基于需求扩散和随机技术进步的连续产品引入过程.ppt

TimingSuccessiveProductIntroductionswithDemandDiffusionandStochasticTechnologyImprovement基于需求扩散和随机技术进步的连续产品引入过程,R.MarkKrankelDepartmentofIndustrialandOperationsEngineering,UniversityofMichigan,IzakDuenyas,RomanKapuscinskiRossSchoolofBusiness,UniversityofMichigan,AnnArbor,MichiganPresentbyLiWei,CONTENTS,IntroductionLiteratureModelOptimalPolicyComputationalStudyandInsightsExtensions,Introduction,Consideraninnovativefirmthatmanagesthedevelopmentandproductionofasingle,durableproduct.Overtime,thefirmsresearchanddevelopment(Rrather,itisdeterminedbythepaceoftechnologyimprovementalongwiththefirmsdynamicdecisionsonwhentointroduce.Analysisiscenteredupontwokeyinfluencesaffectingtheintroductiontimingdecisions:(1)demanddiffusiondynamics,wherefutureproductdemandisafunctionofpastsales(2)technologyimprovementprocess,specificallytheconceptthatdelayingintroductiontoalaterdatemayleadtothecaptureoffurtherimprovementsinproducttechnology.,Introduction,Previousliteratureexaminingincrementaltechnologyintroductionhasfocusedoneither(1)or(2),butnonehaveconsideredbothfactorssimultaneously.Asaresult,thepresentanalysisprovidesnewinsightintothestructureoftheoptimalintroductiontimingpolicyforaninnovativefirm.Usingaproposeddecisionmodelthatincorporatesbothkeyinfluences,weprovetheoptimalityofathresholdpolicy:itisoptimalforthefirmtointroducethenextproductgenerationwhenthetechnologyofthecurrentgenerationisbelowastate-dependentthreshold,inwhichthestateisdefinedbythefirmscumulativesalesandthetechnologylevelinRi.e.,onceanewgenerationisintroduced,salesofthepreviousgenerationimmediatelydroptoandremainatzero.Thispropertyisreferredtolaterasthe“completereplacement”condition.Itisassumedthat(1)availableproducttechnologyimprovesineachperiodaccordingtoastochasticprocess,and(2)salesforanygivengenerationfollowademanddiffusionprocess.,ModelNotationandAssumptions,Boththetechnologylevelandthepriceofanewproductareexpectedtoinfluencetheproductsmarketpotentialandassociateddemanddiffusiondynamics.Tounderstandtheeffectsofprogressingtechnologyindependentofothercompoundingfactors,weassumeaveryspecificbutrealisticpricingstrategythatmaintainsconstantunitprofitmargins.,ModelNotationandAssumptions,Asmentionedabove,salespotentialisassumedtobeanincreasingfunctionofproducttechnologylevel.Moreover,wedonotmodelcapacityconstraintsandassumethatalldemandcanbemetsothatsalesequalsdemand.,ModelFormulation,thefollowingassumptionismadeonthesalesratecurves:,ModelFormulation,(i)ensuresthat,allelseequal,productsalesrateisnondecreasinginproducttechnology.Part(ii)accommodatesrealisticdurable-goodmarketscenariosinwhichthepotentialmarketsizeisfiniteandcurrentperiodsalesdonotexceedtotalremainingmarketpotential.Condition(iii)limitstherateatwhichsalesdecreaseandinadiscrete-timeframeworkguaranteesthatthesalesratefromoneperiodtothenextdoesnotdecreaseatafasterpacethansalesaccumulatedwithintheperiod.,ModelFormulation,ModelFormulation,Theoptimumintroductionpolicyiscomputedfromtheoptimalityequation:,ModelRelationshiptoDemandDiffusion,Forthescenarioconsideredinthispaper,thereisanaturallinkbetweenthissalesmodelandthatofatypical(continuous-time)diffusionmodel.ConsidertheBassdiffusionmodelforasingleinnovativeproduct:,ModelRelationshiptoDemandDiffusion,MahajanandMuller(1996)presentanextensionoftheBassmodelforthecaseofmultipleproductgenerations.,ModelRelationshiptoDemandDiffusion,ModelRelationshiptoDemandDiffusion,whereaandbarecoefficientsofinnovationandimitation,respectively.Becausecumulativesalesistrackedasastatevariable,thedecisionmodel(1)(3)clearlycapturestheinteractionbetweenproductgenerationswhensalescurvesareofthedemanddiffusionform(6).Moreover,anexaminationof(6)showsthatthedemanddiffusionformsatisfiesAssumption1subjecttoamildrestrictiononproblemparameters.,OptimalPolicy,OptimalPolicy,OptimalPolicy,Thefirstresultstatesthatasthetwosystemsprogressovertime,thecumulativesaleslevelofthefirmwithlowerinitialcumulativesaleswillneversurpassthefirmwithhigherinitialcumulativesales.,OptimalPolicy,OptimalPolicy,Theresultstatesthatallelseequal,thediscountedoptimalprofit-to-goforafirmwithlowercumulativesaleswillnotexceedthatofafirmwithhighercumulativesalesbymorethanthenetvalueoftheircumulativesalesdifference.Thatis,futurebenefitscannotmakeupforthecurrentsalesdeficit.,OptimalPolicy,OptimalPolicy,OptimalPolicy,OptimalPolicy,OptimalPolicy,OptimalPolicy,OptimalPolicy,OptimalPolicy,OptimalPolicy,ComputationalStudyandInsights,Thenumericalstudyfocusesontheinfluencesofasimpletechnologydiscoveryrate,fixedproduct-introductioncosts,andmarketparametersincludingthediffusioncoefficientsandaparameterdescribingthesensitivityofproductmarketpotentialtochangesinproducttechnology.,ComputationalStudyandInsights,forpurposesofnumericalinvestigationwebeginwithasimplifiedbaselinescenario.Salesratecurvesforthebaselinescenarioaregeneratedwithinadiscrete-timeframeworktoapproximateademanddiffusionprocessaccordingtotheformgivenin(6).TechnologyimprovementisassumedtofollowasimplifiedstochasticprocessinwhichavailabletechnologyinRi.e.,firmswithahigherexpectedtechnologydiscoveryrateareexpectedtointroducenewproductgenerationsmorefrequentlyandwithlargertechnologygainsbetweengenerations.,ComputationalStudyandInsights,Next,weexaminetheinfluenceofthefirmscoststructureontheoptimalpolicy.AsillustratedinFigure8,adecreaseinthefixedintroductioncostKdecreasestheoptimalintroductionthresholdatanygivencumulativesaleslevel.,ComputationalStudyandInsights,Letusturntotheparametersthatdescribetheproductmarket.Themodeledproductsalesdynamicswillbeaffectedbythediffusioncoefficientsaandbin(6)aswellasthemarketpotentialparameterm,thatdeterminestheproductmarketpotentialforaspecifictechnologylevel.,ComputationalStudyandInsights,MarketPotentialParameterAnincreaseinmtranslatestolargergainsinmarketpotentialperunitgainintechnology.Inturn,thesalesrateatagivenlevelofcumulativesalesismoresensitivetoincreasesinproducttechnologywhenmishigher.,ComputationalStudyandInsights,DemandDiffusionCoefficientsAproductwithahighercoefficientofinnovationawouldexhibitasalesratecurvethatstartswithhigherone-periodsales,peaksearlier,andliescompletelyabovethatofaproductwithlowera.Figure10illustrateshowthecoefficientofinnovationinfluencestheproductsalesratecurves.,ComputationalStudyandInsights,DemandDiffusionCoefficientsWefindthataproductwithhighercoefficientofinnovationisassociatedwithmore-frequentproductintroductions.,ComputationalStudyandInsights,DemandDiffusionCoefficientsSimilartotheeffectofa,ahighercoefficientofimitationbtranslatestoasalesratecurvethatliescompletelyabovethatforlowerb.,ComputationalStudyandInsights,DemandDiffusionCoefficientsAswitha,theintroductionthresholdsaredecreasingintheproductscoefficientofimitationb.Thus,afirmshouldintroducenewproductgenerationsmorefrequentlygivenabasetechnologythatdiffusesthroughitspotentialadopterpopulationfaster.,ComputationalStudyandInsights,ComputationalStudyandInsights,UncertainDemandHere,wedescribetwopossiblescenariosforuncertaindemandalongwiththereviseddecision-modelformulations.,ComputationalStudyandInsights,ComputationalStudyandInsights,Thedecisionmodel(1)(3)isreformulatedasfollowstoaccommodatethissingle-perioddemanduncertainty:,ComputationalStudyandInsights,ComputationalStudyandInsights,Afirmmayalsowishtocapturetheuncertaintyinoverallmarketacceptanceofanewproductgenerationwhiletakingintoaccountpotentialcorrelationbetweenthemarketsuccessoftwosequentialgenerations.Forthedemanddiffusioncase,theestimationofanewgenerationsmarketpotentialN(z)isakeysourceofsuchuncertainty.,ComputationalStudyandInsights,ThepresentmodelframeworkcancapturethemarketsuccessuncertaintyforanewgenerationusingaMarkovmodulateddemandformulation.Apossibleimplementationispresentedhereforillustration.,ComputationalStudyandInsights,ComputationalStudyandInsights,Thedecisionmodel(12)(13)implementsthisMarkovmodulateddemandframeworkforaccommodatinguncertaintyinproductsuccess.,Extensions,Alimitationofouranalysisisthatthemodeldoesnotconsidertheeffectsofintroductiontimingonconsumerspurchasestrategiesandresultingdemandpatterns.Significantincreaseinthefirmspaceofproductintroductionsmaycauseconsumerstopostponepurchasedecisionsinanticipationo