Optimal Control Policy for Capacitated Inventory

Submitted to Operations Research manuscript OPRE-2011-08-429.R2 Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication.

Optimal Control Policy for Capacitated Inventory Systems with Remanufacturing Xiting Gong, Xiuli Chao Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48109, {xitingg, xchao}@umich.edu

This paper studies the optimal control policy for capacitated periodic-review inventory systems with remanufacturing. The serviceable products can be either manufactured from raw materials or remanufactured from returned products; but the system has finite capacities in manufacturing, remanufacturing, and/or total manufacturing/remanufacturing operations in each period. Using L-natural convexity and lattice analysis, we show that, for systems with a remanufacturing capacity and a manufacturing/total capacity, the optimal remanufacturing policy is a modified remanufacture-down-to policy and the optimal manufacturing policy is a modified total-up-to policy. Our study reveals that the optimal policies always give production priority to remanufacturing for systems with a remanufacturing capacity and/or a total capacity; but this priority fails to hold for systems with a manufacturing capacity. Key words : inventory model; product returns; remanufacturing; finite capacity; optimal policy History : Received August 2011; revisions received August 2012, November 2012; accepted February 2013

1. Introduction Remanufacturing is the process of restoring used or broken components/products to like-new condition (Toktay et al. 2000). Examples of remanufactured products include tires, engines, single-use cameras, and toner cartridges, etc. As part of the closed-loop logistics, remanufacturing conserves not only the raw material content but also much of the value added during the production process required to manufacture a new product. Since the profit margins of remanufactured products are 1

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Gong and Chao: Optimal Control Policy for Capacitated Remanufacturing Systems Article submitted to Operations Research; manuscript no. OPRE-2011-08-429.R2

usually higher than those of new products, many industries have increasingly recognized the competitive advantages of the remanufacturing operations (Chao et al. 2009). In fact, remanufacturing has been viewed by many as the largest untapped opportunity for the manufacturing sector, and some even consider it as the next great opportunity for boosting the U.S. productivity (Giuntini and Gaudette 2003). Besides economic incentives, the enforcement of environmental laws is another main motivation for firms to remanufacture used/returned products (van der Laan et al. 1999). In contrast to the large remanufacturing industry, Lund and Hauser (2010) report that “remanufacturing companies in the United States are small to medium in size. Even the very largest would still be considered modest in size.” For example, over 70% of their 1003 surveyed firms had annual sales of $1,000,000 or less; and the average employment in a typical company was 24. Thus, finite production capacity is an important characteristic of most remanufacturing firms. In this paper, we study a capacitated periodic-review manufacturing/remanufacturing inventory system with random demand for serviceable products and random product returns in each period. The serviceable products can be either manufactured from raw materials or remanufactured from returned products, with identical production lead times; but the firm has finite capacities in manufacturing, remanufacturing, and/or total manufacturing/remanufacturing operations in each period. The remaining returned products and unsold serviceable products in each period are carried to the next period, and the unsatisfied demand is backlogged to the next period. The firm’s objective is to find the optimal manufacturing and remanufacturing policies that minimize its expected total discounted cost over a finite planning horizon. We formulate the firm’s problem as a two-dimensional stochastic dynamic program. Using L♮ convexity (L-natural convexity) and lattice analysis, we derive important structural properties that enable us to characterize the firm’s optimal policies for models with finite capacities. For models with both a remanufacturing capacity and a manufacturing/total capacity, we show that the optimal remanufacturing policy is a modified remanufacture-down-to policy and the optimal manufacturing policy is a modified total-up-to policy. That is, it is optimal to remanufacture to


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reduce the inventory level of returned products to as close to the remanufacture-down-to level as possible, subject to the available remanufacturing/total capacity; and it is optimal to manufacture to raise the total inventory level (including both serviceable and returned products) to as close to the total-up-to level as possible, subject to the available manufacturing/total capacity after the remanufacturing operation. In addition, the remanufacture-down-to level is a partly-constant increasing function of the total inventory level with slopes no more than 1; and the total-up-to level is a partly-constant increasing function of the inventory level of returned products with slopes no more than 1. The structures of the optimal policies for other capacitated models are similarly characterized. The extensions to the lost-sales model, the model where disposal of returned products is allowed, and the model with joint control of product acquisition, pricing, and inventory decisions are also discussed. An important insight revealed by our study is on the production priority between manufacturing and remanufacturing operations. It is well known that, for uncapacitated systems, the firm’s optimal policies always give production priority to remanufacturing, i.e., manufacturing is resorted to only after all returned products have been remanufactured (see, e.g., Simpson 1978). This is due to the cost effectiveness of remanufacturing. This property, however, does not always hold for capacitated systems. Specifically, we show that the optimal policies give production priority to remanufacturing for systems with a remanufacturing capacity and/or a total capacity; but it is not true for systems with a manufacturing capacity. We will provide intuitive explanations to these results in §2. This work is closely related to two streams of research literature: inventory systems with remanufacturing and inventory systems with finite capacity. For the first stream of research, Fleischmann et al. (1997) provide a detailed review on early inventory models with remanufacturing. The closest study to ours is Simpson (1978), who considers an uncapacitated periodic-review inventory model with product returns and shows that the optimal policy for each period has a very simple structure which is completely determined by some critical numbers. In the past decades, this work has been extended along various directions, including non-zero/non-identical lead times (Inderfurth


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1997; Kiesm¨ uller 2003), multi-echelon systems (DeCroix and Zipkin 2005; DeCroix 2006), multiple types of returns (Zhou et al. 2011), acquisition and pricing (Zhou and Yu 2011), and random yield (Tao et al. 2012). All of these studies assume infinite capacities on both manufacturing and remanufacturing operations. The second stream of related research is on inventory systems with finite capacity. For pure inventory/manufacturing systems without remanufacturing, the issue of finite capacity has been extensively studied. For example, Federguren and Zipkin (1986) consider a capacitated periodicreview inventory system and show that the optimal inventory policy is a modified base-stock policy (i.e., it is optimal to bring the inventory level to as close to the base-stock level as possible, subject to the available capacity). Shaoxiang (2004) studies a periodic-review two-product inventory system with a total capacity and shows the optimality of a hedging point policy. Recently, Ceryan et al. (2013) study joint inventory and pricing control of two substitutable products with finite capacities; and Chen et al. (2011) study a more general problem with two either substitutable or complementary products. To the best of our knowledge, there has been no study on inventory control for capacitated inventory systems with remanufacturing, and our study is the first one to address this issue. The rest of this paper is organized as follows. In §2, we present the general model, its mathematical formulation, and two important structural properties. In §3, we characterize the firm’s optimal policies for different capacitated models. In §4, we conclude the paper with some extensions and future research directions. All proofs are provided in the online companion. Throughout the paper, we use “increasing” and “decreasing” in a non-strict sense, i.e., they represent “non-decreasing” and “non-increasing”, respectively; and for any real number x, we denote x+ = max{x, 0}.

2. The Model and Structural Properties Consider a hybrid manufacturing/remanufacturing inventory system that produces a single product over a planning horizon of T periods, indexed by t = 1, . . . , T . In each period t, the firm receives random customer demand Dt for serviceable products and random product returns Rt . A serviceable


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product can be either manufactured from raw materials or remanufactured from a returned product; but the firm has finite capacities in manufacturing, remanufacturing, and/or total production operations in each period. At the end of each period, leftover returned products and unsold serviceable products are carried to the next period, and unsatisfied demand is backlogged to the next period. Following Simpson (1978), we assume the manufacturing and remanufacturing operations have identical production leadtimes. Under this and the backlogging assumptions, the system can be transformed into an equivalent system with zero lead times by working on inventory positions (see, e.g., Inderfurth, 1997). Hence for brevity, we focus on the system with zero leadtime. The sequence of events in each period is as follows: First, the firm reviews the starting inventory levels of serviceable products and returned products; second, the firm decides on how many units of serviceable products to manufacture from raw materials and how many units to remanufacture from returned products, subject to its production capacities; then, the random customer demand and product returns are realized; and finally, all costs for this period are calculated. The firm’s cost structure includes a unit manufacturing cost p, a unit remanufacturing cost r, a unit holding cost s for returned products, and a unit holding cost h and a unit backlog cost b for serviceable products. All these unit costs are positive, and it is reasonable to assume that manufacturing is more costly than remanufacturing, i.e., p > r. There is also a discount factor α, 0 < α ≤ 1. The firm’s objective is to minimize its expected total discounted cost over the planning horizon. A main feature of our model is that the firm has finite production capacities in manufacturing and/or remanufacturing operations in each period. Specifically, we assume that the firm has one or more of the following three types of capacities: the remanufacturing capacity Kr , the manufacturing capacity Km , and the total manufacturing/remanufacturing capacity K. Different combinations of these capacities yield in total seven capacitated models, with the most general one having all three capacities. The main focus of this paper is to study the firm’s optimal manufacturing and remanufacturing policies for these different capacitated models. Following the studies on inventory systems with remanufacturing (e.g., Simpson 1978; DeCroix 2006; DeCroix and Zipkin 2005; Zhou et al. 2011), we assume the customer demands and product


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returns in different periods are independent random variables, but they can be correlated within the same period. Thus, (Dt , Rt ) can have an arbitrary joint distribution. The independence assumption of cross-period demands and returns is mainly for simplicity and can serve as an approximation when the cross-period correlations are small, or the lifetime of the product is long thus the returns only depend on sales in the distant past. If this assumption is violated, then the firm will need to keep track of all past demands and returns information so as to predict future demands and returns, giving rise to a high-dimensional stochastic dynamic program, and as a result, the parameters of the optimal control policy for a period will depend on such information. Denote x0t and x1t as the inventory levels of returned products and total products (including both serviceable and returned products) at the beginning of period t, respectively, before any decision is made. Then, x1t − x0t is the initial inventory level of serviceable products in period t. Furthermore, denote yt0 and yt1 as the inventory levels of returned products and total products after manufacturing and remanufacturing decisions are made but before the realization of customer demand and product returns in period t, respectively. Then, the manufacturing and remanufacturing quantities in period t are yt1 − x1t and x0t − yt0 , respectively. Clearly, choosing the manufacturing and remanufacturing quantities in period t is equivalent to choosing yt0 and yt1 ; and for convenience we will use the latter as the firm’s decision variables. Obviously, the decision variables yt0 and yt1 must satisfy 0 ≤ yt0 ≤ x0t ,

and yt1 ≥ x1t ;

t = 1, . . . , T.

(1)

In addition, the production capacities Kr , Km , and K would impose one or more of the following three constraints: for t = 1, . . . , T , x0t − yt0 ≤ Kr ;

(2)

yt1 − x1t ≤ Km ;

(3)

(x0t − yt0 ) + (yt1 − x1t ) ≤ K.

(4)

For convenience, we choose (x0t , x1t ) as the state of the system at the beginning of period t. Then, the system dynamics from period t to period t + 1 are given by x0t+1 = yt0 + Rt ; x1t+1 = yt1 − Dt + Rt .

(5)


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Denote Vt (x0t , x1t ) as the minimum expected total discounted cost from period t to the end of the planning horizon, given the state (x0t , x1t ) at the beginning of period t. Then, we can formulate the firm’s optimization problem as the following dynamic program: for t = 1, . . . , T , Vt (x0t , x1t ) = min {Ht (yt0 , yt1 )} + rx0t − px1t , yt0 ,yt1

(6)

where (yt0 , yt1 ) is subject to constraint (1) and some of the capacity constraints (2), (3), and (4), Ht (u, v) = (s − r)u + pv + Gt (v − u) + αE[Vt+1 (u + Rt , v − Dt + Rt )],

(7)

and Gt (·) is the expected holding and backlog costs for serviceable products in period t, given by Gt (x) = hE[max{x − Dt , 0}] + bE[max{Dt − x, 0}]. Following Simpson (1978), the boundary condition is given as VT +1 (·, ·) ≡ 0. We aim to characterize the optimal manufacturing and remanufacturing policies in each period t that minimize the objective function Ht (yt0 , yt1 ) in problem (6) under different capacity constraints. To this end, we need to first study the structural properties of problem (6). As mentioned earlier, the L♮ -convexity and lattice analysis are the main machinery in our analysis. Let V ⊂ ℜn be a lattice and ℜ the set of real numbers. A function f : V → ℜ is submodular if f (x) + f (y) ≥ f (x ∧ y) + f (x ∨ y) for any x, y ∈ V , where ∧ and ∨ are the component-wise minimum and maximum operators, respectively. Let e = (1, . . . , 1) be the n-dimensional vector of 1s. A function f : V → ℜ is L♮ -convex if the function ψ(v, ζ) = f (v − ζe) is submodular on {(v, ζ)|v − ζe ∈ V }. L♮ -convexity is a much stronger property than convexity: If f is L♮ -convex, then it is convex, submodular, and its Hessian matrix is diagonal dominant. L♮ -convexity was first introduced in discrete convex analysis (Murota 2003) and has been recently used in the study of several inventory control models, e.g., Lu and Song (2005), Zipkin (2008), and Huh and Janakirman (2010). For more details on these concepts we refer the reader to Topkis (1998) for lattice analysis and to Zipkin (2008) for L♮ -convexity. The following important result shows that the value function and the objective function in each period are L♮ -convex functions for the general model with all three capacities.

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Proposition 1. For the general model with capacities Kr , Km , and K, both the value function Vt (x0t , x1t ) and the objective function Ht (yt0 , yt1 ) are L♮ -convex, t = 1, . . . , T . Proposition 1 is the main technical result of the paper; and it plays a key role in characterizing the firm’s optimal policies for all capacitated models. It is noteworthy to point out that this result is established before we obtain the optimal policies, i.e., it is not proved by utilizing the structure of the optimal control policies. As can be seen later, if it has to be proved by exploiting the specific structure of the optimal policies, then the proof will be much more complicated and tedious. There is an important difference among the different capacitated models on the production priority between manufacturing and remanufacturing operations. For the uncapacitated model, Simpson (1978) shows that the optimal policies always give priority to remanufacturing, i.e., manufacturing is resorted to only after all returned products have been remanufactured. This is intuitive, as manufacturing is more costly than remanufacturing. For capacitated models, however, this property does not always hold and it depends on whether there is a manufacturing capacity Km . As will be seen later, the production priority is always given to remanufacturing for models with a remanufacturing capacity and/or a total capacity; but it fails to hold for models with a manufacturing capacity. The result for the former case is intuitive, since for those models, besides being more cost effective, giving production priority to the remanufacturing operation also increases capacity utilization and thus is more desirable. To see the intuition for the latter case, consider the simplest model with only a manufacturing capacity. Recall that, in each period, the supply of serviceable products comes either from returned products or from the manufacturing capacity. Suppose at the beginning of a period there are a lot of backlogs on serviceable products. Remanufacturing all returned products can increase the inventory level of serviceable products; but it will also lose this “potential supply” for future periods. On the other hand, the manufacturing capacity, if not used in this period, will be lost. Hence, in this scenario it is probably better to manufacture some new products while keeping some returned products to satisfy demands in future periods. The following result provides the key structural property that establishes the production priority for the capacitated models without a manufacturing capacity Km .


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Proposition 2. For the model with only capacities Kr and K, Ht (y 0 − ζ, y 1 − ζ) is decreasing in ζ for any fixed y 0 and y 1 , t = 1, . . . , T . To see why Proposition 2 implies that remanufacturing takes priority over manufacturing, note from (6) that y 0 − ζ and y 1 − ζ are the post-production inventory levels for returned products and total products, respectively, while y 1 − y 0 = (y 1 − ζ) − (y 0 − ζ) is the post-production inventory level for the serviceable products which is independent of ζ. Thus, Proposition 2 states that, for the model with only capacities Kr and K, for any fixed post-production inventory level of serviceable products, it is always better to keep fewer returned products. This implies that, to increase the inventory level of serviceable products, remanufacturing should always take priority over manufacturing, leading to the production priority of remanufacturing discussed above. Once the manufacturing capacity Km is added into the model, however, the monotonicity result in Proposition 2 does not hold any more. For each period t, denote (yt0∗ (x0t , x1t ), yt1∗ (x0t , x1t )) as the optimal solution of problem (6). Thus, yt0∗ (x0t , x1t ) is the optimal inventory level of returned products and yt1∗ (x0t , x1t ) is the optimal total inventory level after manufacturing and remanufacturing decisions given the initial state (x0t , x1t ). With the structural properties established in Propositions 1 and 2, we are able to characterize the firm’s optimal policies for different capacitated models in the next section.

3. Optimal Policies In this section, we characterize the firm’s optimal manufacturing and remanufacturing policies for different capacitated models. To begin with, we consider the simplest model where there is only a total capacity K (or when K = Kr = Km ). For this model, the optimization problem (6) is subject to constraints (1) and (4), and the firm’s optimal policies are given in the following theorem. Theorem 1. When K = Kr = Km , the firm’s optimal policies for period t are characterized by a ∗ 1∗ ∗ ∗ critical number x1∗ t and an increasing function δt (·), where δt (xt ) = 0 and the slopes of δt (·) are

no more than one. (i) The optimal remanufacturing policy is a modified remanufacture-down-to policy:


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0 i-1) if x1t ≤ x1∗ t , then remanufacture all returned products if xt ≤ K; and remanufacture K

otherwise; 0 ∗ 1 ∗ 1 i-2) if x1t ≥ x1∗ t , then do not remanufacture if xt ≤ δt (xt ); remanufacture down to δt (xt ) if

δt∗ (x1t ) ≤ x0t ≤ δt∗ (x1t ) + K; and remanufacture K otherwise. (ii) The optimal manufacturing policy is a modified base-stock policy: 1∗ ii-1) if x1t ≤ x1∗ as possible with the t , then raise the total inventory level to as close to xt

leftover capacity (K − x0t )+ ; i.e., if x0t ≥ K, then do not manufacture; if x0t < K and x1t ≤ x1∗ t + x0t − K, then manufacture K − x0t ; and otherwise, manufacture up to x1∗ t ; ii-2) if x1t ≥ x1∗ t , then do not manufacture. Theorem 1 shows that, for the model with only a total capacity K, the optimal remanufacturing policy is to remanufacture the returned products down to the target level δt∗ (max{x1t , x1∗ t }), subject to the total capacity K; while the optimal manufacturing policy is to raise the total inventory up to the base-stock level x1∗ t , subject to the leftover capacity after remanufacturing. Thus, the optimal policies always give production priority to remanufacturing for this model. The firm’s optimal policies for period t in Theorem 1 are illustrated in Figure 1. In this figure, the horizontal and vertical axes are the inventory levels of total products and returned products, respectively; and the state space is divided into four regions separated by solid curves and/or straight lines. In each of regions (I) to (III), we choose a representative state A and depict one or two dotted arrows to indicate the firm’s optimal decisions and the corresponding state transitions. For the initial state AI , it is optimal to remanufacture K (i.e., a downward arrow from AI to B I ) but manufacture nothing. For the initial state AII , it is optimal to remanufacture all returned products to reach B II , and manufacture with the leftover capacity to raise the total inventory to II represents the final state after manufacturing, which can be either as close to x1∗ t as possible (C III , it is optimal to remanufacture fewer than at or on the left of (x1∗ t , 0)). For the initial state A

K to reach B III with coordinates (x1t , δt∗ (x1t )) but manufacture nothing. Finally, in region (IV), it is optimal for the firm to do nothing, thus no arrow is drawn in this region.


Figure 1

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Optimal Policy: Model with Only a Total Capacity

It is interesting to observe how the total capacity constraint impacts the firm’s optimal policies. When there is no capacity constraint (i.e., when K = ∞), Simpson (1978) shows that the optimal strategy is determined by two base-stock levels: One on serviceable products for the optimal remanufacturing policy and the other on total products for the optimal manufacturing policy. In other words, it is optimal to remanufacture to raise the inventory level of serviceable products to a base-stock level (subject to available returned products), and to manufacture to raise the total inventory level to another base-stock level. In contrast, when the total capacity constraint is in force, by Theorem 1, the optimal remanufacturing policy aims to raise the inventory level of serviceable products to x1t − δt∗ (x1t ), an increasing function with slopes no more than 1 and in particular, x1t when x1t ≤ x1∗ t (in this case, it is optimal to remanufacture all returned products); and the optimal manufacturing policy becomes a modified base-stock policy for total products. Therefore, for both models with finite and infinite total capacities, the optimal policies have essentially the same structure when the total inventory level is low, i.e., the firm remanufactures all returned products and then manufactures to raise the total inventory level to a fixed base-stock level. But as the total inventory level gets higher, the optimal remanufacturing policy for the capacitated system becomes more complex. The intuitions for the differences are as follows. When the total inventory level increases, with a finite capacity the firm would remanufacture no less returned products to hedge against possible future capacity shortage, leading to a higher inventory level of


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serviceable products for remanufacturing; with an infinite capacity, however, the target inventory level of serviceable products for remanufacturing would not be changed since the firm can always remanufacture any returned products whenever needed. For the optimal manufacturing policy, its structure is essentially the same after introducing the total capacity constraint, since the inventory level of the leftover returned products must be zero when the firm manufactures new products due to the production priority on the remanufacturing operation, leading to a modified base-stock policy for manufacturing. We next consider the more general model with only a remanufacturing capacity Kr and a total capacity K (or when K = Km ). For this model, the optimization problem (6) is subject to constraints (1), (2), and (4); and the firm’s optimal policies are given in the following theorem. (Without loss of generality, we assume K > Kr , since otherwise the capacity Kr does not have any effect and the resulting model is equivalent to the model with only a total capacity K.) Theorem 2. When K = Km > Kr , the firm’s optimal policies for period t are characterized by a ∗ ∗ ∗ 1∗ 1∗ ∗ critical number x1∗ t and two increasing functions δt (·) and ξt (·), where δt (xt ) = 0, xt = ξt (0) and

the slopes of δt∗ (·) and ξt∗ (·) are no more than one. (i) The optimal remanufacturing policy is a modified remanufacture-down-to policy: 0 i-1) if x1t ≤ x1∗ t , then remanufacture all returned products if xt ≤ Kr ; and remanufacture Kr

otherwise; 0 ∗ 1 ∗ 1 i-2) if x1t ≥ x1∗ t , then do not remanufacture if xt ≤ δt (xt ), remanufacture down to δt (xt ) if

δt∗ (x1t ) ≤ x0t ≤ δt∗ (x1t ) + Kr ; and remanufacture Kr otherwise. (ii) The optimal manufacturing policy is a modified total-up-to policy: ii-1) if 0 ≤ x0t ≤ Kr , then raise the total inventory level to as close to x1∗ t as possible with the leftover capacity K − x0t if x1t ≤ x1∗ t ; and do not manufacture otherwise. ii-2) if x0t ≥ Kr , then raise the total inventory level to as close to ξt∗ (x0t − Kr ) as possible with the leftover capacity K − Kr if x1t ≤ ξt∗ (x0t − Kr ); and do not manufacture otherwise.


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Theorem 2 shows that the optimal policies always give production priority to remanufacturing for the model with only capacities Kr and K. In addition, compared with Theorem 1, it also shows that the structure of the optimal remanufacturing policy for this model is essentially the same as that for the model with only a total capacity K, except that the maximum remanufacturing capacity is Kr instead of K; while the structure of the optimal manufacturing policy is more complex than that for the model with only a total capacity K. This is because, since the remanufacturing capacity Kr is strictly less than the total capacity K, the inventory level of the leftover returned products (x0t − Kr )+ can be positive when the firm turns to manufacture new products, and consequently the optimal manufacturing policy becomes a modified total-up-to policy with the total-up-to level depending on the leftover returned products. The firm’s optimal policies for period t in Theorem 2 are illustrated in Figure 2. In this figure, the state space is divided into five regions separated by solid curves and/or straight lines. Similar to Figure 1, in each of regions (I) to (IV) we choose a representative state A and depict one or two dotted arrows to indicate the firm’s optimal decisions and the corresponding state transitions; and in region (V), it is optimal for the firm to do nothing, thus no arrow is drawn in this region. For example, for the initial state AI with coordinates (x1t , x0t ), it is optimal to first remanufacture Kr (i.e., a downward arrow from AI to B I ) and then manufacture up to ξt∗ (x0t − Kr ) (i.e., a rightward arrow from B I to C I ), subject to the leftover capacity K − Kr . Note that the dotted curve x1t = ξt∗ (x0t ) has slopes no less than one in the x1t -x0t plane, since ξt∗ (x0t ) is increasing in x0t with slopes no more than one; and the solid curve x1t = ξt∗ (x0t − Kr ) is an upward translation of the dotted curve x1t = ξt∗ (x0t ) by Kr . Also note from Theorem 2 that the curve x0t = δt∗ (x1t ) (resp., x0t = δt∗ (x1t ) + Kr ) is always on the right of the curve x1t = ξt∗ (x0t ) (resp., x1t = ξt∗ (x0t − Kr )) and 1∗ they intersect at the point (x1∗ t , 0) (resp., (xt , Kr )). Thus, the solid curves and/or straight lines in

Figure 2 partition the state space into five non-overlapping regions. It is worth mentioning that the solid curve x0t = δt∗ (x1t ) is the control curve for remanufacturing and the dotted curve x1t = ξt∗ (x0t ) is the control curve for manufacturing after the remanufacturing


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Figure 2

Optimal Policy: Model with Only Remanufacturing and Total Capacities

operation. That is, it is optimal to remanufacture if and only if the initial state is above the curve x0t = δt∗ (x1t ) while to manufacture if and only if the state after remanufacturing is on the left of the curve x1t = ξt∗ (x0t ). To see the implication of these two control curves, consider two states, D′ and D, on the two curves respectively. These two states have the same total inventory level, but the state D′ has a higher inventory level for serviceable products. In addition, given the starting state D, it is optimal to manufacture nothing but to remanufacture so that the final state is as close to D′ as possible. This implies that, because of the cost advantage of the remanufacturing operation, it has a higher target inventory level for serviceable products than the manufacturing operation. As a special case of Theorem 2, for the model with only a remanufacturing capacity Kr (i.e., when K = ∞), the firm’s optimal policies are slightly simpler: the optimal remanufacturing policy remains the same as that in Theorem 2, but the optimal manufacturing policy is simplified to a total-up-to policy. This is because, since there is no capacity constraint on the manufacturing operation, the manufacturing total-up-to level can always be reached. Thus, compared with the uncapacitated model, while maintaining the production priority on remanufacturing, the introduction of the remanufacturing capacity alters the structure of the optimal remanufacturing policy as well as that of the optimal manufacturing policy. Both changes are intuitive, since when there is a remanufacturing capacity, the optimal remanufacturing policy and the optimal manufacturing policy need to consider the possible future capacity shortage and the leftover returned products, respectively.


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Now we proceed to study the capacitated models including a manufacturing capacity Km . Since the production priority of remanufacturing fails to hold for these models, the firm’s optimal policies are more involved and our characterization of their structures mainly relies on the L♮ -convexity of the objective function (i.e., Proposition 1). In what follows, we first consider the model with only a manufacturing capacity Km and a remanufacturing capacity Kr (or when K ≥ Kr + Km ). For this model, the optimization problem (6) is subject to constraints (1), (2), and (3); and the firm’s optimal policies are given in the next theorem. Theorem 3. When K ≥ Kr + Km , the firm’s optimal policies for period t are characterized by 1∗ ∗ ∗ 0∗ ∗ 1∗ two critical numbers x0∗ t and xt and two increasing functions δt (·) and ξt (·), where xt = δt (xt ), ∗ 0∗ ∗ ∗ x1∗ t = ξt (xt ), and the slopes of δt (·) and ξt (·) are no more than one.

(i) The optimal remanufacturing policy is a modified remanufacture-down-to policy: 0 ∗ 1 i-1) if x1t ≤ x1∗ t − Km , then remanufacture Kr if xt ≥ δt (xt + Km ) + Kr ; remanufacture down

to δt∗ (x1t + Km ) if 0 ≤ x0t − δt∗ (x1t + Km ) ≤ Kr ; and do not remanufacture otherwise; 1 1∗ 0 0∗ i-2) if x1∗ t − Km ≤ xt ≤ xt , then remanufacture Kr if xt ≥ xt + Kr ; remanufacture down to 0∗ 0 0∗ x0∗ t if xt ≤ xt ≤ xt + Kr ; and do not remanufacture otherwise; 0 ∗ 1 ∗ 1 i-3) if x1t ≥ x1∗ t , then remanufacture Kr if xt ≥ δt (xt ) + Kr ; remanufacture down to δt (xt ) if

δt∗ (x1t ) ≤ x0t ≤ δt∗ (x1t ) + Kr ; and do not remanufacture otherwise. (ii) The optimal manufacturing policy is a modified total-up-to policy: 1 ∗ 0 ∗ 0 ii-1) if 0 ≤ x0t ≤ x0∗ t , then manufacture Km if xt ≤ ξt (xt ) − Km ; manufacture to ξt (xt ) if

ξt∗ (x0t ) − Km ≤ x1t ≤ ξt∗ (x0t ); and do not manufacture otherwise; 1∗ 1∗ 1 0∗ 0 ii-2) if x0∗ t ≤ xt ≤ xt + Kr , then manufacture Km if xt ≤ xt − Km ; manufacture to xt if 1∗ 1 x1∗ t − Km ≤ xt ≤ xt ; and do not manufacture otherwise; 0 ∗ 0 ∗ 1 ii-3) if x0t ≥ x0∗ t + Kr , then manufacture Km if xt ≤ ξt (xt − Kr ) − Km ; manufacture to ξt (xt −

Kr ) if ξt∗ (x0t − Kr ) − Km ≤ x1t ≤ ξt∗ (x0t − Kr ); and do not manufacture otherwise. Theorem 3 shows that, for the model with only two capacities Km and Kr , the optimal remanufacturing policy is to remanufacture the returned products down to the target inventory level


16

δt∗ (max{x1t , min{x1t +Km , x1∗ t }}), subject to the remanufacturing capacity Kr ; and the optimal manufacturing policy is to raise the total inventory up to the target inventory level ξt∗ (min{x0t , max{x0t − Kr , x0∗ t }}), subject to the manufacturing capacity Km . Note that the firm’s optimal policies for this model are more complicated than those for the above models, since the production priority on remanufacturing fails to hold in the presence of a manufacturing capacity Km . The firm’s optimal policies for period t in Theorem 3 are illustrated in Figure 3. In this figure, the state space is divided into nine regions by two parallel solid curves with slopes between 0 and 1 and two parallel solid curves with slopes no less than one. In each of regions (I) to (VIII), we again choose a representative state A and the dotted arrows can be similarly interpreted as those in Figures 1 and 2; and in region (IX), it is optimal for the firm to do nothing, thus no arrow is drawn in this region. For example, for the initial state AII with coordinates (x1t , x0t ), it is optimal to first remanufacture Kr to move the state AII to B II with coordinates (x1t , x0t − Kr ), and then manufacture to move the state B II to the final state C II with coordinates (ξt∗ (x0t − Kr ), x0t − Kr ). Note that in regions (V ), (V II) and (V III), and in part of region (IV ), it is optimal to manufacture some new products while keeping some returned products without using the entire remanufacturing capacity. Thus, remanufacturing does not take priority over manufacturing in these regions.

Figure 3

Optimal Policy: Model with Only Manufacturing and Remanufacturing Capacities

As a special case of Theorem 3, for the model with only a manufacturing capacity Km (i.e., when Kr = ∞), the firm’s optimal remanufacturing policy is simplified to a remanufacture-downto policy, and the optimal manufacturing policy is still a modified total-up-to policy but with a


17

simpler total-up-to level ξt∗ (min{x0t , x0∗ t }). As a result, regions (I)-(III) in Figure 3 do not exist for this model, thus the state space is divided into the six remaining regions (IV)-(IX). Note that, even for this special case with only a manufacturing capacity, remanufacturing does not always take priority over manufacturing. Thus, compared with the uncapacitated model, the introduction of the manufacturing capacity destroys the production priority on remanufacturing as well as alters the structures of the optimal remanufacturing and manufacturing policies. The two remaining models not studied yet are the model with two capacities Km and K, and the most general model with all three capacities Kr , Km and K. For both models, Proposition 1 shows that the objective function in the optimization problem (6) is L♮ -convex. However, since the optimal policies do not always give priority to remanufacturing for either model, the firm’s optimal policies are more complicated and it involves the optimal allocation of the total capacity K. Using L♮ -convexity, we can show that the optimal policies in each period for these two models divide the state space into nine and eleven regions, respectively; and their full characterizations both require two control parameters and three monotone control curves. Since the characterization of the optimal policies are lengthy and less insightful, for brevity we leave the details to the interested reader. We remark that some related studies have been done in other application contexts, e.g., Shaoxiang (2004), Chen et al. (2011), and Ceryan et al. (2013).

4. Conclusion and Discussion In this paper, we study the optimal control policy for capacitated periodic-review inventory systems with remanufacturing. Using L♮ -convexity and lattice analysis, we characterize the optimal manufacturing and remanufacturing policies when one or more of the production capacities are finite. For instance, for systems with a remanufacturing capacity and a manufacturing/total capacity, we show that the optimal remanufacturing policy is a modified remanufacture-down-to policy and the optimal manufacturing policy is a modified total-up-to policy; and the remanufacture-down-to level (resp., the total-up-to level) is a partly-constant increasing function of the total inventory level (resp., the inventory level of returned products) with slopes no more than one. In addition, we


18

show that for systems with a remanufacturing capacity and/or a total capacity, the optimal policies always give production priority to remanufacturing; this priority rule does not hold, however, for systems with a manufacturing capacity. Our analysis and results can be extended in the following three directions. First, all of our results continue to hold under the lost-sales models. When unsatisfied demand in each period is lost, we need to modify the system dynamics on the total inventory level in (5) to x1t+1 = max{yt1 − Dt , yt0 } + Rt . This is because the total inventory level under the lost-sales setting is yt0 + Rt rather than yt1 − Dt + Rt when there is shortage on serviceable products. Accordingly, the expression Vt+1 (u + Rt , v − Dt + Rt ) in (7) needs to be modified to Vt+1 (u + Rt , max{v − Dt , u} + Rt ). It should also be noted that the state space for the lost-sales models is {(x0t , x1t ) | x1t ≥ x0t ≥ 0}, which is different from that of the backlog models. Albeit with the above differences, Propositions 1 and 2 and Theorems 1 to 3 continue to hold under the lost-sales models. It is interesting to note that the optimal manufacturing and remanufacturing policies for the lost-sales models have exactly the same structure as those of the corresponding backlog models, with the only difference lying in their state spaces. Second, for simplicity in this paper we have assumed that disposal of returned products is not allowed. When the firm can dispose returned products in each period, we can show that the structural properties (i.e., Propositions 1 and 2) are still true. Thus, for models with a remanufacturing capacity and/or a total capacity, the production priority result continues to hold when disposal of returned products is allowed. The firm’s optimal policies become more complicated because of the disposal decision. However, with the above structural properties, we are able to partially characterize the structure of the optimal policies. For example, for models with a total capacity, the optimal policies never manufacture new products while dispose returned products in the same period. In addition, we can also study how the optimal policies depend on the initial inventory levels using L♮ -convexity. Third, our analysis and results can also be extended to the models where the firm makes joint product acquisition, pricing, and inventory decisions. Zhou and Yu (2011) study this problem when


19

both manufacturing and remanufacturing operations have infinite capacities. For the capacitated models, after following their assumptions on product acquisition and pricing, we can extend our analysis to study the firm’s optimal inventory, acquisition, and pricing policies. Specifically, the structural properties (i.e., Propositions 1 and 2) continue to hold; and the firm’s optimal manufacturing/remanufacturing policies have the same structure as what is described in Theorems 1 to 3. For brevity, we leave the details to the interested reader. Throughout the paper we have assumed identical lead times for manufacturing and remanufacturing operations (and zero lead time for the lost-sales models). One important future direction is to study the models with non-identical lead times (and the lost-sales models with positive identical lead times). Because of the “curse of dimensionality”, it seems more practical to develop effective heuristic policies (rather than analyze the optimal policies) for such models. For the uncapacitated models, Kiesm¨ uller (2003) and Zhou et al. (2011) have proposed some effective heuristics. It would be interesting to develop some effective heuristic policies for the capacitated models, and we leave this as a future research.

Electronic Companion An electronic companion to this paper is available as part of the online version at http://or.journal.informs.org/.

Acknowledgments The authors are grateful to the Area Editor, Associate Editor, and three anonymous referees for their detailed comments and suggestions, which have helped us to significantly improve both the content and the exposition of the paper. In particular, one referee brought to our attention the preservation result in Chen et al. (2011), which significantly simplified the proof of Proposition 1. This research is supported in part by NSF under CMMI-0927631 and CMMI-1131249.

References [1] Ceryan, O., O. Sahin, I. Duenyas. 2013. Dynamic pricing of substitutable products in the presence of capacity flexibility. Manufacturing & Service Oper. Management, 15(1): 86-101.


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[2] Chao, X., B. Talbot, G. King, C. Koenig. 2009. Remanufacturing at Cummins, Inc. WDI Business Case #1-428-798. Ross School of Business, University of Michigan. [3] Chen, X., P. Hu, S. He. 2011. Preservation of supermodularity in two dimensional parametric optimization problems and its applications. Working paper. [4] DeCroix, G. A. 2006. Optimal policy for a multiechelon inventory system with remanufacturing. Oper. Res., 54(3): 532-543. [5] DeCroix, G. A., P. H. Zipkin. 2005. Inventory management for an assembly system with product or component returns. Management Science, 51(8): 1250-1265. [6] Federgruen A., P. H. Zipkin. 1986. An inventory model with limited production capacity and uncertain demands II: The discounted-cost criterion. Math. Oper. Res., 11(2): 208-215. [7] Fleischmann, M., J. M. Bloemhof-Ruwaard, R. Dekker, E. van der Laan, J. A.E.E. van Nunen, L. N. Van Wassenhove. 1997. Quantitative models for reverse logistics: A review. Eur. J. Oper. Res., 103(1): 1-17. [8] Giuntini, R., K. Gaudette. 2003. Remanufacturing: The next great opportunity for boosting US productivity. Business Horizons, November-December 2003. [9] Huh, W. T., G. Janakiraman. 2010. On the optimal policy structure in serial inventory systems with lost sales. Oper. Res., 58(2): 486-491. [10] Inderfurth, K. 1997. Simple optimal replenishment and disposal policies for a product recovery system with leadtimes. OR Spectrum, 19(2): 111-122. [11] Kiesm¨ uller, G. P. 2003. A new approach for controlling a hybrid stochastic manufacturing/remanufacturing system with inventories and different leadtimes. Eur. J. Oper. Res., 147(1): 62-71. [12] Lu, Y., J. Song. 2005. Order-based cost optimization in assemble-to-order systems. Oper. Res., 53(1): 151-169. [13] Lund, R. T., W. M. Hauser. 2010. Remanufacturing - An American perspective. Proceedings of the 5th International Conference on Responsive Manufacturing - Green Manufacturing, p. 1-6. [14] Murota, K. 2003. Discrete Convex Analysis, Society for Industrial and Applied Mathematics, Philadelphia.


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[15] Shaoxiang, C. 2004. The optimality of hedging point policies for stochastic two-product flexible manufacturing systems. Oper. Res., 52(2): 312-322. [16] Simpson, V. P. 1978. Optimum solution structure for a repairable inventory system. Oper. Res., 26(2): 270-281. [17] Tao, Z., S. X. Zhou, C. S. Tang. 2012. Managing a remanufacturing system with random yield: properties, observations, and heuristics. Production and Operations Management, forthcoming. [18] Toktay, L. B., L. M. Wein, S. A. Zeinos. 2000. Inventory management of remanufacturable products. Management Science, 46(11): 1412-1426. [19] Topkis, D. 1998. Supermodularity and Complementarity. Princeton University Press. [20] Van der Laan, E., M. Salomon, R. Dekker, L. Van Wassenhove. 1999. Inventory control in hybrid systems with remanufacturing. Management Science, 45(5): 733-747. [21] Zhou, S. X., Z. Tao, X. Chao. 2011. Optimal control of inventory systems with multiple types of remanufacturable products. Manufacturing & Service Oper. Management, 13(1): 20-34. [22] Zhou, S. X., Y. Yu. 2011. Optimal product acquisition, pricing, and inventory management for systems with remanufacturing. Oper. Res., 59(2): 514-521. [23] Zipkin, P. 2008. On the structure of lost-sales inventory models. Oper. Res., 56(4): 937-944.

Brief Bio: Xiting Gong is a research fellow of Industrial and Operations Engineering at the University of Michigan, Ann Arbor. His research interests include operations and supply chain management, logistics and inventory control, and game theoretic applications. Xiuli Chao is a professor of Industrial and Operations Engineering at the University of Michigan, Ann Arbor. His recent research interests include stochastic modeling and optimization, inventory control, and game and supply chain applications.

e-companion to Gong and Chao: Optimal Control Policy for Capacitated Remanufacturing Systems

This page is intentionally blank. Proper e-companion title page, with INFORMS branding and exact metadata of the main paper, will be produced by the INFORMS office when the issue is being assembled.

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Electronic Companion to “Optimal Control Policy for Capacitated Inventory Systems with Remanufacturing” Xiting Gong and Xiuli Chao Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48109, {xitingg, xchao}@umich.edu

In this online appendix, we provide proofs for all the technical results. Some preliminaries on lattice and submodularity are needed. Recall that a partially ordered set X is called a lattice if it contains the minimum and maximum of each pairs of its elements; if X ′ is a subset of a lattice X and X ′ contains the minimum and maximum (with respect to X) of each pair of the elements of X ′ , then X ′ is called a sublattice of X. For any subset S ⊂ ℜ2 , we let Sx (Sy ) denote the section of S at x (y). That is, Sx = {y : (x, y) ∈ S },

Sy = {x : (x, y) ∈ S }.

A main tool used in our study is L♮ -convexity. In this paper, we use the following property of an L♮ -convex function. Lemma EC.1. Let S ⊂ ℜ2 be a lattice. If f (x, y) is L♮ -convex in S, then both y ∗ (x) = arg miny∈Sx f (x, y) and x∗ (y) = arg minx∈Sy f (x, y) are increasing functions with slopes between zero and one. Proof of Lemma EC.1

By symmetry, we only need to prove the result for y ∗ (x). The mono-

tonicity of y ∗ (x) follows from Theorem 8.1 of Porteus (2002). Thus we only prove that y ∗ (x) has slopes no more than one, or equivalently, x − y ∗ (x) is increasing in x. By L♮ -convexity, f (x − y, x) is submodular in (x, y). From Lemma 8.4 of Porteus (2002) and that S is a lattice, it follows that Sx is ascending in x. Thus its minimizer, denoted by yˆ(x), is increasing in x. On the other hand, since y ∗ (x) is the minimizer of f (x, y), we have x − yˆ(x) = y ∗ (x). Hence, x − y ∗ (x) = yˆ(x) is increasing in x.


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Proof of Proposition 1. We prove the L♮ -convexity of Ht and Vt by induction on t. Since VT +1 (x0 , x1 ) ≡ 0, it is clearly an L♮ -convex function. Now we assume inductively that Vt+1 (x0 , x1 ) is L♮ -convex, and prove that Vt and Ht are both L♮ -convex functions. We first prove Ht (y 0 , y 1 ) is L♮ -convex. According to (7), we have Ht (y 0 − ζ, y 1 − ζ) = (s − r)(y 0 − ζ) + p(y 1 − ζ) + Gt (y 1 − y 0 ) + αE[Vt+1 (y 0 − ζ + Rt , y 1 − ζ − Dt + Rt )]. Since Gt (·) is a convex function and Vt+1 (x0 , x1 ) is L♮ -convex, it is easily seen that Ht (y 0 − ζ, y 1 − ζ) is submodular in (y 0 , y 1 , ζ), thus Ht (y 0 , y 1 ) is L♮ -convex. We next prove that Vt (x0 , x1 ) is L♮ -convex, or equivalently, Vt (x0 − ζ, x1 − ζ) is submodular in (x0 , x1 , ζ). When there are capacity constraints Kr , Km , and K, by (6) we have Vt (x0 − ζ, x1 − ζ) =

min

0

+

{ 0

0

(x −ζ−Kr ) ≤y ≤x −ζ, 0≤y 1 −x1 +ζ≤Km , 0≤y 1 −x1 +x0 −y 0 ≤K

} Ht (y 0 , y 1 ) + rx0 − px1 + (p − r)ζ.

(EC.1)

According to Theorem 2.6.2 of Topkis (1998), it suffices to prove that Vt (x0 − ζ, x1 − ζ) is submodular in (x0 , x1 ) for any fixed ζ, submodular in (x0 , ζ) for any fixed x1 , and submodular in (x1 , ζ) for any fixed x0 . In what follows, we sequentially prove these three results. We first prove that Vt (x0 − ζ, x1 − ζ) is submodular in (x0 , x1 ) for any fixed ζ. The main idea of the proof is to apply Theorem 1 of Chen et al. (2011) to show the preservation of submodularity under a minimization operation. To apply this theorem, we introduce two additional decision variables z 0 = x0 − y 0 and z 1 = x1 − y 1 . Then, the minimization problem (EC.1) can be rewritten as Vt (x0 − ζ, x1 − ζ) =

min

(y 0 ,y 1 ,z 0 ,z 1 )∈D

{

} Ht (y 0 , y 1 ) : y i + z i = xi , i = 0, 1 + rx0 − px1 + (p − r)ζ, (EC.2)

where D = {(y 0 , y 1 , z 0 , z 1 ) |y 0 ≥ 0, 0 ≤ z 0 − ζ ≤ Kr , −Km ≤ z 1 − ζ ≤ 0, 0 ≤ z 0 − z 1 ≤ K }.

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Since Ht (y 0 , y 1 ) is L♮ -convex, it is convex and submodular in (y 0 , y 1 ). In addition, it can be easily checked that D is a closed convex sublattice. Thus, by applying Theorem 1 of Chen et al. (2011), Vt (x0 − ζ, x1 − ζ) is submodular in (x0 , x1 ). We next prove that Vt (x0 − ζ, x1 − ζ) is submodular in (x0 , ζ) for any fixed x1 . To this end, we introduce the following new decision variables: yˆ0 = y 0 − y 1 + x1 , yˆ1 = x1 − y 1 , zˆ0 = x0 − yˆ0 , zˆ1 = ζ − yˆ1 . Then, the minimization problem (EC.1) can be rewritten as Vt (x0 − ζ, x1 − ζ) =

{ min

ˆ (ˆ y 0 ,ˆ y 1 ,ˆ z 0 ,ˆ z 1 )∈D

Ht (ˆ y 0 − yˆ1 , x1 − yˆ1 ) :

(

yˆ0 +ˆ z0 yˆ1 +ˆ z1

) =

( 0 )} x ζ

+ rx0 − px1 + (p − r)ζ,

where ˆ = {(ˆ D y 0 , yˆ1 , zˆ0 , zˆ1 ) | yˆ0 ≥ yˆ1 , 0 ≤ zˆ0 − zˆ1 ≤ Kr , 0 ≤ zˆ0 ≤ K, 0 ≤ zˆ1 ≤ Km }. Since Ht is L♮ -convex, Ht (ˆ y 0 − yˆ1 , x1 − yˆ1 ) is convex and submodular in (ˆ y 0 , yˆ1 ). In addition, it can ˆ is a closed convex sublattice. Thus, again by applying Theorem 1 of Chen et al. be seen that D (2011), Vt (x0 − ζ, x1 − ζ) is submodular in (x0 , ζ). We finally prove that Vt (x0 − ζ, x1 − ζ) is submodular in (x1 , ζ) for any fixed x0 . Similar to the previous proof, we introduce the following new decision variables: y˜0 = x0 − y 0 , y˜1 = x0 − y 0 + y 1 , z˜0 = ζ − y˜0 , z˜1 = x1 − y˜1 . Then, the minimization problem (EC.1) can be rewritten as Vt (x0 − ζ, x1 − ζ) =

{ min

˜ (˜ y 0 ,˜ y 1 ,˜ z 0 ,˜ z 1 )∈D

Ht (x0 − y˜0 , y˜1 − y˜0 ) :

(

y˜0 +˜ z0 y˜1 +˜ z1

) =

( ζ )} x1

+ rx0 − px1 + (p − r)ζ,

where ˜ = {(˜ D y 0 , y˜1 , z˜0 , z˜1 ) | y˜0 ≤ x0 , 0 ≤ z˜0 − z˜1 ≤ Km , −Kr ≤ z˜0 ≤ 0, −K ≤ z˜1 ≤ 0}. Since Ht is L♮ -convex, Ht (x0 − y˜0 , y˜1 − y˜0 ) is convex and submodular in (˜ y 0 , y˜1 ). In addition, it can ˜ is a closed convex sublattice. Applying Theorem 1 of Chen et al. (2011) again we be seen that D obtain that Vt (x0 − ζ, x1 − ζ) is submodular in (x1 , ζ). The proof is complete.

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Proof of Proposition 2. The proof is by induction on t. First, since VT +1 (·, ·) ≡ 0, it follows from (7) that HT (y 0 − ζ, y 1 − ζ) = (s − r)y 0 + py 1 + GT (y 1 − y 0 ) − (p + s − r)ζ. Since p > r and s > 0, HT (y 0 − ζ, y 1 − ζ) is decreasing in ζ, so the lemma holds for t = T . Now assume inductively that the lemma holds for t + 1. In what follows, we shall prove it also holds for t, which then completes the proof. We first prove Vt+1 (x0 − ζ, x1 − ζ) − (p − r)ζ is decreasing in ζ for any fixed x0 and x1 . Denote ϕ = x1 + y 0 − y 1 and φ = x1 − y 1 . By replacing y 0 and y 1 with ϕ and φ, (6) can be rewritten as Vt+1 (x0 − ζ, x1 − ζ) − (p − r)ζ =

=

{

min

ϕ−x0 +ζ≤φ≤min{ϕ,ζ,ϕ−x0 +ζ+Kr } x0 −K≤ϕ≤x0

min

{ 0

φ≤min{ϕ,ζ,ϕ−x +ζ+Kr } x0 −K≤ϕ≤x0

} Ht+1 (ϕ − φ, x1 − φ) + rx0 − px1

} Ht+1 (ϕ − φ, x1 − φ) + rx0 − px1 ,

(EC.3)

where the second equality follows from the result that Ht+1 (ϕ − φ, x1 − φ) is decreasing in φ. Since the constraint in (EC.3) becomes less restrictive when ζ becomes larger, it follows that Vt+1 (x0 − ζ, x1 − ζ) − (p − r)ζ is decreasing in ζ. Now we prove Ht (y 0 − ζ, y 1 − ζ) is decreasing in ζ. According (7), we have Ht (y 0 − ζ, y 1 − ζ) =(s − r)y 0 + py 1 + Gt (y 1 − y 0 ) ( ) + αE[Vt+1 (y 0 − ζ + Rt , y 1 − ζ − Dt + Rt ) − (p − r)ζ] − s + (1 − α)(p − r) ζ. Since Vt+1 (x0 − ζ, x1 − ζ) − (p − r)ζ is decreasing in ζ, s > 0, p > r, and 0 < α ≤ 1, Ht (y 0 − ζ, y 1 − ζ) is decreasing in ζ. The proof is complete.

Proof of Theorem 1. Note that the model with only a total capacity K is a special case of the model with only capacities Kr and K satisfying Kr = K. Therefore, to prove Theorem 1, it suffices to verify that Theorem 2 reduces to Theorem 1 when K = Kr . Since Theorem 2 (i) is clearly consistent with Theorem 1

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(i) when Kr = K, it remains to verify that Theorem 2 (ii) is consistent with Theorem 1 (ii) when Kr = K. To this end, we consider the following two cases, 0 ≤ x0t ≤ K and x0t > K, separately. Case 1. If 0 ≤ x0t ≤ Kr = K, then Theorem 2 (ii-1) states that it is optimal to use the remaining 1 1∗ capacity K − x0t to raise the total inventory level to as close to x1∗ t as possible when xt ≤ xt and

it is optimal not to manufacture when x1t ≥ x1∗ t . These two statements are consistent with those specified in Theorem 1 (ii-1) and Theorem 1 (ii-2), respectively. Case 2. If x0t ≥ K = Kr , then Theorem 2 (ii) states that it is optimal not to manufacture, which is also consistent with that specified in Theorem 1 (ii-1). In summary, we have shown that, for the model with only a total capacity K, the optimal policies

are given in Theorem 1. The proof is complete.

Proof of Theorem 2. For the model with only capacities Kr and K with Kr ≤ K, the constraints for decisions (yt0 , yt1 ) must satisfy (x0t − Kr )+ ≤ yt0 ≤ x0t , yt1 ≥ x1t , 0 ≤ (yt1 − x1t ) + (x0t − yt0 ) ≤ K. Let ϕt = x1t + yt0 − yt1 , then these constraints are transformed to (x0t − Kr )+ ≤ yt0 ≤ x0t , yt0 ≥ ϕt , x0t − K ≤ ϕ ≤ x0t . Thus, the optimality equation (6) can be rewritten as Vt (x0t , x1t ) =

min

+ 0 0 max{ϕ,(x0 t −Kr ) }≤yt ≤xt 0 x0 −K≤ϕ≤x t t

{

} Ht (yt0 , x1t + yt0 − ϕt ) + rx0t − px1t .

(EC.4)

From Proposition 2, Ht (yt0 , x1t + yt0 − ϕ) is increasing in yt0 , thus the optimal yt0 is achieved at yt0 = max{ϕt , (x0t − Kr )+ }. Substituting this into (EC.4) yields Vt (x0t , x1t ) =

min

0 x0 t −K≤ϕt ≤xt

{

} Ht (max{ϕt , (x0t − Kr )+ }, x1t + ((x0t − Kr )+ − ϕt )+ ) + rx0t − px1t . (EC.5)

0 1 1∗ For each period t, define (x0∗ t , xt ) = arg minx0 ≥0,x1 ∈ℜ {Ht (x , x )}. By Proposition 2, Ht (δ − ζ, ξ −

ζ) is decreasing ζ and thus x0∗ t = 0. In addition, for any δ ≥ 0 and ξ ∈ ℜ, we define


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δt∗ (ξ) = arg minδ≥0 {Ht (δ, ξ)} and ξt∗ (δ) = arg minξ∈ℜ {Ht (δ, ξ)}. ∗ 1∗ ∗ Then, δt∗ (x1∗ t ) = 0 and ξt (0) = xt . In addition, by Proposition 1 and Lemma EC.1, both δt (·) and

ξt∗ (·) are increasing functions with slopes between zero and one. Instead of considering different regions separately, we take a unified approach and show that the firm’s optimal policies have the following simple and unified mathematical expressions: yt0∗ (x0t , x1t ) = min{x0t , max{(x0t − Kr )+ , δt∗ (x1t )}};

(EC.6)

yt1∗ (x0t , x1t ) = max{x1t , min{x1t + K − min{x0t , Kr }, ξt∗ ((x0t − Kr )+ )}}.

(EC.7)

Note that Ht (·, ·) is a convex function due to its L♮ -convexity from Proposition 1 and that the constraint set {(x0 , y 0 , ϕt ) | max{ϕt , (x0t − Kr )+ } ≤ yt0 ≤ x0t }

is a convex set. Thus, it follows that Ht (max{ϕt , (x0t − Kr )+ }, x1t + ((x0t − Kr )+ − ϕt )+ ) =

min

{

0 0 + max{ϕt ,(x0 t −Kr ) }≤yt ≤xt

Ht (yt0 , x1t + yt0 − ϕt )

}

is convex in ϕt . In addition, we have Ht (max{ϕt , (x0t − Kr )+ }, x1t + ((x0t − Kr )+ − ϕt )+ )    Ht ((x0t − Kr )+ , x1t + (x0t − Kr )+ − ϕt ), if ϕt ≤ (x0t − Kr )+ ; =   Ht (ϕt , x1t ), if ϕt ≥ (x0t − Kr )+ .

(EC.8)

Note that Ht ((x0t − Kr )+ , x1t + (x0t − Kr )+ − ϕt ) is convex in ϕt and achieves its minimum value at ϕt = x1t + (x0t − Kr )+ − ξt∗ ((x0t − Kr )+ ); and Ht (ϕt , x1t ) is also convex in ϕt and achieves its minimum value at ϕt = δt∗ (x1t ). In what follows, we consider two cases separately and verify that the optimal manufacturing/remanufacturing policies are given by (EC.6) and (EC.7) in each case. First, if δt∗ (x1t ) > (x0t − Kr )+ , by (EC.8), Ht (max{ϕt , (x0t − Kr )+ }, x1t + ((x0t − Kr )+ − ϕt )+ ) achieves its minimum value at ϕt = δt∗ (x1t ), so the optimal ϕ∗t in (EC.5) is equal to max{x0t − K, min{x0t , δt∗ (x1t )}}. Therefore, when δt∗ (x1t ) > (x0t − Kr )+ , we have yt0∗ (x0t , x1t ) = max{ϕ∗t , (x0t − Kr )+ } = min{x0t , δt∗ (x1t )}; yt1∗ (x0t , x1t ) = x1t + ((x0t − Kr )+ − ϕ∗t )+ = x1t .

ec8


Since ξt∗ (δ) is increasing in δ, it holds that ξt∗ (δt∗ (x1t )) > ξt∗ ((x0t − Kr )+ ), and since δt∗ (x1t ) is increasing + 1 1∗ 0 ∗ 1 in x1t and δt∗ (x1∗ t ) = 0, δt (xt ) > (xt − Kr ) implies xt > xt . In addition, because

∗ 1∗ ξt∗ (δt∗ (x1∗ t )) = ξt (0) = xt

and both δt∗ (·) and ξt∗ (·) are increasing functions with slopes no more than 1, we have x1t ≥ ξt∗ (δt∗ (x1t )) ≥ ξt∗ ((x0t − Kr )+ ). Hence it is seen that (EC.6) and (EC.7) hold when δt∗ (x1t ) > (x0t − Kr )+ . On the other hand, if δt∗ (x1t ) ≤ (x0t − Kr )+ , then by (EC.8), Ht (max{ϕt , (x0t − Kr )+ }, x1t + ((x0t − Kr )+ − ϕt )+ ) is increasing in ϕt when ϕt ≥ (x0t − Kr )+ , and achieves its minimum value at ϕt = (x0t − Kr )+ + min{x1t − ξt∗ ((x0t − Kr )+ ), 0}. Thus, the optimal ϕ∗t in (EC.5) is equal to min{(x0t − Kr )+ , max{x0t − K, x1t + (x0t − Kr )+ − ξt∗ ((x0t − Kr )+ )}. Therefore, when δt∗ (x1t ) ≤ (x0t − Kr )+ , we have yt0∗ (x0t , x1t ) = max{ϕ∗t , (x0t − Kr )+ } = (x0t − Kr )+ ; yt1∗ (x0t , x1t ) = x1t + ((x0t − Kr )+ − ϕ∗t )+ = max{x1t , min{x1t + K − min{x0t , Kr }, ξt∗ ((x0t − Kr )+ )}}. This shows that (EC.6) and (EC.7) also hold when δt∗ (x1t ) ≤ (x0t − Kr )+ . Summarizing the above two cases, we have shown that the firm’s optimal policies can be mathematically expressed by (EC.6) and (EC.7). To complete the proof, it remains to show that the firm’s optimal policies expressed in (EC.6) and (EC.7) are consistent with those in Theorem 2. To see this, we first consider the optimal reman1∗ ∗ 1 ∗ 1 ∗ ufacturing policy. If x1t ≤ x1∗ t , then by the monotonicity of δt (xt ) we have δt (xt ) ≤ δt (xt ) = 0, so


ec9

(EC.6) is simplified to yt0∗ (x0t , x1t ) = (x0t − Kr )+ and it corresponds to case i-1); and if x1t ≥ x1∗ t , then (EC.6) corresponds to case i-2). Thus, (EC.6) is consistent with the firm’s optimal remanufacturing policy in Theorem 2. We next consider the optimal manufacturing policy. First, if 0 ≤ x0t ≤ Kr , since ξt∗ (0) = x1∗ t , then (EC.7) is simplified to yt1∗ (x0t , x1t ) = max{x1t , min{x1t + (K − x0t ), x1∗ t }}. 1∗ 0 1 1 0 1∗ In this case, if x1t ≤ x1∗ t , then yt (xt , xt ) = min{xt + (K − xt ), xt } and it is optimal to use the

remaining capacity K − x0t to raise the total inventory level to as close to x1∗ t as possible; and otherwise, yt1∗ (x0t , x1t ) = x1t and it is optimal not to manufacture. Thus, this first case corresponds to case ii-1) in Theorem 2. Second, if x0t ≥ Kr , then (EC.7) becomes yt1∗ (x0t , x1t ) = max{x1t , min{x1t + K − Kr , ξt∗ (x0t − Kr )}}. In this case, if x1t ≤ ξt∗ (x0t − Kr ), then yt1∗ (x0t , x1t ) = min{x1t + K − Kr , ξt∗ (x0t − Kr )} and it is optimal to use the remaining capacity K − Kr to raise the total inventory level to as close to ξt∗ (x0t − Kr ) as possible; and otherwise, yt1∗ (x0t , x1t ) = x1t and it is optimal not to manufacture. Thus, this second case corresponds to case ii-2) in Theorem 2. Therefore, (EC.7) is consistent with the firm’s optimal manufacturing policy in Theorem 2. The proof is complete.

Proof of Theorem 3. We start the proof by defining the control parameters and functions and proving their properties. 0 1 1∗ For each period t, define (x0∗ t , xt ) = arg minx0 ≥0,x1 ∈ℜ {Ht (x , x )}; and for δ ≥ 0 and ξ ∈ ℜ, define

δt∗ (ξ) = arg minδ≥0 {Ht (δ, ξ)} and ξt∗ (δ) = arg minξ∈ℜ {Ht (δ, ξ)}. ∗ 1∗ ∗ Then, it directly follows from the above definition that x0∗ t = δt (xt ) = arg minδ≥0 {Ht (δ, ξt (δ))} ∗ 0∗ ∗ and x1∗ t = ξt (xt ) = arg minξ∈ℜ {Ht (δt (ξ), ξ)}. In addition, from (7) and Proposition 1, Ht (δ, ξ) is

L♮ -convex. Thus, it follows from Lemma EC.1 that both δt∗ (·) and ξt∗ (·) are increasing functions with slopes between zero and one.

ec10


We now verify the firm’s optimal policies in Theorem 3. Note that the firm’s optimal policies specified in Theorem 3 have the following unified mathematical expressions: yt0∗ (x0t , x1t ) = min{x0t , max{x0t − Kr , δt∗ (max{x1t , min{x1t + Km , x1∗ t }})}};

(EC.9)

yt1∗ (x0t , x1t ) = max{x1t , min{x1t + Km , ξt∗ (min{x0t , max{x0t − Kr , x0∗ t }})}}.

(EC.10)

Thus, to complete the proof of the theorem, we only need to prove that the firm’s optimal policies can be mathematically expressed by (EC.9) and (EC.10). We first prove the firm’s optimal manufacturing policy can be expressed by (EC.10). For con˜ t (x0t , yt1 ) = min(x0 −K )+ ≤y0 ≤x0 {Ht (yt0 , yt1 )}. Then, the optimality equation (6) venience, we denote H r t t t can be rewritten as Vt (x0t , x1t ) =

min

1 1 x1 t ≤yt ≤xt +Km

˜ t (x0t , yt1 )} + rx0t − px1t . {H

(EC.11)

˜ t (x0t , yt1 ) Since Ht (·, ·) is a jointly convex function and the constraint is a convex set, it follows that H is convex in yt1 . Thus, it follows from (EC.11) that { { }} ˜ t (x0t , yt1 )} . yt1∗ = max x1t , min x1t + Km , arg min {H yt1 ∈ℜ

Therefore, to verify the firm’s optimal manufacturing policy in (EC.10), it suffices to show that ˜ t (x0t , yt1 )} = ξt∗ (min{x0t , max{x0t − Kr , x0∗ arg min {H t }}); yt1 ∈ℜ

or equivalently, ˜ t (x0t , yt1 )} = H ˜ t (x0t , ξt∗ (min{x0t , max{x0t − Kr , x0∗ min {H t }})).

yt1 ∈ℜ

(EC.12)

0 0 0 ∗ ˜ t (·, ·), ξt∗ (·) and x0∗ By the definitions of H t and the convexity of Ht (yt , ξt (yt )) in yt , it is easy to

verify that ( ) ˜ t x0t , yt1 } = min {H

yt1 ∈ℜ

min

yt1 ∈ℜ + 0 0 (x0 −K r ) ≤yt ≤xt t

{Ht (yt0 , yt1 )} =

min

0 0 + (x0 t −Kr ) ≤yt ≤xt

{Ht (yt0 , ξt∗ (yt0 ))}

∗ 0 0 0∗ = Ht (min{x0t , max{x0t − Kr , x0∗ t }}, ξt (min{xt , max{xt − Kr , xt }}))


≥

min

0 0 + (x0 t −Kr ) ≤yt ≤xt

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{Ht (yt0 , ξt∗ (min{x0t , max{x0t − Kr , x0∗ t }}))}

˜ t (x0t , ξt∗ (min{x0t , max{x0t − Kr , x0∗ =H t }})) ( ) ˜ t x0t , yt1 }. ≥ min {H yt1 ∈ℜ

Thus, all the inequalities above must hold as equalities and (EC.12) is established. Therefore, the firm’s optimal manufacturing policy can be expressed by (EC.10). We next prove the firm’s optimal remanufacturing policy can be expressed by (EC.9). Similar to ˆ t (x1t , yt0 ) = minx1 ≤y1 ≤x1 +K {Ht (yt0 , yt1 )}. Then, the optimality equation the above proof, we denote H m t t t (6) can be rewritten as Vt (x0t , x1t ) =

min

0 0 + (x0 t −Kr ) ≤yt ≤xt

ˆ t (x1t , yt0 )} + rx0t − px1t . {H

(EC.13)

ˆ t (x1t , yt0 ) is convex in yt0 . Thus, it follows Since, from Proposition 1, Ht (·, ·) is a convex function, H from (EC.13) that { { }} ˆ t (x1t , yt0 )} . yt0∗ = min x0t , max x0t − Kr , arg min{H yt0 ≥0

∗ 1∗ Notice that x0∗ t = δt (xt ). Thus, to complete the proof, it remains to show that

ˆ t (x1t , yt0 )} = δt∗ (max{x1t , min{x1t + Km , x1∗ arg min{H t }}). yt0 ≥0

Since the remaining proof is very similar to that of showing (EC.12), we omit the details for brevity. The proof is complete.

References [1] Chen, X., P. Hu, S. He. 2011. Preservation of supermodularity in two dimensional parametric optimization problems and its applications. Working paper. [2] Porteus, E. L. 2002. Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford, CA. [3] Topkis, D. 1998. Supermodularity and Complementarity. Princeton University Press, Princeton, NJ.