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1、Timing Successive Product Introductions with Demand Diffusion and Stochastic Technology Improvement基于需求扩散和随机技术进步的连续产品引入过程基于需求扩散和随机技术进步的连续产品引入过程 R.Mark KrankelDepartment of Industrial and Operations Engineering,University of Michigan,Izak Duenyas,Roman KapuscinskiRoss School of Business,University of
2、 Michigan,Ann Arbor,MichiganPresent by Li WeiCONTENTSuIntroductionuLiteratureuModeluOptimal PolicyuComputational Study and InsightsuExtensionsIntroductionuConsider an innovative firm that manages the development and production of a single,durable product.uOver time,the firms research and development
3、(R&D)department generates a stochastic stream of new product technology,features,and enhancements for design into successive product generations.IntroductionuThe firm captures the benefits of such advances by introducing a new product generation.uDue to fixed product-introduction costs,it may be unr
4、easonable to immediately release a new product generation after each technology discovery.Rather,the firm may prefer to delay an introduction until sufficient incremental new product technology has accumulated in R&D.uThe objective of this paper is to characterize the firms optimal product-introduct
5、ion policyIntroductionuThe total number of product generations is not pre-specified;rather,it is determined by the pace of technology improvement along with the firms dynamic decisions on when to introduce.uAnalysis is centered upon two key influences affecting the introduction timing decisions:(1)d
6、emand diffusion dynamics,where future product demand is a function of past sales(2)technology improvement process,specifically the concept that delaying introduction to a later date may lead to the capture of further improvements in product technology.IntroductionuPrevious literature examining incre
7、mental technology introduction has focused on either(1)or(2),but none have considered both factors simultaneously.As a result,the present analysis provides new insight into the structure of the optimal introduction timing policy for an innovative firm.uUsing a proposed decision model that incorporat
8、es both key influences,we prove the optimality of a threshold policy:it is optimal for the firm to introduce the next product generation when the technology of the current generation is below a state-dependent threshold,in which the state is defined by the firms cumulative sales and the technology l
9、evel in R&D.IntroductionuRelative papersWilson and Norton(1989)&Mahajan and Muller(1996)These two papers proceed under a demand diffusion framework,but do not model the progression of product technology.Rather,they assume that the next generation product to be introduced is available at all times st
10、arting from Time 0.As a result,they respectively conclude the optimality of“now or never”(the new generation product is introduced immediately or never)and“now or at maturity”(the new generation product is introduced immediately or when the present generation product has reached sufficient sales)rul
11、es governing product introductions.LiteratureuTwo main research areas are directly relevant to the current work.The first centers on models of demand.Papers in this area describe the patterns of demand exhibited by single or multiple product generations,specifically in relation to new innovations.Th
12、ese papers concentrate on system dynamics and/or model fit with empirical data.The second research area examines decision models for technology adoption timing.A subset of this group includes papers that model the introduction of new products subject to demand diffusion.Literature models of demanduB
13、ass(1969)initiates the stream that examines demand diffusion models by formulating a model for a single(innovative)product.The Bass model specifies a potential adopter population of fixed size and identifies two types of consumers within that population:innovators and imitators.uInnovators act indep
14、endently,whereas the rate of adoption due to imitators depends on the number of those who have already adopted.uThe resulting differential equation for sales rate as a function of time describes the empirically observed s-shaped pattern of cumulative sales:exponential growth to a peak followed by ex
15、ponential decay.Literature models of demanduBass,F.M.1969.A new product growth model for consumer durables.Management Sci.15 215227.Prof Dr.Frank M.Bass(1926-2006)was a leading academic in the field of marketing research,and is considered to be among the founders of Marketing science.He became famou
16、s as the creator of the Bass diffusion model that describes the adoption of new products and technologies by first-time buyers.He died on December 1,2006.Literature models of demanduNorton and Bass(1987)extend the original Bass model by incorporating substitution effects to describe the growth and d
17、ecline of sales for successive generations of a frequently purchased product.uJun and Park(1999)examine multiple-generation demand diffusion characteristics by combining diffusion theory with elements of choice theory.uWilson and Norton(1989)propose a multiple-generation demand diffusion model based
18、 on information flow.uKumar and Swaminathan(2003)modify the Bass model for the case in which a firms capacity constraints may limit the firms ability to meet all demand.uUsing their revised demand diffusion model,they determine conditions under which a capacitated firms optimal production/sales plan
19、 is a“build-up policy,”in which the firm builds up an initial inventory level before the start of product sales and all demand is met thereafter.Literaturetechnology adoption timinguGjerde et al.(2002)model a firms decisions on the level of innovation to incorporate into successive product generatio
20、ns.The paper does not consider the diffusion dynamics of the existing products in the market(product sales rates do not depend on cumulative sales).uCohen et al.(1996)assume that product can only be sold during a fixed window of time.Therefore,delaying the product introduction for further developmen
21、t will lead to a better product and higher revenues but over a shorter time.Cohen et al.further assume that the product currently in the market or the newly introduced product both have sales at a constant rate.Thus,they do not consider the diffusion dynamics.They also do not consider the stochastic
22、 nature of the R&D Process.Literaturetechnology adoption timinguBalcer and Lippman(1984)conclude that a firm will adopt the current best technology if its lag in process technology exceeds a certain threshold.The threshold is either nonincreasing or nondecreasing in time,dependent on expectations wi
23、th respect to potential for technology discovery.uFarzin et al.(1998)considers a similar problem under a dynamic programming framework.The paper explicitly addresses the option value of delaying adoption and compares results to those using traditional net present value methods,in which technology ad
24、option takes place if the resulting discounted net cash flows are positive.uIn each of these works,the technology adoption decision does not explicitly consider the effects of adoption timing on product-demand dynamics.Literaturetechnology adoption timinguWilson and Norton(1989)consider the one-time
25、 introduction decision for a new product generation.In their model,product introduction has fixed positive effects on market potential along with negative effects due to cannibalization.They conclude that the optimal policy for the firm is given by a“now or never”rule.That is,it will either be optim
26、al to introduce the improved product as soon as it is available or never at all.uMahajan and Muller(1996)conclude that it will be optimal to either introduce the improved product as soon as it is available or when enough sales have been accumulated for the previous product generation.(“now or at mat
27、urity”rule)uBoth Wilson and Norton(1989)and Mahajan and Muller(1996)implicitly assume that the next product generation is available and remains unchanged regardless of when it is introduced.In contrast,we assume that a firm that delays introduction of the next product generation expects to capture g
28、reater technological advances at a later date.ModeluUnder a discrete-time,infinite-horizon scenario,consider a single base product that progresses through a series of product generations over time.uThe benefits of improved technology are realized only through introduction of a new product generation
29、 that incorporates the latest technology available in R&D.uAn improvement in the incumbent product technology leads to a higher sales potential for the new product generation.However,each new generation requires a fixed introduction cost.The firm seeks an introduction policy that maximizes net profi
30、ts.ModeluIn each period,the firm has the option to either introduce the latest technology or continue selling at the current incumbent technology level(wait).uWe model the level of technology in R&D using a single index,and assume that this level improves stochastically during each period.uOur objec
31、tive is to characterize the firms optimal introduction policy given this stochastic R&D process.ModelNotation and AssumptionsuWe begin with the following definitions under a dynamic-programming framework:ModelNotation and AssumptionsModelNotation and AssumptionsuWe consider a durable base product fo
32、r which product technology is additive and introduction of a new product generation results in complete obsolescence of the previous generation;i.e.,once a new generation is introduced,sales of the previous generation immediately drop to and remain at zero.This property is referred to later as the“c
33、omplete replacement”condition.uIt is assumed that(1)available product technology improves in each period according to a stochastic process,and(2)sales for any given generation follow a demand diffusion process.ModelNotation and AssumptionsuBoth the technology level and the price of a new product are
34、 expected to influence the products market potential and associated demand diffusion dynamics.To understand the effects of progressing technology independent of other compounding factors,we assume a very specific but realistic pricing strategy that maintains constant unit profit margins.ModelNotatio
35、n and AssumptionsuAs mentioned above,sales potential is assumed to be an increasing function of product technology level.uMoreover,we do not model capacity constraints and assume that all demand can be met so that sales equals demand.Model Formulationthe following assumption is made on the sales rat
36、e curves:Model Formulationu(i)ensures that,all else equal,product sales rate is nondecreasing in product technology.uPart(ii)accommodates realistic durable-good market scenarios in which the potential market size is finite and current period sales do not exceed total remaining market potential.uCond
37、ition(iii)limits the rate at which sales decrease and in a discrete-time framework guarantees that the sales rate from one period to the next does not decrease at a faster pace than sales accumulated within the period.Model FormulationModel FormulationuThe optimum introduction policy is computed fro
38、m the optimality equation:ModelRelationship to Demand DiffusionuFor the scenario considered in this paper,there is a natural link between this sales model and that of a typical(continuous-time)diffusion model.Consider the Bass diffusion model for a single innovative product:ModelRelationship to Dema
39、nd DiffusionMahajan and Muller(1996)present an extension of the Bass model for the case of multiple product generations.ModelRelationship to Demand DiffusionModelRelationship to Demand Diffusionwhere a and b are coefficients of innovation and imitation,respectively.Because cumulative sales is tracke
40、d as a state variable,the decision model(1)(3)clearly captures the interaction between product generations when sales curves are of the demand diffusion form(6).Moreover,an examination of(6)shows that the demand diffusion form satisfies Assumption 1 subject to a mild restriction on problem parameter
41、s.Optimal PolicyOptimal PolicyOptimal PolicyuThe first result states that as the two systems progress over time,the cumulative sales level of the firm with lower initial cumulative sales will never surpass the firm with higher initial cumulative sales.Optimal PolicyOptimal PolicyuThe result states t
42、hat all else equal,the discounted optimal profit-to-go for a firm with lower cumulative sales will not exceed that of a firm with higher cumulative sales by more than the net value of their cumulative sales difference.That is,future benefits cannot make up for the current sales deficit.Optimal Polic
43、yOptimal PolicyOptimal PolicyOptimal PolicyOptimal PolicyOptimal PolicyOptimal PolicyOptimal PolicyOptimal PolicyComputational Study and InsightsuThe numerical study focuses on the influences of a simple technology discovery rate,fixed product-introduction costs,and market parameters including the d
44、iffusion coefficients and a parameter describing the sensitivity of product market potential to changes in product technology.Computational Study and Insightsufor purposes of numerical investigation we begin with a simplified baseline scenario.Sales rate curves for the baseline scenario are generate
45、d within a discrete-time framework to approximate a demand diffusion process according to the form given in(6).uTechnology improvement is assumed to follow a simplified stochastic process in which available technology in R&D increases by one in each period with probability p.Computational Study and
46、InsightsComputational Study and InsightsComputational Study and InsightsuThe baseline optimal policy is computed by solving the dynamic program(3).In solving(3)numerically,we use linear interpolation to handle cases in which the current period sales gs z is a noninteger multiple of the indexing unit
47、 used for cumulative sales.Computational Study and InsightsComputational Study and InsightsuNumerical approximation generates the baseline set of technology switching curves illustrated in Figure 6.Computational Study and InsightsuThe switching curves in Figure 6 suggest that optimal introduction of
48、 the next product generation may be triggered in one of two ways:(1)through sufficient product sales at the current technology levels,(2)through significant advances in available product technology.u(1)implies that,even without further gains in R&D,a firm that continues to sell the current generatio
49、n long enough may eventually find it optimal to introduce the technology on hand even though introducing at the same technology level was not profitable in the past.u(2)implies that,regardless of the current generations position along its sales curve,it may be optimal to introduce a new generation w
50、ith large enough gains in R&D technology.Computational Study and InsightsuConsider the expected rate of technology discovery as measured(for the baseline scenario)by the technology discovery probability p.Computational Study and InsightsuIt is natural that under fixed introduction costs,a firm with