Optimizing Replenishment Intervals for Two

Optimizing Replenishment Intervals for Two-Echelon Distribution Systems with Fixed Order Costs Kevin H. Shang • Sean X. Zhou Fuqua School of Business, Duke University, Durham, North Carolina 27708, USA Systems Engineering & Engineering Management, Chinese University of Hong Kong, China [email protected] • [email protected] March 17, 2010

We consider a periodic-review inventory system with one warehouse and N non-identical retailers. Each location has a positive lead time and orders according to a base-stock policy in fixed time interval. A fixed cost is incurred for each inventory reorder. The warehouse fills the retailers’ orders in the same sequence as the original demand. The objective is to minimize the long-run average system cost per period. This paper provides an approach to obtain the optimal base-stock levels and reorder intervals. Specifically, for fixed reorder intervals, we show that the optimal base-stock levels can be obtained by generalizing a result in the literature. To find the optimal reorder intervals, we propose a complete enumeration on feasible solutions. The feasible region is determined based on two analytical results: (1) We decouple the total system cost into each stage and construct a lower bound on the decoupled stage cost; (2) we construct a lower bound on the total system cost by converting the distribution system into a single-location model with the original problem data. These stage and system lower bounds together generate bounds for the optimal reorder intervals. Since the computation of the optimal reorder intervals is quite extensive, we further suggest a simple heuristic by solving a set of serial systems. The heuristic appears to be near-optimal in the tested problems. The primary findings from a numerical study are that the optimal reorder intervals tend to follow integer-ratio relations and that the optimal reorder interval of the warehouse is no less than the minimum of the optimal reorder intervals of the retailers.

1.

Introduction

Materials are often produced and shipped according to fixed schedules in a supply chain. For example, manufacturers often plan production according to material requirement planning (MRP) systems. The MRP system generates replenishment orders weekly, bi-weekly, or monthly, which results in periodic inventory replenishment. Most consumer and finished goods are distributed in the same fashion. Materials are often delivered in fixed days of the week. In these production/distribution systems, fixed costs, such as inventory review costs, shipping costs, and transportation costs, are often associated with each inventory reorder. Since these costs account for a significant portion of

1

total supply-chain cost, it is important for managers to determine effective replenishment schedules and inventory stocking levels across the supply chain. We consider a one-warehouse-multi-retailer system that models the above periodic replenishment practice. There are N non-identical retailers, each facing independent Poisson demand with complete backlogging. The retailers replenish their inventories from the warehouse, which in turn orders from an outside supplier with unlimited stock. Each location implements a stationary echelon (s, T ) policy: Stage j reviews its echelon inventory order position (= inventory on order + inventory on hand + inventory at or in transit to all downstream stages - backorders at the retailers) in every Tj periods and orders up to an echelon base-stock level Sj , j = 0, ..., N . We assume that the order schedules are synchronized, i.e., a retailer, whenever possible, places an order when the warehouse receives a shipment (see §2 for an example). We also assume that the warehouse applies the virtual allocation rule (e.g., Axs¨ater 1993a, Graves 1996), in which the retailers’ orders are filled in the same sequence as the occurrence of the demand at the retailers. The virtual allocation rule may not be the most efficient way for the warehouse to allocate stocks. However, this rule is convenient to implement and thus commonly seen in practice. For example, Wal-Mart’s distribution center can receive real-time demand information from its retail stores. The distribution center assigns replenishment stocks to the demands as they occurred. The assigned stocks will be loaded onto a truck and shipped to the retail stores according to some fixed schedule (Chandran, 2003). We assume that a fixed order cost is incurred for each inventory reorder. In addition, a linear holding cost per period is incurred for each unit of on-hand inventory held at each stage, and a linear backorder cost per period is charged for each unit of backlogs incurred at the retailer. The objective is to obtain the echelon (s, T ) policy such that the average total cost per period is minimized. We first characterize several key inventory variables and their dynamics. We then provide a bottom-up recursion to evaluate the average system cost. It is worth mentioning that our model assumes that the reorder intervals are integer multiples of some base period. In other words, the reorder interval of the warehouse may be shorter than that of the retailer. With fixed reorder intervals, the optimal echelon base-stock levels can be found by generalizing the result of Axs¨ater (1990). We show this generalization in Appendix A. Our main contribution is to provide a method for obtaining the optimal reorder intervals. This is achieved by two results we establish. First, we allocate the system cost into each stage by decomposing the retailer’s cost. We then establish a lower bound to each of the allocated stage cost functions. Specifically, the lower bound for 2

the allocated retailer cost is generated from a single-stage (s, T ) model; the lower bound for the allocated warehouse cost is obtained by regulating the retailers’ reorder intervals. These stage cost bounds are a function of a stage’s control parameters. Second, we construct a lower bound to the system cost by converting the distribution system into a single-stage system with the original problem data. These results together generate bounds for the optimal reorder intervals. The optimal reorder intervals can therefore be obtained by a numerical search over the feasible region. The computation for finding the optimal policy is quite extensive, however. In a numerical study, we observe that the optimal reorder intervals always satisfy integer-ratio relations, i.e., the ratios of retailer’s reorder interval to warehouse’s and to other retailers’ reorder intervals or the reciprocals of the ratios are positive integers. In addition, we find that the optimal reorder interval of the warehouse is always no less than the minimum of the optimal reorder intervals of the retailers. These observations motivate us to propose a simple heuristic by minimizing the sum of the lowerbound cost functions subject to integer-ratio constraints on the reorder intervals. Because there is no order relation on the optimal reorder intervals, solving the above approximate problem is equivalent to solving a set of serial systems. For each of the serial system, we apply Shang and Zhou’s (2009) heuristic to generate a solution. The final heuristic solution is the one that leads to the minimum system cost. A numerical study suggests that the heuristic is near-optimal: The average (maximum) percentage error between the optimal cost and the heuristic cost is 0.28% (2.07%). We also identify the conditions under which the heuristic performs less effectively. Finally, unlike the serial system studied by Shang and Zhou (2009), the ratio of fixed order cost to the holding cost seems to have less impact on the optimal reorder interval. We provide a detailed discussion in §6. Our model is a generalization of the joint replenishment problem (JRP) in the literature. The JRP is described below: Consider a retailer who manages N items. A major fixed cost is incurred if any of the items is ordered and a minor, item-specific fixed cost is incurred if item j is ordered, j = 1, ..., N . The retailer has to decide which item(s) should be replenished and how much to order for each item. Our model reduces to the JRP by assuming that (1) the warehouse lead time is zero, (2) the warehouse does not hold inventory, (3) the reorder intervals satisfy the integer-ratio relations, and (4) the reorder interval of the warehouse is equal to the minimum of those of the retailers. Under these constraints, the warehouse will place an order whenever any of the retailers places an order. The major setup cost in JRP is equivalent to the fixed cost incurred at the warehouse. Consequently, our optimization algorithm may be revised to solve the stochastic JRP.

3

Literature Review The literature related to multi-echelon, distribution systems is quite extensive. We only focus on the most related work, i.e., the stochastic demand models with general reorder intervals. For the deterministic demand model, we refer the reader to Muckstadt and Roundy (1993) for a review. For the stochastic demand model with base-stock policies (i.e., the reorder intervals are one for all stages), we refer the reader to Axs¨ater (1990) for an exact optimization algorithm and Gallego et al. (2007) for an example of heuristic solutions. For the stochastic demand model with reorder point, order quantity policies, we refer readers to Axs¨ater (1993a) and Cachon (2001) for local control systems, and Chen and Zheng (1997) for echelon control systems. See Axs¨ater (2003) for a complete review. For the distribution system with fixed reorder intervals, Graves (1996) considered a general distribution system that may have more than two echelons. The inventory is controlled by a local (s, T ) policy. His model assumes the virtual allocation rule and does not require any pattern of the order schedule other than it is fixed and known. Graves found that most of the safety stock should be held at the retailer sites and that the virtual allocation rule is near optimal in a numerical study. Axs¨ater (1993b) studied a special case of Graves’s model in that the retailers are identical and the order schedule is nested, i.e., the reorder interval of the warehouse is an integer multiple of those of the retailers. He demonstrated that the local (s, T ) policy under the virtual allocation rule in Graves is essentially the same as the echelon (s, T ) policy. In addition, he showed that the algorithm in Axs¨ater (1990) can be applied to find the optimal base-stock levels. Lee et al. (1997) demonstrated that the relationships between ordering schedules and bullwhip effect. They showed that balanced ordering (i.e., the same number of retailers order each period) will lead to the minimum level of order variability to the supplier. Cachon (1999) considered a one supplier, N identical retailer system where retailers’ orders are balanced. He showed that the supplier’s demand variance can be reduced when the retailer order intervals are lengthened. Chen and Samroengraja (2004) compared a staggering order policy (a special case of balanced ordering) with an (r, Q) policy. They found that the staggering policy leads to a higher total cost although it dampens the bullwhip effect. Cheung and Zhang (2008) compared synchronized ordering with balanced ordering and obtained a similar conclusion as Chen and Samroengraja, i.e., synchronized ordering leads to a smaller system cost than balanced ordering. There are papers aiming to find the optimal reorder interval in a context of distribution systems. C ¸ etinkaya and Lee (2000) considered a warehouse that replenishes its stock according to an (s, S) 4

policy and ships stocks to retailers in every fixed interval. They provided a method to find the optimal control parameters. G¨ urb¨ uz et al. (2007) considered a new replenishment policy in which retailers are replenished simultaneously in every fixed interval. They compared this new policy with three existing policies in the literature. The reorder interval decision in these models is a single variable. This is different from ours, where we determine a set of reorder intervals for the entire system. A closely related paper is Marklund (2008), in which a one-warehouse-multi-retailer distribution system is considered under continuous review scheme. The warehouse implements a local (r, Q) policy and allocates the stock to the retailers according to the virtual allocation rule. The retailers implement a (s, T ) policy. There is a fixed order cost for each inventory reorder at the retailer but no fixed order cost at the warehouse. Marklund provided a method to evaluate the average system cost and proposed a heuristic to generate the policy parameters. Our model is different from Marklund’s in that our model assumes (1) echelon control, (2) that the warehouse uses (s, T ) policy, and (3) that there is a fixed order cost incurred at the warehouse. In terms of the result, we develop an approach to obtain the exact optimal solution, while Marklund only proposed an effective heuristic. One important issue for managing distribution systems is how the warehouse allocates stocks to the retailers. This subject has attracted much research attention in the literature. Eppen and Schrage (1981) assumed that the warehouse does not hold stock and the retailers are identical. Under the assumption that the stocks can be freely transmitted between the retailers (the so-called “balanced assumption”), they showed how to obtain the optimal base stock levels. Federgruen and Zipkin (1984) considered a similar model in a discrete time setting. By approximating the dynamic program, they proposed several approaches to generate near-optimal order and allocation policies. Jackson (1988) extended Eppen and Schrage’s model by assuming that the warehouse can hold stocks. Thus, the warehouse can allocate stock during the warehouse order cycle. Schwarz (1989) examined the value of warehouse risk pooling over the warehouse lead time. Chen and Samroengraja (2000) considered a similar system as Cachon (1999) and compared two allocation policy: Past priority allocation (PPA) and current priority allocation (CPA). They found that the CPA, which utilizes the current retailer inventory information, performs slightly better than the PPA on average, but there is no significant dominance. Finally, as stated, our model reduces to JRP under some conditions. A policy suggested to manage JRP in the early literature is called “can-order” policy or an (s, c, S) policy that aims to 5

coordinate orders. This policy were studied by, e.g., Silver (1981) and Federgruen et al. (1984). They found that the policy is considerably more effective than uncoordinated policies. Atkins and Iyogun (1988) studied a periodic-review JRP in which each item is replenished according to an (s, T ) policy. They showed the periodic review (s, T ) policy outperforms the can-order policy especially when the major fixed cost is large. The JRP resulted from our model is the same as that of Atkins and Iyogun. Thus, our optimization method can be used for obtaining the exact optimal solution in Atkins and Iyogun. The rest of the paper is organized as follows. §2 introduces the model and the notation. §3 characterizes the inventory variables and evaluates the total cost per period. §4 provides an approach for finding the optimal reorder intervals. §5 proposes a simple heuristic. §6 conducts a numerical study to examine the optimal solution and the effectiveness of the heuristic. §7 concludes. Appendix A shows how to optimize the base-stock levels when the reorder intervals are fixed. Appendix B provides an evaluation scheme for a serial system with non-nested (s, T ) policies. Appendix C presents the proofs.

2.

Model and Assumptions

We consider a periodic-review, two-echelon distribution system in which a single warehouse supplies N non-identical retailers. Time is divided into equal length of one and periods are indexed 0, 1, 2, .... Let [t, t + τ ) and [t, t + τ ] denote the time interval over periods t, t + 1, ..., t + τ − 1 and periods t, t + 1, ..., t + τ , respectively. We use j as the stage index, where 0 represents the warehouse and j represents the retailer j, j = 1, ..., N . Retailer j faces a Poisson demand with stationary rate λj . The retailers’ demands are independent of each other. Let Dj [t, t + τ ) and Dj [t, t + τ ] denote the cumulative demand over time in [t, t + τ ) and [t, t + τ ]. There is a constant lead times Lj for stage j, j = 0, ..., N . Let L[0,j] = L0 + Lj . Each stage implements a stationary echelon (s, T ) policy, where Sj is the echelon base-stock level and Tj is the reorder interval for stage j. The policy is operated as follows: At the beginning of every Tj periods, stage j orders up to Sj if the echelon inventory order position is less than Sj . For the retailer, the echelon inventory order position is equal to its local inventory order position, i.e., inventory on order plus inventory on hand minus backorders. For the warehouse, the echelon inventory order position is the inventory on order plus inventory on hand plus inventory in transit to and held at the retailers minus backorders at the retailers. We term these Tj -th periods as order periods and the moment of placing an order as order

6

epoch. Let hj be the echelon holding cost rate for stage j, j = 0, ..., N , and the local holding cost rate Hj = h0 + hj for j = 1, ..., N . Unmet demand is fully backlogged at all stages. Let bj be the backorder cost rate for stage j, j = 1, ..., N . Finally, there is a fixed cost Kj associated with each inventory reorder. The objective is to obtain the policy such that the total system cost per period is minimized. We assume that the reorder intervals are an integer multiple of some base period. Thus, T0 may be smaller than Tj , j = 1, ..., N . Without loss of generality, let the base period be one. The ordering activities between the warehouse and the retailers are coordinated in a synchronized manner: Whenever possible, retailer j places an order when the warehouse receives a shipment. For example, consider a one-warehouse-two-retailer system with L0 = 3, T0 = 2, T1 = 1, and T2 = 3. Suppose that the warehouse places an order at the beginning of period t, t + 2, t + 4, ..... Consider the order epoch t. This order placed at t will arrive at the beginning of period t + L0 = t + 3. This is the moment that both retailers place an order. Thus, the order periods for the retailer 1 is t + 3, t + 4, t + 5, ..., and for the retailer 2 is t + 3, t + 6, t + 9, .... We term such a order coordination as synchronized ordering rule. Let T = lcm{T0 , T1 , ..., TN }, where lcm is an operator that generates the least common multiplier of Tj ’s. If we define t and t + L0 as the starting time of a cycle, clearly, the next cycle will start in T periods later under the synchronized ordering rule. (That is, the next time when the warehouse receives a shipment and all retailers place an order occurs in T periods later.) Allowing the possibility of T0 less than Tj implies a non-nested ordering policy. (A ordering policy is nested if each retailer orders every time when the warehouse receives a shipment. The nested ordering policy implies T0 = nj Tj , j = 1, ..., N, nj some positive integer. See Maxwell and Muckstadt 1985). It is interesting to note that for the serial system under the nested policy assumption, Chao and Zhou (2008) showed that the synchronized ordering rule is the best among all possible scheduling rules for the stationary echelon (s, T ) policy; Shang et al. (2009) showed the same result for the stationary local (s, T ) policy. An intuitive explanation for these results is that a shipment arrived at an upstream stage can be immediately transferred to its downstream stage under the synchronized ordering rule. This helps to meet the demand more effectively, which, in turn, results in a smaller total system cost. One critique for synchronized ordering under the distribution system is that it may lead to a significant bullwhip effect when the system is controlled 7

by local policies (Lee et al. 1997). However, Cheung and Zhang (2008) pointed out that high bullwhip effect may not necessarily lead to a high system cost. Moreover, the bullwhip effect has less impact in our model because the demand is learned by the warehouse when it occurs (i.e., echelon control). Under the echelon (s, T ) policy, the retailer fills the incoming demand as if it were a single-stage system. That is, when a unit of demand arrives, the retailer fills the demand and the retailer’s net inventory level (= inventory on hand minus backorders) is reduced by one unit. The retailer does not place an order until its next order epoch. On the other hand, the warehouse immediately learns this arrived demand and assigns one unit to this demand. (In other words, the inventory order position for the retailer is always equal to its base-stock level.) Similarly, the warehouse does not order until its next order epoch. In short, the downstream ordering is for triggering a shipment sent from the warehouse, not for transmitting the demand information. Thus, one may view the retailers’ order epoches as “shipping times” planned by the warehouse. Because of the centralized information available, we assume that the warehouse commits its inventory, if available, to fill the incoming orders according to the sequence of the demands occurring at the retailer site. If the warehouse does not have an uncommitted unit to fill an arrived demand, the warehouse creates a backorder and adds this to the current list of outstanding orders. When inventory becomes available, the outstanding orders are filled in the sequence in which they are created. This is the so-called virtual allocation rule (Graves 1996). We end this section by listing the sequence of events in a period: (1) A stage receives a shipment; (2) the stage places an order if the period is its order period; (3) Poisson demand arrives during the period; (4) costs are evaluated at the end of the period. The objective is to obtain the policy parameters such that the average total cost per period is minimized.

3.

Evaluation

Under the echelon (s, T ) policy, the system forms a regenerative process with a cycle of T periods. Specifically, consider a warehouse order epoch t at which the warehouse orders up to S0 . This order will arrive at the beginning of period t + L0 . We assume that this is the moment that all retailers will simultaneously place an order up to Sj . Thus, the regenerative cycle that we consider in the subsequent analysis is [t, t + T − 1] for the warehouse and [t + L0 , t + L0 + T − 1] for retailer j. We call t and t + L0 regenerative epoch for the warehouse and the retailers, respectively. To evaluate

8

the total cost per period, we only need to characterize the distribution of the inventory variables in the above regenerative cycles. The long-run average total cost per period is equal to the expected total cost incurred in the cycle divided by the cycle length T . We first define inventory variables below: For stage j, j = 0, ..., N , IOPj (n) = echelon inventory order position after ordering at stage j at the beginning of period n, IL− j (n) = echelon inventory level at stage j at the beginning of period n, ILj (n) = echelon inventory level at stage j at the end of period n. We describe the inventory dynamics in the regenerative cycle. Let r be the period index in the considered regenerative cycle, i.e., r = 0, 1, 2, ..., T − 1. Also, define ⌊a⌋ as the roundoff operator, which returns the greatest integer less than or equal to a, a real number. Define ⌊ rj (r) =

⌋ r Tj , Tj

j = 0, 1, 2, ..., N.

Thus, for r = 0, 1, . . . , T − 1, ⌊

⌋ T −1 rj (r) = 0, Tj , 2Tj , ..., Tj . Tj Since t and t + L0 are a regenerative epoch for the warehouse and the retailers, respectively, the warehouse’s order periods are t + r0 (r), and the retailer j’s order periods are t + L0 + rj (r). As we shall see, the order decision for the warehouse at the beginning of period t + r0 (r) will directly or indirectly determine the retailer j’s inventory levels ILj (t + L[0,j] + r). ∑N ∑N Let D0 [s, t) = j=1 Dj [s, t) and D0 [s, t] = j=1 Dj [s, t]. The inventory dynamics for the warehouse are IOP0 (t + r0 (r)) = S0 ,

(1)

IOP0 (t + r) = IOP0 (t + r0 (r)) − D0 [t + r0 (r), t + r)

(2)

= S0 − D0 [t + r0 (r), t + r), IL− 0 (t + L0 + r) = IOP0 (t + r) − D0 [t + r, t + L0 + r),

(3)

IL0 (t + L0 + r) = IOP0 (t + r) − D0 [t + r, t + L0 + r].

(4)

9

Equation (1) means that the warehouse’s echelon inventory order position after ordering is equal to S0 . Equation (2) shows the warehouse’s echelon inventory order position at any given time t + r in the regenerative cycle. Equations (3) and (4) show the echelon inventory level at the beginning and end of period t + L0 + r, respectively. Now, we consider retailer j. Since retailer j may not be able to obtain what it ordered from the warehouse, we define IPj , B0 and B0j to reflect this fact: IPj (n) = inventory in-transit position at the beginning of period n at retailer j, B0 (n) = total number of warehouse backorders at the beginning of period n, B0j (n) = number of warehouse backorders that belong to retailer j. Clearly, B0 (n) =

∑N

j=1 B0j (n).

The corresponding regenerative cycle for stage j includes periods t + L0 + r, r = 0, 1, .., T − 1. Suppose that IPj (t + L0 + r) is known, it will determine ILj (t + L[0,j] + r) as follows: ILj (t + L[0,j] + r) = IPj (t + L0 + r) − Dj [t + L0 + r, t + L[0,j] + r].

(5)

After all ILj in each period of the regenerative cycle is obtained, the total cost per period is C(S, T) =

T −1 N ∑ 1 ∑ [ E h0 IL0 (t + L0 + r) + hj ILj (t + L[0,j] + r) Tj T r=0 j=1 j=0 ] − +(bj + Hj )[ILj (t + L[0,j] + r)] .

N ∑ Kj

+

(6)

The first term in C(S, T) represents the average total fixed cost per period1 . The rest represents the average inventory holding and backorder cost per period. Our remaining task is to characterize IPj (t + L0 + r). First, for retailer j, notice that IPj (t + L0 + r) depends on IPj (t + L0 + rj (r)), i.e., IPj (t + L0 + r) = IPj (t + L0 + rj (r)) − Dj [t + L0 + rj (r), t + L0 + r).

(7)

Since retailer j may not be able to receive the full quantity it ordered in period t + L0 + rj (r), IPj (t + L0 + rj (r)) is not necessary equal to Sj . More specifically, IPj (t + L0 + rj (r)) = IOPj (t + L0 + rj (r)) − B0j (t + L0 + rj (r)) = Sj − B0j (t + L0 + rj (r)).

(8)

∑ Technically, the fixed cost term should be N j=0 Kj Pr(D[Tj ) > 0)/Tj . Following Section 5 of Shang and Zhou (2009), it is easy to show that all of our results can be carried over to the exact cost expression. We ignore Pr(D[Tj ) > 0) in the cost expression here because this probability should be reasonably close to one. 1

10

′

To characterize B0j (t+L0 +rj (r)), we first have to characterize B0 (t+L0 +rj (r)). Define IL0 (n) and ′

IOP0 (n) the local inventory level and inventory order position for the warehouse at the beginning of period n, respectively. By definition, ′

B0 (t + L0 + rj (r)) = [IL0 (t + L0 + rj (r))]− ′

= [IOP0 (t + rj (r)) − D0 [t + rj (r), t + L0 + rj (r))]− N [ ]− ∑ = IOP0 (t + rj (r)) − D0 [t + rj (r), t + L0 + rj (r)) − IOPi (t + rj (r)) [ =

IOP0 (t + rj (r)) − D0 [t + rj (r), t + L0 + rj (r)) −

i=1 N ∑

Si

]−

,

(9)

i=1

where [x]− = min{0, −x}. The last term in (9) is due to the fact that IOPi (t + rj (r)) = Si under the virtual allocation rule. Furthermore, from (2), we can re-write (9) as follows: B0 (t + L0 + rj (r)) [ = S0 − D0 [t + r0 (rj (r)), t + rj (r)) − D0 [t + rj (r), t + L0 + rj (r)) − [

]−

N ∑

]− Sj

j=1

= s0 − D0 [t + r0 (rj (r)), t + rj (r)) − D0 [t + rj (r), t + L0 + rj (r)) [

]−

= s0 − D0 [t + r0 (rj (r)), t + L0 + rj (r)) where s0 = S0 −

∑N

j=1 Sj ,

(10)

the local base-stock level for the warehouse. Note that r0 (rj (r)) ≤ rj (r),

which implies that the time interval in D0 in (10) must be at least L0 periods. Under the virtual allocation rule, we can apply binomial disaggregation on B0 to obtain the distribution of B0j because demand arrival at each retailer follows an independent Poisson process. That is, for any period n, we have )m−k ( ) ( )k ( λj λ0 − λj m P(B0j (n) = k|B0 (n) = m) = , k λ0 λ0 where λ0 =

∑N

j=1 λj .

k = 0, 1, ..., m,

(11)

Since this conditional probability is independent of time, we shall omit the

period index n in the further analysis. Consequently, the distribution of B0j (t + L0 + rj (r)) is P (B0j (t + L0 + rj (r)) = k) =

∞ ( ( ( ) ) ) ∑ P B0 t + L0 + rj (r) = m P (B0j = k|B0 = m) . m=k

11

(12)

From Equations (10)-(12), we can obtain the distribution for B0j at the retailer j’s order period. After B0j is characterized, we can use (7) and (8) to further obtain the distribution of IPj . Finally, we apply (5) to obtain the distribution of ILj for each period in the considered regenerative cycle. The average total cost per period is shown in (6). A Bottom-up Evaluation Scheme We provide a bottom-up recursion to simplify the evaluation of C(S, T). This procedure is to first evaluate the retailer j’s cost by assuming that the retailer has ample supply. Then, we evaluate the echelon cost of the warehouse, which is equivalent to the total system cost. More specifically, let Mx (y) be an operator that returns the remainder of y divided by x, where x is a positive integer and y is a nonnegative integer. Let Dj [τ ) and Dj [τ ] denote the total demand in τ and τ + 1 periods, respectively, at retailer j. For j = 1, ..., N , define Gj (y, Tj , r) = E[hj (y − Dj [Lj + MTj (r)]) + (bj + Hj )(y − Dj [Lj + MTj (r)])− ], and

 G0 (S, T, r) = E h0 (S0 − D0 [L0 + MT0 (r)]) +

N ∑

(13)

 Gj (Sj − B0j (L0 + rj (r)) , Tj , r) , (14)

j=1

where B0j (L0 + rj (r)) is the steady-state distribution for B0j (t + L0 + rj (r)), which can be found by removing the time index t in (10)-(12). Let def

G(S, T) =

T −1 1 ∑ G0 (S, T, r). T r=0

Proposition 1 For given echelon base-stock policies with parameters (S, T), the average total cost per period is C(S, T) =

N ∑ Kj j=0

where

∑N

j=0 (Kj /Tj )

Tj

+ G(S, T),

is the average total fixed order cost per period and G(S, T) is the average

inventory holding and backorder cost per period. Proposition 1 can be proven by comparing the total cost obtained by using the above recursion with the total cost obtained in (6). We omit the details for brevity.

12

4.

Optimization

This section discusses how to obtain the optimal (s, T ) policies. When the reorder intervals are fixed, it can be shown that Axs¨ater’s (1990) algorithm can be revised to obtain the optimal echelon base-stock levels. See Appendix A for the revised algorithm. As a result, our focus is to optimize the reorder intervals. We suggest a complete enumeration to find the optimal reorder intervals. To facilitate the search, we construct bounds for the optimal reorder interval Tj∗ , j = 0, ..., N . These solution bounds can be obtained by constructing a lower-bound function for each retailer’s cost and for the entire system cost. More specifically, in §4.1, we allocate the total system cost into each stage by decomposing the retailer’s cost. In §4.2, we construct a lower bound to the allocated warehouse’s cost and retailer’s cost. These lower bounds are a function of a stage’s control parameters. In §4.3, we show that the total cost for the distribution system is bounded below by the total cost obtained from a single-stage system with the original problem data. Finally, in §4.4, we show how to construct upper and lower bounds for the optimal reorder intervals by using these stage and system cost bound functions.

4.1

Decomposition of the System Cost

We provide an approach to allocate G(S, T) into each stage. This approach is to decompose the retailer’s cost function in each period r in the considered regenerative cycle. More specifically, for j = 1, 2, ..., N and r = 1, 2, ..., T − 1, define gj (y, Tj , r) = Gj (y, Tj , r). Let Sj (Tj , r) = argminy gj (y, Tj , r), and { gjj (y, Tj , r) =

gj (Sj (Tj , r), Tj , r), gj (y, Tj , r),

if y ≤ Sj (Tj , r), otherwise.

Let gj0 (y, Tj , r) = gj (y, Tj , r) − gjj (y, Tj , r). Notice that, for fixed Tj and r, gjj (y, Tj , r) is a constant before Sj (Tj , r) and convex and increasing in y after Sj (Tj , r). The minimum value is gj (Sj (Tj , r), Tj , r). On the other hand, gj0 (y, Tj , r) is convex and decreasing in y with minimum value zero after Sj (Tj , r). 13

We provide an explanation for this decomposition. The gj (y, Tj , r) function is the expected inventory holding and backorder cost for retailer j in period t + L[0,j] + rj (r) + MTj (r), referred to as period r below, when retailer j’s IPj (t + L0 + rj (r)) = y. Sj (Tj , r) represents the IPj level that leads to retailer j’s minimum cost in period r. Notice that Sj (Tj , r) is different in each period r. From retailer j’s perspective, it would be the best if the IPj (t + L0 + rj (r)) level can achieve Sj (Tj , r) in period r. Unfortunately, according to the considered (s, T ) policy, achieving different Sj (Tj , r) in period r is impossible because retailer j is only allowed to place an order in period t + L0 + rj (r). For this reason, we define gjj (y, Tj , r) as minimum period cost for retailer j, and g0j (y, Tj , r) as policy-induced period penalty cost from retailer j or perid penalty cost for short. There are two reasons that attribute to this penalty cost function. The first reason is that the warehouse may be short of stock, making IPj less than a desired level; the second reason is due to the nature of the (s, T ) policy – it is just impossible to have a different IPj for each period r. Note that gj0 (y, Tj , r) is different from the so-called induced-penalty cost function discussed in the multi-echelon literature, e.g., Chen and Zheng (1998), Shang and Zhou (2009), in that the penalty cost function charged to the warehouse here is a function of Sj (Tj , r), not the retailer j’s target base-stock level Sj at its ordering epoch. Since gj (y, Tj , r) = gj0 (y, Tj , r) + gjj (y, Tj , r) for any y with fixed Tj and r, the total inventory and backorder cost can be rewritten as [ T −1 N ∑ 1 ∑ G(S, T) = E h0 (S0 − D0 [L0 + MT0 (r)]) + gj0 (Sj − B0j (L0 + rj (r)), Tj , r) T r=0 j=1 ] N ∑ + gjj (Sj − B0j (L0 + rj (r)), Tj , r) . j=1

We allocate the penalty cost gj0 to the warehouse in each period r, and define the allocated warehouse cost as gb0 (S, T) =

[ ] T −1 N ∑ 1 ∑ E h0 (S0 − D0 [L0 + MT0 (r)]) + gj0 (Sj − B0j (L0 + rj (r)), Tj , r) . (15) T r=0

j=1

Furthermore, we define the allocated retailer cost as [ ] T −1 1 ∑ gbj (S, T) = E gjj (Sj − B0j (L0 + rj (r)), Tj , r) . T

(16)

r=0

Then, we have G(S, T) = gb0 (S, T) +

∑N

bj (S, T). j=1 g

system holding and backorder cost. 14

This completes the decomposition of the total

4.2

A Lower Bound on the Stage Cost

We proceed to construct a lower bound function for gbj (S, T) and gb0 (S, T). As we shall see, the lower-bound cost functions are a function of each stage’s own control parameters and therefore they are independent. The independence feature is important for us to construct the solution bounds. We first construct a lower bound for the retailer j’s cost. Recall (16), we have Tj −1 T −1 1 ∑ 1 ∑ gbj (S, T) ≥ gjj (Sj (Tj , r), Tj , r) = gjj (Sj (Tj , r), Tj , r) T Tj r=0

r=0

Tj −1

=

1 ∑ def gj (Sj (Tj , r), Tj , r) = g j (Tj ). Tj r=0

The inequality holds because gjj (Sj (Tj , r), Tj , r) is the minimum cost for the retailer in period t + L0 + rj (r) + MTj (r). Proposition 2 summarizes this result. Proposition 2 gbj (S, T) ≥ g j (Tj ). We next construct a lower bound to gb0 (S, T) in (15). The first term in (15) is a function of (S0 , T0 ). Our goal is to construct a lower bound that is a function of (S0 , T0 ) to the second term ∑ −1 ∑T0 −1 and then as a result, we can also reduce 1/T Tr=0 to 1/T0 r=0 and the whole function will depend only on (S0 , T0 ). First, for any retailer j, notice that Sj − B0j (t + L0 + rj (r)) = IPj (t + L0 + rj (r)). By definition, IPj (t + L0 + rj (r)) ≤ IL− 0 (t + L0 + rj (r)) = IOP0 (t + rj (r)) − D0 [t + rj (r), t + L0 + rj (r)) = S0 − D0 [t + r0 (rj (r)), t + L0 + rj (r)).

[from (3)]

[from (2)]

Since gj0 (y, Tj , r) is convex and decreasing in y, we have Proposition 3 For fixed Tj and r, E [gj0 (Sj − B0j (L0 + rj (r)), Tj , r)] ≥ E [gj0 (S0 − D0 [L0 + MT0 (rj (r))), Tj , r)] . The right-hand side of the inequality in Proposition 3 is a lower bound to the period penalty cost incurred from retailer j. This lower bound still depends on the retailer j’s reorder interval Tj . The next result develops a lower bound to the above right-hand side term by removing this dependency.

15

Proposition 4 [

]

[

]

E gj0 (S0 − D0 [L0 + MT0 (rj (r))), Tj , r) ≥ E gj0 (S0 − D0 [L0 + MT0 (r0 (r))), T0 , r) . Proposition 4 states that by regulating the retailer j’s reorder interval Tj to T0 , the resulting period penalty cost function is smallest among all other possibilities. Since MT0 (r0 (r)) = 0, we have [ ] [ ] E gj0 (S0 − D0 [L0 + MT0 (r0 (r))), T0 , r) = E gj0 (S0 − D0 [L0 ), T0 , r) .

(17)

It is interesting to compare this result with the one established in Shang and Zhou (2009), who studied a serial system with nested reorder interval policies (i.e., the reorder interval at an upstream stage is an integer multiple of that of its immediate downstream stage) under the synchronized ordering rule. They constructed a lower bound to the induced penalty cost function by regulating the reorder intervals of the downstream stages. Although the idea of constructing bounds is similar, our result here is different from Shang and Zhou’s. First, we construct a lower bound to each period penalty cost by regulating the retailer’s reorder interval, not to the induced penalty cost. Second, the reorder interval of the warehouse may be smaller than the reorder interval of the retailer in our model. We substitute the right-hand-side term in (17) for the exact period penalty cost function in gb0 (S, T), and define the resulting function as g 0 (S0 , T0 ), i.e., [ ] T0 −1 N ∑ 1 ∑ g 0 (S0 , T0 ) = E h0 (S0 − D0 [L0 + MT0 (r)]) + gj0 (S0 − D0 [L0 ), T0 , r) . T0 r=0

j=1

The g 0 (S0 , T0 ) function is the lower bound to the warehouse cost gb0 (S, T). Proposition 5 For fixed T0 , g 0 (S0 , T0 ) is convex in S0 . Set S0 (T0 ) = argminS0 g 0 (S0 , T0 ), and g 0 (T0 ) = g 0 (S0 (T0 ), T0 ). With Propositions 2 - 5, we have G(S, T) =

N ∑

gbj (S, T) ≥

j=0

N ∑

g j (Tj ).

j=0

Now, define def

cj (Tj ) =

Kj + g j (Tj ). Tj

We have C(S, T) ≥

N ∑ j=0

16

cj (Tj ).

(18)

The following unimodality property is useful to construct the solutions bounds for Tj∗ , j ≥ 1 in §4.4. Proposition 6 For j = 1, ..., N , cj (Tj ) is unimodal in Tj . Unfortunately, c0 (T0 ) is not unimodal in T0 . In order to construct bounds for T0∗ , we provide a different approach based on establishing a lower bound to the system cost demonstrated in the next section.

4.3

A Lower Bound to the System Cost

This section provides a lower bound to the average system cost. As we shall see at the end, this lower bound is a function of warehouse’s control parameters (S0 , T0 ). Define h = min{h1 , ..., hN },

b = min{b1 , ..., bN },

H = h0 + h,

IPr (τ ) = the sum of inventory in-transit positions of the retailers in period τ =

N ∑

IPj (τ ).

j=1

Recall the average inventory holding and backorder cost G(S, T) in (6). ( )] [ T −1 N ∑ 1 ∑ − hj ILj (t + L[0,j] + r) + (bj + Hj )[ILj (t + L[0,j] + r)] G(S, T) = E h0 IL0 (t + L0 + r) + T r=0 j=1 ( )] [ T −1 N ∑ 1 ∑ − hILj (t + L[0,j] + r) + (b + H)[ILj (t + L[0,j] + r)] ≥ E h0 IL0 (t + L0 + r) + T r=0 j=1 [ ( ) T −1 N ∑ 1 ∑ ≥ E h0 IL0 (t + L0 + r) + h IPr (t + L0 + r) − Dj [Lj ] T r=0 j=1 ( )− ] N ∑ def +(b + H) IPr (t + L0 + r) − Dj [Lj ] = GS (S, T). (19) j=1

The cost function GS (S, T) in (19) can be viewed as the inventory holding and backorder cost generated from a two-stage serial inventory system, in which the upstream stage has the parameters as those in the warehouse and the downstream stage has holding cost rate h, backorder cost rate b, and inventory in-transit position IPr (t + L0 + r). In Appendix B, we provide a simple recursion to evaluate GS (S, T). Here, we give an example to see the difference between G(S, T) and GS (S, T). Consider a two-retailer system that has the following parameters: h0 = 0.5, h1 = h2 = 1, b1 = b2 = 9, L0 = L1 = L2 = 1, λ1 = λ2 = 3, and 17

(S0 , T0 ) = (45, 6), (S1 , T1 ) = (15, 3), and (S2 , T2 ) = (10, 2). In this example, G(S, T) = 29.74, and GS (S, T) = 24.14. Shang and Zhou (2009) developed a lower bound to the average inventory holding and backorder cost for a serial system. The lower bound cost is obtained from a single-stage (s, T ) system. We show that their result can be extended to the above two-stage cost function. We first define this single-stage system. Consider a single-stage system with holding cost rate ∑ h0 , backorder cost rate b, and the demand during lead time D0 [L0 + MT0 (r)) + N j=1 Dj [Lj ]. The cost function for this single-stage system for a given period r, r = 0, 1..., T0 − 1, is ] [ N N ( ) ( )− ∑ ∑ F (y, r) = E h0 y − D0 [L0 + MT0 (r)) + Dj [Lj ] + (b + h0 ) y − D0 [L0 + MT0 (r)) + Dj [Lj ] . j=1

Let

j=1

T0 −1 1 ∑ G (S0 , T0 ) = F (S0 , r). T0 ℓ

r=0

Following Rao (2003) or Shang and Zhou (2009), by assuming continuous approximation on the demand and control variables, one can show Gℓ (S0 , T0 ) is jointly convex in (S0 , T0 ). def

Let S0ℓ (T0 ) = arg minS0 Gℓ (S0 , T0 ), and Gℓ (T0 ) = Gℓ (S0ℓ (T0 ), T0 ). Also, let π = h0

∑N

j=1 λj Lj .

We have Proposition 7 GS (S, T) ≥ Gℓ (S0 , T0 ) + π ≥ Gℓ (T0 ) + π. Notice that Gℓ (T0 ) is a convex function of T0 and is independent of Tj , j = 1, ..., N .

4.4

Bounds on the Optimal Reorder Intervals

We shall use the system cost bound Gℓ (T0 ) + π to find the bounds for T0∗ , and use the retailer cost bound cj (Tj ) to find the bounds for Tj∗ , j = 1...., N . We start with the construction of bounds for T0∗ . Let C h be the cost from any heuristic policy (see §5 for a heuristic policy). Then, C h ≥ C(S∗ , T∗ ) ≥

K0 + Gℓ (T0∗ ) + π. T0∗

Since Gℓ (T0 ) is convex in T0 and Gℓ goes to infinity when T0 → ∞, a lower bound T 0 and an upper T 0 can be obtained as follows: T0 T0

{ } K0 ℓ h = max T0 | + G (T0 ) ≤ C − π , T0 } { K0 + Gℓ (T0 ) ≤ C h − π . = min T0 | T0 18

We next construct the bounds for Tj∗ , j = 1, ..., N . First, we can obtain the minimum value of c0 (T0 ) by searching T0 ∈ [T 0 , T 0 ]. Denote the resulting minimum value c0 . Then, C

h

∗

∗

≥ C(S , T ) ≥

c0 (T0∗ )

+

N ∑

cj (Tj∗ )

≥ c0 +

j=1

N ∑

cj (Tj∗ ).

j=1

From Proposition 6, we can find the minimal value of cj (Tj ), referred to as cj , j = 1, ..., N . Thus, an upper bound T j and a lower bound T j to Tj∗ can be obtained from the following inequalities:  ∑  ci , T j = max Tj | cj (Tj ) ≤ C h − c0 −  

   ∑  T j = min Tj | cj (Tj ) ≤ C h − c0 − ci .  

 

i̸=j

i̸=j

To find the optimal reorder intervals, we have to evaluate the policies such that Tj ∈ [T j , T j ], and their corresponding optimal base-stock levels. Apparently, the effectiveness of the optimization algorithm depends on C h . In the next section, we provide a heuristic that seems to work well.

5.

Heuristic

In our numerical experience, we observe that the optimal reorder intervals always follow integerratio relations. That is, Tj∗ ∈ R, T0∗

j = 1, ..., N,

(20)

where R is the set of all positive integers and their reciprocals. Based on this observation, we propose a heuristic that minimizes the sum of the lower bound cost functions subject to the integer-ratio constraints. More specifically, we solve the following problem: ( ) N ∑ Kj (P ) min C(T) = + g j (Tj ) T Tj j=0

s.t.

Tj ∈ R, T0

j = 1, ..., N.

Since there are no order relations between Tj ’s, we need to consider all possible combinations. Taking a two-retailer system as an example, we have to consider the following six possible combinations: (1) T0 = nT1 ,

T1 = mT2 ,

(2) T0 = nT2 ,

T2 = mT1 ,

(3) T1 = nT0 ,

T0 = mT2 ,

(4) T1 = nT2 ,

T2 = mT0 ,

(5) T2 = nT1 ,

T1 = mT0 ,

(6) T2 = nT0 ,

T0 = mT1 ,

19

where n and m are positive integers. In general, when there are N retailers, we have to consider (N + 1)! possible combinations. For each of the possible constraints, we solve a problem with a sum of unimodal functions2 subject to integer-ratio constraints. Continuing the two-retailer example, suppose that we consider the first constraint above, the problem is: (P 1)

min

C(T)

s.t.

T0 = nT1 ,

T

T1 = mT2 .

This problem has exactly the same structure as the approximate problem suggested by Shang and Zhou (2009) for generating an effective heuristic solution in serial inventory systems. We therefore apply their two-step procedure to generate an effective solution for (P 1). We refer the reader to Shang and Zhou (2009) for a detailed explanation. After solving the problem with each possible constraint, we will have (N + 1)! feasible solutions. Our final heuristic solution is the one that leads to the smallest system cost.

6.

Numerical Study

This section aims to examine the behaviors of the optimal reorder intervals and the effectiveness of the proposed heuristic.

6.1

Properties of the Optimal Policy

We use the following four two-retailer cases to illustrate the behaviors of the optimal reorder intervals. These four instances have the following common parameters: h0 = h1 = h2 = 1, L0 = L1 = L2 = 1, λ1 = λ2 = 1, and b1 = b2 = 18. The fixed cost parameters and the optimal reorder intervals are shown in Table 1 below. Case 1 2 3 4

K0 16 1 0.5 0.5

K1 4 4 0.5 1

K2 1 16 16 16

(T0∗ , T1∗ , T2∗ ) (4, 2, 2) (3, 3, 3) (1, 1, 3) (2, 2, 4)

Optimal Cost 25.23 25.31 23.81 24.11

Table 1: Optimal solutions for two-retailer systems 2

Although g0 (T0 ) is not unimodal, the cost function tends to be unimodal in our numerical experience.

20

Notice that the observations obtained from these four instances generally hold true for the instances we tested in this section. We first provide four interesting observations for the optimal reorder intervals. (1) The optimal reorder intervals always satisfy the integer-ratio relations. (2) We cannot find any instance such that T0∗ < min{T1∗ , ..., TN∗ }. For example, in Case 2, one would expect T1∗ to be larger than T0∗ since K1 /h1 is four times larger than K0 /h0 . However, the optimal reorder intervals are T0∗ = T1∗ = 3. This seems to suggest that the reorder interval of warehouse should be at least equal to the minimum of those of the retailers at optimality. To further verify this statement, we reduce both K0 and K1 to 0.5 in Case 3 and the optimal reorder intervals are T0∗ = T1∗ = 1. Finally, we increase K1 from 0.5 to 1 in Case 4, T1∗ increases from 1 to 2. Interestingly, T0∗ in Case 4 also increases from 1 to 2. Thus, the findings from these experiments are consistent with the statement. (3) The cost ratios Kj /hj seem to have less impact on the optimal reorder intervals. For example, in Case 1, one would expect that T0∗ > T1∗ > T2∗ since K0 /h0 > K1 /h1 > K2 /h2 . The optimal reorder intervals are T1∗ = T2∗ = 2, although K1 /h1 is four times larger than K2 /h2 . This observation suggests that the optimal reorder intervals of the retailers tend to be the same when their corresponding cost ratios are not significantly different. It is worth noting that this conclusion is different from that observed in Shang and Zhou (2009) for the serial system. In their numerical study, Shang and Zhou found that the cost ratio Kj /hj has direct impact on the optimal reorder interval Tj∗ . More specifically, let j = N (j = 1) represent the most upstream (downstream) stage in an N -stage serial system. ∗ Shang and Zhou found that (i) if Kj /hj ≥ Kj−1 /hj−1 , Tj∗ ≥ Tj−1 for j = 2, ..., N ; (ii) if

Kj /hj ≤ Kj−1 /hj−1 , Tj∗ tends to be the same between stages. Apparently, the impact of the cost ratios on the optimal reorder intervals is less obvious for the distribution system. (4) Based on the observation (3), it is natural to ask how “significant” the difference between K1 /h1 and K2 /h2 should be in order to see that T1∗ is different from T2∗ . In Case 4, we find that when K2 /h2 is 16 times larger than K1 /h1 , we see a difference. In Case 3, we reduce both K0 and K1 to 0.5, and now K2 /h2 is 32 times larger than K0 /h0 . In this case, we see T0∗ = T1∗ = 1, which is smaller than T2∗ = 3. To our surprise, although K2 /h2 is 32 times larger than K0 /h0 , T2∗ is only three times larger than T0∗ . 21

6.2

Effectiveness of the Heuristic

This section examines the effectiveness of the heuristic. We focus on the two-retailer system. We fix the retailer 1’s parameters and vary the parameters for the warehouse and retailer 2. More specifically, the parameters for retailer 1 are K1 = h1 = L1 = λ1 = 1 and b1 = 3. For retailer 2, we set h2 = λ2 = 1. The rest of the parameters for the warehouse and the retailer 2 are chosen from the following sets: h0 ∈ {1, 2},

K0 , K2 ∈ {0.25, 16},

L0 , L2 ∈ {1, 3},

b2 ∈ {3, 18}.

The total number of instances is 64. The percentage error is defined as below: ξ% =

˜ − C(T∗ ) C(T) × 100%, C(T∗ )

˜ = (T˜0 , T˜1 , ..., T˜N ) is the vector of the heuristic reorder intervals. where T In this test, the average percentage error is 0.28% with maximum error of 2.07%. The heuristic generates the optimal solution in 33 instances. The heuristic appears to be effective. We find that the heuristic performs less effectively when (K0 , K1 , K2 ) = (16, 1, 16). For the 16 instances with these fixed order costs, the average percentage error is 0.82%.

7.

Concluding Remarks

This paper studies a one-warehouse, multi-retailer system in which base-stock, reorder interval policies are implemented. The feature of our model is that we explicitly consider fixed order costs. Specifically, we assume that a fixed order cost is incurred for each inventory reorder. We show how to evaluate the system cost. We provide a method to obtain the optimal policy parameters. This result is established by constructing stage independent lower bounds on the cost for each stage and a lower bound to the system cost. We also suggest a heuristic that generates effective solution in a numerical study. We observe that the optimal reorder intervals often follow integer-ratio relations. Finally, unlike the serial inventory model studied by Shang and Zhou (2009), we find that the cost ratio of fixed order cost to holding cost has less impact on the optimal reorder interval.

References [1] Atkins, D., P. Iyogun. 1988. Period versus ‘can-order’ policies for coordinated multi-item inventory systems. Management Science 34 791-796. 22

[2] Axs¨ater, S. 1990. Simple solution procedure for a class of two-echelon inventory problems. Operations Research 38 64-69. [3] Axs¨ater, S. 1993a. Exact and approximate evaluation of batch-ordering policies for two-level inventory systems. Operations Research 41 777-785. [4] Axs¨ater, S. 1993b. Optimization of order-up-to-S policies in two-echelon inventory systems with periodic review. Naval Research Logistics 40 245-253. [5] Axs¨ater, S. 2003. Supply chain operations: serial and distribution inventory systems, Chapter 10 in Graves, S. C. and T. de Kok (Eds.), Handbooks in Operations Research and Management Science, Vol. 11: Supply Chain Management: Design, Coordination and Operation, Elsevier, 525-559. [6] Cachon, G. 1999. Managing supply chain demand variability with scheduled ordering policies. Management Science 45 843-856. [7] Cachon, G. 2001. Exact evaluation of batch-ordering policies in two-echelon supply chains with periodic review. Operations Research. 49(1). 79-98. [8] C ¸ etinkaya, S., C.-Y. Lee. 2000. Stock replenishment and shipment scheduling for vendormanaged inventory systems. Management Science 46 217-232. [9] Chandran, M. 2003. Wal-Mart’s supply chain management practice. ICFAI Center for Management Research, Nagarjuna Hills, Hyderabad, India. [10] Chao, X., S. Zhou. 2009. Optimal policy for multi-echelon inventory system with batch ordering and fixed replenishment intervals. Operations Research 57 377-390. [11] Chen, F., R. Samroengraja. 2000. A staggered ordering policy for one-warehouse multi-retailer systems. Operations Research 48 281-293. [12] Chen, F., R. Samroengraja. 2004. Order volatility and supply chain costs. Operations Research 52 707-722. [13] Chen, F., Y.-S. Zheng. 1997. One-warehouse multi-retailer systems with centralized stock information. Operations Research 45, 275-287. [14] Cheung, K.-L., S.-H. Zhang. 2008. Balanced and synchronized ordering in supply chains. IIE Transactions 40 1-11. [15] Eppen, G., L. Schrage. 1981. Centralized ordering policies in a multi-warehouse system with lead times and random demand. TIMS Studies in the Management Sciences 16 51-67. [16] Federgruen, A., H. Groenevelt, H. Tijms. 1984. Coordinated replenishments in a multi-item inventory system with compound Poisson demands. Management Science 30 344-357. [17] Federgruen, A., P. Zipkin. 1984. Allocation policies and cost approximations for multilocation inventory systems. Naval Research Logistics Quarterly 31 97-129. 23

¨ Ozer, ¨ [18] Gallego, G., O. P. Zipkin. 2007. Bounds, heuristics and approximations for distribution systems. Operations Research 55, 503-517. [19] Graves, S. 1996. A multiechelon inventory model with fixed replenishment intervals. Management Science 42 1-18. [20] G¨ urb¨ uz, M., K., Moinzadeh, Y.-P. Zhou. 2007. Coordinated replenishment strategies in inventory/distribution systems. Management Science 53 293-307. [21] Jackson, P. L. 1988. Stock allocation in a two-echelon distribution system or ‘what to do until your ship comes in’. Management Science 34 880-895. [22] Lee, H., V. Padmanabhan, S. Whang. 1997. Information distortion in a supply chain: the bullwhip effect. Management Science 43 546558. [23] Marklund, J. 2008. Inventory control in divergent supply chains with time based dispatching and shipment consolidation. Working paper. Lund University, Lund, Sweden. [24] Maxwell, W., J. Muckstadt. 1985. Establishing consistent and realistic reorder intervals in production-distribution systems. Operations Research 33 1316-1341. [25] Muckstadt, J., R. Roundy. 1993. Analysis of multistage production systems. S. C. Graves, A. H. G. Rinnooy Kan, P. H. Zipkin (Eds.), Handbook in Operations Research and Management Science, Vol. 4: logistics of Production and Inventory. North Holland, Amsterdam 59-131. [26] Rao, U. S. 2003. Properties of the period review (R, T ) inventory control policy for stationary, stochastic demand. Manufacturing & Service Operations Management 5 37-53. [27] Schwarz, L. 1989. A model for assessing the value of warehouse risk-pooling: Risk-pooling over outside-supplier leadtimes. Management Science 35 828-842. [28] Shang, K., S. Zhou. 2009. Optimal and heuristic (r, nQ, T ) policies in serial inventory systems with fixed costs. Forthcoming in Operations Research. [29] Shang, K., S. Zhou, G.-J. van Houtum. 2009. Improving supply chain performance: realtime demand information and flexible deliveries. Forthcoming in Manufacturing and Service Operations Management. [30] Silver, E. 1981. Establishing reorder points in the (S, c, s) coordinated control system under compound Poisson demand. International Journal of Production Research 9 743-750.

24

Appendix A: Optimization of Base-Stock Levels with Fixed T This appendix shows how to optimize the optimal echelon base-stock levels when the reorder intervals T are fixed. It is worth mentioning that Axs¨ater (1993) provides an approach (which is a generalization of Axs¨ater (1990) on base-stock systems) for finding the optimal local base-stock levels with fixed reorder intervals. Axs¨ater’s model is a special case of ours because he assumes identical retailers and that the reorder interval of the warehouse is an integer multiple of that of the retailer. Below we show that his approach can be generalized to our model. We present the analysis from the local policy perspective, because B0 is a function of s0 , and Sj is the same as the local base-stock levels sj . Define the local inventory holding and backorder cost for the retailer j as follows: fj (y, r) = E[Hj (y − Dj [Lj + MTj (r)]) + (bj + Hj )(y − Dj [Lj + MTj (r)])− ], and



f0 (y, r) = h0 E y − D0 [L0 + MT0 (r)] +

N ∑

Dj [Lj + MTj (r)] +

j=1

N ∑ λj j=1

λ0

j = 1, ..., N,

 (y − D0 [L0 + MT0 (rj (r))))−  .

Proposition 8 C(S, T) =

N ∑ Kj j=0

Tj

+

T −1 N N ) ∑ ∑ 1 ∑( f0 (s0 , r) + E[fj (sj − B0j (L0 + rj (r)), r)] + h0 (λj Lj ). T r=0

j=1

j=1

Note that f0 (s0 , r) is the warehouse inventory holding cost; fj (sj − B0j (L0 + rj (r)), r) is the retailer j’s inventory holding and backorder cost; the last term is the average pipeline cost per period, which is constant. For convenience, let us define T −1 1 ∑ fb0 (s0 ) = f0 (s0 , r), T r=0

T −1 1 ∑ fbj (s0 , sj ) = E[fj (sj − B0j (L0 + rj (r)), r)]. T r=0

Here, fbj (·, ·) is a function of s0 because B0j is a function of s0 . Proposition 9 For fixed T and s0 , fbj (s0 , sj ) is convex in sj . With Proposition 9, we can find the best local base-stock level sj (s0 ) for each retailer j. That is, sj (s0 ) = arg min fbj (s0 , sj ). sj

25

Substituting sj (s0 ) for sj in C(S, T), the objective function becomes a function of s0 , i.e., C(s0 ) = ∑ b fb0 (s0 ) + N j=1 fj (s0 , sj (s0 )). Unfortunately, C(s0 ) is not convex in s0 , so we have to construct bounds for the optimal s0 , denoted as s∗0 , and conduct a search over the feasible interval. Following Axs¨ater’s (1990) approach, we next provide bounds for s∗0 . Let sℓj = sj (∞) and suj = sj (0). Define su0 = arg min s0

  

fb0 (s0 ) +

N ∑ j=1

  fbj (s0 , sℓj ) 

  N   ∑ sℓ0 = arg min fb0 (s0 ) + fbj (s0 , suj ) . s0  

and

j=1

Then, Proposition 10 (1) sℓj ≤ s∗j ≤ suj ; (2) sℓ0 ≤ s∗0 ≤ su0 . With Proposition 10, we can search over all possible s0 between sℓ0 and su0 . After s∗0 is found, the optimal base-stock level for retailer j is s∗j = sj (s∗0 ).

Appendix B: A Recursion to Evaluate GS (S, T) We provide a simple recursion to evaluate GS (S, T). Let SR = GS (S, T) =

∑N

j=1 Sj .

Then,

T −1 1 ∑ F0 (S, T, r), T r=0

where

[

F0 (S, T, r) = E h0 (S0 − D0 [L0 + MT0 (r)])   ] N   ∑ +F1 min SR , S0 − D0 [L0 + MT0 (r)) + Dj [MTj (r)) , T, r ,   j=1     −  N N ∑ ∑ F1 (y, T, r) = E h y − Dj [Lj + MTj (r)] + (b + H) y − Dj [Lj + MTj (r)]  . 

j=1

j=1

We omit the proof for brevity. A complete proof is available from the authors. Notice that the recursion is similar to the one presented in Shang and Zhou (2009) to evaluate the inventory holding and backorder cost for a serial system with nested echelon (s, T ) policies. In fact, the above recursion can be used to evaluate a two-stage serial system with nonnested echelon (s, T ) policies by setting N = 1 (one retailer) in the recursion. 26

Appendix C: Proofs Proposition 3 Note that, for any order period t of stage j, Sj − B0j (L0 + rj (r)) = IPj (t + L0 + rj (r)), S0 − D0 [L0 + MT0 (rj (r))) = IL− 0 (t + L0 + rj (r)), and IL− 0 (t + L0 + rj (r)) ≥ IPj (t + L0 + rj (r)). For simplicity, we suppress (t + L0 + rj (r)) from IPj and IL− 0 since we also consider the same time period. Thus,   N ∑ E gj0 (IPj , Tj , r) j=1

=

N ∑ j=1

≥

 

gj0 (n, Tj , r)P(IPj = n)

n=−∞

S0 N ∑ ∑ j=1 n0 =−∞

=



Sj ∑

S0 N ∑ ∑

[

n0 ∑

[ gj0 (n0 , Tj , r) P(IPj =

n=−∞

[

gj0 (n0 , Tj , r)P(IL− 0

j=1 n0 =−∞



= E

N ∑

]]

= n0 )

n|IL− 0

n0 ∑

=

n0 )P(IL− 0

= n0 ) ]

P(IPj = n|IL0 = n0 )

n=−∞



 gj0 (IL− 0 , Tj , r) ,

j=1

where the inequality follows from that gj0 is nonincreasing. Proposition 4 By definition E[gj0 (y − D0 [L0 + MT0 (r0 (r))), T0 , r)] = E [gj (min{y − D0 [L0 ), Sj (r, T0 )}, T0 , r) − gj (Sj (r, T0 ), T0 , r)] and

[

(

[

(⌊

⌋ )) )] r E gj0 y − D0 L0 + MT0 Tj , Tj , r Tj [ ( { [ (⌊ ⌋ )) } ) ] r = E gj min y − D0 L0 + MT0 Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) . Tj

So to prove the result, we need to show, for r = 0, 1, ..., T − 1, [ ( { [ } ) ] (⌊ ⌋ )) r E gj min y − D0 L0 + MT0 Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) Tj ≥ E [gj (min{y − D0 [L0 ), Sj (r, T0 )}, T0 , r) − gj (Sj (r, T0 ), T0 , r)] . 27

(21)

It is clear that we just need to verify r = 0, 1, ..., lcm(T0 , Tj ) − 1. Let lcm(T0 , Tj ) = nT0 = mTj . Consider the following two cases differentiated by n ≥ m or n < m. Case 1: n < m and so T0 > Tj . Thus we can write T0 =

⌊m⌋ n

Tj +

(m n

−

⌊ m ⌋) n

Tj ,

⌊m⌋ where Tj must be divisable by n as m and n are relatively prime. Note that ( m n − n ) could take ⌊ ⌋ 1 value 1/n, . . . , (n − 1)/n. For ease of presentation, we assume it is 1/n or T0 = m n Tj + n Tj . Other cases can be similarly proved using the following procedure. To prove the inequality (21) for all r = 0, 1, ..., lcm(T0 , Tj ) − 1, we consider different regions of r. Subcase 1. For r = 0, .., Tj − 1,

⌊

r Tj

⌋ Tj = 0 and from the definition of gj (y, T, r) and Sj (r, T ),

E [gj (min {y − D0 [L0 ), Sj (r, T0 ))} , T0 , r) − gj (Sj (r, T0 ), T0 , r)] } ) ] [ ( { [ (⌊ ⌋ )) r Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) , = E gj min y − D0 L0 + MT0 Tj because Sj (r, Tj ) = Sj (r, T0 ) and gj (y, Tj , r) = gj (y, T0 , r) in this case. Subcase 2. We next consider r = Tj , . . . , T0 − 1. Note that, for r < T0 , Sj (r, T0 ) is increasing in r because gj (y, T0 , r) = E[hj (y − Dj [Lj + r]) + (bj + Hj )(Dj [Lj + r] − y)+ ] is submodular in y and r as ∂gj (y, T0 , r)/∂y = hj − (bj + Hj )P r(Dj [Lj + r] > y) is decreasing in r. Moreover, in this case,

⌋ ) ⌊ ⌋ r r MT0 Tj = Tj , Tj Tj ⌊ ⌋ from the definition of gj (y, T0 , r) and MT0 (r) = MTj (r) + Trj Tj , (⌊

} )] { [ ⌊ ⌋ ) r Tj , Sj (r, Tj ) , Tj , r E gj min y − D0 L0 + Tj [ ( { [⌊ ⌋ ) } )] r ≥ E gj min y − D0 [L0 ) − Dj Tj , Sj (r, Tj ) , Tj , r Tj )] [ ( { [⌊ ⌋ )} r Tj , T0 , r = E gj min y − D0 [L0 ), Sj (r, Tj ) + Dj Tj ≥ E[gj (min{y − D0 [L0 ), Sj (r, T0 )}, T0 , r)], [

(

(22)

where the first inequality follows from that gj (min{y, Sj (r, Tj )}, Tj , r) is decreasing in y and the second inequality from the convexity of gj (y, T0 , r) and the optimality of Sj (r, T0 ) for gj (y, T0 , r). 28

Meanwhile, as gj (Sj (r, T0 ), T0 , r) ≥ gj (Sj (r, Tj ), Tj , r) for Tj ≤ r < T0 beacause MT0 (r) > MTj (r), the following inequality is valid, } ) ] [ ( { [ ⌊ ⌋ ) r Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) E gj min y − D0 L0 + Tj ≥ E [gj (min{y − D0 [L0 ), Sj (r, T0 )}, T0 , r) − gj (Sj (r, T0 ), T0 , r)] , and so is inequality (21). Subcase 3. Now we examine r = ℓT0 , ℓT0 + 1, ..., (ℓ + 1)T0 − 1 for 1 ≤ ℓ ≤ n − 1. (a) First consider r = ℓT0 + Tj , ℓT0 + Tj + 1, ..., (ℓ + 1)T0 − 1. Note that for r = ℓT0 + (k − ⌊ ⌋ 1)Tj , . . . , ℓT0 + kTj − 1 for 1 < k ≤ m n Tj , MT0 (r) = (k − 1)Tj , (k − 1)Tj + 1, . . . , kTj − 1,

=

MTj (r) { ℓ n Tj ,

(23)

(24)

ℓ n Tj + 1, 0, 1, . . . , nℓ Tj −

. . . , Tj − 1, r = ℓT0 + (k − 1)Tj , . . . , T0 + (k − 1)Tj + r = ℓT0 + (k − 1)Tj +

1,

n−ℓ n Tj , . . . , ℓT0

n−ℓ n Tj

− 1,

+ kTj − 1,

and (⌊ MT0 { =

⌋ ) r Tj Tj

(25)

(k − 1)Tj − nℓ Tj , , . . . (k − 1)Tj − nℓ Tj , r = ℓT0 + (k − 1)Tj , . . . , ℓT0 + (k − 1)Tj +

n−ℓ n Tj

− 1,

kTj − nℓ Tj , . . . , kTj − nℓ Tj ,

r = ℓT0 + (k − 1)Tj + n−ℓ n Tj , . . . , ℓT0 + kTj − 1. (⌊ ⌋ ) With these, we can show that MT0 (r) = MTj (r) + MT0 Trj Tj . Following the arguments as those in deriving (22), [ ( { [ (⌊ ⌋ )) } )] r Tj , Sj (r, Tj ) , Tj , r E gj min y − D0 L0 + MT0 Tj ≥ E[gj (min{y − D0 [L0 ), Sj (r, T0 )}, T0 , r)]. Morover, as in this range, MT0 (r) ≥ MTj (r) and so gj (Sj (r, T0 ), T0 , r) ≥ gj (Sj (r, Tj ), Tj , r). Thus, (21) is true. For r = ℓT0 +

⌊m⌋ n

Tj , ℓT0 +

⌊m⌋ n

Tj + 1, ..., (ℓ + 1)T0 − 1,

MT0 (r) =

⌊m⌋

Tj ,

⌊m⌋

Tj + 1,

⌊m⌋

Tj +

1 Tj − 1, n

n n n ℓ ℓ ℓ+1 MTj (r) = Tj , Tj + 1, . . . , Tj − 1, n n n (⌊ ⌋ ) ⌊ ⌋ m ℓ r MT0 Tj = Tj − Tj . Tj n n 29

As MT0 (r) = MTj (r) + MT0

(⌊

r Tj

⌋

) Tj , the inequality (21) is also valid following the preceding

argument. (b) For r = ℓT0 , ℓT0 + 1, ..., ℓT0 + Tj − 1. Note that in this case, MT0 (r) and MTj (r) are given in (23) and (24) respectively with k = 1. Therefore, it can be seen that when r = ℓT0 , ℓT0 +1, . . . , ℓT0 + n−ℓ n Tj −1,

MT0 (r) ≤ MTj (r) and so Sj (r, Tj ) ≥ Sj (r, T0 ) but gj (Sj (r, T0 ), T0 , r) ≤ gj (Sj (r, Tj ), Tj , r).

However, we can show that [ ( { [ (⌊ ⌋ )) } ) ] r E gj min y − D0 L0 + MT0 Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) Tj ≥ E [gj (min {y − D0 [L0 ), Sj (r, Tj )} , Tj , r)] − gj (Sj (r, Tj ), Tj , r) ≥ E [gj (min {y − D0 [L0 ), Sj (r, T0 )} , Tj , r)] − gj (Sj (r, T0 ), Tj , r) ≥ E [gj (min{y − D0 [L0 ), Sj (r, T0 ))}, T0 , r)] − gj (Sj (r, T0 ), T0 , r), where the first inequality follows from that gj0 (y, Tj , r) is decreasing in y; the second inequality follows from the convexity of gj (y, Tj , r) and the optimality of Sj (r, Tj ) for gj (y, Tj , r); the third inequality follows from that E[hj (y − Dj [Lj + r]) + (bj + Hj )(Dj [Lj + r] − y)+ ] is submodular in (y, r) and MT0 (r) ≤ MTj (r). For r = ℓT0 + n−ℓ n Tj , . . . , ℓT0 + Tj − 1, from (23) and (24), MT0 (r) ≥ MTj (r). Moreover, as (⌊ ⌋ ) ⌊ ⌋ MT0 Trj Tj = Tj − ℓ/nTj , MT0 (r) = MTj (r) + MT0 ( Trj Tj ) and so the inequality (21) follows the same argument as we prove the case r = ℓT0 + Tj , . . . , (ℓ + 1)T0 − 1. Case 2: n ≥ m and so Tj ≥ T0 and Tj =

⌊n⌋ m

T0 +

(n m

−

⌊ n ⌋) m

T0 .

⌊n⌋ n can take Similar to the analysis of Case 1, we discuss different ranges of r. And although m − m ⌊n⌋ 1 m−1 1 values m , . . . , m , we again only consider the case where it is 1/m, or Tj = m T0 + m T0 . Subcase 1. For r = 0, 1, ..., T0 − 1, the inequality (21) still holds as equality. Subcase 2. For r = T0 , T0 + 1..., Tj − 1, then MT0 (r) < MTj (r) and so } ) ] [ ( { [ (⌊ ⌋ )) r Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) E gj min y − D0 L0 + MT0 Tj = E[gj (min{y − D0 [L0 ), Sj (r, Tj )}, Tj , r) − gj (Sj (r, Tj ), Tj , r)] ≥ E [gj (min{y − D0 [L0 ), Sj (r, T0 )}, Tj , r) − gj (Sj (r, T0 ), Tj , r)] ≥ E [gj (min{y − D0 [L0 ), Sj (r, T0 ))}, T0 , r) − gj (Sj (r, T0 ), T0 , r)] , 30

where the first inequality follows from the convexity of gj (y, Tj , r) and the optimality of Sj (r, Tj ) while the second inequality follows from the submodularity of gj (y, T, r) on (y, r) when r < T . ⌊n⌋ Subcase 3. For r = ℓTj + (k − 1)T0 , . . . , ℓTj + kT0 − 1, for 1 ≤ ℓ ≤ m − 1 and 1 ≤ k ≤ m , { ℓ ℓ m−ℓ m T0 , m T0 + 1, . . . , T0 − 1, r = ℓTj + (k − 1)T0 , . . . , ℓTj + (k − 1)T0 + m T0 − 1 (26) MT0 (r) = ℓ 0, 1, ..., m T0 − 1, r = ℓTj + (k − 1)T0 + m−ℓ m T0 , . . . , ℓTj + kT0 − 1, MTj (r) = (k − 1)T0 , (k − 1)T0 + 1, . . . , kT0 − 1, (⌊ ⌋ ) r ℓ MT0 Tj = T0 . Tj m

(27) (28)

In this case, note that for k > 1, as MT0 (r) ≤ MTj (r), we can use the same arguments as in the subcase 2 of this case to verify (21). For k = 1, if r = ℓTj , . . . , ℓTj + m−ℓ m T0 − 1, MT0 (r) ≥ MTj (r) and MT0 (r) = MTj (r) + (⌊ ⌋ ) MT0 Trj Tj . So gj (Sj (r, T0 ), T0 , r) ≥ gj (Sj (r, Tj ), Tj , r) and [ ( { [ (⌊ ⌋ )) } ) ] r E gj min y − D0 L0 + MT0 Tj , Sj (r, Tj ) , Tj , r − gj (Sj (r, Tj ), Tj , r) Tj [ ( ] { [ (⌊ ⌋ )]} ) r ≥ E gj min y − D0 [L0 ), Sj (r, Tj ) + Dj MT0 Tj , T0 , r − gj (Sj (r, Tj ), Tj , r)) Tj ≥ E [gj (min{y − D0 [L0 ), Sj (r, T0 ))}, T0 , r) − gj (Sj (r, T0 ), T0 , r)] , where the last inequality follows also from the optimality of Sj (r, T0 ). Otherwise, if r = ℓTj +

m−ℓ m T0 , . . . , ℓTj

+ T0 − 1, MT0 (r) < MTj (r) and the inequality (21) can

be proved similarly as subcase 2. ⌊n⌋ ⌊n⌋ ⌊n⌋ Finally for r = ℓTj + m T0 , ℓTj + m T0 + 1, . . . , ℓTj + m T0 +

1 m T0

− 1, MT0

(⌊

r Tj

⌋

) Tj

is

the same as (28) while ℓ ℓ ℓ+1 T0 , T0 + 1, . . . , T0 − 1 m m m ⌊n⌋ ⌊n⌋ ⌊n⌋ 1 MTj (r) = T0 , T0 + 1, . . . , T0 + T0 − 1, m m m m

MT0 (r) =

which shows that MT0 (r) < MTj (r) and so again the inequality (21) can be proved similarly as subcase 2. Therefore, the proof is complete. Proposition 5 It is clear that, for each Tj and r, gj0 (y, Tj , r) is convex in y from its definition. Thus, from its definition, the convexity of g 0 (S0 , T0 ) in S0 follows. 31

Proposition 6 To show cj (Tj ) is unimodal, we need to show cj (Tj + 1) − cj (Tj ) is positive after some Tj . It can be shown that gj (Sj (Tj ), Tj ) is increasing in Tj . Thus, ) ( Tj −1 ∑ 1 cj (Tj + 1) − cj (Tj ) = Tj gj (Sj (Tj ), Tj ) − gj (Sj (r), r) − Kj Tj (Tj + 1)

(29)

r=0

Let

Tj −1

def

∆(Tj ) = Tj gj (Sj (Tj ), Tj ) −

∑

gj (Sj (r), r).

r=0

Notice that ∆(Tj + 1) = (Tj + 1)gj (Sj (Tj + 1), Tj + 1) −

Tj ∑

gj (Sj (r), r)

r=0

≥ gj (Sj (Tj + 1), Tj + 1) + Tj gj (Sj (Tj ), Tj ) −

Tj ∑

gj (Sj (r), r)

r=0

= [gj (Sj (Tj + 1), Tj + 1) − gj (Sj (Tj ), Tj )] + ∆(Tj ) ≥ ∆(Tj ). So the bracketed terms in (29) are increasing in Tj , which implies the unimodality of cj (Tj ). Proposition 7 For simplicity, let Dj′ (r) = Dj [Lj + MTj (r)) and Dj′ [r] = Dj [Lj + MTj (r)] for j = 0, ..., N . From Appendix B, GS (S, T) =

=

≥

≥

  ] N   ∑ 1 E h0 (S0 − D0′ [r]) + F1 min SR , S0 − D0′ (r) + Dj [MTj (r)) , r   T r=0 j=1     [ T −1 N N   ∑ ∑ ∑ 1 E h0 (S0 − D0′ [r]) + h min SR , S0 − D0′ (r) + Dj [MTj (r)) − Dj′ [r]   T r=0 j=1 j=1    − ] N N   ∑ ∑ ′  +(b + H) min SR , S0 − D0 (r) + Dj [MTj (r)) − Dj′ [r]   j=1 j=1 [ T −1 1 ∑ E h0 (S0 − D0′ [r]) T r=0    − ] N N   ∑ ∑ +(b + h0 ) min SR , S0 − D0′ (r) + Dj [MTj (r)) − Dj′ [r]   j=1 j=1  − ] [ T −1 N N ∑ ∑ ∑ 1 E h0 (S0 − D0′ [r]) + (b + h0 ) S0 − D0′ (r) + Dj [MTj (r)) − Dj′ [r] T T −1 ∑

[



r=0

j=1

32

j=1

[ T0 −1 N ∑ 1 ∑ = Dj [Lj ]) E h0 (S0 − D0′ (r) − T0 r=0 j=1  − ] N N ∑ ∑ +(b + h0 ) S0 − D0′ (r) − Dj [Lj ] + h0 λj (Lj + 1) − h0 λ0 j=1

= Gℓ (S0 , T0 ) + h0

N ∑

j=1

λj Lj ,

j=1

where the first inequality follows by dropping a positive term h()+ and the second inequality from that ()− is decreasing. Proposition 8 Note that Gj (y, Tj , r) = E[hj (y − Dj [Lj + MTj (r)]) + (bj + Hj )(y − Dj [Lj + MTj (r)])− ], and G0 (S, T, r)  = E h0 (S0 − D0 [L0 + MT0 (r)]) +

N ∑

 Gj (Sj − B0j (L0 + rj (r)) , r)

j=1

= E[h0 (S0 − D0 [L0 + MT0 (r)])] +

N ∑

E[hj (sj − B0j (L0 + rj (r)) − Dj [Lj + MTj (r)])

j=1

+(bj + Hj )(sj − B0j (L0 + rj (r)) − Dj [Lj + MTj (r)])− ] = E[h0 (S0 − D0 [L0 + MT0 (r)])] −

N ∑

E[h0 (sj − B0j (L0 + rj (r)) − Dj [Lj + MTj (r)])

j=1

+

N ∑

E[Hj (sj − B0j (L0 + rj (r)) − Dj [Lj + MTj (r)])

j=1

+(bj + Hj )(sj − B0j (L0 + rj (r)) − Dj [Lj + MTj (r)])− ] = E[h0 (s0 − D0 [L0 + MT0 (r)])] +

N ∑

E[h0 (B0j (L0 + rj (r)))] −

j=1

+

N ∑

N ∑

E[h0 (−Dj [Lj + MTj (r)])

j=1

E[fj (sj − B0j (L0 + rj (r)), r)]

j=1

 N N ∑ ∑ λ j (s0 − D0 [L0 + rj (r) − r0 (rj (r))))−  = h0 E (s0 − D0 [L0 + MT0 (r)] + (Dj [MTj (r)])) + λ0 

j=1

j=1

33

+

N ∑

h0 (λj (Lj )) +

j=1

N ∑

E[fj (sj − B0j (L0 + rj (r)), r)].

j=1

As h0 = H0 , so from Proposition 1, the result follows. Proposition 9 It is straightforward to prove the convexity from the definition of fbj (s0 , sj ). Proposition 10 We need to first show that fbj (s0 , sj ) is supermodular in s0 and sj , or fbj (s0 , sj + 1) − fbj (s0 , sj ) ≤ fbj (s0 + 1, sj + 1) − fbj (s0 + 1, sj ).

(30)

For notational simplicity, we denote Dj [Lj + MTj (r)] by Dj . In addition, to emphasize the dependency of B0j on s0 and for brevity, let B0j (s0 ) denote B0j (L0 + rj (r)). Note that fbj (s0 , sj + 1) − fbj (s0 , sj ) = Hj + ≤ Hj +

T −1 1 ∑ (bj + Hj )E[(sj + 1 − B0j (s0 ) − Dj )− − (sj − B0j (s0 ) − Dj )− ] T

1 T

r=0 T −1 ∑

(bj + Hj )E[(sj + 1 − B0j (s0 + 1) − Dj )− − (sj − B0j (s0 + 1) − Dj )− ]

r=0

= fbj (s0 + 1, sj + 1) − fbj (s0 + 1, sj ) because ()− is convex and B0j (s0 + 1) is smaller than B0j (s0 ). So the inequality (30) is valid. With this result and the convexity of fbj (s0 , sj ) is sj , part (1) follows. To show part (2), from (30), we have, for any sj ≥ sℓj , 0 ≥ fb0 (su0 ) +

N ∑

fbj (sℓj , su0 ) − [fb0 (su0 + 1) +

j=1

≥ fb0 (su0 ) +

N ∑

N ∑

fbj (sℓj , su0 + 1)]

j=1

fbj (sj , su0 ) − [fb0 (su0 + 1) +

j=1

N ∑

fbj (sj , su0 + 1)]

j=1

where the first inequality follows from the definition of su0 . As the optimal s∗j will be greater than sℓj , su0 is an upper bound for s∗0 . Similarly, we can verify the lower bound sℓ0 .

34