A Series System with Returns: Stationary Analysis

Transcription

1 OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp issn X eissn informs doi /opre INFORMS A Series System with Returns: Stationary Analysis Gregory DeCroix, Jing-Sheng Song, Paul Zipkin The Fuqua School of Business, Duke University, Durham, North Carolina {decroix@duke.edu, jssong@duke.edu, zipkin@duke.edu} This paper analyzes a series inventory system with stationary costs and stochastic demand over an infinite horizon. A distinctive feature is that demand can e negative, representing returns from customers, as well as zero or positive. We oserve that, as in a system with nonnegative demand, a stationary echelon ase-stock policy is optimal here. However, the steadystate ehavior of the system under such a policy is different from that in systems with nonnegative demands. We present an exact procedure and several approximations for computing the operating characteristics and system costs for any stationary echelon ase-stock policy, and also descrie an algorithm for computing a good policy. As an alternative to the echelon ase-stock policy, we discuss a policy that uses only local information. Finally, we descrie how to extend the analysis to the case where returns occur at multiple stages instead of just at the stage closest to demand, and the case where returns require a recovery lead time. Suject classifications: inventory/production: multiechelon; environment: product recovery and remanufacturing. Area of review: Manufacturing, Service, and Supply Chain Operations. History: Received May 2002; revision received July 2003; accepted January Introduction This paper analyzes a series inventory system with stochastic demand. The standard model, first studied y Clark and Scarf (1960), assumes that demand is nonnegative; here we relax this assumption. The purpose is to model systems that receive returns (i.e., negative demands) as well as the usual positive demands from customers. Customers may return items for a variety of reasons. Some returns occur shortly after the time of purchase, when a customer changes his mind aout wanting the item. This is common for goods that may not e easy to assess fully at the time of purchase, such as clothing or gifts. In mail-order and e-usiness channels, returns are especially prolematic; if a customer is unale to touch an item, he is even less ale to assess it at the time of purchase, and is thus more likely to return it. (See, for example, Tedeschi 2001.) More recently some firms have started taking ack items after customers use them. In several countries, particularly in Europe, such take-acks are legally required for products such as automoiles, electronic goods, and packaging, due to environmental concerns. (See, for example, Frankel 1996, Diem 1999, Schenkman 2002, Thorn and Rogerson 2002.) In other cases, companies voluntarily collect used products to recover residual value, and may even design the products to maximize this value. Examples include singleuse cameras (Kodak, Fuji), toner cartridges (Xerox, Canon, Hewlett-Packard), personal computers (IBM), and communication network equipment (Lucent). Regardless of why they occur, product returns complicate the management of an inventory system y introducing an uncertain reverse flow of materials. Our goal is to develop methods for choosing and evaluating inventory management policies for systems facing this additional complexity, and to provide insights into how the ehavior of such systems differs from traditional nonnegativedemand systems. Specifically, we study a series system with possily negative demand and stationary data over an infinite horizon. The system has linear costs and employs a stationary echelon ase-stock policy. We develop an exact procedure for computing the operating characteristics and system costs for any such policy, and also propose approximations that perform well in numerical trials. We present an algorithm, ased on one of the approximations, for computing a good echelon ase-stock policy. This policy is optimal for that approximation, and appears to e nearly optimal for the exact system ased on a numerical study. As an alternative to the echelon ase-stock policy, we discuss a ase-stock policy that uses only local information. In the nonnegative demand case, this local policy is equivalent to the echelon policy; here this no longer is true, and in general the local policy is suoptimal. Finally, we show how to extend the analysis to the case where returns occur at multiple stages instead of just the stage closest to demand, and the case where returns require a recovery lead time. A key assumption is that demands in different periods are independent. In particular, current returns are independent of past demands. This is an approximation in some situations. Returns, y definition, must have gotten to the customers y some means. One might therefore specify returns as a function of the cumulative sales to date. This approach, however, adds a layer of complexity to the model. Indeed, a similar issue arises for demand itself. No product has 350

2 Operations Research 53(2), pp , 2005 INFORMS 351 an unlimited market. So, one could argue, current demand should always depend on cumulative demand. In practice, we rarely do that. The effect is small in most cases, so the added modeling complexity yields little enefit. For similar reasons, the independence assumption is common in the literature on systems with returns, although the rationale for dependence is stronger here. (See Fleischmann 2000 for a more detailed justification of this assumption in that context, and Zipkin 2000, 6.3, for a discussion of these issues for nonnegative-demand systems. Also, see Kiesmüller and van der Laan 2001 for a single-stage model where returns depend on past demands.) We allow current demand and returns to e correlated, ut otherwise assume independence over time. Hence, our model applies when the product has een in the market for some time, and the lag etween sale and return is either negligile or large. Results for the standard nonnegative-demand model are well known. Clark and Scarf (1960) showed that, for a finite horizon, an echelon ase-stock policy is optimal, and Federgruen and Zipkin (1984) extended the result to an infinite horizon with stationary data and descried how to compute optimal echelon ase-stock levels. Chen and Zheng (1994) streamlined the proof and the notation. See Zipkin (2000), 8.3 and 9.8 for a summary of these results. Results for systems allowing negative demands are more limited. For a single-stage system with a finite horizon, Heyman and Soel (1984) pointed out that Scarf s (1960) proof of the optimality of an (s S) policy still works for possily negative demand. Fleischmann (2000) and Fleischmann et al. (2002) extended this result to the infinite-horizon case. (For a linear order cost, of course, a ase-stock policy is optimal.) Cohen et al. (1980) estalished conditions under which a ase-stock policy is optimal for a system where a fixed fraction of demands in any period is returned after a fixed numer of periods. Kelle and Silver (1989) developed a heuristic approach for managing a similar system that also includes fixed order costs and stochastic return times. There has also een research on systems where returns are not returned directly to stock as assumed here, ut instead are kept in a separate uffer until they are processed or disposed of. Simpson (1978) and Inderfurth (1997) showed that a three-parameter generalization of the ase-stock policy is optimal for the single-stage, finite-horizon case with linear costs. DeCroix (2005) extended this result to a series system, again with a separate return uffer, assuming disposal is not allowed at downstream stages. Mahadevan et al. (2003) developed heuristic methods for determining policy parameters for similar systems. For reviews of other research on reverse logistics, see Fleischmann et al. (1997) and Fleischmann (2000). Apart from the work cited aove, the form of the optimal policy for a series system facing possily negative demand has not een studied. However, careful examination of the proof in Clark and Scarf (1960) reveals that their result does not require the assumption that demand is nonnegative. This result extends to an infinite horizon with stationary data, following the approaches of Parker and Kapuscinski (2004) and Fleischmann (2000). Thus, an echelon asestock policy is optimal for the model considered here. While the optimal policy has the same form as in the nonnegative-demand case, policy evaluation and optimization are more difficult here, ecause the optimal policy is not myopic. For instance, the simple argument of Veinott (1965) for the single-stage system reaks down, ecause demand can raise the inventory position aove the target ase-stock level, and thus the myopic ase-stock level is not always reachale. Similarly, the lower-ound approach of Chen and Zheng (1994) for the multistage system no longer works. Thus, new techniques are necessary for evaluating and choosing policies. The rest of the paper is organized as follows. Section 2 introduces the asics of the model for a single-stage system. Section 3 extends the model to a multistage system and presents a procedure for computing system performance for a given echelon ase-stock policy. This procedure requires solving a system of equations; a method for doing so is presented in 4. Section 5 discusses the local ase-stock policy. Section 6 presents three approximations, and 7 descries an approximate optimization algorithm ased on one of these. Section 8 extends the results to returns at multiple stages, and 9 extends them to the case of a lead time for product recovery. Section 10 provides some concluding remarks. 2. Single Stage To estalish concepts and notation, let us review the case of a single-stage system. Time is discrete, indexed y t = 0 1 and stockouts are acklogged. Denote L = order lead time, a positive integer, D t = demand in period t, etween times t and t + 1, D t u = demand in periods t through u 1, IN t = net inventory (inventory minus acklog) at time t, IP t = inventory position after shipment at time t = IN t plus stock in transit at time t. IP t = inventory position efore shipment at time t. In each period (i) the eginning state IP t is reviewed and an order is placed; (ii) orders arrive and demands occur; and (iii) costs are assessed on the ending state. Following Zipkin (2000, 9.6), the shortest possile lead time is L = 1, when a shipment arrives y the end of the period in which it is placed. Demands in different periods are independent, and demand can e negative as well as zero or positive. Still, the standard flow-conservation equation holds: IN t + L = IP t D t t + L Now consider a system with stationary data, operating over an infinite horizon under a stationary ase-stock policy

3 352 Operations Research 53(2), pp , 2005 INFORMS with ase-stock level s. Let the following denote equilirium quantities: IP = inventory position, IN = net inventory, D = 1-period demand, LD = D 0 L = lead time demand. Assumption. E D > 0 and E D 2 is finite. We need the first condition to prevent returns from flooding the system. Later, we point out that we need the second also to ensure that the equilirium quantities aove exist. A ase-stock policy in this setting can e descried in the usual way. Before shipment, if IP t <s, order the difference s IP t, to ring IP t up to s; otherwise, order nothing. When demand can e negative, the ehavior of the state variales is more complicated than usual. With nonnegative demand, once IP t falls elow s, it will stay at or elow s in the future, so that IP t = s from that point on. In that case, a ase-stock policy is a demandreplacement policy. Here, demands can e negative, ut orders are still always positive. So the ase-stock policy is clearly not a demand-replacement policy. Also, negative demands may cause IP t (and thus IP t ) to exceed s, so some additional work is required to descrie the ehavior of IP t. Define Z t = IP t s, i.e., the amount the inventory position exceeds s due to negative net demands. Then, Z t + 1 = Z t D t +, and the equilirium random variale Z satisfies Z = Z D + This is the Lindley equation that descries the waiting time in a GI/G/1 queue. For example, suppose that D = D P D N, where D P and D N are independent, nonnegative random variales. Then, D N corresponds to the service time and D P to the interarrival time. The assumption aove is needed to ensure that Z exists and its mean is finite. (See Wolff 1989.) Later we discuss how to compute its distriution. The equilirium random variales IP and IN can e expressed as IP = s + Z IN = IP LD = s + Z LD In each period, inventory-holding and acklog-penalty costs are assessed on IN t + and IN t, respectively. Using the equilirium quantities we can then compute the average cost per period in the usual way. Letting h = holding cost rate, = acklog penalty cost rate, and C x = h x + + x = hx + + h x, the average cost per period of any ase-stock policy is C s = E C IN = E C s + Z LD This is a convex function, and its minimum yields the optimal s, denoted s. It is interesting to compare this system to one with nonnegative demands. Let D = D P D N, where D P denotes actual demands and D N returns. Then, LD = LD P LD N. Let C P s e the average-cost function for the system with demand D P, and s P the optimal ase-stock level. Then, C P s = E C s LD P Because oth LD N and Z are nonnegative, LD Z = LD P LD N + Z st LD P, where st stands for stochastically smaller. (That is, P LD Z x P LD P x for all x.) Therefore, s s P. (See Song 1994.) Thus, not surprisingly, returns lower the optimal ase-stock level. More generally, y a similar argument, a system with more returns (stochastically larger D N ) has a lower ase-stock level than one with fewer returns. 3. Series System Now consider a series system, and let j = stage index, j = 1 J, L j = lead time for shipments to stage j, IN j t = echelon net inventory (stock at stage j plus all stock downstream minus ackorders at stage 1), IP j t = echelon inventory position after shipment (IN j t plus all shipments currently in transit to stage j), IP j t = echelon inventory position efore shipment = IP j t 1 D t 1. Stage 1 is closest to demand, and stage J to the external supplier. Note that IP j t is equal to IN j t plus actual shipments in transit from stage j + 1 i.e., there are no ackorders etween stages. We assume that the data are stationary and demand is discrete. Just as in a series system with nonnegative demand, IN j t + L j = IP j t D t t + L j There are linear-ordering and inventory-holding costs at all stages and a linear ackorder cost. Using the approaches in Clark and Scarf (1960), Parker and Kapuscinski (2004), and Fleischmann (2000), we can show the following: Proposition 1. An echelon ase-stock policy is optimal for the finite-horizon system with the ojective of minimizing total expected discounted cost. A stationary echelon ase-stock policy is optimal for the infinite-horizon system, under either the total expected discounted-cost criterion or the long-run average-cost criterion. Accordingly, our next goal is to evaluate a stationary echelon ase-stock policy with parameters s j.forj = J, the policy sets IP J t = max s J IP J t, as in the singlestage system. For j<j, IP j t is constrained y echelon net inventory at stage j +1, so the policy compares IP j t to s j and IN j+1 t. Because IP j t IN j+1 t, there are three cases to consider: IP j t IN j+1 t < s j set IP j t = IN j+1 t IP j t <s j IN j+1 t set IP j t = s j s j IP j t IN j+1 t set IP j t = IP j t

4 Operations Research 53(2), pp , 2005 INFORMS 353 This rule can e expressed as IP j t = max min s j IN j+1 t IP j t or equivalently, IP j t = min max s j IP j t IN j+1 t (1) To facilitate evaluation of the key quantities, we rewrite the equations at shifted times. Set L j = i>j L i Then, IP J t = max s J IP J t IN j t + L j 1 = IP j t + L j D t + L j t+ L j 1 IP j t + L j = max min s j IN j+1 t + L j IP j t + L j j<j Now, consider the system in equilirium. Denote demand during stage j s lead time y LD j = D L j L j 1 These random variales are independent. Also, let IN j and IP j denote the equilirium versions of IN j t + L j 1 and IP j t + L j. In the case of nonnegative demands, IP J = s J IN j = IP j LD j (2) IP j = min s j IN j+1 j < J This recursion can e used to compute the distriutions of all the random variales, and from those the average cost. Things are not quite so simple for the current case, where demand can e negative. Analogous to the single-stage system, define Z t = IP J t s J. Then, for j = J,asinthe single-stage system, IP J = s J + Z where Z is independent of the LD j. Also, the relations IN j = IP j LD j remain valid. What remains is to evaluate the distriution of IP j given that of IN j+1. Let p y = Pr IP j = y and p x = Pr IN j+1 = x. We distinguish three cases: Case 1. y<s j. In the case IN j+1 <s j,wehaveip j = IN j+1. This is the only case, moreover, where IP j <s j.so for y<s j, p y = p y. Case 2. y>s j. Let IP j denote the equilirium version of IP j t = IP j t 1 D t 1, D the equilirium oneperiod demand D t 1, and g d the proaility mass function of D. In the case IP j >s j,wehaveip j = IP j, and this is the only case where IP j >s j.sofory>s j, the p y satisfy the equations p y = Pr IP j = y = Pr IP j D = y = g x y p x x= = x<s j g x y p x + x s j g x y p x Given the distriution of IN j+1, we have all the terms in the first sum; it is a constant. The p x in the second sum remains unknown. Case 3. y = s j. We do not need a separate equation for p s j. Its value is determined y the normalization equation p y = 1, given the other p y. In summary, we can directly compute p y, y<s j, and we have a system of simultaneous linear equations for p y, y s j. The system is not triangular, and there seems to e no way to solve it directly. The next section shows how to solve it numerically. Note that the distriution of Z is the solution to the special case of this system with s j = 0 and p x = 0, x<0. Next, we argue that the system and the alance equations are stale. Denote Z j t = IP j t s j W j+1 t = IN j+1 t s j Then, (1) is equivalent to j J j <J Z J t = Z J t 1 D t 1 + and Z j t = min Z j t 1 D t 1 + W j+1 t j < J If we ignore the W j+1 t term in the minimum, we otain precisely the dynamics of Z t, i.e., a random walk truncated at 0. Thus, Z j t Z t. An inductive argument estalishes that Z j t has a stale equilirium for j = 1 J. We know that Z J t = Z t has a stale equilirium, and so W J t = IN J t s J 1 = s J +Z t L J D t L J t s J 1 does also. Now assume (inductively) that W j+1 t has a stale equilirium. Then so does Z j t, and y the previous argument so does W j t. The overall policy-evaluation algorithm, then, works as follows: Start at J, and work downstream. Set IP J = s J +Z. For each j, givenip j, set IN j = IP j LD j. Then, decrement j, and given IN j+1, determine the distriution of IP j y solving the linear system aove. This last step makes the procedure more intricate than in the nonnegative-demand case.

5 354 Operations Research 53(2), pp , 2005 INFORMS Given the distriutions of IN j, the average cost of a policy can e computed in the usual way. Let h j e the unit echelon inventory holding cost rate for stage j, and h 1 the sum of those quantities, the unit local holding cost rate at stage 1. Again, is the unit ackorder cost rate. Note that the system ackorders is B = IN 1. The average cost, then, is J h j E IN j + + h 1 E B j=1 This quantity charges shipments in transit from j to j 1 the same holding cost as inventory at j. This approach for solving a series system with returns is more involved than that presented in Fleischmann et al. (2002) and Fleischmann (2000) for a single-stage system with returns. They show that the single-stage model can e transformed to have the same form as a model with nonnegative demands. Such a transformation does not appear to work in a series system, however. The difficulty lies in the dependence etween the inventory position at a given stage and the net inventory at the previous stage, comined with the way that returns affect oth quantities. 4. Solving the Balance Equations Next, we descrie an approach for solving the alance equations to otain p y = Pr IP j = y, y s j, given IN j+1. Consider = p s j a parameter for now. Define the column vectors p = p s j + k k>0, v = l>0 g l k p s j l k>0, and g = g k k>0, and the matrix g 0 g 1 g 2 g 3 g 1 g 0 g 1 g 2 = g l k kl = g 2 g 1 g 0 g 1 For any fixed value of, the alance equations for p have the form I p = v + g (3) The solution to (3) can e written p = u + f, where I u = v and I f = g. Now, sustitute this into the normalization equation 1 = e p + + e u + f = e p + e u e f or = 1 e p e u (4) 1 + e f (Here, e is a row-vector of ones, and p = p s j k k>0.) Once is determined, we can recover p. Oserve that g and f depend only on the demand distriution, not on j or s j. So, we can solve the linear system I f = g once, efore anything else, and use e f from then on. However, p, v, and u do depend on j and s j. So, the system I u = v must e solved anew for each policy and every j. Think of u as an infinite vector. Several methods are availale to solve these systems. One is the recursion u 0 = v u n+1 = v + u n n 0 This is just the familiar successive-approximation algorithm. The u n are increasing, and so the corresponding n (otained from (4) using u n in place of u) are decreasing. Consequently, the convergence of the n can e used as a stopping criterion. (Of course, to keep each iteration finite, one must truncate the calculations somehow. But this truncation can e controlled y similar stopping criteria, as usual, in dealing with infinite systems.) A faster method is Jacoi s: u 0 = u n+1 = 1 1 g 0 v 1 1 g 0 v + off u n n 0 Here, off = g 0 I, the matrix of off-diagonal elements of. 5. A Local Policy Consider now a local ase-stock policy with parameters s j 0: Stage 1 sends orders to stage 2, stage 2 orders from stage 3, etc. Every stage j monitors its own local inventory order-position IOP j t. (This includes stock on order from stage j + 1, not just stock already shipped from there.) Its order at time t is s j IOP j t +. Each stage j>1 treats the orders from stage j 1 as its demands. In the nonnegative-demand case, this local ase-stock policy and the echelon ase-stock policy aove with s j = i j s i are equivalent. Here they differ in two respects, oth involving loss of information. First, under the local policy stages j>1 do not see the actual demand process, ut rather a filtered version of it. At stage 1, when IOP 1 t 1 = s 1 and D t 1 0, the order at time t is just D t 1. Otherwise, the order is not D t 1. However, assuming that IOP j 0 = s j, all stages j>1 see the same demand process, namely, stage 1 s order process. Second, the local policy responds differently to upstream shortages. Suppose that Z t 1 = 0 and IN j+1 t 1 s j for all j<j, so that IP j t 1 = s j for all j. Then, suppose that a positive demand D t 1 >0 occurs such that IN j+1 t <s j for some j<j. Under the local policy, each stage (including stage j) orders D t 1 at time t to replace demand, ut only IN j+1 t IP j t <D t 1 is actually shipped to stage j. Under the echelon policy, this same quantity is shipped; the difference is that no order is

6 Operations Research 53(2), pp , 2005 INFORMS 355 placed for the remaining units. In effect, the echelon policy delays orders for additional units until the upstream stage has the units availale, whereas the local policy orders immediately. If returns (negative demand) arrive in the next period, these units can partly offset the shortfall at stage j under the echelon policy. Under the local policy, however, stage j has already committed to an order that cannot e retracted, so returns will cause that stage to have excess units. In essence, stage j does not take advantage of additional demand (return) information that could e otained costlessly y waiting another period to place the order. We know that the echelon policy is optimal. Because the local policy can result in different material flows (due to the informational differences descried aove), it generally is not. Still, there may e situations where a local policy is easier to implement, ecause it requires only local information. For that reason we riefly descrie an approach for analyzing this policy. Now, define the echelon quantities s j = i j s i and IOP j t = IP j t +B j+1 t. (Here, B j+1 t is the local ackorders at j + 1, that is, stock on order ut not yet shipped to j.) It is clear from the mechanics of the policy that IOP j t = s j + Z t, where we define Z t = IP J t s J. This would e IP j t too, if IP j t were not constrained y the stock availale at stage j + 1. That constraint, however, requires IP j t IN j+1 t. Thus, IP j t = min s j + Z t IN j+1 t This is the analogue of (1) for the local policy. We have IP J t = max s J IP J t = s J + Z t Suppose that we have the joint distriution of IP j t + L j and Z t + L j Define X j t + L j = IP j t + L j, and for u t + L j, X j u + 1 = X j u D u Z u + 1 = Z u + D u + In this way we can recursively compute the joint distriution of X j u and Z u. Then, for u = t + L j 1,wehave X j u = IN j u. Now, decrement j y 1. The calculation aove thus gives the joint distriution of IN j+1 t + L j and Z t + L j. We can use this to otain the distriution of IP j t + L j = min s j + Z t + L j IN j+1 t + L j From this we can egin the calculation for the next stage. The equilirium analogue of this algorithm just sets Z t at its equilirium distriution and proceeds as aove. This technique is more complex than that of 3. It involves ivariate distriutions, and it requires a period-y-period evaluation within each stage. 6. Approximations Sections 3 and 4 descrie a method for evaluating a stationary echelon ase-stock policy. The method is exact, up to the convergence tolerance. This section presents three approximations for (1), each of which leads to an approximate method for policy evaluation. There are several reasons to pursue approximations. First, the exact approach requires the numerical solution of a system of equations for each stage j; we seek approximations that involve simpler calculations. Second, ecause an exact approach for evaluating a local ase-stock policy is even more involved, approximations that work well for that policy are even more valuale. Finally, the exact approach does not appear to lend itself to efficient computation of an optimal policy. We seek approximations that are more amenale to optimization Approximation 1 In determining IP j for j<j, replace max s j IP j y s j + Z j, where the random variales Z j are all distriuted as Z. Assume that they are independent of each other and the LD j, so that the Z j and IN j+1 are also independent. This leads to the recursion IP J = s J + Z J IN j = IP j LD j (5) IP j = min s j + Z j IN j+1 j < J Note that the distriution of Z j can e computed once and then used for each stage j. As a result, this approximate policy-evaluation scheme is almost as simple as the nonnegative-demand case s (2). We call this Approximation 1; it applies to the local as well as the echelon policy. Indeed, the expression for IP j in (5) looks much like the local policy s. This approach would e exact for the local policy, except that the actual Z j and IN j+1 are not independent Approximation 2 The next approach is a refinement of Approximation 1. It replaces the formula for IP j in (5) y IP j = IN j+1 1 INj+1 s j + s j + Z j 1 INj+1 >s j = min s j IN j+1 + Z j 1 INj+1 >s j j <J (6) where 1 is the indicator function. Approximations 1 and 2 oth follow the exact formula for IN j+1 s j. But, for IN j+1 >s j,ifz j happens to e large, Approximation 1 still sets IP j = IN j+1. Approximation 2 uses IP j = s j + Z j instead, which seems closer to the spirit of the exact calculation.

7 356 Operations Research 53(2), pp , 2005 INFORMS Tale 1. Evaluation of approximations. = P = 4 (fixed) J = 2 Linear holding costs Echelon ase-stock levels s j Cost Exact App App App % Error App App App Approximation 3 The third approximation sets IP j = min s j IN j+1 + Z j j <J (7) It is somewhat simpler than the other two. It agrees with Approximation 2 for IN j+1 >s j, ut it does not follow the exact formula for IN j+1 s j. This approach provides the asis for the approximate optimization algorithm of Evaluation We next evaluate and compare these approximations, using a few examples. The examples all have D = D P D N, where D P and D N are independent, Poisson random variales with P = E D P, = E D N, and = P = E D > 0. All the L j = 1. Tale 1 shows the results for several systems with linear holding costs (every stage s echelon holding cost h j = 1/J ), J = 2 and = 4, ut with three different values of (and hence P ), and three values of the ackorder penalty cost. For each case the tale shows the optimal policy and its average cost. For = 0 (no returns) these were computed as in Zipkin (2000, Chapter 8). For positive the optimal policies were computed using exhaustive search, and the average costs were computed using the approach of 3. All the costs reported exclude holding costs for goods in transit. Tale 2 gives the same results for = 16. Tales 3 and 4 give the same results for an affine holding-cost pattern, i.e., h J = + 1 /J and h j = 1 /J j < J, for some 0 1. Here, we used = Tale 5 shows similar results for systems with J = 4, = 4, and linear holding costs. In these examples, for positive the tale evaluates a reasonale ut not necessarily optimal policy. We selected the policy y fixing the differences s j s 1 at their optimal values for = 0 and searching over s 1. Tale 6 gives the same results for = 16. The approximations all give accurate estimates of the exact system cost, with a maximum relative error of less than 3%. Tale 2. Evaluation of approximations. = P = 16 (fixed) J = 2 Linear holding costs Echelon ase-stock levels s j Cost Exact App App App % Error App App App

8 Operations Research 53(2), pp , 2005 INFORMS 357 Tale 3. Evaluation of approximations. = P = 4 (fixed) J = 2 Affine holding costs Echelon ase-stock levels s j Cost Exact App App App % Error App App App Tale 4. Evaluation of approximations. = P = 16 (fixed) J = 2 Affine holding costs Echelon ase-stock levels s j Cost Exact App App App % Error App App App Tale 5. Evaluation of approximations. = P = 4 (fixed) J = Echelon ase-stock levels s j Cost Exact App App App % Error App App App

9 358 Operations Research 53(2), pp , 2005 INFORMS Tale 6. Evaluation of approximations. = P = 16 (fixed) J = Echelon ase-stock levels s j Cost Exact App App App % Error App App App Optimization Up to this point we have focused on policy evaluation; we now turn to the task of identifying a good echelon asestock policy. Consider the following recursion: C 0 x = + h 1 x C j x = h jx + C j 1 x C j y = E C j y + Z j LD j (8) s j = arg min C j y j J C j x = C j s j x j < J Also, for any given policy parameters s j, let C j x, C j y, and C j x denote the corresponding functions, skipping the optimization step and using s j in place of sj. It is not hard to show that this calculation correctly evaluates any policy, according to Approximation 3. The average cost is C J s J. For nonnegative demand, the recursion aove (with Z j = 0) is precisely the one that determines an optimal policy. (See Chen and Zheng 1994 and Zipkin 2000, ) Here, given Approximation 3, the sj aove comprise an optimal policy. Proposition 2. Under Approximation 3, the sj determined y the recursion (8) comprise an optimal policy for the series system. Proof. To show this we prove y induction (following Lemma in Zipkin 2000) that, for any other policy, C j y Cj y, etc., and also, all the functions C j y, etc. are convex. Evidently, C0 = C 0 is convex. Assume that Cj 1 is convex and C j 1 Cj 1. Then, C j and Cj are convex, and C j C j and C j Cj. Also, C j x = C j s j x C j s j x We want to show that this is C j s j x = C j x Consider two cases: Case 1. s j <s j.forx s j, the assertion holds with equality. For s j <x<s j, C j s j x = C j s j C j x = C j s j x The inequality follows from the fact that Cj x is convex and attains its minimum at sj, so it is decreasing in x on s j x<sj.forx s j, C j s j x = C j s j C j s j = C j s j x y the definition of sj. Case 2. s j sj.forx s j, the assertion holds with equality. For x>sj, C j s j x C j s j = C j s j x y the definition sj. So, C j Cj. To show convexity, note that for x s j, Cj x = C j x, which is convex and decreases on that region to the value Cj s j.forx>s j, C j x = C j s j. It then follows that CJ s J C J y C J y for all y, i.e., the average cost of the sj policy is less than or equal to the cost of any other policy. We have not found a corresponding method for the exact calculation or the other approximations. We have seen that Approximation 3 is fairly accurate, and so we would expect this method to produce a good policy. To evaluate this method, we tested it on the same set of examples used in 6. For the examples in Tales 1 4 with J = 2, the algorithm correctly identified the optimal

10 Operations Research 53(2), pp , 2005 INFORMS 359 Tale 7. Evaluation of approximate optimization. = P = 4 (fixed) J = Algorithm Cost Optimal Cost policy in all cases. For the examples in Tales 5 and 6 with J = 4, an exhaustive search was performed to find the optimal policy. Tales 7 and 8 show those optimal policies, as well as the approximate policies for those cases. The costs of the two policies were computed using the exact policy-evaluation method. As can e seen from the tales, the approximate procedure performs very well for these cases as well. The cells where the true optimal policy differs from the algorithm s are printed in old. The approximate policy matches the optimal one in 10 out of 12 cases (not counting cases with no returns), and in the other two cases it is very close. The relative cost penalty from using the approximate policy is never more than 0.02%. Comparing Tales 5 and 7, and also Tales 6 and 8, one can see that the heuristic method used in 6 does not work very well. The policies it selects have costs that are significantly higher than those of the algorithm s policies; and, in fact, it requires more computational work. Keeping the mean net demand constant, increasing returns increases the variance of net demand. The tales Tale 8. Evaluation of approximate optimization. show that this effect increases the ase-stock levels and costs. This finding is consistent with what we know (e.g., Zipkin 2000, Chapter 6) aout the effects of other sources of system variance. Examining the aove algorithm, and following the arguments in Shang and Song (2003), we can otain a singlestage heuristic for the optimal echelon ase-stock levels. Specifically, let = P = 16 (fixed) J = Algorithm Cost Optimal Cost L j = j L i i=1 = total lead time in the susystem consisting of stages 1 through j D j = D 0 L j = total lead time demand in the susystem consisting of stages 1 through j. Then, for each j, Cj y is ounded aove and elow y the cost functions of two single-stage systems, which differ only in their holding-cost rates. The upper-ound system has holding cost rate j i=1 h i, while the lower-ound system has holding cost rate h j. Both systems have lead time L j, ackorder-cost rate + N i=j+1 h i, and the same demand

11 360 Operations Research 53(2), pp , 2005 INFORMS process as the original system. Let the resulting averagecost functions of the two systems e C u j y and Cl j y, respectively. Define Z as in 2. Then, C l j y C j y Cu j y where C u j y = j + G u j y [( j ) G u j y = E h i y + Z D j + i=1 ( ) N + + h i y + Z D j ] j 1 j = h i+1 E D i i=1 i=j+1 = average in-transit holding cost from stage j to stage 1 C l j y = j + G l j y [ G l j y = E h j y + Z D j + ( + + N i=j+1 h i ) y + Z D j ] The optimal ase-stock levels of these single-stage ounding systems are sj l = arg min Gu j and su j = arg min Gl j. Then, sj l s j sj u. Moreover, ased on experience with nonnegative-demand systems we expect the simple average of these ounds, sj l + su j /2, to e a good approximation for sj. This calculation is much simpler than the recursion aove. (See Shang and Song 2003 for other heuristics along these lines.) As a result, the qualitative property discussed at the end of 2 holds for the multistage system as well, that is, more returns imply lower (approximately) optimal echelon asestock levels. 8. Returns at Several Stages Now, suppose that returns can arrive at any stage or at several stages. Here, the returning goods may require some processing through the system efore they can e used to meet customer demands. The higher the stage of arrival, the more processing is required. Returns to several stages represent goods with heterogeneous processing needs. Actual demands still occur only at stage 1. Let D P t denote the demand at time t, and Dj N t the returns at stage j at time t. These are all nonnegative. They may e dependent, ut they are independent over t. Let D j t = D P t i j Di N t, the net echelon demand at stage j. Assume that E D J t > 0. The entire development of the policy-evaluation scheme of 3 goes through, with D j t replacing D t. Thus, we redefine LD j = D j L j L j 1, and the D in the alance equations ecomes D j, the equilirium version of D j t. The approximations of 6 and the optimization recursion of 7 extend similarly, with one modification. Now the Z j are not identically distriuted as Z. Instead, each one satisfies Z j = Z j D j +. The preceding oservations indicate that upstream returns in a series system can e accommodated just as easily as downstream returns. This is not the case for upstream (positive) demands. The key reason is that upstream returns preserve the relations IP j t IN j+1 t, while upstream demands do not. Note also that Z j st Z j+1. This implies that the optimization scheme of 7 could yield sj+1 <s j. While such an outcome is possile with no returns, the cause there is different (e.g., h j < 0). Also, in that case there is no loss in optimality from using s j = sj+1, ecause stage j s echelon inventory position can never exceed sj+1, regardless of how large s j is. With returns, this is no longer the case. Because returns are (stochastically) larger at stage j + 1, IN j+1 may exceed IP j even if sj+1 <s j. In fact, s j+1 is chosen so that, when returns to stage j + 1 are taken into account, stage j + 1 has the proper amount of stock relative to the needs of stage j, considering oth holding and implied shortage costs. 9. Lead Time for Product Recovery Fleischmann (2000) shows that the results for the singlestage prolem can e extended to the case of a productrecovery lead time L R, provided L R L. (Such a lead time might arise if returned items require an off-line recovery process efore they can e used to satisfy demand.) Following a similar approach, we can handle a recovery lead time in the series system, under the additional assumption that demands and returns in each period are independent. The idea is similar to advance demand information (Hariharan and Zipkin 1995). For j = 1 J, assume that L R L j for all j. To make use of the advance return information, we need to incorporate that information into the key state variale, the inventory position. Following Fleischmann (2000), redefine the inventory position to include returns in transit, i.e., ĨP j t = IP j t + returns in transit at time t Also, ĨP j t denotes the same quantity efore shipment. Reinterpret the ase-stock policy to utilize the new information: If ĨP j t <s j, then ship up to s j. Let D P t = demand in period t and D N t = returns we learn aout in period t. With zero recovery lead time, D N t units entered inventory immediately; with a positive recovery lead time those items do not enter inventory until the end of period t + L R. Define D P t u and D N t u analogously to D t u. With zero recovery lead time we had D t = D P t D N t and lead time demand was D t t + L j = DP t t + L j DN t t + L j

12 Operations Research 53(2), pp , 2005 INFORMS 361 With a positive recovery lead time, we have ĨP j t = IP j t + D N t L R t and we define a modified version of lead time demand, D t t + L j = DP t t + L j DN t t + L j L R (This incorporates all positive demands during the lead time, plus those new returns we learn aout during the lead time and that will also arrive during the lead time. Returns we already know aout that will arrive during the lead time have een captured in ĨP j t.) With these new definitions we again have the standard relationship IN j t + L j = ĨP j t D t t + L j Next, we want to otain the distriution of ĨP j t, given that of IN j+1 t. Without recovery lead times, we made use of the fact that IP j t IN j+1 t. Here, the analogous relationship is ĨP j t IN j+1 t + D N t L R t Note that IN j+1 t depends on D N k for k t L R 1 and D P k for k t 1. Thus, the two terms on the righthand side aove are independent, assuming that D N k and D P k are independent. So, define ĨN j+1 t = IN j+1 t + D N t L R t Then, we otain the analogue of (1), ĨP j t = min max s j ĨP j t ĨN j+1 t Starting with this relationship we can follow an approach analogous to that descried in 3 and 4 to compute steadystate system performance. 10. Conclusions In this paper, we analyze the steady-state ehavior of a series inventory system with possily negative stochastic demands, i.e., the system may receive returns as well as demands from customers. The presence of negative demands significantly complicates the task of policy evaluation and optimization. Because traditional arguments estalish the optimality of an echelon ase-stock policy, we focus on methods for evaluating such policies and for identifying a near-optimal policy. We develop an exact procedure and several approximations for computing the operating characteristics and system costs for any echelon ase-stock policy, and also present an algorithm for computing a good policy. Also, we discuss a ase-stock policy that uses only local information. Finally, we descrie how to extend the analysis to the case where returns occur at several stages, and also the case of a recovery lead time. In addition to providing practical methods for managing inventories in a multi-echelon system with possily negative demands, our analysis reveals some fundamental ways that such systems differ from traditional nonnegativedemand systems. For example, echelon ase-stock policies are no longer equivalent to local policies. Also, returns (negative demands) can e handled just as easily upstream as downstream, quite unlike nonnegative demands. In addition, upstream returns may cause upstream stages to have lower echelon ase-stock levels than downstream stages. Acknowledgments The authors thank Yue Dai for the numerical results. They also thank her and Roin Roundy for helpful discussions on this topic. Finally, they thank the reviewers for many useful suggestions. 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