Integrated production and allocation policies with one direct shipping option

European Journal of Operational Research 181 (2007) 716–732 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Integrated production and allocation policies with one direct shipping option Wooseung Jang *, Dongwon Kim Department of Industrial and Manufacturing Systems Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Department of Industrial and Information Systems Engineering, Chonbuk National University, Chonbuk 561-756, Republic of Korea Received 4 October 2004; accepted 30 June 2006 Available online 29 September 2006

Abstract We consider a single period inventory problem in which a supplier faces stochastic demands and customer specific waiting costs from multiple customers. The objective is to develop integrated production, allocation, and distribution policies so that the total production and customer waiting costs are minimized. We present an optimal policy for the two customer problem and derive a heuristic for a general problem based on the structural results of the two customer case. We show, numerically, that the heuristic performs very well with error bounds of less than 2% on average, while typical approximations may lead to significant sub-optimality. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Inventory; Production; Allocation; Stochastic models

1. Introduction This paper addresses the integrated problem of production, inventory allocation, and distribution in a two-echelon system, where customer and policy dependent waiting costs exist. The higher echelon is a single location, referred to as a supplier, who produces a single commodity. The lower echelon consists of many geographically dispersed customers, each of which faces customer specific demands. Customers might be retail outlets facing stochastic demands or distribution centers having deterministic orders from retailers. The supplier is located overseas or in a remote area, so transportation cost is assumed to be very high. Our analytical model provides optimal or efficient coordinated decision policies from the view of a supplier. We now describe the envisioned scenario in greater detail: first, the supplier produces a certain number of an item for the upcoming sales period based on uncertain customer demands and other parameters. The demand information may be obtained directly from customers or estimated from their past ordering history. At the beginning of the sales period, the supplier receives the exact ordering quantities from his customers. If *

Corresponding author. Fax: +1 573 882 2693. E-mail address: [email protected] (W. Jang).

0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.06.038

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

717

the supplier has previously produced a sufficient quantity of the product to satisfy all of the demands, the items are immediately shipped. Otherwise, the supplier allocates the current inventory to customers and immediately ships the product to only those customers whose demands can be completely met. The items are delivered to other customers once the supplier has produced enough of the product to satisfy their demands. There is one direct delivery to each customer because of the high fixed cost of transportation. The supplier pays a customer specific penalty cost, called the customer waiting cost, according to the delivery time, which may be determined by the actual production speed and the distribution sequence of additional production. Under this scenario, production, allocation, and distribution decisions are no longer separated. One of the key ideas is that the inventory allocations partially determine the distribution strategy. The customer waiting cost is dependent on the distribution sequence and any additional production quantities for other customers. In addition, the supplier’s optimal production quantity is affected by the distribution policy in use, and thus the supplier’s production and distribution decisions should not be made in isolation from each other. Our objective is to compute (1) an optimal production quantity before the sales period under the unknown stochastic demands, (2) the inventory allocation at the beginning of the period under the known deterministic demands, and (3) the distribution sequence of any additional production during the period, so that the total cost for a supplier, including the production cost and the cost of waiting associated with orders of not-yetdelivered items, is minimized. Applications of this type can be found in the automotive industry (Jayaram and Vickery, 1996), the semiconductor industry (Hicks and Brown, 1995), and the furniture industry (Vickery et al., 1995). The management of customer waiting time has been an important concept to most manufacturers and the value of time, with respect to customer satisfaction, has been well established on the conceptual level. However, this important strategy has not been well analyzed in a supply chain setting. Most of the inventory models under stochastic demands consider fixed waiting (shortage) cost. The shortage cost is the same when items are backlogged, regardless of the actual delivery time. A limited number of publications are concerned with the determination of customer waiting time. Furthermore, many of them assume waiting times (or replenishment lead times) as exogenously given, that is, independent of the demand and replenishment processes. However, in many settings, waiting times should be determined endogenously when they result from production lead times in a capacitated production facility: the time to fill an order depends on the workload in the production facility, which, in turn, is caused by other order streams. A good review of inventory models can be found in Zipkin (2000) and Silver et al. (1998). Our study captures the characteristics of the endogeneous waiting time explicitly, and models its effect on optimal decisions. The remainder of the paper is organized as follows. A review of the relevant literature and the comparison with our model are provided in Section 2. A precise optimization problem is sated in Section 3, and the optimal coordinated decisions minimizing total production and waiting costs when there are two customers are presented in Section 4. The results are extended to multiple customers in Section 5. Effective algorithms, which utilize the properties of the two customer problem are also provided. To illustrate the performance of the proposed algorithms, a number of numerical tests are given in Section 6. Conclusions are given in Section 7. 2. Related literature We note that the literature on production and distribution decisions is abundant. In the interest of brevity, our discussion reviews and compares only those models most closely related to our research. Recently, a new approach integrating decisions of different functions such as production planning, inventory management, allocation and distribution, facility location, etc. into a single optimization model has been identified and become significant. Sarmiento and Nagi (1999) and Thomas and Griffin (1996) present good reviews of these integrated models. Most of these models focus on transportation issues such as vehicle routing and shipment size. They rarely consider inventory allocation and customer waiting issues, which are often avoided in deterministic demand models and simple independent and identically distributed stochastic demand models. In addition, the classification tables in Sarmiento and Nagi (1999) show that most of the work is focusing on deterministic models leaving the stochastic area underdeveloped.

718

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

On the other hand, we pay more attention to inventory allocation and customer waiting issues than transportation issues. In our model, we only consider direct shipping from the supplier to each customer. A direct shipping policy is often considered in similar systems (for example, see Barnes-Schuster and Bassok, 1997; Blumenfeld et al., 1991; and Jones and Qian, 1997), and it is very effective especially when a lot size is large (Gallego and Simchi-Levi, 1990). There are several studies similar to ours in many aspects. Jang (2006) considers the same problem as ours with different assumptions. He considers production and allocation policies in a two-class inventory system with time and quantity dependent waiting costs. His work identifies the value of the integrated policy and compare it with typical approximations. We extend his work considering multiple customers in a setting where high cost of transportation limits only one delivery to each customer. This also makes the customer waiting cost solely dependent upon his waiting time. Swaminathan and Srinivasan (1999) try to determine only the production quantity and allocation policy without considering any backlogging or customer waiting. However, even with the simple allocation policy constraining the percentage of the time in which demand is satisfied completely, the model becomes very complicated and an iterative solution procedure using a Monte Carlo simulation is suggested. They also provide an example showing that a production decision which is not integrated with the allocation policy may not be efficient. So and Song (1998) examine the relationship between pricing and customer delivery in an environment where demand is sensitive to both price and delivery time. An optimization model is derived to jointly determine pricing and capacity expansion decisions subject to the reliability requirement such that the probability of meeting the delivery time guarantee must be at least a prespecified value. They do not distinguish customers, however. Geunes and Zeng (2001) investigate the impacts of inventory shortage policies in one-to-one base-stock distribution system under uncertain demand. They demonstrate that the reduction in total cost resulting from a backlogging policy can be significant. Chien (1993) addresses the problem of finding profit maximizing production and shipping quantities under one-to-one direct shipping and stochastic demand assumptions. An iterative procedure is presented and used to find nearly optimal solutions assuming a uniform demand distribution. Federgruen and Zipkin (1984) integrate the allocation and routing problems within the same model by treating a single-period, single-item problem with random demands at the retailers. Their method is the first to decompose the problem into two portions: inventory allocation and vehicle routing. The solution then follows by constructive and/or interchange heuristics. The stock allocation problem is considered in several inventory models. For example, Jackson (1988) and McGavin et al. (1993) study optimal stock allocation problems for periodic review systems, but they assume simpler settings such as identical retailers and constant lead times. Several articles address the allocation issue in capacitated inventory models. Gavirneni (2001) models inventory allocation mechanisms with or without co-operation where one capacitated supplier distributes a single product to many identical retailers. Deshpande et al. (2003) consider a threshold inventory rationing policy for two demand classes with different arrival rates and shortage costs. Cachon and Lariviere (1999) investigate different types of allocation mechanisms, and show that a broad class of mechanisms is prone to manipulation. However, a truth-inducing mechanism can lower profits. It is shown that the linear allocation, which awards each retailer his order minus a common deduction, achieves a reasonably good allocation. Lee et al. (1997) outlines an allocation model with identical retailers served by a supplier experiencing stochastic capacity shocks. Inventory rationing problems in a capacitated production facility (Ha, 2000) are also related to our topic. The work by de Vericourt et al. (2001) compares different allocation policies such as a simple FCFS rule and a priority allocation rule when the same item is demanded by several classes of customers. In addition, Wein (1992), Pena-Perez and Zipkin (1997), and de Vericourt et al. (2002) present queueing systems mainly focusing on dynamic scheduling and stock allocation when waiting times result from production lead times in a capacitated production facility. While Topkis (1968), Ha (1997) and de Vericourt et al. (2002), among others, consider a more difficult multi-period problem by reserving some stocks for the coming periods, they neither consider a fixed cost for additional stock allocation nor restrict the number of allocations. This leads to a rather conceptually simple threshold policy that on the one hand always privileges customers with higher waiting costs in allocation decisions. On the other hand, in our paper, we limit the number of allocation/delivery due to the high fixed cost associated with it. The optimal policy, based on this assumption, sometimes favors less important customers, satisfying their demands first.

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

719

3. Problem formulation Our objective is to make the following three sequential decisions optimally for a supplier so that the expected cost is minimized. 1. Production quantity: A supplier decides the number of the item to produce before knowing the real demands from customers. 2. Inventory allocation: A supplier allocates items and delivers them to customers whose demands are fully met, after seeing the actual demands. 3. Distribution sequence: A supplier determines the delivery sequence of any unsatisfied demands while producing additional items. The first decision should be made based on the estimated stochastic demands while the second and the third decisions rely on the realized deterministic demands. In addition, the third decision is necessary only if the current inventory cannot satisfy all the demands from customers. Each customer receives one delivery, either from the inventory allocation at the beginning or from the distribution during additional production. Hence, a customer, whose first inventory allocation is not enough, receives direct delivery as soon as the supplier produces enough number to meet her total demand and it is her turn in the distribution sequence. The transportation cost, which is a constant value under our assumptions, is not considered explicitly because it does not affect the control policy. We take into account the following parameters: Q Qi Di yi wi si c

original production quantity of a supplier quantity allocated to customer i random demand from customer i with density fi(Æ) quantity that is not immediately allocated to customer i waiting cost per unit time for customer i position in the delivery sequence for customer i production cost per item

We assume all parameters are non-negative. Throughout the paper we assume that wi P wj, if i < j, without loss of any generality. Although Di is random, we treat it as if it is a constant once the demand is realized. P Observe that Qi 6 Q and yi 6 Di. The waiting cost includes a loss-of-goodwill penalty as well as an opportunity loss in postponed receipt of revenues. If demands are delivered as they arrive (without backlogging), then no waiting cost accumulates. The value of si is 0 if the demand for customer i is satisfied from the inventory allocation because the distribution sequence from additional production is not applicable in this case. Items are produced in a facility that can produce one unit at a time, and production time per item is assumed to be one unit time. The precise optimization problem is given below. Three sets of policy parameters such as production quantity, Q, allocation quantity, Qi, distribution sequence, si, should be determined so that the total expected cost for a supplier is minimized Minimize s:t:

cQ þ E½K 1 ðQ1 Þ þ    þ K n ðQn Þ si X K i ðQi Þ ¼ wi ðD½k  Q½k Þ i ¼ 1; . . . ; n; k¼1

0 6 Qi 6 Di n X Qi 6 Q

i ¼ 1; . . . ; n;

ð1Þ

i¼1

where [k] represents the customer occupying kth position in the distribution sequence. That is, if si = k then [k] = i. Note that the first constraint computes the waiting cost for each customer based on the given distribution sequence si and waiting time based on the additional production quantity D[k]  Q[k] of all the

720

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

customers placed ahead in the distribution sequence. The value of Ki(Qi) is 0 for a customer whose demand is fully satisfied from inventory allocation without waiting because we set si = 0 in such a case. 4. Two customer problem In this section, we solve a special case of the problem when there are two customers, and derive an optimal solution. This solution is used as a basis for designing efficient solutions to the multiple customer problem in the next section. Instead of solving the integrated problem formulation (1) directly, we derive production, allocation, and distribution decisions sequentially in the reverse order from the third decision to the first one. Suppose that the supplier’s original production is not enough to satisfy demands from two customers. Then, after inventory allocation, y1 and y2 items should be produced for deliveries to customers 1 and 2, respectively. The supplier needs to decide the distribution sequence, i.e., to deliver customer 1 first or customer 2 first, during his additional production. Because the customer waiting cost is determined by the actual delivery time, this decision obviously affects the overall cost and other decisions. The next lemma determines the distribution sequence. Lemma 1. The supplier should distribute items to customer 1 first if and only if w1y2 P w2y1. Proof. If the supplier delivers items to customer 1 first, the total waiting cost is given by w1 y 1 þ w2 ðy 1 þ y 2 Þ: If customer 2 receives items first, it is equal to w2 y 2 þ w1 ðy 1 þ y 2 Þ: The supplier should choose the distribution sequence minimizing the total waiting cost. Therefore, items should be delivered to customer 1 first if and only if w1 y 1 þ w2 ðy 1 þ y 2 Þ 6 w2 y 2 þ w1 ðy 1 þ y 2 Þ () w2 y 1 6 w1 y 2 : This completes the proof.

h

For a given Q value, we can decide an optimal allocation policy according to the realized customer demands. If D1 + D2 6 Q, obviously Q1 = D1 and Q2 = D2. The following lemma provides an optimal allocation policy when the demands from customers exceed the current inventory. Lemma 2. Suppose that D1 + D2 > Q. Then, optimal allocation quantities are given as follows: 8 if D1 < Q; D1 > > > > > 2 > Q if Q 6 D1 < ww12 Q and D2 P w1ww ðD1  QÞ; > > 2 > > > < or if D1 P ww12 Q and D2 P ww21 D1 ; Q1 ¼ > 2 > Q  D2 if Q 6 D1 < ww12 Q and D2 < w1ww ðD1  QÞ; > 2 > > > w1 > > or if D1 P w2 Q and D2 6 Q; > > > : 0 otherwise; Q2 ¼ Q  Q1 : Proof. Because D1 + D2 > Q, a supplier cannot satisfy both demands. He has to allocate Q items to two customers. Hence, we have Q ¼ Q1 þ Q2 ; y 1 ¼ D1  Q1 ; and y 2 ¼ D2  Q þ Q1 ; where Q1 6 D1 and Q2 6 D2. The total waiting cost W(Q1), is given by W ðQ1 Þ ¼ w1 y 1 þ w2 y 2 þ minfw2 y 1 ; w1 y 2 g ¼ w1 ðD1  Q1 Þ þ w2 ðD2  Q þ Q1 Þ þ minfw2 ðD1  Q1 Þ; w1 ðD2  Q þ Q1 Þg:

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

721

We first prove that Q1 is equal to either min[Q, D1] or max[Q  D2, 0]. Let hðQ1 Þ ¼ w2 ðD1  Q1 Þ  w1 ðD2  Q þ Q1 Þ ¼ ðw1 þ w2 ÞQ1 þ w2 D1 þ w1 ðQ  D2 Þ: Observe that h(Q1) is a decreasing function of Q1. That is, if hð^qÞ 6 ðPÞ0 then h(q1) 6 (P) 0 for all ^ q1 P ð6ÞQ. If h(Q1) 6 0, then W ðQ1 Þ ¼ w1 Q1 þ w1 D1 þ w2 ðD1 þ D2  QÞ: Because the largest possible value of Q1 minimizes W(Æ) and still satisfies the condition h(Q1) 6 0, Q1 = min[Q, D1]. If h(Q1) > 0, we have W ðQ1 Þ ¼ w2 Q1 þ w1 ðD1 þ D2  QÞ þ w2 ðD2  QÞ: Hence, the smallest possible value of Q1 = max[Q  D2, 0] is feasible and minimizes W(Æ). Next, we determine the optimal value of Q1 for given D1 and D2. Suppose that D1 < Q and D2 < Q. From the above argument, Q1 should be equal to either D1 or Q  D2. If Q1 = D1, then hðD1 Þ ¼ w1 ðD1 þ D2  QÞ < 0; W ðD1 Þ ¼ w2 ðD1 þ D2  QÞ:

and

On the other hand, if Q1 = Q  D2, then hðQ  D2 Þ ¼ w2 ðD1 þ D2  QÞ > 0;

and

W ðQ  D2 Þ ¼ w1 ðD1 þ D2  QÞ: Because w1 P w2, W(D1) 6 W(Q  D2). Therefore, Q1 = D1. If D1 < Q and D2 P Q, Q1 should be equal to either D1 or 0. If Q1 = 0 is the solution, then hð0Þ ¼ w2 D1  w1 ðD2  QÞ should be positive, and W ð0Þ ¼ w1 ðD1 þ D2  QÞ þ w2 ðD2  QÞ should be less than W(D1) computed previously. However, W ð0Þ  W ðD1 Þ ¼ ðw1  w2 ÞðD1 þ D2  QÞ þ w2 ðD2  QÞ P 0: Therefore, Q1 is again equal to D1, and this proves the first case of the lemma. Because the other cases can be proven similarly, we omit proofs for them. h The result of Lemma 2 is graphically represented in Fig. 1. Using the equation yi = Di  Qi and the values of Qi’s given in the figure, the total expected cost can be written as cQ þ

Z

Q

Z

0

þ

Z

Q

Z

Z

þ

Q

Z Q

Z

Q

w1 w2 w2 d 2

Z

Q

ðc þ w2 Þðd 1 þ d 2  QÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2

0

ðw1 ðd 1  QÞ þ ðc þ w2 Þðd 1 þ d 2  QÞÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2

1 Qþ

1

Z

w1 w2 w2 d 2

w1 w2 d 2

ðc þ w1 Þðd 1 þ d 2  QÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2

ðw1 ðd 1  QÞ þ ðc þ w2 Þðd 1 þ d 2  QÞÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2

Q

Q

þ

Qþ

1

Q

0

Z

ðc þ w2 Þðd 1 þ d 2  QÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2 þ

Z

Qd 2

0

þ

Q

1

Z

1

w1 w2 d 2

ððc þ w1 Þðd 1 þ d 2  QÞ þ w2 ðd 2  QÞÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2 :

722

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

Fig. 1. Inventory allocation.

Note that c(D1 + D2  Q) represents the additional production cost, if there is any. The variable portion of the total expected cost, which is dependent upon the operating policy, can be written as follows after arranging and removing constant terms such as c(E[D1] + E[D2]) from the original cost. We call it K(Q) Z Q Z Qd 2 Z 1 KðQÞ ¼ cðQ  d 1  d 2 Þf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2 þ w1 ðd 1  QÞf1 ðd 1 Þ dd 1 0

þ

0 1

Z

w2 ðd 2  QÞf2 ðd 2 Þ dd 2 þ

Z

1

Z

1 Q

Z Z

Q

Z

Q 1

Qþ

1

w1 w2 d 2

Q

þ

Q 0

Q

þ

Z

w1 d 2 f1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2 þ

w1 w2 d 2

w1 w2 w2 d 2

Z

1

Z

Q

w2 d 1 f1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2

0

Q

w2 d 1 f1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2 þ

w1 d 2 f1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2

Z

Q 0

Z

Qþ

w1 w2 w2 d 2

w2 ðd 1 þ d 2  QÞf1 ðd 1 Þf2 ðd 2 Þ dd 1 dd 2 : ð2Þ

Qd 2

Observe that the second and third integrations are waiting costs caused by individual demands without considering interactions when each demand exceeds the current inventory level. The fourth and fifth integrations represent waiting costs of customer 1 when the product distribution is delayed due to the delivery to customer 2. This occurs only when the demand from customer 1 is larger than the current inventory. The same costs for customer 2 are given in the next two integrations. The waiting cost for customer 2 even when her demand is lower than the current inventory level is given in the last integration. This situation occurs because the supplier often allocates items to customer 1 first due to her higher unit waiting cost even though it is possible to satisfy the demand from customer 2. Our final decision is to calculate the optimal production quantity, Q*, minimizing K(Q). Taking the derivative with respect to Q based on the Leibniz’s rule and after considerable algebraic simplification, we obtain Z Q dKðQÞ ¼ ðc þ w2 Þ F 1 ðQ  d 2 Þf2 ðd 2 Þ dd 2  w1 ½1  F 1 ðQÞ  w2 ½1  F 2 ðQÞ dQ 0  Z Q  w1  w2 F1 Q þ d 2 f2 ðd 2 Þ dd 2 : ð3Þ  w2 w2 0

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

Similarly, the second derivative can be computed as follows:    Z Q d2 KðQÞ w1 ¼ ðc þ w2 Þ f1 ðQ  d 2 Þf2 ðd 2 Þ dd 2 þ w1 f1 ðQÞ þ w2 f2 ðQÞ 1  F 1 Q w2 dQ2 0  Z Q  w1  w2  w2 f1 Q þ d 2 f2 ðd 2 Þ dd 2 : w2 0

723

ð4Þ

The value of this function may be negative for a certain set of parameters. Hence, the convexity of the total cost function may not be generally proven. We show that the convexity can be determined by the probability density function of customer 1, and present a sufficient condition, which often holds for common distributions such as uniform, exponential, and normal distributions. Lemma 3. K(Q) is convex in Q if w1f1(Q) P w2f1(Q + (w1/w2  1)x) for any given x 2 [0, Q]. Proof. Observe that from (4)    Z Q o2 KðQÞ w1 ¼ ðc þ w Þ f ðQ  d Þf ðd Þ dd þ w f ðQÞ þ w f ðQÞ 1  F Q 2 1 2 2 2 2 1 1 2 2 1 w2 oQ2 0   Z Q  Z Q  w1  w2 w1  w2  w2 f1 Q þ d 2 f2 ðd 2 Þ dd 2 P w1 f1 ðQÞ  w2 f1 Q þ d 2 f2 ðd 2 Þ dd 2 w2 w2 0 0 Z Q     w1  w2 P w1 f1 ðQÞ  w2 f1 Q þ d2 f2 ðd 2 Þ dd 2 : w2 0   2 Hence, if w1 f1 ðQÞ  w2 f1 Q þ w1ww d 2 P 0 then o2K(Q)/oQ2 P 0. This completes the proof. h 2 Corollary 1. If f1(d1) follows either uniform or exponential distribution, then K(Q) is convex. Proof. From Lemma 3, K(Q) is convex if f1(d1) is a non-increasing function. This completes the proof.

h

Corollary 2. Suppose that f1(d1) follows a normal distribution. If w1 P c then K(Q) is convex in [l, 1] for any l P E[d1], where Q* 2 [l, 1]. Proof. Assume that f1 ðd 1 Þ  N ðl1 ; r21 Þ. If w1 P c, the optimal production quantity in a single customer problem is equal to   w1 F 1 P F 1 ð0:5Þ ¼ l1 c þ w1 from Eq. (8). Because the optimal quantity for multiple customers is larger than that for a single customer, Q P l1 : On the other hand, from Lemma 3, K(Q) is convex if 2 2 xl   ðQþ w1ww 1Þ ðQl1 Þ2 2   w1  w2 1 1 2 2r2 1 x () pffiffiffiffiffiffi e 2r1 P pffiffiffiffiffiffi e f1 ðQÞ P f1 Q þ w2 2pr1 2pr1  2 w1  w2 2 () ðQ  l1 Þ 6 Q  l1 þ x w2

for any positive x. Because the inequality holds when Q P l1, the region around the optimal quantity Q* is always convex. This completes the proof. h As Lemma 3 shows, the convexity of the function K(Q) can be determined by the characteristics of the customer with the largest unit waiting cost. If K(Q) is convex the optimal Q* is easily found by searching the equation dK(Q)/dQ = 0. The conditions given in Lemma 3 and Corollary 2 are very strong conditions. In fact, the function K(Q) was always convex in our numerical tests, which used randomly generated normal distribution data sets.

724

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

We can observe several properties of Q* using Eq. (3). We can compute a lower bound on Q* by proving that the optimal production quantity for multiple customers has to be greater than the quantity necessary for one customer (see Appendix for the proof). This lower bound can significantly reduce the computational time. The relationships between Q* and other parameters can be easily observed if the cost function is convex. For example, the optimal production quantity is increasing in w1 and w2 and decreasing in c. The proof using the Implicit Function Theorem is given in Appendix. In addition, the optimal production quantity is increasing in customer demands if all the other parameters remain unchanged (see Appendix for the proof). The following example demonstrates how to apply the procedures developed in this section to a given problem. Example 1. Suppose there are two customers with demands D1  N(100, 302) and D2  N(100, 202). The unit production cost and unit waiting costs are assumed to be c = 1, w1 = 1, and w2 = 0.5. First, the supplier must decide how many items to produce based on random variables, D1 and D2. After solving dK(Q)/dQ = 0 using Eq. (3), the supplier produces Q* = 185 items. When the sales period begins, the supplier receives the exact order information from the two customers. If D1 = 150 and D2 = 150 the supplier immediately ships 150 items to customer 1, i.e., Q1 = 150, and allocates 35 items to customer 2, i.e., Q2 = 35 according to the first case of Lemma 2. The supplier produces 115 more items and delivers total 150 items to customer 2, i.e., y2 = 115, as soon as production is finished. On the other hand, if D1 = 200 and D2 = 10 then Q1 = 175 and Q2 = 10 according to the third case of Lemma 2. The value of y1 is equal to 25, and it is delivered after additional production. As a final example, consider a case when D1 = 200 and D2 = 50. Because Q1 = 185 and Q2 = 0, the unsatisfied demands become y1 = 15 and y2 = 50. In this case, the supplier needs to decide the sequence of distribution of his additional production. Because w1y2 = 50 is larger than w2y1 = 7.5, the supplier should distribute items to customer 1 first based on Lemma 1. 5. Extension to multiple customers In this section, the same model discussed is extended to multiple customers. However, as we can imagine from the complexity of the two customer problem, it is very difficult to characterize the exact optimal solutions for the multiple customer case. Hence, we present heuristic algorithms based on structural properties and results from the previous section. The following definitions are used throughout this section. Definition 1. The weighted smallest quantity first (WSQ) rule sequences the customers in decreasing order of the ratio wi/yi. Definition 2. Let P contain all possible sequences of the n customers. In a sequence p 2 P represented by ([1], [2], . . . , [i], . . . , [n]), let [i] indicate the customer occupying ith position in that sequence. Definition 3. The probability density function f1+  +j(Æ) represents the convolution of random variables D1, . . . , Dj. There are again three sequential decisions such as production quantity, inventory allocation, and distribution sequence for a supplier to make. The supplier needs to decide the distribution sequence when his original production is not enough to satisfy demands from n customers. Next lemmas are used to determine an optimal distribution sequence and help us derive heuristic solutions. Lemma 4. The WSQ rule generates an optimal distribution sequence. Proof. Suppose a distribution sequence p1, which is not WSQ, is optimal. In this schedule, there must be at least two adjacent customers, say customer j followed by customer k, such that wj =y j < wk =y k : Generate a new sequence p2 by performing a so-called adjacent pairwise interchange on customers j and k. All other customers remain in their original positions. Thus, the difference in the values of the waiting

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

725

costs under sequences p1 and p2 is due only to customers j and k. Under p1 the waiting cost for customers j and k is wj ðy ½1 þ    þ y ½i þ y j Þ þ wk ðy ½1 þ    þ y ½i þ y j þ y k Þ for some i, whereas under p2 it is wk ðy ½1 þ    þ y ½i þ y k Þ þ wj ðy ½1 þ    þ y ½i þ y k þ y j Þ: It is easily verified that if wj/yj < wk/yk the waiting cost under p2 is strictly less than under p1. This contradicts the optimality of p1 and proves the lemma. h Lemma 5. At most one customer gets partial inventory allocation (i.e., 0 < Qi < Di). Proof. We prove this lemma by showing that the cost can be always improved by transferring items between customers if there are more than one customer with partial allocations. Suppose that inventory allocation Qi’s, i = 1, . . . , n, which cause more than one partial allocation, are optimal for given Di’s. In this allocation, assume that the demands from customers j and k are partially satisfied. That is, y j ¼ Dj  Qj > 0

and

y k ¼ Dk  Qk > 0:

Without loss of generality, assume that wj/yj P wk/yk. Then, there exist i1 and i2, i1 < i2, such that [i1] = j and [i2] = k in the optimal distribution sequence generated by the WSQ rule. The total waiting cost W can be written as ! ! !! ! i1 1 i2 1 i1 1 n s s s n s X X X X X X X X X w½s y ½t ¼ w½s y ½t þ w½s y ½t þ y j þ y ½t w½s y ½t : W ¼ þ s¼1

t¼1

t¼1

s¼1

s¼i1

t¼1

t¼i1 þ1

s¼i2

t¼1

Generate a new inventory allocation scheme by transferring the small positive allocation amount  from customer k to customer j. Then, the unsatisfied demands are given as follows: y 0j ¼ Dj  Qj   P 0 and

y 0k ¼ Dk  Qk þ  > 0:

All the other yi’s are not changed. The new waiting cost W 0 can be computed under this new allocation scheme. The difference in the values of the waiting costs between these two schemes is given as follows: W W0 ¼

i2 1 X

w½s ðy j  y 0j Þ ¼ 

s¼i1

i2 1 X

w½s > 0:

s¼i1

This contradicts the optimality of the original inventory allocation and proves the lemma.

h

Lemma 5 does not provide an optimal inventory allocation scheme, but it allows us to compute the optimal scheme at least numerically. The following algorithm determines the optimal allocation by evaluating all possible cases. For given Q and Di, i = 1, . . . , n, suppose that D1 +    + Dn > Q. Otherwise, the allocation is trivial. IA-opt: Optimal inventory allocation algorithm Step 0. Let S = P and W* = 1. Step 1. Select a sequence p = ([1], [2], . . . , [n]) 2 S, and determine j such that D½1 þ    þ D½j 6 Q

and D½1 þ    þ D½jþ1 > Q:

726

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

Let

Q½i ¼

8 D½i > > > < > > > :

Q

if i 6 j; j P

D½k

if i ¼ j þ 1;

k¼1

0

otherwise

and let yi = Di  Qi for all i. Step 2. Generate a new sequence based on the WSQ rule. Call it p 0 = ([1], [2], . . . , [n]). Let ! ! n s n s X X X X 0 W ðp Þ ¼ w½s y ½t ¼ w½s y ½t : s¼1 0

t¼1

W*

s¼jþ1

W*

t¼jþ1 0

If W(p ) < then set = W(p ) and p* = p. Step 3. S = Sn{p}. If S 5 ; go to Step 1. The above algorithm generates the optimal inventory allocation, but it requires the complete enumeration of all possible sequences. Hence, it is not computationally possible to use the algorithm when the number of customers becomes large. The next algorithm focuses on the relationship between two customers instead of seeking a global optimal solution. That is, the proposed heuristic is a greedy type myopic algorithm. By applying the optimal results developed in the previous section, this approach provides efficient solutions, which are validated later through numerical tests. ^ and demands Di and Dj, i < j, customer i has Definition 4. For given values of unallocated inventory level Q ^ (ii) Q ^ 6 Di < wi Q ^ or (iii) Di P wi Q ^ and Dj P wj ðDi  QÞ, ^ and priority over customer j if (i) Di < Q, wj wi wj wj wj Dj P wi Di . IA-app: Approximate inventory allocation algorithm ^ ¼ Q, and j = 1. Step 0. Let p = (1, 2, . . . , n), Q Step 1. If customer [j] 2 p has priority over customer [j + 1] 2 p or if customer [j] is the last element in the sequence p, set ^ D½j g; Q½j ¼ minfQ; ^ ¼Q ^  Q½j ; Q p ¼ p n f½jg; and 1 j¼ maxfj  1; 1g

if ½j is the last element; otherwise:

Otherwise, let j = j + 1. ^ > 0 and p 5 ; go to Step 1. Step 2. If Q If we consider a special case of the problem when unit waiting costs are identical (i.e., wj = wk, for all j and k), the optimal inventory allocation can be easily determined as follows. P P Lemma 6. Without loss of generality assume that Dj 6 Dk when j < k. If ji¼1 Di 6 Q and jþ1 i¼1 Di > Q, then optimal allocation quantities are given as follows: 8 Di > > > < j P Qi ¼ Q  Dk > k¼1 > > : 0

if i 6 j; if i ¼ j þ 1; otherwise:

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

727

Proof. The proof is omitted because it is similar to that of Lemma 5. h Based on Lemma 6 the expected cost function K(Q) can be written as follows when all the unit waiting costs are identical. KðQÞ ¼

Z



Z

cðQ  d 1      d n Þdd 1    dd n

d 1 þþd n 6Q

þ ¼

Z

" X Z p2P





Z

n X

w½s

d 1 þþd n >Q;d ½1 6d ½2 66d ½n s¼1

Z

s X

#

y ½t dd ½1    dd ½n

t¼1

cðQ  d 1      d n Þdd 1    dd n

d 1 þþd n 6Q

þ

!

" X Z



Z

n X

d 1 þþd n >Q;d ½1 6d ½2 66d ½n s¼1

p2P

w½s

s X

(

(

D½t  max min Q 

t¼1

t1 X

) )!! D½i ; D½t ; 0

# dd ½1    dd ½n :

i¼1

Unfortunately, this cost function is too complicate to perform further analysis. Moreover, we even cannot write the expected total cost as a closed form function when unit waiting costs are not the same. Thus, we must design a heuristic method to compute an appropriate production quantity. From (3) the optimal production quantity of the two customer problem should satisfy the following equation: cF 1þ2 ðQÞ ¼ w1 ½1  F 1 ðQÞ þ w2 ½1  F 2 ðQÞ   Z Q  w1  w2 F1 Q þ d 2  F 1 ðQ  d 2 Þ f2 ðd 2 Þ dd 2 : þ w2 w2 0

ð5Þ

That is, the expected cost of over-production should be equal to the expected waiting cost due to under-production. The waiting cost occurs when (i) customer 1 orders too many items, (ii) customer 2 orders too many items, and (iii) the order from customer 2 is not delivered immediately, even though it is small, because of the order from customer 1. The costs associated with these three cases are given in the right hand side of (5). This concept can be extended to the n customer problem as follows: cF 1þþn ðQÞ ¼

n X

wi ½1  F i ðQÞ

i¼1

þ

n X i¼2

Z

Q

wi 0

    wi  wi F 1þþði1Þ Q þ d i  F 1þþði1Þ ðQ  d i Þ fi ðd i Þ dd i ; wi

ð6Þ

where wi implicitly represents the average unit waiting cost of customers 1, . . . , i  1. Again, the left hand side of (6) is the expected over-production cost while the right hand side represents the expected waiting cost. The first summation expressed by wi[1  Fi(Q)] is the cost when individual customers order too many items. The next summation represents cases when small orders are not delivered immediately due to other customers’ orders. Due to the complexity of Eq. (6), an approximation is needed. As shown through numerical tests in the next section, the following approximate solution performs very well. Because the optimal production quantity, Q*, increases if an additional customer is added or any customer demand is increased as proved in Proposition 3 in Appendix, Q* may well be relatively large from the view of individual demands. In other words, F1+  +i(x) can be close to 1 if x is larger than Q*. While this claim may not be accurate as i gets larger, the effect of errors in such cases is diminished because the coefficient wi in (6) becomes smaller. Based on the argument we use the approximation such that F 1þþi ðxÞ ¼ 1 in (6), if i < n and x > Q.

728

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

If we apply this approximation to the term in Eq. (6), then    Z Q Z Q wi  wi F 1þþði1Þ Q þ d i  F 1þþði1Þ ðQ  d i Þ fi ðd i Þ dd i ¼ ð1  F 1þþði1Þ ðQ  d i ÞÞfi ðd i Þ dd i wi 0 0 ¼ F i ðQÞ  F 1þþi ðQÞ: Hence, Eq. (6) becomes, under the approximation, n X

wi F 1þþi ðQÞ þ cF 1þþn ðQÞ ¼

i¼1

n X

wi :

ð7Þ

i¼1

Because the left hand side of (7) is an increasing function with a value 0 when Q = 0 and a value larger than Pn i¼1 wi when Q = 1, there is a unique solution satisfying (7). 6. Computational study The magnitude of errors from the proposed algorithms and the value of the integrated approach over the typical non-integrated methods are not amenable to exact analysis. Thus, we present numerical examples that illustrate results within the following settings: Each example consists of 2, 5 or 10 customers. The demand of each customer is normally distributed. The means of demands (li) are real numbers, randomly and independently drawn from the uniform distribution U[100, 500]. The variances are also randomly generated, yet with the restriction that requires 99% of demands to be positive. The production cost and unit waiting costs can be anywhere between 1 and 10. For a given number of customers, 100 examples are generated and 1000 replications are conducted for each example. Data in each row of Tables 2 and 3 are computed by 100 examples, i.e., 100,000 simulations. We evaluate the performance of our recommended approximate production quantity (PQ-app), which is computed by (7), combined with the optimal inventory allocation (IA-opt) and the optimal distribution sequence (DS-opt), and compare the performance both to the simple benchmark schemes and to the overall optimal solution. Two well-known approximation schemes are used to compute the production quantities such as aggregating multiple demands (PQ-agg) or solving the single demand problem multiple times (PQ-sep), which we will shortly describe in greater detail. The aggregation heuristic combines multiple demands in order to use the single demand model. When the multiple customer model is not available, the first approximation we can think of might be to treat multiple customers as an aggregated demand with some averaged waiting cost. Thus, the production quantity of this approach, Q, satisfies, by Eq. (8), w F 1þþn ðQÞ ¼ ; cþw P P where the average waiting cost w is given by w ¼ ni¼1 li wi = ni¼1 li . The separation heuristic treats each demand independently. The single demand problem is solved n times, and the production quantities of those problems are added to represent the total quantity. The production quantity of this approach, Q, is given as follows:   n X wi 1 Q¼ Fi : c þ wi i¼1 We also test two simple inventory allocation schemes such as a priority rule (IA-prt), which always allocates items to customers with higher waiting costs first, and a random rule (IA-rnd), which randomly determines the allocation sequence. Note that this is essentially a First Come First Served rule, which is frequently used in practice. To determine the distribution sequence, we apply the same two rules such as a priority rule (DS-prt) and a random rule (DS-rnd). Several heuristics for our integrated problem are created by combining the production, allocation, and distribution rules described above. Table 1 lists the heuristics we tested. Note that Heuristics 1 and 1* are recommended methods based on our work. We first establish the superiority of Heuristic 1 over typical approximate

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

729

Table 1 List of heuristics

Production quantity Inventory allocation Distribution sequence

Heuristic 1

Heuristic 1*

Heuristic 2

Heuristic 3

Heuristic 4

Heuristic 5

Heuristic 6

PQ-app IA-opt DS-opt

PQ-app IA-app DS-opt

PQ-agg IA-opt DS-opt

PQ-sep IA-opt DS-opt

PQ-app IA-prt DS-prt

PQ-app IA-rnd DS-rnd

PQ-opt IA-app DS-opt

Table 2 Deviation of heuristics from optimal solution

n=2

Q K(Q)

n=5

Q K(Q)

Heuristic 1 (%)

Heuristic 2 (%)

Average Worst case Average Worst case

0.1 1.3 0.0 0.2

5.6 32.9 11.9 169.4

Heuristic 3 (%) 3.7 18.8 5.6 81.4

Heuristic 4 (%) 0.1 1.3 0.1 3.6

Heuristic 5 (%) 0.1 1.3 107.6 1629

Average Worst case Average Worst case

0.6 3.4 0.3 2.7

8.1 25.5 55.9 412.1

6.5 28.7 30.8 350.0

0.6 3.4 1.0 24.7

0.6 3.4 315.5 3869

methods, and then show that Heuristic 1*, which is computationally much simpler, is comparable to Heuristic 1 in further analysis. The performance comparison of Heuristics 1–5 with the optimal solution is given in Table 2. The optimal production quantity (PQ-opt) when n = 2 is analytically computed based on (3). When n = 5, we find an optimal solution using an extensive search and simulated demands. That is, for a given production quantity, Q, we compute the expected cost, K(Q), using 1000 sets of simulated demands per example. After evaluating all the possible Q values, we select one that yields the smallest cost. We also use the optimal allocation and distribution algorithms in these cases. From Table 2, we can see that our Heuristic 1 clearly outperforms the other heuristics. It works very well in that its average deviation from the optimal solution is less than 1% and the worst case error is about 3%. However, Heuristics 2 and 3, which compute the production quantity using PQ-agg and PQ-sep, respectively, are not good approximations especially with a large number of customers. When n is 5, their average cost errors range from 30% to 55% with the worst case error reaching about 400%. Apparently, these methods should not be used for large n values. Heuristic 5, which uses PQ-app and IA-prt works pretty well. Its performance is very close to Heuristic 1 although it deteriorates as the number of customers increases. From this analysis, we see that the priority based inventory allocation rule is very good and can be practically used if the appropriate production quantity is given. However, with incorrect production quantity, as seen from Heuristics 2 and 3, the performance is poor even when the optimal inventory allocation is used. The last column, Heuristic 5, shows terrible performance of random allocation, something that should be avoided. We conduct further analysis to test the approximate inventory allocation algorithm (IA-app) and to observe the behavior of our heuristic when the number of customers increases. On the one hand, although Heuristic 1 is very good, it takes a considerable amount of time to use the optimal allocation (IA-opt) due to the complete enumeration of customer sequences. On the other hand, the execution of the approximate allocation (IA-app) is almost instantaneous. Hence, in the following tests, we want to show that Heuristic 1*, which uses IA-app instead of IA-opt as an inventory allocation module, is almost as good as Heuristic 1. Table 3 presents average values of Q and K(Q) obtained from different approaches. The first two, optimal solution and Heuristic 6, use PQ-opt based on the extensive search to find the optimal production quantity, while the next two use PQ-app based on (7). The optimal solution and Heuristic 1 use IA-opt, while the other two use IA-app to determine the inventory allocation. When n = 5, all three heuristics are comparable to the optimal solution. Their error bounds are always less than 1.6%. As expected, the production quantity of Heuristic 6, which is generated by PQ-opt, is close to the optimal production quantity with an error 0.2%, while

730

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

Table 3 Evaluation of proposed algorithms

n=5

Q K(Q)

n = 10

Q K(Q)

Optimal solution

Heuristic 6

Heuristic 1

Heuristic 1*

1374.4 339.5

1377.1 344.3

1381.9 340.7

1381.9 344.8

2795.8 388.6

2799.9 384.6

2799.9 388.9

Heuristic 1, which uses IA-opt, generates the best approximation in terms of the expected cost with an error 0.3%. Our recommended solution, Heuristic 1* using PQ-app and IA-app, provides answers almost instantaneously with average errors 0.5% and 1.6% for Q and K(Q), respectively. While IA-opt may be used as an one-time evaluation tool, it cannot be used as a sub-module of a search algorithm when the number of customers is ten because of its huge computation time. Therefore, we compute decisions for only three heuristics when n = 10. As seen from Table 3, the performance of these methods is again very similar. Although we cannot directly compare our recommended approach, Heuristic 1*, to the optimal solution, the difference of production quantity from Heuristic 6 with PQ-opt is 0.2%, and the difference of expected cost from Heuristic 1 with IA-opt is 1.1%. Based on all of these results, we believe that our Heuristic 1* is actually a very good approximation scheme generating near optimal solutions even when there are many customers. 7. Conclusions In this work, we developed an algorithm, which provides integrated production, allocation, and distribution decisions. The model included a non-traditional waiting cost derived to acknowledge the fact that customers evaluate response time differently. The results presented here allow us to evaluate and to provide appropriate decisions when multiple customers with different priorities and ordering quantities exist. An optimal policy for a two customer problem is presented, but analytical solutions are difficult or impossible to obtain when there are more than two customers. Our algorithm constructed based on the structural results of the two customer problem provides a viable alternative. The computational study shows that the actual error bound of our algorithm is less than 2% on average. Appendix Proposition 1. The optimal production quantity for multiple customers is greater than the quantity necessary for one customer. Proof. Note that the expected cost function for a single customer case can be written as Z 1 cQ þ ðc þ wi Þ ðd i  QÞfi ðd i Þ dd i Q

and the optimal production quantity, qi , satisfies F ðqi Þ ¼

wi : c þ wi

ð8Þ

Hence, we need to show that Q P maxfq1 ; q2 g. The optimal production quantity Q* makes the Eq. (3) be equal to 0. Because the following terms: w2 ½1  F 2 ðQÞ  w2

Z

Q 0

    w1  w2 F1 Q þ d 2  F 1 ðQ  d 2 Þ f2 ðd 2 Þ dd 2 ; w2

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

731

in (3) is always non-positive, the remaining terms should satisfy Z Q F 1 ðQ  d 2 Þf2 ðd 2 Þ dd 2  w1 ½1  F 1 ðQ Þ P 0: c 0

Because Z Q



F 1 ðQ  d 2 Þf2 ðd 2 Þ dd 2 6

0

Z

Q

F 1 ðQ Þf2 ðd 2 Þ dd 2 6 F 1 ðQ Þ; 0

we have ðc þ w1 ÞF 1 ðQ Þ  w1 P 0; which is equivalent to w1 : F ðQ Þ P c þ w1 Therefore, from (8), Q P q1 : Because Q P q2 can be proven similarly, the proof is complete.

h

Proposition 2. The optimal production quantity is increasing in w1 and w2 and decreasing in c. Proof. We only prove the increasing property in w1 because the other cases can be shown similarly. If we let Z Q gðQ; w1 Þ ¼ ðc þ w2 Þ F 1 ðQ  d 2 Þf2 ðd 2 Þ dd 2  w1 ½1  F 1 ðQÞ  w2 ½1  F 2 ðQÞ 0  Z Q  w1  w2 F1 Q þ d 2 f2 ðd 2 Þ dd 2 ;  w2 w2 0 the optimal production quantity satisfies g(Q, w1) = 0 for a given value w1 from (3). By the Implicit Function Theorem,  RQ  w1 w2 1  F ðQÞ þ f Q þ d d 2 f2 ðd 2 Þ dd 2 1 1 2 0 w2 oQ og=ow1 ¼ ¼ 2 2 ow1 og=oQ d KðQÞ=dQ using the notation in Eq. (4). To show that Q is increasing in w1, we only have to prove that the above equation is positive. This is clear thanks to the convexity assumption of K(Q). Hence, the proof is complete. h Proposition 3. The optimal production quantity is non-decreasing in customer demands. Proof. Let Q* be the optimal production quantity for a given set of customer demands. Suppose that a customer demand with associated waiting cost w is added. If the value of w is equal to zero, then the optimal production quantity is unchanged because any new demand can be delayed without penalty. Based on Proposition 2 the optimal production quantity becomes larger than Q* when w is positive. This completes the proof. h References Barnes-Schuster, D., Bassok, Y., 1997. Direct shipping and the dynamic single-depot/multi-retailer inventory system. European Journal of Operational Research 101, 509–518. Blumenfeld, D.E., Burns, L.D., Daganzo, C.F., 1991. Synchronizing production and transportation schedules. Transportation Research 25B, 23–37.

732

W. Jang, D. Kim / European Journal of Operational Research 181 (2007) 716–732

Cachon, G.P., Lariviere, M.A., 1999. Capacity choice and allocation: Strategic behavior and supply chain performance. Management Science 45, 1091–1108. Chien, T.W., 1993. Determining profit-maximizing production/shipping policies in a one-to-one direct shipping, stochastic demand environment. European Journal of Operational Research 64, 83–102. Deshpande, V., Cohen, M.A., Donohue, K., 2003. A threshold inventory rationing policy for service-differentiated demand classes. Management Science 49, 683–703. de Vericourt, F., Karaesmen, F., Dallery, Y., 2001. Assessing the benefits of different stock allocation policies for a make-to-stock production system. Manufacturing & Service Operations Management 3, 105–121. de Vericourt, F., Karaesmen, F., Dallery, Y., 2002. Optimal stock allocation for a capacitated supply system. Management Science 48, 1486–1501. Federgruen, A., Zipkin, P., 1984. A combined vehicle routing and inventory allocation problem. Operations Research 32, 1019–1037. Gallego, G., Simchi-Levi, D., 1990. On the effectiveness of direct shipping strategy for the one-warehouse multi-retailer R-systems. Management Science 36, 240–243. Gavirneni, S., 2001. Benefits of co-operation in a production distribution environment. European Journal of Operational Research 130, 612–622. Geunes, J., Zeng, A.Z., 2001. Impacts of inventory shortage policies on transportation requirements in two-stage distribution systems. European Journal of Operational Research 129, 299–310. Ha, A.Y., 1997. Inventory rationing in a make-to-stock production system with several demand classes and lost sales. Management Science 43, 1093–1103. Ha, A.Y., 2000. Stock rationing in an M/Ek/1 make-to-stock queue. Management Science 46, 77–87. Hicks, D.A., Brown, S. 1995. Evaluating time-based performance in the U.S. semiconductor industry. In: Proceedings, Industrial Engineering Research Conference, Norcross, GA, pp. 102–112. 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. Jang, W., 2006. Production and allocation policies in a two-class inventory system with time and quantity dependent waiting costs. Computers and Operations Research 33, 2301–2321. Jayaram, J., Vickery, S., 1996. Exploratory study of time-based competition in the automotive supplier industry. In: Proceedings – Annual Meeting of the Decision Science Institute, Atlanta, GA, pp. 1508–1514. Jones, P.c., Qian, T., 1997. Fully loaded direct shipping strategy in one warehouse N retailer systems without central inventories. Transportation Science 31, 193–195. Lee, H., Padmanabhan, V., Whang, S., 1997. Information distortion in a supply chain: The bullwhip effect. Management Science 43, 546– 558. McGavin, E.J., Schwarz, L.B., Ward, J.E., 1993. Two-interval inventory allocation policies in a one-warehouse n-identical retailer distribution system. Management Science 39, 1092–1107. Pena-Perez, A., Zipkin, P., 1997. Dynamic scheduling rules for a multiproduct make-to-stock queue. Operations Research 45, 919–930. Sarmiento, A.M., Nagi, R., 1999. A review of integrated analysis of production–distribution systems. IIE Transactions 31, 1061–1074. Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, 3rd ed. J. Wiley, New York, NY. So, K.C., Song, J., 1998. Price, delivery time guarantees and capacity selection. European Journal of Operational Research 111, 28–49. Swaminathan, J.M., Srinivasan, R., 1999. Managing individual customer service constraints under stochastic demand. Operations Research Letters 24, 115–125. Thomas, D.J., Griffin, P.M., 1996. Coordinated supply chain management. European Journal of Operational Research 94, 1–15. Topkis, D.M., 1968. Optimal ordering and rationing policies in a nonstationary dynamic inventory model with n demand classes. Management Science 15, 160–176. Vickery, S.K., Droge, C.L.M., Yeomans, J.M., Markland, R.E., 1995. Time-based competition in the furniture industry. Production and Inventory Management Journal 36, 14–21. Wein, L.M., 1992. Dynamic scheduling of a multiclass make-to-stock queue. Operations Research 40, 724–735. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill, Boston, MA.