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Good buy? Delaying end-of-life purchases
European Journal of Operational Research 146 (2003) 216–228 www.elsevier.com/locate/dsw
Good buy? Delaying end-of-life purchases Kyle D. Cattani a
a,*
, Gilvan C. Souza
b,1
The Kenan-Flagler Business School, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3490, USA b The Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA Received 30 October 2000; accepted 7 December 2001
Abstract We study the effects of delaying an end-of-life buy. Manufacturers sometimes are required to place a final, end-of-life buy for a component that the supplier will no longer provide. The manufacturer performs a newsvendor analysis with the possible result of significant expected overage and underage costs. If the decision can be delayed, the expected overage and underage costs can be reduced. We model the effects of a delay of the final purchase under various scenarios of remaining demand. We contrast the manufacturing benefits with the costs incurred by the supplier and show that the supplier, who benefits greatly from the end-of-life buy, likely will require an incentive to enact a delay. Our results provide an insight to the observation of increasing numbers of end-of-life buys and provides a framework for analysis as manufacturers strive to cope with the issue. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Inventory; Supply chain management
1. Introduction The final stages of the product life cycle can create undesirable and unavoidable issues for inventory management of a product. Demand for the product is declining, yet total remaining demand can be very difficult to predict. Demand is uncertain due to inherent variability as well as external factors such as introductions of competing products. The challenges of managing product end of life have intensified with shorter product *
Corresponding author. Tel.: +1-919-962-3273; fax: +1-919962-6949. E-mail addresses: [email protected] (K.D. Cattani), [email protected] (G.C. Souza). 1 Tel.: +1-301-405-0628; fax: +1-301-405-8655.
life cycles and can lead to disastrous results. Many firms have been faced with large write-offs of excess inventory after the product life ultimately ends. IBM disclosed a $1 billion loss from its personal-computer business in 1998, attributed to excess PCs in dealer channels that had to be sold at a steep discount (Bulkeley, 1999). Product components, particularly those used only in one product, inherit inventory management complexities from their dying parents. For the productÕs end of life, manufacturers strive to reduce or eliminate all buffers and excess stock of such components. Concurrent with the manufacturerÕs struggles to manage product end of life, the components suppliers (who consider the component to be their product) are also concerned with managing this stage of their productÕs life cycle. If
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 2 ) 0 0 2 1 2 - 6
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a supplierÕs customer base is well-defined and concentrated, the supplier may opt to do a final build of the component and require that its customers (e.g., manufacturers) make a final, end-of-life buy, after which production of the component is discontinued. Thus, the manufacturer is required to make a potentially large purchase of a part that faces uncertain and disappearing demand but remains crucial for current business. 2 From the manufacturerÕs perspective, stocking such a part in inventory is quite undesirable. One of the authors worked as a purchasing manager at one of Hewlett-PackardÕs computer manufacturing divisions, where there consistently are tens of millions of dollars of inventory resulting from end-of-life buys for dozens of components. Much of this inventory eventually is scrapped, reducing profits by up to 1% of revenue each year. These write-offs are significant, especially as pressure on margins increase. All other things being equal, the manufacturer would prefer to eliminate, or at the least postpone, this final end-of-life buy. Postponing the end-of-life buy is helpful to the extent that demand uncertainty can be partially or fully resolved during the delay, reducing the size of the inventory buffer of parts that mostly will be scrapped. In this paper we study the effects of delaying end-of-life purchases. We first model the benefits to the manufacturer of a delay of the final end-of-life purchase. We contrast these benefits to the costs the supplier incurs in delaying its final build and thus determine the optimal timing of the final buy, from an aggregated supply chain perspective. We show that a supplier must be provided an incentive for a delay to be feasible. Our results provide an insight to the observation of increasing numbers of endof-life buys and provides a framework for analysis as manufacturers strive to cope with the issue. While the example above involves sales by a component supplier to a manufacturer, the problem is identical for other settings where there is a 2 This scenario is apparently becoming more common. An IBM manager reports, for example, that recently there has been an increase in the frequency of technology suppliers ending production of parts earlier than in the past. (July 22, 1999 interview with Holly Hach, Director of Reengineering and Planning, Personal Systems Group, IBM.)
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proposed end-of-life buy or build. For example, a manufacturer may wish to shut down an assembly line dedicated to a dying product but may be unable to require that customers place final orders, either because customers are too diverse, too powerful, or not known in advance. In this case, an end-of-life build to finished goods inventory may be proposed. Then, both the costs and benefits of delaying the final build accrue to the manufacturer, although the decision is of similar form to the first example. We sometimes frame the discussion in terms of an end-of-life build as it is easier to articulate the cost of underage and overage since there is only one player, and since bill of material relationships between products and components that might confound the discussion are not present. The paper proceeds as follows: In Section 2 we discuss previous work in this area. In Section 3, we use a result from Cattani (1997) to develop a general multi-period model of the benefits from a delay of an end-of-life build of a product that faces declining prices and demand over its remaining life, although the results hold for end-of-life buys as well. We contrast the expected profits of the end-of-life build at time zero with the profits if there is a delay of d periods. In Section 4, we use continuous time and we use the traditional newsvendor analysis to consider an end-of-life buy of a component that has a constant price (over the remaining lifetime) and make additional assumptions that allow us to derive a closed-form solution for delay benefits. We model the benefit of delaying the end-of-life buy under different demand scenarios and identify key properties of the delay benefits. In Section 5, we consider the perspective of each individual player in the supply chain. We summarize and conclude in Section 6.
2. Related research Much has been written on managing product obsolescence, where we define obsolescence as a high likelihood that demand will drop substantially from its current level. In the inventory literature, a typical focus determines stocking levels over time for a product or a purchased part. The traditional single-period newsvendor problem is
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directly relevant to the end-of-life buy and is a basis for much of the research related to product obsolescence. Multi-period inventory problems raise additional issues, many of which also are relevant to product obsolescence. Zipkin (2000) provides a good review of inventory theory. We review here only papers that are more closely related to this research, concentrating on obsolescence as defined above. A stream of literature analyzes the obsolescence problem by finding optimal stocking levels, assuming that the demand rate is constant but an exogenous process causes obsolescence – and so the lifetime of the product is a random variable, with a known distribution. Brown (1971), Masters (1991), and Joglekar and Lee (1993) assume that the lifetime is exponentially distributed. David et al. (1997) generalize this problem to other lifetime distributions. Another stream of obsolescence literature assumes that each periodÕs demand is stochastic, increasing the complexity of the models. Many problems are formulated as discrete-time dynamic programs and solved numerically. Pierskalla (1969) formulates a finite-horizon inventory problem with independent and identically distributed demand, but where there is a prior probability, pj , that the product will be obsolete in a future period j. He shows that for various probability distributions of pj , the difference in the optimal solutions (total discounted costs) considering and not considering obsolescence decreases as the number of periods increases. Brown et al. (1964) model the demand distribution as a function of a demand-generating state, and these states form a discrete-time Markov Chain. They formulate the problem of minimizing total discounted costs over an infinite horizon as a dynamic program and recommend its solution by a value iteration procedure. Song and Zipkin (1993, 1996) consider a model where demand is a Poisson Process, with parameter ki dependent on the state of the world i, which behaves like a continuoustime Markov Chain. There are fixed ordering costs, and lead times are stochastic. The problem is formulated as a dynamic program that minimizes total expected discounted costs. The works described above highlight the issues and challenges to managing product obsolescence
but do not focus specifically on end-of-life buys. Even more closely related to our research is the work by Fortuin (1980) and Teunter (1998) on lifetime buys of service parts. A firm has a legal commitment to stock service parts for its customers for a period of N years after end of regular production. In Fortuin (1980), demand in each period (year) of analysis is assumed to be independent, normally distributed, and with an exponentially decaying mean. Based on a desired service level, Fortuin calculates how many parts are needed in order to cover service requests during this N-years period. In Teunter (1998), holding, penalty and disposal costs are introduced, demand in each period is a Poisson Process, with period-dependent rate, and the problem is formulated as a multiperiod inventory problem, but the decision (final order) occurs at t ¼ 1 only. In addition to solving the problem exactly, he proposes an approximation which is shown to be very good for high levels of service (high penalty costs). Teunter and Fortuin (1998) validate this approach with real data. Our work is inspired also by Handfield and Pannesi (1994), who provide a framework for managing the purchasing of components of products in dynamic technological environments. Their framework considers the number of suppliers, contractual agreements, and so forth, for a component depending on the stage of the product life. Our research supports and facilitates contractual agreements with suppliers regarding end-of-life buys. Our research differs from the literature on fashion buys (for example, see Fisher and Raman, 1996), where early sales are used to improve forecasts for the entire life of the product. At the end of the product life cycle, individual demand events are less helpful as an indicator of future demand. We focus first on end-of-life builds of products that face non-constant, declining prices, and other relatively general assumptions. We apply results from Cattani (1997) that determine optimal profit of a universal product used to satisfy N distinct demand classes. In addition to determining optimal stocking levels, we focus on the profit benefits of a delay. Next, we consider end-of-life buys of a component used in a product and we assume constant
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prices. We develop insights into the structure of delay benefits under various demand scenarios, deriving a closed-form solution for the delay benefits. Demand is non-stationary with high overage costs at the end of the time horizon, where nonstationary is defined as having the mean of the demand distribution change over time. In all of our models we make the simplifying assumption of negligible lead times, consistent with most of the related literature. To the extent that demand can be backlogged with negligible penalty, this assumption is not significant, and its primary effect is to make the end-of-life decision occur earlier by the length of the lead time.
3. End-of-life build of a component facing declining prices Consider an end-of-life build to finished goods inventory for a product facing random demand and non-increasing prices over discrete time. There are N periods of remaining demand, distributed normally with known means and covariances. In this section we use results from Cattani (1997) to determine the optimal expected profits from an end-of-life build scenario in a fairly general setting. We determine delay benefits by calculating the optimal expected profits from the end-of-life if it occurs in period 1 versus in alternate periods. The model Cattani (1997) develops is for a very different setting that can be translated into our setting. He examines a single-period stochastic inventory problem where N distinct demand classes can be satisfied with a single universal product. Demands are from a joint normal probability distribution, and revenue is differentiated by demand class. There is a hierarchical policy for filling the individual demands. If demand exceeds supply, priority is given first to demand class 1, then to demand class 2, and so on. The model is directly applicable to our problem when demand is independent between periods, if we consider the N distinct demands to be the demands for the same product in N distinct periods. When demand is not independent between periods, the model can be made applicable by computing the conditional demand distribution after the delay.
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3.1. Notation and assumptions Consider a product sold over N remaining periods. Customers wish to be supplied in the period of their demand. This problem leads to a natural hierarchy for filling demand: first priority is to period 1 customers, then to period 2 customers, etc. Demand is assumed to be distributed with normal marginals Nðli ; r2i Þ for each period i. The marginal densities for demand are denoted by fi ðxÞ. The QNjoint density is f ðx1 ; x2 ; . . . ; xN Þ which equals i¼1 fi ðxi Þ if demands are independent across periods. While we assume that demand is normally distributed with known means and variances in each period, we make no restrictions on their respective values over time, although the coefficient of variation is assumed to be small enough that the possibility of negative demands is negligible. Let pi and Xi denote the selling price and single-period Pk demand, respectively, for period i. Let Xðj;kÞ ¼ i¼j Xi represent total demand during periods j through k, and denote the c.d.f. of Xðj;kÞ by Fðj;kÞ ðÞ and Var½Xðj;kÞ ¼ r2ðj;kÞ . Let Q denote the production level, the decision variable. Denote profits from period j to k as Pðj;kÞ ðÞ. All prices and costs are non-negative, a denotes the one-period discount factor, and h denotes the additional holding cost per unit per period (in addition to opportunity cost of capital – for example, warehousing, insurance, etc. – the holding cost due to opportunity cost of capital is captured by the discount factor). The following is a key result from Cattani (1997) (translated to our setting). Theorem 1. If demand is multivariate-normally distributed, expected profit of an end-of-life build of Q units in period j for demand from periods j through k is Pðj;kÞ ðQÞ ¼
k1 X
ai1 ½pi li ðpi apiþ1 Þnðj;iÞ ðQÞ
i¼j
þ ak1 pk ½lk nðj;kÞ ðQÞ aj1 cQ " !# k i X X i1 a Q ll þ nðj;iÞ ðQÞ ; h i¼j
l¼j
ð1Þ
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where nðj;kÞ ðQÞ ¼ the expected lost sales in periods j to k ! ! k X Q li ¼ rðj;kÞ L rðj;kÞ ; i¼j
rðj;kÞ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k k1 X k X uX ¼t r2i þ 2 qim ri rm i¼j
i¼j m¼iþ1
and
pffiffiffiffiffiffi Z LðzÞ ¼ 1= 2p
1
2 =2
ðu zÞeu
du
z
is the standardized normal loss function. The optimal stocking level is determined by setting the first derivative of (1) to zero. If p1 P p2 P P pN , as is reasonable for many products at end-of-life, then Cattani shows that the problem is concave and any Q such that ðdPð1;N Þ ðQ Þ=dQ ¼ 0 is necessarily optimal, where dPð1;N Þ ðQ Þ dQ N X ¼ ai1 ðpi piþ1 Þð1 Fð1;iÞ ðQÞÞ hFð1;iÞ ðQÞ c i¼1
¼ 0: The non-increasing prices are sufficient to ensure a concave profit function. Other price patterns as well are possible that cause the second derivative to be negative, ensuring concavity. Straightforward extensions to this model account for discounted salvage values or disposal costs and for penalty costs (such as lost goodwill). Setup costs for the builds may also be considered directly. The theorem can be used directly to determine optimal end-of-life build quantities and profits, as shown in the following example. Example 1. Suppose that N ¼ 5, demands are independent across time, there is a constant coefficient of variation cv ¼ 0:4, l1 ¼ 50; l2 ¼ 26; l3 ¼ 14; l4 ¼ 7; l5 ¼ 3; p1 ¼ $100; p2 ¼ $85; p3 ¼ $70; p4 ¼ $55; p5 ¼ $40; c ¼ $50; a ¼ 1; and h ¼ 0: Then the optimal (denoted with a ‘‘*’’) end-
of-life build is found by numerically solving (1) to find Q ¼ 88 units and optimal expected profit of $3194. 3.2. Determining delay benefits under independent demand When demand is independent across time, the benefit of delaying an end-of-life build can be determined by comparing the expected profit calculated above with the expected profit for each possible value of the delay. We use (1) to determine Pð1;N Þ ðQ 1 Þ and Pðdþ1;N Þ ðQ dþ1 Þ, the expected profits given no delay and a delay of d, respectively. To compare these results we must add to Pðdþ1;N Þ ðQ dþ1 Þ the expected profits achieved before the delay Pð1;dÞ ðÞ, which are determined directly (and not by (1)). Example 2. Using the cost and demand parameters from Example 1, assuming no underage or overage costs before the end-of-life build, c ¼ $50, the benefits of delay are shown in Fig. 1 for oneperiod discount factors of 1, 0.9, and 0.8, and h ¼ 0 (left frame) and h ¼ 5 (right frame). With h ¼ 0; delay benefits under no discount (a ¼ 1) are increasing and concave in d varying from 10% increase in profits with a one period delay to 17% for a four period delay. With a discount of a ¼ 0:8, profits increased 66% with a one period delay and 99% with a four period delay. Delay benefits are slightly higher with h ¼ 5: In Section 4 we determine general properties of a delay by using more restrictive assumptions. 3.3. Determining delay benefits if demand is not independent If demand is not independent across time, the benefit of delaying an end-of-life build still is determined by comparing the expected profit of an end-of-life build under no delay with the expected profit for each possible value of the delay. We use (1) to determine Pð1;N Þ ðQ Þ. In calculating optimal end-of-life builds at time t ¼ d þ 1, however, we must use the distribution of demand for periods d þ 1 through n conditioned on actual demands realized in periods 1 through d. We denote this
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Fig. 1. Delay benefits in a discrete time example under various discount factors.
Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ and it is determined using (1) with conditional demand distributions: Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ " N X N 1 d ¼ pN a E½Xi j x1 ; . . . ; xd ca i¼dþ1
#
dictable after realization of initial demand. At time t ¼ 1, a calculation of total expected profits over the N periods with a delay of d must consider all possible values for demand in the first d periods, taking the expectation over all possible realizations of demand before d: Pð1;N Þ ðQ j x1 ; . . . ; xd Þ
nðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ ¼
" þ cad Q
N X
#
þ nðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ N 1 X
a
þ
Z
1
0
Z
1
Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ 0
fXð1;dÞ ðx1 ; . . . ; xd Þ dx1 dxd :
½ai1 pi aN 1 pN E½Xi j x1 ; . . . ; xd
i¼dþ1 i1
ai1 ðpi cÞli
i¼1
E½Xi j x1 ; . . . ; xd
i¼dþ1
þ
d X
½pi apiþ1 nðdþ1;iÞ ðQ j x1 ; . . . ; xd Þ :
We add to the profit Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ the expected profits achieved before the delay (from time t ¼ 1 until time t ¼ d), which should include expected margin as well as expected underage and overage costs, if any. The calculation for Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ has as its lower bound the delay benefits calculated assuming independence; a stronger auto-correlation creates greater benefit from delay since remaining demand is more pre-
4. Properties of delay benefits In this section, we develop further insights into the structure of delay benefits of an end-of-life buy by considering relatively tighter assumptions that allow us to derive a closed-form solution for the delay benefits in a continuous-time model. Let X denote the random variable representing the remaining lifetime demand for the component as of time t ¼ 0 (X is demand in the interval ½0; 1Þ). We now let X1 denote demand for the component in the interval ½0; d; and X2 the
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demand in the interval ½d; 1Þ. That is, X ¼ X1 þ X2 : We make several simplifying assumptions. We assume that X1 and X2 are independent, that the price is constant over time, and that fixed ordering costs are negligible. We consider that the component is used in products that are made-to-order (i.e., the components are not purchased and assembled absent a customer order for the product) with full backlogging (at no penalty). Thus, under these assumptions, before the end-of-life buy the manufacturer realizes negligible inventory shortage or excess costs for the component. Finally, we consider relatively short time periods and ignore discounting and holding costs. These assumptions of independence and constant prices are tighter than those of Section 3 but allow us further insights into the structure of the problem. The assumption of a made-to-order and full backlogging environment focuses the modelÕs results on the uncertainty reduction effects of delaying an end-of-life buy, since the effects of variability before the buy are controllable. The rationale for the assumption is that the unit underage and overage costs after the buy are significantly greater than before the buy because there is no longer chance for recourse. After the final buy, backlogging is not relevant (there is no future supply), but the manufacturer incurs inventory underage or overage costs. The assumptions of no discounting or holding costs are strong, but have the effect of making the modelÕs results conservative. Delaying the final purchase will lower inventory levels and postpone a cash outflow – benefits not captured in this model. 4.1. Basic costs of initial problem The basic costs of an end-of-life buy at time t ¼ 0 are calculated using the newsvendor model. Following the newsvendor formulation, let cu be the cost of being under by one unit and let co be the cost of being over by one unit, including disposal costs and/or salvage values. Also, let m represent the unit margin on the product, e.g. p c (which is equivalent to cu if there are no underage penalties in addition to lost margin). The optimal order quantity is
y ¼ F 1 ðcu =ðcu þ co ÞÞ;
ð2Þ
where F ðxÞ is the c.d.f. of X. Total expected profits are ^ ðy; X Þ ¼ mE½X co E½ðy X Þþ P cu E½ðX yÞþ ¼ mE½X co ðy E½X Þ ðcu þ co ÞnX ðyÞ; ð3Þ R1 where nX ðyÞ ¼ x¼y ðx yÞ dF ðxÞ is the loss function, the mean number of customers lost if the order quantity is y. With an end-of-life buy at time t ¼ d; the problem is the same format as with t ¼ 0, but the remaining demand (for the end-of-life buy) is now X2 : Because this component is used in products that are made to order, at t ¼ d the component inventory is zero. Then, total expected profits in the interval ½d; 1Þ are a direct application of (3), ^ ðy2 ; X2 Þ, where y2 is the optimal order quantity at P t ¼ d. Since we assume underage and overage costs in the period ½0; d to be negligible, total expected ^ ðy2 ; X2 Þ. profits for ½0; 1Þ are mE½X1 þ P 4.2. Evaluation of delay benefits Let BðdÞ denote the benefits from a delay of d. Then ^ ðy2 ; X2 Þ P ^ ðy; X Þ BðdÞ ¼ mE½X1 þ P ¼ co ðy E½X Þ þ ðcu þ co ÞnX ðyÞ ðco ðy2 E½X2 Þ þ ðcu þ co ÞnX2 ðy2 ÞÞ ¼ co ððy y2 Þ ðE½X E½X2 ÞÞ þ ðcu þ co ÞðnX ðyÞ nX2 ðy2 ÞÞ:
ð4Þ
The following proposition formalizes an intuitive property of d, that the delay benefits are nondecreasing in d. For the proof, we need the following concept. Given two random variables X and Y, we say that X is more variable than Y, denoted by X P v Y , if E½hðX Þ P E½hðY Þ for all increasing, convex functions h. From this, it follows that if X P v Y and E½X ¼ E½Y ; then VarðX Þ P VarðY Þ.
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Property 1. B is non-decreasing in d. Proof. We can rewrite (4) as BðdÞ ¼ TCð0Þ TCðdÞ, where TC are the total costs of the newsvendor problem. It suffices to show that TCðdÞ are non-increasing in d. We denote X2 ¼ X2 ðdÞ to make the dependence on d clearer. Consider two delays a and b, such that a > b, and denote by X½b;a the demand in the interval ½b; a. Then X2 ðbÞ ¼ X2 ðaÞ þ X½b;a :
ð5Þ
Ross (1996, p. 455), shows that if there exists a random variable Z such that E½Z P 0, and such that X has the same distribution as Y þ Z; then X P v Y . Thus, (5) and the fact that E½X½b;a P 0 implies that X2 ðbÞ P v X2 ðaÞ. Song (1994, Proposition 4.10), shows that in the newsvendor model, a sufficient condition for total costs to be higher is that the demand distribution be more variable. Then, X2 ðbÞ P v X2 ðaÞ implies TCðbÞ P TCðaÞ, which completes the proof. Property 1 states that a longer delay is always as good or better than a shorter delay. 4.3. Delay benefits under normal demand
Pðl; rÞ ¼ ml rðco z þ ðcu þ co ÞLðzÞÞ:
ð6Þ
From (2) and using z ¼ ðy lÞ=r; we rewrite z in terms of cu and co and the c.d.f. of the standard normal distribution (U): z ¼ U1 ðcu =ðcu þ co ÞÞ:
in delaying the final buy d time units. We discuss this issue in depth in Section 5. A key issue determining the benefits of delay is the structure of demand over time. If demand uncertainty is expected to be resolved early on, we would expect a significant benefit for a delay of the end-of-life buy until after the uncertainty is resolved. Under a relatively mild condition, we next show that the delay benefits are concave in d. This property is of managerial relevance: most benefits of a delay are obtained early. We model the mean demand rate over time, which we formally define as follows. We assume that demand for the component in the interval ½0; t; denoted by X ðtÞ; is distributed normally with mean lðtÞ and variance r2 ðtÞ. Then the mean demand rate, denoted by rðtÞ is defined as rðtÞ ¼ dlðtÞ=dt. Further, let us assume we can describe rðtÞ as a function of lðtÞ, that is, rðtÞ ¼ gðlðtÞÞ. For example, if gðuÞ ¼ u the coefficient of variation is constant, and if gðuÞ ¼ u1=2 the variance-tomean ratio is constant. (Constant coefficients of variation arise under exponential distributions while constant variance-to-mean ratios arise under Poisson distributions.) Then BðdÞ ¼ ð1 gðl2 ðdÞÞ=gðlÞÞrAðzÞ:
When demands are distributed normally (as in Fortuin, 1980), we are able to derive a closed-form solution for the delay benefits. If demand is from a normal distribution X Nðl; r2 Þ, and X2 Nðl2 ; r22 Þ, then nðyÞ ¼ rLðzÞ, where z ¼ ðy lÞ=r; and LðzÞ is the standard loss function defined in Theorem 1. Then (3) can be rewritten as
ð7Þ
Denote, for convenience AðzÞ ¼ ðco z þ ðcu þ co Þ LðzÞÞ, the total underage and overage costs in the newsvendor model with a demand variance of one. Then, the delay benefits (4) are B ¼ ðr r2 ÞAðzÞ. From a supply-chain perspective, delaying the end-of-life buy is attractive only if the delay benefits B are greater than the cost the supplier incurs
223
ð8Þ
Property 2. B is concave in d if rðÞ is non-increasing and g is increasing and concave. Proof. First, note that oB ol ¼ g0 ðl2 ðdÞÞ 2 : od od
Rd We rewrite l2 ¼ l u¼0 rðuÞ du: Then, ol2 =od ¼ rðdÞ. Thus, oB=od ¼ g0 ðl2 ðdÞÞrðdÞ. Taking the second derivative, o2 B=od 2 ¼ g00 ðl2 ðdÞÞrðdÞ þ r0 ðdÞg0 ðl2 ðdÞÞ 6 0, if gðÞ is increasing (a mild assumption) and concave, and r is non-increasing. B is concave, for example, if the varianceto-mean-ratio or the coefficient of variation is constant. From Property 2 we conclude that most benefits of delaying an end-of-life buy are obtained early: even a short delay may help. Since the product is at the end of its life cycle, it typically has a decaying demand rate according to the traditional product life-cycle curve, and rðÞ non-increasing is
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generally a reasonable assumption. The assumption that g is increasing is also reasonable: it only requires that as the mean increases, the variance also increases. The assumption that g is concave requires that the variance increases at a constant or decreasing return to scale with the mean. If demand is an aggregation of independent increments, for example, this assumption is satisfied. 4.4. Examples of different demand profiles for remaining demand We examine two examples of demand during the product end of life: a constant mean demand subject to a sudden death (which is identical to an exponential decay of the mean demand rate), and an exponential decay of the mean demand rate subject to a rate change. In both examples we assume a constant coefficient of variation, that is, in (8), gðuÞ ¼ u. Example 3. Sudden death (or exponential decay of mean demand rate). Suppose that the mean demand rate is constant but subject to a sudden death. This scenario represents a possible change in the state of the world, such as a new product introduction. As in Song and Zipkin (1993, 1996), we assume that this change occurs at a random time T, where T expðqT Þ. That is, E½T ¼ 1=qT . The mean demand rate is lqT ; 0 6 t 6 T ; rht j T i ¼ 0; otherwise:
demand as of t ¼ 0 is R 1expected remaining R 1The rht j T iqT eqT T dt dT ¼ l. The expected T ¼0 t¼0 remaining demand (as of t ¼ 0) for t ¼ d is Z 1 Z 1 l2 ¼ rht j T i dt qT eqT T dT T ¼0
¼
Z
1
T ¼d
t¼d
Z
T
lqT dt qT eqT T dT ¼ leqT d :
t¼d
The delay benefits (8) are BðdÞ ¼ ð1 eqT d Þ rAðzÞ. An intuition of the result can be developed as follows. At t ¼ 0 we have the choice of making an end-of-life buy immediately, or deciding to delay the end-of-life buy until t ¼ d. If we delay the buy until t ¼ d, sudden death will have occurred (with probability 1 eqT d ) or not occurred (with probability eqT d ). If sudden death has occurred, we have zero underage and overage costs (we meet our demand exactly), and cost savings (compared to having made an end-of-life buy at time t ¼ 0Þ are rAðzÞ. If it has not occurred, we must make an end-of-life buy. Here, because of the memoryless property, time until the sudden death is still distributed as expðqT Þ, and our expected costs are just as if we had made the end-of-life buy at t ¼ 0. That is, in this case our cost savings are zero. Thus, our delay benefits are ð1 eqT d ÞrAðzÞ. Notice that in this intuitive derivation, we need no assumptions with respect to the variance or mean, only that there is independence of demand. Suppose now that mean demand rate decays exponentially with time (Fig. 2, middle frame), that is, rðtÞ ¼ lqR eqR t , where qR is the obsolescence rate, a parameter. This is equivalent to saying that the mean of the remaining demand is
Fig. 2. Delay benefits under constant demand rate or exponential decay rate.
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leqR t . Then l2 ¼ leqR d . The delay benefits (8) are BðdÞ ¼ ð1 eqR d ÞrAðzÞ. This scenario is identical to a sudden death, if qT ¼ qR . That is, the exponential decay of demand rate is mathematically equivalent to a constant demand rate subject to an exponentially distributed sudden death. Fig. 2 (right frame) shows a graph of BðdÞ for cu ¼ 1000, co ¼ 45, r ¼ 300, and two values of q (where q could be either qT or qR ). We note the following property of B. The sooner the sudden death is expected to occur, the more attractive a delay becomes. Alternatively, the higher the obsolescence rate, the more attractive a delay. Example 4. Exponential decay of demand rate subject to obsolescence rate change. We now consider a situation where the obsolescence rate is subject to change. Mean demand rate decays with obsolescence rate qR1 , as in Example 3. At random time T, mean demand rate changes suddenly to crðT Þ, and the obsolescence rate becomes qR2 , where qR2 6¼R1 . This scenario represents a possible change in the state of the world, such as in scenario 1. For example, a competitor announces a new product and demand drops, i.e., qR2 > qR1 . Alternatively, a competitive product is delayed, and demand increases, i.e., qR2 < qR1 . We assume, as in scenario 1, that T expðqT Þ. The expression for demand rate is
rht j T i ¼
lqR1 eqR1 t ; clqR1 e
225
t < T;
qR1 T qR2 ðtT Þ
e
;
t PT:
To compute r2 , we need to compute the expected remaining demand for t ¼ d Z 1 Z 1 l2 ¼ rht j T idt qT eqT T dT T ¼0
t¼d
cqT qR1 eqR2 d eðqR1 þqT Þd ðqR1 þqT Þd ¼l þe qR2 qR1 þ qT qR2 ¼ bðdÞl: The delay benefits (8) are BðdÞ ¼ ð1 bðdÞÞrAðzÞ. Fig. 3 shows this relationship for various values of qR2 and qT , using the parameter values from Example 3 (and qR1 ¼ qR ). The benefits are greater when there are more chances of resolving the uncertainty by a delay, that is, when there is a larger difference between qR1 and qR2 , a lower value of c, and a higher value of qT (the obsolescence rate change is expected to occur sooner). In Fig. 3, the benefits are greatest for qT ¼ 1; c ¼ 0 – a sudden death scenario. The least benefits occur for qR2 ¼ 1; qT ¼ 0:5; c ¼ 0:8. In this case, there is only a 20% sudden change in the mean demand rate with no obsolescence rate change (qR1 ¼ qR2 ), and it is expected to occur relatively far in the future (2 years ¼ 1/0.5).
Fig. 3. Delay benefits for exponential demand rate subject to obsolescence rate chance.
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5. The individual player’s perspective
Pm ðdÞ ¼ mm l r2 ðdÞAðzÞ:
The analysis to here has focused on the benefits of a delay of an end-of-life purchase from the perspective of the manufacturer, which we suggested should be aggregated with the delay costs incurred by the supplier to determine the delay effects from a supply chain perspective. (For a manufacturer considering an end-of-life build, no supply chain aggregation is necessary since both benefits and costs accrue to the manufacturer.) In this section, we present an analysis of both individual playerÕs perspectives. There are two players: the supplier and the manufacturer. We consider the assumptions of the continuous-time model (Section 4), and add subscripts m and s to indicate the manufacturer and supplier, respectively. Generally, the aggregate supply chain effects of a delay include a benefit to the manufacturer offset partly by a loss to the supplier. The supplier would prefer no delay for at least two reasons: the costs of maintaining a line for production of a declining product, and because an end-of-life buy forces the manufacturer to buy more units (as safety stock) than otherwise would have been required with no end-of-life buy. As a secondary benefit to the supplier, the revenue for all of the units corresponding to the end-of-life buy is received earlier than it would otherwise have arrived. From a supply chain perspective, an end-of-life buy causes increased underage costs and overage costs to the manufacturer, which is partially offset by the increased margin to the supplier. To gain insights to the dynamics between the supplier and manufacturer, we consider two games. In the first, the manufacturer does not provide any incentives to the supplier in delaying the final buy decision. We show that there is no equilibrium, since the two players have directly opposite interests. In the second, the manufacturer allocates a portion of the delay benefits to the supplier to compensate for the supplierÕs losses.
Consider that the supplier has a non-decreasing maintenance cost CðdÞ of delaying the end-of-life purchase by d. CðdÞ includes only the costs associated with maintaining the production line (overhead, floor space, etc.). The payoff to the supplier Ps ðdÞ is ms l1 þ ms y2 CðdÞ. Under the assumptions of normality, y2 ¼ l2 þ zr2 , and thus
5.1. A game with no incentives to the supplier Given a delay of d, the payoff (i.e., profit) to the manufacturer Pm ðdÞ is mm l1 þ Pm ðl2 ; r2 Þ, or:
Ps ðdÞ ¼ ms l þ ms r2 ðdÞz CðdÞ:
ð9Þ
ð10Þ
In an end-of-life situation where l2 ðdÞ is nonincreasing in d, then r2 ðdÞ is non-increasing in d, given the reasonable assumption that demand variance increases with the mean. Thus (9) is monotonically increasing in d, and (10) is monotonically decreasing in d. The two players have diametrically opposite interests, and thus there is no equilibrium. 5.2. A game with supplier incentive to delaying the end-of-life purchase Consider now the same game from Section 5.1, but now the manufacturer compensates the supplier for any profit losses incurred by delaying, that is, the manufacturer pays to the supplier ðPs ð0Þ Ps ðdÞÞ. In addition, as an incentive, the manufacturer also allocates to the supplier a portion a of the supply-chain benefits of delaying the end-of-life buy BSC ðdÞ, given by: BSC ðdÞ ¼ ðPm ðdÞ Pm ð0ÞÞ þ ðPs ðdÞ Ps ð0ÞÞ ¼ ðr r2 ðdÞÞðAðzÞ ms zÞ CðdÞ:
ð11Þ
Thus, the manufacturerÕs and supplierÕs profits after allocation are: Pam ðdÞ ¼ Pm ð0Þ þ ð1 aÞBSC ðdÞ;
ð12Þ
Pas ðdÞ ¼ Ps ð0Þ þ aBSC ðdÞ:
ð13Þ
Note that for any a > 0; there is an incentive to the supplier in delaying the end-of-life buy, since Pas ðdÞ > Pas ð0Þ, for d > 0: Also, this allocation assures that the maximum of (12) and (13) occur at the same d ; which is the d that maximizes (11). (If BSC ðdÞ 6 0 for all d, then d ¼ 0.)
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Fig. 4. Delay benefits – playersÕ perspective.
Fig. 4 illustrates the two games for the parameter values from Example 3 (and with ms ¼ 20; a ¼ 0:1; and CðdÞ ¼ 4000dÞ. The top and bottom curves represent the delay benefits (PðdÞ Pð0Þ) for the manufacturer and supplier, respectively, under no incentives. In this example, the net supply chain benefits of a delay (second curve from top) are small compared to the net gain by the manufacturer (top curve) and the net loss by the supplier (bottom curve). In the absence of significant incentives from the manufacturer to the supplier, the supplier will want end-of-life buys as early as possible in the product life cycle. Under scenarios such as this, manufacturers might want to consider creating incentives that manage endof-life buy considerations for key suppliers of critical components other than through incentives at the end of the product life cycle. For example, the issue might be raised and addressed during initial sourcing negotiations. Otherwise, suppliers
have significant incentives to require end-of-life buys, and as early as possible.
6. Conclusion We have studied end-of-life purchases, and modeled the effects of delaying an end-of-life buy or build. Under various scenarios of remaining demand, we observe that benefits of a delay to the manufacturer of an end-of-life buy are nondecreasing and concave in the delay time. A longer delay is always as good as or better than a shorter delay, and most benefits of a delay are obtained early. In general, the sooner the uncertainty is expected to be resolved, the more attractive to the manufacturer a delay becomes. If there are two players involved, as is the case when a critical component of the product is obtained from a supplier, we show that the parties
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have directly opposite interests with respect to the delay decision. Thus, it is not surprising to find anecdotal evidence of an increasing number of suppliers requiring end-of-life buys from their customers. For the delay decision to be implemented, it is necessary that the manufacturer compensate the supplier for their losses incurred. From a supply chain perspective, delaying the endof-life buy is attractive if the delay benefits are greater than the cost the supplier incurs under the delay. The end-of-life purchase is not always a good buy.
Acknowledgements This research was partly funded by the Burress Faculty Development and Support Fund. The authors would like to thank Kyle Christensen of Hewlett-Packard for information on end-of-life inventory.
References Brown, G., Lu, J., Wolfson, R., 1964. Dynamic modeling of inventories subject to obsolescence. Management Science 11 (1), 51–63. Brown, R.G., 1971. Economic order quantities for material subject to engineering change. Production & Inventory Management (2nd Quarter), 89–91. Bulkeley, W.M., 1999. IBM Had Õ98 PC pretax loss of nearly $1 billion. The Wall Street Journal, March 25, 1999. Cattani, K.D., 1997. Demand pooling effects of a universal product when demand is from distinct markets. Ph.D.
Dissertation, Department of Industrial Engineering and Engineering Management, Stanford University. David, I., Greenshtein, E., Mehrez, A., 1997. A dynamicprogramming approach to continuous-review obsolescent inventory problems. Naval Research Logistics 44, 757–774. Fisher, M., Raman, A., 1996. Reducing the cost of demand uncertainty through accurate response to early sales. Operations Research 44 (1), 87–99. Fortuin, L., 1980. The all-time requirement of spare parts for service after sales – theoretical analysis and practical results. International Journal of Operations and Production Management 1 (1), 59–70. Handfield, R., Pannesi, R., 1994. Managing component life cycles in dynamic technological environments. International Journal of Purchasing and Materials Management (Spring), 20–27. Joglekar, P., Lee, P., 1993. An exact formulation of inventory costs and optimal lot size in face of sudden obsolescence. Operations Research Letters 14, 283–290. Masters, J.M., 1991. A note on the effect of sudden obsolescence on the optimal lot size. Decision Sciences 22, 1180–1186. Pierskalla, W., 1969. An inventory problem with obsolescence. Naval Research Logistics Quarterly 16, 217–228. Ross, S., 1996. Stochastic Processes, second ed. Wiley, New York. Song, J., 1994. The effect of leadtime uncertainty in a simple stochastic inventory model. Management Science 40 (5), 603–613. Song, J., Zipkin, P., 1993. Inventory control in a fluctuating inventory environment. Operations Research 41 (2), 351–370. Song, J., Zipkin, P., 1996. Managing inventory with the prospect of obsolescence. Operations Research 44 (1), 215–222. Teunter, R., 1998. Inventory control of service parts in the final phase. Theses on Systems, Organisations and Management, University of Groningen, The Netherlands. Teunter, R., Fortuin, L., 1998. End of life service: A case study. European Journal of Operational Research 107, 19–34. Zipkin, P., 2000. Foundations of Inventory Management. McGraw-Hill, New York.
Good buy? Delaying end-of-life purchases Kyle D. Cattani a
a,*
, Gilvan C. Souza
b,1
The Kenan-Flagler Business School, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3490, USA b The Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA Received 30 October 2000; accepted 7 December 2001
Abstract We study the effects of delaying an end-of-life buy. Manufacturers sometimes are required to place a final, end-of-life buy for a component that the supplier will no longer provide. The manufacturer performs a newsvendor analysis with the possible result of significant expected overage and underage costs. If the decision can be delayed, the expected overage and underage costs can be reduced. We model the effects of a delay of the final purchase under various scenarios of remaining demand. We contrast the manufacturing benefits with the costs incurred by the supplier and show that the supplier, who benefits greatly from the end-of-life buy, likely will require an incentive to enact a delay. Our results provide an insight to the observation of increasing numbers of end-of-life buys and provides a framework for analysis as manufacturers strive to cope with the issue. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Inventory; Supply chain management
1. Introduction The final stages of the product life cycle can create undesirable and unavoidable issues for inventory management of a product. Demand for the product is declining, yet total remaining demand can be very difficult to predict. Demand is uncertain due to inherent variability as well as external factors such as introductions of competing products. The challenges of managing product end of life have intensified with shorter product *
Corresponding author. Tel.: +1-919-962-3273; fax: +1-919962-6949. E-mail addresses: [email protected] (K.D. Cattani), [email protected] (G.C. Souza). 1 Tel.: +1-301-405-0628; fax: +1-301-405-8655.
life cycles and can lead to disastrous results. Many firms have been faced with large write-offs of excess inventory after the product life ultimately ends. IBM disclosed a $1 billion loss from its personal-computer business in 1998, attributed to excess PCs in dealer channels that had to be sold at a steep discount (Bulkeley, 1999). Product components, particularly those used only in one product, inherit inventory management complexities from their dying parents. For the productÕs end of life, manufacturers strive to reduce or eliminate all buffers and excess stock of such components. Concurrent with the manufacturerÕs struggles to manage product end of life, the components suppliers (who consider the component to be their product) are also concerned with managing this stage of their productÕs life cycle. If
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 2 ) 0 0 2 1 2 - 6
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a supplierÕs customer base is well-defined and concentrated, the supplier may opt to do a final build of the component and require that its customers (e.g., manufacturers) make a final, end-of-life buy, after which production of the component is discontinued. Thus, the manufacturer is required to make a potentially large purchase of a part that faces uncertain and disappearing demand but remains crucial for current business. 2 From the manufacturerÕs perspective, stocking such a part in inventory is quite undesirable. One of the authors worked as a purchasing manager at one of Hewlett-PackardÕs computer manufacturing divisions, where there consistently are tens of millions of dollars of inventory resulting from end-of-life buys for dozens of components. Much of this inventory eventually is scrapped, reducing profits by up to 1% of revenue each year. These write-offs are significant, especially as pressure on margins increase. All other things being equal, the manufacturer would prefer to eliminate, or at the least postpone, this final end-of-life buy. Postponing the end-of-life buy is helpful to the extent that demand uncertainty can be partially or fully resolved during the delay, reducing the size of the inventory buffer of parts that mostly will be scrapped. In this paper we study the effects of delaying end-of-life purchases. We first model the benefits to the manufacturer of a delay of the final end-of-life purchase. We contrast these benefits to the costs the supplier incurs in delaying its final build and thus determine the optimal timing of the final buy, from an aggregated supply chain perspective. We show that a supplier must be provided an incentive for a delay to be feasible. Our results provide an insight to the observation of increasing numbers of endof-life buys and provides a framework for analysis as manufacturers strive to cope with the issue. While the example above involves sales by a component supplier to a manufacturer, the problem is identical for other settings where there is a 2 This scenario is apparently becoming more common. An IBM manager reports, for example, that recently there has been an increase in the frequency of technology suppliers ending production of parts earlier than in the past. (July 22, 1999 interview with Holly Hach, Director of Reengineering and Planning, Personal Systems Group, IBM.)
217
proposed end-of-life buy or build. For example, a manufacturer may wish to shut down an assembly line dedicated to a dying product but may be unable to require that customers place final orders, either because customers are too diverse, too powerful, or not known in advance. In this case, an end-of-life build to finished goods inventory may be proposed. Then, both the costs and benefits of delaying the final build accrue to the manufacturer, although the decision is of similar form to the first example. We sometimes frame the discussion in terms of an end-of-life build as it is easier to articulate the cost of underage and overage since there is only one player, and since bill of material relationships between products and components that might confound the discussion are not present. The paper proceeds as follows: In Section 2 we discuss previous work in this area. In Section 3, we use a result from Cattani (1997) to develop a general multi-period model of the benefits from a delay of an end-of-life build of a product that faces declining prices and demand over its remaining life, although the results hold for end-of-life buys as well. We contrast the expected profits of the end-of-life build at time zero with the profits if there is a delay of d periods. In Section 4, we use continuous time and we use the traditional newsvendor analysis to consider an end-of-life buy of a component that has a constant price (over the remaining lifetime) and make additional assumptions that allow us to derive a closed-form solution for delay benefits. We model the benefit of delaying the end-of-life buy under different demand scenarios and identify key properties of the delay benefits. In Section 5, we consider the perspective of each individual player in the supply chain. We summarize and conclude in Section 6.
2. Related research Much has been written on managing product obsolescence, where we define obsolescence as a high likelihood that demand will drop substantially from its current level. In the inventory literature, a typical focus determines stocking levels over time for a product or a purchased part. The traditional single-period newsvendor problem is
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directly relevant to the end-of-life buy and is a basis for much of the research related to product obsolescence. Multi-period inventory problems raise additional issues, many of which also are relevant to product obsolescence. Zipkin (2000) provides a good review of inventory theory. We review here only papers that are more closely related to this research, concentrating on obsolescence as defined above. A stream of literature analyzes the obsolescence problem by finding optimal stocking levels, assuming that the demand rate is constant but an exogenous process causes obsolescence – and so the lifetime of the product is a random variable, with a known distribution. Brown (1971), Masters (1991), and Joglekar and Lee (1993) assume that the lifetime is exponentially distributed. David et al. (1997) generalize this problem to other lifetime distributions. Another stream of obsolescence literature assumes that each periodÕs demand is stochastic, increasing the complexity of the models. Many problems are formulated as discrete-time dynamic programs and solved numerically. Pierskalla (1969) formulates a finite-horizon inventory problem with independent and identically distributed demand, but where there is a prior probability, pj , that the product will be obsolete in a future period j. He shows that for various probability distributions of pj , the difference in the optimal solutions (total discounted costs) considering and not considering obsolescence decreases as the number of periods increases. Brown et al. (1964) model the demand distribution as a function of a demand-generating state, and these states form a discrete-time Markov Chain. They formulate the problem of minimizing total discounted costs over an infinite horizon as a dynamic program and recommend its solution by a value iteration procedure. Song and Zipkin (1993, 1996) consider a model where demand is a Poisson Process, with parameter ki dependent on the state of the world i, which behaves like a continuoustime Markov Chain. There are fixed ordering costs, and lead times are stochastic. The problem is formulated as a dynamic program that minimizes total expected discounted costs. The works described above highlight the issues and challenges to managing product obsolescence
but do not focus specifically on end-of-life buys. Even more closely related to our research is the work by Fortuin (1980) and Teunter (1998) on lifetime buys of service parts. A firm has a legal commitment to stock service parts for its customers for a period of N years after end of regular production. In Fortuin (1980), demand in each period (year) of analysis is assumed to be independent, normally distributed, and with an exponentially decaying mean. Based on a desired service level, Fortuin calculates how many parts are needed in order to cover service requests during this N-years period. In Teunter (1998), holding, penalty and disposal costs are introduced, demand in each period is a Poisson Process, with period-dependent rate, and the problem is formulated as a multiperiod inventory problem, but the decision (final order) occurs at t ¼ 1 only. In addition to solving the problem exactly, he proposes an approximation which is shown to be very good for high levels of service (high penalty costs). Teunter and Fortuin (1998) validate this approach with real data. Our work is inspired also by Handfield and Pannesi (1994), who provide a framework for managing the purchasing of components of products in dynamic technological environments. Their framework considers the number of suppliers, contractual agreements, and so forth, for a component depending on the stage of the product life. Our research supports and facilitates contractual agreements with suppliers regarding end-of-life buys. Our research differs from the literature on fashion buys (for example, see Fisher and Raman, 1996), where early sales are used to improve forecasts for the entire life of the product. At the end of the product life cycle, individual demand events are less helpful as an indicator of future demand. We focus first on end-of-life builds of products that face non-constant, declining prices, and other relatively general assumptions. We apply results from Cattani (1997) that determine optimal profit of a universal product used to satisfy N distinct demand classes. In addition to determining optimal stocking levels, we focus on the profit benefits of a delay. Next, we consider end-of-life buys of a component used in a product and we assume constant
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prices. We develop insights into the structure of delay benefits under various demand scenarios, deriving a closed-form solution for the delay benefits. Demand is non-stationary with high overage costs at the end of the time horizon, where nonstationary is defined as having the mean of the demand distribution change over time. In all of our models we make the simplifying assumption of negligible lead times, consistent with most of the related literature. To the extent that demand can be backlogged with negligible penalty, this assumption is not significant, and its primary effect is to make the end-of-life decision occur earlier by the length of the lead time.
3. End-of-life build of a component facing declining prices Consider an end-of-life build to finished goods inventory for a product facing random demand and non-increasing prices over discrete time. There are N periods of remaining demand, distributed normally with known means and covariances. In this section we use results from Cattani (1997) to determine the optimal expected profits from an end-of-life build scenario in a fairly general setting. We determine delay benefits by calculating the optimal expected profits from the end-of-life if it occurs in period 1 versus in alternate periods. The model Cattani (1997) develops is for a very different setting that can be translated into our setting. He examines a single-period stochastic inventory problem where N distinct demand classes can be satisfied with a single universal product. Demands are from a joint normal probability distribution, and revenue is differentiated by demand class. There is a hierarchical policy for filling the individual demands. If demand exceeds supply, priority is given first to demand class 1, then to demand class 2, and so on. The model is directly applicable to our problem when demand is independent between periods, if we consider the N distinct demands to be the demands for the same product in N distinct periods. When demand is not independent between periods, the model can be made applicable by computing the conditional demand distribution after the delay.
219
3.1. Notation and assumptions Consider a product sold over N remaining periods. Customers wish to be supplied in the period of their demand. This problem leads to a natural hierarchy for filling demand: first priority is to period 1 customers, then to period 2 customers, etc. Demand is assumed to be distributed with normal marginals Nðli ; r2i Þ for each period i. The marginal densities for demand are denoted by fi ðxÞ. The QNjoint density is f ðx1 ; x2 ; . . . ; xN Þ which equals i¼1 fi ðxi Þ if demands are independent across periods. While we assume that demand is normally distributed with known means and variances in each period, we make no restrictions on their respective values over time, although the coefficient of variation is assumed to be small enough that the possibility of negative demands is negligible. Let pi and Xi denote the selling price and single-period Pk demand, respectively, for period i. Let Xðj;kÞ ¼ i¼j Xi represent total demand during periods j through k, and denote the c.d.f. of Xðj;kÞ by Fðj;kÞ ðÞ and Var½Xðj;kÞ ¼ r2ðj;kÞ . Let Q denote the production level, the decision variable. Denote profits from period j to k as Pðj;kÞ ðÞ. All prices and costs are non-negative, a denotes the one-period discount factor, and h denotes the additional holding cost per unit per period (in addition to opportunity cost of capital – for example, warehousing, insurance, etc. – the holding cost due to opportunity cost of capital is captured by the discount factor). The following is a key result from Cattani (1997) (translated to our setting). Theorem 1. If demand is multivariate-normally distributed, expected profit of an end-of-life build of Q units in period j for demand from periods j through k is Pðj;kÞ ðQÞ ¼
k1 X
ai1 ½pi li ðpi apiþ1 Þnðj;iÞ ðQÞ
i¼j
þ ak1 pk ½lk nðj;kÞ ðQÞ aj1 cQ " !# k i X X i1 a Q ll þ nðj;iÞ ðQÞ ; h i¼j
l¼j
ð1Þ
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where nðj;kÞ ðQÞ ¼ the expected lost sales in periods j to k ! ! k X Q li ¼ rðj;kÞ L rðj;kÞ ; i¼j
rðj;kÞ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k k1 X k X uX ¼t r2i þ 2 qim ri rm i¼j
i¼j m¼iþ1
and
pffiffiffiffiffiffi Z LðzÞ ¼ 1= 2p
1
2 =2
ðu zÞeu
du
z
is the standardized normal loss function. The optimal stocking level is determined by setting the first derivative of (1) to zero. If p1 P p2 P P pN , as is reasonable for many products at end-of-life, then Cattani shows that the problem is concave and any Q such that ðdPð1;N Þ ðQ Þ=dQ ¼ 0 is necessarily optimal, where dPð1;N Þ ðQ Þ dQ N X ¼ ai1 ðpi piþ1 Þð1 Fð1;iÞ ðQÞÞ hFð1;iÞ ðQÞ c i¼1
¼ 0: The non-increasing prices are sufficient to ensure a concave profit function. Other price patterns as well are possible that cause the second derivative to be negative, ensuring concavity. Straightforward extensions to this model account for discounted salvage values or disposal costs and for penalty costs (such as lost goodwill). Setup costs for the builds may also be considered directly. The theorem can be used directly to determine optimal end-of-life build quantities and profits, as shown in the following example. Example 1. Suppose that N ¼ 5, demands are independent across time, there is a constant coefficient of variation cv ¼ 0:4, l1 ¼ 50; l2 ¼ 26; l3 ¼ 14; l4 ¼ 7; l5 ¼ 3; p1 ¼ $100; p2 ¼ $85; p3 ¼ $70; p4 ¼ $55; p5 ¼ $40; c ¼ $50; a ¼ 1; and h ¼ 0: Then the optimal (denoted with a ‘‘*’’) end-
of-life build is found by numerically solving (1) to find Q ¼ 88 units and optimal expected profit of $3194. 3.2. Determining delay benefits under independent demand When demand is independent across time, the benefit of delaying an end-of-life build can be determined by comparing the expected profit calculated above with the expected profit for each possible value of the delay. We use (1) to determine Pð1;N Þ ðQ 1 Þ and Pðdþ1;N Þ ðQ dþ1 Þ, the expected profits given no delay and a delay of d, respectively. To compare these results we must add to Pðdþ1;N Þ ðQ dþ1 Þ the expected profits achieved before the delay Pð1;dÞ ðÞ, which are determined directly (and not by (1)). Example 2. Using the cost and demand parameters from Example 1, assuming no underage or overage costs before the end-of-life build, c ¼ $50, the benefits of delay are shown in Fig. 1 for oneperiod discount factors of 1, 0.9, and 0.8, and h ¼ 0 (left frame) and h ¼ 5 (right frame). With h ¼ 0; delay benefits under no discount (a ¼ 1) are increasing and concave in d varying from 10% increase in profits with a one period delay to 17% for a four period delay. With a discount of a ¼ 0:8, profits increased 66% with a one period delay and 99% with a four period delay. Delay benefits are slightly higher with h ¼ 5: In Section 4 we determine general properties of a delay by using more restrictive assumptions. 3.3. Determining delay benefits if demand is not independent If demand is not independent across time, the benefit of delaying an end-of-life build still is determined by comparing the expected profit of an end-of-life build under no delay with the expected profit for each possible value of the delay. We use (1) to determine Pð1;N Þ ðQ Þ. In calculating optimal end-of-life builds at time t ¼ d þ 1, however, we must use the distribution of demand for periods d þ 1 through n conditioned on actual demands realized in periods 1 through d. We denote this
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221
Fig. 1. Delay benefits in a discrete time example under various discount factors.
Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ and it is determined using (1) with conditional demand distributions: Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ " N X N 1 d ¼ pN a E½Xi j x1 ; . . . ; xd ca i¼dþ1
#
dictable after realization of initial demand. At time t ¼ 1, a calculation of total expected profits over the N periods with a delay of d must consider all possible values for demand in the first d periods, taking the expectation over all possible realizations of demand before d: Pð1;N Þ ðQ j x1 ; . . . ; xd Þ
nðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ ¼
" þ cad Q
N X
#
þ nðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ N 1 X
a
þ
Z
1
0
Z
1
Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ 0
fXð1;dÞ ðx1 ; . . . ; xd Þ dx1 dxd :
½ai1 pi aN 1 pN E½Xi j x1 ; . . . ; xd
i¼dþ1 i1
ai1 ðpi cÞli
i¼1
E½Xi j x1 ; . . . ; xd
i¼dþ1
þ
d X
½pi apiþ1 nðdþ1;iÞ ðQ j x1 ; . . . ; xd Þ :
We add to the profit Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ the expected profits achieved before the delay (from time t ¼ 1 until time t ¼ d), which should include expected margin as well as expected underage and overage costs, if any. The calculation for Pðdþ1;N Þ ðQ j x1 ; . . . ; xd Þ has as its lower bound the delay benefits calculated assuming independence; a stronger auto-correlation creates greater benefit from delay since remaining demand is more pre-
4. Properties of delay benefits In this section, we develop further insights into the structure of delay benefits of an end-of-life buy by considering relatively tighter assumptions that allow us to derive a closed-form solution for the delay benefits in a continuous-time model. Let X denote the random variable representing the remaining lifetime demand for the component as of time t ¼ 0 (X is demand in the interval ½0; 1Þ). We now let X1 denote demand for the component in the interval ½0; d; and X2 the
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demand in the interval ½d; 1Þ. That is, X ¼ X1 þ X2 : We make several simplifying assumptions. We assume that X1 and X2 are independent, that the price is constant over time, and that fixed ordering costs are negligible. We consider that the component is used in products that are made-to-order (i.e., the components are not purchased and assembled absent a customer order for the product) with full backlogging (at no penalty). Thus, under these assumptions, before the end-of-life buy the manufacturer realizes negligible inventory shortage or excess costs for the component. Finally, we consider relatively short time periods and ignore discounting and holding costs. These assumptions of independence and constant prices are tighter than those of Section 3 but allow us further insights into the structure of the problem. The assumption of a made-to-order and full backlogging environment focuses the modelÕs results on the uncertainty reduction effects of delaying an end-of-life buy, since the effects of variability before the buy are controllable. The rationale for the assumption is that the unit underage and overage costs after the buy are significantly greater than before the buy because there is no longer chance for recourse. After the final buy, backlogging is not relevant (there is no future supply), but the manufacturer incurs inventory underage or overage costs. The assumptions of no discounting or holding costs are strong, but have the effect of making the modelÕs results conservative. Delaying the final purchase will lower inventory levels and postpone a cash outflow – benefits not captured in this model. 4.1. Basic costs of initial problem The basic costs of an end-of-life buy at time t ¼ 0 are calculated using the newsvendor model. Following the newsvendor formulation, let cu be the cost of being under by one unit and let co be the cost of being over by one unit, including disposal costs and/or salvage values. Also, let m represent the unit margin on the product, e.g. p c (which is equivalent to cu if there are no underage penalties in addition to lost margin). The optimal order quantity is
y ¼ F 1 ðcu =ðcu þ co ÞÞ;
ð2Þ
where F ðxÞ is the c.d.f. of X. Total expected profits are ^ ðy; X Þ ¼ mE½X co E½ðy X Þþ P cu E½ðX yÞþ ¼ mE½X co ðy E½X Þ ðcu þ co ÞnX ðyÞ; ð3Þ R1 where nX ðyÞ ¼ x¼y ðx yÞ dF ðxÞ is the loss function, the mean number of customers lost if the order quantity is y. With an end-of-life buy at time t ¼ d; the problem is the same format as with t ¼ 0, but the remaining demand (for the end-of-life buy) is now X2 : Because this component is used in products that are made to order, at t ¼ d the component inventory is zero. Then, total expected profits in the interval ½d; 1Þ are a direct application of (3), ^ ðy2 ; X2 Þ, where y2 is the optimal order quantity at P t ¼ d. Since we assume underage and overage costs in the period ½0; d to be negligible, total expected ^ ðy2 ; X2 Þ. profits for ½0; 1Þ are mE½X1 þ P 4.2. Evaluation of delay benefits Let BðdÞ denote the benefits from a delay of d. Then ^ ðy2 ; X2 Þ P ^ ðy; X Þ BðdÞ ¼ mE½X1 þ P ¼ co ðy E½X Þ þ ðcu þ co ÞnX ðyÞ ðco ðy2 E½X2 Þ þ ðcu þ co ÞnX2 ðy2 ÞÞ ¼ co ððy y2 Þ ðE½X E½X2 ÞÞ þ ðcu þ co ÞðnX ðyÞ nX2 ðy2 ÞÞ:
ð4Þ
The following proposition formalizes an intuitive property of d, that the delay benefits are nondecreasing in d. For the proof, we need the following concept. Given two random variables X and Y, we say that X is more variable than Y, denoted by X P v Y , if E½hðX Þ P E½hðY Þ for all increasing, convex functions h. From this, it follows that if X P v Y and E½X ¼ E½Y ; then VarðX Þ P VarðY Þ.
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Property 1. B is non-decreasing in d. Proof. We can rewrite (4) as BðdÞ ¼ TCð0Þ TCðdÞ, where TC are the total costs of the newsvendor problem. It suffices to show that TCðdÞ are non-increasing in d. We denote X2 ¼ X2 ðdÞ to make the dependence on d clearer. Consider two delays a and b, such that a > b, and denote by X½b;a the demand in the interval ½b; a. Then X2 ðbÞ ¼ X2 ðaÞ þ X½b;a :
ð5Þ
Ross (1996, p. 455), shows that if there exists a random variable Z such that E½Z P 0, and such that X has the same distribution as Y þ Z; then X P v Y . Thus, (5) and the fact that E½X½b;a P 0 implies that X2 ðbÞ P v X2 ðaÞ. Song (1994, Proposition 4.10), shows that in the newsvendor model, a sufficient condition for total costs to be higher is that the demand distribution be more variable. Then, X2 ðbÞ P v X2 ðaÞ implies TCðbÞ P TCðaÞ, which completes the proof. Property 1 states that a longer delay is always as good or better than a shorter delay. 4.3. Delay benefits under normal demand
Pðl; rÞ ¼ ml rðco z þ ðcu þ co ÞLðzÞÞ:
ð6Þ
From (2) and using z ¼ ðy lÞ=r; we rewrite z in terms of cu and co and the c.d.f. of the standard normal distribution (U): z ¼ U1 ðcu =ðcu þ co ÞÞ:
in delaying the final buy d time units. We discuss this issue in depth in Section 5. A key issue determining the benefits of delay is the structure of demand over time. If demand uncertainty is expected to be resolved early on, we would expect a significant benefit for a delay of the end-of-life buy until after the uncertainty is resolved. Under a relatively mild condition, we next show that the delay benefits are concave in d. This property is of managerial relevance: most benefits of a delay are obtained early. We model the mean demand rate over time, which we formally define as follows. We assume that demand for the component in the interval ½0; t; denoted by X ðtÞ; is distributed normally with mean lðtÞ and variance r2 ðtÞ. Then the mean demand rate, denoted by rðtÞ is defined as rðtÞ ¼ dlðtÞ=dt. Further, let us assume we can describe rðtÞ as a function of lðtÞ, that is, rðtÞ ¼ gðlðtÞÞ. For example, if gðuÞ ¼ u the coefficient of variation is constant, and if gðuÞ ¼ u1=2 the variance-tomean ratio is constant. (Constant coefficients of variation arise under exponential distributions while constant variance-to-mean ratios arise under Poisson distributions.) Then BðdÞ ¼ ð1 gðl2 ðdÞÞ=gðlÞÞrAðzÞ:
When demands are distributed normally (as in Fortuin, 1980), we are able to derive a closed-form solution for the delay benefits. If demand is from a normal distribution X Nðl; r2 Þ, and X2 Nðl2 ; r22 Þ, then nðyÞ ¼ rLðzÞ, where z ¼ ðy lÞ=r; and LðzÞ is the standard loss function defined in Theorem 1. Then (3) can be rewritten as
ð7Þ
Denote, for convenience AðzÞ ¼ ðco z þ ðcu þ co Þ LðzÞÞ, the total underage and overage costs in the newsvendor model with a demand variance of one. Then, the delay benefits (4) are B ¼ ðr r2 ÞAðzÞ. From a supply-chain perspective, delaying the end-of-life buy is attractive only if the delay benefits B are greater than the cost the supplier incurs
223
ð8Þ
Property 2. B is concave in d if rðÞ is non-increasing and g is increasing and concave. Proof. First, note that oB ol ¼ g0 ðl2 ðdÞÞ 2 : od od
Rd We rewrite l2 ¼ l u¼0 rðuÞ du: Then, ol2 =od ¼ rðdÞ. Thus, oB=od ¼ g0 ðl2 ðdÞÞrðdÞ. Taking the second derivative, o2 B=od 2 ¼ g00 ðl2 ðdÞÞrðdÞ þ r0 ðdÞg0 ðl2 ðdÞÞ 6 0, if gðÞ is increasing (a mild assumption) and concave, and r is non-increasing. B is concave, for example, if the varianceto-mean-ratio or the coefficient of variation is constant. From Property 2 we conclude that most benefits of delaying an end-of-life buy are obtained early: even a short delay may help. Since the product is at the end of its life cycle, it typically has a decaying demand rate according to the traditional product life-cycle curve, and rðÞ non-increasing is
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generally a reasonable assumption. The assumption that g is increasing is also reasonable: it only requires that as the mean increases, the variance also increases. The assumption that g is concave requires that the variance increases at a constant or decreasing return to scale with the mean. If demand is an aggregation of independent increments, for example, this assumption is satisfied. 4.4. Examples of different demand profiles for remaining demand We examine two examples of demand during the product end of life: a constant mean demand subject to a sudden death (which is identical to an exponential decay of the mean demand rate), and an exponential decay of the mean demand rate subject to a rate change. In both examples we assume a constant coefficient of variation, that is, in (8), gðuÞ ¼ u. Example 3. Sudden death (or exponential decay of mean demand rate). Suppose that the mean demand rate is constant but subject to a sudden death. This scenario represents a possible change in the state of the world, such as a new product introduction. As in Song and Zipkin (1993, 1996), we assume that this change occurs at a random time T, where T expðqT Þ. That is, E½T ¼ 1=qT . The mean demand rate is lqT ; 0 6 t 6 T ; rht j T i ¼ 0; otherwise:
demand as of t ¼ 0 is R 1expected remaining R 1The rht j T iqT eqT T dt dT ¼ l. The expected T ¼0 t¼0 remaining demand (as of t ¼ 0) for t ¼ d is Z 1 Z 1 l2 ¼ rht j T i dt qT eqT T dT T ¼0
¼
Z
1
T ¼d
t¼d
Z
T
lqT dt qT eqT T dT ¼ leqT d :
t¼d
The delay benefits (8) are BðdÞ ¼ ð1 eqT d Þ rAðzÞ. An intuition of the result can be developed as follows. At t ¼ 0 we have the choice of making an end-of-life buy immediately, or deciding to delay the end-of-life buy until t ¼ d. If we delay the buy until t ¼ d, sudden death will have occurred (with probability 1 eqT d ) or not occurred (with probability eqT d ). If sudden death has occurred, we have zero underage and overage costs (we meet our demand exactly), and cost savings (compared to having made an end-of-life buy at time t ¼ 0Þ are rAðzÞ. If it has not occurred, we must make an end-of-life buy. Here, because of the memoryless property, time until the sudden death is still distributed as expðqT Þ, and our expected costs are just as if we had made the end-of-life buy at t ¼ 0. That is, in this case our cost savings are zero. Thus, our delay benefits are ð1 eqT d ÞrAðzÞ. Notice that in this intuitive derivation, we need no assumptions with respect to the variance or mean, only that there is independence of demand. Suppose now that mean demand rate decays exponentially with time (Fig. 2, middle frame), that is, rðtÞ ¼ lqR eqR t , where qR is the obsolescence rate, a parameter. This is equivalent to saying that the mean of the remaining demand is
Fig. 2. Delay benefits under constant demand rate or exponential decay rate.
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leqR t . Then l2 ¼ leqR d . The delay benefits (8) are BðdÞ ¼ ð1 eqR d ÞrAðzÞ. This scenario is identical to a sudden death, if qT ¼ qR . That is, the exponential decay of demand rate is mathematically equivalent to a constant demand rate subject to an exponentially distributed sudden death. Fig. 2 (right frame) shows a graph of BðdÞ for cu ¼ 1000, co ¼ 45, r ¼ 300, and two values of q (where q could be either qT or qR ). We note the following property of B. The sooner the sudden death is expected to occur, the more attractive a delay becomes. Alternatively, the higher the obsolescence rate, the more attractive a delay. Example 4. Exponential decay of demand rate subject to obsolescence rate change. We now consider a situation where the obsolescence rate is subject to change. Mean demand rate decays with obsolescence rate qR1 , as in Example 3. At random time T, mean demand rate changes suddenly to crðT Þ, and the obsolescence rate becomes qR2 , where qR2 6¼R1 . This scenario represents a possible change in the state of the world, such as in scenario 1. For example, a competitor announces a new product and demand drops, i.e., qR2 > qR1 . Alternatively, a competitive product is delayed, and demand increases, i.e., qR2 < qR1 . We assume, as in scenario 1, that T expðqT Þ. The expression for demand rate is
rht j T i ¼
lqR1 eqR1 t ; clqR1 e
225
t < T;
qR1 T qR2 ðtT Þ
e
;
t PT:
To compute r2 , we need to compute the expected remaining demand for t ¼ d Z 1 Z 1 l2 ¼ rht j T idt qT eqT T dT T ¼0
t¼d
cqT qR1 eqR2 d eðqR1 þqT Þd ðqR1 þqT Þd ¼l þe qR2 qR1 þ qT qR2 ¼ bðdÞl: The delay benefits (8) are BðdÞ ¼ ð1 bðdÞÞrAðzÞ. Fig. 3 shows this relationship for various values of qR2 and qT , using the parameter values from Example 3 (and qR1 ¼ qR ). The benefits are greater when there are more chances of resolving the uncertainty by a delay, that is, when there is a larger difference between qR1 and qR2 , a lower value of c, and a higher value of qT (the obsolescence rate change is expected to occur sooner). In Fig. 3, the benefits are greatest for qT ¼ 1; c ¼ 0 – a sudden death scenario. The least benefits occur for qR2 ¼ 1; qT ¼ 0:5; c ¼ 0:8. In this case, there is only a 20% sudden change in the mean demand rate with no obsolescence rate change (qR1 ¼ qR2 ), and it is expected to occur relatively far in the future (2 years ¼ 1/0.5).
Fig. 3. Delay benefits for exponential demand rate subject to obsolescence rate chance.
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5. The individual player’s perspective
Pm ðdÞ ¼ mm l r2 ðdÞAðzÞ:
The analysis to here has focused on the benefits of a delay of an end-of-life purchase from the perspective of the manufacturer, which we suggested should be aggregated with the delay costs incurred by the supplier to determine the delay effects from a supply chain perspective. (For a manufacturer considering an end-of-life build, no supply chain aggregation is necessary since both benefits and costs accrue to the manufacturer.) In this section, we present an analysis of both individual playerÕs perspectives. There are two players: the supplier and the manufacturer. We consider the assumptions of the continuous-time model (Section 4), and add subscripts m and s to indicate the manufacturer and supplier, respectively. Generally, the aggregate supply chain effects of a delay include a benefit to the manufacturer offset partly by a loss to the supplier. The supplier would prefer no delay for at least two reasons: the costs of maintaining a line for production of a declining product, and because an end-of-life buy forces the manufacturer to buy more units (as safety stock) than otherwise would have been required with no end-of-life buy. As a secondary benefit to the supplier, the revenue for all of the units corresponding to the end-of-life buy is received earlier than it would otherwise have arrived. From a supply chain perspective, an end-of-life buy causes increased underage costs and overage costs to the manufacturer, which is partially offset by the increased margin to the supplier. To gain insights to the dynamics between the supplier and manufacturer, we consider two games. In the first, the manufacturer does not provide any incentives to the supplier in delaying the final buy decision. We show that there is no equilibrium, since the two players have directly opposite interests. In the second, the manufacturer allocates a portion of the delay benefits to the supplier to compensate for the supplierÕs losses.
Consider that the supplier has a non-decreasing maintenance cost CðdÞ of delaying the end-of-life purchase by d. CðdÞ includes only the costs associated with maintaining the production line (overhead, floor space, etc.). The payoff to the supplier Ps ðdÞ is ms l1 þ ms y2 CðdÞ. Under the assumptions of normality, y2 ¼ l2 þ zr2 , and thus
5.1. A game with no incentives to the supplier Given a delay of d, the payoff (i.e., profit) to the manufacturer Pm ðdÞ is mm l1 þ Pm ðl2 ; r2 Þ, or:
Ps ðdÞ ¼ ms l þ ms r2 ðdÞz CðdÞ:
ð9Þ
ð10Þ
In an end-of-life situation where l2 ðdÞ is nonincreasing in d, then r2 ðdÞ is non-increasing in d, given the reasonable assumption that demand variance increases with the mean. Thus (9) is monotonically increasing in d, and (10) is monotonically decreasing in d. The two players have diametrically opposite interests, and thus there is no equilibrium. 5.2. A game with supplier incentive to delaying the end-of-life purchase Consider now the same game from Section 5.1, but now the manufacturer compensates the supplier for any profit losses incurred by delaying, that is, the manufacturer pays to the supplier ðPs ð0Þ Ps ðdÞÞ. In addition, as an incentive, the manufacturer also allocates to the supplier a portion a of the supply-chain benefits of delaying the end-of-life buy BSC ðdÞ, given by: BSC ðdÞ ¼ ðPm ðdÞ Pm ð0ÞÞ þ ðPs ðdÞ Ps ð0ÞÞ ¼ ðr r2 ðdÞÞðAðzÞ ms zÞ CðdÞ:
ð11Þ
Thus, the manufacturerÕs and supplierÕs profits after allocation are: Pam ðdÞ ¼ Pm ð0Þ þ ð1 aÞBSC ðdÞ;
ð12Þ
Pas ðdÞ ¼ Ps ð0Þ þ aBSC ðdÞ:
ð13Þ
Note that for any a > 0; there is an incentive to the supplier in delaying the end-of-life buy, since Pas ðdÞ > Pas ð0Þ, for d > 0: Also, this allocation assures that the maximum of (12) and (13) occur at the same d ; which is the d that maximizes (11). (If BSC ðdÞ 6 0 for all d, then d ¼ 0.)
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227
Fig. 4. Delay benefits – playersÕ perspective.
Fig. 4 illustrates the two games for the parameter values from Example 3 (and with ms ¼ 20; a ¼ 0:1; and CðdÞ ¼ 4000dÞ. The top and bottom curves represent the delay benefits (PðdÞ Pð0Þ) for the manufacturer and supplier, respectively, under no incentives. In this example, the net supply chain benefits of a delay (second curve from top) are small compared to the net gain by the manufacturer (top curve) and the net loss by the supplier (bottom curve). In the absence of significant incentives from the manufacturer to the supplier, the supplier will want end-of-life buys as early as possible in the product life cycle. Under scenarios such as this, manufacturers might want to consider creating incentives that manage endof-life buy considerations for key suppliers of critical components other than through incentives at the end of the product life cycle. For example, the issue might be raised and addressed during initial sourcing negotiations. Otherwise, suppliers
have significant incentives to require end-of-life buys, and as early as possible.
6. Conclusion We have studied end-of-life purchases, and modeled the effects of delaying an end-of-life buy or build. Under various scenarios of remaining demand, we observe that benefits of a delay to the manufacturer of an end-of-life buy are nondecreasing and concave in the delay time. A longer delay is always as good as or better than a shorter delay, and most benefits of a delay are obtained early. In general, the sooner the uncertainty is expected to be resolved, the more attractive to the manufacturer a delay becomes. If there are two players involved, as is the case when a critical component of the product is obtained from a supplier, we show that the parties
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have directly opposite interests with respect to the delay decision. Thus, it is not surprising to find anecdotal evidence of an increasing number of suppliers requiring end-of-life buys from their customers. For the delay decision to be implemented, it is necessary that the manufacturer compensate the supplier for their losses incurred. From a supply chain perspective, delaying the endof-life buy is attractive if the delay benefits are greater than the cost the supplier incurs under the delay. The end-of-life purchase is not always a good buy.
Acknowledgements This research was partly funded by the Burress Faculty Development and Support Fund. The authors would like to thank Kyle Christensen of Hewlett-Packard for information on end-of-life inventory.
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