MANAGEMENT SCIENCE Vo. 51, No. 8, August 2005, pp. 1250 1265 issn 0025-1909 eissn 1526-5501 05 5108 1250 informs doi 10.1287/mnsc.1050.0394 2005 INFORMS Inventory Management for an Assemby System wh Product or Component Returns Gregory A. DeCroix, Pau H. Zipkin The Fuqua Schoo of Business, Duke Universy, Box 90120, Durham, North Caroina 27708-0120 {decroix@mai.duke.edu, zipkin@mai.duke.edu} This paper considers an inventory system wh an assemby structure. In addion to uncertain customer demands, the system experiences uncertain returns from customers. Some of the components in the returned products can be recovered and reused, and these uns are returned to inventory. Returns compicate the structure of the system, so that the standard approach (based on reduction to an equivaent series system) no onger appies in genera. We identify condions on the em-recovery pattern and restrictions on the inventory poicy under which an equivaent series system does exist. For the specia case where ony the end product (or a ems used to assembe the end product) is recovered, we show that the system is equivaent to a series system wh no poicy restrictions. For the genera case, we expain how and why the system becomes more probematic and propose two heuristic poicies. The heuristics are easy to compute and practica to impement, and they perform we in numerica trias. Based on these numerica trias, we obtain insights into the impact of various factors on system performance. For exampe, we find that hoding and backorder costs tend to increase when the average return rate, the variabiy of returns, or the number of components recovered increases. However, neher the product archecture nor the specific set of components being recovered seems to have a significant impact on these costs. Whether product recovery reduces tota system costs depends on the magnude of the addiona hoding and backorder costs reative to potentia procurement cost savings. Key words: assemby systems; inventory poicies; reverse ogistics; remanufacturing; environment History: Accepted by Fangruo Chen, suppy chain management; received Juy 13, 2002. This paper was wh the authors 1 year and 8 months for 4 revisions. 1. Introduction In recent years there has been an increasing need for companies to manage reverse fows of materias in their suppy chains. One reason is the increased frequency wh which customers change their minds and return goods shorty after purchases. Firms have deat wh such returns for many years, but growth in mai-order and e-business traffic has increased the voume of such returns customers unabe to see and touch the ems they are purchasing are more ikey to return them. (See, for exampe, Tedeschi 2001.) Another contributor to the return fow of materias is product take-back the recovery of products after customers use them. Because of environmenta concerns, severa countries egay require manufacturers to take back certain used products, incuding automobies, eectronic goods, and packaging. (See, for exampe, Diem 1999 and Franke 1996.) Even when not required to, some companies vountariy coect used products from their customers. Exampes of such products incude singe-use cameras (Kodak, Fuji), toner cartridges (Xerox, Canon, Hewett-Packard), persona computers (IBM), and communication network equipment (Lucent). Whie this practice may have environmenta benefs, the primary incentive is often economic gain companies prof from recovering the residua vaue in the products. In some cases companies may even design products to maximize this vaue. The introduction of uncertain return fows into a suppy chain can compicate the management of the system by increasing variabiy, thus reducing the precision wh which managers can contro inventory eves. The insights and soution methods for tradiona inventory systems (whout returns) may no onger appy. Moreover, entirey new research questions regarding product design, returns network design, returns handing, etc., may arise. This paper expores the impact on inventory management of introducing returns into an assemby system where components are assembed into subassembies, etc., unti a finished product is produced. These returns may consist of finished goods that can be used immediatey to satisfy new customer demand, or used goods from which components or subassembies can be harvested. Our objectives are to deveop good poicies for managing inventories in this context and to gain insight into the factors affecting the performance of the system. 1250
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1251 Specificay, we anayze an infine-horizon, periodic-review system wh stationary data, fu backordering of unsatisfied demands, and inear hoding and backorder costs. We identify two primary ways that returns disrupt the structure of the system, so that the standard approach (i.e., conversion to an equivaent series system) no onger appies. We identify condions on the cass of poicies and the emrecovery pattern under which these difficuties can be avoided. For the specia cases of recovery of the finished good or recovery of a ems used to assembe the finished good, the assemby system can be reduced to an equivaent series system wh no poicy restrictions. As a resut, an echeon base-stock poicy is optima, and methods for soving a series system wh returns can be appied. Finay, for a genera assemby system whout these condions, we expain how and why the system becomes more probematic and present two heuristic methods for computing a good poicy. These heuristics perform we in numerica trias. For a fine-horizon, two-component assemby system, Schmidt and Nahmias (1985) show that the probem can be decomposed into ordering decisions for the components and an assemby decision for the finished good (simiar to the resut of Cark and Scarf 1960 for a series system). They show, however, that the optima poicy has a compex structure, wh the optima order for one component depending on the inventory of the other. Rosing (1989) studies a genera assemby system over an infine horizon. He shows that under an optima poicy inventories in the system satisfy a condion caed ong-run baance, so that the system can be reduced to an equivaent series system. As a resut, an optima poicy can be computed using the series-system method of Federgruen and Zipkin (1984) and Chen and Zheng (1994). See Zipkin (2000) for a more detaied discussion of these resuts. For a singe-ocation, fine-horizon inventory system, Heyman and Sobe (1984) point out that Scarf s (1960) proof of the optimay of an s S poicy sti works when the system faces uncertain returns in addion to demands. Feischmann et a. (2002) extend this resut to the infine-horizon case. When there is no fixed order cost, a base-stock poicy is optima. Cohen et a. (1980) estabish condions under which a base-stock poicy is optima when a fixed fraction of demands in each period is returned after a fixed number of periods. Kee and Siver (1989) deveop a heuristic approach for managing a simiar system that aso incudes fixed order costs and stochastic return times. There has aso been research on singe-stage systems where returns are not sent directy to stock (as assumed here), but instead are kept in a separate buffer unti they are processed or disposed of. Simpson (1978) shows that a three-parameter (remanufacture-up-to, order-up-to, and dispose-down-to) poicy is optima for the singe-stage, fine-horizon case wh inear costs and zero ead times. Inderfurth (1997) extends this resut to the case of posive and equa ead times for deivery of new ems and remanufacturing of used ems. Mahadevan et a. (2003) deveop heuristic poicies for systems wh more genera ead times. Research on mutiecheon systems facing product returns has been rather imed. DeCroix (2001) extends the resuts of Simpson (1978) and Inderfurth (1997) to a series system, assuming disposa is not aowed at downstream stages. For an infine-horizon series system where returns go directy to stock, DeCroix et a. (2005) show that an echeon base-stock poicy is optima and present exact and approximate methods for evauating any such poicy. The authors aso propose an approximate optimization agorhm for computing a good poicy. For reviews of other research on reverse ogistics, see Feischmann et a. (1997) and Dekker et a. (2004). This paper extends existing knowedge about management of assemby systems to those wh product or component returns. It aso contributes to the recent erature on mutiecheon inventory systems wh returns particuary by addressing issues, such as recovery of parts of products and the baancing of component inventories, that do not arise in previous research. The rest of the paper is organized as foows. Section 2 introduces the mode and notation, whie 3 expores condions under which the assemby system is equivaent to a series system. Section 4 presents a heuristic approach for computing good poicies for a genera assemby system, and 5 presents the resuts of a numerica study of that heuristic. Section 6 provides some concuding remarks. 2. Mode The mode buids on that of Rosing (1989), and where possibe his notation is used. Consider an assemby system consisting of N ems (finished product, components, subassembies, etc.) indexed by i = 1 N. Each em i has a unique immediate successor, denoted s i (where s 1 = 0, and may have a number of immediate predecessors denoted P i (where P 0 = 1 ). Assembing a un of em i that has predecessors requires combining one un each of a immediate predecessors of i. Items whout predecessors are ordered from an outside suppier. There is a ead time i 1 for deivery or assemby of em i. Let A i be the set of a successors of em i, and B i be the set of a predecessors of em i. Figure 1 iustrates an exampe of such a system.
1252 Management Science 51(8), pp. 1250 1265, 2005 INFORMS Figure 1 Exampe of an Assemby System 1 = 1, 2 = 1, 3 = 2, 4 = 2, 5 = 3, 6 = 3, 7 = 4 6 7 We mode time as discrete. In each period t, the system experiences stochastic demand D t for end ems, and stochastic returns R t of end ems. A fixed (deterministic) subset J of the ems is recovered from each of the returned uns these ems are immediatey paced in inventory and can be used to satisfy future demand. (This impicy assumes 100% recovery yied and negigibe refurbishment time.) Because we wi work wh echeon inventories, we say an em is recovered if a return causes that em s echeon inventory to increase. For exampe, a component is recovered if that component aone is recovered, or if the end product or any subassemby containing that component is recovered. As a resut, if i J, then j J for a j B i. For brevy, we sometimes describe J by just referring to the most assembed eements of J. For exampe, saying that ony em i is recovered means that J = i B i. Let K = 1 N \J be the set of ems that are not recovered. Demands and returns in different periods are independent and identicay distributed. This impies that returns are independent of past demands, which is an approximation in some cases. One coud argue that returns shoud instead be modeed as a function of past demands. In fact, a simiar issue arises for demand sef. Because no product has an unimed market, one coud argue that current demand shoud aso depend on past demands. In practice we rarey do that the effect is sma in most cases, so the added compexy yieds te benef. For simiar reasons, the independence assumption is common in the erature on systems wh returns. (See Feischmann 2000 for a more detaied justification of this assumption in that context. Aso, see Kiesmueer and van der Laan 2001 for a singe-stage mode where returns depend on past demands.) Let = E D t, = E R t, and et D u and R u be u-period demand and returns, respectivey. 4 5 2 3 1 Let M i = tota ead time for em i and a s successors, where M 0 = 0 and M i = i + j A i j for i = 1 N. (If necessary, reindex the ems so that M i M i 1 for a i and j<ifor a j A i.) Aso, et L i = M i M i 1. For i J, et ˆk i = max k A i k K, i.e., the argest-indexed em among i s successors that is not recovered, and for i K define ˆk i = i. (If 1 J, define ˆk i = 0 for a i.) Aso, et j = min j j J be the smaest-indexed em recovered. The system incurs a cost c i for each un of em i purchased or assembed (due upon deivery). Shortages of the end em are backordered at a un cost of p per period, and each un of em i in inventory or in the process of being assembed into s successor em incurs a (physica) instaation hoding cost of H i per period. Define the echeon hoding cost as the addiona hoding cost incurred when the predecessors of an em move (are assembed), i.e., h i = H i k P i H k. We assume that the system must accept a returned ems, and that the system incurs a un recovery cost of r for each of these ems. Future costs are discounted using a discount factor 0 < 1. A events occur at the beginning of the period in the foowing order: (1) outstanding orders arrive, (2) ordering decisions are made, (3) backorders are fied, (4) customer demands and returns occur, (5) costs are charged. Foowing standard practice, we work wh echeon inventory, which has essentiay the same definion here as in a series system. Onhand echeon inventory is the inventory in stock at a given stage, pus a inventory downstream of that stage (successor ems), whie echeon inventory posion is equa to that quanty pus ems that have been ordered, but have not arrived. Define the foowing variabes: X = echeon inventory posion of em i in period t before ordering/assemby decisions are made; Y = echeon inventory posion of em i in period t after ordering/assemby decisions are made; X = echeon inventory on hand of em i in period t before ordering/assemby decisions are made, but after orders arrive. Note that X = Y i t 1 D t 1 + R t 1 if i J em i recovered (1) X = Y i t 1 D t 1 if i K em i not recovered (2) and Y X (disposa is not aowed). Aso, the assemby decision for each em i is constrained by the on-hand inventory of s predecessors, i.e., Y X for a k P i,so t t X D t + R t = Y i t i D a + R a a=t i a=t i if i J (em i recovered) (3)
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1253 X = Y i t i t 1 a=t i D a if i K (em i not recovered) (4) We seek a poicy (represented by Y ) that minimizes the expected discounted cost of operating the system over an infine horizon. In each period, each em i 2 incurs physica hoding costs on on-hand inventory, i.e., [( H i X D ) ( t + R t X s i t D )] t + R t if i J and s i J [( H i X D ) ( t X s i t D )] t if i K and s i K and [( H i X D ) ( t + R t X s i t D )] t if i J and s i K Simiary, hoding and shortage costs associated wh em 1 are [ H 1 X 1t D ] t + R [ + t + p X 1t D ] t + R t if 1 J and [ H 1 X 1t D + t] [ + p X 1t D t] if 1 K Finay, the system incurs cost c i Y X for ordering/assembing uns of em i and rr t for recovering returned ems. Decisions in period t affect costs for em i in period t + i, so we charge period t + i costs to period t, discounted by i. (Because hoding/shortage costs for em i in periods t i cannot be infuenced, they are omted. Aso, because recovery costs are not associated wh a decision, for simpicy we charge those costs to the period in which the returns occur.) For the case of 1 K, summing the cost terms and converting to echeon hoding costs yieds a tota cost in period t of N c i i Y X + ( h i X D ) t i=1 i K + ( h i X D ) t + R t + p + H 1 X 1t D t + rr t i J Now use (1) and (2) to substute for X and (3) and (4) to substute for X. Taking expected vaues, summing over a periods, and coecting ike terms yieds the probem formuation. Assemby Probem { ( N min E t 1 i h i + 1 c Y i Y t=1 i=1 + 1 p + H 1 [ D 1+1 Y 1t ] + )} + constant (5) subject to X Y X for a k P i and a i and t (6) If instead 1 J, the probem is the same, except that D 1+1 Y 1t + in (5) becomes D 1+1 R 1+1 Y 1t +. The case = 1 corresponds to the average-cost probem. The objective function is derived by mutipying (5) by 1 / and taking the im as goes to 1. Regardess of whether 1 K or 1 J, the constant in (5) is given by N i c i h i i + 1 / 1 i=1 i c i h i i + 1 / 1 + r / 1 i J if <1 and N c i h i i + 1 c i h i i + 1 + r if = 1 i=1 i J (7) The cost expression in (5) consists of four different categories of costs. In ater sections we wi find convenient to refer to these categories, so we define them now. Hoding/backorder costs: the (physica and financia) costs of hoding inventory in stock pus the cost of end-product backorders. (These costs do not incude hoding costs on inventory in trans.) Pipeine costs: the (physica and financia) hoding costs of inventory in trans from an em to s successor. In any given period, the expected pipeine costs are equa to N i=1 j P i i J ( H j + 1 j P i j P j c i ) i ( H j + 1 j P j c i ) i Procurement/assemby costs: the costs of purchasing new components and assembing components into their successor ems. In any given period, the expected procurement/assemby costs are equa to N i=1 c i i J c i. Recovery costs: the costs of acquiring a returned end product and harvesting usabe ems from. In each period the expected recovery costs are r. To see how these categories reate to the cost expressions above, consider, for simpicy, the case = 1. The constant term (7) ceary contains the procurement/assemby and recovery costs. Hoding/backorder costs and pipeine costs make up the rest of the cost expression in (5). To see this, note that the portion of (5) before the constant term consists of backorder costs pus hoding costs on echeon inventory posion. However, the system ony
1254 Management Science 51(8), pp. 1250 1265, 2005 INFORMS incurs em i hoding costs on that em s echeon inventory not on uns of em i that have not yet arrived. The remainder of the constant term i.e., N i=1 h i i + 1 + i J h i i + 1 adjusts the costs to refect this. By charging hoding costs on echeon inventory, the expression in (5) thus incudes hoding costs for on-hand inventory (i.e., the other haf of hoding/backorder costs) pus pipeine costs e.g., hoding costs associated wh em i whie is in trans from stage i to s successor. Note that the constant term can be dropped whout affecting the optima poicy. If we do so and redefine the cost parameters h i = h i + 1 c i, H i = h i + k P i H k and p = p 1 N i=1 c i, we see that (5) is equivaent to { ( N mine t 1 i h i Y + 1 p+h 1 [ ] D 1+1 + Y 1t )} Y t=1 i=1 (8) Whout oss of generay, to simpify exposion, we use the form in (8) for a theoretica anaysis regarding the optima poicy for the assemby probem. For the numerica studies in 5, we incude those of the four cost components (from the fu cost expression (5)) that are most reevant to the question being addressed. Athough un procurement costs are constant regardess of the choice of poicy, and thus have been dropped from (8) (and are not incuded in hoding/backorder or pipeine costs), note that the redefined h i (and thus (8) and hoding/backorder and pipeine costs) incude financia hoding costs 1 c i that are based party on those procurement costs. Foowing common practice, we continue to incude a financia hoding-cost term in h i even when using the average-cost mode. It is interesting to note that the cost expressions (5) and (8) charge the same physica and financia hoding costs on both new and returned ems. This may seem odd, because the procurement cost c i is not incurred when em i is recovered from a used product, but instead the recovery cost r is incurred (and shared among a ems that can be harvested). It may seem more natura to have two different hodingcost rates one for new ems and one for returned ems. In fact, a singe hoding-cost rate for both ems actuay makes sense. What matters when making a of the ordering/assemby decisions is the margina (hoding) cost of increasing the stock of each em. Because the system has no decision to make regarding returns (rejection and disposa are not aowed), the recovery cost rr t is fixed, and so r shoud not become part of the financia hoding cost. On the other hand, if a un (eher new or used) of em i is hed in inventory at the end of some period, then that un of inventory coud have been avoided by ordering one ess new un of em i at some point in the past, thus deaying the procurement cost c i. In sum, is ogica that both new and used ems are charged a financia hoding cost of 1 c i. Assume that the cost parameters satisfy h i > 0 for a i and N i=1 h i M s i <p+ H1. The atter assumption assures that is aways optima to fi existing backorders, whie the former refects higher physica and financia hoding costs typicay associated wh ems that have progressed farther through the system. Define X M to be the echeon inventory posion at time t of em i ordered M i periods ago or earier. This quanty is an upper bound, based on the current echeon inventory and uns on order of em i, onthe amount of end product that coud be made avaiabe whin time periods. Letting s = M i, we have if i K, then X M if i J, then if Mˆk i if <Mˆk i = Y i t s X M X M t 1 a=t s D a = Y i t s = Y i t s s= 0 1 M i t 1 a=t s D a + t 1 a=t s s = 0 1 M i t 1 a=t s D a + R a t Mˆk i 1 a=t s and R a s = 0 1 M i This definion adjusts the one in Rosing (1989) to incude returns. Note that recovered uns of em i require Mˆk i periods to be converted into a un of finished product, so recenty recovered uns have been omted from the expression for X M in the case of <Mˆk i. The foowing emma estabishes some usefu properties of X M. Lemma 1. (a) X M and (c) X M M s i = X. X M +1, (b) X = X M M i 1, A proofs are contained in a technica appendix avaiabe at http://mansci.pubs.informs.org/ecompanion. htm. 3. Poicy Properties Our best hope for obtaining an efficient method of soving the system in (8) and (6) is to show that the system is equivaent to a series system, as Rosing (1989) does for an assemby system whout returns. Rosing s resuts depend on a condion he cas ongrun baance, and this condion wi aso be key to our anaysis. We use the same definion, athough
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1255 our definion appies to the modified X M defined above. Definion: Long-Run Baance. We say the system is in ong-run baance in period t if for i = 1 2 N 1 X M X M i+1 t for = 0 1 M i 1 Intuivey, this condion says that the number of uns avaiabe to satisfy end-em demand whin time periods is increasing in i, i.e., as we ook farther from the end em (in terms of tota ead time). This property aways hods for a series system, and that ink is the key to the equivaence between the two systems. Unfortunatey, returns can disrupt ong-run baance in two main ways. The goa of this section is to expain the reasons for this and to identify condions under which ong-run baance is preserved so that series-system methods can be appied. Consider the system shown in Figure 1, and suppose first that ony Item 6 can be recovered. If there are many returns in some period, the echeon inventory of Item 6 coud exceed that of Item 7, vioating ong-run baance. A condion that avoids this possibiy is that J be of the form J = j N i.e., if any em i is recovered, then a ems j>iare aso. We say that such a recovery pattern satisfies ongest-ead-time recovery. This condion is pausibe: Items vauabe enough to be recovered tend to be compex, high-cost ems, which tend to require ong ead times to procure or produce. In such cases, the ongest-ead-time condion may hod. (For exampe, Toay et a. 2000 report that the reusabe circu board from a Kodak singe-use camera is the primary cost driver for the product, and that the board is manufactured overseas, resuting in a ong deivery ead time.) Now suppose that Items 5, 6, and 7 are recovered. Whie this recovery pattern exhibs ongest-ead-time recovery, may sti vioate ong-run baance. The difficuty here is that recovered uns of Item 5 can be converted to finished product more quicky (in 1 + 2 periods) than recovered uns of Item 6 (which take 1 + 3 periods), which coud resut in X M 5t >X M 6t for = 1 + 2. A condion that avoids this possibiy is Mˆk i Mˆk i+1 for j i N 1. One natura type of recovery pattern that satisfies this atter condion we ca singe-modue recovery. This hods if just a singe modue is recovered (e.g., just em i for any i = 1 7 in Figure 1, so that J = i B i, or if precisey those ems required to assembe a singe modue are recovered (e.g., Items 2 and 3, or Items 4 and 5, or Items 6 and 7 in Figure 1). One interpretation of the atter case is that the modue is taken apart into subassembies and ceaned or tested before being returned to inventory. Of the singe-modue recovery patterns, the ones corresponding to J = 1 7, J = 2 7, and J = 6 7 aso satisfy ongest-ead-time recovery. Because the recovery pattern is a function of engineering and design choices, avaiabe recovery technoogy, etc., nothing in the inventory management poicy can prevent the system from moving out of baance so anaytica resuts for systems that do not satisfy these condions seem unikey. We present a heuristic approach for soving such systems in the next section. Even if the recovery-pattern condions are satisfied, the optima ordering/assemby poicy may move the system out of ong-run baance. Consider again the system in Figure 1, and suppose that ony Item 7 is recovered. When deciding how much of Item 6 to order, we may try to anticipate future recovery of Item 7. If those returns do not materiaize, the system wi fa out of ong-run baance. One way to avoid this possibiy is to prohib anticipatory orders i.e., infated orders of a shorter-ead-time em in anticipation of recovery of a onger-ead-time em. The definion of nonanticipatory poicies can be formaized as foows. Define b to be the index such that X M M i bt = min k>i X M M i Aso, for ems i K, define the set J i = j J j >i j B i and Mˆk j M i A poicy Y is nonanticipatory if satisfies the foowing condions. Nonanticipatory Poicy (a) If i K, then if X > min X M M i jt then Y = X j J i if X min X M M i j J i jt then X Y min (b) If i b J and Mˆk i >Mˆk b, then j J i if X >X M M i bt then Y = X if X X M M i bt then X Y X M M i bt X M M i jt Case (a) corresponds to the exampe described above, whie Case (b) addresses the temptation to order excess of em i (even though that em is recovered) when another recovered em is coser to the finished product. In genera, the restriction to nonanticipatory poicies is suboptima. In some systems, however, the restriction imposes few constraints. Consider the system in Figure 1, and suppose that J = 2 4 5 7 and K = 1 3 6. For i b J, is easy to verify that Mˆk i > Mˆk b can never occur, whie for i K we have J 1 =,
1256 Management Science 51(8), pp. 1250 1265, 2005 INFORMS J 3 = 4 5, and J 6 = 7. Thus, the restriction to nonanticipatory poicies ony ims orders of Item 3 (reative to Items 4 and 5) and orders of Item 6 (reative to Item 7). We expore the cost impact of the nonanticipatory ordering restriction as part of a numerica study in 5. The foowing resut summarizes the preceding discussion and estabishes condions under which an assemby system is equivaent to a series system. Proposion 1. Suppose that an assemby system starts in ong-run baance, the system experiences ongestead-time recovery, and Mˆk i Mˆk i+1 for j i N 1. Then, under the restriction to nonanticipatory poicies, the assemby system is equivaent to a series system wh returns at stage j, the same cost coefficient p + H 1, echeon hoding costs i L ihi, and ead times L i. One specific recovery pattern that satisfies the condions of Proposion 1 is of particuar interest. That is the case where the end product or a of s immediate predecessors are recovered. For an assemby system wh eher of these recovery patterns, the optima poicy among a poicies (whout the restriction to nonanticipatory poicies) can be determined by soving an equivaent series system. Coroary 1. Suppose that an assemby system recovers the end product or a of s immediate predecessors. Aso, suppose that the system starts in ong-run baance. Then, the system is equivaent to a series system wh returns at Stage 1 (in the case of end-em recovery) or Stage 2 (if the immediate predecessors are recovered). ComputingOptima Poicies. Proposion 1 provides condions under which an assemby system is equivaent to a series system wh returns at a singe stage. For the average-cost case = 1, DeCroix et a. (2005) show that an echeon base-stock poicy is optima for the series system wh returns. They aso provide an optimization agorhm that uses approximate cost functions to compute near-optima basestock eves S i for that system. That poicy can then be transated for use in the origina assemby system as foows: { ( min Si X M M ) i i+1 t if X S i Y = if X >S i X For recovery of the end product or s immediate predecessors, Y is the optima poicy for the assemby system, whie for other systems satisfying the condions of Proposion 1, this is true whin the cass of nonanticipatory poicies. 4. Heuristic Poicies The preceding section describes how to compute optima poicies (eher among a poicies or whin Figure 2 A Simpe Assemby System 1 = 1, 2 = 1, 3 = 2 the cass of nonanticipatory poicies) for assemby systems wh recovery patterns satisfying particuar condions. This section presents two heuristic approaches for a genera recovery pattern. 2 3 1 4.1. Numerica Exporation of Optima Poicies To guide the design of heuristic poicies (and to provide a benchmark against which to test them), we constructed a set of test probems based on the system in Figure 2. This set contained 64 probems 32 wh recovery of Item 2, and 32 wh recovery of Item 3. (We describe the parameters in detai in 5.) We computed the optima ordering poicy for each probem by dynamic programming, using the agorhm in Ding et a. (1988), which is a variation of the poicy-eration method of Howard (1960). Visua inspection of the optima poicies reveaed some interesting patterns. A typica pattern for the case of recovery of Item 3 is iustrated in Figure 3. As defined in 2, X3 1 is the echeon inventory of Item 3 ordered one period ago or earier. (Here we suppress the time-period subscript t.) From Figure 3 is easy to see that Item 3 foows an echeon base-stock poicy, wh base-stock eve 13. The optima poicy for Item 2 is somewhat more compex. Item 2 foows a type of modified base-stock poicy order up to a target echeon inventory posion Y2 (given any starting echeon inventory posion X2 1 Y 2 ), but that target eve changes wh X3 1. The poicy for Item 1 is a base-stock poicy, modified as necessary to refect avaiabiy of Items 2 and 3. Note that the optima poicy for Item 2 exhibs anticipatory ordering. A nonanticipatory poicy woud restrict Y 2 X3 1. In Figure 3, however, for ow vaues Figure 3 Order up to 30 25 20 15 Exampe of Optima Ordering Poicies, Item 3 Recovered 10 X 1 3 5 Y 2 Y * 3 0 0 5 10 15 20 25 30 X 1 3
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1257 of X3 1 we have Y 2 = X1 3 + 3, for medium vaues Y 2 = X3 1 + 2, and for high vaues Y 2 = 16, a constant basestock eve. When Item 2 is recovered, the optima poicy has some simiar patterns, but is a b more compex. The optima Y2 is again a function of X3 1, but in this case Y2 X1 3, refecting anticipation of recovery of Item 2. However, if returns exceed demands for a whie, Item 2 s echeon inventory X2 1 may become arger than Y2. We observed cases where this effect made optima for Item 3 s order to increase a deviation from a pure base-stock poicy. (This interaction between Item 2 s inventory and Item 3 s order never occurs in systems whout returns.) Whie the preceding discussion yieds some insights regarding the optima poicy, that poicy does not appear to have enough structure to indicate an efficient agorhm. Computing the optima poicy directy using a genera dynamic programming method is feasibe for three-em systems wh short ead times and sma demand and returns distributions, but not for arger probems. (A three-em probem wh ead times 1 = 1, 2 = 1, and 3 = 2, maximum demand of 8, and maximum returns of 5 took approximatey 32 hours to sove on a desop PC wh a 933 MHz processor. The state space, and thus computation time, grows very rapidy wh onger ead times, arger distributions, and more ems.) However, we can use the insights above to construct tractabe heuristics. 4.2. Description of Heuristic Poicies We propose two heuristic poicies: Heuristic A and Heuristic B. Both are modified base-stock poicies, simiar in structure to the optima poicy described earier. Each poicy can be described by a base-stock eve S i for each em i, and a rue for modifying the base-stock poicy based on information about the returns distribution and the pipeine inventories of higher-indexed ems. The poicies differ in how they determine the base-stock eves and the modification rues. Heuristic A uses just the mean of the returns distribution to make simpe adjustments to the optima poicy for a system wh no returns. Heuristic B is somewhat more invoved, and impicy makes use of information about the entire returns distribution. Concise specifications of the two heuristics are given beow, foowed by some iustrative exampes. Heuristic A Base-Stock Leves (1) Compute the optima base-stock eves S i for the assemby system assuming no returns, using the techniques of Federgruen and Zipkin (1984) and Chen and Zheng (1994). Then, for each em i K, set S i = S i. (2) For i J, define N i M i Mˆk i and set S i = S i N i. Modification Rue (3) For i K, k J, and k>i, define P ik M i Mˆk k. Then, in any period t, the order-up-to quanty Y for each em i K is { Y = min S i min k>i k K X M M { i M M min X i + P ik }} k>i k J (9) (4) For i J, k J, and k>i, define Q ik Mˆk k Mˆk i. Then, in any period t, the order-up-to quanty Y for each em i J is { Y = min S i min k>i k K min k>i k J { M M X i N i } { X M M i + Q ik }} (10) Step 2 of Heuristic A adjusts the optima no-returns base-stock eves for each em i J by the expected amount by which an order for that em wi be suppemented as passes through the system. For exampe, for the system in Figure 1, suppose that Item 2 is recovered (i.e., J = 2 4 5 ) and consider an order for Item 4. This order wi be suppemented by recovered uns unti reaches Location 2, i.e., for N 4 = M 4 Mˆk 4 = M 4 M 1 = 3 periods, so we set S 4 = S 4 3. Rosing (1989) shows that for a system whout returns the optima poicy uses the modification rue Y = min k>i S i XM M i i.e., em i s echeon inventory posion is constrained by the pipeine inventories of higher-indexed ems. There is no advantage to ordering more, because those ems woud have to wa to be matched wh higher-indexed ems farther out in the pipeine. The modification rues in Steps 3 and 4 generaize this to refect returns. Consider again the exampe mentioned above. For any pair i k K wh k>i, e.g., i = 6 and k = 7, Rosing s ogic sti appies, so that is aways best to restrict Y min S i X M M i. On the other hand, if i = 3 and k = 4 (i.e., i K and k J ), then returns in future periods wi suppement the X M M 3 4t uns currenty in Item 4 s pipeine. Because those uns are M 3 = 3 periods away from the end em at the time an order for Item 3 is paced, and they are being suppemented by recovery of Item 2, which is Mˆk 4 = M 1 = 1 period away from the end em, Item 4 s pipeine wi be suppemented by P ik M i Mˆk k = M 3 M 1 = 2 periods of returns. So, for i K and k J, Step 3 incudes the restriction Y X M M i + P ik. If i = 4 and k = 6 in our exampe (i.e., i J and k K), then an order for Item 4 is suppemented by N i M i Mˆk i = M 4 M 1 = 3 periods of returns on s path to the end em, whie Item 6 receives no suppement. So, for i J and k K, Step 4 incudes the restriction Y min S i X M M i N i. Finay, suppose that we modify our exampe so that Item 7 is aso recovered (i.e., J = 2 4 5 7. Ifi = 4 and k = 7
1258 Management Science 51(8), pp. 1250 1265, 2005 INFORMS (i.e., i J and k J ), then an order for Item 4 is suppemented for M 4 Mˆk 4 = 3 periods on s path to the end em, whie the X M M 4 7t uns currenty in Item 7 s pipeine are suppemented for the M 4 Mˆk 7 = M 4 M 3 = 1 period. As a resut, Item 4 is recovered for Q ik Mˆk k Mˆk i = M 3 M 1 = 2 more periods than Item 7. So, for i J and k J, Step 4 incudes the restriction Y X M M i Q ik. Heuristic B is simiar in spir to Heuristic A, but uses a different method to compute the base-stock eves and modification rue. Heuristic B Base-Stock Leves (1) For i K, set S i = S i as in Heuristic A. (2) For i J, define k i min k i A i k J. Now construct a subnetwork of the origina assemby system consisting of em k i and a of s successors A k i and predecessors B k i. (Note that a eements of B k i wi aso be eements of J.) The resuts of 3 impy that this smaer assemby system is equivaent to a series system wh returns arriving at stage k i. Compute the optima base-stock eves S i for this system using the approach of DeCroix et a. (2005). Then, for each i k i B k i, set S i = S i. Modification Rue (3) For i K, k J, and k>i, choose an adjustment quanty V ik Y X M M i such that Pr V ik R Pik = for some 0 < <1 (or, if the returns distribution is discrete, choose the smaest V ik such that Pr V ik R Pik. Then, in any period t, the orderup-to quanty Y for each em i K is { Y = min S i min X M M { i M M min X i } } + V ik k>i k K k>i k J (4) For i J, k K, and k>i, define U ik N i, and for i J, k J, and k>i, define U ik Q ik.ifk J and Q ik 0, choose an adjustment quanty V ik such that Pr V ik R Uik = for some 0 < <1. Otherwise, choose an adjustment quanty V ik <0 such that Pr R Uik V ik = for some 0 < <1. Then, in any period t, the order-up-to quanty Y for each em i J is { { M M Y = min S i min X i } } + V ik k>i To iustrate Heuristic B, consider the system in Figure 1 and suppose that Items 3 and 4 are recovered. To determine the base-stock eves for ems i J = 3 4 6 7, Step 2 computes the optima base-stock eves for the series systems 4 2 1 wh returns to Stage 4, and 7 6 3 1 wh returns to Stage 3. The adjustment quanty V ik in Steps 3 and 4 is chosen so that (adjusting for returns) when the current order for em i arrives, there wi be enough uns of that em to match a uns of em k wh probabiy, i.e., Pr Y + R M i Mˆk i X M M i + R M i Mˆk k =, or Pr V ik R M i Mˆk k R M i Mˆk i =. The different cases in Steps 3 and 4 provide aternate (simper) versions of this expression, depending on whether em i or em k is suppemented by more periods of returns. For exampe, if i = 2 and k = 3 (i.e., i K and k J ), then Item 3 wi be suppemented by P 23 = 1 period of returns. Therefore, we can order Item 2 up to the quanty of Item 3 in the pipeine pus a posive adjustment V 23 > 0 that soves Pr V 23 R P23 =. Suppose instead that i = 3 and k = 4 (i.e., i J and k J ). Then, Item 3 wi be suppemented by returns for Q 34 = Mˆk 4 Mˆk 3 = M 2 M 1 = 1 more period than Item 4, so we can order Item 3 up to the quanty of Item 4 in the pipeine, pus a negative adjustment V 34 < 0 that soves Pr R U34 V 34 = Pr R Q34 V 34 = Pr R 1 V 34 =. The parameter is user specified. The numerica studies in 5 use = p/ p + h i, the crica fractie for the newsvendor probem that woud resut if a shortage of em i (reative to em k) resuted in a backorder. Computing the base-stock eves for Heuristic A requires essentiay the same effort as computing the optima eves for a series system whout returns. More extensive computations are required for Heuristic B, because in addion we must sove some number of series systems wh returns. For Heuristic A, computing the parameters associated wh the modification rues in Steps 3 and 4 invoves ony a few simpe computations. For Heuristic B these steps are again somewhat more invoved, but they ony require constructing the mutiperiod returns distributions and computing the appropriate fracties. For both heuristics, impementation of the poicy requires the same kind of pipeine information as in an assemby system whout returns. However, because returns can disrupt ong-run baance, is necessary to track the pipeine inventory of a higher-indexed ems when pacing an order. (Rosing 1989 shows that, whout returns, when ordering em i, is necessary to check the pipeine of em i + 1 ony.) Numerica studies (discussed in the next section) revea that eher Heuristic A or Heuristic B may yied ower hoding/backorder costs, depending on system parameters. Whie those studies shed some ight on when one heuristic might be expected to perform better than the other, a third option is to compute the costs of both heuristic poicies and use the one wh the ower cost. We refer to this approach as the combined heuristic. Finay, whie the theoretica resuts of the preceding section depend on the assumption of a fixed recovery pattern, both heuristics can be generaized to settings where the recovery pattern is stochastic (e.g., systems wh random recovery yieds). To that end, et
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1259 R i = uns of em i recovered, J = i Pr R i > 0 >0, and i = E R i. (Note that the R i may or may not be correated.) Then, for Heuristic A, the base-stock eves for i J become S i = S i N i i. Aso, the foowing modifications are made in (9) and (10): P ik P ik k in (9); N i N i i and Q ik M i Mˆk i i M i Mˆk k k in (10). For Heuristic B, for each i J we modify Step 2, soving a smaer series system consisting of just i A i, i.e., the path from em i to the end em. The modification rues are determined using ogic simiar to that above, but now they are based on R i, i.e., choose V ik Y X M M i X M M i R M i Mˆk k k so that Pr Y + R M i Mˆk i i + R M i Mˆk k k =, or equivaenty, Pr V ik R M i Mˆk i i =. In 5.1, we report resuts of a numerica test of the heuristics in this more genera setting. 5. Numerica Study In this section, we present a two-part numerica study. The first part focuses on sma probems for which the optima poicy can be computed. Here we expore three questions: (1) How we do the heuristic poicies perform reative to the optima poicy and a naïve poicy (i.e., the optima poicy assuming no returns, but appying, naivey, to the system that does experience returns), (2) How do returns affect hoding/backorder costs, and (3) How do factors such as the expected returns, the variance of returns, and the recovery pattern affect system performance and the performance of the heuristics? The second part of the study considers arger probems for which is impractica to compute the optima poicy. Here we expore system behavior using ony the combined heuristic poicy. We compare hoding/backorder costs under that poicy to two benchmarks hoding/backorder costs for a system whout returns, and hoding/backorder costs for a system wh returns under the naïve poicy that ignores returns. We aso expore how different component-recovery patterns and system structures affect system behavior. We concude wh some comments about the cost of restricting the system to nonanticipatory poicies. In both studies we focus on average cost per period, but foowing common practice we interpret the un hoding costs h i to incude both physica and financia components. We iniay focus on hoding/backorder costs. For some questions (e.g., the performance of the heuristic poicies reative to the optima and naïve poicies) these are the ony costs that matter, because these are the ony costs that can be infuenced by the ordering poicy. For other questions (e.g., the cost impact of increasing the average return rate), these are the ony costs that require invoved cacuations to compute the other three cost components can be easiy computed using the expressions in 3. We provide a few exampes to iustrate the impact of these factors on these other three cost components. 5.1. Optima Hoding/Backorder Cost and Heuristic Performance We investigate optima hoding/backorder cost and heuristic poicy performance using the 64 three-em probems mentioned in 4. Specificay, we consider two recovery patterns recovery of Item 2 ony, and recovery of Item 3 ony. (Note that the other two possibe recovery patterns for this system recovery of Items 2 and 3, or recovery of Item 1 satisfy the condions of Coroary 1. As a resut, the methods of DeCroix et a. 2005 can be used to compute the optima poicy, so there is no need to use the heuristic poicies.) For each recovery pattern we consider 32 probems by setting the echeon inventory hoding costs h 1 h 2 h 3 to 1 1 1, 4 1 1, 1 4 1, and 1 1 4, the un backorder cost p = 10, and using the 8 demand/returns distributions in Tabe 1. In a cases, returns in a period are independent of demand in that period. Note that Cases 1 through 3 represent increasing mean return rates whie return variance is hed constant. Cases 2 and 4 through 6 represent increasing return variabiy whie hoding the mean return rate constant. Case 7 represents a skewed return distribution, whie Case 8 represents a skewed demand distribution. Tabe 1 Demand/Returns Distributions Demand Returns Case 1 D = 0 1 2 3 4 5 6 7 8 R= 0 1 2 Prob = 0 04 0 08 0 12 0 16 Prob = 0 2 0 6 0 2 0 2 0 16 0 12 0 08 0 04 E D = 4, var D = 4 E R = 1, var R = 0 4 Case 2 Same as Case 1 R = 1 2 3 Prob = 0 2 0 6 0 2 E R = 2, var R = 0 4 Case 3 Same as Case 1 R = 2 3 4 Prob = 0 2 0 6 0 2 E R = 3, var R = 0 4 Case 4 Same as Case 1 R = 0 1 2 3 4 Prob = 0 1 0 2 0 4 0 2 0 1 E R = 2, var R = 1 2 Case 5 Same as Case 1 R = 0 1 2 3 4 Prob = 0 2 0 2 0 2 0 2 0 2 E R = 2, var R = 2 Case 6 Same as Case 1 R = 0 1 2 3 4 Prob = 0 3 0 15 0 1 0 15 0 3 E R = 2, var R = 2 7 Case 7 Same as Case 1 R = 0 1 2 3 4 5 6 7 Prob = 0 002 0 405 0 306 0 207 0 058 0 008 0 008 0 006 E R = 2, var R = 1 2 Case 8 D = 1 2 3 4 5 6 7 8 R= 0 1 2 3 4 Prob = 0 055 0 25 0 18 0 14 Prob = 0 1 0 2 0 4 0 2 0 1 0 12 0 103 0 09 0 062 E D = 4, var D = 4 E R = 2, var R = 1 2
1260 Management Science 51(8), pp. 1250 1265, 2005 INFORMS For each of the 64 test probems, we compute expected hoding/backorder cost per period for both heuristic poicies and the naïve poicy using successive approximations, and then compare those to the hoding/backorder cost of the optima poicy computed by dynamic programming as described in 4. Performance is measured by the reative error: reative error avg. cost of heuristic avg. cost of optima poicy = avg. cost of optima poicy Both heuristic poicies perform we reative to the true optima poicy the average reative errors across a 64 test probems were 1.46% for Heuristic A and 1.65% for Heuristic B. For Heuristic A, the average reative error was smaer for recovery of Item 3 (1.22%) than for recovery of Item 2 (1.70%), whie the oppose hed for Heuristic B (2.23% for Item 3 versus 1.068% for Item 2). The maximum error was 8.70% for Heuristic A and 6.08% for Heuristic B. By comparison, the naïve poicy performs reativey poory, yieding an average reative error across the 64 test probems of 10.72% and a maximum error of 44.23%. For two-tier systems consisting of just the end product and a set of components (ike the test probems considered here), is possibe to theoreticay address the second question by comparing the optima hoding/backorder costs for a system wh recovery of some of the components to that of a system whout returns. The foowing resut provides such a comparison. Proposion 2. If P 1 = 2 N, then the optima hoding/backorder cost for the system whout returns is a ower bound for the optima hoding/backorder cost of a system wh returns and any recovery pattern satisfying J 2 N. To expore the magnude of the cost difference identified in Proposion 2, for each of our 64 test probems we compare the hoding/backorder cost under the optima poicy to the optima hoding/backorder cost for the same system whout returns. On average, introducing returns increased optima hoding/backorder costs by 23.4%, wh a range of 6.3% to 63.2%. For Heuristic A (B) the average increase was 25.3% (25.5%), wh a range of 6.4% to 68.4% (6.3% to 63.2%). Note that if the end em is recovered, returns may resut in eher higher or ower hoding/backorder costs. For exampe, if demand and returns in a given period are independent, then returns cause the average (net) demand for each em in the system to be ower, but the variance of (net) demand to be higher. This increased variance can make harder to match suppy wh demand, resuting in higher hoding/backorder costs. (This can occur even if Pr D > R = 1, i.e., when net demand is aways nonnegative.) On the other hand, if demand and returns in a given period are correated, returns may reduce (net) demand variance, yieding ower hoding/backorder costs. The third question expores the impact of higher return rates or higher return variance. Figure 4 iustrates the impact of higher average returns when Item 3 is recovered and h 1 h 2 h 3 = 1 1 1. (The graphs for recovery of Item 2 and the other hodingcost vaues are simiar.) The figure shows the effect of the return rate on the optima poicy, both proposed heuristics, and aso the naïve poicy. As E R increases from 1 to 3.75 wh E D fixed at 4 (i.e., demand/returns distribution Cases 1 through 3, suppemented by two addiona cases wh E R = 3 5 and E R = 3 75 to expore behavior as E R approaches E D, hoding/backorder costs increase at an increasing rate. The absoute and reative heuristic errors tend to grow as E R increases, but there are some exceptions to this pattern. However, even for very high return rates neary 94% both heuristics sti perform reasonaby we. In that case, the average reative error for Heuristic A is 5.0% when Item 2 is recovered and 2.3% when Item 3 is recovered, whie for Heuristic B the errors are 5.7% and 3.6%, respectivey. Note aso that the cost advantage of both heuristics over the naïve poicy grows wh E R. In fact, when E R = 3 75, the naïve poicy yieds average errors of 20.3% (Item 2 recovered) and 72.9% (Item 3 recovered). Wh the insights from Figure 4, is easy to determine how E R affects tota system costs in any given suation. Reca that the sum of procurement/assemby, pipeine, and recovery costs is inear in E R. If the sope of this sum is posive, then more returns wi aways ead to higher tota system costs. Figure 4 Hoding/Backorder cost 40 35 30 25 20 15 10 5 0 Impact of Return Rate on Hoding/Backorder Costs No returns Recover 3: Optima Recover 3: Heuristic A Recover 3: Heuristic B Recover 3: Naive 0 0.25 0.50 0.75 1.00 Avg. returns/avg. demand
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1261 This woud be the case, for exampe, if returns consisted of recenty purchased (new) products, where customers received a fu refund of the retai price. The recovery cost woud then equa the retai price pus any addiona costs of ceaning/testing/restocking the em. Because profabiy requires that the retai price is greater than the sum of the pipeine and procurement/assemby costs associated wh producing a singe un, the sope of the inear term must be posive. If instead the sope is negative (which may occur if te or no payment is made for the returned product and usabe ems can be harvested at sufficienty ow cost), then a higher return rate wi reduce tota system costs at first. However, if as the return rate rises the sope of the hoding/backorder cost curve in Figure 4 becomes equa to the negative of the sope of the inear term, then any further increase in the return rate woud increase tota system costs. Figures 5a and 5b iustrate this reationship between E R and Figure 5 (a) Tota system cost (b) Tota system cost 385 380 375 370 365 360 355 (a) Impact of Return Rate on Tota System Costs: Low Un Recovery Cost; (b) Impact of Return Rate on Tota System Costs: High Un Recovery Cost No returns Recover 3: Optima Recover 3: Heuristic Recover 3: Naive 350 0 0.25 0.50 0.75 1.00 396 394 392 390 388 386 384 382 380 378 376 Avg. returns/avg. demand No returns Recover 3: Optima Recover 3: Heuristic Recover 3: Naive 0 0.25 0.50 0.75 1.00 Avg. returns/avg. demand tota system costs for two sets of exampes. Both figures are based on the same recovery structure (i.e., Item 3 is recovered), hoding costs, and sequence of returns distributions as depicted in Figure 4. In addion, we assume un procurement/assemby costs of c i = 30 for both components and the finished product. Figure 5a depicts a un recovery cost of r = 0 7 c 3 = 21, whie Figure 5b depicts a un recovery cost of r = 0 9 c 3 = 27. (For simpicy, both figures show the combined heuristic the better of Heuristics A and B rather than graphing them separatey.) In Figure 5a, a higher return rate continues to reduce tota system costs for the entire range of exampes considered. In contrast, in Figure 5b higher returns start to increase tota system costs once the return rate goes beyond about 75% of average demand. Even at these higher rates, however, tota system costs wh returns are ower than wh no returns. To give a sense of the reative magnudes of the four cost components in these exampes, when r = 0 9 c 3 = 27 and E R is 75% of E D, the optima hoding/backorder cost is 19.2, pipeine cost is 8, procurement cost is 270, and recovery cost is 81. Buiding on this comparison, is straightforward to identify the factors that infuence when a higher return rate is beneficia. If materia and abor cost savings on recovered ems are arge (i.e., i J c i r is arge), then these savings woud tend to outweigh any addiona hoding/backorder costs associated wh the addiona returns. On the other hand, a arge un backorder cost p woud tend to yied the oppose resut. Large un hoding costs h i coud resut in eher outcome. On one hand, they woud ampify the rate at which higher returns increase hoding/shortage costs. At the same time, however, they coud aso increase the amount of pipeine-cost savings resuting from returned ems. The net effect woud depend on the specific setting in question. (For exampe, in the settings in Figures 5a and 5b, pipeine costs are ony incurred in trans to Item 1, so the return rate has no impact on these costs. As a resut, higher vaues of h i woud ony increase the hoding/ shortage costs.) Note that in Figures 5a and 5b, not ony does the naïve poicy yied higher costs than the combined heuristic, but aso sends miseading signas regarding the profabiy of higher return rates. In Figure 5a, the naïve poicy suggests that increasing the return rate beyond about 87% of average demand woud ead to increased tota system costs, whereas in Figure 5b that cutoff point is around 50%. In fact, in the atter case, the naïve poicy suggests that a return rate above about 70% 75% is actuay more costy than no returns. Because the combined heuristic tracks the optima cost function much more
1262 Management Science 51(8), pp. 1250 1265, 2005 INFORMS Figure 6 Hoding/Backorder cost 25 20 15 10 5 Impact of Return Variabiy on Hoding/Backorder Costs No returns Recover 3:Optima Recover 3:Heuristic A Recover 3:Heuristic B Recover 3:Naive 0 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Coefficient of variation cosey, provides a much more accurate assessment of returns profabiy. Figure 6 shows the impact of returns variance when Item 3 is recovered, h 1 h 2 h 3 = 1 1 1, E D is fixed at 4, E R is fixed at 2, and the coefficient of variation increases from 0.32 to 1.53 (i.e., demand/returns distribution Cases 2 and 4 through 6, suppemented by an addiona case wh var R = 9 4 to expore behavior wh high variance). (Again, the graphs for recovery of Item 2 and for other hoding costs are simiar.) As returns variance increases, the optima hoding/backorder costs increase at a neary inear rate. Heuristic A tends to perform better when the returns variance is ow, whie Heuristic B performs better in high-variance cases. This reative performance pattern makes sense given the way the two heuristics are specified. By soving subsystems (rather than the entire assemby system) to compute the base-stock eves, Heuristic B distorts the structure of the assemby system somewhat, introducing some deviations from optimay. When returns variance is ow, this distortion causes arger errors than Heuristic A, which is based on the origina assemby system. (Attempts to identify aternative heuristics that incorporate variance information but do not introduce this kind of distortion did not yied any heuristics wh stronger overa performance.) When returns variance is very high, the performance of Heuristic A deteriorates somewhat because does not make use of variance information. The average reative error for that heuristic in the high-variance case is 6.23% when Item 2 is recovered (wh a worst-case error of 14.57%) and 6.78% when Item 3 is recovered (wh a worstcase error of 13.92%). The advantage that Heuristic B gains by making use of that information when variance is high outweighs the distortions associated wh that heuristic, and aows to outperform Heuristic A. The average reative error for Heuristic B in the high-variance case is 0.60% when Item 2 is recovered and 1.95% when Item 3 is recovered. Interestingy, the performance of the naïve poicy appears to improve as the variance of returns increases. By ignoring returns, this poicy uses higher base-stock eves than are optima when returns are considered. Because greater variabiy makes higher base-stock eves attractive (when un backorder costs are arger than hoding costs, as is the case here), the naïve poicy parameters are not as far off when returns variance is high. Because returns variance does not affect the other three cost components, higher variance aways reduces the attractiveness of product recovery in terms of tota system costs. At some point, the higher hoding/shortage costs may outweigh any net savings associated wh procurement/assemby, pipeine, and recovery costs. For returns variances up to that threshod eve, recovery is attractive, but at higher variance eves woud actuay be better not to recover the product at a. This phenomenon is iustrated in Figure 7, which is based on the demand/returns distributions used in Figure 6 combined wh the cost parameters used in Figure 5b. Figure 7 suggests that, for these parameters and coefficients of variation beow around 1.25, product recovery is attractive, whereas this is not the case for more variabe returns distributions. (Note that if r = 0 7 c 3 = 21 as in Figure 5a, product recovery is attractive at a variance eves considered.) Finay, as stated in 4, the heuristic poicies can aso be appied to systems wh stochastic recovery patterns. To test the heuristics in such a setting, we modified our mode so that R j uns of em j, j = 2 3, are recovered each period, wh R j assumed to be identicay distributed and independent. (The atter assumption contrasts wh our origina mode, where the fixed recovery pattern impies that the quanties of different ems recovered are perfecty Figure 7 Tota system cost 385 384 383 382 Impact of Return Variabiy on Tota System Costs: High Un Recovery Cost 381 No returns Recover 3:Optima 380 Recover 3: Heuristic Recover 3:Naive 379 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Coefficient of variation
Management Science 51(8), pp. 1250 1265, 2005 INFORMS 1263 correated. Most rea systems probaby ie somewhere in-between.) We constructed eight test probems by combining a four hoding-cost variations wh demand/returns distribution Cases 1 and 2 in Tabe 1. The higher overa return rate and addiona variabiy in the recovery pattern increased hoding/backorder costs under the optima and heuristic poicies. The hoding/backorder costs increased by an average of 18.2% for the optima poicy, 22.4% for Heuristic A, and 18.7% for Heuristic B. As a resut, the average reative error for Heuristic A was 3.56% (compared to 0.50% for fixed recovery pattern) and for Heuristic B was 1.8% (compared to 1.6% for fixed recovery pattern). Athough the addiona variabiy caused the reative errors to grow, the heuristics sti performed reasonaby we even for the extreme case of independent recovery. The naïve poicy resuted in an average reative error of 8.65%. Figure 8 Exampe of a Two-Tier Assemby System 1 = 1, 2 = 1, 3 = 2, 4 = 3, 5 = 4, 6 = 5, 7 = 6 2 3 4 5 6 7 1 5.2. System Behavior For probems wh more than three ems, the computationa demands of dynamic programming wh arge state spaces make intractabe to compute the optima poicy. To gain some insights into the impact of different component recovery patterns and system structures for arger systems, we performed a numerica study exporing the effects of these factors under the combined heuristic poicy. (It is interesting to note that, whie Heuristic A contributed the ower cost in 57% of the cases studied, neher heuristic dominated. Heuristic B tended to yied better performance in the three-tier cases described beow, whie tended to perform significanty worse in two-tier cases wh recovery of a arge number of ems. This is not surprising Heuristic B yieds ess distortion of the system in the former cases, and more in the atter.) To provide some estimate of the effectiveness of the heuristic poicy in each setting, we aso computed two benchmark cost measures. The first is the optima hoding/backorder cost for each system when there are no returns. The second is the expected hoding/backorder cost for the naïve poicy. (For both the heuristic and the naïve poicy, we estimated average costs by simuating the poicies for 2,000,000 periods, after an inia burn in of 200,000 periods.) A probems in this tria consisted of seven ems, wh em i having tota ead time M i = i. We considered two different system structures: a two-tier system, as shown in Figure 8, and a three-tier system, as shown in Figure 1. A probems had the demand/returns distribution of Case 4 in Tabe 1. We considered four hoding/shortage-cost scenarios by combining h i = 1 for a i and h i = 4 for a i wh p = 20 and p = 50. We investigated the foowing questions: (1) How do hoding/backorder costs behave as more ems are recovered? (2) How do hoding/backorder costs behave as higher-indexed ems are recovered? (3) How do hoding/backorder costs for a two-tier system compare to those for a three-tier system? To answer Question 1, we computed hoding/backorder costs for two-tier systems wh recovery of Items 2 through j for j = 2 3 7, and aso systems wh recovery of ems j through 7 for j = 7 6 2. Figure 9 shows the hoding/backorder costs (naïve and heuristic poicies) for both sequences of probems for the case of h i = 1 for a i and p = 20. (The resuts were quaativey simiar for the other cost scenarios.) As can be seen in Figure 9, recovering a arger number of ems causes hoding/backorder costs to increase for both poicies. However, recovering more ems increased both the absoute and reative cost advantage of the heuristic poicy. Indeed, in one case the heuristic saved 44%, so the heuristic can provide significant cost savings compared to a poicy that does not adjust for em recovery. Figure 9 Hoding/Backorder cost 100 75 50 Impact of Number of Items Recovered on Hoding/Backorder Costs Heuristic: 2 j Naïve:2 j 25 Heuristic: j 7 Naïve: j 7 No returns 0 1 2 3 4 5 6 Number of ems recovered
1264 Management Science 51(8), pp. 1250 1265, 2005 INFORMS Another way to measure heuristic performance is to compare hoding/backorder costs to those of a simiar system whout returns. As we have seen, in some cases the atter can be shown to provide a ower bound on the optima cost wh returns, but the reative gap (heuristic cost) (optima no-returns cost) (optima no-returns cost) can be arge. For three-em probems wh demand/ returns distribution Case 4 (which was used for a of the seven-em probems), the average gap was 24.9%, wh a range of 13.9% to 48.1%. Across a sevenem probems considered (incuding those described above, as we as those in the remainder of the trias described beow), the average gap was 28.3%, wh a range of 6.1% to 62.7%. This comparison represents ony an indirect measure of heuristic performance. However, does provide some evidence that athough performance may be somewhat weaker in arger systems, the combined heuristic may sti perform reasonaby we. To answer Question 2, we computed costs for probems where ems j, j + 1, and j + 2 were recovered, for j = 2 3 4 5. The consistent pattern that appeared was that hoding/backorder costs first increased, then decreased in j. However, the effect was que sma the difference between hoding/backorder costs of the highest- and owest-cost recovery patterns was aways smaer than 4.3%. Thus, appears that the number of ems recovered significanty affects hoding/backorder costs, but these costs are reativey insensive to which ems are recovered. To answer Question 3, we identified seven recovery patterns that are possibe in both two-tier and threetier systems. These patterns were J = 2 3 4 5 6 7, 2 4 5, 3 6 7, 4, 5, 6, and 7. For each pattern, we computed hoding/backorder costs under the combined heuristic poicy for the two- and three-tier systems. In most (but not a) cases, the costs were ower in the three-tier system. However, the cost differences were sma ranging from 6.7% ower in the three-tier system to 1.8% higher in the three-tier system which suggests that the system structure does not have a strong impact on the hoding/backorder costs. Finay, the theoretica resuts in 3 invove restricting attention to nonanticipatory poicies. This raises the question of how costy such a restriction is. The answer depends on the structure of the assemby system and the recovery pattern. For exampe, consider the three-em probem in Figure 2, and restrict attention to recovery patterns satisfying ongest-eadtime recovery. If J = 1 2 3 or J = 2 3, Coroary 1 impies that the restriction to nonanticipatory poicies has no cost. If J = 3, however, the nonanticipatory restriction increases hoding/backorder costs by a te over 3%. Now consider the two-tier, seven-em probem in Figure 8 wh J = 7. Across the four cost-parameter scenarios, the hoding/backorder cost of the best nonanticipatory poicy ranged from 9.7% to 15% higher than the cost of the combined heuristic poicy and so at east that much higher than the optima cost. The key difference appears to be the number of periods of anticipation prohibed. In the seven-em probem, Item 6 woud ike to anticipate P 67 = 5 periods of returns. In the three-em probem, Item 2 woud ike to anticipate ony P 23 = 1 period of returns so there is ess of a restriction in this case. 6. Concusions In this paper, we studied an assemby system experiencing uncertain returns/recovery of end products, components, or subassembies as we as uncertain customer demands. We showed that returns may disrupt the property of ong-run baance by directy increasing inventory of an em above that of a higherindexed em, or by inducing anticipatory orders. We identified condions on the em-recovery pattern and restrictions on the inventory poicy under which ongrun baance is preserved, so that the system can be soved using known techniques for series systems wh returns. For the specia case where end products (or a ems used to assembe the end product) are recovered, we showed that the system is equivaent to a series system whout any restrictions on the inventory poicy. For genera assemby systems, we proposed two heuristic poicies. The heuristics are easy to compute and practica to impement, and in numerica trias they were shown to perform we. We aso performed numerica trias using the heuristics (and, when possibe, the optima poicy) to obtain insights into the impact of various factors on system performance. We found that hoding and backorder costs tend to increase when the average return rate, the variabiy of returns, or the number of components recovered increases. 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