecreases.As c t m increases,Iseethat p 2 increasesinordertobalancethetotal marginofoeringGeneration2. Asthepopulationsizesincrease,thecorrespondinggeneration'spriceincreasesin ordertotakeadvantageofthelargermarket.Forexample,as M 2 increases, p 2 increases. Thecomplimentarygeneration'spricehasmorevariation,thatisitislessstableovertime. Forthesameexample,if M 2 increases, p 1 becomeslessstableovertime.AsinCASE1, as a 1 or b 1 increase,thecorrespondinggeneration'spricedecreasessincesalesbecome moresensitivetoprice.Asthediusioncoecients a 2 and b 2 increase,thecorresponding generation'spricesincrease.Thismaybemotivatedbytheincreasedsalesfromdiusion. Ialsolookatthesensitivityofoverallprottotheseparameters.ForCASE3,prot isdecreasingincosts.Thisisclearfor c 1 and c 2 sinceanincreaseinthesevaluesdecreases 69

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theircorresp onding generation'sprotmargin.ForthecostofintroducingGeneration2to themarket,sinceanincreasein c t m increases t m ,thismeansGeneration2willbeoered forlesstime,andthereforetotalprotsdecrease.Protisincreasinginpopulationsize. Thisisintuitivesincealargerpopulationsizewillincreasethesalesrateandthusthe correspondinggeneration'sprotwillalsoincrease,increasingtotalprot.Anincreasein thediusioncoecients a 2 and b 2 alsoincreasesprot.Sincethesecoecientsdirectly increasethediusionrate,thecorrespondinggeneration'sprotcomponentalsoincreases. Anincreaseinthepricecoecients a 1 and b 1 decreasesprotduetotheirnegativeeect onsales. Overall,thesenumericalresultssupportmyanalyticalconclusionsfor t m ,andin CASE3,for p 1 and p 2 .Thesensitivityofmydecisionvariablestotheselectedparameters seemsfairlyintuitive.CASE3doesoeramoreinterestingsetofresults.Icanobserve theimpliciteectoftimetomarketonoptimalprices,andtheeectonpricevariation inthecomplimentarygenerations.ForCASE2,changesintheseparametersmayeect whetheroneortwogenerationswillbesold.ForCASE3,usuallytwogenerationsare alwaysoptimal,althoughthetimetomarketmaychangesignicantly.Amanagermay considerthesensitivityoftimetomarketandpricingwhenchoosingornegotiatinghis unitcosts,decidingwhatpopulationsizetomarketto,orinstudyingtheweightofprice anddiusioninthemarket. PleaserefertoChapter6fortherelatedconclusionsandfutureresearchextensions. 70

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CHAPTER5 OPTIMAL NUMBER OFGENERATIONSFORAMULTI-GENERATIONPRICING ANDTIMINGMODELINNEWPRODUCTDEVELOPMENT 5.1Introduction Whenexamininganewproductdevelopment(NPD)marketingscenario,there areseveraloperationsmanagement(OM)andmarketingdecisionstobeconsidered. Operationsdecisionsmayincludeproductiontimingandquantity,capacityinvestments andconstraints,andtime-to-marketdecisions.Themarketingperspectivemayconsider pricingdecisions,andinvestmentsinqualityandinnovationspeed.TheNPDliterature hasexaminedthesedecisionsinaprimarilydisjointfashion.Thischaptercontributes totheOM/MarketingInterfaceliteraturebysolvingbothpricingandtime-to-market decisionssimultaneously.Inaddition,therehasbeenlimitedliteraturediscussingthe optimalinnovationspeed,orclockspeed,forNPDproducts.Inthischapter,Isolvefor theoptimalnumberofgenerationsofanNPDproduct,whensalesareafunctionofboth pricinganddiusion. Theremainderofthischapterisorganizedasfollows.InSection2,Idescribeindetail mymodelassumptionsandmathematicalobjective.InSection3,Iperformmyanalysis usingoptimalcontroltheorytoderiveexpressionsfortheoptimalnumberofgenerations andoptimalpricepergeneration.InSection4,Iperformnumericalexperimentsto examinethebehavioroftheseoptimalexpressionsaswellasasensitivityanalysistoall parameters. PleaserefertotheChapter2forareviewoftherelatedliterature. 5.2Model Iconsidermultiplegenerationsofanewproductsoldoveranitetimehorizon, T Eachproductofgeneration i isproducedataunitcost c i (t )andsoldataprice p i (t )to apopulationofsize M i .Eachnewgenerationisintroducedtothemarketataxedcost c t m i .Iconsiderasinglerolloverscenario,inwhichsalesofGeneration i willstoponce Generation i +1isintroducedtothemarket. 71

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Ourob jectiv eistodeterminetheoptimalpriceforeachgeneration, p i (t),theoptimal introductiontimeforeachgeneration, t m i ,andtheoptimalnumberofgenerations, n suchthattotalprotismaximizedoverthetimehorizon.Theobjectivefunctionisstated mathematicallyinEquation3.1. = n X i =1 Z t m i 0 x i (t )(p i (t) c i (t))dt n X i =1 c t m i v t m i (5{1) WhereIhavesalesratedenedas: x i = a 0 i a 1 i p i + a 2 i ( ( M i x i )+ M i (M i x i ) x i ) (5{2) I denetotal protasthenetrevenueforeachgenerationearnedoverthetime horizon,minusthecostofintroductiontomarketforeachgeneration.Theobjective functionmaybeconsideredasanextensionofthetwo-generationpricingandtiming modelfromChapter4.SinceIassumeanitetimehorizon T ,Idonotincludediscounting. Thesalesrateisanadditivemodelofbothpriceanddiusioneectsonsales.The diusioncomponentisbasedontheBass(1969)[ 10 ]diusionmodel.Ihave a 0 i a 1 i ,and a 2 i ascoecientsforinitialsales,pricingeect,anddiusioneect,respectively,foreach generation i Thevariable v t m i isanimpulsevariablewhichisusedtodeterminewhetherornot Generation i isintroducedtothemarket.Ifitisintroduced,thecorrespondingxedcost c t m i isincurred. Tosimplifythisnon-linearoptimizationproblem,letusassumeallgenerationstobe equal.ThusIhave: p i = p, c i = c c t m i = c t m ,_ x i =_ x whichimplies M i = M and x i = x.I thenhavethefollowingprotexpression: = n Z t m 0 x(t )(p(t) c(t))dt nc t m (5{3) Noticethattheimpulsevariable v t m i hasbeentransformedimplicitlyinthenumberof generationstobesold, n.Ourdecisionvariablesarenow p(t), t m ,and n. 72

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Basedon results fromthebenchmarkscenarioofbothCASE2andCASE3from Chapter4,Iassumethatallgenerations(ofequalcharacteristics)willbespacedout equallyinthetimehorizon.Thatis,Iassume t m = T=n.Thisassumptionisalsomade inCarrillo(2004)[ 22 ].Morgan,MorganandMoore(2001)[ 21]notethatequalspacingof generationsmaynotbeoptimal,butstillyieldsgoodsolutions.Carrillo(2005)[ 23]points outthattheempiricalliteraturesupportspacingnewproductsatregulartimeintervals. Iwouldliketonotethatmyassumptionsaremadebasedforatop-levelstrategic perspective.Iamconsideringanaggregateplanningmodeltodetermineanoptimal time-pacingstrategy.Thus,assumingequalgenerationspacedatequalintervalsprovides insightintothistop-levelstrategyforboththeoptimalnumberofgenerationsandthe optimalpricingpergeneration.Theseassumptionsalsoallowustoderiveclosedform expressionformyoptimaldecisions. Thus,Ihavethefollowingmodel: Max= n Z T=n 0 x(t )(p(t) c(t))dt nc t m (5{4) s.t. (t):_ x(t)= f (x (t );p (t)) (5{5) Wherethesalesratefunctionis: f (p (t );x (t ))= a 0 a 1 p(t)+ a 2 ( (M x(t))+ M (M x( t ))x(t))(5{6) A summary of mynotationisgivenbelow. 73

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n num b erofgenerationstointroduce t m time at whic hanewgenerationisintroducedtothemarket p (t ) dynamic price oered pergeneration T length of the planninghorizon c (t ) unit cost at time t ofthecurrentgeneration c t m xed cost of introducingthenextgenerationtothemarketattime t m x(t) sales rate at time t ofthecurrentgeneration x(t) cum ulativ e salesinthetimeinterval(0 ;t)ofthecurrentgeneration (t) marginal v alue ofsellingoneunitofthecurrentgeneration a 0 p ositiv e constantforinitialsales a 1 p ositiv e constantforpriceeectonsalesrate a 2 p ositiv e constantfordiusioneectonsalesrate M mark et size ofeachgeneration co ecien t ofinnovation co ecien t ofimitation 5.3 Analysis I solv emymodelusingOptimalControlTheory,sinceIhavethedynamicnonlinear optimizationfor p(t).OptimalControlTheoryisanon-linearoptimizationmethodology utilizedfordynamiceconomicproblems.Forasummaryofthemethodologyand applications,refertoSethiandThompson(2000)[ 15].Formymodel,thecontrolvariable isthepriceforeachgeneration, p (t ).Thestatevariableisthecumulativesalesforeach generation, x( t ).Theadjointvariableisthemarginalvaluedenedintheproblem constraint: (t).SimilartoGaimonandMorton(2005)[ 35],Iwillsolvefortheoptimal numberofgenerations, n ,usingrstorderconditions(f.o.c.)oftheprotexpression.I denetheHamiltonianasfollows.(Pleasenote:Fromthispointforward,Iremovethe 74

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timev ariable forclarity.) H =( + n(p c ))(a 0 a 1 p + a 2 ( ( M x)+ M (M x)x)))(5{7) T o nd theoptimalprice p (t ),Irstusethef.o.c.typicallyassociatedwithOptimal ControlTheorytoderiveanexpressionfor p intermsofcumulativesales x andthe marginalvalueofsales .Ithensolvefor x and usingsimultaneousdierential equationsandOptimalControlmethods.Ithensubstitutebackinthederived x and expressionstondtheexpressionfor p Theorem1:OptimalPriceperGeneration Forequalgenerationsandsalesrateasa functionofbothpriceanddiusion,theoptimalpricepergeneration p (t) is: p (t)= 0 B B B B B B B B @ 2a 2 1 a 2 ct 2 T 2 n ( T 2 n 2 + a 2 ( )t T 2 n 2 + t 2 (2 + a 2 ( + ) T n )) r a 2 1 M n 2 (4 a 2 ( a 0 a 1 c ) T 2 n 2 + M (4 + a 2 T n (4 4 + a 2 ( + ) 2 T n ))) + a 1 n (2a 0 a 2 T 2 n 2 ( t 2 + T 2 n 2 ) + M ( a 2 ( ) t T 2 n 2 (2 + a 2 ( ) T n ) + T 2 n 2 (2 + a 2 T n ( + 2 a 2 T n )) t 2 (4 + a 2 T n (4 4 + a 2 ( 2 + 2 ) T n )))) 1 C C C C C C C C A 2a 2 1 a 2 T 4 n 3 (5{8) Here Iha veassumedequalgenerationsoeredforequaltime,andthus t m = T=n .I mustnotethat a 1 a 2 ,and shouldneverbezero.[PleaserefertoAppendixCforproofs ofalltheorems.] Iamalsoabletoshowthattheoptimalprice p (t)isconcaveovertimeforconstant costsandrelativelylargeinitialsales.ThisresultissimilartothoseshowninKalish (1983)[ 12]forasinglegenerationofanewproduct. Corollary1:OptimalPriceConcaveoverTime Forequalgenerationsandsalesrate asafunctionofbothpriceanddiusion,assumingconstantcosts c ( t )= c andinitialsales relativelylargerthancostsuchthat a 0 a 1 c 0,theoptimalpricepergeneration p (t) isconcave overtime. 75

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Using p (t)as deriv edabove,Icanrestatemymaximumprotobjectivefunctionas follows. (p (t))= n Z T=n 0 x(p (t ))( p (t) c(t))dt nc t m (5{9) Again,assumingequalgenerationsoeredforequaltimetomarketforeach generation,Ihave t m = T=n .Makingthissubstitutionin,whens.o.c.forconcavity ofprotaremet,Icannd n fromthef.o.c.tobethefollowing. Theorem2:OptimalNumberofGenerations Forequalgenerationsandsalesrateas afunctionofbothpriceanddiusion,andassumingthateachgenerationisoeredforanequal amountoftime( t m = T=n),whenthesecondorderconditionsforprothold,theoptimalnumber ofgenerationstooer, n ,isthefollowing: n = ((8(3 (2=3) )a 2 0 a 1 a 2 c t m M 2 ( )+8(3 (2=3) ) a 3 1 a 2 cc t m M ( cM ( )+6 c t m ) 16(3 (2=3) ) a 0 a 1 a 2 c t m M (a 2 M 2 ( )+ a 1 (cM ( )+3c t m )) +3 (2 =3) a 2 1 a 2 2 c t m M 2 (16cM ( )+3 c t m ( 2 18 + 2 )) +3 (1 =3) K (2=3) + a 1 a 2 c t m M ( )(8(3 (2 =3) )a 2 2 M 3 2 9K (1 = 3) )) T ) 24a 1 c t m M K (1 = 3) (5{10) Wher eIdene K as: K = (a 2 1 a 2 c 2 t m M ( 192a 3 1 c 3 M +2 p 6 v u u u u u u u u u u t 0 B B B B B B B @ 1 a 1 c t m ((M ( a 0 a 1 c + a 2 M )(8( a 0 a 1 c ) 2 a 2 (9 a 1 c t m 16a 0 M +16 a 1 cM ) +8 a 2 2 M 2 2 )+9 a 1 a 2 c t m (a 1 (2 c t m cM ) + M (a 0 + a 2 M)) ) 2 (a 2 M 2 ( ) 3 24a 1 c t m 2 )) 1 C C C C C C C A 24M (a 0 + a 2 M) 2 (8a 0 + a 2 M (3 2 14 +3 2 )) 72a 2 1 ( 8a 0 c 2 M + a 2 (3 cc t m M ( ) +6 c 2 t m 2 + c 2 M 2 ( 2 10 + 2 ))) +9a 1 M (64 a 2 0 c +8 a 0 a 2 (3 c t m ( ) +2 cM ( 2 10 + 2 )) + a 2 2 M (16cM ( 2 6 + 2 )+ c t m ( )( 2 +22 + 2 ))))) (5{11) 76

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Notethat this resultwillonlybeoptimalifthes.o.c.ofprothold;thatis,Imust showthatprotisconcavein n inordertousef.o.c.tondtheoptimal n which maximizestheprotfunction.Iamnotabletoshowthatthiss.o.c.(specically 2 n 2 0) alw aysholdsnorderiveanalyticalconditionsunderwhichitholds.However,for specicnumericalinstances,Icaneasilycheckthiss.o.c.toensurethatthe n valuefound fromthef.o.c.isindeedoptimal.PleaserefertotheAppendixforacompleteexpressionof thiss.o.c.forprottobeconcavein n. Thisanalyticalexpressionallowsanymanagerwhocansetreasonableparameter valuestodeterminetheoptimalnumberofgenerationstooerinagiventimehorizon. Theoptimalnumberofgenerations, n ,willalsodeterminetheoptimalamountoftime eachgenerationshouldbeonthemarket: t m = T=n 5.4NumericalExperiments Inowperformnumericalexperimentstodrawfurtherinsightsfrommyanalytical results.Iconsiderabenchmarkscenarioofequalgenerationswiththefollowingparameter values. T = 100 c = 3 c t m = 50 a 0 = 25 a 1 = 1 a 2 = 1 M = 100 = 0 : 05 = 0 : 5 Note that I assumethexedcostpergeneration, c t m ,ismuchhigherthanthe unitcost, c.Iwouldalsoliketoexplainthatbasedontheotherparametervalues,the parameter a 0 musthavearelativelyhighvalueinordertohavepositivesalesvaluesover timeforthepricecomponentofthesalesrate( a 0 a 1 p (t ) 0, 8t). 77

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Usingthese parameter values,Ifoundtheoptimalnumberofgenerationstobe n =24:015,yieldinganoptimaltimetomarket t m = T=n =4:164.Thus,underthese parametervaluesforequalgenerations,amanagerwouldsell25generationsfor4.164time unitseach.Theprotachievedwiththisoptimalstrategyis$28,679. Protasafunctionof n isgraphedinFigure 5-1 .Iamabletocheckthes.o.c.of protundertheseparametervaluesandndprottobeconcavein n for n 41: 25.Thus, my n valueof24 :015foundfromthef.o.c.resultisindeedoptimal. Figure 5-1. Prot asafunctionof n. TheoptimalpriceovertimeisshowninFigure 5-2 .Theoptimalpriceattime t =0 is p (0)=$13: 07.Pricethenincreasestoamaximumvalueof ^ p (t)=$23:19,and decreasesto p (t m )=$20:21beforethenextgenerationisintroduced.Thisconcave increasing/decreasingbehaviorofoptimalpricereectsaninitialattempttoattract customersfollowedbyanincreaseinpriceonceasolidmarketpositionisattained nishingwithadecreasingpriceasthediusionofsalesdeclines. Icanbetterunderstandthebehavioroftheoptimalprice p (t)byexaminingthe behaviorofthesalesrate f (p(t);x(t))overtime.RecallthatfromEquation3.2thatthe 78

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Figure 5-2. Optimal Price p (t),overtime. salesrate f (p;x )iscomprisedofanegativepricecomponentandapositivediusion component.IgraphthesetwocomponentsseparatelyinFigure 5-3. Figure 5-3. Price ComponentandDiusionComponentofSalesRate,overtime. Overtime,thediusioncomponentofthesalesrateisincreasingthendecreasing. ThisiscomparabletotheliteratureonNPDdiusionandtheBassModel(seefor 79

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exampleBass (1969) [ 10]andNortonandBass(1987)[ 11 ]).Ibelievethatthisdiusion behaviorofsalesaectstheoptimalprice,suchthatitisalsoincreasingthendecreasing. Iobservethatthetimeatwhichthemaximumoptimalpriceisoered, ^ t ,isequaltothe timeatwhichpeaksalesfromthediusioncomponentoccur, t p =2:7.SinceIhave anegativepriceeectonsalesplusapositivediusioneect,mysalesratebecomes constantat f (p;x )=16:93(seeFigure 5-4 ). Figure 5-4. Sales Rate f (p;x ),overtime. Iperformasensitivityanalysisonseveralparametersaectingtheoptimalnumberof generations(andthusalsotheoptimaltimetomarket),theoptimalpriceovertimeand maximumpriceoered,andthetotalprot.Ivarythefollowingparametervaluesoverthe rangespeciedbelow: T = (10,150) c t m = (1,150) c = (0,10) M = (10, 150) a 0 = (22, 40) a 1 = (1,6) a 2 = (1,6) 80

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T able5-1. SummaryofSensitivityAnalysisResults. Asummaryofmyresultsisshowninthetablebelow. Iwilldiscusstheseresultsoneparameteratatime.Foreachparameter,Iwill discussthesensitivityofthedecisionvariableforthechangesinthatparametervalue. ReferringtoTable5-1,Iwillexamineonerowatatimeanddiscusstheresultsshownin thecorrespondingcolumnvariables.Asaquicknote,amongthevariablesthatIexamine (columnsinTable5-1),theprice p(t)istheonlydynamicvariable.Isummarizethe overallchangeinbehaviorforpriceovertime.Iexplaininmoredetailsbelowwhateachof thesedescriptionsimply. Letusrstdiscusstheparameter T ,thetimehorizon.Asthetimehorizonincreases, theoptimalnumberofgenerationstooer, n ,increases.Thatis,themoretimeIhaveto sell,themoregenerationsIwilloer.SinceIassume t m = T=n ,andsince n isincreasing linearlyin T ,Ind t m toremainconstantas T increases.Thatis,thetimeIoereach generationisnotaectedbythelengthofthehorizonsinceIamincreasingthenumberof generationsoered. Indprotincreasingin T ;thisisexpectedsinceasthenumberofgenerations increasessodomytotalsales.Indtheoptimalpriceovertimetobeunaectedby changesin T ,andthusthemaximumpriceisconstant. Forincreasingvaluesof c t m ,Ind n tobedecreasing.Thatis,themorecostlyit istointroduceanewgeneration,thefewernumberofgenerationswillbeintroduced. 81

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SinceI ha veaninverserelationshipbetween n and t m (t m = n =T ),Ithusnd t m to beincreasing.Thatis,ifIoerfewernumberofgenerationsoverthetimehorizon,then eachgenerationwillbeoeredformoretime.Asdevelopmentcostsincrease,protis decreasing. As c t m increases,Indanincreasingshiftintheoptimalprice.Thatis,theconcave behaviorofpricestillholds,buttheentirepricecurveshiftsupwardascostsincrease. Thiscreatesalargermaximumpricetocounterthehighercosts.Iillustratethisaecton priceinFigure 5-5.Noticetheoverallincreaseinthepricecurvefromthesolidlinetothe dashedline.Althoughpriceisincreasingindevelopmentcosts,itdoesnotseemtobevery sensitive;thatis,overalargeincreasein c t m thereisaslightincreasein p (t )overtime. Figure 5-5. Increasing ShiftinPriceCurve: p ( t )vs. c t m Similarly,asunitcosts, c ,increase,thenumberofgenerationsoered, n ,decreases andthusthetimeeachgenerationisoered, t m ,increases.Intuitively,ascostsincrease, protdecreases.Inresponsetoincreasing c ,Indthatpricesalsoincreasethroughouttm; thatis,thepricecurveshiftsup.Themaximumpricethusalsoincreases. 82

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For mark etsize, M ,thelargermymarket,thefewernumberofgenerationsIshould oer.Thisimpliesthat t m isincreasingin M while n isdecreasing.Thisresultreectson theimpactofdiusiononsales.Thatis,thelargermymarketsize,themoretimeIshould allowmyproducttodiuseintothemarket.Thus,Indahigher t m value.Eventhough thenumberofgenerationsdecreasesinmarketsize,mytotalprotincreasesfromthe positiveeectofincreased t m onsales.Indpricetohavemorevariationovertimeas M increases.Thatis,thepeakinthepricecurveissteeperwithlargervaluesof M ,meaning thatforlargerpopulationsizethechangeinpriceovertimeismoredramatic.Thislarger variationinpricesmaybepossiblesincealargermarketmayincreasetheprobabilityof sales.Thismayalsoexplaintheincreaseinthemaximumpriceoered. Lastly,Iexaminethesalesratecoecientparameters.For a 0 ,theinitialsales,an increasingvalueleadstomoregenerationsandadecreasingtimeonthemarketforeach generation.Theprotisincreasing,asisprice.For a 1 ,thecustomer'ssensitivityto price,anincreaseleadstofewergenerationsbeingoeredformoretimeeach.Thatis,as customersbecomemoresensitivetopricetheyarelesswillingtobuyatthesameprice andthusfewerproductsareoered.Bothprotandpricedecreaseas a 1 increases.For a 2 ,thespeedofdiusion,anincreaseyieldsalargernumberofgenerationsoeredforless timeeach.Thatis,asdiusionspeedincreases,theamountoftimenecessarytogainsales isless,andthus t m decreases,drivingup n .Sincethenumberofgenerationsincreases, andthustotalsales,protisalsoincreasingin a 2 .As a 2 increases,Indmorevariationin theoptimalprice.Thatis,thechangeinpriceovertimeismorenoticeable.Thiscreates alargermaximumprice.IillustratethisaectonpriceinFigure 5-6 .Asthepricecurve changesfromthesolidlinetothedashedline,variationinthepricevalueincreases.(In Figure 5-6,notethatthepricecurvefor a 2 =6actuallyendsat t =15,whichisthe correspondinglengthoftimeonthemarket, t m ,forthis a 2 value.) Overall,thesenumericalresultssupportmyanalyticalconclusions.Thesensitivityof mydecisionvariablestotheselectedparametersseemsfairlyintuitive.Icanobservethe 83

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Figure 5-6.Increase inPriceVariation: p (t)vs. a 2 eectoftheoptimalnumberofgenerations,orlikewisetheoptimaltimetomarket,onthe optimalpriceovertime,variationinprice,andmaximumpriceoered.Amanagermay considerthesereportedsensitivitieswhendetermininghistimehorizon,negotiatinghis unitcosts,decidingwhatpopulationsizetomarketto,orinstudyingtheweightofprice anddiusioninthemarket. PleaserefertoChapter6fortherelatedconclusionsandfutureresearchextensions. 84

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CHAPTER6 CONCLUSIONS In this chapter,Ireviewtheconclusionsofeachoftheresearchchaptersandhighlight howtheresultsofeachresearchcontributetotherelatedliterature.Ialsodiscuss futureresearchextensionsforeachoftheseresearchtopicsaswellasforthegeneral OM/MarketingInterfaceresearch,whichisthefocusofthisdissertation. 6.1InventoryManagementunderAdvanceSelling:OptimalOrderand PricingPolicies Inthispaper,Iconsiderinventoryorderandpricingdecisionsforanadvanceselling marketingstrategy.Intheadvancesellingstrategy,therearetwoperiods:theadvance salesperiod,succeededbythespotsalesperiod.Iassumethatconsumptionoccursatthe endofthespotperiod.Iassumethatanorderquantity Q isplacedatthebeginningof theadvancesalesperiodandsomeportionofthisinventory X a isallocatedapriorifor advancesales.Iannounce Q, X a ,andtheprices p a and p s beforetheadvancesalesperiod. Customersdecidewhetherornottopurchasebasedontheirvaluation V ofthe product.Iassumethisvaluationisnotrealizeduntilthebeginningofthespotperiod.In theadvancesalesperiod,customerscomparetheirexpectedutilityofadvancepurchasing versuswaitingtospotpurchase.Theseexpectedutilityexpressionsincludetheexpected futurevaluationandtheprobabilityofndinginventory( ).Ideriveamaximumadvance salesprice^ p a forwhichtheexpectedutilityofanadvancepurchaseisgreaterthanor equaltotheexpectedutilityofwaitingtopurchaseinthespotperiod.IndthatifIset myadvancesalespricetothismaximumprice,Icaninducealladvancesalescustomers topurchase.Ithusassumethatupondecidingtheadvancesalesinventorylevel, X a ,I willadvancesellto N a = X a customers,ofwhomallwilladvancepurchaseattheprice p a =^ p a Ithusseektodeterminetheoptimalorderpolicy( Q ;X a )andoptimalpricingpolicy (p a ;p s )whichmaximizedtotalexpectedprot E []attainedbybothadvancesalesand spotsales. 85

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T able 6-1. SummaryofSensitivityAnalysisTrials AssumingaBernoullicustomervaluationdistribution,Indtheoptimalorder quantity Q foragiven X a forboththecasewhen p s = H and p s = L.When p s = L, Q = M underagivenconditionand Q =0,otherwise.When p s = H Q hasastandard newsvendorcomponentplustheadvancesalesinventory X a Ithenndtheoptimaladvancesalesinventorylevel X a foragiven Q forboth p s cases.Inbothcases,Indanextremepointsolutionfor X a .When p s = L,Ind X a equaltoitsupperbound Q underagivenconditionand X a =0,otherwise.When p s = H Ind X a equaltoitsupperbound M k 2 (1 ) < Q under a givenconditionand X a =0, otherwise. Indtheoptimalorderpolicy( Q ;X a )bysolvingfor X a aftersubstituting Q = Q intotheexpectedprotexpression.Indtheoptimalpolicywhen p s = L tobe (Q ;X a )=(M; 0);thatisIneveradvancesellwhenthespotpriceislow.For p s = H ,I nd( Q ;X a )=(M;M k 2 (1 ) ) (adv ance selltoalmosteveryone)underagivencondition and(Q ;X a )=(F 1 D s ( H c H ); 0) otherwise. I determine theoptimalpricingpolicytobe( p a ;p s )=( L +(H L);H )foragiven conditionand( p a ;p s )=(L;L )otherwise. 86

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Ip erform severalnumericalexperimentstoanalyzethesensitivityofmyanalytical resultstothecustomervaluationparameters H ,and L,andtobetterunderstand thebehaviorofexpectedprotintheadvancesalesinventorydecision, X a .Iperforma sensitivityanalysisontheeectofthevaluationprobability andthespreadbetween thehighandlowvaluationlevels( H L spread)ontheoptimalvalues Q X a ,andthe expectedprot.IalsoperformexperimentsforaUniformcustomervaluationdistribution. AsummaryofalloftheseexperimentsisshowninTable 6-1. Mycontributionstotheliteratureincludeanalyzingtheadvancesalesinventory decision X a aswellasthetotalorderdecision Q.Ishowanextremepointsolutionforthe advancesalesinventory X a leadingtoa"go/no-go"advancesalesdecisions.Ialsoperform extensivenumericalexperimentsonthesensitivityanalysisofthecustomervaluation parameters H ,and L.Indathresholdbehaviorinthe and H L Spreadvaluesin determiningwhetherornottoadvancesell.Ialsoextendmystudytoconsideradierent customervaluationdistribution. Ibelievefurtherresearchcanbedonetoexamineothercustomervaluationdistributions. Futureresearchmayalsoconsideradditionalinventorycostparameters,suchasgoodwill penaltyandsalvagevalue. 6.2Multi-GenerationPricingandTimingDecisionsinNewProduct Development Inmyresearch,Istudytheoptimaltimetomarketandpricingdecisionsofatwo generationproduct.Iconsiderthreecasesformysalesfunction:CASE1,priceeectonly; CASE2,diusioneectonly;andCASE3,priceanddiusioneect. Idevelopanalyticalexpressionsfortheoptimaldecisionsinallthreecases.ForCASE 1,Inda"now"or"never"resultfortimetomarket.Theoptimaltimetomarketis alsoafunctionoftheoptimalpriceswhicharelinearincosts.ForCASE2,Iderivea thresholdvalueforthetimehorizonwhichdetermineswhetherGeneration2willbe oeredsometimewithinthehorizonornever.Ialsondthatwhentwogenerationsare 87

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sold,the optimal introductiontimeforGeneration2occursafterpeaksalesofGeneration 1.ForCASE3,Ideriveoptimalpriceexpressionsandndthatatwo-generationscenario seemstobeoptimalalmostallofthetime. Iperformnumericalanalysisforanequalgenerationbenchmarkcase,which supportsmyanalyticalresults.Iexaminethesensitivityofmydecisionvariablesto costparameters,populationsize,andpriceanddiusionweightparametersforCASE2 andCASE3. Iamconsideringsomeextensionstothisresearch.Onesuchextensionwouldbeto modifymysalesfunctionsuchthatpriceanddiusioneectsarenotsimplyadditive. Thisshouldresultinasalesratethatisnotconstant,asIfoundinCASE3.Toadjust mymodel,Imayconsiderreplacingthediusioncoecient, a 2 and b 2 ,withsomefunction ofprice: A 2 (p 1 )and B 2 ( p 2 ).Anotherextensionwouldbetoconsidertheproblemof determiningtheoptimalnumberofgenerationstointroduceinagiventimehorizonunder myjointpricingandtimingdecisionmodel. 6.3OptimalNumberofGenerationsforaMulti-GenerationPricingand TimingModelinNewProductDevelopment InChapter5,Ilookatstrategicdecisionsforamulti-generationnewtechnology productwithsalesasafunctionofbothpriceanddiusion.Ideriveananalytical expressionfortheoptimaldynamicprice p (t)andalsoshowthatpriceisconcaveover time.Ialsoderiveananalyticalexpressionfortheoptimalnumberofgenerations n given thatthesecondorderconditionshold.Iassumeequalgenerationsandeachgenerationto beoeredforanequalamountoftime,thusfrommyresultfor n ,Iamabletodetermine t m = T=n .Myanalyticsemployoptimalcontroltheory.Anextensivenumericalanalysis isalsoperformed. Ibelievemyanalyticalresultsaswellasmynumericalexperimentscontributeto thenewproductdevelopment(NPD)literatureandOperationsManagement(OM)/ MarketingInterfaceliteraturebyhighlightingthedecisionoftheoptimalnumberof 88

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generationsto in troduce.Incomparisonwithinnovationspeed,orclockspeed,papers, Iconsideraspecicadditivemodelofsaleswithbothpricinganddiusioneects andsimultaneouslysolveforoptimalpricing,timing,andnumberofgenerationsfora maximumprotobjective. Asanextension,Imayconsidertheclockspeed n asafactorinmysalesratefunction. Thisisbasedonsomeoftheliterature(mostlyempirical)whichhascommentedthatan increasedclockspeedcancreateapositiveperceptionoftheproductonthemarket.Imay haveasalesrateincreasinginclockspeedupuntilsomethresholdandthendecreasing beyondsomenumberofgenerations.Considerthefollowingsalesrate: x = a 0 a 1 p + a 2 ( (M x)+ M (M x)x) + S (n ) (6{1) Where S ( n)issomefunctionof n.Imayalsoconsiderreplacingtheconstant a 2 witha function A 2 (p)dependentonprice.Thismayreplacetheadditivecontributionofpriceto saleswithafunctioninwhichthediusioneectincludesamultiplicativepriceeect. Mymodelmayalsobere-examinedtoconsiderthescenariowhengenerationsare notoeredforequaltime,thussolvingfor n and t m separately.Themodelcanalsobe expandedtoconsideruniquegenerationsandtheoptimalvaluesfor t m i ,and p i (t)foreach generation i Onecouldalsoconsideracapacityconstraintonsales.Thatis,thesalesrate_ x( t ) Z ,where Z issomexedcapacity.Anotherextensionwouldbetoexaminethemodel underaninnitetimehorizon T = 1.Thiswouldrequireaddingdiscountingtotheprot objective. 6.4OM/MarketingInterface TheOM/MarketingInterfaceresearchareaisnewandquicklyexpandingresearch area.Therehasbeensubstantialevidencefortheneedtoconsidermulti-disciplinary perspectiveswhenmakingsupplychaindecisions.Thegoalofthisdissertationhasbeen toexaminevariousresearchquestionswhichcombinebothoperationsmanagement 89

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andmark eting decisions.TheOMquestionsIconsiderincludeinventorymanagement andNPDtimingandproductiondecisions.Iincludethemarketingdecisionofpricing ineachoftheseOMscenarios.Ifeelthatmyworkhasmadeacontributiontothe growingOM/MarketingInterfaceresearchliterature,andIhopethatmyresultscanbe implementedbysupplychainmanagers. 90

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APPENDIXA PR OOFS FORCHAPTER3:INVENTORYMANAGEMENTUNDERADVANCE SELLING:OPTIMALORDERANDPRICINGPOLICIES ProofofTheorem 1 :Optimal Q forBernoulliCustomerValuationswith p s = L Iderivetheoptimalinventory Q asfollows. E [(L)]=(L +( H L) ( M Q M X a ) c a )X a +L(Q X a ) cQ (A{1) s.o.c. E [( L )] Q ] = X a (H L ) M X a + L c (A{2) 2 E [(L)] Q 2 ] = 0 (A{3) Pro ofofTheorem 2 :Optimal Q forBernoulliCustomerValuationswith p s = H IapproximatethespotdemandwithaNormaldistributionwithmean D s = (M X a ) andstandarddeviation D s = p (M X a ) (1 ). I deriv etheoptimal 91

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inv en tory Q asfollows. E [(H )]=( L +( H L) c a ) X a +H ((M X a ) D s (Q X a )) cQ (A{4) s.o.c. E [(H )] Q ] = H ( F D s (Q X a ) 1) c (A{5) 2 E [(H )] Q 2 ] = H f D s ( Q X a ) 0 (A{6) f.o.c. 0= H (F D s (Q X a ) 1) c (A{7) F D s (Q X a )= H c H (A{8) Q = F 1 D s ( H c H ) + X a (A{9) Pro of ofTheorem 3 :Optimal X a forBernoulliCustomerValuationswith p s = L Ishowexpectedprot E [(L)]tobeconvexinoptimaladvancesalesinventory X a asfollows. E [(L)]=(L +( H L ) ( M Q M X a ) c a )X a +L(Q X a ) cQ (A{10) s.o.c. E [(L )] X a ] = ( H L ) ( M Q) (M X a ) (1 + X a M X a ) c a (A{11) 2 E [(L)] X 2 a ] = 2( H L ) ( M Q) (M X a ) 2 (1 + X a M X a ) 0(A{12) Assuming H L and X a Q M the s.o.c. isalwayspositive,andthustheexpected prot E [(L)]isconvexin X a 92

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Ithen compare theextremepointsolutionsof X a = Q and X a =0toderivethe followingcondition. E [(L;X a = Q)] E [(L;X a =0)] (L +(H L ) c a )Q cQ Q ( L c) ((H L) c a )Q 0 (H L) c a (A{13) ProofofTheorem 4 :Optimal X a forBernoulliCustomerValuationswith p s = H Ishowanextremepointsolutionfor X a Ideterminethat X a mustbeanextremepointiftheexpectedprotfunctionisnever increasingthendecreasing. Claim: If E [(Q ;X a +1)] E [(Q ;X a )] then E [(Q ;X a +2)] E [(Q ;X a +1)], 8X a (A{14) WhereIdenethefollowing: E [(H )]=( L +(H L) c a )X a +H ((M X a ) D s (Q X a )) cQ D s (Q X a ))= Z 1 Q X a (t Q + X a )f D s (t)dt f D s (t )= 1 D s p 2 e (t D s ) 2 = 2 2 D s D s = ( M X a ) D s = p (M X a ) (1 ) 93

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SinceI kno w Q = F 1 D s (X a ) H c H + X a ,I can furtherdenethefollowing: D s (Q X a ))= D s (F 1 D s (X a ) H c H )) (A{15) F 1 D s ( X a ) H c H = D s + D s k (A{16) Where I dene k = p 2er f 1 (2( H c H ) 1) as a constant. Examiningtheconjectureofmyclaim,Ihave: E [(Q ;X a +1)] E [(Q ;X a )] (A{17) H D s ( X a +1) (F 1 D s (X a +1) H c H ) H D s ( X a ) (F 1 D s ( X a ) H c H ) +ck X a +1 ( L (1 ) c a + c (1 )) +ck X a (A{18) Since D s isalwaysdecreasingin X a anditisreasonabletoassumethat c a L + c thisconjecturewillbetruewhen D s ( X a +1) D s (X a ) .Thatis,expectedprotis increasingwhen D s ( X a ) isdecreasing. Thus,toprovemyClaim,IshowthefollowingLemmatobetrue. Lemma: If D s (X a +1) (F 1 D s (X a +1) H c H )) D s (X a ) (F 1 D s (X a ) H c H ) then (A{19) D s (X a +2) (F 1 D s ( X a +2) H c H )) D s (X a +1) (F 1 D s (X a +1) H c H ) 94

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Ideriv e thefollowingconditionsfor D s (X a ) tobedecreasingin X a X a D s ( X a ) ( F 1 D s ( X a ) H c H ) = X a D s ( X a ) ( D s + D s k ) 0 (A{20) if (1 ) D s k 1=2 (A{21) and 1 D s (2 D s + 3 k D s ) 2 k 2 D s (A{22) and 1 2 2 D s D s + k D s (A{23) These conditions reducetothefollowing: c 0:31H (A{24) D s 1 4 k 1 8 (A{25) LB U B (A{26) where LB = 8 > < > : 1 q 1 D s k 1=2 2 for c H 0:00169 1 D s (2 D s +3k D s ) 2 k 2 D s otherwise (A{27) U B = 1 + q 1 D s k 1=2 2 (A{28) Therefore, for within the boundsdenedabove,Ihave D s (X a ) tobedecreasing in X a ,andthusexpectedprottobeincreasingin X a .FrommyClaim,iftheexpected protisincreasing(thatis meetstheconditionsdenedabove,thenitisalways increasing.Thus,Ihaveanextremempointsolutionfor X a 95

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Ithen compare theextremepointvaluesof X a todeterminewhichpolicywillyield thelargestexpectedprot. E [(H )]=( L +(H L) c a )X a +H ((M X a ) D s (Q X a )) cQ (A{29) E [(H;X a = Q)]=(L +(H L) c a )Q +H ((M Q ) D s (0)) cQ (A{30) E [(H;X a =0)]= H ((M ) D s (Q)) cQ (A{31) (A{32) E [(H;X a = Q )] E [(H;X a =0)] (A{33) Q[L (1 ) c a ] H [ D s D s (Q)] (A{34) Since D s (0) (0)= R 1 0 tf D s (t )dt = D s ProofofTheorem 5 :OptimalPolicy (Q ;X a ) forBernoulliCustomer Valuationswith p s = L Iderivetheoptimalorderpolicy( Q ;X a )for p s = L and Q (L)= M asfollows. E [(Q (L)= M )]=( L c a )X a + L(M X a ) cM (A{35) s.o.c. E [(Q (L)= M )] X a ] = c a (A{36) 2 E [( Q ( L)= M )] X 2 a ] = 0 (A{37) Since thef.o.c.isnegative,expectedprotisdecreasingin X a ;thus, X (L)=0.I comparetheexpectedprotforthe( M; 0)and(0 ; 0)policiestodeterminetheoptimal 96

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policy Iclearlyndthe( M; 0)policytobeoptimal. E [(L;Q = M;X a =0)]=( L c)M (A{38) E [(L;Q =0;X a =0)]=0 (A{39) (A{40) ProofofTheorem 6 :OptimalPolicy (Q ;X a ) for p s = H Indanextreme pointsolutionfor X a asdoneinProof A. Indtheupperboundof X a (H )asfollows,assuming X a Q andthemaximum Q valueis M Q = F 1 D s ( H c H ) + X a = D s + D s k + X a (A{41) Where I dene k = p 2er f 1 (2( H c H ) 1) as a constant,independentof X a .AndIhave D s =(M X a ) and D s = p (M X a ) (1 ). If Q = M its maxim umvalue,thenthemaximumvaluesof X a isfoundasfollows. M = D s + D s k + X a =( M X a ) + p (M X a ) (1 ) k + X a X a = M k 2 1 (A{42) If X a is at its otherextreme,0,then Q = F 1 D s H c H 97

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Ino w comparethe( M;M k 2 1 )and ( F 1 D s H c H ; 0)p olicies asfollows. E [(Q = M;X a = M k 2 1 )] = ( L +(H L ) c a )(M k 2 1 ) +H ( 2 k 2 1 D s (F 1 D s H c H )) cM (A{43) E [(Q = F 1 D s H c H ; X a = 0)] = H (M D s ( F 1 D s H c H )) cF 1 D s H c H (A{44) E [(Q = M ; X a = M k 2 1 )] E [(Q = F 1 D s H c H ; X a = 0)] (A{45) L(1 )+ c c a (A{46) ProofofTheorem7:OptimalSpotPrice p s BernoulliCustomerValuations Icomparetheexpectedprotundertheoptimalorderpoliciesfor p s = L and p s = H E [(L )]=(L +(H L ) ( M Q M X a ) c a ) X a + L(Q X a ) cQ (Q (L); X a ( L ))=(M; 0) E [(Q (L);X a (L))]=(L +(H L ) ( M M M 0 ) c a )0 + L ( M 0) cM = (L c )M (A{47) E [(H )]=(L +(H L) c a )X a + H ((M X a ) D s ( Q X a )) cQ (Q (H );X a (H ))=(F 1 D s ( H c H ) + X a ( H ) ;X a (H )) E [(Q (H );X a (H ))]=(L +(H L) c a )X a (H )+ H ((M X a (H )) D s (F 1 D s ( H c H ))) c(F 1 D s ( H c H ) + X a (H )) (A{48) 98

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Proof of Theorem 8 :OptimalOrderQuantity Q forUniformCustomer Valuations Ishowexpectedprottobeconcavein Q forUniformcustomervaluations. E []=( H + L 2 (H p s ) 2 2(H L) + D s (Q X a )(H p s ) 2(M X a ) c a ) X a +p s ((M X a ) H p s H L D s (Q X a )) cQ (A{49) E [] Q = X a ( H p s ) 2(M X a ) p s [F D s ( Q X a ) 1] c (A{50) 2 E [] Q 2 = X a ( H p s ) 2(M X a ) p s f D s (Q X a ) 0 (A{51) I can observethatthes.o.c.isconcavefor Xa(H p s ) 2( M X a ) p s 0. Assuming this condition tohold,Isolvethef.o.c.tondtheoptimal Q valueasfollows. Q = F 1 D s ( p s c X a ( H p s ) 2( M X a ) p s X a ( H p s ) 2(M X a ) ) + X a (A{52) 99

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APPENDIXB PR OOFS FORCHAPTER4:MULTI-GENERATIONPRICINGANDTIMING DECISIONSINNEWPRODUCTDEVELOPMENT ProofofTheorem1: GiventhesalesrateexpressionsforCASE1,Iderivethe following: f p 1 = a 1 (B{1) g p 2 = b 1 (B{2) Since Ihave f x 1 = g x 2 = 0, thefollowingsolutionsformarginalsales 1 and 2 directly follow. 1 =0, 8t (B{3) 2 =0, 8t (B{4) Icanthenderivetheoptimalpricesasfollows: p 1 : f (x 1 ;p 1 )+( p 1 c 1 + 1 ) f p 1 = 0 (B{5) a 0 a 1 p 1 a 1 ( p 1 c 1 ) =0 (B{6) p 1 = 1 2 ( a 0 a 1 + c 1 ) (B{7) p 2 : g ( x 2 ; p 2 ) + (p 2 c 2 + 2 ) g p 2 = 0 (B{8) b 0 b 1 p 2 b 1 ( p 2 c 2 )=0 (B{9) p 2 = 1 2 ( b 0 b 1 + c 2 ) (B{10) Pro of of Theorem2: Todeterminetheoptimaltimetomarket, t m ,Iderive 3 (t m ) asfollows. 3 =(p 1 c 1 + 1 )f (x 1 ;p 1 ) ( p 2 c 2 + 2 )g (x 2 ;p 2 )(B{11) =(p 1 c 1 )(a 0 a 1 p 1 ) (p 2 c 2 )(b 0 b 1 p 2 ) (B{12) 100

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Letus dene K 1 = (p 1 c 1 )(a 0 a 1 p 1 )and K 2 =( p 2 c 2 )(b 0 b 1 p 2 )astheprot rateforeachgeneration.Icanthensolvefor 3 (t m )usingbackwardsintegration(given 3 (T )=0). 3 (t m )= 3 (T ) Z T t m 3 dt (B{13) = Z T t m (K 1 K 2 )dt (B{14) Icanthendeterminetheoptimalvalueof t m fromthecondition 3 (t m ) c t m ProofofTheorem3: GiventhesalesrateexpressionsforCASE2,Icanderivethe following: f x 1 = a 2 [ 2 M 1 x 1 ] (B{15) g x 2 = b 2 [ 2 M 2 x 2 ] (B{16) Since I ha ve f p 1 = g p 2 = 0, theoptimalpricesimmediatelyfollow. p 1 = 8 > < > : ^ p 1 ;f (x 1 ;p 1 ) 0, 0; otherwise. (B{17) p 2 = 8 > < > : ^ p 2 ;g (x 2 ;p 2 ) 0, 0; otherwise. (B{18) Where^ p 1 and^ p 2 bedenedassomemaximumpricelevelssetbythemarket.SinceI dene f (x 1 ;p 1 )and g (x 2 ;p 2 )aspositivefunctions,theaboveconditionswillalwayshold. ProofofTheorem4: Tosolvefortheoptimaltimetomarket, t m ,Iderivethe followingdierentialequationsolutionsforthemarginalvalueofsalesofeachgeneration andthecumulativesalesofeachgeneration: 1 = 8 > < > : (p 1 c 1 ) e a 2 ( )( tt m ) ( e a 2 ( + )t + ) 2 (e a 2 ( + )t m + ) 2 1 ; t < t m 1 ( t )= 1 (T )=0 ; t t m (B{19) 101

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2 = 8 > > > > < > > > > : 2 (0) = ( p 2 (t m ) c 2 ) e b 2 ( + )T ( + ) 2 (e b 2 (+ ) T + ) 2 1 ; t < t m (p 2 c 2 ) e b 2 ( + )(t T ) ( e b 2 ( + )t +e b 2 ( + )t m ) 2 (e b 2 ( + )T +e b 2 ( + )t m ) 2 1 ; t m t < T 2 (T )=0; t = T (B{20) x 1 = 8 > < > : M 1 (1 e a 2 t( + ) ) 1+ e a 2 t( + ) ; t < t m x 1 (t m )= M 1 (1e a 2 t m ( + ) ) 1+ e a 2 t m ( + ) ; t t m (B{21) x 2 = 8 > < > : x 2 (0) = 0 ;t < > : ( p 1 c 1 )f (x 1 (t m )) (p 2 c 2 + 2 (t m ))( b 2 M 2 );t = t m ( p 1 c 1 )f (x 1 (t m )) (p 2 c 2 + 2 (t))g (x 2 (t));t m
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Ican then write 3 ( t m )asfollows. 3 (t m )= ( + ) 2 a 2 M 1 e a 2 ( + )t m (p 1 c 1 ) (e a 2 (+ )t m + ) 2 + b 2 M 2 e b 2 ( + )( T + t m ) (p 2 c 2 ) (e b 2 ( + )T + e b 2 ( + )t m ) 2 (T t m ) (B{26) This can b esimpliedtothefollowingexpression: 3 (t m )=[(p 2 c 2 )g ( x 2 (T t m )) (p 1 c 1 )f (x 1 (t m ))](T t m )(B{27) Ithennd t m fromtheconditionthat 3 (t m ) c t m ProofofTheorem5: Isolveforthisthresholdvaluebydeterminingthesmallest valueof T forwhich 3 (t m =0) c t m 3 ( t m =0)= ( + ) 2 ( (a 2 M 1 (p 1 c 1 )) ( + ) 2 + (b 2 e b 2 (+ )T M 2 (p 2 c 2 )) ( e b 2 (+ )T + ) 2 )T c t m (B{28) When the cost ofbringingthesecondgenerationtomarketisnegligible( c t m =0),Ican deriveavaluefor T ProofofTheorem6: Thefollowingrelationshipsarederivedfromtherstorder conditionsofoptimality: f x 1 = a 2 [ 2 M 1 x 1 ] (B{29) f p 1 = a 1 (B{30) g x 2 = b 2 [ 2 M 2 x 2 ] (B{31) g p 2 = b 1 (B{32) 103

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Ib egin byexaminingtheoptimalpriceexpressiontond p 1 (x 1 ; 1 ). p 1 : f (x 1 ;p 1 )+(p 1 c 1 + 1 ) f p 1 = 0 a 0 a 1 p + a 2 [ ( M 1 x 1 )+ M 1 (M 1 x 1 )x 1 ] a 1 (p 1 c 1 + 1 ) = 0 (B{33) p 1 ( x 1 ; 1 )= M 1 (a 0 + a 1 c 1 + a 2 M 1 )+ a 1 M 1 1 + a 2 x 1 (M 1 ( )+ x 1 ) 2a 1 M 1 (B{34) I replace p 1 with p 1 ( x 1 ; 1 )intheexpressionsfor_ x 1 and 1 .NowIsolvesimultaneous dierentialequationstond x 1 and 1 x 1 = f (x 1 ;p 1 (x 1 ; 1 )) = a 0 a 1 p 1 (x 1 ; 1 )+ a 2 [ (M 1 x 1 )+ M 1 (M 1 x 1 )x 1 ](B{35) 1 = (p 1 (x 1 ; 1 ) c 1 + 1 ) f x 1 1 = ( p 1 (x 1 ; 1 ) c 1 + 1 )a 2 [ 2 M 1 x 1 ] (B{36) x 1 = 0 B @ t [a 1 M 1 (2 + a 2 ( + ) t m ) p a 2 1 M 1 (4a 2 (a 0 a 1 c 1 ) t 2 m + M 1 (4 + a 2 t m (4 4 + a 2 ( + ) 2 t m )))] 1 C A 2a 1 a 2 t 2 m (B{37) 1 = 0 B B B B B B B B B B @ ( t + t m )[2 a 2 1 a 2 c 1 t 2 m ( t + t m ) +(2t m + t (2+ a 2 ( )t m )) p a 2 1 M 1 (4a 2 (a 0 a 1 c 1 ) t 2 m + M 1 (4+ a 2 t m (4 4 + a 2 ( + ) 2 t m ))) a 1 (2a 0 a 2 t 2 m (t + t m )+ M 1 (2t m (2+ a 2 t m ( + a 2 t m )) +t (4+ a 2 t m (4 4 + a 2 ( 2 + 2 )t m ))))] 1 C C C C C C C C C C A 2a 2 1 a 2 t 4 m (B{38) 104

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Ican no wreplacethesesolutionsfor x 1 and 1 inmyexpressionfor p 1 (x 1 ; 1 )tond p 1 Likewise,Icanrepeatthisanalysistond x 2 2 ,and p 2 p 2 : g (x 2 ;p 2 )+(p 2 c 2 + 2 ) g p 2 =0 b 0 b 1 p 2 + b 2 [ (M 2 x 2 )+ M 2 (M 2 x 2 )x 2 ] b 1 ( p 2 c 2 + 2 ) = 0 (B{39) p 2 ( x 2 ; 2 )= M 2 (b 0 + b 1 c 2 + b 2 m 2 )+ b 1 M 2 2 + b 2 x 2 (M 2 ( )+ x 2 ) 2b 1 M 2 (B{40) x 2 = g (x 2 ; p 2 ( x 2 ; 2 )) = b 0 b 1 p 2 (x 2 ; 2 )+ b 2 [(M 2 x 2 )+ M 2 (M 2 x 2 )x 2 ](B{41) 2 = (p 2 ( x 2 ; 2 ) c 2 + 2 ) g x 2 = ( p 2 ( x 2 ; 2 ) c 2 + 2 )b 2 [ 2 M 2 x 2 ] (B{42) 105

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x 2 = 0 B B B B @ (t t m )[b 1 M 2 (2 b 2 ( )(T t m )) v u u u t b 2 1 M 2 (M 2 (4 + 4 b 2 ( )(T t m )+ b 2 2 ( + ) 2 (T t m ) 2 ) +4 b 2 (b 0 b 1 c 2 ) (T t m ) 2 ) ] 1 C C C C A 2b 1 b 2 (T t m ) 2 (B{43) 2 = 0 B B B B B B B B B B B B B B B B B B @ (T t)[2b 2 1 b 2 c 2 (t + T 2t m )(T t m ) 2 v u u u t b 2 1 M 2 (M 2 (4 + 4 b 2 ( )(T t m )+ b 2 2 ( + ) 2 (T t m ) 2 ) +4 b 2 (b 0 b 1 c 2 ) (T t m ) 2 ) (t(2+ b 2 ( )(T t m ))+2(T 2t m ) b 2 ( )(T t m )t m ) b 1 (2 b 0 b 2 (t + T 2t m )(T t m ) 2 + m 2 (t(4+4b 2 ( )(T t m ) +b 2 2 ( 2 + 2 )(T t m ) 2 )+4( T 2t m ) b 2 (T t m )(2T ( + b 2 T ) +(6 6 + b 2 ( 2 +4 + 2 )T )t m b 2 ( + ) 2 t 2 m )))] 1 C C C C C C C C C C C C C C C C C C A 2b 2 1 b 2 (T t m ) 4 (B{44) Pro of of Theorem7: Iderivetheexpressionfor 3 (t m )asfollows. 3 (p 1 ;p 2 )=( p 1 c 1 + 1 )f (x 1 ;p 1 ) (p 2 c 2 + 2 ) g (x 2 ;p 2 ) (B{45) Iknowthat 3 (T )=0andfor t t m ,Ihave 1 (t )=0, x 1 (t)= x 1 (t m ),and p 1 (t )= p 1 (t m ). Inowsolvefor 3 (t m )asfollows: 3 (T ) Z T t m 3 ( )d = Z T t m [(p 1 (t m ) c 1 )f (x 1 (t m );p 1 (t m )) (p 2 ( ) c 2 + 2 ( ))g ( x 2 ( ) ;p 2 ( ))] d 106

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APPENDIXC PR OOFS FORCHAPTER5:OPTIMALNUMBEROFGENERATIONSFORA MULTI-GENERATIONPRICINGANDTIMINGMODELINNEWPRODUCT DEVELOPMENT ProofofTheorem1:OptimalPrice Tondtheoptimalprice p (t ),werstuse OptimalControltoderiveanexpressionfor p intermsofcumulativesales x andthe marginalvalueofsales H p = a 1 ( + n(p c ))+ n(a 0 a 1 p + a 2 ((M x)+ M (M x)x)) (C{1) p = M n ( a 0 + a 2 M)+ a 1 M (nc ) a 2 nx(M ( )+ x) 2a 1 M n (C{2) W e thensolvefor x and usingsimultaneousdierentialequationsandOptimalControl methods. H x =( n(p c )+ )(M ( )+2 x) a 2 M (C{3) x (t) = t 0 B @ a 1 Mn ( 2+ a 2 ( )t m )+ v u u u t a 2 1 M n 2 (4 a 2 ( a 0 a 1 c ) t 2 m +M (4+ a 2 t m (4 4 + a 2 ( + ) 2 t m ))) 1 C A 2a 1 a 2 n t 2 m (C{4) ( t )= 0 B B B B B B B @ (t m t )(2 a 2 1 a 2 cnt 2 m (t + t m )+(2 t m + t(2+ a 2 ( )t m )) p a 2 1 M n 2 (4 a 2 ( a 0 a 1 c )t 2 m + M (4+ a 2 t m (4 4 + a2( + ) 2 t m ))) a 1 n (2a 0 a 2 t 2 m (t + t m )+ M (2t m (2+ a 2 t m ( + a 2 t m )) +t (4+ a 2 t m (4 4 + a 2 ( 2 + 2 )t m ))))) 1 C C C C C C C A 2a 2 1 a 2 t 4 m (C{5) W e thensubstitutebackinthederived x and expressionstondtheexpressionfor p 107

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Proof of Corollary1:OptimalPriceConcaveoverTime Thesecond derivativeoftheoptimalprice p (t)overtimeisthefollowing: 2 p (t) t 2 = (2 a 2 1 a 2 cn t 2 m + (2+ a 2 ( )t m ) p ( a 2 1 M n 2 (4 a 2 ( a 0 a 1 c )t 2 m + M (4+ a 2 t m (4 4 + a 2 ( + ) 2 t m )))) a 1 n(2a 0 a 2 t 2 m + M (4+ a 2 t m (4 4 + a 2 ( 2 + 2 )t m )))) a 2 1 a 2 n t 4 m (C{6) W e wanttoshowthat 2 p (t) t 2 0 in ordertomeetthes.o.c.forconcavity.Letusdene thefollowing: Y =2a 1 a 2 nt 2 m (a 1 c a 0 ) (C{7) J =2+ a 2 t m ( ) (C{8) = a 1 nM (C{9) =2a 2 2 t 2 m (C{10) Thenwecanrewritethes.o.c.asfollows: Y + J p (J 2 + ) a 2 1 a 2 n t 4 m 0 (C{11) Where = ( ( J 2 +2 ) 2Y ). Wecanrearrangethes.o.c.expressionasfollows: J p (J 2 + ) Y (C{12) If we assume a 0 a 1 c,wehave Y 0.Thisassumptionisreasonablefortypicalvalues of a 1 a 2 ,and c basedontheempiricalliterature.If Y 0,then Y 0andtheRHS ofexpressionC.12ispositive.Nowwemustnotethatitispossiblefor J tobenegative. Fromtheempiricalliterature,typicalvaluesof aremuchsmallerthantypicalvaluesof .Thus,weusuallyhave << whichimpliesthat( )wouldbenegative.Thus,if 108

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t m 2 a 2 ( ) ,then J 0. If thisisthecase,and J isnegative,thentheLHSofexpression C.12isnegativeandthes.o.c.forconcavityof p(t)in t alwaysholds.However,ifthe condition t m 2 a 2 ( ) is not true and J ispositive,thenwecanperformthefollowing analysis: J 2 ( (J 2 + ) Y ) 2 (C{13) J 2 ( 2 J 2 + 2 2 2Y ) 2 (J 4 +2J 2 + 2 ) 2Y (J 2 + )+ Y 2 (C{14) 0 ( Y ) 2 (C{15) SincetheexpressionC.15isalwaystrue,thes.o.c.forconcavityof p(t)in t alwaysholds. Thus,regardlessofwhether J ispositiveornegative,thes.o.c.alwaysholds. Wemustalsoensurethatthesquarerootexpressionisrealbycheckingthat 0. Thisrequiresthefollowingconditiontobetrue. 0 ( J 2 +2 ) 2Y (C{16) Sincewehave Y 0when a 0 a 1 c 0,thentheexpressioninC.16isalwaystrueand thereforeisalwayspositive. Thus,for a 0 a 1 c 0,thes.o.c.istrueand p ( t )isconcavein t. ProofofTheorem2:OptimalNumberofGenerations Substitutingthe optimalpriceexpressionfoundinTheorem1intoourprotexpressionyieldsthe 109

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following: ( p ( t )) = n Z T=n 0 x( p (t))(p ( t ) c (t ))dt nc t m (C{17) = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ (a 1 Mn (2+ a 2 ( )t m ) + v u u u t a 2 1 M n 2 (4 a 2 ( a 0 a 1 c ) t 2 m +M (4+ a 2 t m (4 4 + a 2 ( + ) 2 t m ))) ) (((cT )=n)+1 =(12a 2 1 a 2 n 4 t 4 m )(T (4a 2 1 a 2 cnT 2 t 2 m +(6n 2 t 2 m +3a 2 n ( )Tt 2 m +2T 2 (2+ a 2 ( )t m )) v u u u t a 2 1 M n 2 (4 a 2 ( a 0 a 1 c )t 2 m +M (4+ a 2 t m (4 4 + a 2 ( + ) 2 t m ))) +a 1 n(4a 0 a 2 t 2 m (T 2 +3n 2 t 2 m ) +M (3a 2 n ( ) Tt 2 m (2+ a 2 ( )t m ) +6n 2 t 2 m (2+ a 2 t m ( +2a 2 t m )) 2T 2 (4+ a 2 t m (4 4 + a 2 ( 2 + 2 )t m ))))))) 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A 2a 1 a 2 t s m 2 nc t m (C{18) Assuming a benchmarkscenarioofequalgenerationsoeredforequaltimetomarketfor eachgeneration,wereplace t m with T=n.Makingthissubstitutionin,wecannd n usingthef.o.c.: n =0. n = (16M n 2 +10 a 2 Mn ( )T + a 2 (4a 0 + a 2 M ( + ) 2 ) T 2 4a 1 a 2 cT 2 ) p a 2 1 M (4M n 2 +4 a 2 Mn ( )T + a 2 (4(a 0 a 1 c) + a 2 M ( + ) 2 )T 2 ) +a 1 M ( 32Mn 3 36a 2 Mn 2 ( )T 12a 2 n (2a 0 + a 2 M ( 2 + 2 ))T 2 a 2 2 ( )(6a 0 + a 2 M ( 2 +4 + 2 ))T 3 ) +6a 2 1 a 2 T 2 (2a 2 c t m T + cM (4 n + a 2 ( )T ))) 12a 2 1 a 2 2 2 T 3 (C{19) 110

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Theoptimal expression for n isderivedbysettingtheaboveexpressiontozeroand solvingfor n SecondOrderConditionforProttobeConcavein n Inordertohave n fromTheorem2,derivedfromthef.o.c.ofprotin n,tobeoptimal,wemustrstshow thatthes.o.c.ofprotin n holds.Thatis,wemusthave 2 n 2 0 true inorderforprot tobeconcavein n andthus n tobethe n valuewhichmaximizestheprotfunction.The expressionforthiss.o.c.isasfollows: 2 n 2 0 0 B B B B @ 8M n 2 (K 2Mn)+2a 2 Mn ( 2K +10 Mn( )T a 2 (2(a 0 a 1 c)(K 6Mn) + a 2 M (4Mn (2 2 + 2 2 ) +K ( 2 + 2 ))) T 2 + a 2 2 M ( )(4( a 0 a 1 c ) + a 2 M ( ) 2 )T 3 1 C C C C A 0 (C{20) Wherewedene K = p M (4M n 2 + 4 a 2 Mn( )T + a 2 (4(a 0 a 1 c ) + a 2 M ( + ) 2 )T 2 ) Althoughwecannotshowanalyticallythatthisexpressionisalwaystrue,norderive analyticalconditionsunderwhichthisexpressionwouldhold,givennumericalvaluesfor theexpressionparameters,thisconditioncaneasilybechecked.Ifthisconditionholds, then n foundinTheorem2isindeedtheoptimalnumberofgenerationstooer. 111

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BIOGRAPHICALSKETCH Mic helle M.H.SerefreceivedherBachelorofScienceinindustrialandsystems engineeringin2002,andherMasterofScienceinindustrialandsystemsengineeringin 2004,bothfromtheUniversityofFlorida.UponcompletionofherPh.D.inOperations ManagementfromtheUniversityofFlorida,Michelleplanstopursueanacademiccareer. MichellehaspublishedvariousworksintheOperationsManagementandIndustrial Engineeringliterature.IntheOperationsManagementarea,shehascompleteda papertitled"OptionRequirementsPlanningforMassCustomizedProducts"with Drs.AnandPaulandAsooVakharia,andisworkingonareviewpaperofextensionsto thenewsvendorproblemwithDr.Vakhariaandseveralothercolleagues.IntheIndustrial Engineeringarea,Michellehaspublishedthejournalarticle"Decisionsupportsystems development:AnessentialpartofOReducation"withDr.RavindraAhuja,inaddition toachaptertitled"Spreadsheet-BasedDecisionSupportSystems"publishedinthe book HandbookonDecisionSupportSystems .Michellehasalsoco-authoredatextbook DevelopingSpreadsheet-BasedDecisionSupportSystems(DSS)UsingExcelandVBA forExcel withDr.AhujaandDr.WayneWinston.Michelleplanstosubmitthethree researchchaptersofthisdissertationaspublicationsintheOperationsManagement (OM)literature.Herresearchfocusesonjointoperationsandpricingdecisionsinthe OM/Marketinginterfacearea. MichellehastaughtseveralcoursesinboththeIndustrialEngineeringandInformation SystemsandOperationsManagementDepartmentsattheUniversityofFlorida.These coursesincludeProjectManagementcourse,ManagerialOperationsAnalysis,and DevelopingDecisionSupportSystemsforbothundergraduateandgraduatestudents. Shehasreceivedhighevaluationsinallofherteaching. Michelleenjoysexaminingadvanced-levelresearchproblems,teachingcoursesto studentsofalllevels,andinteractingwithcolleaguesatnationalandinternational conferences.Shelooksforwardtoanexcitingacademiccareer. 115