E±cient and Robust Design for Transshipment Networks

Transcription

1 E±cient and Robust Design for Transshipment Networks Transshipment, the sharing of inventory among parties at the same echelon level, can be used to reduce costs in a supply chain. The e ectiveness of transshipment is in part determined by the con guration of the transshipment network. We introduce chain con gurations in transshipment settings and show their superiority over grouping con gurations that are suggested in the literature. Further, we extend our research to general con gurations and develop properties of low cost and robust transshipment networks. In addition, we provide managerial insights regarding preferred con gurations based on problem parameters.. Introduction According to the th Annual State of Logistics Report [], logistic costs in the United States are rising, from $90 billion in 00 to $9 billion in 00. Inventory costs account for a third of this total. The report identi es \the ability to respond faster to changing customer needs" and \the exibility to adjust manufacturing and delivery cycles" as keys to success in this competitive environment. Uncertainty in customer demand and operating costs can lead to major supply chain ine±ciencies, causing lost revenue, poor customer service, high inventory levels and unrealized pro ts. Retailer A Retailer B Bidirectional transshipment link Warehouse Replenishment link Retailer C Retailer D Figure : Supply Chain With Transshipments Inventory transshipment is a promising strategy to provide operational exibility to mitigate the e ect of demand uncertainty. Transshipment is the sharing of inventory among locations at the same echelon level of a supply chain. In Figure, four retailers are supplied from one warehouse. Rather than relying solely on their own inventory or costly emergency replenishment from the warehouse, retailers can collaborate to address demand uncertainty.

2 Transshipment achieves the bene ts of risk pooling to meet uncertain demand while maintaining low inventory levels at individual locations. The use of transshipment has been facilitated by advances in information technology and information sharing. Companies using transshipments include Footlocker, Macy's, and a group of chip manufacturers (NEC, Toshiba) sharing a common supplier, ASML. Recent papers in transshipment models have considered restrictions on transshipment network connectivity since it is not always possible to transship directly among all locations; see [8] and []. Establishing a link between locations requires investments in bidirectional communication channels that enable information sharing, physical distribution systems and nancial and administrative arrangements. Alternatively, transshipment networks without direct links between all locations tend to consolidate transshipment ows on a few routes, which can reduce the demand for communication channels and transportation (vehicles and drivers) and lower the overall complexity of a system. This is important in the case of outsourced transportation between locations that is negotiated in advance or in the case of multiple products which share common transshipment methods. This paper considers the e±ciency and robustness of a range of transshipment network con gurations. E±cient networks achieve low expected inventory and transshipment costs. Robust networks maintain e±ciency even with changes to cost or demand parameters. We present analytical results to show that the chain con guration is more e±cient than the group con guration that appears in the literature. Using a numerical study, we analyze the characteristics of con gurations under a range of cost and demand parameters, and provide insights on the parameters that a ect the con guration choice. Section reviews related literature. Section describes operational and strategic transshipment problems. Section presents comparisons of basic con gurations. Section discusses research extensions.. Literature review The transshipment literature has focused on operational decisions for a xed network design, such as the transshipment amount between locations and the order amount at each location. Most work considers two locations ([], [8] and [9]), or locations that are identical in cost parameters ([] and [8]). [] and [7] consider locations that are non-identical in demand and cost parameters. [7] and [9] allow replenishment lead times larger than one; in the others, transshipment lead times are negligible and replenishment lead times are one period.

3 The literature on transshipment network design is more recent. In addition to the complete pooling network in which any location may transship to any other location, [8] and [] consider group con gurations, in which locations are divided into groups and transshipment is allowed only within groups. It is shown that while group con gurations cannot achieve all of the savings of complete pooling, the savings are considerable, and the number of links is smaller. [7] compare ve transshipment con gurations, di ering in the number and cost of links. The paper quanti es the value of transshipments, but does not analyze network con gurations. [] consider a transshipment network with one supplier and three retailers with six network design options which they refer to as operational exibility levels. They nd the retailers' optimal order quantities for any given exibility level with the newsvendor network model of [0] and analyze the interaction between optimizing order quantities and increasing operational exibility. In this paper we consider con gurations with one supplier and N retailers, determining optimal order quantities with the method of [7]. While the transshipment literature considers only a limited number of network con gurations, other network con gurations have been considered in the manufacturing and service operations literature. The chain con guration has been shown to be an e±cient structure in supply chains and labor cross-training. [] and [] study the chain structure in supply chains. They show that by assigning the capacity of factories to products according to a chain structure in which each factory only produces two types of products, as in Figure, one can achieve most of the potential bene t of complete pooling in which all factories are able to produce all products. [8], [], [], [9], [0], [] and [] highlight the properties of the chain structure in di erent production and repair/maintenance environments. FACTORIES PRODUCTS Retailer Retailer S D S D FACTORY WAREHOUSE S D S D Retailer Retailer Figure : (left) Chaining in a manufacturing setting; (right) Chaining in a transshipment setting Our research addresses the gap between the transshipment literature and network design

4 by studying a wide range of network con gurations, including the chain con guration. Like [], we investigate the chain analytically.. Transshipment problems Section. reviews the operational transshipment problem from the literature and Section. introduces the strategic transshipment network design problem studied in this paper.. Existing work on operational transshipment problems The objective of the operational transshipment problem is to minimize inventory and transshipment costs per period for a given network design with known demand and cost parameters. Given N retailers, facing independent, stationary demand, events occur as follows:. Replenishment from the warehouse arrives from orders made in the previous period; backlogged demand is satis ed. The inventory level is equal to the base-stock level.. Demand is observed.. Transshipment decisions are made and occur immediately. Transshipment costs are incurred.. Demand is satis ed or backlogged. Holding and shortage costs are incurred.. Inventory level is updated.. Replenishment orders are made. Thereplenishmentleadtimefromthewarehouseisonetimeperiod,anditisassumed that the warehouse has su±cient capacity to respond to all orders. Transshipments serve as a quicker source of supply if demand exceeds available inventory. Retailers follow a base-stock policy which [7] prove minimizes costs in a transshipment setting. At each location, basestock level and periodic transshipment decisions are made to minimize the total long-run expected costs, which is the sum of inventory holding, shortage and transshipment costs of all locations. With a base-stock policy, the system regenerates itself every period; minimizing the expected cost for one period is equivalent to minimizing the long-run expected costs. In the literature, locations are often assumed to be identical, as in this paper. Identical locations incur the same costs and observe demand from an identical distribution, as in cases of retailers serving near-homogenous populations over moderately sized geographic regions. Further, analysis of nonidentical locations is case-dependent and does not provide general

5 insights, see Section. Since variable replenishment costs are the same across locations and unmet demands are backlogged and eventually satis ed, these costs are ignored. Similar to [], [8], [7] and other works, xed costs of orders and transshipments are assumed to be incurred every period or negligible when compared with other costs. The operational transshipment problem uses the following parameters: N set of retail locations (also called \nodes"), i f::n g K set of directed transshipment links (i; j) K, de ned by con guration, Kµ(N N) d i observed demand at location i N per period c t cost of transshipping one unit along one link c s cost of one unit of shortage for one period cost of holding one unit in inventory for one period. c h We de ne decision variables S i and X ij, and auxiliary variables I + i and I i as follows: S i X ij I + i base-stock level at location i N number of items transshipped on link (i; j) K net surplus at end of time period (after transshipment) at location i N I i netshortageatendoftimeperiod(aftertransshipment)atlocationin. Given a base-stock level vector S, the following linear program presented by [7] is solved to determine transshipment ows for each period of observed demand. X min z(k; S) =c t I i I i + (i;j)k in in X ij + c s X + c h X subject to X X ij X X ji + I i + I i = S i d i 8i N (b) j:(i;j)k j:(j;i)k (a) X X ij S i 8i N (c) j:(i;j)k X ij 0 8(i; j) K (d) I + i ;I i 0 8i N (e) The objective function (a) minimizes z(k; S), the sum of transshipment costs, shortage costs and holding costs, given K, S and the demand realization. Constraints (b) state that demand must be satis ed by inventory and/or transshipments, or backlogged. Constraints (c) state that retailers cannot transship more than the base-stock level. Node i N may both receive units from location j and transship units to location j for some j ;j such that (j ;i) Kand (i; j ) K, up to its base-stock level; i.e., if j has a surplus and j has a shortage. Constraints (d) and (e) are non-negativity constraints. We use the in nitesimal

6 perturbation analysis (IPA) procedure from [7] to nd optimal base-stock levels and the optimal expected cost for a given con guration. The objective (a) is linear in holding, shortage and transshipment costs. A transshipment involving one or more links deducts one unit of surplus at a node to ful ll one unit of shortage at another. This action incurs transshipment costs to avoid one unit of holding cost and one unit of shortage cost. We de ne a shift to be a transshipment along a single link. The number of pro table shifts is the number of links along which units are transferred to meet shortage and still reduce costs. The number of pro table shifts, J, is the oor of the ratio between the sum of holding and shortage costs and transshipment cost; J = b c h+c s c t c. Our modeling allows for multiple shifts, yet unless the transshipment cost is low and demand uncertainty is high, results indicate that more than two shifts are rare. In such cases, con gurations which depend on multiple shifts to balance shortages and surpluses are less e±cient than those in which locations are directly linked to one node that acts as a warehouse. Formulation () with c s =andc h = c t = 0 is similar to the manufacturing model in []. Retailers satisfy demand at other retailers by shifting capacity (i.e., inventory) along multiple links with no penalty. Transshipment and inventory costs in the transshipment setting limit this exibility. Capacity in manufacturing models in [] is independent of network design; in transshipment, retailers adjust base-stock levels to minimize system costs.. The strategic transshipment network design problem The strategic transshipment network design problem determines the optimal network con- guration (i.e., the link set K) given a limit, P, on the number of transshipment links. To compare the e±ciency of transshipment networks, we introduce Z(K; S) todenotetheop- timal expected cost (over demand realizations) of a network with transshipment link set K and base-stock level vector S. The transshipment network design problem is formulated as: min Z(K; S) (a) subject to jkj P Kµ(N N) (b) (c) The objective function (a) minimizes the expected cost. Constraint (b) limits the size of the link set, and constraint (c) de nes the possible choices of links between nodes. E±cient

7 networks are those that have low values of Z(K; S) when compared to other networks. Robust networks are those that, after being selected over other networks under one set of cost and demand parameters, still perform better given deviations from the initial parameters.. Network con gurations Figure shows the group con gurations from the literature ((a) and (b)) and chain con gurations ((c) and (d)). Let group(l) denote a group con guration of M groups of l nodes and l(l ) directed links each, for any positive integer M such that lm = N. Group()in Figure (a) and the unidirectional chain in Figure (c) have links while group() in Figure (b) and the bidirectional chain in Figure (d) have links or bidirectional links. (a) Group() (b) Group() (c) Unidirectional chain (d) Bidirectional chain Figure : Basic network con gurations with N = locations Section. presents analytical comparisons of chain and group con gurations, and Section. presents numerical comparisons of the chain con guration with a range of con gurations.. Analytical results for chain and group con gurations We show that the group con guration incurs higher expected costs than the chain con guration under the identical location assumption. The unidirectional chain outperforms group(), both with N = M directed links, and the bidirectional chain outperforms group(), both with N = M bidirectional links. Proofs for the lemmas (needed to prove the theorems) and Theorems and are provided in the online appendix. Lemma Given a network with identical nodes, a vector of base-stock levels, S, cost parameters that allow for only one pro table shift, and a xed con guration, then: minimizing the expected number of units in inventory, minimizing the expected number of units in shortage or maximizing the number of units transshipped will yield the minimum total expected cost. 7

8 Lemma Given an identical node network with nodes facing independent demand, linked for transshipments either as a chain or in a group(l) con guration, the optimal base-stock levels are identical across locations in a con guration. Lemma Consider three identical nodes linked in the following manner: where the nodes face independent demand, their base-stock levels are identical, and where cost parameters allow for only one pro table shift. The expected ending inventory/shortage after transshipments at nodes and are positively correlated. Theorem Given an identical node network with N =M locations (M =; ; ), the optimal expected cost in a unidirectional chain con guration is lower than or equal to the optimal expected cost in a group() con guration. Theorem states that the bidirectional chain outperforms group() in expectation for networks with six nodes. Theorem extends the result to networks with multiples of three nodes. Theorem Given a network with six identical locations facing independent demand, the optimal expected cost in a bidirectional chain con guration is lower than or equal to the optimal expected cost in a group() con guration. Proof: From Lemma, the optimal base-stock levels at all locations within each con- guration are identical. We prove that for the same base-stock level at each location, S, the bidirectional chain outperforms group() for one pro table shift. With more than one pro table shift, the chain con guration can only improve while the group() con guration sees no bene t. Recall that K is the set of transshipment links in formulation (). Let Z(K c ) and Z(K g ) denote the optimal expected inventory, shortage and transshipment costs for a network with chain links, K c, and group() links, K g, respectively. The argument S is omitted from Z(K; S) since the base-stock levels are the same. We prove that Z(K c ) Z(K g ), according to the outline in Figure. 8

9 Step : Optimize flow on dark links; no flow on dashed links Step : Optimize flow on dark links; fixed flow on gray links obtained from Step Step : Compare solution with optimal flow on all arcs Figure : Outline of proof of Theorem Step : Remove two links from each network, such that the same links remain in both networks. Let K 0 = f(; ); (; ); (; ); (; )g be the set of remaining links. Given a demand instance, the transshipment quantities are found with formulation (). The solutions for both networks are the same since the link set is K 0 for both. The optimal ows on these networks, assuming no ow on the removed links, is denoted X 0 with cost Z(K 0 ). Step : Add back the removed links. The resulting link sets are K c = f(; ); (; ); (; ); (; ); (; ); (; )g and K g = f(; ); (; ); (; ); (; ); (; ); (; )g, for chain and group(), respectively. Fix the ows on the links in K 0 at the levels of X 0, allow ows on the reintroduced links to lower costs. Denote the costs after Step as Z(K c jx 0 )andz(k g jx 0 ). From Lemma, the expected net inventory at the corner nodes (i.e., nodes and, and nodes and ) are positively correlated. The expected net inventory at the endpoints of connecting links in the chain network (i.e., nodes and, and nodes and ) are independent. There will be less transshipment ow on the new links in the group() network than on the new links in the chain network. Since only one shift is allowed, maximizing transshipments will minimize cost, from Lemma. Hence, Z(K c jx 0 ) Z(K g jx 0 ). Step : Attempt to improve the solution of each con guration by changing the link ows. We show that the solution of group() cannot be improved while the solution to the chain may be improved. Consider six exhaustive cases for nodes,, and in group().. All nodes are short. No transshipments are possible.. No nodes are short. No transshipments are needed.. Only one corner node (e.g. node ) is short. InStep,themiddlenodeshipstonode. If node is still short, then node ships in Step. Since holding and transshipment costs are identical at each node, redistributing ows cannot lower the total cost. 9

10 . Only the middle node is short. Corner nodes do not ship to each other; the ows after Step are optimal.. Corner node (e.g. node ) is not short and other nodes are short. In Step, node ships to the middle node, and, if possible, ships to node in Step. Again since shortage and transshipment costs are identical, the ows after Step are optimal.. Middle node is not short and all other nodes are short. Since both corner nodes are short, the ows after Step are optimal. In all six cases, changing transshipment ows among nodes, and after Step cannot lower the total cost. The same holds for nodes,, and, thus Z(K g jx 0 )=Z(K g ). For the chain, we show by example that Z(K c ) Z(K c jx 0 ). Consider inventory before transshipment= (-0,,-, 0, 0, 0). An optimal solution in Step is x =and0otherwise. In Step, x = and 0 otherwise. The total inventory of this solution is, the shortage is, and the number of transshipments is 0. However, the optimal solution, x =and x = 0, results in no inventory or shortage with transshipments. From Steps and, Z(K c ) Z(K c jx 0 ) Z(K g jx 0 )=Z(K g ). As a result, Z(K c ) Z(K g ) for one pro table shift. With more than one pro table shift, the chain con guration bene ts while the group con guration does not. Therefore, Z(K c ) Z(K g ). Theorem Given a network with N =M identical locations (M =; ; ) facing independent demand, the optimal expected cost in a bidirectional chain con guration is lower than or equal to the optimal expected cost in a group() con guration.. Numerical results for chain and general con gurations We extend the previous results by comparing the bidirectional chain con guration with con- gurations with the same number of bidirectional links, using network con gurations with N nodes and N transshipment links, where N =;, and 8. The numerical tests incorporate a range of parametric values. We x the sum of holding and shortage costs at with three cases for holding and shortage costs, denoted by =(c h ;c s ) f(; ); (; 9); (; 7)g, and consider six transshipment costs: c t =; ; ; 8; 0;, resulting in four pro table shifts: J =; ; ;. Demand at each node follows a Gamma distribution with mean of 00 and ve coe±cients of variation: = 0:; 0:; ; :;. For each scenario, we perform pairwise comparisons of the chain with the other con gurations. 0

11 Consider a pairwise comparison of the chaincon gurationwithcon gurationf. The optimal expected cost for each con guration is obtained with the IPA method using its optimal base-stock level vector. Let Z(K c js c ) be the optimal expected cost of the chain and Z(K f js f ) be the optimal expected cost of con guration f. Let ( ;c t ; )denotethe percent of pairwise comparisons in which the chain has a lower expected cost than the other con guration with scenario values, c t,and. Let ( ;c t ; ) denote the deviation of the expected cost of the chain from the expected cost of the lower cost con guration in pairwise comparisons in which the chain is not the lower cost con guration: ( ;c t ; )= Z(K cjs c ) Z(K f js f ) 00%: Z(K f js f ) Let ( ;c ¹ t ; ) be the average deviation of all pairwise comparisons in which the chain does not yield the lower cost and ( ;c t ; ) be the maximum deviation of these comparisons. We compare the relative magnitude of ( ;c t ; )with ( ;c t ; ), the di erence between the highest and lowest optimal expected costs for all con gurations for scenario values, c t,and. We denote the set of all con gurations, including the chain, as F,andcalculate ( ;c t ; )asfollows: µ maxff fz(k f js f )g min ff fz(k f js f ) ( ;c t ; )= 00%: min ff fz(k f js f )g For networks with N =, there are 0 possible con gurations (other than chain), shown in the Appendix. Table presents the results as a function of coe±cient of variation, the factor that was observed to have the most signi cant impact on the results. The table presents the number of total comparisons(#) at each value of and the value of the e±ciency metrics for a xed : ( ; ; ), ( ; ; ¹ ), ( ; ; ), and ( ; ; ). We use,, ¹,and when the values of, c t,and are constant, or when these values are unambiguous. # ( ; ; ) ( ; ; ¹ ) ( ; ; ) ( ; ; ) % % % % 0 00% - -.% %.%.% 7.% %.% 7.% 0.% Total %. % 7.% 8.% Table : E±ciency of the chain network for N =asafunctionof The chain is more e±cient in 97.7% of all comparisons. The chain is the most e±cient con guration for values of, implying that chain remains e±cient regardless of the cost

12 parameters for low to moderate levels of demand uncertainty. The chain is still highly e±cient at =. (high value of and low values of and ¹ ). At =, other con gurations maybemoree±cientthanthechain,althoughthechainismoree±cientin90%ofthe comparisons. The maximum deviation between the chain and more e±cient con gurations is 7.%, which is small relative to the maximum cost spread of 0%. Values of decrease as increases, suggesting that when demand is highly uncertain, the con guration is less important since the high costs to accommodate this uncertainty are unavoidable. In Tables and, we explore the e±ciency of the chain with respect to cost parameters for = (i.e., the fth row of Table ). Table presents results as a function of holding and shortage costs, and Table presents results as a function of transshipment costs. c h c s # (; ; ) (; ¹ ; ) (; ; ) (; ; ) 0 9.0%.%.8% 0.% %.9% 7.%.9% %.% 7.0% 8.7% Total %.% 7.% 0.% Table : E±ciency of the chain network for N =for = asafunctionofc h and c s Table suggests that while the e±ciency of the chain con guration in terms of decreases with higher holding costs and lower shortage costs, the deviation between the chain and other con gurations does not change signi cantly. The maximum deviation among all con gurations decreases with higher holding costs and lower shortage costs. c t # (;c t ; ) (;ct ¹ ; ) (;c t ; ) (;c t ; ) 0 00% % % - -.% % - -.% 0 98.% 0.0% 0.0% 7. % 0 78.% 0.7%.%. % 0.%.% 7.% 0. % Total %.% 7.% 0.% Table : E±ciency of the chain network for N =and = asafunctionofc t Table suggests that the cases in which the chain is less e±cient correspond to scenarios with high demand variability and low transshipment costs (i.e., =andc t =and). When demand variability is high and transshipment costs are low relative to holding and shortage costs, con gurations that centralize inventory to maximize pooling are desirable.

13 In these cases, the most e±cient con guration is Con guration in the Appendix, which we refer to as the star network (an extra link exists compared to a typical star since all con gurations must include links). The star network stores additional inventory at one centralized node and transships items to other nodes as needed. This results in a lower cost than the chain since centralized inventory pooling reduces the relatively high holding costs with highly variable demand. We make the following observations. Observation When demand uncertainty is low or transshipment costs are high relative to holding and shortage costs, multiple shifts are not signi cantly bene cial in all con gurations. Under these conditions, balanced base-stock levels among all locations are desirable and the chain is likely to be the most e±cient con guration for N=. Observation When demand uncertainty is high and transshipment costs are low relative to holding and shortage costs, con gurations that utilize multiple shifts are not the most e±cient. Rather, these conditions promote centralized risk pooling and the star con guration is likely to be the most e±cient con guration for N=. We con rm Observations and in networks with N =and8. Sincethenumber of possible unique con gurations for and 8 location/link scenarios are much greater, we consider randomly generated unique networks in the pairwise comparisons with the chain and star con gurations. The networks and result tables are presented in the Online Appendix. The chain is more e±cient in over 97% of the comparisons with the randomly generated con gurations. For, the chain is most e±cient when compared to the randomly generated con gurations. When the chain is inferior, the average value of isless ¹ than % with a maximum deviation below 7%. When compared to the star con guration, the chain is more e±cient in all cases with 0:. Consistent with Observation, the star performs signi cantly better than the chain when :andc t is small relative to c s +c h.in pairwise comparisons of the star with the randomly generated networks at :andc t, the star con guration is the most e±cient. In conclusion there are many scenarios in which the chain is the most e±cient con guration; however, in cases in which centralized inventory is desirable (high demand uncertainty and low transshipment cost relative to shortage and holding costs), the star con guration is more e±cient. Based on the above analysis, the chain is a robust, e±cient con guration. We present the three key desirable properties for an e±cient and robust transshipment network. While we have identi ed other desirable properties, they are all related to those mentioned below.

14 ² Appropriate pool size. To achieve the bene ts of risk pooling, nodes should have access to the inventory of other nodes. The number of nodes that should be accessible (i.e., the pool size) is impacted by demand variation. As variation increases, the pool size should increase. ² Short Transshipment Paths. As expected, reducing the path length (number of shifts) between locations in a transshipment pool can lower transshipment costs. Note that only paths of length less than the number of pro table shifts are included in the pool. ² Balanced Node Degree. The degree of a node is equal the number of incident links. Networks with balanced node degrees have well-distributed transshipment links throughout the network in identical networks. Balancing links among all nodes leads to low inventory levels at all nodes.. Discussion Our work is a rst step in transshipment network design research. Several of the initial assumptions may be relaxed in future research. In particular, it is important to investigate the transshipment network design problem in which the unit transshipment cost between di erent locations depends on the speci c pair of locations (non-homogeneous transshipment costs). Such transshipment cost structure can represent the distance between the locations, or other location-pair characteristics. Unfortunately, the above extension complicates the analysis quite considerably. A chain con guration with non-homogeneous transshipment costs must be further de ned by the sequence in which the locations are ordered within the chain, and there are (n-)! possible sequences. Each sequence is characterized by di erent total transshipment costs along the chain, and nding the sequence with minimal total transshipment costs is equivalent to the well known TSP problem. However, ordering the locations according to the optimal TSP sequence does not necessarily minimize total inventory and transshipment costs, and therefore, may not provide the best chain sequence. The locations will di er in their accessibility due to di erent (transshipment) costs of connected links and varying base stock levels. With the grouping con guration and non-homogeneous transshipment costs, similar and additional issues arise. While equal group sizes are preferable for the case of homogeneous costs, this may not necessarily be the case with non-homogeneous costs. In the latter case, the group size may be signi cantly a ected by the closeness of a group of locations to other

15 locations, as expressed in terms of the transshipment costs. Thus, the group size is yet another factor to be determined. It appears that with non-homogeneous transshipment costs the best con guration depends heavily on the speci c cost parameters. Thus we believe that our analytical results cannot be extended to this more general case. Another important and interesting case is when the demand parameters di er among locations. This again will lead to unequal base stock levels a ecting the required or available stock going into or out of each location, thus complicating again the desired con guration problem. Future research can build on the framework and basic fundamental analysis in this paper to study particular cases of these more general settings. Acknowledgments This research has been supported by grants DMI-08 and DMI-008 from the National Science Foundation. References [] Aksin,O.Z.andF.Karaesmen.00.Characterizing the Performance of Process Flexibility Structures. Working paper. [] Chou, M., C. Teo and H. Zheng. 00. Process exibility revisited: graph expander and food-from-the-heart. Department of Decision Sciences, National University of Singapore, Singapore. [] Graves, S.C. and B.T. Tomlin. 00. Process exibility in supply chains. Management Science [] Gurumurthi S. and S. Benjaafar. 00. Modeling and analysis of exible queueing systems. Naval Research Logistics 7{78. [] Herer, Y. T. and A. Rashit Lateral stock transshipments in a two-location inventory system with xed and joint replenishment Costs. Naval Research Logistics {7.

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17 [8] Tagaras, G Pooling in multi-location periodic inventory distribution systems. Omega 7 9{9. [9] Tagaras, G. and D. Vlachos. 00. E ectiveness of stock transshipment under various demand distributions and nonnegligible transshipment times. Production and Operations Management 8{98. [0] Van Mieghem, J. A. and N. Rudi. 00. Newsvendor networks: Inventory management and capacity investment with discretionary activities. Manufacturing & Service Operations Management {. [] Yu, D.Z., S.Y. Tang, H. Shen and J. Niederho. 00. On bene ts of operational exibility in a distribution network with transshipment. John M. Olin School of Business, Washington University, St. Louis, Missouri. [] Council of Logistics Management. 00. th Annual State of Logistics Report. Appendix Configuration Configuration Configuration Configuration Configuration Configuration Configuration 7 Configuration 8 Configuration 9 Configuration 0 Configuration Configuration Configuration Configuration Configuration Configuration Configuration 7 Configuration 8 Configuration 9 Configuration 0 Configuration Figure A.: All node - link network con gurations 7