# Optimizing Strategic Safety Stock Placement in Supply Chains

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Optimizing Strategic Safety Stock Placement in Supply Chains Stephen C. Graves • Sean P. Willems Leaders for Manufacturing Program and A. P. Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, sgraves@mit.edu College of Business Administration, University of Cincinnati, Cincinnati, Ohio 45221-0130 M anufacturing managers face increasing pressure to reduce inventories across the supply chain. However, in complex supply chains, it is not always obvious where to hold safety stock to minimize inventory costs and provide a high level of service to the ﬁnal customer. In this paper we develop a framework for modeling strategic safety stock in a supply chain that is subject to demand or forecast uncertainty. Key assumptions are that we can model the supply chain as a network, that each stage in the supply chain operates with a periodic-review base-stock policy, that demand is bounded, and that there is a guaranteed service time between every stage and its customers. We develop an optimization algorithm for the placement of strategic safety stock for supply chains that can be modeled as spanning trees. Our assump- tions allow us to capture the stochastic nature of the problem and formulate it as a determin- istic optimization. As a partial validation of the model, we describe its successful application by product ﬂow teams at Eastman Kodak. We discuss how these ﬂow teams have used the model to reduce ﬁnished goods inventory, target cycle time reduction efforts, and determine component inventories. We conclude with a list of needs to enhance the utility of the model. (Base-Stock Policy; Dynamic Programming Application; Multi-echelon Inventory System; Multi-Stage Supply-Chain Application; Safety Stock Optimization) 1. Introduction rewards. Using a supply chain as a focusing mecha- nism challenges an organization to examine cross- Manufacturing ﬁrms are subject to pressure to do ev- functional solutions to address some of the barriers erything faster, cheaper, and better. Firms are expected that inhibit improvements. to continue to improve customer service by increasing The primary intent of this research is to develop a on-time deliveries and reducing delivery lead-times. At the same time, they must provide this service more tactical tool to help cross-functional teams in their ef- cheaply and utilize fewer assets. Increasingly, ﬁrms forts to model and improve a supply chain. We pro- need to do this for a global marketplace. vide a framework for modeling a supply chain and This pressure to improve forces companies to look develop an approach, within the framework, to opti- at their operations from a supply-chain perspective mize the inventory in a supply chain. More speciﬁ- and to seek improvements from better coordination cally, we provide an optimization algorithm for ﬁnd- and communication across the supply chain. A supply- ing the optimal placement of safety stock in a supply chain perspective is essential to avoid some of the local chain, modeled as a spanning tree and subject to un- suboptimization that occurs when each step in a pro- certain demand. Key assumptions for the optimization cess operates independently with its own metrics and are that each stage of the supply chain operates with a 1523-4614/00/0201/0068$05.00 Manufacturing & Service Operations Management 䉷2000 INFORMS 1526-5498 electronic ISSN Vol. 2, No. 1, Winter 2000, pp. 68–83 68

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains periodic-review, base-stock policy, that each stage assumptions about the demand process and about the quotes a guaranteed service time to its customers, and internal control policies for the supply chain. Our work that demand is bounded. is also closely related to that of Inderfurth (1991, 1993), We refer to this effort as the placement of “strategic” Inderfurth and Minner (1998), and Minner (1997), who safety stock. As will be seen, the optimization model also build off Simpson’s framework for optimizing leads to the determination of where to place decou- safety stocks in a supply chain. We extend the work of pling inventories that protect one part of the supply Simpson and of Inderfurth and Minner by treating a chain from another. In particular, a decoupling safety more general network, namely spanning trees. We also stock is an inventory large enough to permit the down- provide a different, and we believe richer, interpreta- stream portion of the supply chain to operate indepen- tion of the framework and its applicability to practice. dently from the upstream, provided that the upstream We provide new results in the appendix on the form portion replenishes the external demand. In this sense, of the optimal policies when we relax a constraint on the determination of where to place these decoupling the internal control policy for the supply chain. points in a supply chain is a major design decision and Second, our work is closely related in intent to Lee is “strategic” in nature. Furthermore, this terminology and Billington (1993), Glasserman and Tayur (1995), is consistent with that used in industry. and Ettl et al. (2000). Each of these papers examines the In order to have an opportunity to test the research determination of the optimal base-stock levels in a sup- and validate its utility for industry, we have built a ply chain, and tries to do so in a way that is applicable commercial-quality software application to implement to practice. Glasserman and Tayur (1995) show how to the model described in this paper. The software can be use simulation and inﬁnitesimal perturbation analysis downloaded from our website, http://web.mit.edu/ to ﬁnd the optimal base-stock levels for capacitated lfmrg3/www/. multi-stage systems. Both Lee and Billington (1993) In the remainder of this section we brieﬂy discuss and Ettl et al. (2000) develop performance evaluation related literature. In §2, we present our framework for models of a multi-stage inventory system, where the modeling a supply chain by stating and discussing the key challenge is how to approximate the replenish- key assumptions. We introduce the model for a single ment lead-times within the supply chain. They then stage in §3; this serves as the building block for the formulate and solve a nonlinear optimization problem multi-stage model described in §4. In §5 we develop that minimizes the supply chain’s inventory costs sub- the optimization algorithm for safety stock placement ject to user-speciﬁed requirements on the customer ser- in a supply chain modeled as a spanning tree. We pres- vice level. Our work is similar in that we also assume ent an overview of our application experience with the base-stock policies and focus on minimizing the inven- model in §6, and conclude in §7 with thoughts on how tory requirements in a supply chain. The resulting to improve the tool. models and algorithms are much different, though, Related Literature. due to different assumptions about the demand pro- There is an extensive literature on inventory models cess and different constraints on service levels within for multi-stage or multi-echelon systems with uncer- the supply chain. tain demand; much of this literature is applicable to supply chains as now deﬁned. We refer the reader to 2. Assumptions the survey articles by Axsäter (1993), Federgruen (1993), Inderfurth (1994), van Houtum et al. (1996), and Multi-Stage Network. Diks et al. (1996). Within this vast literature, we men- We model a supply chain as a network where nodes tion two sets of papers that are most related to our are stages in the supply chain and arcs denote that an work. upstream stage supplies a downstream stage. A stage First, we note the work by Simpson (1958) who de- represents a major processing function in the supply termined optimal safety stocks for a supply chain mod- chain. A stage might represent the procurement of a eled as a serial network. Our work is based on similar raw material, or the production of a component, or the Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 69

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains manufacture of a subassembly, or the assembly and which we term demand nodes or stages. For each de- test of a ﬁnished good, or the transportation of a ﬁn- mand node j, we assume that the end-item demand ished product from a central distribution center to a comes from a stationary process for which the average regional warehouse. Each stage is a potential location demand per period is lj. for holding a safety-stock inventory of the item pro- An internal stage has only internal customers or suc- cessed at the stage. cessors; its demand at time t is the sum of the orders We associate with each arc a scalar ij to indicate placed by its immediate successors. Since each stage how many units of the upstream component i are re- orders according to a base-stock policy, the demand at quired per downstream unit j. If a stage is connected internal stage i is: to several upstream stages, then its production activity is an assembly requiring inputs from each of the up- di(t) ⳱ 兺 (i,j)僆A ij dj(t) stream stages. A stage that is connected to multiple downstream stages is either a distribution node or a where dj(t) denotes the realized demand at stage j in production activity that produces a common compo- period t and A is the arc set for the network represen- nent for multiple internal customers. tation of the supply chain. For every arc (i, j) 僆 A, stage j orders an amount ij dj(t) from upstream stage i in Production Lead-Times. time period t. The average demand rate for stage i is: For each stage, we assume a known deterministic pro- duction lead-time; call it Ti. When a stage reorders, the li ⳱ 兺 (i,j)僆A ij lj. production lead-time, is the time from when all of the inputs are available until production is completed and We assume that demand at each stage j is bounded available to serve demand. The production lead-time by the function Dj(s), for s ⳱ 1, 2, 3, . . . Mj, where Mj includes the waiting and processing time at the stage, is the maximum replenishment time for the stage.1 plus any transportation time to put the item into in- That is, for any period t and for s ⳱ 1, 2, 3, . . . Mj, we ventory. For instance, suppose stage k requires inputs have from stage i and j; then for a production request made at time t, stage k completes the production at time t Ⳮ Dj(s) ⱖ dj(t ⳮ s Ⳮ 1) Ⳮ dj(t ⳮ s Ⳮ 2) Ⳮ • • • Ⳮ dj(t). Tk, provided that there are adequate supplies of i and We deﬁne Dj(0) ⳱ 0 and assume that Dj(s) is increasing j at time t. and concave on s ⳱ 1, 2, 3, . . . Mj. We assume that the production lead-time is not im- pacted by the size of the order; hence, in effect, we Discussion of Assumption of Bounded Demand. assume that there are no capacity constraints that limit The assumption of bounded demand is contrary to production at a stage. most of the literature on stochastic-demand inventory models, and as such, is controversial. We need to dis- Periodic-Review Base-Stock Replenishment Policy. cuss this assumption in the context of the intent of the We assume that all stages operate with a periodic- research, namely to provide tactical guidance for review base-stock replenishment policy with a com- where to position safety stock in a supply chain. mon review period. For each period, each stage ob- We presume that it is possible to establish a mean- serves demand either from an external customer or ingful upper bound on demand over varying horizons from its downstream stages, and places orders on its for each end item. By meaningful, we mean in the con- suppliers to replenish the observed demand. There is no time delay in ordering; hence, in each period the text of setting safety stocks: the safety stock is set to ordering policy passes the external customer demand cover all demand realizations that fall within the upper back up the supply chain so that all stages see the cus- bounds. If demand exceeds the upper bounds, then the tomer demand. safety stock, by design, is not adequate. In such ex- traordinary cases, a manager resorts to other tactics to Demand Process. Without loss of generality, we assume that external de- The maximum replenishment time for node j is deﬁned as Mj ⳱ Tj 1 mand occurs only at nodes that have no successors, Ⳮ max {Mi | i:(i,j) 僆 A}. Manufacturing & Service Operations Management 70 Vol. 2, No. 1, Winter 2000, pp. 68–83

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains handle the excess demand. A manager might use ex- pediting, subcontracting, premium freight transporta- tion, and/or overtime to accommodate the windfall of Di(s) ⳱ sli Ⳮ p 冪兺 (i,j)僆A {ij(Dj(s)⬘ ⳮ slj)}p (2) demand. In specifying the demand bounds, a manager where p ⱖ 1 is a given constant. Larger values of p indicates explicitly a preference for how demand vari- correspond to more risk pooling. Setting p ⳱ 1 models ation should be handled—what range is covered by the case of no risk pooling. If we were to model the safety stock and what range is handled by other actions end-item demand bounds by Equation (1), then setting or responses. p ⳱ 2 equates to combining standard deviations of in- As an example, consider a typical assumption where dependent demand streams. demand for end item j is normally distributed each We do not attempt to model what happens when period and i.i.d., with mean l and standard deviation actual demand exceeds the bounds. When this hap- r. Then, for the purposes of positioning safety stock, a pens, we assume that the supply chain responds with manager might specify the demand bounds at the de- an equally extraordinary measure, as noted above. We mand node by: regard this as beyond the scope of the model, given Dj(s) ⳱ sl Ⳮ kr冪s (1) the stated intention to provide tactical decision sup- where k reﬂects the percentage of time that the safety port. See Kimball (1988), Simpson (1958), and Graves stock covers the demand variation. The choice of k in- (1988) for further discussion of this assumption. dicates how frequently the manager is willing to resort Finally we note that there are no assumptions made to other tactics to cover demand variability. about the demand distribution. In some contexts there may be natural bounds on the end-item demand. For instance, suppose the end Guaranteed Service Times. item is a component or subassembly for a manufac- We assume that each demand node j promises a guar- turing process whose production is limited by capacity anteed service time Sj by which the stage j will satisfy constraints or by a frozen master schedule. An exam- customer demand. That is, the customer demand at ple is a supply chain that supplies components to an time t, dj(t), must be ﬁlled by time t Ⳮ Sj. Furthermore, automobile assembly line or an OEM subassembly to we assume that stage j provides 100% service for the a system integrator. In these cases, bounded demand speciﬁed service time: stage j delivers exactly dj(t) to for the component corresponds to the maximum con- the customer at time t Ⳮ Sj. sumption by the manufacturing process over various Similarly, an internal stage i quotes and guarantees time horizons. a service time Sij for each downstream stage j, (i, j) 僆 For each internal stage we assume that we can also A. Given a base-stock policy, stage j places an order establish meaningful demand bounds. If stage i has a equal to ij dj(t) on stage i at time t; then stage i delivers single successor, say stage j, then Di(s) ⳱ ij Dj(s) for exactly this amount to stage j at time t Ⳮ Sij. all relevant s. For stages with more than one successor, For the initial development, we assume that stage i we require some judgment for deciding how to com- quotes the same service time to all of its downstream bine the demand bounds for the downstream stages to customers; that is, Sij ⳱ Si for each downstream stage obtain a relevant demand bound for the upstream j, (i, j) 僆 A. Graves and Willems (1998) describe how stage for the purposes of positioning the safety stock. to extend the model to permit customer-speciﬁc ser- One possibility is just to sum the downstream demand vice times. In brief, if there is more than one down- bounds; however, this approach assumes that there is stream customer, we can insert zero-cost, zero- no risk pooling from combining the demand of mul- production lead-time dummy nodes between a stage tiple end items. An alternative approach is to assume and its customers to enable the stage to quote different that there will be some relative reduction in variability service times to each of its customers. The stage quotes as we combine demand streams, i.e., some risk pool- the same service time to the dummy nodes and each ing. For instance, we might infer the demand bounds dummy node is free to quote any valid service time to for internal stages by means of an expression like its customer stage. Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 71

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains The service times for both the end items and the in- 3. Single-Stage Model ternal stages are decision variables for the optimization In this section we present the single-stage model (see model, as will be seen in §4. However, as a model in- Kimball 1988 or Simpson 1958) that serves as the build- put, we may impose bounds on the service times for ing block for modeling a multi-stage supply chain. each stage. In particular, we assume that for each end item we are given a maximum service time, presum- Inventory Model ably set by the marketplace. We assume the inventory system starts at time 0 with initial inventory Ij(0). Given our assumptions, we can Discussion of Assumption of Guaranteed Service express the ﬁnished inventory at stage j at the end of Times. period t as We assume that there are no violations of the guar- Ij(t) ⳱ Bj ⳮ dj(t ⳮ SIj ⳮ Tj, t ⳮ Sj) (3) anteed service times; each stage provides perfect or 100% service to its customers. As such, we do not ex- where Bj ⳱ Ij(0) ⱖ 0 denotes the base stock, dj(a, b) plicitly model a tradeoff between possible shortage denotes the demand at stage j over the time interval costs and the costs for holding inventory. Rather, we (a, b], and SIj is the inbound service time for stage j. pose the problem as being how to place safety stocks Since we assume a discrete-time demand process, we across the supply chain to provide 100% service for the understand dj(a, b) to be assumed bounded demand with the least inventory dj(a, b) ⳱ dj(a Ⳮ 1) Ⳮ dj(a Ⳮ 2) Ⳮ • • • Ⳮ dj(b) holding cost. for a b, where dj(t) ⳱ 0 for t ⱕ 0. When a ⱖ b, we In defense of this assumption, it is often very difﬁ- deﬁne dj(a, b) ⳱ 0. cult in practice to assess shortage costs for an external The inbound service time SIj is the time for stage j customer. Similarly, when we have asked managers to get supplies from its immediate suppliers and to for their desired service level, more often than not the commence production. In period t, stage j places an response is that there should be no stock-outs for ex- order equal to ij dj(t) on each upstream stage i for ternal customers. We have found that managers seem which ij 0. Stage j cannot start production to re- more comfortable with the notion of 100% service for plenish dj(t) until all inputs have been received; thus some range of demand; they accept the fact that if de- we have SIj ⱖ max {Si| i:(i, j) 僆 A}. mand exceeds this range, they will have shortages un- We permit SIj max {Si | i:(i, j) 僆 A} to allow for less they can somehow expand the response capability the possibility that the replenishment time for the in- of their supply chain. The assumptions for the model ventory at stage j is less than its service time Sj; that is, presented herein are consistent with this perspective. the case when For an internal customer, guaranteed service times need not be optimal in terms of least inventory costs. max {Si|i:(i, j) 僆 A} Ⳮ Tj Sj. Indeed we show in the Appendix how to relax this In this case we would delay the orders to the suppliers assumption for a serial network, and report the cost by Sj ⳮ max {Si| i:(i, j) 僆 A} ⳮ Tj periods, so that the impact of this assumption for a set of 36 test problems: supplies arrive exactly when needed. To account for the safety stock holding cost is 26% higher on average, this case, we set the inbound service time so that the while the total inventory cost is 4% higher on average. effective replenishment time for the inventory at stage However, guaranteed service times are very practical j, namely SIj Ⳮ Tj, equals the service time Sj, i.e., SIj Ⳮ in contexts where there is a need to coordinate replen- Tj ⳱ Sj. Thus, we deﬁne the inbound service time as ishments. For instance, any assembly or subassembly stage requires the concurrent availability of multiple SIj ⳱ max{Sj ⳮ Tj, max {Si|i:(i, j) 僆 A}}. components, not all of which might be explicitly in- If the inbound service time is such that SIj Si for cluded in the model. When we assume guaranteed ser- some (i, j) 僆 A, then by convention stage j delays or- vice times, we make the challenge of coordinating the ders from stage i by SIj ⳮ Si periods to avoid unnec- availability of these components much easier. essary inventory. Manufacturing & Service Operations Management 72 Vol. 2, No. 1, Winter 2000, pp. 68–83

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains Now, to explain Equation (3), we observe that in pe- the demand bound is given by Equation (1); then the riod t stage j completes the replenishment of the de- safety stock is E[Ij] ⳱ kr冪SIj Ⳮ Tj ⳮ Sj. mand observed in period t ⳮ SIj ⳮ Tj. By the end of Pipeline Inventory. period t, the cumulative replenishment to the inven- In addition to the safety stock, we may want to account tory at stage j equals dj(0, t ⳮ SIj ⳮ Tj). In period t, for the in-process or pipeline stock at the stage. Follow- stage j ﬁlls the demand observed in time period t ⳮ Sj ing the development for Equation (3), we observe that from its inventory. By the end of period t the cumu- the work-in-process inventory at time t is given by lative shipment from the inventory at stage j equals dj(0, t ⳮ Sj). The difference between the cumulative Wj(t) ⳱ dj(t ⳮ SIj ⳮ Tj, t ⳮ SIj). replenishment and the cumulative shipment is the in- That is, the work-in-process corresponds to Tj periods ventory shortfall, dj(t ⳮ SIj ⳮ Tj, t ⳮ Sj). The on-hand of demand. The expected work-in-process depends inventory at stage j is the initial inventory or base stock only on the lead-time at stage j and is not a function minus the inventory shortfall, as given by Equation (3). of the service times: Determination of Base Stock. E[Wj] ⳱ Tj lj. In order for stage j to provide 100% service to its cus- tomers, we require that Ij(t) ⱖ 0; from (3) we see that Hence, in posing an optimization problem in the next this requirement equates to section, we ignore the pipeline inventory and only model the safety stock. This is not to say that the work- Bj ⱖ dj(t ⳮ SIj ⳮ Tj, t ⳮ Sj). in-process is not a signiﬁcant part of the inventory in Since demand is bounded, we can satisfy the above a supply chain. But for the purposes of this work, we requirement with the least inventory by setting the assume that the lead-time of a stage, as well as the base stock as follows: demand rate, are input parameters and thus the pipe- line stock is predetermined. Nevertheless, in any ap- Bj ⳱ Dj(s) where s ⳱ SIj Ⳮ Tj ⳮ Sj. (4) plication, we account for both the safety stock and the Any smaller value does not assure that Ij(t) ⱖ 0, and pipeline stock as both will contribute to the total sup- thus cannot guarantee 100% service. ply chain inventory. In words, the base stock equals the maximum pos- sible demand over the net replenishment time for the 4. Multi-Stage Model stage. The net replenishment time for stage j is the re- To model the multi-stage system, we use Equation (5) plenishment time (SIj Ⳮ Tj) minus its service time (Sj). for every stage, but where the inbound service time is At any time t, stage j has ﬁlled its customers’ demand a function of the outbound service times for the up- through time t ⳮ Sj, but has only been replenished for stream stages; to wit, the model for stage j is demand through time t ⳮ SIj ⳮ Tj. The base stock must cover this time interval of exposure, namely the net E[Ij] ⳱ Dj(SIj Ⳮ Tj ⳮ Sj) ⳮ (SIj Ⳮ Tj ⳮ Sj)lj, (6) replenishment time. SIj Ⳮ Tj ⳮ Sj ⱖ 0, (7) Safety Stock Model. SIj ⳮ Si ⱖ 0 for all (i, j) 僆 A. (8) We use Equations (3) and (4) to ﬁnd the expected in- ventory level E[Ij]: Equation (6) expresses the expected safety stock as a function of the net replenishment time. Equation (7) E[Ij] ⳱ Bj ⳮ E[dj(t ⳮ SIj ⳮ Tj, t ⳮ Sj)] assures that the net replenishment time is nonnegative. Equation (8) constrains the inbound service time to ⳱ Dj(SIj Ⳮ Tj ⳮ Sj) ⳮ (SIj Ⳮ Tj ⳮ Sj)lj. (5) equal or exceed the service times for the upstream The expected inventory represents the safety stock stages. held at stage j, and depends on the net replenishment We assume that the production lead-times, the time and the demand bound. As an example, suppose means and bounds of the demand processes, and the Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 73

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains maximum service times for the demand nodes are under more traditional assumptions where demand is known input parameters. This suggests the following not bounded. optimization problem P for ﬁnding the optimal service Graves (1988) observed that the serial-line problem times: can be solved as a shortest path problem. In a series of N papers. Inderfurth (1991, 1993), Inderfurth and Minner P min 兺 j⳱1 hj{Dj(SIj Ⳮ Tj ⳮ Sj)ⳮ(SIj Ⳮ Tj ⳮ Sj)lj} (1998), and Minner (1997) show how to solve problem P by dynamic programming when the supply chain is an assembly network or a distribution network. s. t. Sj ⳮ SIj ⱕ Tj for j ⳱ 1, 2, . . . , N, Graves and Willems (1996) developed similar results SIj ⳮ Si ⱖ 0 for all (i, j) 僆 A, for assembly and distribution networks. In the next section we present a dynamic programming algorithm Sj ⱕ sj for all demand nodes j, for the more general case of a spanning tree. Sj, SIj ⱖ 0 and integer for j ⳱ 1, 2, . . . , N, where hj denotes the per-unit holding cost for inven- 5. Algorithm for Spanning Tree tory at stage j, and sj is the maximum service time for We describe in this section how to solve P by dynamic demand node j. The objective of problem P is to min- programming when the underlying network for the imize the holding cost for the safety stock in the supply supply chain is a spanning tree, like in the Figure 1. chain. The constraints assure that the net replenish- We solve P by decomposing the problem into N ment times are nonnegative, the inbound service time stages where N is the number of nodes in the spanning equals the maximum supplier service time, and the tree and there is one stage for each node. For a span- end-item stages satisfy their service guarantee. The de- ning tree, there is not a readily-apparent ordering of cision variables are the service times. the nodes by which the algorithm would proceed. In- Problem P is a nonlinear optimization problem. The deed, we label the nodes in a spanning tree (and thus objective function is a concave function, provided that sequence the algorithm) so that there will be a single the demand bound Dj( ) is a concave function for each state variable for the dynamic programming recursion. stage j. The feasible region is convex but not necessarily However, the state variable for the dynamic program bounded; however, one can show that the optimal ser- will be either the inbound service time at a stage or its vice times need not exceed the sum of the production outbound service time, where the determination de- lead-times, provided that Dj( ) is a nondecreasing func- pends on the topology of the network. tion for each stage j. Thus, problem P is the minimi- We ﬁrst present the algorithm for labeling the nodes. zation of a concave function over a closed, bounded Next we present the functional equations for the dy- convex set. An optimum for such problems is at an namic programming recursions, and then state the extreme point of the feasible region, e.g., Luenberger algorithm. 1973. Simpson (1958) considered a serial-line supply Labeling the Nodes. chain, where he assumed that the guaranteed service The algorithm for labeling or re-numbering the nodes time for the external customer is zero. Simpson is as follows: showed that there is an optimal extreme point solution 1. Start with all nodes in the unlabeled set, U. for P for which Si ⳱ 0 or Si ⳱ SiⳭ1 Ⳮ Ti, where stage 2. Set k :⳱ 1. i Ⳮ 1 supplies stage i. Thus, there is an “all or nothing” 3. Find a node i 僆 U such that node i is adjacent to optimal solution; a stage either has no safety stock (Si at most one other node in U. That is, the degree of node ⳱ SiⳭ1 Ⳮ Ti) or has sufﬁcient safety stock (Si ⳱ 0) to i is 0 or 1 in the subgraph with node set U and arc set de-couple it from its downstream stage. Gallego and A deﬁned on U. Zipkin (1999) provide supporting evidence that “all or 4. Remove node i from set U and insert into the la- nothing” policies can be near optimal in serial systems beled set L; label node i with index k. Manufacturing & Service Operations Management 74 Vol. 2, No. 1, Winter 2000, pp. 68–83

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains 5. Stop if U is empty: otherwise set k :⳱ k Ⳮ 1 and Functional Equations. repeat steps 3– 4. The dynamic program evaluates a functional equation For a spanning tree, there is always an unlabeled node for each node in the spanning tree, where we have re- in step 3 that is adjacent to at most one other unlabeled numbered the nodes as described above. There are two node. As a consequence, the algorithm will eventually forms for the functional equation. First, the function label all of the nodes in N iterations. Indeed, each node fk(S) is the minimum holding cost for safety stock in a labeled in the ﬁrst N ⳮ 1 steps is adjacent to exactly subnetwork with node set Nk, where we assume that one other node in set U. Thus, the nodes with labels 1, the outbound service time for stage k is S. Second, the 2, . . . , N ⳮ 1 have one adjacent node with a higher function gk(SI) is the minimum holding cost for safety label, denoted by p(k) for k ⳱ 1, 2, . . . , N ⳮ 1. Node stock in a subnetwork with node set Nk, where we as- N has no adjacent nodes with larger labels. sume that the inbound service time for stage k is SI. As an illustration, we renumber the nodes in Figure At node k (or stage k) for 1 ⱕ k ⱕ N ⳮ 1, the algo- 1 to produce Figure 2. Note that the labeling is not rithm determines either fk(S) or gk(SI), depending upon unique as there may be multiple choices for node i in the location of the node with higher label that is adja- step 3. cent to k. If p(k) is downstream [upstream] of node k, For each node k we deﬁne Nk to be the subset of then we evaluate fk(S) [gk(SI)]. For node N, we can eval- nodes {1, 2, . . . , k} that are connected to k on the sub- uate either functional equation. graph with node set {1, 2, . . . , k}. We will use Nk to To develop the functional equations, we ﬁrst deﬁne explain the dynamic programming recursion. We can the minimum inventory holding cost for the subnet- determine Nk by the following equation: work with node set Nk as a function of both the in- bound service time and the outbound service time at Nk ⳱ {k} Ⳮ 艛 Ni Ⳮ 艛 Nj. ik,(i,k)僆A jk,(k,j)僆A node k: For instance, in Figure 2, Nk is {3} for k ⳱ 3, {1, 2, 3, 9} ck(S, SI) ⳱ hk{Dk(SI Ⳮ Tk ⳮ S) ⳮ (SI Ⳮ Tk ⳮ S)lk} for k ⳱ 9, {1, 2, 3, 4, 5, 9, 11} for k ⳱ 11, and {6, 7, 8, 10, 12} for k ⳱ 12. We can compute Nk as part of the Ⳮ 兺 (i,k)僆A fi(SI) Ⳮ 兺 (k,j)僆A gj(S). labeling algorithm. ik jk The ﬁrst term is the holding cost for the safety stock at Figure 1 Spanning Tree node k as a function of S and SI. The second term corresponds to the nodes in Nk that are upstream of k. For each node i that supplies node k, we include the minimum inventory holding costs for the subnetwork with node set Ni, as a function of SI. The inbound service time to node k, SI, is an upper bound for the outbound service time for node i. We can show that fi( ), the inventory holding costs for the subnetwork with node set Ni, is nonincreasing in the service time at node i. Hence, we equate the outbound Figure 2 Renumbered Spanning Tree service time at i to the inbound service time at k with- out loss of generality. The third term corresponds to the nodes in Nk that are downstream of k. For each node j, j 僆 Nk and (k, j) 僆 A, we include the minimum inventory holding costs for the subnetwork with node set Nj, as a function of S. The outbound service time for node k, S, is a lower bound for the inbound service time for node j. We can Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 75

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains show that gj( ), the inventory holding costs for the sub- the rest of the network. When the connecting arc orig- network with node set Nj, is nondecreasing in the in- inates in Nk, then the state variable is the outbound bound service time to node j; and thus we equate the service time (step 2); otherwise, the state variable is the inbound service time at j to the outbound service time inbound service time (step 3). We number the nodes at k without loss of generality. so that we have previously determined the functions We solve the following optimization by enumera- required to evaluate either fk(S) or gk(SI). At stage N tion to ﬁnd the functional equation fk(S): (step 4), we determine the inventory costs for the entire network as a function of the inbound service time to fk(S) ⳱ min{ck(S, SI)} node N. At step 5, we optimize over the inbound ser- SI vice time to ﬁnd the optimal inventory cost. s. t. max (0, S ⳮ Tk) ⱕ SI ⱕ Mk ⳮ Tk, and SI integer, The computational complexity of the algorithm is of order NM2 where M is the maximum service time, where Mk is the maximum replenishment time for which is bounded by the sum of the production lead- node k. The lower bound on SI comes from P, while the deﬁnition of Mk gives the upper bound. 兺 N times j⳱1Tj. We have implemented the algorithm for a PC in the CⳭⳭ programming language. The run The functional equation for gk(SI) is very similar in times for real problems with 25 to 30 nodes are effec- structure: tively instantaneous on a Pentium PC with a 100 meg- gk(SI) ⳱ min{ck(S, SI)} ahertz Intel processor. S s. t. 0 ⱕ S ⱕ SI Ⳮ Tk, and S integer. 6. Application If node k is a demand node, then we also constrain S This section presents an application of the model at the by its maximum service time, i.e., S ⱕ sk. The mini- Eastman Kodak Company. Starting in 1995, Kodak has mization can be done by enumeration. applied the model to more than eleven ﬁnished prod- ucts across two of its assembly sites within its equip- Dynamic Program. ment division. We ﬁrst present the model’s application The dynamic programming algorithm is now as to the internal supply chain for a high-end digital cam- follows: era,2 and then summarize Kodak’s ﬁnancial results, as 1. For k :⳱ 1 to N ⳮ 1 of 1997 year-end. 2. If p(k) is downstream of k, evaluate fk(S) for S Product Background. ⳱ 0, 1, . . ., Mk. The key subassemblies for the digital camera are a tra- 3. If p(k) is upstream of k, evaluate gk(SI) for SI ⳱ ditional 35 mm camera, an imager, and a circuit-board 0, 1, . . ., Mk ⳮ Tk. assembly. The 35mm camera is procured from an out- 4. For k :⳱ N evaluate gk(SI) for SI ⳱ 0, 1, . . . , Mk side vendor. The imager (a charge-coupled device) and ⳮ Tk. the circuit-board assembly are produced internally. 5. Minimize gN(SI) for SI ⳱ 0, 1, . . . , MN ⳮ TN to The 35mm camera supplies the lens, shutter, and focus obtain the optimal objective function value. functions for the digital camera. The imager captures This procedure ﬁnds the optimal objective function and digitizes the picture, and the circuit-board assem- value; we can ﬁnd an optimal set of service times by bly processes and stores the image. To produce the dig- the standard backtracking procedure for a dynamic ital camera, the back of the 35mm camera is removed program. and replaced with a housing containing the imager To summarize, at each stage of the dynamic pro- gram, we ﬁnd the minimum inventory holding costs 2 The data presented in this section has been altered to protect pro- for the subnetwork with node set Nk, as a function of prietary information. However, the resulting qualitative relation- a state variable. The state variable depends on the di- ships and insights drawn from this example are the same as they rection of the arc that connects the subnetwork Nk to would be from using the actual data. Manufacturing & Service Operations Management 76 Vol. 2, No. 1, Winter 2000, pp. 68–83

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains and circuit board. The camera is then tested to make department. The circuit-board stage entails circuit sure that there are no defects in the imager. Once the board assembly and test. The imager stage consists of camera passes the quality tests, the product is shipped a semiconductor operation to produce wafers, fol- to the distribution center. From the distribution center, lowed by packaging and testing of the semiconductors, the camera is shipped to the ﬁnal customers, which for followed by an assembly operation. our purposes are high-end photography shops and computer superstores. Implementation Approach. In Figure 3, we provide a high-level depiction of this The product’s supply chain crosses several functional supply chain. In addition to the three key subassem- boundaries within Kodak. Functional areas like circuit- blies, we include the remaining parts in order to ac- board assembly and imager assembly are separate de- curately represent the product’s cost structure; since partments and act as suppliers to an assembly group there are nearly 100 additional parts in a camera, mod- that performs ﬁnal assembly and test. Distribution is a eling them in any level of detail would greatly expand separate organization and owns the product once it the size of the model. Hence, we group these parts into leaves the ﬁnal assembly area. To improve coordina- two aggregate stages of the supply chain, where one tion across the departments, the equipment division at stage represents all of the parts with long procurement Kodak has set up product ﬂow teams with the general lead-times (greater than 60 days) and the other stage charge to optimize their supply chains. represents the short lead-time parts (less than 60 days). The product ﬂow team for the digital camera relied We also aggregate certain operations. As seen in Fig- on the model to identify opportunities for better co- ure 3, we combine the build operation for a camera ordination and improved asset utilization. The team with the test operation and the packing operation. The implemented the model in phases. The implementa- imager stage and circuit board stage are also aggre- tion strategy was to start simple and get experience gates as each represents the ﬂow through a separate with the model; once there was some evidence of the utility of the model, the team extended the application in increments to capture more and more of the supply chain. Figure 3 Implemented Safety Stock Policy for Digital Camera. All Phase One. Stages Have a Circle That Denotes the Processing Activity at the Stage. A Triangle Denotes That the Stage Holds a The initial goal was to optimize the safety stock levels Safety Stock of Its Finished Goods for the stages that were under the direct control of the ﬁnal assembly area. The decision to start with the ﬁnal assembly area was based on the product’s high mate- rial cost and its relatively simple supply chain struc- ture, as described above. The (disguised) costs and production lead-times are: Table 1 Phase One Digital Camera Information Stage Name Production Lead-Time Cost Added Camera 60 750 Imager 60 950 Circuit Board 40 650 Other Parts LT 60 days 60 150 Other Parts LT 60 days 150 200 Build/Test/Pack 6 250 Transfer to DC 2 50 Ship to Customer 3 0 Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 77

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains The demand bound was estimated by Equation (1) The team also investigated a policy in which the dis- where l ⳱ 11, r ⳱ 7 and k ⳱ 1.645. From looking at tribution center would hold safety stock but the manu- historical demand and future demand estimates, Ko- facturing site would not. The safety stock cost for this dak felt that this function realistically captured the policy was $81,000, which was deemed to be accept- range of demand for which they wanted to use safety able as it was quite close to the unconstrained opti- stock. mum and satisﬁed distribution’s desire to hold inven- This demand characterization excluded large one- tory. This policy, as shown in Figure 3, was time orders from the government and some large cor- implemented at the end of phase one of the porations. These orders are typically for 200 – 300 units application. with delivery scheduled less than a month from when Phase Two. the order is placed. However, since there is advance After the initial phase, the product ﬂow team ex- warning about these orders and they are independent panded the model to incorporate the internal supply of the other demand for the product, we developed a chain for the imager. The resulting supply chain is separate anticipatory stock policy to deal with large, shown in Figure 4. infrequent orders. Prior to this study, safety stocks of (in-process) im- Marketing determined that the maximum service agers had been held at each stage of the supply chain. time to the ﬁnal customer is ﬁve days. By application of the model, the team decided to re- Finally, the assembly group imposed the constraint move safety stocks from two stages in the supply chain that a safety stock of imagers must be held on-site at for the imagers, as shown in the ﬁgure. This required ﬁnal assembly. Therefore, we set the service time for some increase in the downstream safety stocks of ﬁn- the imager stage to be zero; the effect of this constraint ished imagers, but overall the supply chain’s safety increased the total safety stock cost by 8.7%. stock of imagers (measured in terms of ﬁnished ima- In the optimal solution, the subassembly stages, the gers) was more than halved. aggregate parts stages, and the build/test/package Now that the model has been successfully piloted stages hold safety stocks and quote zero service times. with an internal supplier, the product ﬂow team is in The ship-to-distribution and ship-to-customer stages the process of extending the model to incorporate each quote their maximum feasible service times, two other key internal and external suppliers. and ﬁve days, respectively. The annual holding cost for the safety stock is $78,000. Thus, the optimal solu- Financial Results. tion holds an inventory of components, subassemblies, Table 2 contains the ﬁnancial summary for two assem- and completed cameras at the manufacturing site, but bly sites that use the model. Site A has applied the holds no inventory in the distribution center. In effect, the distribution center would act only as an order pro- cessing and transshipment center. This is feasible since Figure 4 Digital Camera Supply Chain it is possible to get the product from the assembly area through the distribution center and to the ﬁnal cus- tomer within the maximum service time of ﬁve days. The product ﬂow team decided to explore some near-optimal solutions because they felt that there were some additional organizational constraints not captured in the model; in particular, distribution would want to hold safety stock on-site. To ameliorate the situation, the team suggested that both manufac- turing and distribution hold safety stock and quote zero service times. However, the model showed that the cost for the safety stock would increase to $89,000. Manufacturing & Service Operations Management 78 Vol. 2, No. 1, Winter 2000, pp. 68–83

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains Table 2 Financial Summary for Assembly Sites A and B the ﬁnal customer. With the model, the supply-chain team can accurately estimate the costs of a one-day, Assembly Site A Y/E 95 Y/E 96 Y/E 97 one-week, or two-week guaranteed service time to the customer, and weigh the costs of the policy against the Worldwide FGI $6.7m $3.3m $3.6m Raw Material & WIP $5.7m $5.6m $2.9m marketing beneﬁts of the policy. Delivery Performance 80% 94% 97% Another beneﬁt of the model is that it provides a Manufacturing Operation MTS RTO RTO common, objective framework with which a cross- Assembly Site B functional supply-chain team can work. In particular, Worldwide FGI $4.0m $4.0m $3.2m we note that it provides a standard terminology and Raw Material & WIP $4.5m $1.6m $2.5m set of assumptions for these teams to use as they work Delivery Performance Unavailable 78% 94% together to improve or optimize a supply chain. As Manufacturing Operation MTS RTO RTO such the model has been a very effective communica- tion vehicle or platform. model to each of its eight products and Site B has ap- plied the model to each of its three product families. 7. Conclusion The sales volume has remained relatively constant In this paper we introduce and develop a model for over the three years. positioning safety stock in a supply chain. We model At the start of 1996, the sites moved from a make- the supply chain as a network, where the nodes of the to-schedule (MTS) to a replenish-to-order (RTO) sys- network are the stages of a supply chain. We assume tem. The modeling effort began at the end of 1995 and that each stage uses a base-stock policy to control its was used to help guide the transition to replenish-to- inventory. We also assume that each stage quotes a order. The increase in worldwide ﬁnished goods in- service time to its customers, both internal and exter- ventory for 1997 is due to a marketing promotion that nal, and that each stage provides 100% service for these was underway in Europe. By our estimate, this pro- quoted service times. Finally we assume that external motion increased the ﬁnished goods inventory by as customer demand is bounded. much as $0.5 million. In the ﬁrst year of the project, We show how to evaluate the inventory require- the emphasis was on reducing the areas directly under ments at each stage as a function of the service times. the control of ﬁnal assembly. Over the following year, For supply chains that can be modeled as spanning the effort was on reducing the raw material costs and trees, we develop an optimization algorithm for ﬁnd- WIP in the manufacturing supply chain. The total ing the service times that minimize the holding cost for value of the inventory for these products has been re- the safety stock in the supply chain. duced by over one third over the two years. As a form of validation, we describe an application Kodak’s product ﬂow teams have also used the of the model at Kodak to an internal supply chain for model for a variety of purposes other than setting a digital camera. This application helped Kodak to re- safety stocks. Some products have tens of components position its inventories in this supply chain to reduce with long procurement lead-times. The model has its inventory and increase its service performance. In helped to prioritize the suppliers with whom to work particular, Kodak realized the beneﬁt from creating a to reduce these lead-times. The teams have used the few strategic locations to hold safety stocks rather than model to determine the cost effectiveness of lead-time spreading the safety stock across the entire supply reduction efforts in manufacturing. One can compare chain. We have also applied the model to a number of the investment required to reduce a lead-time versus other related products at Kodak and at two other com- the cost savings from the reductions in pipeline and panies (Black 1998, Coughlin 1998, Felch 1997, Wala safety stock cost. Finally, manufacturing and market- 1999). ing personnel have used the model to help quantify As with any research, we end with a number of un- the cost of quoting a speciﬁc maximum service time to resolved issues and new questions. We discuss these Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 79

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains in the relative order of importance, based on our ex- implement this extension, as it is a major programming perience in applying the research to date. task and it may only be a partial ﬁx to the issue. Stochastic Lead-Times. We assume that associated Capacity Constraints. In the model we ignore ca- with each stage is a known, deterministic lead-time. In pacity constraints. For certain stages in a supply chain, practice, this is often not true; for instance, component the consideration of a capacity limit may be necessary procurement times are often highly uncertain. It will in order to get a credible model for determining safety be of value to capture this in the model. We know how stock requirements. At this time, we do not have a to extend the model in an approximate way for stages good idea of how to add this to the model. General Networks. In this paper, we have devel- that procure raw materials or components from an out- oped and implemented an optimization algorithm for side vendor. In effect, for such a stage we just need to supply chains that can be modeled as spanning trees. approximate its inventory requirements as a function We describe in Graves and Willems (1998) how to ex- of the outbound service time quoted by the stage and tend this algorithm to general networks. However, we the stochastic procurement time. But it is less clear how have not done a systematic study of this extension be- to extend the model, either exactly or approximately, yond some exploratory work. More research is needed to permit stochastic lead-times at stages whose func- to test and reﬁne these ideas as well as uncover better tion is not procurement. approaches.3 Non-Stationary Demand. We assume that the Appendix end-item demand processes are stationary. Yet virtu- In this appendix we examine the assumption that each internal stage ally all of the products with which we have worked quotes a guaranteed service time to its customers. To get some in- have short lifetimes over which demand is never really sight, we consider a serial system for which we can determine the optimal policy when we relax the assumption of guaranteed service stationary. In practice, one runs the model under vari- times for internal customers. We then compare the inventory hold- ous (stationary) scenarios to see how sensitive the ing costs for the optimal policies with and without this assumption safety stock is to the demand characterization for a small set of test problems. (Coughlin 1998). Fortunately, we have found empiri- Consider a serial supply chain with N stages where stage 1 is the demand node and stage i supplies stage i ⳮ 1 for i ⳱ 2, . .. , N. The cally that where the model locates safety stock in the same assumptions hold as in the original model, except that we do supply chain is fairly insensitive to the demand. The not require guaranteed service times to internal customers. There size of the safety stock, though, does depend directly are no restrictions on the service level that stage i provides to its on the demand characterization. We currently are con- customer, stage i ⳮ 1 for i ⳱ 2, . . . , N; rather, these internal service levels depend on the base stocks, which are chosen to minimize the ducting research to understand these observations bet- inventory holding costs for the entire supply chain. We do assume ter, to extend the model to treat non-stationary that stage 1 provides a 100% service level to the external customer, demand. and, without loss of generality, we assume that the service time Different Review Periods. We assume that each quoted to the external customer is zero. For ease of presentation, we assume that i,iⳮ1 ⳱ 1 for i ⳱ 2, . . . , stage operates with a base-stock policy with a common N. We let d(t) denote the end-item demand in period t; d(a, b]; denote review period. In many supply chains different stages the end-item demand over the time interval (a, b]; and D(s) denote will operate with different reorder frequencies. That is, the maximum possible end-item demand over a time interval of s periods. whereas one stage may place replenishment orders on a daily basis, another stage may do this weekly. In 3 This research has been supported in part by the Eastman Kodak other cases, a stage may operate with a continuous- Company; by the MIT Leaders for Manufacturing Program, a part- review policy so that the time between orders varies. nership between MIT and major U.S. manufacturing ﬁrms; and by We can extend the model to evaluate nested periodic- the MIT Integrated Supply Chain Management consortium. The au- review base-stock polices in which whenever one stage thors acknowledge and thank Dr. John Ruark who contributed sig- niﬁcantly to this research effort; John played a lead role in devel- reorders, all stages downstream also reorder. That is, oping the software application for implementing the results of this the review period for an upstream stage is an integer research. We also wish to thank the editors and referees for their multiple of the review period of its immediate custom- very helpful and constructive feedback on earlier versions of the ers. However, we have not yet built the software to paper. Manufacturing & Service Operations Management 80 Vol. 2, No. 1, Winter 2000, pp. 68–83

GRAVES AND WILLEMS Optimizing Strategic Safety Stock Placement in Supply Chains For each stage i, we deﬁne Qi(t) to be the shortfall or backlog at nonnegativity constraints on the base stocks. After dropping con- time t, namely the amount that has been ordered by the stage’s cus- stant terms in Equation (A5) and noting that Q1(t) ⳱ 0 for any fea- tomer but not yet delivered. We assume at t ⳱ 0, Ii(t) ⳱ Bi ⱖ 0 and sible solution, we write the optimization as Qi(t) ⳱ 0 for all stages. N N We can show for i ⳱ 1, 2, . . . , N that the on-hand inventory and min 兺 hiBi ⳮ i⳱2 i⳱1 兺 eiⳮ1E[Qi] backlog at time t are: Ii(t) ⳱ [Bi ⳮ d(t ⳮ Ti, t) ⳮ QiⳭ1(t ⳮ Ti)]Ⳮ, P* s.t. Qi(t) ⳱ [d(t ⳮ Ti, t) Ⳮ QiⳭ1(t ⳮ Ti) ⳮ Bi]Ⳮ, (A1) B1 Ⳮ B2 Ⳮ • • • Ⳮ Bi ⱖ D(T1 Ⳮ T2 Ⳮ • • • Ⳮ Ti) where [x]Ⳮ ⳱ max(0, x), and QNⳭ1(t) ⳱ 0 by deﬁnition. Equation for i ⳱ 1, 2, . . . , N, (A1) requires that each stage has a deterministic lead-time and that each stage follows a base-stock policy in which, for each period, each Bi ⱖ 0 for i ⳱ 1, 2, . . . , N, stage observes end-item demand and places a replenishment order where ei ⳱ hi ⳮ hiⳭ1 is the echelon holding cost. We note from for this amount. The essence of the argument is to observe that the Equation (A2) that E[Qi] is a nonlinear function of Bi, . . . , BN for i ⳱ net inventory on hand at a stage equals the stage’s base stock minus 1, 2, . . . , N. the inventory on order. For stage i, the inventory on order at time t Our main result is that there is an optimal solution to P* in which is the backlog as of time t ⳮ Ti, plus all of the demand over the all the constraints in Equation (A3) are binding. More formally we interval (t ⳮ Ti, t]. state the following: From Equation (A1) we can show by induction that for i ⳱ Result. If the echelon holding costs are nonnegative and if D( ) 1,2, . . . , N, is a nondecreasing function, then an optimal solution to P* is given Qi(t) ⳱ max[0, d(t ⳮ Ti, t) ⳮ Bi, d(t ⳮ Ti ⳮ TiⳭ1, t) by B1 ⳱ D(T1), ⳮ Bi ⳮ BiⳭ1, . . . , d(t ⳮ Ti ⳮ TiⳭ1 ⳮ • • • ⳮTN, t) ⳮ Bi ⳮ BiⳭ1 ⳮ • • • ⳮBN]. (A2) Bi ⳱ D(T1 Ⳮ • • • Ⳮ Ti) ⳮ D(T1 Ⳮ • • • Ⳮ Tiⳮ1) In order for the supply chain to provide 100% service to the external for i ⳱ 2, . . . , N. (A6) customer, we must never have a backlog at stage 1; thus, we must Proof. We note that the solution given by Equation (A6) is non- select base stocks so that Q1(t) ⳱ 0 for all t. From Equation (A2) we negative and satisﬁes the constraints in Equation (A3) as equalities; see that Q1(t) ⳱ 0 is assured if the base stocks satisfy the following thus it is a feasible solution to P*. To prove that this is also an optimal constraints: solution, we will argue that there must be an optimal solution in B1 Ⳮ B2 Ⳮ • • • ⳭBi ⱖ D(T1 Ⳮ T2 Ⳮ • • • ⳭTi) which the constraints in Equation (A3) are binding. Suppose we have a solution B*1 , . . . , B*N such that Equation (A3) for i ⳱ 1, 2, . . . , N. (A3) holds as a strict inequality for one or more constraints. Suppose the Thus, if the base stocks satisfy Equation (A3), there will never be a kth constraint is the ﬁrst constraint that is not binding and that k shortfall at stage 1 and end-item demand will be satisﬁed with 100% N; we will treat the case when k ⳱ N later. Thus, we assume service. As we assume that the demand bounds can be realized, then B*1 Ⳮ B*2 Ⳮ • • • Ⳮ B*i ⳱ D(T1 Ⳮ T2 Ⳮ • • • Ⳮ Ti) the constraint set (A3) provides not just sufﬁcient but also necessary conditions for assuring 100% service for end-item demand. for i ⳱ 1, 2, . . . , k ⳮ 1 and In order to select the base stocks to minimize the inventory hold- B*1 Ⳮ B*2 Ⳮ • • • Ⳮ B*k D(T1 Ⳮ T2 Ⳮ • • • Ⳮ Tk). ing costs for the supply chain, we must develop an expression for the inventory holding costs; we note from Equation (A1) that the net We now deﬁne a new solution B** 1 , . . . , B** N in which the ﬁrst k con- inventory on hand at stage i is given by: straints are satisﬁed as equalities, and show that its objective value is no worse than that for B*1 , . . . , B*N: Ii(t) ⳮ Qi(t) ⳱ Bi ⳮ d(t ⳮ Ti, t) ⳮ QiⳭ1(t ⳮ Ti). (A4) B** i ⳱ B*i for i ⳱ 1, . . . , N and i ⬆ k, k Ⳮ 1, From Equation (A4) we can write the inventory holding costs for the supply chain as: k ⳱ B* B** k ⳮ D N N 兺 hiE[Ii(t)] ⳱ i⳱1 i⳱1 兺 hi{Bi ⳮ lTi Ⳮ E[Qi(t)] ⳮ E[QiⳭ1(t ⳮ Ti)]} (A5) kⳭ1 ⳱ B* B** kⳭ1 Ⳮ D, where where l is the expected demand rate, and E[ ] denotes expectation. k kⳮ1 We now pose an optimization problem to select the base stocks; namely, we minimize Equation (A5) subject to Equation (A3) and D ⳱ B*k ⳮ D 冢 兺 T 冣 Ⳮ D冢 兺 T 冣. i⳱1 i i⳱1 i Manufacturing & Service Operations Management Vol. 2, No. 1, Winter 2000, pp. 68–83 81

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