Joint cyclic production and delivery scheduling in a two-stage supply chain

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ARTICLE IN PRESS Int. J. Production Economics 119 (2009) 55–74

Contents lists available at ScienceDirect

Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Joint cyclic production and delivery scheduling in a two-stage supply chain Guruprasad Pundoor a, Zhi-Long Chen b, a b

ILOG, Inc., 1195 West Fremont Avenue, Sunnyvale, CA 94087, USA Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA

a r t i c l e in fo

abstract

Article history: Received 19 March 2008 Accepted 22 January 2009 Available online 3 February 2009

We study an integrated production and distribution scheduling model in a two-stage supply chain consisting of one or more suppliers, a warehouse, and a customer. Each supplier produces a different product at a constant rate. There is a setup time and a setup cost per production run for each supplier. Completed products are ﬁrst delivered from the suppliers to the warehouse and are then sent from the warehouse to the customer. The customer’s demand for each product is constant over time. The problem is to ﬁnd jointly a cyclic production schedule for each supplier, a cyclic delivery schedule from each supplier to the warehouse, and a cyclic delivery schedule from the warehouse to the customer so that the customer demand for each product is satisﬁed without backlog at the least total production, inventory and distribution cost. We consider two production and delivery scheduling policies. We derive either an exact or a heuristic solution algorithm for the problem under each policy. Heuristics are evaluated computationally. We also evaluate the value of the warehouse by comparing our model with a model that does not have the warehouse in the supply chain (i.e. the products are delivered directly from the suppliers to the customer). Various managerial insights based on an extensive set of computational tests are reported, including how the value of warehouse and the delivery frequency from the suppliers to the warehouse are inﬂuenced by various problem parameters including production rates of the suppliers, unit inventory costs, delivery costs, and the warehouse location. & 2009 Elsevier B.V. All rights reserved.

Keywords: Two-stage supply chain Production–distribution integration Cyclic scheduling Value of warehouse

1. Introduction Production and distribution operations are the two most important operational functions in a supply chain. It is critical to plan and schedule these two functions in a coordinated manner in order to achieve optimal operational performance of the supply chain. In this paper, we study an integrated production and distribution scheduling model in a two-stage supply chain, illustrated in Fig. 1, which consists of one or more suppliers, a warehouse, and a customer. Each supplier manufactures a unique item at a Corresponding author.

E-mail addresses: [email protected] (G. Pundoor), [email protected] (Z.-L. Chen). 0925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2009.01.007

constant rate. The customer’s demand for each item is constant and known in advance. Each supplier manufactures its item in batches and there is a setup time and setup cost incurred for every production batch. Manufactured items are shipped directly from the suppliers to the warehouse, and from the warehouse to the customer. In the delivery from the warehouse to the customer, different products from the suppliers are consolidated and shipped together. There are inventory holding costs at all the facilities and transportation costs for deliveries from the suppliers to the warehouse and from the warehouse to the customer. The objective is to ﬁnd a joint cyclic production and delivery schedule over an inﬁnite planning horizon to minimize the total production, inventory and transportation cost per unit time.

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Supplier 1

Supplier 2 Warehouse

Customer

Supplier m Fig. 1. A two-stage supply chain.

We deﬁne the following notation: m : number of suppliers. i : supplier and product index, i ¼ 1; . . . ; m (Supplier i produces product i). Di : customer demand rate for product i, for i ¼ 1; . . . ; m. pi : unit processing time of product i, for i ¼ 1; . . . ; m. si ; Si : setup time and setup cost per production batch at supplier i, respectively, for i ¼ 1; . . . ; m. hsi ; hwi ; hci : unit inventory holding cost for product i at supplier i, at the warehouse, and at the customer, respectively, for i ¼ 1; . . . ; m. Ai : transportation cost per delivery batch from supplier i to the warehouse, for i ¼ 1; . . . ; m. Aw : transportation cost per delivery batch from the warehouse to the customer. For ease of presentation, we assume that both the delivery time from a supplier to the warehouse and that from the warehouse to the customer are negligible, and hence they are set to zero. This assumption can be easily relaxed. At each supplier, we have pi Di p1 in order to satisfy the customer demand subject to the capacity constraint (the inequality is strict unless the setup time is zero). We also assume that the unit inventory holding cost of a product at the customer is the highest whereas that at the supplier is the lowest, i.e. hsi phwi phci , for i ¼ 1; . . . ; m. This assumption reﬂects the situation in many supply chains where the customers (e.g. retailers) are located at the most populated areas and hence have the highest unit inventory holding cost because of the tight space limit, whereas the suppliers (e.g. plants) are located at places with very low holding costs. Given these parameters, we need to ﬁnd a joint cyclic production and delivery schedule, which is equivalent to ﬁnding the following cycle times and the relative positions of these cycles: T i : time between successive production setups at supplier i, for i ¼ 1; . . . ; m. Ri : time between successive deliveries from supplier i to the warehouse, for i ¼ 1; . . . ; m. Rw : time between successive deliveries from the warehouse to the customer. Since the schedules are cyclic, in each production or delivery cycle, exactly the same amount will be produced

or delivered. Consequently, for i ¼ 1; . . . ; m, in each production cycle at supplier i, T i Di units of product i need to be produced, and in each delivery cycle from supplier i to the warehouse, Ri Di units of product i need to be delivered. Clearly, there is only one product involved in a production batch at each supplier and in a delivery from a supplier to the warehouse. However, all the m products are included in a delivery from the warehouse to the customer. That is, in each delivery cycle from the warehouse to the customer, Rw D1 units of product 1, Rw D2 units of product 2, . . ., and Rw Dm units of product m need to be delivered. It can be easily shown that an optimal cyclic schedule to our model satisﬁes the property: T i XRi XRw , for i ¼ 1; . . . ; m: This implies that there is no need to consider schedules in which the delivery cycle time from the warehouse to the customer is greater than that from a supplier to the warehouse or the delivery cycle time from a supplier to the warehouse is greater than the production cycle time at that supplier. Based on this observation, we consider the following two policies: (i) Production cycle time at each supplier is the same as the delivery cycle time from that supplier to the warehouse, i.e. T i ¼ Ri , for i ¼ 1; . . . ; m. (ii) Production cycle time at each supplier is an integer multiple of the delivery cycle time from that supplier to the warehouse, and the delivery cycle time from a supplier to the warehouse is an integer multiple of the delivery cycle time from the warehouse to the customer, i.e. T i ¼ Msi Ri and Ri ¼ M wi Rw , for i ¼ 1; . . . ; m, where M s1 ; . . . ; M sm and Mw1 ; . . . ; M wm are all positive integers. As we will see later, these policies are similar to some commonly considered policies for similar models in the literature. We note that an optimal cyclic schedule that satisﬁes policy (i) or (ii) may not be optimal to our model. However, schedules that satisfy these policies are easier to implement in practice than those that satisfy the property: T i XRi XRw , for i ¼ 1; . . . ; m, but not these policies. The remainder of this paper is organized as follows. We review related literature in Section 2 where we highlight the differences between our model and results and existing models and results in the literature. We then study our model under policy (i) and that under policy (ii) in Sections 3 and 4, respectively. In Section 3, we will show that in an optimal cyclic schedule under policy (i), the delivery cycle time from each supplier to the warehouse is an integer multiple of the delivery cycle time from the warehouse to the customer, i.e. Ri ¼ M wi Rw , for i ¼ 1; . . . ; m and some positive integers M w1 ; . . . ; Mwm . We will show that for the model with a single supplier, an optimal cyclic schedule can be obtained by closed-form formulas. For the model with multiple suppliers, we propose a heuristic and evaluate the performance of the heuristic computationally. In Section 4, we propose and computationally evaluate a heuristic for the problem under policy (ii) with multiple suppliers. We then

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evaluate the value of warehouse in our two-stage supply chain by comparing this supply chain with a single-stage supply chain without the warehouse. Managerial insights are derived based on an extensive set of computational tests. It is conceptually well-understood that a warehouse plays an important role in a supply chain; it consolidates different products and positions the inventory closer to customers, and hence saves on transportation and inventory costs. Our study here quantiﬁes these beneﬁts for the supply chain we consider. In addition, our results show that the suppliers’ production rates, among other parameters, have a signiﬁcant impact on the delivery frequency from the suppliers to the warehouse. This means that our model which has a ﬁnite production rate cannot be approximated by any of the existing multistage models with an inﬁnite production rate (reviewed in Section 2.2) or by the joint replenishment problem (JRP) (reviewed in Section 2.3). Finally, we conclude the paper in Section 5. 2. Literature review Our model is related to some existing production–distribution models, some existing multistage lotsizing models for assembly systems, and the JRP. We provide a review on those models, respectively, in the following subsections. 2.1. Production–distribution models Many of the so-called production–distribution models in the literature oversimplify production operations by assuming that production can be done instantaneously without production time or/and that the production capacity is unlimited. Most of these models study joint inventory replenishment decisions across the supply chain without considering explicit production decisions. The models that do consider production time and capacity constraints can be divided into two broad classes. One class of models (e.g. Dogan and Goetschalckx, 1999; Sabri and Beamon, 2000; Jayaraman and Pirkul, 2001; Kaminsky and Simchi-Levi, 2003) deals with dynamic demand patterns over a ﬁnite planning horizon and seeks to ﬁnd a joint dynamic production and distribution schedule at a minimum total cost over the planning horizon. It is unlikely that closed-form optimal solutions exist for this class of models. Often, mathematical programming based solution approaches are used. The other class of models assumes constant production and demand rates and an inﬁnite planning horizon, and seeks to ﬁnd a joint cyclic production and delivery schedule at a minimum total cost per unit time. Closed-form optimal solutions may be available for this class of models under some policies on the relationship between the production and delivery cycles. See Chen (2004, 2009) for comprehensive reviews on production–distribution models involving explicit production decisions. Since the model we study in this paper belongs to the second class of models mentioned here, we provide a review of the related literature in this area below.

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All existing models in this area involve a single-stage supply chain consisting of one or more suppliers producing products and one or more customers ordering products directly from the supplier(s) without going through a warehouse. Most models are variations of the one-supplier–one-customer model where a single product is produced at a single supplier and delivered directly from the supplier to the customer and the production, transportation and inventory characteristics are the same as in our model. Most models are concerned with ﬁnding an optimal cyclic schedule from a given class of policies. The following two classes of policies, which are similar to our policies (i) and (ii) deﬁned in Section 1, are commonly considered: (a) production cycle time and delivery cycle time are identical, and (b) production cycle time is an integer multiple of delivery cycle time. Hahm and Yano (1992) consider the one-supplier–onecustomer model mentioned above. They assume that the unit inventory holding cost at the supplier is the same as that at the customer. They show that production and delivery cycles in the optimal solution satisfy policy (b) and formulate the problem as a nonlinear mixed integer program which is solved by a heuristic approach. They also consider the problem where the delivery cost per order is dependent on the order size. Benjamin (1989) studies the same problem except thatthe inventory cost at the supplier is calculated differently. The problem studied by Hahm and Yano (1992) is extended by Hahm and Yano (1995a–c) to include multiple products, each with a constant production and demand rate. The ﬁrst paper considers policy (a), and the second and the third ones consider policy (b). Jensen and Khouja (2004) consider the same problem studied by Hahm and Yano (1995a). Khouja (2000) considers the same problem studied by Hahm and Yano (1995a) with the added model characteristic that the production rates are decision variables and the product quality depends on the production rates and lot sizes. Single-stage models with multiple suppliers or/and multiple customers are studied by Benjamin (1989), Blumenfeld et al. (1985, 1991), and Hall (1996). Benjamin (1989) considers a model with multiple suppliers and multiple customers where only a single product is involved. Blumenfeld et al. (1985) consider various delivery options from suppliers to customers including direct shipping, shipping via a consolidation terminal, and a combination of terminal and direct shipping. Problems with one or multiple suppliers and one or multiple customers are considered under various assumptions. Blumenfeld et al. (1991) study a model with one supplier and multiple customers where the supplier produces multiple products, one for each customer. Hall (1996) considers various scenarios: one or more suppliers, one or more customers, one or more machines at each supplier, and one or more products that can be processed by each machine. He derives the cost formulas for many scenarios under policy (a). As in our model, in all the models reviewed above, there are a setup time and a setup cost associated with each production cycle. A set of similar single-stage models but with no production setup times are studied by Goyal

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(1988), Aderohunmu et al. (1995), Hill (1999), and Chan and Kingsman (2007), among others. In summary, our model is more complex and more general in structure than the models considered in the above-reviewed literature because our model involves a two-stage supply chain whereas all of the existing models involve a single-stage supply chain. Furthermore, as discussed later, the study of this two-stage supply chain enables us to evaluate the value ofwarehouse in the supply chain and obtain related managerial insights.

2.2. Multistage lotsizing models The structure of our model may be viewed as a multistage assembly system if we view the warehouse as an assembly stage. In this context, the production (i.e. assembly) at the warehouse would be instantaneous because it does not really assemble the products; it merely puts all the products together for joint delivery to the customer. Therefore, our model may be viewed as a special lotsizing model for a multistage assembly system. In the following, we compare our model and solution approaches with existing ones in the area of lotsizing for multistage assembly systems with an inﬁnite planning horizon and constant demand. First of all, to our knowledge, none of the existing multistage lotsizing models with an inﬁnite planning horizon and constant demand explicitly consider delivery from stage to stage (i.e. products are transferred from stage to stage at zero cost), and none of them consider production setup times and hence production capacity constraints due to setup times. To see other differences, we consider existing models with a ﬁnite production rate at each facility separately from existing models with an inﬁnite production rate (i.e. instantaneous production) at each facility. Comparing to the existing models with an inﬁnite production rate at each facility (Crowston et al., 1973; Blackburn and Millen, 1982; Moily and Matthews, 1987), our model has different and more complex inventory functions at the suppliers because the production rates at the suppliers in our model are ﬁnite, which leads to the requirement of inventory accumulation prior to each delivery to the warehouse. Given this and the fact that there are capacity constraints in our model but not in those existing models, the solution approaches used in those papers cannot be applied to our model. All the existing models with a ﬁnite production rate at each facility (Schwarz and Schrage, 1975; Moily, 1986; Atkins et al., 1992) assume that the production rates are non-increasing across the system (i.e. from components to ﬁnal products), whereas this assumption does not hold in our model if our model is viewed as a multistage assembly system. Crowston et al. (1973) show the property that under this assumption and without the capacity constraint due to setup times, the lot size at each facility is an integer multiplier of that at each immediately succeeding facility. This property is similar to one of the results we prove in this paper (see Section 3.1). However, our result is proved without this assumption and with the capacity constraint. Schwarz and Schrage (1975) use this property

to formulate the problem as an integer program. They propose a branch-and-bound algorithm for getting optimal solutions and a heuristic procedure that optimizes the system as a collection of two-stage systems by ignoring multistage interaction effects. In addition to the nonincreasing-production-rates assumption, the models studied by Moily (1986) and Atkins et al. (1992) assume that the product is transferred from one stage to the next immediately and continuously upon its completion, whereas in our model all the units in a delivery shipment are transferred together. Because of this difference, how inventory accumlates and hence the inventory function at various facilities in our model are different from the models considered in these existing papers. Atkins et al. (1992) derive some theoretical results for which the nonincreasing-production-rates assumption is a key. The solution approach used by Moily (1986) is different from the approach we use. His approach is based on a one-time rounding of any non-integer multipliers obtained without taking into account its effects on the other participants of the system.

2.3. Joint replenishment problem Our model is also related to the extensively studied JRP. The JRP is an inventory problem concerning a customer ordering multiple products from a single supplier where economies exist for replenishing several products together. There are several versions of JRP considered in the literature. In the classical version of JRP, the customer’s demand rate for each product is constant over an inﬁnite planning horizon, and each order placed by the customer incurs a major setup cost independent of the number of products ordered and a minor setup cost per product ordered. The problem is for the customer to develop an ordering policy that minimizes the total setup cost and inventory holding cost per time unit. See Aksoy and Erenguk (1988), and Goyal and Satir (1989) for surveys of existing results up to 1989, Khouja and Goyal (2008) for a survey of existing results between 1989 and 2005, and Bayindir et al. (2006), Moon and Cha (2006), and Nilsson and Silver (2008), among others, for most recent results. The other versions of JRP that have been studied in the literature involve dynamic or stochastic demand. In the dynamic demand version, the demand is deterministic but not constant over a planning horizon with a ﬁnite number of periods and the objective is to minimize the total cost over all the periods, whereas in the stochastic demand version, the demand is stochastic but stationary in the mean and the objective is to minimize the total expected cost per time unit. See the survey paper by Khouja and Goyal (2008) for results up to 2005, and Robinson et al. (2007), and Viswanathan (2007), among others, for more recent results. It can be seen that the stage from the warehouse to the customer in our model has a similar structure to JRP. However, there are three major differences between our model and JRP. First of all, our model consists of two stages whereas JRP has only one stage. Second, JRP is a pure inventory problem without involving production

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operations, whereas our model considers production capacity constraints and production setup times. As it is shown later in our computational study, the production capacity has a signiﬁcant impact on the solution to our problem. Third, there is no minor setup cost associated with each delivery in our model. In our model, each delivery incurs a ﬁxed transportation cost, regardless of the number of products carried. This is easily justiﬁed as in many practical situations, the transportation cost for a shipment is indeed independent of the number of products carried by the shipment. Our requirement for a cyclic schedule implies that in our model, each delivery from the warehouse to the customer carries a certain amount of each of the m products, whereas in JRP, a shipment from the supplier to the customer may only carry a subset of the products. These differences mean that our problem is different from JRP and cannot be solved as JRP.

denoted as DC, are given below: SC ¼

DC ¼

Theorem 1. In an optimal cyclic schedule under policy (i), the delivery cycle time from each supplier to the warehouse is an integer multiple of the delivery cycle time from the warehouse to the customer, i.e. Ri ¼ Mwi Rw for some positive integer M wi, for i ¼ 1; . . . ; m.

where the ﬁrst, second, and third terms correspond to the average inventory cost per unit time at the suppliers, at the warehouse, and at the customer, respectively. The average total production setup cost per unit time, denoted as SC, and the average total distribution cost per unit time,

m X Ai

Ri

¼

m X Si i¼1

þ

(2)

Ri

Aw Rw

(3)

Therefore, the average total cost per unit time, denoted as TC, is given as TC ¼ IC þ SC þ DC

! ! m m 1 X 1 X 2 þ h pD R þ h D ðR Rw Þ ¼ R 2 i¼1 si i i i 2 i¼1 wi i i i¼1 i ! m m X 1 X Ai Aw hci Di Rw þ þ þ R Rw 2 i¼1 i¼1 i m X Si

¼

m X ðSi þ Ai Þ i¼1 m X i¼1

Ri

þ

m X

ai Ri þ bRw þ

i¼1

Aw Rw

m Qi X A þ ai Ri þ bRw þ w Ri Rw i¼1

(4)

ai ¼ ðhsi pi Di þ hwi ÞDi =2, and where Q i ¼ Si þ Ai , P b¼ m ðh h ÞD =2. ci wi i i¼1 Our objective is to ﬁnd the values for Rw and R1 ; . . . ; Rm that minimizes TC subject to the production constraint. The production constraint requires that the production cycle time T i at each supplier i is sufﬁcient to produce the required quantity along with the setup time, i.e. si þ pi Di T i pT i . This means that T i Xti , where ti ¼ si =ð1 pi Di Þ, or equivalently Ri Xti, as Ri ¼ T i under policy (i). So we can formulate our problem under policy (i) as follows: Minimize

TC

(5)

Subject to

Ri Xti ; 8i 2 f1; . . . ; mg Ri is a positive integer; 8i 2 f1; . . . ; mg Rw Rw X0

(6)

Proof. See Appendix. By Theorem 1, we can focus on schedules where Ri =Rw is integer valued for each supplier i ¼ 1; . . . ; m. In the following, we derive the average total cost per unit time in such a schedule. We have seen in the proof of Theorem 1 how to calculate the average inventory costs at the supplier and at the warehouse for the single-supplier case when R=Rw is integer valued. Extending that to the case with multiple suppliers, we get the following equation for the average total inventory cost per unit time, denoted as IC: ! ! m m 1 X 1 X hsi pi D2i Ri þ hwi Di ðRi Rw Þ IC ¼ 2 i¼1 2 i¼1 ! m X 1 h D Rw (1) þ 2 i¼1 ci i

Ti

i¼1

¼

3.1. An optimality property

m X Si i¼1

3. The model with T i ¼ Ri In this section we consider the model under policy (i) which requires that T i ¼ Ri for i ¼ 1; . . . ; m. We ﬁrst prove an optimality property and derive the various cost components for this model in Section 3.1. We then give an optimal solution to the case with a single supplier in Section 3.2, and propose a heuristic for the case with multiple suppliers and evaluate its performance in Section 3.3.

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(7) (8)

The average total cost per unit time TC given in Eq. (4) is a separable function of the variables Rw and R1 ; . . . ; Rm , and corresponding to each variable, the function is convex. Hence without constraint (7), the above formulation can be solved optimally using the ﬁrst order Karush–Kuhn–Tucker (KKT) conditions. But the presence of the integrality constraint makes this problem more complicated. In the next two sections, we will show a way to obtain the optimal solution in the case of a single supplier and propose a heuristic for the multiple-supplier case. 3.2. Optimal solution for the single-supplier case In this section, we derive the optimal solution in the case where there is only one supplier. For simplicity, we drop the supplier subscript from our notation. The total cost given in (4) can be rewritten as TC ¼

Q Aw þ aR þ bRw þ Rw R

(9)

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where Q ¼ S þ A, a ¼ ðhs pD þ hw ÞD=2, and b ¼ ðhc hw ÞD=2. Our objective is to ﬁnd the values of R and Rw that minimize TC subject to the constraints that RXt and R=Rw is a positive integer, ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ppDÞ. pﬃﬃﬃﬃﬃﬃﬃﬃﬃwhere t ¼ s=ð1 Deﬁne R0 ¼ maxf Q =a; tg and R0w ¼ Aw =b. If we ignore the integrality constraint, it can be shown by the ﬁrst order KKT conditions that the optimal solution to this problem is given by R ¼ R0 and Rw ¼ R0w . Let M 0w ¼ R0 =R0w . Then we have the following result: Theorem 2. Let R and Rw be the optimal values of R and Rw for the single-supplier problem under policy (i). Then, (sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ) Q þ M w Aw t ; (10) Rw ¼ Max M w ðaM w þ bÞ M w R ¼ M w Rw

(11)

where M w 2 fmaxfbM 0w c; 1g; dM 0w eg. Proof. See Appendix. By Theorem 2, if MaxfbM0w c; 1g ¼ dM 0w e, then the optimal solution is uniquely deﬁned by (10) and (11). Otherwise, we only need to compare two solutions, one with M w ¼ MaxfbM 0w c; 1g and the other with M w ¼ dM0w e, and the one with a lower objective value TC is the optimal solution to the problem. 3.3. A heuristic solution for the multiple-supplier case We ﬁrst give an alternative representation for the problem deﬁned in (5)–(8), where we substitute the variables Ri by Mwi Rw, for i ¼ 1; . . . ; m: Minimize

m X i¼1

Subject to

m X Qi A þ a M Rw þ bRw þ w M wi Rw i¼1 i wi Rw

M wi Rw Xti ;

8i 2 f1; . . . ; mg

M wi is a positive integer; Rw X0

(12) (13)

8i 2 f1; . . . ; mg (14) (15)

The heuristic in this section tries to ﬁnd a near optimal solution ðM w1 ; . . . ; M wm ; Rw Þ to the above formulation. Since there are multiple suppliers involved, the choice of M wi at one supplier can inﬂuence the choice of M wi at another supplier. If we just try two values of M wi at each supplier as in the optimal solution for the single-supplier case shown in the previous subsection, the resulting solution is unlikely to be optimal or even locally optimal. Our heuristic keeps trying different values of Mwi ’s for the suppliers until a locally optimal solution is found. More speciﬁcally, in each iteration, the heuristic chooses one supplier and ﬁxes the M wi values for all the other m 1 suppliers. Then it ﬁnds the values for Rw ; Ri, and M wi for the chosen supplier using an approach similar to the one used for the single-supplier problem. If the resulting total cost is lower, then the value of Mwi for the chosen supplier is updated and ﬁxed in the next several iterations along with the Mwi values at ðm 2Þ other suppliers. The procedure stops when no improvement is found in one round of iterations across all the suppliers.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Heuristic Set R0i ¼ Maxf Q i =ai ; ti g, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H1. 0 Step 0 1:0 0 Rw ¼ Aw =b, M wi ¼ Ri =Rw , for i ¼ 1; . . . ; m. Let j ¼ 1 be the index of the supplier to be considered next. Set the iteration counter c ¼ 0, and the non-improvement counter n ¼ 0. Set the total cost TC 0 ¼ 1. Step 2: Set c ¼ c þ 1. For supplier j, let M 0wj ¼ R0j =R0w where ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8v 9 uP > > Qi >u > m > > ( ) > > i¼1;iaj 0 þ Aw > M wi > > i¼1;iaj ai M wi þ b > > > > : ; (16) ¯ wj ¼ Set M follows:

maxfbM 0wj c; 1g

and get the corresponding R¯ w as

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8v u P > Qj Qi u > m > u þ Aw >u i¼1;iaj 0 þ < ¯ wj M tj M wi ; R¯ w ¼ Max tPm ; 0 ¯ > ¯ M M a M þ a þ b > wj j wj i¼1;iaj i wi > > : 9 > > ( )> > = ti ; 8iaj max 0 > Mwi > > > ;

(17)

Calculate the total cost TC of the solution ðM w1 ; . . . ; ¯ wj , M wi ¼ M 0wi for iaj, and M wm ; Rw Þ with M wj ¼ M ¯ wj ¼ dM 0wj e and get the correRw ¼ R¯ w . Similarly, set M sponding R¯ w and the total cost, TC, of the corresponding solution. Choose the solution with a lower total cost. Let the total cost of this solution be TC j . ¯ wj correspondStep 3: If com, let M 0wj be equal to the M ing to the solution generated in Step 2. If cXm and ¯ wj TC j oTC 0 , let TC 0 ¼ TC j and M 0wj be equal to the M corresponding to the solution generated in Step 2, and reset n ¼ 0. If cXm and TC j XTC 0 , let n ¼ n þ 1. Step 4: If n ¼ m, there has been no improvement for any supplier in the last m iterations, and hence STOP. Otherwise, if j ¼ m, set j ¼ 1, else set j ¼ j þ 1. Go to Step 2. In the ﬁrst m iterations, the heuristic ﬁnds an integer solution for the variables Mwj at each supplier. After these iterations, the heuristic tries to improve the existing feasible solution, choosing one supplier at a time. By the ﬁrst order KKT conditions, it can be shown that the values R01 ; . . . ; R0m and R0w deﬁned in Step 1 of the heuristic are optimal to the problem (5)–(8) without the integrality requirement (7). Hence the values M 0w1 ; . . . ; M0wm and R0w deﬁned in Step 1 are optimal to the problem (12)–(15) if the integrality constraint (14) is relaxed. Similarly, under the condition that M wi is ﬁxed as M 0wi for all iaj, it can be shown that the solution ðM 0wj ; R0w Þ deﬁned in Step 2 is optimal to the remaining problem (12)–(15) with the integrality constraint relaxed. Since M 0wj may not be an integer, in Step 2, two integer solutions of M wj rounded from M 0wj are evaluated. It can be easily veriﬁed that if M 0wi

ARTICLE IN PRESS G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

is an integer for all iaj, then the solution generated in Step 2 is feasible to the problem (12)–(15). Before evaluating the performance of the heuristic H1 computationally, we derive a lower and an upper bound on the M wi ’s in an optimal solution to the problem (12)–(15). For i ¼ 1; . . . ; m, deﬁne Y i ¼ ai R0i þ Q i =R0i , where R0i is deﬁned in Step 1 of the heuristic. Clearly, Ri ¼ R0i is the optimal solution and Y i is the optimal objective value of the problem minfai Ri þ Q i =Ri jRi Xti g. Let Z H1 denote the objective value of the solution obtained by heuristic H1 for the problem (12)–(15). Then in any optimal solution of the problem (12)–(15), we have

bRw þ Aw =Rw pZ H1

m X

Yi

i¼1

Since the left-hand side of the above inequality is a convex function of Rw , it implies that in any optimal solution of the problem (12)–(15), Rw 2 ½RLw ; RU w , where rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 P Pm H1 Z H1 m Y Y 4 b A Z w i¼1 i i¼1 i RLw ¼ 2b ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 P Pm H1 Z H1 m Y Y 4 b A þ Z w i¼1 i i¼1 i RU w ¼ 2b

bR0w

Aw =R0w ,

R0w

Deﬁne Y w ¼ þ where is deﬁned in Step 1 of the heuristic. Clearly Y w is the optimal objective value of the problem minfbRw þ Aw =Rw g. In any optimal solution of the problem (5)–(8), we have

ai Ri þ Q i =Ri pZ H1

m X

Yj þ Yi Yw

for i ¼ 1; . . . ; m

j¼1

Since the left-hand side of the above inequality is a convex function of Ri , it implies that in any optimal for solution of the problem (5)–(8), Ri 2 ½RLi ; RU i , i ¼ 1; . . . ; m, where

61

given the values of M wi for i ¼ 1; . . . ; m, the optimal value of Rw is given by ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 9 8v uP > > m Qi > > >u > þ Aw u i¼1 = < ti t M wi Rw ¼ Max Pm ; 8i 2 f1; . . . ; mg ; max > > M wi > > i¼1 ai M wi þ b > > ; : (18) Since the enumerative approach for generating the optimal solution is computationally very demanding, we only test problems with two and four suppliers. In all the test problems, the following parameters are kept constant: demand rate Di ¼ 10 units per time unit, setup time si ¼ 1, and setup cost Si ¼ 100, for i ¼ 1; . . . ; m. We test two production rates at the suppliers as follows: (i) one with lower production rates where each pi is uniformly generated from the interval ½0:02; 0:08; and (ii) the other with higher production rates where each pi is uniformly generated from the interval ½0:002; 0:008. So, the higher production rate considered is on average 10 times that of the lower production rate. The other parameters involved are generated randomly as follows for i ¼ 1; . . . ; m. Unit holding cost of product i at supplier i, hsi ¼ hs 2 f0:1; 1; 10g 8i. Unit holding cost of product i where at the warehouse hwi ¼ hw ¼ g1 hs ; 8i, g1 2 f1; 5; 10g. Unit holding cost of product i at the customer hci ¼ hc ¼ g2 hw ; 8i, where g2 2 f1:1; 5; 10g. The transportation cost per delivery Ai ¼ F i þ V i and Aw ¼ F w þ V w where F i and F w represent ﬁxed costs and V i and V w variable costs determined by the corresponding travel distances. The ﬁxed cost components F i ¼ F w ¼ rF 0 ; 8i and w, where the transportation cost index F 0 2 f1; 10; 100g and the ﬁxed cost factor r 2 f0:01; 0:1; 1g. Thus a value of 0:01 for the ﬁxed cost factor implies very low ﬁxed cost, while a

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 9 r 2 P Pm > > H1 > > Y þ Y Y Y þ Y Y 4 a Q Z < Z H1 m = w w j i j i i i j¼1 j¼1 RLi ¼ max ; ti > > 2ai > > : ; r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 9 2 Pm P > > > 4ai Q i > Z H1 m < Z H1 j¼1 Y j þ Y i Y w þ = j¼1 Y j þ Y i Y w U ; ti Ri ¼ max > > 2 a i > > : ;

Based on the above-derived lower and upper bounds of Rw and R1 ; . . . ; Rm , we can conclude that in any optimal solution to the problem (12)–(15), the value of M wi is L L U within the interval ½M Lwi ; M U wi , where M wi ¼ dRi =Rw e, and U L ¼ bR =R c. MU wi i w Now we are ready to test the performance of the heuristic H1. We compare the solution generated by the heuristic with the optimal solution obtained through an enumerative approach which enumerates all possible positive integer values of M wi within its lower and upper bounds ½MLwi ; MU wi , for i ¼ 1; . . . ; m, and for each possible combination of the values of ðMw1 ; . . . ; M wm Þ, solves the rest of the problem with one variable Rw . We note that

value of 1 implies relatively high ﬁxed cost. The variable cost components V i and V w are generated as follows. We assume that the suppliers are symmetrically arranged along a vertical line above and below a horizontal line through the customer so that the distance between neighboring suppliers is 0:2 units, and the horizontal distance from each supplier to the customer is one unit. For example, if there are two suppliers, then the ðx; yÞ coordinates of the two suppliers are fð0; 0:1Þ; ð0; 0:1Þg while those of the customer are ð1; 0Þ. We consider three cases of the warehouse location: the warehouse is located at ðx; 0Þ, where x 2 f0:2; 0:5; 0:8g. The distance between any two

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locations is calculated as the Euclidean distance. The variable cost V i and V w are calculated as the product of the transportation cost index F 0 and the Euclidean distance between the respective origin and destination. For a given problem instance, while all the suppliers have the same holding costs, setup costs, and ﬁxed transportation costs, they differ in their processing times (and hence capacity) and the variable transportation costs. We calculate the relative percentage gap between the total cost of the heuristic solution and that of the optimal solution. The relative gap (%) is deﬁned as ðZ H1 Z Þ=Z 100%, where Z H1 is the objective of the solution provided by the heuristic H1 and Z is the optimal objective value. For each combination of parameters (pi ; hs ; g1 ; g2 ; F 0 ; r; x), we test ﬁve different random instances. Table 1 shows the results of our computational tests. Due to space restrictions, we aggregate the results corresponding to the two different ranges of production rates (pi ), three different values of the holding costs at the supplier (hs ), and the three different ﬁxed cost factors (r). We provide both the average and maximum gaps for each entry corresponding to a combination of (g1 ; g2 ; F 0 ; x). Thus each entry corresponds to 90 different problem instances. The overall average gap has a very small value of 0:18%. The average gap with two suppliers is 0:13% while that with four suppliers is 0:23%. The maximum among all the random test instances is 7:83%. The heuristic is very fast. In most cases, the convergence was achieved within one round of iterations between all the suppliers. In none of the cases, the CPU time was more than one second. Therefore, we can conclude that the heuristic is capable of generating near optimal solutions quickly for almost all the problems tested. In general, the performance of the heuristic improves as we increase the value of g2 , the holding cost multiplier for the customer. This is intuitive since when the holding cost at the customer increases, the frequency of deliveries from the warehouse to the customer relative to the frequency of delivery from the suppliers to the warehouse goes up. In other words, the values of Mwi ’s increase. Hence the difference in the solution value for a small deviation from the optimal value for M wi ’s would be small. Also, when we increase the value of x (i.e. move the warehouse closer to the customer), the gap in general decreases. The reason for this is the same as the earlier one. When the warehouse iscloser to the customer, the values of M wi ’s increase and the heuristic performs better. In summary, the heuristic seems to perform very well for most of the cases tested. 4. The model with T i ¼ M si Ri and Ri ¼ M wi Rw In this section, we study the model under policy (ii) which requires that T i is an integer multiple of Ri and Ri is an integer multiple of Rw , for i ¼ 1; . . . ; m. It should be noted that when multiple deliveries per production cycle are allowed at the suppliers, Theorem 1 does not necessarily hold. However, Policy (ii) requires a feasible schedule to satisfy that theorem, i.e. the delivery cycle time from each supplier to the warehouse has to be an

integer multiple of the delivery cycle time from the warehouse to the customer. We ﬁrst formulate the problem under policy (ii) as a mathematical program. The inventory calculations at the warehouse and the customer remain the same as before (see (1) and the explanations thereafter), i.e. the average inventory cost per unit time at the warehouse is Pm 1 2 ðPi¼1 hwi Di ðRi Rw ÞÞ and that at the customer is m 1 i¼1 hci Di ÞRw . The inventory calculations at the suppliers 2ð are more complicated. Fig. 2 shows an example inventory cycle for a supplier i where T i ¼ 3Ri . When T i is an integer multiple of Ri , the average inventory at the supplier can be shown to be 12 ð1 pi Di ÞDi T i þ ðpi Di 12ÞDi Ri . For details on the derivation, refer to Hahm and Yano (1992). The average total cost per time unit TC is thus given as follows: TC ¼

m X Si i¼1

Ti

þ

m X

ai T i þ

m X Ai

i¼1

i¼1

Ri

þ

m X

bi R i þ

i¼1

Aw þ gRw Rw

(19)

where ai ¼ 12 ð1 pi Di ÞDi hsi , bi ¼ ðhsi ðpi Di 12Þ þ hwi =2ÞDi , P and g ¼ 12 m i¼1 ðhci hwi ÞDi . Our problem under policy (ii) can be formulated as follows: Minimize Subject to

TC T i Xti ; 8i 2 f1; . . . ; mg Ti is a positive integer; 8i 2 f1; . . . ; mg Ri Ri is a positive integer; 8i 2 f1; . . . ; mg Rw Rw X0

(20) (21) (22) (23) (24)

We propose a heuristic in Section 4.1 to solve this problem, and use this heuristic to study the value of warehouse in Section 4.2. Some other insights obtained from our computational experiments are provided in Section 4.3. 4.1. A heuristic solution The idea of the heuristic is similar to that of heuristic H1 proposed in Section 2. The formulation (20)–(24) can be written as follows with Ms1 ; . . . ; Msm , M w1 ; . . . ; Mwm , and Rw as the decision variables: Minimize

m X i¼1

m m X X Si Ai þ a M M Rw þ M si M wi Rw i¼1 i si wi M wi Rw i¼1

þ

m X

bi Mwi Rw þ

i¼1

Subject to

Aw þ gRw Rw

M si Mwi Rw Xti ; 8i 2 f1; . . . ; mg M si and M wi are positive integers,

(25) (26)

8i 2 f1; . . . ; mg

(27)

Rw X0

(28)

The heuristic tries to ﬁnd a near optimal solution ðM s1 ; . . . ; M sm ; M w1 ; . . . ; M wm ; Rw Þ to the formulation (25)–(28) through the following iterative procedure: In each iteration, the heuristic chooses one supplier and ﬁxes the M si and M wi values for all the other m 1 suppliers. Then it solves the remaining problem with three decision variables Rw ; Msi , and M wi for the chosen supplier using an

ARTICLE IN PRESS G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

63

Table 1 Average and maximum relative gaps (%) between the optimal solution and the solution provided by heuristic H1.

g1

g2

F0

m¼2

m¼4

Average gap

1

1.1

5

10

5

1.1

5

10

10

1.1

5

10

Maximum gap

Maximum gap

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

1 10 100 1 10 100 1 10 100

0.22 0.45 0.00 0.01 0.05 0.03 0.01 0.02 0.02

0.14 0.41 0.05 0.02 0.04 0.00 0.01 0.01 0.01

0.12 0.14 0.23 0.02 0.03 0.01 0.01 0.02 0.01

1.77 4.87 0.00 0.16 0.50 0.63 0.10 0.27 0.60

1.81 4.09 1.55 0.23 0.82 0.06 0.12 0.26 0.13

1.36 1.80 4.32 0.26 0.27 0.13 0.13 0.20 0.22

0.25 0.58 0.67 0.04 0.07 0.06 0.02 0.04 0.04

0.20 0.47 0.91 0.03 0.05 0.03 0.02 0.04 0.02

0.12 0.43 0.66 0.03 0.04 0.03 0.01 0.01 0.02

1.34 3.42 4.40 0.23 0.42 0.72 0.13 0.29 0.47

1.30 3.33 6.24 0.17 0.33 0.48 0.10 0.27 0.23

0.86 3.54 4.96 0.17 0.24 0.31 0.11 0.11 0.23

1 10 100 1 10 100 1 10 100

0.20 0.32 0.69 0.03 0.06 0.07 0.02 0.03 0.03

0.15 0.36 0.43 0.02 0.05 0.04 0.02 0.04 0.03

0.14 0.24 0.49 0.02 0.04 0.02 0.01 0.02 0.03

1.26 2.29 4.93 0.18 0.52 0.80 0.11 0.21 0.33

0.67 2.00 5.13 0.14 0.35 0.68 0.09 0.28 0.25

1.12 2.06 4.39 0.15 0.27 0.40 0.08 0.16 0.28

0.35 0.60 1.14 0.03 0.08 0.07 0.02 0.04 0.05

0.21 0.50 1.40 0.03 0.05 0.05 0.03 0.03 0.04

0.23 0.45 1.09 0.03 0.05 0.04 0.02 0.03 0.03

1.25 2.64 7.83 0.13 0.61 0.90 0.11 0.22 0.44

0.93 2.87 7.82 0.17 0.30 0.40 0.13 0.15 0.49

0.89 1.93 7.80 0.15 0.30 0.42 0.07 0.13 0.38

1 10 100 1 10 100 1 10 100

0.30 0.70 0.65 0.02 0.05 0.10 0.02 0.04 0.05

0.18 0.51 0.75 0.03 0.06 0.06 0.02 0.03 0.04

0.18 0.29 0.24 0.02 0.04 0.06 0.02 0.03 0.03

1.34 3.75 4.20 0.12 0.36 0.77 0.11 0.17 0.44

1.39 2.42 5.15 0.17 0.56 0.85 0.09 0.19 0.34

1.08 2.09 2.16 0.20 0.36 0.38 0.10 0.19 0.28

0.34 0.72 1.15 0.05 0.10 0.11 0.02 0.05 0.07

0.27 0.62 1.13 0.04 0.07 0.10 0.02 0.04 0.06

0.18 0.55 0.92 0.04 0.06 0.06 0.02 0.03 0.03

1.11 2.79 5.69 0.15 0.37 0.63 0.09 0.24 0.52

0.97 2.64 5.46 0.14 0.23 0.58 0.07 0.17 0.41

0.68 2.29 5.78 0.12 0.33 0.34 0.08 0.16 0.15

for i ¼ 1; . . . ; m. Set supplier index j ¼ 1, the iteration counter c ¼ 0, and the non-improvement counter n ¼ 0. Set the total cost TC 0 ¼ 1. ¯ sj ¼ Step 2: Set c ¼ c þ 1. For supplier j, let M maxfbM 0sj c; 1g. Let M 0wj ¼ R0j =R0w , where ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8v 9 u > > Sj >u > > > u þ Aj > > > aj M > > > > > > : ;

2RiDi Inventory

Average gap

RiDi

0 Ri

Ri Time

Ri

Fig. 2. Inventory level at supplier i when T i ¼ 3Ri .

approach similar to the one in heuristic H1. In solving this problem, we try out four different combinations of M si and M wi values for the chosen supplier, and the best solution is used. If the resulting total cost is lower, then the values of M si and M wi for the chosen supplier are updated and ﬁxed in the next several iterations. The procedure stops when no improvement is found in one round of iterations across all the suppliers. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 Si =ai ; ti g, Heuristic pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H2. Step ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃSet 0 T i 0¼ Maxf p1: 0 0 0 0 Ri ¼ Ai =bi , and Rw ¼ Aw =g, M si ¼ T i =Ri , M wi ¼ R0i =R0w ,

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 9 8v ! u u > > Si 1 > > > >uPm > > ( ) þ A þ A w > > i i¼1;iaj 0 0 = > Msi Mwi > > i¼1;iaj ðai M si þ bi ÞM wi þ g > > > > > > ; :

(30) ¯ wj ¼ Set M

maxfbM 0wj c; 1g,

and deﬁne R¯ w as follows:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 8v ! ! u uP > Sj Si 1 1 > n u > > þ Ai þ Aw þ þ Aj > ¯ ¯ M > i¼1;iaj ðai M si þ bi ÞM wi þ g þ ðaj M sj þ bj ÞM wj > > > : ( )) max

ti

M 0si M 0wi

; 8iaj

(31)

ARTICLE IN PRESS 64

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Calculate the total cost TC of the solution ðMs1 ; . . . ; ¯ sj , M wj ¼ M ¯ wj at supM sm ; Mw1 ; . . . ; M wm ; Rw Þ with M sj ¼ M plier j, M si ¼ M 0si and M wi ¼ M 0wi for iaj, and Rw ¼ R¯ w . Similarly, calculate the total costs of the other three ¯ wj Þ are deﬁned ¯ sj ; M solutions where only the values of ðM differently to be ðmaxfbM 0sj c; 1g; dM 0wj eÞ, ðdM 0sj e; max fbM 0wj c; 1gÞ, and ðdM 0sj e; dM0wj eÞ, respectively. Choose the solution with the lowest total cost. Let the total cost of this solution be TC j . ¯ sj and Step 3: If cpm, let M 0sj and M 0wj be equal to the M ¯ wj corresponding to the solution chosen in Step 2. If M cXm and TC j oTC 0 , let TC 0 ¼ TC j and let M 0sj and M 0wj be ¯ wj corresponding to the solution ¯ sj and M equal to the M chosen in Step 2, and reset n ¼ 0. If cXm and TC j XTC 0 , let n ¼ n þ 1. Step 4: If n ¼ m, there has been no cost improvement for any supplier in the last set of iterations, and hence STOP. Otherwise, if j ¼ m, set j ¼ 1, else set j ¼ j þ 1. Go to Step 2.

Similar to heuristic H1, in the ﬁrst m iterations, we ﬁnd an integer solution for the two variables ðM sj ; M wj Þ at each supplier. After these iterations, we try to improve the existing feasible solution, choosing one supplier at a time. We evaluate the performance of heuristic H2 by comparing the solution generated by it with the optimal solution obtained by an enumerative approach which enumerates all possible positive integer values of M si and M wi within their valid lower and upper bounds, for i ¼ 1; . . . ; m. We will discuss how to generate a valid lower and upper bounds for each of these variables in the paragraphs that follow. Given the values of these variables, the optimal value of Rw is given by RLj ¼

RU j ¼

Z H2

Pm

i¼1 ðU i

A lower and upper bound of each M variable in the formulation (25)–(28) can be derived in the same way as what we have done for the M variables in the formulation (12)–(15) in Section 2. For i ¼ 1; . . . ; m, deﬁne U i ¼ ai T 0i þ Si =T 0i , and V i ¼ bi R0i þ Ai =R0i , and deﬁne W ¼ gR0w þ Aw =R0w , where T 0i , R0i , and R0w are deﬁned in Step 1 of heuristic H2. Let Z H2 denote the objective value of the solution obtained by heuristic H2 for the problem (25)–(28). Then in any optimal solution of the problem, we have

gRw þ Aw =Rw pZ H2

m X ðU i þ V i Þ i¼1

bj Rj þ Aj =Rj pZ

H2

m X ðU i þ V i Þ þ V j W

aj T j þ Sj =T j pZ H2

i¼1 m X

ðU i þ V i Þ þ U j W

for j ¼ 1; . . . ; m for j ¼ 1; . . . ; m

i¼1

which imply that in any optimal solution of the problem L U L U (25)–(28), Rw 2 ½RLw ; RU w , Rj 2 ½Rj ; Rj and T j 2 ½T j ; T j , for j ¼ 1; . . . ; m, where

RLw ¼

RU w ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 P P Z H2 m Z H2 m 4gAw i¼1 ðU i þ V i Þ i¼1 ðU i þ V i Þ 2g rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 P P Z H2 m Z H2 m 4gAw i¼1 ðU i þ V i Þ þ i¼1 ðU i þ V i Þ 2g

Based on the above-derived lower and upper bounds of Rw , R1 ; . . . ; Rm , and T 1 ; . . . ; T m , we can conclude that in any optimal solution to the problem (25)–(28), the values of M si and M wi , for every i ¼ 1; . . . ; m, are within the interval

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ r 2 P þ V iÞ þ V j W Z H2 m 4bj Aj i¼1 ðU i þ V i Þ þ V j W 2bj

ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 P P Z H2 m Z H2 m 4bj Aj i¼1 ðU i þ V i Þ þ V j W þ i¼1 ðU i þ V i Þ þ V j W 2bj

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 9 r 2 Pm P > > H2 > Z H2 m 4aj Sj > < Z i¼1 ðU i þ V i Þ þ U j W = i¼1 ðU i þ V i Þ þ U j W L T j ¼ max ; tj > > 2 a j > > : ; ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 9 r 2 Pm P > > H2 > Z H2 m 4aj Sj > < Z i¼1 ðU i þ V i Þ þ U j W þ = i¼1 ðU i þ V i Þ þ U j W U ; tj T j ¼ max > > 2 a j > > : ; ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8v uP S > 1 u n i > > þ A þ Aw u > i¼1 ðai M si þ bi ÞM wi þ g > : Max

ti Msi M wi

; i ¼ 1; . . . ; m

(32)

L U L L U ½M Lsi ; M U si and ½M wi ; M wi , respectively, where M si ¼ dT i =Ri e, U U L L L U U U L M si ¼ bT i =Ri c, M wi ¼ dRi =Rw e, and M wi ¼ bRi =Rw c. Now we are ready to test the performance of heuristic H2. The parameter settings for the experiments are exactly the same as the ones for heuristic H1. Since the computational complexity for the enumeration procedure for policy (ii) is much higher than the enumeration

ARTICLE IN PRESS G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

procedure under policy (i), it is not possible to test for four suppliers within reasonable amounts of computational time. Hence we test cases with two and three suppliers. The results are shown in Table 2. Here again, the results are very good. The overall average gap is 0:16% while the maximum gap is 4:61%. The average and maximum for the two supplier cases are 0:19% and 4:61%, respectively, while that for the three supplier cases are 0:12% and 3:88%, respectively. The heuristic is very fast and none of the test instances took more than one second of CPU time. We see a trendsimilar to that of H1. When the multiplier values (M si ’s and M wi ’s) are large (such as low holding cost at the warehouse with high holding cost at the customer), the gap is very close to zero. 4.2. The value of warehouse In the supply chains we consider, there is a warehouse between the suppliers and the customer and the products from the suppliers are consolidated at the warehouse for delivery to the customer. It is well-understood conceptually that the presence of a warehouse can lower the transportation and inventory costs compared to a singlestage supply chain where there is no warehouse between the suppliers and the customer. There are several simulation studies that compare freight consolidation through a warehouse and direct shipments based on transportation

65

costs. For example, the study by Bagchi and Davis (1988) shows that direct shipments from vendors are almost always more expensive. Cooper (1984) compares freight consolidation across time and customers, use of warehouses, and direct less-than-truckload distribution systems on the basis of distribution costs and delivery times for selected product characteristics and demand patterns. She concludes that in general consolidation lowers costs but this may lead to an increase in the delivery time. These existing studies focus on transportation costs only and do not consider production operations and costs in the system. To our knowledge, no existing studies have investigated the value of consolidation or warehouses from a total system cost point of view. In this section, we computationally evaluate the typical reduction of total production, inventory and transportation cost that can be achieved by the use of a warehouse in the supply chain we consider. More speciﬁcally, we compare the total costs per unit time for the following two supply chains: (1) The two-stage supply chain considered in this paper where there are m suppliers, one warehouse, and one customer. 2) A single-stage supply chain where there are also m suppliers and one customer as in our supply chain, but with no warehouse between the suppliers and the customer, and the product at each supplier is directly delivered to the customer.

Table 2 Average and maximum relative gaps (%) between the optimal solution and the solution provided by heuristic H2.

g1

g2

F0

m¼2

m¼3

Average gap

1

1.1

5

10

5

1.1

5

10

10

1.1

5

10

Maximum gap

Average gap

Maximum gap

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

1 10 100 1 10 100 1 10 100

1.03 1.45 0.38 0.41 0.30 0.06 0.14 0.04 0.01

0.37 0.16 0.00 0.11 0.02 0.02 0.02 0.00 0.00

0.17 0.04 0.12 0.02 0.02 0.01 0.01 0.00 0.00

2.31 3.49 2.32 1.32 1.47 0.81 0.64 0.50 0.34

1.39 0.74 0.03 0.34 0.58 0.71 0.26 0.00 0.03

0.83 0.47 4.21 0.40 0.16 0.13 0.16 0.18 0.01

1.06 0.86 0.26 0.19 0.06 0.02 0.08 0.03 0.00

0.61 0.19 0.17 0.06 0.01 0.00 0.03 0.01 0.00

0.51 0.18 0.00 0.02 0.00 0.00 0.01 0.00 0.00

2.72 2.70 1.47 0.79 0.51 0.18 0.37 0.44 0.11

2.31 1.03 3.88 0.35 0.17 0.06 0.14 0.19 0.06

2.35 3.06 0.06 0.24 0.02 0.00 0.05 0.01 0.00

1 10 100 1 10 100 1 10 100

1.31 1.64 0.82 0.28 0.22 0.05 0.17 0.11 0.02

0.24 0.20 0.03 0.02 0.00 0.00 0.01 0.00 0.00

0.04 0.02 0.01 0.02 0.01 0.00 0.01 0.00 0.00

2.63 4.47 3.22 1.01 1.27 0.63 0.57 0.62 0.15

0.46 0.46 0.37 0.06 0.05 0.03 0.03 0.06 0.05

0.20 0.22 0.43 0.14 0.13 0.01 0.02 0.01 0.01

0.67 0.73 0.33 0.13 0.04 0.00 0.13 0.05 0.00

0.10 0.07 0.00 0.09 0.05 0.01 0.03 0.00 0.00

0.02 0.00 0.01 0.01 0.00 0.00 0.01 0.00 0.00

1.30 1.72 1.18 0.41 0.21 0.03 0.37 0.24 0.05

0.30 0.26 0.00 0.26 0.16 0.12 0.09 0.02 0.05

0.17 0.02 0.36 0.05 0.03 0.01 0.10 0.01 0.01

1 10 100 1 10 100 1 10 100

1.55 1.65 0.81 0.30 0.21 0.08 0.18 0.11 0.03

0.28 0.23 0.03 0.01 0.00 0.00 0.02 0.01 0.00

0.04 0.01 0.00 0.02 0.01 0.01 0.01 0.00 0.00

3.38 4.61 2.54 1.08 1.14 0.49 0.57 0.45 0.30

0.52 0.58 0.27 0.05 0.05 0.04 0.06 0.05 0.00

0.22 0.15 0.03 0.13 0.21 0.11 0.03 0.06 0.04

0.75 0.78 0.34 0.12 0.05 0.01 0.13 0.06 0.00

0.10 0.08 0.01 0.07 0.03 0.02 0.02 0.01 0.00

0.01 0.00 0.04 0.00 0.00 0.00 0.01 0.00 0.00

1.44 1.74 0.93 0.40 0.25 0.18 0.40 0.35 0.09

0.20 0.21 0.09 0.14 0.11 0.23 0.06 0.06 0.00

0.10 0.04 1.93 0.04 0.03 0.00 0.15 0.04 0.00

ARTICLE IN PRESS 66

G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

We deﬁne the relative cost reduction with the addition of a warehouse as ðZ 1 Z 2 Þ=Z 1 100%, where Z 1 and Z 2 are the optimal total cost per unit time in the single-stage and two-stage supply chains, respectively. Since there is no delivery consolidation in the single-stage supply chain, that problem can be viewed as m separate single-supplier problems, each equivalent to the model considered by

Hahm and Yano (1992). Therefore, we solve the m separate single-stage single-supplier problems optimally by applying the solution approach of Hahm and Yano, and get the optimal total cost Z 1 . We use heuristic H2 to solve the problem with the two-stage supply chain, and use the total cost of the solution obtained by H2, denoted as Z H2 , to replace Z 2 in calculating the relative cost reduction.

Table 3 Relative cost reductions (%) due to the warehouse when there are two suppliers. pi

U[0.02,0.08]

g1

1

g2

F0

1.1

5

10

5

5.5

25

50

10

11

50

100

U[0.002,0.008]

1

1.1

5

10

5

5.5

25

50

10

11

50

100

r ¼ 0:01

r ¼ 0:1

r¼1

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

1.25 4.23 10.97 4.60 10.45 16.55 7.82 14.71 20.51 3.49 8.41 14.65 8.62 14.55 18.71 12.62 18.92 22.56 4.68 10.29 15.90 10.41 16.07 19.40 14.49 20.24 23.17

0.96 3.48 6.73 5.65 11.76 17.53 9.32 17.78 24.67 2.97 6.27 9.65 9.67 16.12 20.29 14.91 22.75 27.12 3.85 7.49 10.36 11.76 17.67 21.06 17.60 24.50 27.86

0.05 1.19 2.25 7.74 16.00 24.31 13.01 24.52 33.70 1.23 2.44 3.63 12.92 21.59 27.29 21.03 31.37 37.47 1.55 2.89 3.92 15.81 23.79 28.29 24.38 33.80 38.44

1.07 3.42 8.73 4.19 9.21 14.21 7.11 13.48 18.43 2.99 6.92 11.88 7.80 12.88 16.29 11.23 16.98 20.23 3.88 8.44 12.78 9.42 14.14 16.89 13.26 18.27 20.75

0.53 2.66 4.91 4.99 10.34 15.59 8.79 16.41 22.72 2.17 4.61 7.19 8.56 14.17 17.75 14.22 20.82 24.72 2.81 5.48 7.67 10.42 15.52 18.41 16.37 22.43 25.35

0.25 0.47 0.71 6.77 13.93 20.95 11.88 22.35 30.38 0.56 1.16 1.70 11.53 18.73 23.35 18.85 28.06 33.23 0.74 1.34 1.85 13.81 20.56 24.25 21.67 29.97 34.11

1.44 1.90 2.47 1.72 2.89 4.10 4.70 7.99 9.96 1.53 2.19 2.52 2.33 3.68 4.38 6.72 9.48 10.79 1.79 2.42 2.66 2.77 3.92 4.55 7.60 9.98 10.99

1.75 3.02 4.57 1.94 3.92 6.04 5.75 10.23 12.34 2.25 3.83 5.13 3.24 4.88 5.79 8.53 11.88 13.52 2.82 4.45 5.40 3.79 5.25 6.04 9.67 12.54 13.80

1.99 4.24 6.94 2.79 5.44 6.92 7.21 12.69 15.66 3.20 5.75 8.01 4.18 6.45 7.91 10.62 14.81 16.85 4.00 6.61 8.30 4.90 6.87 7.89 12.17 15.65 17.20

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

0.72 2.83 9.82 5.19 12.06 19.87 7.64 15.84 24.13 2.59 6.86 12.94 7.43 13.76 18.72 10.53 17.64 22.37 3.57 8.50 14.63 8.95 15.06 19.23 12.52 19.08 22.85

0.47 2.06 6.15 6.77 15.56 24.59 10.17 21.13 30.76 2.18 5.05 8.27 8.34 15.26 20.72 13.14 21.79 27.48 2.96 6.32 9.68 10.05 16.67 21.02 15.40 23.29 27.76

0.60 2.12 2.35 9.46 21.53 32.71 13.92 29.14 41.36 0.85 1.94 3.14 11.34 20.74 28.00 18.19 30.11 37.96 1.17 2.40 3.54 13.60 22.53 28.36 21.39 32.25 38.38

0.68 2.40 8.21 5.14 11.74 18.64 7.59 15.69 23.03 2.17 5.72 11.09 6.74 12.21 16.29 9.99 16.39 20.43 3.00 6.97 11.88 8.12 13.34 16.77 11.67 17.49 20.76

0.30 1.70 4.74 6.61 14.97 22.70 9.94 20.52 28.94 1.63 3.86 6.27 7.57 13.68 18.35 12.31 20.18 25.16 2.20 4.69 7.02 9.04 14.79 18.53 14.33 21.42 25.35

0.45 1.68 1.16 8.89 20.12 29.43 13.27 27.32 37.90 0.42 0.92 1.53 10.04 18.16 24.40 16.59 27.18 34.01 0.57 1.16 1.71 11.93 19.55 24.47 19.39 28.92 34.23

0.19 0.49 1.96 4.92 10.44 11.62 7.63 14.54 16.62 1.04 1.72 2.40 2.64 4.42 5.62 6.69 10.11 12.05 1.39 1.89 2.34 2.61 4.14 4.96 7.31 10.17 11.56

0.34 1.06 4.16 6.01 12.52 13.28 9.23 17.45 19.55 1.57 3.13 5.33 3.52 5.79 7.40 8.23 12.45 14.70 2.15 3.75 4.96 3.75 5.67 6.64 9.02 12.54 14.29

0.41 1.66 6.47 7.08 14.68 15.76 11.03 20.37 22.97 2.30 4.91 7.75 4.57 7.60 9.29 10.18 15.38 18.12 3.08 5.68 7.71 4.86 7.37 8.77 11.25 15.61 17.78

ARTICLE IN PRESS G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

Thus the relative cost reduction we get in our computational test, ðZ 1 Z H2 Þ=Z 1 100%, is a lower bound of the actual relative cost reduction. In our computational experiment, we test three sets of problems with 2, 4, and 8 suppliers, respectively. All other parameters are generated exactly the same way as in the earlier experiments used for testing the performance of heuristics H1 and H2. For each set of the test problems,

67

there are 2 36 ¼ 1458 possible combinations of the parameters (pi ; hs ; g1 ; g2 ; F 0 ; r; x), and for each combination of these parameters, we run 10 random problem instances. For every test problem, the relative gap (%) between Z 1 and Z H2 is computed. Tables 3, 4, and 5 show the results for the two cases of the supplier production rates for the two-, four-, and eight-supplier cases, respectively. The tables aggregate the results for the three

Table 4 Relative cost reductions (%) due to the warehouse when there are four suppliers. pi

U[0.02,0.08]

g1

1

g2

F0

1.1

5

10

5

5.5

25

50

10

11

50

100

U[0.002,0.008]

1

1.1

5

10

5

5.5

25

50

10

11

50

100

r ¼ 0:01

r ¼ 0:1

r¼1

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

2.07 8.43 17.33 8.58 18.49 27.93 12.72 24.32 33.79 6.60 14.83 22.94 14.90 25.17 31.88 20.71 31.36 37.40 8.80 17.47 24.65 18.35 27.65 33.12 24.29 33.81 38.41

1.07 5.05 10.41 8.87 18.26 27.56 13.81 25.96 35.69 4.54 9.66 14.37 15.04 24.91 31.45 22.03 33.17 39.64 6.00 11.27 15.49 18.06 27.36 32.60 25.56 35.78 40.71

0.03 1.82 3.68 9.83 20.34 30.49 15.94 29.97 41.39 1.76 3.61 5.48 16.75 27.60 34.76 25.51 38.33 45.67 2.25 4.28 5.87 19.83 30.18 36.04 29.56 41.13 46.91

2.03 7.62 14.41 8.28 17.48 26.16 12.66 23.46 32.30 6.06 13.37 19.95 14.20 23.59 29.55 20.07 30.13 35.54 7.98 15.49 21.42 16.93 25.49 30.55 23.50 32.26 36.48

1.04 4.68 8.49 8.50 17.45 25.94 13.53 25.13 34.32 3.99 8.39 12.37 14.36 23.56 29.49 21.55 31.87 37.82 5.19 9.71 13.13 17.27 25.89 30.55 24.75 34.36 38.80

0.16 1.34 2.24 9.27 19.07 28.33 15.33 28.50 39.07 1.37 2.86 4.22 15.77 25.46 31.93 24.18 36.18 42.73 1.81 3.33 4.53 18.77 28.11 33.11 28.29 38.60 43.83

0.29 2.08 4.35 6.97 13.23 18.32 11.81 20.33 26.39 2.14 4.21 5.85 11.04 16.72 19.95 17.69 24.64 28.01 2.76 4.85 6.29 12.89 17.99 20.49 20.19 26.01 28.58

0.47 0.06 0.89 6.98 13.31 18.50 12.50 21.45 27.25 0.75 1.47 2.08 11.35 17.07 20.36 18.67 25.83 29.46 0.98 1.72 2.22 13.27 18.37 20.94 21.13 27.30 30.05

1.44 1.83 2.52 7.36 13.66 18.78 13.22 22.53 29.32 0.90 1.62 2.32 11.62 17.58 20.85 19.63 27.39 31.30 1.10 1.78 2.32 13.64 18.90 21.53 22.38 28.84 31.82

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

1.08 4.27 14.00 8.36 19.25 31.79 12.08 25.14 37.67 4.83 11.91 20.46 12.47 23.06 31.28 17.72 29.35 36.99 6.73 14.72 22.86 15.38 25.68 32.42 21.10 31.81 37.87

0.98 3.02 9.02 9.42 21.59 34.33 13.76 28.58 41.89 3.39 7.69 12.74 12.72 23.30 31.41 19.08 31.55 39.73 4.61 9.55 14.39 15.36 25.57 32.21 22.56 33.96 40.40

0.89 2.84 4.36 11.15 25.53 39.04 16.32 34.06 48.60 1.29 2.90 4.30 14.33 26.16 35.31 22.14 36.56 46.02 1.76 3.64 5.29 17.25 28.50 35.88 26.10 39.25 46.68

0.93 3.85 12.75 8.41 19.19 30.90 12.22 25.12 36.90 4.50 10.69 18.11 11.96 21.87 29.35 17.37 28.38 35.49 6.18 13.09 19.93 14.56 24.05 30.18 20.48 30.55 36.15

0.91 2.93 8.04 9.42 21.26 33.04 13.76 28.46 40.63 2.97 6.75 10.66 12.29 22.21 29.71 18.64 30.48 38.00 4.04 8.27 12.22 14.81 24.23 30.35 21.91 32.66 38.60

0.80 2.65 3.22 10.90 24.81 36.81 16.03 33.01 46.23 1.02 2.36 3.74 13.55 24.45 32.80 21.19 34.61 43.29 1.40 2.84 4.17 16.22 26.49 33.13 24.89 37.01 43.77

0.59 1.52 4.78 9.41 19.96 25.55 13.71 25.99 32.34 1.67 3.52 5.76 10.03 16.62 21.09 16.04 24.20 28.84 2.20 4.30 5.76 11.54 17.51 20.73 18.24 25.32 28.84

0.50 1.10 1.72 10.00 21.17 26.52 14.70 27.84 34.08 0.60 1.29 2.06 10.27 17.09 21.59 16.94 25.60 30.42 0.80 1.49 2.06 11.87 17.94 21.30 19.22 26.66 30.32

0.36 0.56 1.74 10.71 22.66 27.54 15.82 29.90 36.07 0.61 1.44 2.61 10.70 17.79 22.31 18.02 27.24 32.25 0.82 1.52 2.07 12.29 18.55 22.05 20.42 28.33 32.23

ARTICLE IN PRESS 68

G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

different values of holding costs at the supplier (hs ), and hence each entry corresponds to an average over 30 random test instances for the given combination of the six parameters(pi ; g1 ; g2 ; F 0 ; r; x). The warehouse serves as a place for pooling deliveries for commodities from the various suppliers in addition to acting as a place for holding inventory. Hence we would expect the relative gap to increase as we increase the

number of suppliers. This is supported in the results obtained. The overall average gap values for the two, four, and eight supplier cases are 10:75%, 18:18%, and 22:71%, respectively. The few negative values in the tables indicate instances where adding a warehouse increases the total costs. Again we notice that the magnitude and the occurrence of negative values in general go down when we increase the number of suppliers. For all the results, in

Table 5 Relative cost reductions (%) due to the warehouse when there are eight suppliers. pi

U[0.02,0.08]

g1

1

g2

F0

1.1

5

10

5

5.5

25

50

10

11

50

100

U[0.002,0.008]

1

1.1

5

10

5

5.5

25

50

10

11

50

100

r ¼ 0:01

r ¼ 0:1

r¼1

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

1.03 7.88 16.10 11.08 22.85 34.33 16.35 30.82 42.31 4.72 13.12 21.14 19.01 31.20 39.06 26.28 39.25 46.65 6.81 15.65 22.73 22.56 34.05 40.43 30.64 42.11 47.83

0.82 5.40 10.71 11.06 22.83 33.86 17.06 31.86 43.77 4.01 9.34 14.15 18.87 30.87 38.58 27.33 40.49 48.07 5.50 10.85 15.07 22.43 33.70 39.96 31.69 43.52 49.35

0.61 2.16 4.49 11.55 23.66 35.09 18.31 34.16 47.03 1.85 4.15 6.21 19.35 31.93 39.99 29.15 43.43 51.60 2.45 4.87 6.68 23.52 34.88 41.42 34.04 46.64 52.94

1.06 7.03 14.26 10.89 22.36 33.10 16.44 30.37 41.37 4.82 12.14 19.31 18.43 30.16 37.43 25.96 38.61 45.41 6.61 14.34 20.53 22.05 32.78 38.74 30.20 41.29 46.53

0.88 4.82 9.25 10.92 22.25 32.80 17.06 31.31 42.64 3.76 8.46 12.66 18.53 29.94 37.16 26.72 39.61 46.78 5.06 9.86 13.52 21.76 32.69 38.48 31.07 42.40 47.99

0.24 1.57 2.81 11.26 22.74 33.72 18.09 33.24 45.09 1.59 3.48 5.01 18.76 30.57 38.04 28.48 42.00 49.56 2.09 4.04 5.49 22.64 33.39 39.39 33.22 45.14 50.81

0.65 3.22 6.28 10.60 20.09 27.72 16.95 28.87 37.13 3.05 6.04 8.44 16.92 25.50 30.41 25.30 34.99 39.78 3.91 6.94 8.95 19.71 27.37 31.20 28.72 36.93 40.57

0.21 1.57 3.40 10.70 20.13 27.88 17.32 29.46 37.96 1.84 3.58 4.86 16.87 25.56 30.40 25.90 35.74 40.65 2.33 4.07 5.21 19.82 27.43 31.23 29.32 37.69 41.44

0.78 0.51 0.12 10.73 20.20 27.66 17.74 30.25 38.65 0.13 0.37 0.50 16.86 25.49 30.32 26.36 36.52 41.52 0.21 0.45 0.59 19.79 27.39 31.14 29.98 38.54 42.32

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

1.15 4.48 13.55 10.79 24.58 39.50 15.51 32.01 47.04 3.55 10.84 19.20 15.89 28.78 38.61 22.61 37.02 46.32 5.47 13.62 21.27 19.38 31.79 39.81 26.77 39.97 47.33

1.29 3.97 9.52 11.54 26.20 41.04 16.73 34.48 49.79 3.16 7.62 12.52 15.91 28.81 38.64 23.47 38.43 48.10 4.38 9.41 14.12 19.29 31.59 39.56 27.73 41.35 48.96

1.12 3.48 5.28 12.65 28.81 43.78 18.46 38.07 53.99 1.55 3.60 5.89 16.70 30.19 40.54 25.31 41.46 51.92 2.05 4.28 6.36 20.11 32.93 41.23 29.83 44.49 52.67

1.18 4.19 12.66 10.96 24.66 38.88 15.80 32.24 46.55 3.74 10.31 17.44 15.60 28.00 37.26 22.48 36.42 45.26 5.48 12.56 19.28 18.90 30.74 38.28 26.51 39.16 46.11

1.26 3.84 8.67 11.65 26.28 40.19 16.89 34.50 48.95 2.94 6.92 10.96 15.71 28.12 37.42 23.31 37.77 46.94 4.05 8.51 12.53 18.94 30.69 38.18 27.42 40.49 47.68

1.06 3.40 4.55 12.63 28.44 42.27 18.39 37.61 52.38 1.34 3.00 5.00 16.28 29.11 38.82 24.83 40.23 50.05 1.75 3.60 5.34 19.51 31.62 39.33 29.13 43.02 50.66

1.14 2.94 6.83 12.70 26.61 35.17 18.18 34.28 43.29 2.41 5.10 8.21 15.05 24.93 31.34 22.72 34.15 40.54 3.20 6.15 8.28 17.57 26.45 31.42 25.95 35.83 40.70

0.93 2.42 4.01 13.13 27.52 35.58 18.91 35.58 44.37 1.44 3.02 4.81 15.16 25.09 31.49 23.26 34.98 41.47 1.90 3.55 4.81 17.64 26.56 31.51 26.55 36.66 41.61

0.80 1.86 0.47 13.58 28.38 35.99 19.65 36.91 45.53 0.17 0.36 0.57 15.26 25.21 31.50 23.85 35.84 42.39 0.22 0.42 0.57 17.66 26.58 31.59 27.17 37.50 42.61

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general, the relative gap increases as the holding cost at the customer goes up, or when the variable transportation cost goes up. The gap also increases with a reduction in the ﬁxed cost factor value. The explanations for these are straightforward. All these changes make the warehouse an inexpensive transit point. The effect of the location of the warehouse on total cost is more involved. When both the holding cost at the customer and the variable transportation cost are high, we would like to have small and frequent shipments to save on the holding costs at the customer, but would like to avoid traveling long distances to contain the transportation expense. So it is advantageous to have the warehouse at close proximity to the customer site. On the other hand, if the holding cost at the customer is low when the transportation cost is high, we would like to make large shipments from the warehouse to the customer anyway, and hence placing the warehouse close to the supplier would reduce the delivery expenses from the supplier to the warehouse. Hence we see that in the tables, the relative gap increases with the value of x for the ﬁrst case, and the gap decreases with x for the second. Finally, we can see from these tables that the production rates of the suppliers have some impact on the relative gap, but not as signiﬁcant as the other parameters. Hence the suppliers’ production rates do not seem to play a critical role in deciding whether to use a warehouse or not in the supply chain.

4.3. Other insights In this subsection, we discuss some other insights obtained from our computational experiments. Theoretically, policy (ii) should always give solutions that are at least as good as the ones under policy (i) in terms of total cost. Since our heuristics do not guarantee to generate optimal solutions, we cannot assert that H2 will always dominate H1. But in general, since both the heuristics have been shown to perform well in the computational experiments, we can expect H2 to provide better solutions than H1 in most cases. That is the reason why we chose H2 as the benchmark for the two-stage supply chain while evaluating the value of warehouse in the previous subsection. In fact, a comparison between the solutions obtained by H1 and H2 justify this choice. Due to the lack of space, we do not give any detailed reports on this. Instead, we provide a summary of our ﬁndings. Out of a total of 14 580 test problems for the two supplier case, there were only six cases where H1 beat H2. Even in those cases, the gaps are very small in magnitude. The same holds true for the four supplier case. With the eight supplier problem instances, there was not even a single test problem for which H1 gave a total cost that was lower than the one given by H2. On average, the objective values from H2 were 28:36%; 29:85%, and 30:24% lower than the objective values from H1 for the two-, four-, and eightsupplier cases, respectively. Another interesting insight obtained is about the multipliers M si in the model under policy (ii). A multiplier M si represents the number of delivery cycles from supplier i to the warehouse per production cycle at supplier i.

69

Table 6 shows the average number of M si ’s aggregated over the three cases of the number of suppliers (m) and the three cases of the holding cost at the suppliers (hsi ). We would expect the multipliers to depend on the location of the warehouse. For example, if the warehouse is closer to the suppliers, then the transportation cost from the suppliers to the warehouse is going to be lower and hence there will be more shipments from the suppliers per production cycle. This is supported in the results as the multipliers drop when the warehouse location is moved from x ¼ 0:2 to 0:5 or 0:8. Similarly, we would expect the multiplier to decrease with the variable or ﬁxed transportation cost. This is reﬂected in the table as the multipliers drop with r or F 0. We would expect the shipments from the suppliers to become more frequent (hence smaller) as the holding cost at the warehouse is increased. This observation is also supported in the table by the fact that the multipliers increase with g1 . Production rates can also play a crucial role. Production rates inﬂuence both the production batch size and the inventory holding costs. If the production is too slow along with a positive setup time, this may place restrictions on the batch size hence leading to higher costs. The effect of production rates on the inventory holding cost is slightly more complicated. A higher production rate may in fact be undesirable. This is because in the case of higher production rates, an item that is meant for a future shipment gets ready at an earlier time compared to a system with a lower production rate. This leads to increased waiting time for that item before getting shipped, thus resulting in an increased inventory holding cost at the supplier. Hence a higher production rate has both positive and negative impacts. As mentioned in Section 2, many studies in the past assumed inﬁnite production rates at the suppliers. But Table 6 shows that changes in the production rate can signiﬁcantly impact the number of delivery cycles from the suppliers per production cycle. When the production rates are increased by a factor of 10, the production–delivery cycle time ratio drops signiﬁcantly. In many cases, the ratio drops by more than 75% of its original value.

5. Conclusions In this study, we have studied a joint cyclic production and distribution scheduling problem in a two-stage supply chain with one or more suppliers, one warehouse, and one customer. We have given either optimal approaches or heuristic methods to solve the problem under two policies on production and delivery cycles. For the case with common production and delivery cycle at each supplier (policy (i)), we have proved that there exists an optimal solution where the delivery cycle time from a supplier to the warehouse is an integer multiple of the delivery cycle time from the warehouse to the customer. Based on this property, we have shown that there is a closed-form optimal solution to the problem with a single supplier under policy (i), and developed an efﬁcient heuristic for the general problem under policy (i). The

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Table 6 Average number of deliveries from the suppliers to the warehouse per production cycle at the suppliers. pi

U[0.02,0.08]

g1

1

g2

F0

1.1

5

10

5

5.5

25

50

10

11

50

100

U[0.002,0.008]

1

1.1

5

10

5

5.5

25

50

10

11

50

100

r ¼ 0:01

r ¼ 0:1

r¼1

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

x ¼ 0:2

x ¼ 0:5

x ¼ 0:8

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

28.88 8.23 2.53 28.47 8.52 2.84 28.42 9.23 2.76 61.12 19.33 5.82 62.48 19.65 5.97 62.39 19.37 6.32 85.99 27.13 8.34 85.64 27.27 8.63 85.88 27.79 8.86

19.98 6.47 1.89 20.37 6.32 2.00 20.12 6.35 2.05 43.84 13.63 4.22 43.52 14.27 4.37 43.68 14.14 4.47 62.00 19.84 6.14 62.70 19.93 6.33 63.37 19.74 6.43

16.39 5.19 1.67 16.21 5.21 1.69 16.97 5.14 1.73 36.15 11.48 3.74 36.65 11.25 3.70 36.30 11.47 3.73 52.11 16.07 4.98 51.96 15.99 5.10 51.94 16.25 5.23

24.04 7.26 2.22 24.06 7.38 2.34 23.67 7.56 2.53 51.57 16.41 4.98 53.32 16.47 5.19 54.38 16.83 5.32 74.76 23.72 7.17 75.98 23.45 7.44 73.93 23.48 7.54

18.68 5.50 1.82 18.84 6.11 1.89 18.81 5.72 1.97 41.48 12.68 3.95 41.92 13.06 4.14 40.42 13.17 4.19 57.64 18.04 5.52 59.41 18.40 5.91 59.35 18.24 5.88

15.67 4.77 1.57 15.79 4.96 1.64 15.75 4.90 1.55 34.43 11.21 3.54 34.25 10.80 3.43 34.06 10.97 3.51 49.01 15.34 4.92 48.11 15.33 4.81 50.08 15.43 5.00

12.81 4.09 1.21 12.93 4.10 1.44 13.50 4.30 1.39 29.04 9.02 2.72 29.34 9.52 3.11 30.43 9.36 3.02 40.00 12.58 3.87 41.98 13.22 4.39 41.30 13.16 4.09

11.90 3.81 1.16 11.86 3.79 1.35 12.02 3.83 1.34 26.17 8.09 2.50 26.80 8.54 2.62 26.79 8.76 2.68 37.87 11.55 3.62 37.08 11.76 3.70 38.11 12.09 3.79

11.22 3.57 1.14 11.45 3.62 1.25 10.64 3.70 1.27 24.67 7.56 2.39 24.97 7.95 2.50 24.67 7.93 2.53 34.74 10.74 3.43 34.87 11.20 3.55 35.52 11.29 3.65

1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

5.91 1.74 1.00 5.87 1.85 1.00 5.89 1.96 1.00 37.76 11.69 3.58 37.82 11.76 3.83 37.63 11.62 3.54 56.40 17.78 5.43 56.38 17.78 5.59 56.80 18.31 5.92

4.18 1.31 1.00 4.28 1.44 1.00 4.24 1.42 1.00 27.33 8.50 2.62 27.67 8.74 2.86 27.33 8.76 2.87 40.74 12.65 4.00 40.82 12.92 4.08 40.73 12.58 3.96

3.48 1.01 1.00 3.47 1.05 1.00 3.51 1.03 1.00 22.29 7.18 2.33 22.58 6.95 2.00 22.57 6.93 2.00 33.35 10.59 3.33 33.29 10.72 3.08 33.24 10.61 3.05

5.04 1.53 1.00 5.09 1.63 1.00 5.04 1.64 1.00 32.83 10.25 3.00 32.86 10.17 3.11 32.89 9.99 3.33 49.04 15.41 4.58 49.08 15.47 4.75 49.06 15.53 4.69

3.87 1.18 1.00 3.94 1.35 1.00 3.99 1.27 1.00 25.35 7.92 2.33 25.74 8.37 2.78 25.33 7.86 2.80 37.84 11.71 3.79 38.01 12.01 3.83 38.10 11.78 3.88

3.37 1.00 1.00 3.32 1.00 1.00 3.32 1.00 1.00 21.29 6.57 2.00 21.32 6.88 2.00 21.39 6.86 2.00 31.73 9.98 3.00 31.98 10.12 3.00 31.65 9.90 3.00

2.74 1.00 1.00 2.70 1.00 1.00 2.81 1.00 1.00 17.81 5.18 1.97 17.90 5.49 2.00 17.90 5.70 2.00 26.63 8.01 2.16 26.90 8.06 2.77 26.91 8.37 2.92

2.49 1.00 1.00 2.56 1.00 1.00 2.58 1.00 1.00 16.20 5.00 1.72 16.59 5.32 2.00 16.23 5.00 2.00 24.41 7.48 2.00 24.80 7.92 2.83 24.93 7.89 2.77

2.34 1.00 1.00 2.51 1.00 1.00 2.41 1.00 1.00 15.01 4.54 1.43 15.30 5.00 1.98 15.27 4.99 2.00 22.51 7.00 2.00 22.73 7.03 2.00 22.70 7.33 2.00

problem under policy (ii), which is more general than policy (i), is solved by a heuristic approach. Both heuristics are shown to perform very well for an extensive set of test problems. We have also computationally evaluated the value of warehouse in our two-stage supply chain. Various managerial insights have been reported. An important use of this study is to make operational decisions regarding the delivery intervals in a two-stage

supply chain. The approaches provided in this paper are easy to implement. Moreover, computationally they are very efﬁcient. The models can also be used to make strategic decisions related to conﬁguring or making changes to a supply chain. For example, we could use the heuristics to choose between a single-stage and a twostage supply chain. Given that a warehouse has to be built, we could use this study to analyze the total costs

ARTICLE IN PRESS G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

corresponding to various locations of the potential warehouse. We could use the heuristics to analyze the tradeoffs involved in moving an existing warehouse to a new location. This model can also be used to analyze the effect of reducing the setup cost or setup time on the performance of the entire supply chain. For example, reducing the setup cost or setup time at the supplier would enable more frequent deliveries from the supplier to the warehouse, thus saving on the average inventory costs. A trade-off between this savings and the increase in the total transportation costs and the expenses related to reducing the setup time and costs can be used to analyze whether it is worth trying for a reduction in the setup cost or time.

Acknowledgment This research was supported in part by the National Science Foundation under Grant DMI-0421637. Appendix A. Proofs of Theorems 1 and 2

Proof of Theorem 1. We prove the theorem for the single-supplier case ﬁrst and then extend the result to the case with multiple suppliers. When we have only one supplier, we discard the supplier subscript in our notation. Hence the production cycle time and the delivery cycle time from the supplier to the warehouse are denoted as T and R, respectively. Under policy (i), we have T ¼ R. Suppose that M w ¼ R=Rw is not an integer. We show that the total cost of a new schedule where both the production cycle time at the supplier and the delivery cycle time from the supplier to the warehouse are increased to dM w eRw is not greater than that of the original schedule. Before giving a formal analysis, we explain through a diagram the various categories of inventory at the warehouse when M w is not an integer. The inventory level over time at the warehouse repeats every kR units of time, where k is the smallest integer such that kR=Rw is an integer. The period of kR time units is hence called an inventory cycle of the warehouse. For illustration purposes, let us assume Mw ¼ 2 13. Fig. 3 shows the inventory level at the warehouse over one entire inventory cycle, i.e. over 3R time units (k ¼ 3 here). In Fig. 3, the solid vertical lines below the horizontal axis (i.e. at times 0, R, 2R, 3R) indicate deliveries from the supplier to the warehouse while the dotted lines (i.e. at times e1 , e1 þ Rw , e1 þ 2Rw ; . . .) indicate deliveries from the warehouse to the customer. The earliness parameter ei, for i 2 f1; 2; 3g represents the gap between the time of the ith delivery from the supplier to the warehouse (i.e. time ði 1ÞR) and the time of the ﬁrst delivery from the warehouse to the customer after time ði 1ÞR. For the third shipment from the supplier, the delivery from the supplier to the warehouse coincides with a delivery from the warehouse to the customer. Hence e3 ¼ 0. Note that the ﬁrst delivery from the warehouse to the customer does not take place until time e1 ¼ 23 Rw . Without this

71

intentional delay, the warehouse will not have sufﬁcient inventory on time for some of the future deliveries. We divide the inventory into three categories: (i) Fractional inventory represented by the areas with the horizontal line shading in Fig. 3. This is the inventory that has to wait till the next shipment from the supplier before getting delivered to the customer. This inventory is always a fraction of Rw D, the demand corresponding to one delivery period from the warehouse to the customer. (ii) Earliness inventory represented by the areas shaded with lines sloping downwards. This corresponds to the earliness e1 , e2 , and e3 described earlier. (iii) Integral inventory represented by all the remaining areas with a vertical shading. We note that if R=Rw was an integer, this would be the only kind of inventory at the warehouse, as there would be no fractional or earliness inventory. Now we look at the general case of non-integer Mw. Let d ¼ Mw bMw c. Clearly, 0odo1. We ﬁrst calculate the minimum value of earliness e1 , as minimizing this minimizes the total inventory cost at the warehouse. Let us assume e1 ¼ lRw for some l 2 ð0; 1Þ. Let k be the smallest integer such that kM w is an integer. We can express d as (a=k), where a ¼ kM w kbM w c is an integer, aok, and a and k are relatively prime to each other. As shown in Fig. 3, deliveries from the supplier occur at time points iR ¼ iM w Rw , where i 2 f0; 1; . . . ; k 1g. The ﬁrst delivery to the customer from the warehouse containing orders from shipment ði þ 1Þ from the supplier occurs at time ðbiM w c þ lÞRw . In order that the shipment from the supplier has been delivered by this time, we should have ðbiM w c þ lÞRw XiR ¼ iM w Rw , for i ¼ 0; . . . ; k 1. The difference between these two numbers is the earliness for delivery ði þ 1Þ from the supplier, denoted as eðiþ1Þ . This implies that eðiþ1Þ ¼ ðbiM w c þ lÞRw iM w Rw ¼ ðbia=kc þ l ia=kÞRw X0, or lXia=k bia=kc, for i ¼ 0; . . . ; k 1. We show in the following that the minimum value of l that satisﬁes the above condition is ðk 1Þ=k. For every i 2 f0; . . . ; k 1g, we can express ia=k as: ia=k ¼ bia=kcþ r=k, r 2 f0; . . . ; k 1g. We argue that ia=k has a unique remainder r=k, for every i ¼ 0; . . . ; k 1. Suppose that there exist i and j with 0piojpk 1 such that the remainders of ia=k and ja=k are identical. Then we have

ja ia ðj iÞa ja ia ¼ ¼ k k k k k

(33)

Since the difference of two integers is an integer, Eq. (33) implies that ðj iÞa is an integer multiple of k. This is not possible since ðj iÞok, and a and k are relatively prime to each other. Hence, each value of i 2 f0; . . . ; k 1g corresponds to a unique remainder. This implies that there is exactly one i with a remainder value of zero, one with 1=k, one with 2=k, and so on. Hence the maximum remainder is ðk 1Þ=k. This means that the minimum possible value of l is ðk 1Þ=k.

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3RwD e2 = 1/3Rw Inventory

e1 = 2/3Rw e3 = 0

2RwD

RwD

0.67 0.33 e1

e2 Rw

Rw

Rw

Rw

R Time

R

Rw

Rw R

Fig. 3. Inventory level at the warehouse over one inventory cycle when M w ¼ 2 13.

We now look at the three categories of inventory over the entire inventory cycle of kR time units at the warehouse. Over each interval of R time units, the number of deliveries to the customer is either bM w c or ðbM w c þ 1Þ. Hence the earliness inventory corresponding to the ði þ 1Þth delivery from the supplier to the warehouse is a rectangle with a height of either bMw cRw D or ðbM w c þ 1ÞRw D and width of eðiþ1Þ ¼ ðbia=kc þ l ia=kÞRw , for i 2 f0; 1; . . . ; k 1g. The total time in a cycle is kR ¼ kM w Rw units. Therefore, the average earliness inventory per unit time Ie is

Pk1 ia ia l þ Rw bMw cRw D i¼0 k k Ie X kM w Rw ! k1

X ia ia bMw cRw D ¼ kl þ k k kMw i¼0 k 1 bM w cRw D ¼ kl 2 kM w ðk 1ÞbM w cRw D X (34) 2kM w where

the

last

inequality

is

obtained

by

letting

l ¼ ðk 1Þ=k. The fractional inventory level during period ½iR; ði þ 1ÞRÞ can be calculated as the difference between the cumulative quantity delivered to the warehouse and the cumulative quantity delivered out of the warehouse by the end of the period. This is given by ði þ 1ÞM w Rw D bði þ 1ÞM w cRw D

ði þ 1Þa ði þ 1Þa ¼ Rw D k k

(35)

Each period ½iR; ði þ 1ÞRÞ is for a duration of R ¼ M w Rw time units, and each inventory cycle is for a duration of kM w Rw time units. Therefore, the average fractional inventory per unit time during a cycle of kR time units is

given by

Pk1 ði þ 1Þa ði þ 1Þa Rw DðMw Rw Þ i¼0 k k If ¼ kM w Rw k1 2 M w Rw D ðk 1ÞRw D 2 ¼ ¼ kM w Rw 2k

(36)

where we have used the fact observed earlier that there is a distinct remainder of ia=k for each i ¼ 0; . . . ; k 1, which Pk1 Pk1 implies that i¼0 ðði þ 1Þa=k bði þ 1Þa=kcÞ ¼ i¼0 i=k ¼ ðk 1Þ=2. Next we calculate the remaining part of the inventory, the integral inventory. If eðiþ1Þ oða=kÞRw for some i 2 f0; . . . ; k 1g, there will be dM w e deliveries from the warehouse to the customer during the period ½iR; ði þ 1ÞRÞ. Otherwise, there will be ðdM w e 1Þ deliveries. The ﬁrst of these dM w e or ðdM w e 1Þ deliveries from the warehouse has already been counted in the form of earliness inventory (see Fig. 3). Hence the integral inventory begins at a level of bMw cRw D or ðbM w c 1ÞRw D depending on the value of eðiþ1Þ . And it reduces by Rw D every Rw time units, ﬁnally reaching zero. Based on our analysis following Eq. (33), there exists an i 2 f0 . . . k 1g, for which ia=k bia=kc ¼ k 1=k ¼ l. For this value of i, eðiþ1Þ ¼ ðbia=kc þ l ia=kÞRw ¼ 0oða=kÞRw . Therefore we have at least one instance where the integral inventory begins at a higher level of bM w cRw D. Hence alower bound on the total integral inventory over an inventory cycle is k

bM wc X

ðbM w c iÞRw DRw þ bM w cRw DRw

i¼1

¼

kðbM w c 1ÞbM w cR2w D þ bM w cR2w D 2

(37)

ARTICLE IN PRESS G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

Here, the second term on the left-hand-side accounts for the instance where the integral inventory begins at the higher level. The average integral inventory Ii per unit time satisﬁes bM w cR2w D

kðbM w c 1ÞbM w c 2 Rw D þ 2kM w Rw kM w Rw ðbM w c 1ÞbM w cRw D bM w cRw D ¼ þ 2M w kM w

Ii X

(38)

Combining all the three parts of the inventory, we can see that the average inventory holding cost per unit time at the warehouse is ðIe þ If þ Ii Þhw . It can be easily shown that the average inventory holding cost per unit time at the supplier over one production cycle is ð1=2ÞpMw Rw D2 hs . Now consider a new schedule where both the production cycle time at the supplier and the delivery cycle time from the supplier to the warehouse are increased to dM w eRw . In this schedule, there is no fractional or earliness inventory at the warehouse, as dM w e is an integer. Hence, the average inventory per unit time at the warehouse is ð1=2ÞbM w cRw D. The average inventory per unit time at the supplier is ð1=2ÞpdM w eRw D2 hs . Clearly, the average inventory per unit time at the customer, and the transportation cost from the warehouse to the customer in this new schedule remain the same as in the original schedule. Both transportation and production setup costs per unit time at the supplier are lower in this new schedule than in the original schedule as these activities are carried out less frequently. Therefore, the difference between the average total costs per unit time for the two schedules, denoted as D, satisﬁes 1 2

DXðIe þ If þ Ii Þhw þ pMw Rw D2 hs 1 ððdM w e 1Þhw þ pdM w eDhs ÞRw D 2 ðk 1ÞbM w cRw D ðk 1ÞRw D ðbM w c 1ÞbM w cRw D X þ þ 2kM w 2k 2M w bM w cRw D bM w cRw D þ hw kM w 2 1 1 þ pMw Rw D2 hs pdM w eRw D2 hs 2 2 hw Rw D ð1 aÞbM w c k 1 pDhs ðk aÞRw D ¼ þ (39) 2 kM w k 2k Since by model assumptions hs phw and pDp1, we have pDhs phw . Therefore (39) implies h R D ð1 aÞbM w c 1 a DX w w þ1 1 2 kM w k k hw Rw D ða 1Þ bM w c X0 (40) 1 ¼ 2 k Mw This proves that the total cost of the new schedule is not greater than that of the original schedule. In the multiplesupplier case, we can generate a new schedule by increasing both T i and Ri to Rw dRi =Rw e whenever ðRi =Rw Þ is not an integer. By the above proof, the total cost related to each product i 2 f1; . . . ; mg is not greater than that in

73

the original schedule. Thus the average total cost in this new schedule is not greater than that in the original schedule. & Proof of Theorem 2. We prove this by transforming our problem to an equivalent problem studied by Hahm and Yano (1992). Their supply chain consists of one supplier and one customer, with no warehouse in-between. We use the subscript y to denote the parameters involved in their problem. Their objective is to ﬁnd the production cycle time T y at the supplier and the delivery cycle time Ry from the supplier to the customer such that the average total cost per unit time is minimized. They show that in an optimal schedule, T y is an integer multiple of Ry , and formulate their problem as the following optimization model: Minimize Subject to

Sy 1 Ay þ ð1 py Dy ÞDy hy T y þ py D2y hy Ry þ Ty 2 Ry sy TyX 1 p y Dy Ty is a positive integer Ry Ry X0

where T y and Ry are the decision variables and every other notation represents a problem parameter in the same way as the corresponding notations in our problem. In their problem, the unit holding costs at the supplier and at the customer are assumed to be equal and is represented by hy. This formulation is identical to our formulation with the following substitutions: Sy ¼ S þ A

(41)

Ay ¼ Aw 1 2ð1

(42)

py Dy ÞDy hy ¼

py D2y hy

1 2ðhs pD

þ hw ÞD

¼ 12ðhc hw ÞD

(43) (44)

sy s ¼ 1 py Dy 1 pD

(45)

If we are able to ﬁnd non-negative Sy ; sy ; py ; Dy ; Ay , and hy satisfying (41)–(45) and the capacity constraint py Dy p1, then we can use the optimal solution from the Hahm and Yano model as the optimal solution for our model. The optimal solution for their problem is what we have given in Eqs. (10) and (11) (with the corresponding substitutions of the parameters). It can be easily shown that Sy ; Ay ; Dy ; py ; hy , and sy that are deﬁned by (41), (42), and the following equations, respectively, satisfy (41)–(45) and py Dy p1: Dy ¼ D py ¼

hc hw

2hs pD2 þ hc D þ hw D hy ¼ hs pD þ 12ðhc þ hw Þ sy ¼

2s ðhs pD þ hw Þ ð1 pDÞ ð2hs pD þ hc þ hw Þ

This completes the proof.

&

ARTICLE IN PRESS 74

G. Pundoor, Z.-L. Chen / Int. J. Production Economics 119 (2009) 55–74

References Aderohunmu, R., Mobolurin, A., Bryson, R., 1995. Joint vendor-buyer policy in JIT manufacturing. Journal of the Operational Research Society 46, 375–385. Aksoy, Y., Erenguk, S., 1988. Multi-item inventory models with coordinated replenishments: a survey. International Journal of Operations and Production Management 8, 63–73. Atkins, D., Queyranne, M., Sun, D., 1992. Lot sizing policies for ﬁnite production rate assembly systems. Operations Research 40, 126–141. Bagchi, P.K., Davis, F.W., 1988. Some insights into inbound freight consolidation. International Journal of Physical Distribution and Materials Management 18 (6), 27–33. Bayindir, Z.P., Birbil, S.I., Frenk, J.B.G., 2006. The joint replenishment problem with variable production costs. European Journal of Operational Research 175, 622–640. Benjamin, J., 1989. An analysis of inventory and transportation costs in a constrained network. Transportation Science 23 (3), 177–183. Blackburn, J.D., Millen, R.A., 1982. Improved heuristics for multi-stage requirement planning systems. Management Science 28, 44–56. Blumenfeld, D.E., Burns, L.D., Daganzo, C.F., 1991. Synchronizing production and transportation schedules. Transportation Research 25B, 23–37. Blumenfeld, D.E., Burns, L.D., Diltz, J.D., Daganzo, C.F., 1985. Analyzing tradeoffs between transportation, inventory and production costs on freight networks. Transportation Research 19B, 361–380. Chan, C.K., Kingsman, B.G., 2007. Coordination in a single-vendor multibuyer supply chain by synchronizing delivery and production cycles. Transportation Research, Part E: Logistics and Transportation Review 43, 90–111. Chen, Z.-L., 2004. Integrated production and distribution operations: taxonomy, models, and review. In: Simchi-Levi, D., Wu, S.D., Shen, Z.-J. (Eds.), Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era. Kluwer Academic Publishers, Dordrecht. Chen, Z.-L., 2009. Integrated production and outbound distribution scheduling: review and extensions. Operations Research, to appear. Cooper, M., 1984. Cost and delivery time implications of freight consolidation and warehousing strategies. International Journal of Physical Distribution and Materials Management 14 (6), 47–67. Crowston, W.B., Wagner, M., Williams, J.F., 1973. Economic lot size determination in multi-stage assembly system. Management Science 19, 517–527. Dogan, K., Goetschalckx, M., 1999. A primal decomposition method for the integrated design of multi-period production–distribution systems. IIE Transactions 31, 1027–1036. Goyal, S.K., 1988. A joint economic-lot-size model for purchase and vendor: a comment. Decision Sciences 19, 236–241. Goyal, S.K., Satir, A.T., 1989. Joint replenishment inventory control: deterministic and stochastic models. European Journal of Operational Research 38, 2–13.

Hahm, J., Yano, C.A., 1992. The economic lot and delivery scheduling problem: the single item case. International Journal of Production Economics 28, 235–252. Hahm, J., Yano, C.A., 1995a. Economic lot and delivery scheduling problem: the common cycle case. IIE Transactions 27, 113–125. Hahm, J., Yano, C.A., 1995b. Economic lot and delivery scheduling problem: models for nested schedules. IIE Transactions 27, 126–139. Hahm, J., Yano, C.A., 1995c. Economic lot and delivery scheduling problem: powers of two policies. Transportation Science 29, 222–241. Hall, R.W., 1996. On the integration of production and distribution: economic order and production quantity implications. Transportation Research 30B, 387–403. Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem. International Journal of Production Research 37, 2463–2475. Jayaraman, V., Pirkul, H., 2001. Planning and coordination of production and distribution facilities for multiple commodities. European Journal of Operational Research 133, 394–408. Jensen, M.T., Khouja, M., 2004. An optimal polynomial time algorithm for the common cycle economic lot and delivery scheduling problem. European Journal of Operational Research 156, 305–311. Kaminsky, P., Simchi-Levi, D., 2003. Production and distribution lot sizing in a two stage supply chain. IIE Transactions 35, 1065–1075. Khouja, M., 2000. The economic lot and delivery scheduling problem: common cycle, rework, and variable production rate case. IIE Transactions 32, 715–725. Khouja, M., Goyal, S., 2008. A review of the joint replenishment problem literature: 1989–2005. European Journal of Operational Research 186, 1–16. Moily, J.P., 1986. Optimal and heuristic procedures for component lotsplitting in multistage manufacturing systems. Management Science 32, 113–125. Moily, J.P., Matthews, J.P., 1987. Procedures for determining relative frequencies of production/order in multistage assembly systems. Decision Sciences 18, 279–291. Moon, I.K., Cha, B.C., 2006. The joint replenishment problem with resource restriction. European Journal of Operational Research 173, 190–198. Nilsson, A., Silver, E.A., 2008. A simple improvement on silver’s heuristic for the joint replenishment problem. Journal of the Operational Research Society 59, 1415–1421. Robinson, E.P., Narayanan, A., Gao, L.-L., 2007. Effective heuristics for the dynamic demand joint replenishment problem. Journal of the Operational Research Society 58, 808–815. Sabri, E., Beamon, B.M., 2000. A multi-objective approach to simultaneous strategic and operational planning in supply chain design. Omega 28, 581–598. Schwarz, L.B., Schrage, L., 1975. Optimal and system myopic policies for multi-echelon production-inventory assembly systems. Management Science 21, 1285–1294. Viswanathan, S., 2007. An algorithm for determining the best lower bound for the stochastic joint replenishment problem. Operations Research 55, 992–996.

Joint cyclic production and delivery scheduling in a two-stage supply chain

Joint cyclic production and delivery scheduling in a two-stage supply chain

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