Myopic Analysis for Multi-Echelon Inventory Systems with Batch Ordering and Nonstationary/ Time-Correlated Demands*

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1 Vol. 26, No. 1, January 2017, pp ISSN EISSN DOI /poms Production and Operations Management Society Myopic Analysis for Multi-Echelon Inventory Systems with Batch Ordering and Nonstationary/ Time-Correlated Demands* Yi Yang Department of Management Science and Engineering, Zhejiang University, Hangzhou, , China, Yimin Yu Department of Management Sciences, College of Business, City University of Hong Kong, Hong Kong, Tingliang Huang Operations Management Department, Carroll School of Management, Boston College, Chestnut Hill, Massachusetts 02467, USA, W e provide an exact myopic analysis for an N-stage serial inventory system with batch ordering, linear ordering costs, and nonstationary demands under a finite planning horizon. We characterize the optimality conditions of the myopic nested batching newsvendor (NBN) policy and the myopic independent batching newsvendor (IBN) policy, which is a single-stage approximation. We show that echelon reorder levels under the NBN policy are upper bounds of the counterparts under both the optimal policy and the IBN policy. In particular, we find that the IBN policy has bounded deviations from the optimal policy. We further extend our results to systems with martingale model of forecast evolution (MMFE) and advance demand information. Moreover, we provide a recursive computing procedure and optimality conditions for both heuristics which dramatically reduces computational complexity. We also find that the NBN problem under the MMFE faced by one stage has one more dimension for the forecast demand than the one faced by its downstream stage and that the NBN policy is optimal for systems with advance demand information and stationary problem data. Numerical studies demonstrate that the IBN policy outperforms on average the NBN policy over all tested instances when their optimality conditions are violated. Key words: serial system; multiechelon; myopic policies; forecasting; MMFE History: Received: December 2014; Accepted: July 2016 by Panos Kouvelis, after 2 revisions. 1. Introduction The classical Clark Scarf (1960) model plays an important role in supply chain management. It is well known that the optimal echelon base-stock levels can be obtained by minimizing a sequence of nested convex functions recursively. However, in many cases, the material flows from one stage to another are in fixed batch sizes. For this reason, Chen (2000) and Huh and Janakiraman (2012) extend the Clark Scarf approach to settings with batch ordering by showing the optimality of the echelon (r, nq) policy for the infinite-horizon setting and the finite-horizon nonstationary-demand setting, respectively. However, the Clark Scarf approach suffers from the curse of dimensionality for systems with demand forecasting updates. It is well known that the optimal inventory policy even for the single-stage inventory *All authors contributed equally. model with dynamic forecasting is very complicated. For this case, myopic policies may be used to reduce computational complexity. That is why myopic policies have received considerable attention particularly for single-stage inventory models. For example, due to tractability, myopic policies have been used to gain various insights such as the value of information or information sharing (e.g., G ull u 1997, Lee et al. 2000), and quantifying bullwhip effect (e.g., Chen et al. 2000) with demand forecasting updates. The optimality of a myopic policy can validate its use in this type of analysis. In this study, we analyze myopic policies for a serial inventory system under a finite horizon with batch ordering and nonstationary demands. The system has N stages where demands occur at stage 1, and each stage i is replenished from stage i + 1 with a fixed lead time. The order/replenishment quantity of stage i in each period is required to be of an integral multiple of a specific batch size and batch sizes at upstream stages 31

2 32 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society are integral multiples of batch sizes at downstream stages. Without causing confusion, we use order and replenish/replenishment interchangeably to refer to the material flow throughout the study. Unfulfilled demand is fully backlogged. There is a unit holding cost for leftover inventory at each stage and a unit backordering cost for backordered demand at stage 1. Although the structure of the optimal policy is simple for independent demand, we have to solve a dynamic program recursively in a nested fashion in order to obtain the optimal echelon reorder levels (an echelon i refers to stage i and all its downstream stages). With demand forecast updates, the optimal policy no longer has a simple form. To simplify the computation procedure, we instead focus on the optimality of myopic policies through an exact analysis. We consider two types of myopic policies. We refer to the first type as the nested batching newsvendor (NBN) policy, that is, we provide the newsvendor-like exact cost formula for each stage which may depend on the echelon reorder levels of the downstream stages in future periods. Moreover, the exact cost is derived from an explicit convex function modified by the echelon (r, nq) policy. Notably, the myopic analysis for systems with batch ordering is quite different from the counterpart without batching ordering due to the following two effects: (i) the effect of batch ordering, making the objective function nonconvex; (ii) the effect of different batch sizes in different stages. We characterize the necessary and sufficient conditions under which the NBN policy is optimal, and show that the echelon reorder levels under the NBN policy are upper bounds of the optimal echelon reorder levels. Further, if demands are stochastically nondecreasing, then the NBN policy is in fact optimal; if demands have a positive minimum support, then the NBN policy can be optimal even for some stochastically decreasing demands. Moreover, the optimal echelon reorder level for one stage is nondecreasing in the optimal echelon reorder levels of the downstream stages in future periods. We refer to the second type of myopic policy as the independent batching newsvendor (IBN) policy. Under this myopic policy, the echelon reorder level of each stage can be obtained by solving a single-stage problem without knowing those of downstream stages. We also provide the optimality condition for the IBN policy, which is more restrictive than that for the NBN policy. We show that the echelon reorder levels under both the optimal policy and the IBN policy are bounded above by those of the NBN policy and below by those derived from the upper-bound system of Shang (2012). This implies that the IBN policy has bounded deviations from the optimal policy. We also extend our results to systems with the martingale model of forecast evolution (MMFE), which is proposed by Hausman (1969). We refer the readers to Hausman (1969), Aviv (2002, 2007), Lu et al. (2006), and Chen and Lee (2009) for detailed discussions concerning the MMFE. We first show that the optimal policy can be described by a state-dependent echelon (r, nq) policy. We then show a similar sufficient and necessary condition to ensure the optimality of the myopic policies and the related monotone properties. Note that the optimal policy is extremely hard to obtain under MMFE due to the curse of dimensionality. Hence, the myopic policies can reduce the computational complexity significantly. The dimension of the forecast demands of the NBN problem under MMFE faced by one stage is one more than the one faced by its downstream stage. Furthermore, for the single-stage problem if the updating process is stationary, we show that the NBN policy is optimal when the forecast demand in each period is nonnegative, which is plausible for demand updating. This result is similar to the result obtained by Iida and Zipkin (2006) regarding the optimality of myopic policies for the single stage model under the MMFE. For systems with advance demand information, we find that for stationary data the NBN policy is in fact optimal as the advance demand information does not involve forecast updating. Finally, we investigate the performance of these two myopic policies for a two-echelon problem with independent but stochastically decreasing demands. Overall, the average relative performance gaps for the two myopic policies are around 1% 4% under our settings, which is quite good. We also observe that the average performance of the IBN policy is better than the counterpart of the NBN policy when their optimality conditions are violated. We find that as the batch sizes increase, the performance of the two heuristics tends to improve. We also find the demand variability has different impacts on the performance of the two heuristics: as variability increases, the performance of the NBN policy tends to be stable, while the IBN policy performs much better. Hence, the IBN policy appears to be suitable for problems with high demand variability. Our contributions to the existing literature include: (i) We propose two types of myopic policies, the NBN policy and the IBN policy, for a general serial inventory system with batch ordering under a finite horizon and time-varying demands. We characterize the sufficient and necessary optimality conditions for the NBN policy. The IBN policy is easier to calculate but it has a more restrictive optimality condition. Numerical studies demonstrate that the average performance of the IBN policy is better than that of the NBN policy when their optimality conditions are violated. (ii) We show that, under the NBN policy, the optimal echelon reorder levels is nondecreasing in those of the downstream stage in future periods. We also show that

3 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society 33 echelon reorder levels under the NBN policy are the upper bounds of the counterparts of both the optimal policy and the IBN policy. More interestingly, we find that the echelon reorder levels under both the IBN policy and the optimal policy are sandwiched by the counterparts under the NBN policy and the upperbound system in Shang (2012). (iii) The myopic analysis for the multi-echelon model is quite different from the myopic analysis without batch ordering. Specifically, there are two unique effects with batch ordering: the effect of different batch sizes on different stages and the effect of batch ordering. For example, in contrast to the traditional multi-echelon models, the IBN policy may not be the upper bounds of optimal solutions. Our study is organized as follows. Section 2 reviews the related literature. In section 3, we provide a general model and preliminary results for the serial system with batch ordering. In section 4, we analyze the optimality of myopic policies for a basic model. In section 5, we conduct an extensive numerical study. Finally, we provide concluding remarks in section Literature Review There are primarily three streams of literature related to this study: multi-echelon inventory models, inventory problems with batch ordering, and myopic policies for inventory systems Multi-Echelon Inventory Models The stream of research on multi-echelon inventory models originates from Clark and Scarf (1960), who show the optimality of echelon base-stock policies by decomposing the value functions into the sum of cost functions for individual stages. Federgruen and Zipkin (1984) consider an infinite horizon model with i.i.d. demands. They develop an exact approach for both average cost and discounted cost criteria with normally distributed demands for a two-echelon system. Chen and Zheng (1994) present an alternative approach to show the optimality of echelon basestock policies and an algorithm to facilitate the computation. Chen and Song (2000) consider the Clark Scarf model with Markov modulated demands. van Houtum et al. (1996) provide an exact formula to compute the optimal echelon base-stock levels recursively, under the average cost setting with i.i.d. demands. Their analysis does not build upon Clark and Scarf (1960), instead they recursively optimize over the average cost. Gallego and Ozer (2003) consider a serial system with advance demand information. They show the optimality of a nested newsvendor solution for a two-echelon serial inventory model with i.i.d. demands under both finite and infinite horizons. They show that result by constructing a myopic cost function for the second stage. Dong and Lee (2003) consider the optimal policy for a serial inventory system with time-correlated demand. They develop a lower bound for the optimal echelon inventory position and an upper bound for the total system cost for a serial inventory model in the infinite horizon setting with a discounted-cost criterion. Shang and Song (2003) develop simple lower and upper newsvendor bounds and heuristics for the optimal policies for each stage. Gallego and Ozer (2005) provide a heuristic that requires solving one newsvendor problem per stage. They also provide a closed-form approximate upper bound that allows for accurate sensitivity analysis. Shang and Song (2007) study two serial supply chain models for which the optimal policies are echelon (r, q) policies and develop effective single-stage and closed-form bounds and approximations for the optimal policy parameters. Chao and Zhou (2007) obtain the probabilistic solution and newsvendor bounds for a continuous review serial system under the discounted infinite-horizon setting with i.i.d. demands. They utilize a recursion which is different from the Clark Scarf type of recursion but yields optimal echelon base-stock levels. Shang (2012) develops lower and upper bounds on the optimal base-stock level for each echelon by solving upper- and lower-bound systems, respectively. Both auxiliary systems are independent and single-stage systems. Based on these results, he also proposes a single-stage heuristic, as well as a myopic policy, by solving N independent single-stage problems with a weighted average of the cost parameters Inventory Models with Batch-Ordering Batch-ordering models have been extensively investigated in the extant literature. Veinott (1965a) demonstrates the optimality of the (r, nq) policy for the single-stage, periodic-review inventory problem with batch ordering. Chen (2000) extends Veinott s results to the multi-echelon model under the long-run average criterion. We show that under Chen (2000), the optimal echelon reorder levels can be obtained by a myopic policy by developing the exact analysis. Recently, Chao and Zhou (2009) generalize the results in Chen (2000) by exploring the structural properties of a serial system with batch ordering and fixed replenishment intervals which may contain multiple review periods. All of the above papers require the system to be stationary. Huh and Janakiraman (2012) extend the work of Chen (2000) to a nonstationary setting by proposing strong Q-jump-convexity. Our analysis on the optimality condition of myopic policies is also facilitated by the concept of strong Q-jump-convexity. Yang et al. (2014) further study the inventory-pricing problems with batch ordering.

4 34 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society However, to the best of our knowledge, the myopic analysis for inventory problems with batch ordering has not been addressed yet in the existing literature Myopic Policies Our study is also related to the myopic policies for the single-stage inventory models. Karlin (1960) and Veinott (1965b, c) show the optimality of myopic policies under independent demands. Graves (1999) and Aviv (2003) propose myopic policies for inventory models with specific demand forecasting methods. Johnson and Thompson (1975), Miller (1986), and Lovejoy (1990, 1992) establish sufficient conditions under which a myopic policy can be optimal under various demand forecasting models or time-dependent demands. Our study is closely related to Iida and Zipkin (2006) and Lu et al. (2006), both of which consider the single-stage inventory model under the MMFE. In particular, Iida and Zipkin (2006) show that if the forecast updating is stationary and the forecast demand in each period is nonnegative, then the myopic policy is optimal under MMFE. Lu et al. (2006) derive a sufficient and necessary condition under which the myopic policy is optimal under MMFE. However, the myopic policies for inventory systems with batch ordering have not been addressed in the existing literature. Gallego and Ozer (2003) show that a myopic policy is optimal for serial systems with advance demand information and stationary data. Our paper extends this literature by determining sufficient and necessary conditions for and proving the optimality of myopic policies for serial inventory systems with batch ordering. There are also several studies using the MMFE to investigate production planning and inventory control. G ull u (1996, 1997) uses the MMFE to assess the value gained from using a dynamic demand-forecasting model. Graves et al. (1998) analyze how to adjust the material requirement schedule when the safetystock plans are modified with the demand forecasts. Toktay and Wein (2001) focus on a forecast corrected inventory policy for an inventory model with finite capacity and obtain closed-form approximations. 3. Model and Preliminaries In this section, we first present our model and formulation. Then, we review some relevant results which facilitate our subsequent analysis. Throughout the study, we define R as a real number set, Z þ as a nonnegative integer set, and E½Š as the expectation operator. All proofs are relegated to the Appendix S Problem Description Consider an N-stage periodic-review serial inventory system, where Stage 1 replenishes its inventory from Stage 2, Stage 2 from Stage 3, and so on, Stage N replenishes its inventory from an outside supplier (denoted by Stage N + 1) that has ample stock. The planning horizon consists of T periods with a discount factor a 2 (0, 1]. The order quantity of Stage i in each period is required to be of integral multiples of a specific batch size (denoted by Q i ) and the partial shipment is not allowed. We assume that batch sizes at upstream stages are integral multiples of batch sizes at downstream stages, that is, there exists an integer m i which satisfies Q iþ1 ¼ m i Q i. The upstream fulfills the order from the downstream to the extent possible. There is a fixed lead time l i, which is nonnegative integer, between Stage i + 1 and Stage i, i = 1, 2,..., N 1. In other words, the order placed at Stage i + 1 at period t will be received by Stage i at period t þ l i. External demand only occurs at Stage 1 and unfulfilled demand is fully backordered. The demand in period t is a nonnegative random variable D t, t = 1,..., T, with the probability distribution function F t and the mean d t. Demands in different periods are independent but not necessarily identically distributed. Let D½i; tš ¼R tþl i k¼t D k be the total demand between period t and period t þ l i, that is, it is the lead time demand for Stage i, for i = 1, 2,..., N. Let H i be the stage holding cost rate at Stage i and let b be the backordering cost rate. Consistent with the existing literature of multi-echelon inventory theory, we assume that H 1 [ H 2 [ [ H N. Then, we can define the echelon holding cost rates as h i ¼ H i H iþ1, i = 1,..., N 1, and h N ¼ H N. Let c i be the unit ordering cost for Stage i. The sequence of events in each period is as follows: At the beginning of a period, (i) each Stage i places an order to Stage i + 1; (ii) each Stage i receives the order placed l i periods ago from Stage i + 1 which is due to arrive in this period; (iii) demand is realized. Consistent with the literature, we proceed to introduce the following key concepts. The echelon inventory level at Stage i is the inventory on hand at Stage i plus inventories at or in transit to all its downstream stages minus the total amount of backorders at Stage 1. The echelon inventory position at Stage i is the sum of the echelon inventory level at Stage i and the inventories in transit to Stage i. The echelon inventory stock at Stage i is the sum of the echelon inventory position at Stage i and the current order quantity for Stage i Existing Results With discretionary order sizes, Clark and Scarf (1960) demonstrate that the value function for such a serial inventory problem can be decomposed into the sum of N convex functions linked through the inducedpenalty cost functions, which are essentially penalty costs charged to an upstream stage if it cannot fulfill an order from its immediate downstream stage. Based

5 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society 35 on this result, they show that the echelon base-stock policy is optimal for each stage. Chen (2000) and Huh and Janakiraman (2012) extend this result to the setting with batch ordering and show that the echelon (r, nq) policy is optimal which operates as follows: If the echelon inventory position falls below a critical number r, then order enough amount to bring the echelon inventory stock (echelon inventory position plus inventory on order) into the interval [r, r + Q) oras close to it as possible; otherwise, do not order. The analysis in Huh and Janakiraman (2012) is facilitated by the following key concept. DEFINITION 1 (STRONG Q-JUMP-CONVEXITY). A function fðxþ : R! R is strong Q-jump-convex if for any z 1 2 R, z 2 2 R, and z 1 z 2, D Q fðz 1 ÞD Q fðz 2 Þ; where D Q fðxþ ¼fðx þ QÞ fðxþ. The following lemma summarizes some properties of strong Q-jump-convex functions (please refer to Huh and Janakiraman (2012) and Yang et al. (2014) for the proofs). LEMMA 1 (PROPERTIES OF STRONG Q-JUMP-CONVEXITIES). (a) If Q 2 ¼ mq 1 for some m 2 Z þ, then a strong Q 1 - jump-convex function is also strong Q 2 -jumpconvex. (b) If f(x) is strong Q-jump-convex and a is a positive scalar, then so is af(x). (c) The sum of any two strong Q-jump-convex functions is also strong Q-jump-convex. (d) If v(x) is strong Q-jump-convex and W is a random variable that takes only nonnegative values, then GðyÞ ¼E W fvðy WÞg is also strong Q-jump-convex. For any i = 1, 2,..., N, define the following inventory variables at the beginning of period t: x i;t = echelon inventory position for Stage i before orders are placed; z i;t = echelon inventory stock: echelon inventory position for Stage i after an order is placed but before demand is realized. We may drop the time index t for simplicity. Next, we briefly go through the procedure on how to decompose the original system into N single-stage systems, which is the basis of our subsequent analysis. Stage 1: For period t, define P t ðxþ ¼E½h 1 ðx D½1; tšþ þ ðb þ H 1 ÞðD½1; tš xþ þ Š ð1þ as the expected cost incurred at period t þ l 1 when the echelon inventory position at period t is x. Define A Q ðxþ ¼fyjy ¼ x þ mq; m 2 Z þ g. Since the order size of Stage 1 must be a nonnegative integral multiple of Q 1, the replenishment decision space in terms of z 1;t can be characterized by A Q 1 ðx 1;t Þ. We adopt the following cost accounting scheme: At time t, we charge the inventory holding/backordering costs of Stage 1 incurred at time t þ l 1. This cost accounting scheme only shifts costs across time points, which does not affect the total inventory holding and backordering costs over the entire planning horizon. It has an intuitive interpretation: an order placed by Stage 1 at time t does not have an effect on the inventory holding/backordering cost of Stage 1 until time t þ l 1. Following the definition, the net inventory level of Stage 1 at time t þ l 1 is z 1;t D½1; tš, and hence, the corresponding expected holding/backordering cost is a l 1 P t ðz 1; t Þ. Assume that Stage 1 has ample supply from Stage 2. Under this assumption, the inventory problem for Stage 1 becomes a single-stage problem with the following dynamic programming formulation: C 1;t ðx 1 Þ¼ min c 1 x 1 þ G 1;t ðz 1 Þ ; ð2þ z 1 2A Q 1 ðx 1 Þ where G 1;t ðz 1 Þ¼c 1 z 1 þ a l 1 P t ðz 1 ÞþaE½C 1;tþ1 ðz 1 D t ÞŠ. We assume that C 1;Tþ1 ðx 1 Þ¼ c 1 x 1. Recall A Q 1 ðx 1 Þ¼ fyjy ¼ x 1 þ mq 1 ;m 2 Z þ g. Define r 1;t ¼ maxfzjd Q 1 G 1;t ðzþ ¼0g: Huh and Janakiraman (2012) show that C 1;t ðx 1 Þ is strong Q 1 -jump-convex for any t. Therefore, at period t, the echelon ðr 1;t ; nq 1 Þ policy is optimal for Stage 1. Stage 2: However, Stage 1 may not be able to obtain ample supply from Stage 2, namely, the echelon inventory stock of Stage 1 is constrained by the echelon inventory level of Stage 2. If Stage 2 is not able to raise the inventory stock of Stage 1 to the interval ½r 1;t ; r 1;t þ Q 1 Þ, then the following induced-penalty cost C 1;t ðx 2 Þ is charged to Stage 2: C 1;t ðxþ ¼ G 1;tðxÞ G 1;t ðkxk Q 1 r 1;t Þ if x\r 1;t ; ð3þ 0 otherwise; where kxk Q r ¼fry\r þ Qjy ¼ x þ mq; for some m 2 Z þ g. Similarly, at time t, we charge the inventory holding costs and the induced-penalty cost of Stage 2 incurred at time t þ l 2. Then, the optimal cost function C 2;t of Stage 2 can be expressed as C 2;t ðx 2 Þ¼ min f c 2 x 2 þ G 2;t ðz 2 Þg; z 2 2A Q 2 ðx 2 Þ ð4þ where G 2;t ðz 2 Þ¼c 2 z 2 þ a l 2 h 2 ðz 2 d 2;t Þþa l 2 E½C 1;tþl2 ðz 2 D½2; tšþš þ ae½c 2;tþ1 ðz 2 D t ÞŠ and d 2;t ¼ E½D ½2; tšš (note

6 36 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society that z 2 D½2; tš is the echelon inventory level for Stage 2 at time t þ l 2 ). Similarly, we assume that C 2;Tþ1 ðx 2 Þ¼ c 2 x 2. As a result, the optimal echelon reorder level for Stage 2 in period t can be defined as r 2;t ¼ maxfzjd Q 2 G 2;t ðzþ ¼0g: Moreover, the objective functions C 2;t ðx 2 Þ are strong Q 2 -jump-convex and hence an echelon ðr 2;t ; nq 2 Þ policy is optimal. Stage i, 2< i N: Define d i;t ¼ E½D½i; tšš. Similarly, we assume that Stage i faces an ample supply from Stage i + 1. We can recursively define ( C i 1;t ðxþ¼ G i 1;tðxÞ G i 1;t ðkxk Q i 1 r i 1;t Þ if x\r i 1;t ; 0 otherwise ; ð5þ C i;t ðx i Þ¼ min c i x i þg i;t ðz i Þ ; z i 2A Q i ðx i Þ where G i;t ðz i Þ¼c i z i þ a l i h i ðz i d i;t Þþa l i E½C i 1;tþli ðz i D½i; tšþš þ ae½c i;tþ1 ðz i D t ÞŠ. Note that A Q i ðx i Þ¼fyjy¼ x i þ mq i ; m 2 Z þ g. Moreover, we set the terminal condition as C i;tþ1 ðxþ ¼ c i x for i = 1,..., N. Moreover, the echelon reorder level can be characterized by r i;t ¼ maxfzjd Q i G i;t ðzþ ¼0g: Then, an echelon ðr i;t ; nq i Þ policy is optimal for Stage i. The following theorem summarizes the main results from Huh and Janakiraman (2012.) THEOREM 1 (HUH AND JANAKIRAMAN 2012). Let C t ðx 1 ;...; x N Þ be the optimal cost from period t and onward. For any period t = 1, 2,..., T, and Stage i = 1, 2,..., N, we have (a) C t ðx 1 ;...; x N Þ¼R N i¼1 C i;tðx i Þ. (b) C i;t ðx i Þ and C i;t ðxþ are strong Q i -jump-convex. (c) The echelon ðr i;t ; nq i Þ policy is optimal for Stage i. Based on Equations (2), (4), and (5), the optimal echelon reorder levels r i;t ; i ¼ 1;...; N, can be obtained by solving N nested functions sequentially. Even with the decomposition property, Huh and Janakiraman s procedure still suffers from the curse of dimensionality, especially for systems with high dimensions. 4. Analysis of Myopic Policies In this section, we first consider systems with independent demands and then systems with demand forecast updates Systems with Independent Demands In this subsection, we propose two types of myopic policies, the myopic NBN policy and the myopic IBN policy, which are simple policies that are amenable to exact analysis. To facilitate our analysis, we first analyze when the myopic policy is optimal for the single-stage inventory model with batch ordering. Let g t ðxþ ¼ E½Hðx D t Þ þ þ bðd t xþ þ Š be the expected inventory cost in period t, which is convex. Then the dynamic recursion for the single-stage model is V t ðxþ ¼ min f cx þ G t ðzþg; z2a Q ðxþ ð6þ with G t ðzþ ¼cz þ g t ðzþ þae½v tþ1 ðz D t ÞŠ and the boundary condition V Tþ1 ðxþ ¼ cx. Let r M t be the optimal reorder level under the myopic policy. The idea of constructing the myopic policy is to use cq to approximate D Q V tþ1 ðzþ, the Q-difference of the cost-to-go function. Define r t as the optimal reorder levels of the problem (6). Note that for a generic period t, D Q V t ðxþ ¼ cq if x\r t; cq þ D Q ð7þ G t ðxþ otherwise. Based on Equation (7), the approximation has no gap if the initial inventory level is always below the reorder level. We can show that if r M t D t r M tþ1, then the starting inventory level in each period t is always no more than r M t given x 1 r M 1 under the myopic policy. Thus, we can show the optimality of the myopic policy based on an inductive argument. THEOREM 2 (OPTIMALITY OF MYOPIC POLICIES IN A SINGLE STAGE). Let r M t ¼ maxfzjð1 aþcq þ D Q g t ðzþ ¼ 0g. The sufficient and necessary condition for the optimality of the myopic ðr M t ; nqþ policy for the dynamic program (6) is: r M t D t r M tþ1 with probability 1 for all t. By the definition, r M t only depends on the singleperiod cost function g t ðzþ, but is independent of the cost-to-go function V tþ1 ðzþ. Theorem 2 shows that under the batch ordering setting, the condition that the optimal reorder levels satisfy r M t D t r M tþ1 with probability 1 for all t is a sufficient and necessary condition for the optimality of the myopic policy of the single-stage model. This result generalizes the one under the discretionary ordering size setting in Lu et al. (2006). By applying the same approach, we construct the myopic policy for the general serial inventory system with batch ordering stated in section 3. In particular, for each Stage i, we use c i Q i to approximate

7 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society 37 D Q i C i;tþ1 ðzþ in each period t. Replacing C i;tþ1 ðzþ with c i z in Equation (5) and taking Q-difference, we can recursively define D Q 1 ^G1;t ðz 1 Þ¼ð1 aþc 1 Q 1 þ a l 1 D Q 1 P t ðz 1 Þ; r M 1;t ¼ maxfzjdq 1 ^G1;t ðzþ ¼0g: and for i = 2,..., N, ð8þ D Q i ^C i 1;t ðzþ¼ 8P mi 1 j¼1 DQ i 1 ^Gi 1;t ðzþðj 1ÞQ i 1 Þ if z\r >< M i 1;t Q i; P ^mi 1 j¼1 DQ i 1 ^Gi 1;t ðzþðj 1ÞQ i 1 Þ if r M i 1;t Q i z\r M i 1;t >: ; 0 ifzr M i 1;t ; D Q i ^Gi;t ðz i Þ¼ð1 aþc i Q i þa l i h i Q i þa l i E½D Q i ^C i 1;tþli ðz i D½i;tŠÞŠ; r M i;t ¼ maxfzjdq i ^Gi;t ðzþ¼0g; kzk Q i 1 r M z i 1;t Q i 1 ð9þ where ^m i 1 ¼, and refer to Equation (1) for the definition of P t ðxþ. Under the approximation scheme, ^Ci;t ðzþ and ^G i;t ðzþ, as well as their Q-difference, only depend on the one-period cost of Stage i and possibly the oneperiod cost of its all downstream stages. Note that D Q i ^Gi;t ðzþ also depends on the echelon reorder levels in downstream stages, which implies that the echelon reorder level of Stage i is nested upon those of its downstream stages. Therefore, we refer to ðr M i;t ; nq iþ policy as the nested myopic policy or the nested batching newsvendor (NBN) policy since they are the solutions of NBN problems. For simplicity, we may refer to it as the NBN policy. Notably, Equation (9) is different from the myopic analysis for serial systems without batch ordering (see Gallego and Ozer 2003): (i) the effect of different batch sizes on different stages, that is, Q iþ1 ¼ m i Q i ; (ii) the effect of batch ordering, that is, even for z r M i 1;t, we need to know whether the distance from z to r M i 1;t is less than Q i. Further, C i 1;tþli also depends on the echelon reorder levels of downstream stages with these two aforementioned effects. As a result, the analysis of the optimality of the myopic policy in the presence of batch ordering is more complicated. We impose the following assumption on the myopic echelon reorder levels in different periods. ASSUMPTION 1 (OPTIMALITY CONDITION). For any t = 1, 2,..., T and any i = 1, 2,...,N, r M i;t D t r M i;tþ1 with probability 1. This condition mimics that for the single-stage model. Surprisingly, in spite of the two aforementioned unique effects, the following theorem indicates that this simple assumption is the sufficient and necessary condition for the optimality of the NBN policy. THEOREM 3 (OPTIMALITY OF THE NBN POLICY). (a) The myopic echelon reorder levels are upper bounds of the optimal echelon reorder levels, that is, r M i;t r i;t. (b) For each Stage i, the myopic echelon ðr M i;t ; nq iþ policy is optimal if and only if Assumption 1 holds. (c) r M i;t is increasing in r M j;tþl i j, j = 1, 2,...,i 1, where L i j ¼ R i m¼jþ1 l m. From the approximation scheme, the myopic policy simply uses a lower bound c i Q i to approximate D Q i C i;tþ1 ðzþ in each period t. Thus, combined with strong Q-jump convexity, Part (a) indicates that the myopic echelon reorder levels are upper bounds of the optimal echelon reorder levels. Although the decomposition algorithm introduced in section 3.2 significantly simplifies computation by converting the original N-dimension problem into solving a series of N single-dimension problems, the computation is still quite involved because one needs to solve a multiple-period inventory problem for each stage whose echelon reorder level depends on all of its downstream ones. However, Theorem 3(b) demonstrates that if Assumption 1 holds, one only needs to sequentially solve N nested newsvendor problems. Theorem 3(c) demonstrates the monotone property of the optimal echelon reorder levels under the NBN policy. If it is optimal to stock more inventories at the downstream stages in the future, then correspondingly the optimal echelon reorder level for this stage should also be higher. However, Assumption 1 is not straightforward to verify, as one has to first calculate the myopic echelon reorder levels. Next, we shall further provide an explicit sufficient condition that ensures this assumption to be satisfied. THEOREM 4. If the demand is stochastically nondecreasing in time and the initial echelon inventory level at each stage i is below r M i;1, then the myopic echelon reorder levels r M i;t ; i ¼ 1; 2;...; N; are nondecreasing in t. As a result, the NBN policy is optimal. The stochastically nondecreasing demands are particularly suitable for a product in the growth stage of its product life cycle or daily consumables with relatively stable demands. Note that Theorem 4 implies that the NBN policy is optimal if the demands are independent and identically distributed. The optimal

8 38 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society echelon reorder levels increase in time, which is consistent with the results for the single-stage inventory models. We have the following two corollaries. COROLLARY 1. If the demands are independent and identically distributed and the initial echelon inventory level at stage i is below r M i;1, then the NBN policy is optimal. COROLLARY 2. If the support of D tþ1 is ½d min ; d max Š such that 0 \ d min \ d max, then the NBN policy in period t is optimal if D t D tþ1 þ d min and x i is below r M i;t. Hence, if D t follows the the same distribution as D T þðt tþd min for all t, where d min the minimum support of D T, then the NBN policy is optimal in each period t. Corollary 2 is due to the fact that under the NBN policy, with demand D t D tþ1 þ d min, we must have r M t þ d min ¼ r M tþ1, since it is not optimal to intentionally backorder a demand. Note that for a supply firm, usually it has regular customers possibly under a long-term supply contract and ad hoc customers. Then d min can be viewed as the demand rate for regular customers. Note that even if demands are stochastically decreasing as in Corollary 2, the NBN policy can still be optimal. Indeed, we numerically observed that the NBN policy continues to be optimal for some instances with stochastically decreasing normal demands Independent Batching Newsvendor Policy. Although we can solve the NBN problems efficiently, in order to obtain the optimal myopic echelon reorder levels, we still have to solve a series of nested newsvendor problems in a recursive fashion. Next, we introduce the IBN policy. Mathematically, the idea of this myopic policy is to use the term c i Q i to approximate D Q i C i;tþ1 ðzþ and the term R m i 1 j¼1 DQ i 1 ^Gi 1;t ðz þðj 1ÞQ i 1 Þ to approximate D Q i ^Ci 1;t ðzþ in Equation (9). In particular, for Stage 1, D Q 1 G1;t ~ ðz 1 Þ¼D Q 1 ^G1;t ðz 1 Þ, and for Stage i, i = 2,..., N, we can by simplifying the terms obtain D Q i ~ Gi;t ðz i Þ¼ð1 aþc i Q i þ Xi þ Xi j¼2 j¼2 P i a l k¼j k ð1 aþc j 1 Q i P i P i a l k¼j k h j Q i þ a l k¼1 k D Q i EPtþ Pi P ðz i X i l k¼2 k D tþj Þ: j¼0 l k k¼2 ð10þ It is clear that this approximation is rougher than the approximation scheme in Shang (2012), but with the added advantage of simplicity and transparency. However, unlike the traditional newsvendor policy, with batch ordering we have to deal with the effects of batch ordering and different batch sizes at different stages. For i = 1, 2,..., N, we define the independent myopic echelon reorder levels as ~r M i;t ¼ maxfzjdq i ~ Gi;t ðzþ ¼0g: Unlike the NBN policy, ~r M i;t are independent of the echelon reorder levels in downstream stages which is appealing, since we can obtain the echelon reorder levels for different stages independently. Essentially, the IBN problem is a single-stage problem. To obtain the echelon reorder level at stage j, we only need to minimize the expected cost incurred during the echelon lead time, that is, the total lead time from stage j to stage 1. In the following, we present a sufficient condition under which the IBN policy is optimal. Consequently, we can reduce the problem of finding the optimal echelon reorder levels into a series of IBN problems. First, we shall show the sufficient conditions for the IBN policy and it provides lower bounds for the NBN policy. THEOREM 5 (OPTIMALITY OF THE IBN POLICY). (a) For each Stage i, the IBN policy is optimal, that is, the myopic echelon ð~r M i;t ; nq iþ policy is optimal if Assumption 1 holds and ~r M i;t þ Q i D½i; tš ~r M i 1;tþl i with probability 1 for all i and t. (b) The echelon reorder level under the IBN for each stage is a lower bound of the counterpart under the NBN policy, that is, ~r M i;t rm i;t. In general, the echelon reorder levels under the IBN policy can be either larger or smaller than the optimal echelon reorder levels (see Table 2 for a counterexample). According to the approximation scheme, it is not surprising to see that the optimality condition of the IBN policy is more restrictive than that of the NBN policy. Theorem 5(b) indicates that the echelon reorder level under the IBN policy for each stage is a lower bound of the counterpart under the NBN policy, that is, ~r M i;t rm i;t for any i and t. Recall that Theorem 3 shows that the echelon reorder levels under the NBN policy are upper bounds of the optimal echelon reorder levels, i.e, r M i;t r i;t for any i and t. When the optimality condition for the NBN policy holds, the above results imply r M i;t ¼ r i;t ~r M i;t for any i and t. In this case, the NBN policy definitely outperforms the IBN policy. Suppose the optimality condition for the NBN policy does not hold. However, the numerical study in section 5 demonstrates that ~r M i;t could be

9 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society 39 Table 1 Comparion with the Lower-Bound System of Shang (2012) c 1 h 1 s2;t u r2;t M 2 3 {261, 167, 101, {243, 165, 99, 61, 37} 59, 35} {261, 167, 101, {279, 185, 111, 61, 37} 67, 40} {261, 167, 101, {284, 189, 113, 61, 37} 68, 41} larger or smaller than r i;t (see Table 3). If ~r M i;t r i;t for any i and t, then the IBN policy outperforms the NBN policy; otherwise, it is not clear which myopic policy performs better. Notice that the otherwise setting includes two cases: (i) r M i;t \ r i;t for any i and t; (ii) there may exist some i and t such that r M i;t r i;t, while there may exist other i and t such that r M i;t \ r i;t. Shang (2012) provides both lower and upper bounds on the optimal echelon reorder level for each stage by solving a series of independent single-stage systems with batch size 1 (his result can be generalized to a fixed batch size Q i for stage i). Specifically, when considering the ordering policy for Stage j, in his upper-bound system, Stage i always orders up to x iþ1 for i < j; in his lower-bound system, when considering the policy for Stage j, set H i ¼ H j and c i ¼ 0 for all i < j. Define s l i;t and s u i;t as the solutions to the upper-bound and lower-bound systems, respectively. Under both lower and upper-bound systems, Stage 1 is always optimally solved, that is, s l 1;t ¼ s u 1;t ¼ r 1;t. Recall that ~r M 1;t ¼ r M 1;t. Interestingly, the following proposition shows that the echelon reorder levels under the NBN and IBN policy are upper bounds of the echelon reorder levels for the upper-bound system. PROPOSITION 1 (COMPARISON WITH SHANG 2012). Both the echelon reorder levels under the IBN policy and the NBN policy are upper bounds of the counterparts under the upper-bound system of Shang (2012), that is, r M i;t ~rm i;t ; r i;t s l i;t, for any i and t. As a result, j~r M i;t r i;t jr M i;t s l i;t. ~r M 2;t {181, 113, 68, 41, 24} {258, 164, 99, 60, 36} {269, 173, 104, 63, 38} a = 0.95, h 2 ¼ 0:05, c 2 ¼ 1, b = 1, c = 0.6, r 0 l 0 ¼ 0:4, L 1 ¼ L 2 ¼ 1, T = 10, D m ¼ 100, Q 1 ¼ Q 2 ¼ 1 and l 0 ¼ 100. The implication of Proposition 1 is that the echelon reorder levels under the IBN policy and the optimal policy are sandwiched by the counterparts under the NBN policy and the upper-bound system. In this sense, the IBN policy has bounded deviations from the optimal policy, that is, the maximum deviation for stage i is r M i;t s l i;t. However, we find that our two policies have no clear relationship with the policy induced by the lowerbound system of Shang (2012). Table 1 presents the echelon reorder levels of the first five periods and illustrates that s u 2;t canbeeitherlargerorsmallerthanrm 2;t and ~r M 2;t. Consider a two-stage serial system in which thedemanddistributionineachperiodt is D t c t D 0, t = 1,..., T, whered 0 follows the truncated normal distribution Nðl 0 ; r 0 Þ with the mean l 0 and variance r 2 0. In particular, in the first instance with c 1 ¼ 2and h 1 ¼ 3, s u 2;t [ rm 2;t [ ~rm 2;t ; in the second instance, r M 2;t [ su 2;t [ ~rm 2;t ; in the last instance, rm 2;t [ ~rm 2;t [ su 2;t Systems with Demand Forecast Updates In this subsection, we extend our results to a serial inventory model with demand forecast updates. For simplicity, we assume that the demand forecast update follows the MMFE, which is proposed by Hausman (1969), and is further studied in Graves et al. (1986), Heath and Jackson (1994) and Oh and Ozer (2013). As mentioned in Lu et al. (2006), the MMFE is quite general and flexible as it can represent nonstationary and time-correlated demands and accommodate judgmental forecasts. It can also incorporate commonly used time series models such as the autoregressive moving average model. With demand forecast updates, the demand process fd t ; t ¼ 1; 2;...; Tg can be nonstationary and correlated over time. At each period t, the firm generates forecasts of the demands for all future periods during the entire horizon and makes the ordering decisions based on the current inventory state and the demand forecast. Let D t;tþi be the forecast made at the end of period t for the demand in period t+i, i = 0,..., T i. Clearly, D t;t ¼ D t. Let D t ¼ðD t;tþ1 ;...; D t;t Þ be the demand forecast vector made at the end of period t, Table 2 Performance Comparison of Heuristic Policies Parameter settings Fixed parameters L 1 ¼ 1, L 2 ¼ 1, h 1 ¼ 0:1, c 1 ¼ 2, Q 1 ¼ 4 l 0 ¼ 100, a = 0.95, T = 10 Viariable parameters m 1 2f1; 2; 3; 4g, h 2 2f0:05; 0:1; 0:2; 0:3g, c 2 ¼f1; 1:5; 2; 4g b 2 {1, 2, 4, 6}, c 2 {0.5, 0.6, 0.7, 0.8}, r 0 l 2f0:1; 0:2; 0:3; 0:4g 0 Overall performance of two policies Myopic policies No. of instances No. of optimality No. of suboptimality No. of dominating instances The NBN policy The IBN policy Myopic policies Aver. ξ (%) ξ 1% 1% < ξ 5% ξ > 5% The NBN policy (57%) 1598 (42%) 33 (1%) The IBN policy (89%) 456 (11%) 0

10 40 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society Table 3 Comparisons of Echelon Reorder Levels under Different Policies Relationship r2;t M ~r 2;t M r 2;t r2;t M r 2;t ~r 2;t M Mixed No. of instrances Specific examples c r 2;10 r M 2;10 ~r M 2;10 ξ(nbn) ξ(ibn) 0.8 {338, 273, 218, 174, 139} {338, 273, 218, 174, 139} {335, 270, 216, 172, 138} {303, 212, 148, 103, 72} {305, 213, 149, 104, 72} {301, 211, 147, 103, 71} {192, 76, 30, 11, 4} {209, 83, 32, 13, 4} {199, 79, 31, 12, 4} m 1 ¼ 1, h 2 ¼ 0:05, c 2 ¼ 2, b = 2, r l ¼ 0:1 and other parameter values are set in Table 2. where D 0 represents the initial forecast vector. We consider the additive forecast updates and the multiplicative forecast updates, respectively. For additive updates, define e t;tþi ¼ D t;tþi D t 1;tþi as the forecast update made at the end of period t for demand in period t + i. Denote by e t ¼ðe t;tþ1 ;...; e t;t Þ the demand forecast update vector made at the end of period t. We assume that the forecasts are unbiased, that is, E½e s;t Š¼0; s t. We also assume that the forecast updates e t are independent over time. The forecast updates within a period may be correlated because they might rely on the same or related information. For multiplicative updates, we similarly define e t;tþi ¼ D t;tþi =D t 1;tþi as the demand forecast update. Here, E½e t;tþi Š¼1. We assume that both D t;tþi and e t;tþi are positive and the forecast updates are independent over time t. As before, the forecast updates within a period are not necessarily independent. For ease of analysis, we mainly focus on the additive model throughout the study. However, most of the results are also valid for the multiplicative model. For i = 1, 2,..., N, we denote by D s ½i; tš ¼R tþl i k¼t D s;k and e s ½i; tš ¼R tþl i k¼t e s;k the total updated demand and the forecast updates from period t and period t þ l i under the aggregated information at period s t, respectively. Stage 1: At period t, we again charge the inventory holding/backordering costs of Stage 1 incurred at time t þ l 1 and hence, the corresponding expected holding/backordering cost is P t ðx; D t 1 Þ¼E½h 1 ðx Xl 1 m¼0 þðb þ H 1 Þ D tþm Þ X l 1 m¼0 D tþm x! þ jd t 1 Š: ð11þ Under the assumption that Stage 1 has ample supply from Stage 2, the inventory problem of Stage 1 becomes a single-stage problem with the following dynamic formulation: C 1;t ðx 1 ; D t 1 Þ¼ min c 1 x 1 þ G 1;t ðz 1 ; D t 1 Þ ; z 1 2A Q 1 ðx 1 Þ where G 1;t ðz 1 ; D t 1 Þ¼c 1 z 1 þ a l 1 P t ðz 1 ; D t 1 ÞþaE ½C 1;tþ1 ðz 1 D t 1;t e t;t ; D t ÞŠ. Define r 1;t ðd t 1 Þ¼maxfzjD Q 1 G 1;t ðz; D t 1 Þ¼0g: Stage i 2 i N: Similarly, we assume that Stage i faces an ample supply from Stage i + 1. We can recursively define C i 1;t ðx; D t 1 Þ¼ ( G i 1;t ðx; D t 1 Þ G i 1;t ðkxk Q i 1 r i 1;t ; D t 1 Þ if x\r i 1;t ; 0 otherwise; C i;t ðx i ; D t 1 Þ¼min zi 2A Q i ðx i Þ c ix i þ G i;t ðz i ; D t 1 Þg; ð12þ where G i;t ðz i ;D t 1 Þ¼c i z i þ a l i h i " #! z i E Xl i D tþm jd t 1 m¼0 þ a l i E½C i 1;tþli ðz i Xl i m¼0 þ ae½c i;tþ1 ðz i D t ;D t ÞjD t 1 Š: D tþm ;D tþli 1ÞjD t 1 Š Moreover, the echelon reorder levels can be characterized by r i;t ðd t 1 Þ¼maxfzjD Q i G i;t ðz; D t 1 Þ¼0g: THEOREM 6. For any period t = 1, 2,..., T, and Stage i = 1, 2,..., N, we have (a) C i;t ðx i ; D t 1 Þ and G i;t ðx i ; D t 1 Þ are strong Q i - jump-convex for any given D t 1 ; (b) an echelon ðr i;t ðd t 1 Þ; nq i Þ policy is optimal for Stage i. It can be seen from Theorem 6 that the optimal policy can be described as a state-dependent echelon (r, nq) policy where the optimal echelon reorder levels depend on D t 1 in period t. It is clear that the corresponding dynamic recursion even for the individual stages is multi-dimensional. Hence, it is extremely challenging to obtain the optimal solution.

11 Production and Operations Management 26(1), pp , 2016 Production and Operations Management Society 41 To construct the myopic policy, we again use c i Q i to approximate D Q i C i;tþ1 ðz; D t 1 Þ in each period t. We can recursively define D Q 1 ^G1;t ðz 1 ; D t 1 Þ¼ð1 aþc 1 Q 1 þ a l 1 D Q 1 P t ðz 1 ; D t 1 Þ; r M 1;t ðd t 1Þ ¼maxfzjD Q 1 ^G1;t ðz; D t 1 Þ¼0g; ð13þ and for i = 2,..., N, D Q i ^C i 1;t ðz;d t 1 Þ¼ 8P mi 1 j¼1 DQ i 1 ^Gi 1;t ðzþðj 1ÞQ i 1 ;D t 1 Þ if z\r M i 1;t Q i; >< P ^mi 1 j¼1 DQ i 1 ^Gi 1;t ðzþðj 1ÞQ i 1 ;D t 1 Þ if r M i 1;t Q i z \r M i 1;t >: ; 0 ifzr M i 1;t ;; D Q i ^Gi;t ðz i ;D t 1 Þ¼ð1 aþc i Q i þa l i h i Q i þa l i E½D Q i ^C i 1;tþli ðz i D½i;tŠ;D tþli 1ÞjD t 1 Š: r M i;t ðd t 1Þ¼maxfzjD Q i ^Gi;t ðz;d t 1 Þ¼0g; ð14þ kzk Q i 1 r M z i 1;t Q i 1. where ^m i 1 ¼ By the same argument as in the independent demand case, we can show the following results THEOREM 7. (a) The echelon reorder levels of the myopic policy are upper bounds of the optimal echelon reorder levels, that is, r M i;t ðd t 1Þ r i;t ðd t 1 Þ. (b) For each Stage i, the myopic echelon ðr M i;t ðd t 1Þ; nq i Þ policy is optimal if and only if r M i;t ðd t 1Þ D t r M i;tþ1 ðd tþ; i ¼ 1; 2;...; N with probability 1 for all t. (c) The NBN policy is optimal if given any D t 1, D tþm is stochastically nondecreasing in m, m 0. Theorem 7 extends the single-stage result given by Lu et al. (2006) to the multiple stages with batch ordering. Unlike the independent demand case, the safety stock levels depend on not only the echelon reorder levels in downstream stages, but also the forecasted demand. With the correlated demand, the curse of dimensionality is exacerbated because the state space has to contain more information about demand forecast. Thus, to calculate the NBN policy, we have to sequentially solve N nested newsvendor problems with high dimensions. However, the following theorem provides a recursive expression of the NBN policy that reduces the computation effort. THEOREM 8. The echelon reorder level for each stage at period t under the NBN policy is given by r M i;t ðd t 1Þ ¼ Xi 1 D t 1 ½i j; t þ Xj j¼0 where h i;t is defined as: m¼1 l i mþ1 Š þ h i;t ðh i 1;tþli ;...; h 1;tþ P i j¼2 l jþ; h i;t ¼ maxfzjd Qi^g i;t ðz; h i 1;tþli ;...; h 1;tþ P i j¼2 l jþ¼0g: ð15þ The expressions of function ^g i;t ðþ is given in the proof of Theorem 8. To better understand the expressions stated in Equation (15), we look into those of Stages 1 and 2. In particular, for Stage 1, r M 1;t ðd t 1Þ ¼D t 1 ½1; tšþh 1;t : ð16þ The expression stated in Equation (16) explains why the myopic policy is popular and desirable, although it might be suboptimal. At each period t, its optimal echelon reorder level can be represented by the sum of two terms: the latest forecast of the subsequent l 1 -period demand D t 1 ½1; tš and a safety stock, h 1;t, that depends only on the distribution of the forecast update. Hence, the safety stock of Stage 1 can be calculated independently without updating the demand information. If we further assume that the cost parameters are stationary, then the safety stock is a constant. Therefore, in order to calculate the echelon reorder level in each period, we only need to adjust the latest demand forecast according to the realized demand in the previous period. Such a result significantly reduces the computation efforts. For Stage 2, the optimal echelon reorder level is given by r M 2;t ðd t 1Þ ¼D t 1 ½2; tšþd t 1 ½1; t þ l 2 Šþh 2;t ðh 1;tþl2 Þ: ð17þ It can seen from Equation (17) that the optimal echelon reorder levels of Stage 2 can also be represented by the sum of two parts: the latest forecast of the subsequent l 1 þ l 2 -period demand D t 1 ½2; tšþ D t 1 ½1; t þ l 2 Š, and a safety stock, h 2;t. Note that the latest demand forecast consists of two terms, D t 1 ½2; tš and D t 1 ½1; t þ l 2 Š, which represent the future demand forecast from Stages 2 and 1, respectively. Unlike Stage 1, the safety stock level in Stage 2 also depends on that in Stage 1, which increases computation efforts. From Equation (15), the upstream stage will face a one-more dimension for the forecast demands than the one faced by its downstream stage. However, the nested myopic policy has significantly reduced the problem dimension.