The joint economic lot sizing problem: Review and extensions

European Journal of Operational Research 185 (2008) 726–742 www.elsevier.com/locate/ejor

O.R. Applications

The joint economic lot sizing problem: Review and extensions M. Ben-Daya *, M. Darwish, K. Ertogral Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 15 August 2005; accepted 11 December 2006 Available online 11 January 2007

Abstract With the growing focus on supply chain management, firms realize that inventories across the entire supply chain can be more efficiently managed through greater cooperation and better coordination. This paper presents a comprehensive and up-to-date review of the joint economic lot sizing problem (JELP) and also provides some extensions of this important problem. In particular, a detailed mathematical description of, and a unified framework for, the main JELP models are given. Additionally, a comparative empirical study of the main policies proposed for JELP is conducted. The focus of this study is on assessing the deviation of these policies from the optimal solution. Studying the performance of different models provides additional insights that will help in justifying their use in more complex supply chain models that involve more stages or other practical considerations of interest.  2007 Elsevier B.V. All rights reserved. Keywords: The integrated single-vendor single-buyer problem; Joint economic lot sizing problem; Supply chain management; Inventory coordination

1. Introduction Rapid market changes fueled by the explosion of product varieties with short life cycles have increased competition in today’s global markets. In order to compete effectively, companies must provide better products and services at a reduced cost for customers with heightened expectations. This environment has forced companies to increase the efficiency of their operations in order to reduce cost and become more responsive to changes. As a natural result, companies are pushed towards not only integrating different decision processes within their operational borders but also towards closely collaborating with their customers and suppliers. With the growing focus on supply chain management, firms are increasingly realizing that inventories across the entire supply chain can be more efficiently managed through greater cooperation and better coordination. Consider a two-layer supply chain consisting of a manufacturer and a retailer. The retailer (buyer) observes a deterministic demand and orders lots from the manufacturer (vendor). The vendor manufactures the requested product in lots. Each produced lot is shipped to the buyer in batches. The manufacturer and retailer

*

Corresponding author. E-mail address: [email protected] (M. Ben-Daya).

0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.12.026

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work in a cooperative manner to synchronize the supply with the actual customer demand. In this scenario, it will be more cost-effective to determine the order quantity and the delivery schedule based on their integrated total cost function, rather than using the buyer’s or the supplier’s individual cost functions. Each produced lot is shipped to the buyer in batches. The problem is to find the number of shipments and the size of each batch such that the joint manufacturer–retailer cost is minimized. The integrated single-vendor single-buyer problem is called the joint economic lot sizing problem (JELP). This problem received a lot of attention in recent years as it is the building block for the wider supply chain. The global supply chain can be very complex and link-by-link understanding of joint policies can be very useful. In fact, for vertically integrated supply chains or chains owned by the same company, this problem can provide insights and optimal decisions that lead to global optimization. When the different entities of a supply chain have different owners, global optimization may not benefit all of them equally. In some cases, some of the parties in the supply chain may see an increase in their costs. In these cases, sharing the benefits of coordination becomes a major issue. As discussed in more detail in the next section, different shipment policies have been suggested in the literature for JELP. These policies range from the simple lot-for-lot policy to more complex policies. There is a need for an up-to-date review of this area given the wide interest in the problem shown by researchers and business people alike. The purpose of this paper is to provide a comprehensive and up-to-date review for JELP. In addition, we provide a comparative empirical study amongst the main policies proposed of JELP. The focus of this comparative study is on assessing the deviation of these policies from the optimal solution. Some of these policies have a simple structure. Therefore, having a good idea about the performance of these policies will help in justifying their use in more complex supply chains that involve more stages or introduce other practical considerations of interest. This paper is organized as follows. The next section provides a comprehensive review of the deterministic JELP and some of its extensions. Section 3 deals with a brief description of key JELP models proposed in the literature. A general framework that relates most JELP models to each other is presented in Section 4. Section 5 presents a comparative study of seven selected JELP models. Finally, Section 6 contains some concluding remarks and directions for future research. 2. Literature review In this literature review, we focus on deterministic JELP. In the earlier literature, this problem’s variations were mainly in terms of the production rate assumption for the vendor and shipment policy between two stages. The literature on the deterministic JELP evolved from a simple model with infinite production rate and lot-for-lot assumption (Goyal, 1977), to a general model with a finite production rate and a shipment policy that is not restricted in any way (Hill, 1999). More recently other dimensions were explored in the literature. These dimensions include investing on setup cost reduction, variable production cost, quality and process failure issues, stochastic demand, consideration of a lead time between buyer and supplier, multiple buyers case, transportation cost and capacity, and three-layer systems. In this section we follow the chronological developments of JELP and show how it evolved from simple shipment policies and simple models to more complex policies and models until the optimal solution to the problem was suggested by Hill (1999). A brief review of the main extensions of JELP is also presented to show how this important problem evolved in different directions. Below, we give the details of the relevant literature. The basic policies considered in the literature for JELP were based on equal-shipment sizes, geometrically increasing shipment sizes, or some combination of the two. The reason for considering these two policies is that the requirement of equal-shipment sizes is good from the buyer’s perspective. However, having a small first shipment followed by larger ones leads to a small minimum system stock, which means a reduced average system inventory as described in Section 3. We give detailed mathematical descriptions of these basic policies in Section 3. One of the early works related to JELP was due to Goyal (1977). He suggested a solution to the problem under the assumption of having an infinite production rate for the vendor and lot-for-lot policy for the shipments from the vendor to the buyer. In this policy, each production lot is sent to the buyer as a single shipment. This implies that the entire production lot should be ready before shipment. Banerjee (1986) relaxed the

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infinite production rate assumption of Goyal (1977) while retaining his lot-for-lot policy. This study is the one that coined the term JELP. Goyal (1988) contributed to the efforts of generalizing the problem by relaxing the lot-for-lot policy. He assumed that the production lot is shipped in a number of equal-size shipments, but only after the entire lot has been produced. An extensive review on JELP and its variants up to 1989 can be found in Goyal and Gupta (1989). Many other studies eliminated the restriction of requiring the completion of the production lot before starting the shipments. Lu (1995) relaxed the assumption of Goyal (1988) about completing a batch before starting shipments and explored a model which allowed shipments to take place during production. Other JELP models considering equal-shipment policies include those proposed by Banerjee and Kim (1995), Ha and Kim (1997), and Kim and Ha (2003). Goyal (1995) proposed one of the geometric policies where successive shipments from a production batch, increase by a factor equal to the ratio of production rate to the demand rate. In this paper, the author formulated the problem and gave the optimal expression for the first shipment size as a function of the number of shipments. Viswanathan (1998) called Goyal’s policy (1995) as the ‘‘deliver what is produced (DWP)’’ policy, and he named Lu’s as ‘‘identical delivery quantity (IDQ)’’ policy. Viswanathan also showed the best policy for a particular model depends on problem parameters. Hill (1997) further generalized the model of Goyal (1995) by taking the geometric growth factor as a decision variable. He suggested a solution method that does not in general guarantee optimality. This method is based on exhaustive search for both the growth factor and the number of shipments in certain ranges. He showed numerically that his policy outperforms both the equalshipment-size policy and the policy adopted by Goyal (1995). Another simple geometric-then-equal policy that produces good results was suggested in Goyal and Nebebe (2000). They proposed a policy which calls for a small shipment followed by a series of larger and equal-sized shipments. The ratio of the first shipment size to the size of the remaining equal shipments is set to be the production rate divided by the demand rate as was done by Goyal (1995). Thus, this policy tries to exploit the benefit of both the equal size and the geometric policies. Goyal (2000) suggested a method to improve the solutions obtained by the method given in Hill (1997). In his method, based on first shipment size, the following shipment sizes are increased by the ratio of production rate to demand rate as long as it is feasible to do so. For the remaining shipments, the rest of the production run is equally distributed. The resulting improvement was demonstrated with a small number of experiments. It was unclear whether the improvement was in general significant, or for what kind of problems it is so. All the work listed above looks at the problem under a certain assumption on shipment policy. Hill (1999) found the optimal solution to the problem without any assumptions about the shipment policy. He showed that the structure of the optimal policy includes shipments increasing in size according to a geometric series followed by equal-sized shipments. He also suggested an exact iterative algorithm for solving the problem. Hill and Omar (2006) revisited JELP by relaxing an assumption regarding holding costs, that has been used in all previous literature, where the holding costs were allowed to decrease down the supply chain. This version of JELP was also looked at by Zhou and Wang (2007). They showed that when the vendor’s unit holding cost is greater than the buyer’s, the optimal shipment policy consists only of unequal-sized shipments with all successive shipment sizes increasing by a fixed factor equal to the ratio of the production rate to the demand rate. The basic JELP has been extended in many different directions, as we have pointed out earlier in the beginning of the section. Nasri et al. (1991) investigated the impact of simultaneous investment in setup cost and order cost reduction, on Banerjee’s JELP model (1986). Ben-Daya and Zamin (2002b) considered the effect of an unreliable process on JELP decisions. Huang (2004) also considered JELP under process unreliability considerations. A number of authors have investigated the effect of quality on lot size for JELP. Ben-Daya and Zamin (2002a) considered the effect of imperfect processes that might shift to out-of-control state and produce non-conforming item of lot-sizing decisions under the unequal-shipment policy proposed by Goyal and Nebebe (2000). Huang (2002) considered an integrated vendor–buyer cooperative inventory model for items with imperfect quality under equal-shipment policy. Also, Darwish (2005) incorporated quality issues into JELP by considering process targeting. Other extensions of this problem considered stochastic demand. The stochastic demand dimension of this problem has not yet been studied extensively. One of the rare studies in this regard was carried out by

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Ben-Daya and Zamin (2002c). They considered a JELP problem under equal-shipment policy where the demand is stochastic. Chang et al. (2004) looked at lead-time and ordering cost reductions. They assume that buyer lead-time can be shortened with an extra crashing cost that depends on the lead-time duration to be reduced and the ordering lot size. Additionally, the buyer ordering cost can be reduced through further investment. Yang and Wee (2000), and Pan and Yang (2002) also developed integrated inventory model with controllable lead time. Ouyang et al. (2004) considered a stochastic lead time and assumed that shortage during the lead time is permitted, and lead time can be reduced at an added cost. Hoque and Goyal (2006) developed a heuristic solution procedure to minimize the total cost of setup, inventory holding and lead-time crashing for an integrated inventory system under controllable lead-time between a vendor and a buyer. Another major research direction considered the case of one vendor multi-buyer supply chains. Joglekar and Tharthare (1990) is one of these studies. They refined the JELP model by relaxing the lot-for-lot assumption, and separating the traditional setup cost into two independent costs. The first is the standard manufacturing setup cost per production run, and the second is the vendor’s cost of handling and processing an order from a buyer such that they can consider multiple identical and not identical buyers in the model. Based on these changes, they develop a refined JELP model for both one-vendor-many-identical-buyers case, and the more general one-vendor-many non-identical buyers situations. Multiple buyers issue is also considered in Affisco et al. (1988, 1991, 1993), along with costs reduction aspects. They investigated the one-vendor, many non-identical buyers JELP model with vendor setup cost reduction and buyer order cost reduction. The results indicate that there are significant cost savings for the JELP over independent optimization when such investments are made. This suggests that when an environment of cooperation between the parties has been established, the JELP is a superior policy. A more recent study that assumes multi buyer case is Yau and Chiou (2004). They considered an integrated supply chain model where the vendor supplies items to multiple buyers. The objective of the model is to minimize the vendor’s total annual cost subject to the maximum cost that the buyer is willing to incur. The authors have developed a very efficient search algorithm to solve the optimal cost curve which turned out to be piece-wise convex. Chan and Kingsman (2007) proposed a coordinated singlevendor multi-buyer supply chain model by synchronizing delivery and production cycles. The synchronization is achieved by scheduling the actual delivery days of the buyers and coordinating them with the vendor’s production cycle whilst allowing the buyers to choose their own lot sizes and order cycles. A mathematical model was developed and analyzed. The results of the numerical examples show that the synchronized-cycles policy works better than an independent optimization and restricts buyers to adopt a common order cycle. A new methodology to obtain the joint economic lot size in the case where multiple buyers are demanding one type of item from a single vendor was presented in Siajadi et al. (2006). The production is organized in such a way that the first shipment for each buyer is done in a sequence. Following this sequence, the first delivery starts from the first buyer followed by the second, the third and so on. The duration from one delivery to the next is fixed for each buyer, with equal cycle time for buyers and the vendor. Viswanathan and Piplani (2001) took into account the multi-buyer issue in a game theoretic treatment of the problem. They considered a one-vendor, multi-buyer supply chain for a single product. Under the proposed strategy, the vendor specifies common replenishment periods and requires all buyers to replenish only at those time periods. The vendor offers a price discount to entice buyers to accept this strategy. The optimal replenishment period and the price discount to be offered by the vendor are determined as a solution to a Stackelberg game. Stackelberg game is a sequential game where a leader makes a first game decision considering the possible actions of the follower who determines a decision in the second stage. In this study the vendor is assumed to be the leader in the game. Other studies that use a game theoretic approach on a related problem include Bylka (2003) and Hsiao and Lin (2005). They discussed an EOQ model involving lead time as a decision variable on a Stackelberg game in the supply chain. That is, the distribution channel system contains one supplier and a single retailer and the supplier in the channel holds a monopolistic status, in which he does not only own cost information about the retailer, but also has the decision-making right of the lead time. The optimal lead-time of the supplier and order cycle time of the retailer, are investigated. A recent review focusing on the coordination mechanisms between vendor and buyer through quantity discount schemes is presented in Sarmah et al. (2006). Hoque and Goyal (2000) extended the literature by assuming capacitated transport equipment. They developed an optimal solution procedure for the single-vendor single-buyer production–inventory system with

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unequal and equal-sized shipments from the vendor to the buyer, and under the capacity constraint of the transport equipment. Consideration of the capacitated transport equipment is the main deviation of this study from the earlier literature on the problem. Transportation cost in JELP was brought into the picture in Ertogral et al. (2007). The authors integrated transportation cost explicitly into a JELP model under equal-shipment policy. They have also considered an over-declaration option that exploits the assumed all-unit-discount transportation cost structure. Another paper that considers transportation cost is Abad and Aggarwal (2005). They studied the problem of determining the reseller’s lot size and pricing assuming that the reseller is responsible for paying for freight. They assumed that the final demand for the product is sensitive to the selling price that the reseller sets. More recent extensions of JELP considered more than two-stage supply chains. An integrated Inventory control model, making joint economic lot sizes of manufacturer’s raw material ordering, production batch, and buyer’s ordering, was proposed by Lee (2005). Compared to the basic JELP problem, this paper takes into consideration raw material ordering from the supplier of the manufacturer. Muson and Rosenblatt (2001) considered a three-level chain (supplier–manufacturer–retailer) and explored the benefits of using quantity discounts at both ends of the supply chain to reduce costs. Khouja (2003) looked at a three-stage supply chain where a firm can supply many customers. Three inventory coordination mechanisms between chain members were considered. Ben-Daya and Al-Nassar (in press) further generalized Khouja’s work (2003) to the case where shipment between stages can be made before a whole lot is completed. Ben-Daya and As’ad (2006) also investigated different shipment policies between members of a three-layer supply chain. The above literature review traces the development of JELP from a model of simple lot-for-lot shipment policies to more complex policies until the optimal solution to this problem was derived. We also showed how JELP was extended in different directions to incorporate more practical considerations. In the next section we focus on selected problems that represent distinct shipment policies and present their mathematical models. The purpose of this exercise is twofold. First, this is needed to generate the unified framework developed in Section 4 which presents a common platform that will help in comparing the different models to be studied here. Second, this is needed to provide the necessary background for the comparative experiments conducted in Section 5. 3. Description of basic JELP models In this section, we focus on the main JELP models. These models are selected to provide a representative set of JELP models that presents distinct shipment policies. We provide a more detailed mathematical description of these basic deterministic JELP models and an empirical comparison of these models will also be conducted in Section 5. First, we present some of the common assumptions made in the literature for the JELP and introduce a notation that will be used throughout this paper. Then, a general formulation of JELP is given followed by descriptions of basic JELP models including the lot-for-lot model (Banerjee, 1986), equal-sized shipments models (Goyal, 1988; Lu, 1995) and geometric shipments policy (Goyal, 1995; Hill, 1997). Next, we present geometric-then-equal-sized shipments policy which is a hybrid between Lu’s policy (1995) and Goyal’s (1995). Finally, the optimal policy derived by Hill (1999) is outlined. 3.1. Assumptions and notation We consider a vendor that manufactures a product in lots at a rate P and incurs a setup cost Av per lot. Each production lot is delivered to a buyer in n shipments. The buyer incurs a fixed order/delivery cost Ab per shipment. The vendor and buyer incur holding costs hv and hb, respectively. The objective is to view the system as an integrated whole and determine the production lot size and shipments schedule which minimize the average total cost per unit time. The following assumptions are made in deriving JELP models: 1. The demand rate is deterministic and constant. 2. Shortages are not allowed. 3. Time horizon is infinite.

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4. The produced lot is transferred from the vendor to buyer in n shipments. 5. It is also assumed that P > D and hb > hv . The former assumption ensures that the problem is not trivial while the latter is a reasonable assumption because stock value usually increases as a product moves down the supply chain and the associated holding costs increase accordingly. The following notation is used: D P Av Ab hv hb Iv Ib Is qi n Q T TP TC

demand rate production rate for the vendor production setup cost buyer ordering cost holding cost for the vendor holding cost for the buyer average vendor inventory average buyer inventory average system inventory size of shipment i number of shipments P production lot size ðQ ¼ ni¼1 qi Þ inventory cycle length ðT ¼ Q=DÞ production cycle length ðT P ¼ Q=P Þ average total cost per unit time

3.2. Generic model Fig. 1 shows the vendor’s inventory, buyer’s inventory and the system’s inventory for a general JELP. These three inventories are represented by wide-dash line, narrow-dash line, and solid line, respectively. In order to ensure that the buyer has enough inventory until the vendor produces the first shipment, the system inventory is q1 D=P at the start of production. The system inventory increases at a rate of ðP  DÞ during the production phase and decreases at a rate of D during depletion period during downtime. Therefore, the average system inventory can be set out as Is ¼

q1 D ðP  DÞQ þ : P 2P

ð1Þ

Before finding the average buyer inventory, we have to point out that, because hb > hv , the vendor delivers a shipment only when the buyer’s inventory is just about to run out. Note that the buyer takes qi =D time units to consume the ith shipment. Therefore, the average buyer inventory is as follows:

System inventory Buyer inventory Vendor inventory -D

P-D P

-D time

TP T

Fig. 1. Inventory level against time for n ¼ 3 for JELP general policy.

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Pn Ib ¼

2 i¼1 qi

2Q

ð2Þ

:

The average total cost of the system is given by the sum of setup cost (vendor), ordering cost (buyer) and the inventory costs of both the vendor and the buyer. It can be expressed as TC ¼

ðAv þ nAb ÞD þ hv I s þ ðhb  hv ÞI b : Q

ð3Þ

In order to dispatch the (i + 1)th shipment on time, the time to consume all available items up to the first i shipments, at a rate of D, must be at least as great as the time needed to produce the first (i + 1) shipments, at a rate of P, that is qi 6 q1 þ ðk  1Þ

i1 X

qj ;

i ¼ 2; 3; . . . ; n:

ð4Þ

j¼1

Therefore, the general JELP can be formulated as follows: ðAv þ nAb ÞD Dq þ hv 1 Q P Pn 2 ðP  DÞQ q þ ðhb  hv Þ i¼1 i þ hv 2P 2Q n X subject to : Q ¼ qi ; min TC ¼

i¼1

qi 6 ðk  1Þ

i1 X

qj þ q1 ;

i ¼ 2; 3; . . . ; n:

j¼1

The decision variables in this general model are q1 ; q2 ; . . . ; qn and n. We have to point out that JELP policies differ in their assumptions regarding the relations between q1 ; q2 ; . . . ; qn . 3.3. Lot-for-lot policy This model was proposed by Banerjee (1986) to generalize Goyal’s (1977) model by relaxing the assumption of infinite replenishment rate for the vendor. Also, it is assumed that each production lot is dispatched as a single shipment, that is n ¼ 1. Fig. 2 illustrates the inventory patterns for the vendor, buyer and system for this model. Using Eqs. (1) and (2) with n ¼ 1 and Q ¼ q1 , we can find the system and buyer inventories, as reported in Table 1.

System inventory Buyer inventory Vendor inventory

-D

P-D P -D

time

TP T

Fig. 2. Inventory level against time for lot-for-lot policy.

Policy

Q

Generic model

n P

Is

qi

i¼1

q1 D ðP  DÞQ þ P 2P

Ib

Optimal qðq1 )

Decision variable(s)

NA

n, qis

Pn

2 i¼1 qi

2Q

q

  nq D 1þ 2 P

q 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA þ nAb ÞD  D v1 hv 2P þ 2 þ ðhb  hv Þ 12

q

Delayed equal-size shipments (Goyal, 1988)

nq

  D n q þ P 2

q 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA þ nAb ÞD   v   n2 hv DP þ n2 þ ðhb  hv Þ 12

n; q

None-delayed equal-size shipments (Lu, 1995)

nq

  D ðP  DÞn q þ P 2P

q 2

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ÞD u h ðAv þ nA i b

t þ ðhb  hv Þ 12 n2 hv DP þ ðP DÞn 2P

n; q

Geometric policy (Goyal, 1995; Hill, 1997)

q1 ðkn  1Þ k1

q1 W1 ðn; kÞ

q1 ðkn þ 1Þ 2ðk þ 1Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ðk  1ÞðAv þ nAb ÞD u

t ðkn þ1Þ n ðk  1Þ hv W1 ðn; kÞ þ ðhb  hv Þ 2ðkþ1Þ

n; q1 ; k

Lot-for-lot (Banerjee, 1986)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q1 ðAv þ nAb ÞD W4 ðn; m; kÞ 2 ½W2 ðn; m; kÞ2 ðhv W3 ðn; m; kÞ þ ðhb  hv ÞW4 ðn; m; kÞÞ h i h m i h h ii h 2m i n 1Þ 1Þ DÞ ðkm 1Þ k 1þðnmÞk2m ðk2 1Þ m ; W2 ðn; m; kÞ ¼ ðkk1 ; W4 ðn; m; kÞ ¼ ðkþ1Þ þ ðn  mÞkm ; W3 ðn; m; kÞ ¼ DP þ ðP2P W1 ðn; kÞ ¼ DP þ ðP DÞðk ½km 1þðnmÞkm ðk1Þ . 2P ðk1Þ k1 þ ðn  mÞk Geometric-then-equal (Goyal and Nebebe, 2000; This paper)

q1 W2 ðn; m; kÞ

q1 W3 ðn; m; kÞ

n; m; q1 ; k

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Table 1 Different policies and model components

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3.4. Equal-sized shipments policies In these policies, the size of shipments are equal, therefore, Q = nq. These policies are attractive because they are easy to implement. Two types of equal-sized shipments models are found in literature. We call the first type ‘‘delayed equal-sized shipment policy’’. In this case, shipments are delayed until the production lot is completed. We call the other type ‘‘non-delayed equal-sized shipment policy’’, where the vendor delivers shipments to the buyer during the production phase. 3.4.1. Delayed equal-sized shipments policy Goyal (1988) considered a model with a finite production rate. He relaxed the lot-for-lot assumption by considering that each production lot is delivered to the buyer in n shipments of equal size, q. However, the vendor has to produce the entire lot before delivering any shipment to the buyer for this policy, the inventory patterns for the vendor, the buyer and the system are shown in Fig. 3. The inventories of the system and the buyer are found by Eqs. (1) and (2) and are also reported in Table 1. 3.4.2. Non-delayed equal-sized shipments policy This policy is developed by Lu (1995). It allows the vendor to deliver shipments during production, which is the relaxation of the assumption made by Goyal (1988) about completing a lot before starting shipments. In this policy, the vendor delivers a lot to the buyer in n shipments of equal size, q. From Fig. 4, the average system and buyer inventories can be found. 3.5. Geometric shipments policy Goyal (1995) developed a policy which involves successive shipments within a production lot increasing by a factor equal to the production rate divided by the demand rate, that is

System inventory Buyer inventory Vendor inventory -D

P-D P -D

time

TP T

Fig. 3. Inventory level against time for n ¼ 3 for equal-sized shipments policy.

System inventory Buyer inventory Vendor inventory

-D

P-D P -D TP

time T

Fig. 4. Inventory level against time for n ¼ 5 for equal-sized shipments policy.

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qi ¼ ki1 q1 ;

735

i ¼ 2; 3; . . . ; n;

where k ¼ P =D. Using Eqs. (1) and (2), one can find the average inventory of both the system and buyer for this policy, the inventory patterns for the system, the buyer and the vendor are shown in Fig. 5. We have to point out that the policy presented by Hill (1997) is a generalization of the geometric shipments policy and equal-sized-shipments policies. He considered the case where k is a decision variable in the geometric shipments model. Therefore, the decision variables in this model are q1, n and k. The cost function for this policy is similar to that for the geometric shipments policy. The range of values of k is 1 6 k 6 P/D. This policy yields geometric shipments and equal-sized shipments policies if k ¼ P =D and k ¼ 1, respectively. 3.6. Geometric-then-equal-shipments policy This policy has not been considered before, although a similar idea already exists in the literature in the context of a serial multi-stage production system (Goyal and Szendrovits, 1986). This production and delivery policy involves successive shipments within a production lot such that the first m shipments increase according to a geometric series with factor k, the remaining ðn  mÞ shipments are equal to the last of non-equal shipments, that is, ( ki1 q1 ; i ¼ 2; 3; . . . ; m; qi ¼ km1 q1 ; i ¼ m þ 1; . . . ; n: The production lot size can be found as follows:  m  m X ðk  1Þ i1 m m þ ðn  mÞk : Q¼ k q1 þ ðn  mÞk q1 ¼ q1 k1 i¼1 Fig. 6 illustrates the inventory patterns for the system, the buyer and the vendor for m ¼ 3 and n ¼ 5. The system and buyer inventories can be set out as      q1 D ðP  DÞ q1 ðkm  1Þ D ðP  DÞ ðkm  1Þ m m þ þ ðn  mÞk q1 ¼ q1 þ þ ðn  mÞk Is ¼ P 2P k1 P 2P k1 and Pm Ib ¼

2 2 i1 q1 Þ þ ðn  mÞðkm q1 Þ i¼1hðk i m 2 q1 ðkk11Þ þ ðn  mÞkm q1

  q1 k2m  1 þ ðn  mÞk2m ðk2  1Þ ¼ : 2 ðk þ 1Þ½km  1 þ ðn  mÞkm ðk  1Þ

The parameter k in the above equations may be taken equal to P/D or as a decision variable. These options will be explored in Section 4.

System inventory Buyer inventory Vendor inventory

P-D

-D P -D

time

TP T

Fig. 5. Inventory level against time for n ¼ 3 for geometric shipments policy.

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System inventory Buyer inventory Vendor inventory

-D

P-D P -D TP

time T

Fig. 6. Inventory level against time for m ¼ 2 and n ¼ 4 for geometric-then-equal-sized shipments policy.

3.7. Optimal policy The structure of the optimal policy has been derived by Hill (1999). He used Lagrange method recursively to obtain the structure of the optimal policy. He found that the optimal policy involves n shipments, the first m of which are unequal and increase as a geometric series with factor k ¼ P =D, the remaining ðn  mÞ ones are all of equal size, qe, therefore ( i1 k q1 ; i ¼ 2; 3; . . . ; m; Pm qi ¼ Q qk qe ¼ nmk¼1 ; i ¼ m þ 1; . . . ; n: Fig. 7 shows the inventory levels for the system, vendor and buyer. In this figure, the production lot is delivered to the buyer in four shipments, two of which are of equal size. The only difference between this policy and the geometric-then-equal-sized shipments policy is that the optimal policy allows the equal-sized shipments to take values which may not be the same as the size of the last unequal shipment. Also, Figs. 6 and 7 are similar, except that in Fig. 6, we have q3 ¼ q4 ¼ q2 while in Fig. 7, we have q3 ¼ q4 6¼ q2 . The decision variables for this policy are the size of the first shipment, q1, the size of equal shipments, qe, the number of shipments, n, and the number of unequal shipments, m. Hill (1999) devised an iterative algorithm to find the optimal solution for this policy. 3.8. Summary of JELP basic models Table 1 summarizes the models considered in this section except for the optimal policy (Hill, 1999) where the optimal solution is found recursively. Closed-form solutions for the optimal size of the first shipment, q(q1)

System inventory Buyer inventory Vendor inventory -D P-D P -D TP

time T

Fig. 7. Inventory level against time for m ¼ 2 and n ¼ 4 for optimal policy.

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737

are reported in column 5 of Table 1. In addition, we include the decision variables associated with each model in column 6 of this table. Also, the cost functions for the models discussed in this section can be derived directly from Eq. (3) and they all satisfy condition (4). 4. United framework The previous section shows that the optimal shipment policy is to start with batches increasing by a fixed factor equal to the ratio of production rate to demand rate, followed by shipments of equal size. A close look at the structure of the optimal shipment policy reveals that all previously developed policies can be considered as special cases of this general optimal policy. This result can be used to provide the following general formulation to the problem that accommodates all the policies discussed in the previous section: Pn 2 ðAv þ nAb ÞD Dq1 ðP  DÞQ q ðGFÞ min TC ¼ þ hv þ ðhb  hv Þ i¼1 i þ hv ð5Þ Q 2P P 2Q subject to : n X Q¼ qi ; ð6Þ i¼1

qi 6 ðk  1Þ (

i1 X

qj þ q1 ;

i ¼ 2; 3; . . . ; n;

ð7Þ

j¼1

ki1 q1 for i ¼ 2; . . . ; m; qe for i ¼ m þ 1; . . . ; n; P 16k6 ; D q1 ; qi ; qe ; Q P 0; i ¼ 2; . . . ; n: qi ¼

ð8Þ ð9Þ ð10Þ

Fig. 8 shows how the different policies discussed in Section 4 can be obtained as special cases of the general formulation (GF) given by Eqs. (5)–(10).

General Formulation Equations (5)-(10)

qe = qm This paper (Section 3.6)

m=2

m=n

Lu (1995)

qf = qm and λ = Pi / Pi +1

This paper (Section 3.6)

Hill (1997)

Goyal et al. (1986)

n = 1 (lot-for-lot)

λ = P/D Goyal and Nebebe (2000)

λ = P/D Goyal et al. (1995)

m = 1 (equal qi’s)

Goyal (1977)

Fig. 8. Model classification.

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Note that if qe ¼ qm ¼ km1 q1 in (GF), we obtain a policy similar to that proposed by Goyal and Szendrovits (1986) in the context of a multi-stage production system. This policy calls for shipments between stages to increase by fixed factors equal to the ratios of the production rates of adjacent production stages and then followed by equal shipments equal to last shipment in the increasing series of batches. In Section 3, we derived the cost function of a similar policy for JELP where the factor, k, is a decision variable and 1 6 k 6 P =D. This policy is denoted by GEV. The remaining policies can be all derived from GEV as follows: • The equal-shipment policy first proposed by Lu (1995) can be derived from GEV by letting m ¼ 1 and k ¼ P =D. • The policy proposed by Hill (1997) calling for shipment sizes increasing by a geometric factor can be obtained by letting m ¼ n. • If we let m ¼ n and k ¼ P =D, we obtain the policy proposed by Goyal (1995). • The policy calling for a small shipment q1 followed by equal shipments of sizes (P/D)q1, proposed by Goyal and Nebebe (2000) can be obtained by letting m ¼ 2 and k ¼ P =D. • In this paper, we explore a version of the policy by Goyal and Nebebe (2000) where k is a decision variable. • The lot-for-lot policy (Goyal, 1977) can be obtained by letting n ¼ 1. 5. Numerical results In this section, we provide a comparison of the following seven policies proposed in the literature for this problem: 1. The equal-shipment policy with k ¼ P =D is denoted by E in Tables 2–4. 2. The policy, where shipment sizes are increasing with a factor equal to k ¼ P =D, is denoted by GC. 3. The policy, where shipment sizes are increasing with a factor 1 6 k 6 P =D taken as a decision variable, is denoted by GV. 4. The policy, where shipments are increasing by a factor k ¼ P =D followed by equal shipments, is denoted by GEC. 5. The policy, where shipments are increasing by a factor 1 6 k 6 P =D, taken as a decision variable followed by equal shipments, is denoted by GEV. 6. The policy starting with a small shipment of size q1, followed by equal shipments of size ðP =DÞq1 , is denoted by SLC. 7. The policy starting with a small shipment of size q1, followed by equal shipments of size kq1, where 1 6 k 6 P =D is a decision variable, is denoted by SLV. Table 2 Effect of the buyer’s holding cost hb

Percent deviation from optimal solution E

GC

GV

GEC

GEV

SLC

SLV

5 7 9 11 13 15 17 19 21

6.17 3.58 2.60 2.07 1.68 1.42 1.22 1.05 0.93

1.42 7.75 13.96 19.55 24.53 29.02 33.02 36.73 40.13

1.21 1.72 1.55 1.35 1.18 1.04 0.90 0.80 0.77

0.02 0.16 0.07 0.03 0.02 0.02 0.02 0.01 0.02

0.02 0.16 0.07 0.03 0.02 0.01 0.01 0.01 0.00

0.84 0.16 0.07 0.03 0.02 0.02 0.02 0.01 0.02

0.84 0.16 0.07 0.03 0.02 0.01 0.01 0.01 0.00

Mean St. dev.

2.30 1.68

22.90 13.29

1.17 0.33

0.04 0.05

0.04 0.05

0.13 0.27

0.13 0.27

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Table 3 Effect of vendor’s setup cost Av

Percent deviation from optimal solution E

GC

GV

GEC

GEV

SLC

SLV

50 100 150 200 300 400 500 750 1000

6.30 6.30 6.57 6.57 6.20 6.17 5.74 5.37 4.95

0.00 0.00 0.45 0.47 0.75 1.42 1.87 2.45 2.90

0.00 0.00 0.45 0.46 0.75 1.21 1.21 1.47 1.49

0.00 0.00 0.45 0.46 0.01 0.02 0.01 0.03 0.04

0.00 0.00 0.45 0.46 0.01 0.02 0.01 0.03 0.04

0.00 0.00 0.46 0.46 0.75 0.84 0.84 1.02 1.04

0.00 0.00 0.46 0.46 0.75 0.84 0.84 1.02 1.04

Mean St. dev.

6.02 0.55

1.15 1.07

0.78 0.59

0.11 0.19

0.11 0.19

0.60 0.40

0.60 0.40

Table 4 Effect of production rate P

Percent deviation from optimal solution E

GC

GV

GEC

GEV

SLC

SLV

1500 2000 2500 3000 3500 4000 5000 6000 7000 8000 10,000

13.53 9.96 7.85 6.57 5.51 4.66 3.77 3.11 2.48 2.00 1.30

0.19 0.59 0.89 1.53 1.39 1.52 2.17 2.39 2.25 2.18 2.14

0.19 0.59 0.88 1.18 1.24 1.20 1.15 1.09 0.96 0.80 0.50

0.04 0.03 0.15 0.01 0.06 0.29 0.04 0.01 0.01 0.01 0.01

0.04 0.03 0.15 0.01 0.06 0.25 0.04 0.01 0.01 0.00 0.00

8.10 3.98 2.11 1.11 0.56 0.30 0.04 0.01 0.01 0.01 0.01

8.10 3.98 2.11 1.11 0.56 0.30 0.04 0.01 0.01 0.00 0.00

Mean St. dev.

5.52 3.73

1.57 0.75

0.89 0.34

0.06 0.09

0.05 0.08

1.48 2.52

1.47 2.52

The deviation of these seven policies from the optimal policy developed by Hill (1999) will now be assessed. In the first part of this empirical study, we perform a sensitivity analysis with respect to key parameters for the example presented in most papers dealing with deterministic JELP (e.g. Hill, 1997, 1999; Goyal and Nebebe, 2000). This example has the following parameters: Av ¼ 400;

Ab ¼ 25;

hv ¼ 4;

hb ¼ 5;

P ¼ 3200;

D ¼ 1000:

The numerical results of the sensitivity analysis with respect to the difference between hv and hb, Av and Ab, and P and D, are presented in Tables 2–4, respectively. Note that the GEC and GEV policies performed very well with a maximum deviation of less than half a percent across all ranges of parameters considered. This comes as no surprise since its solution structure is close to that of the optimal solution. However, the simple policy of one small shipment followed by equal shipments performed very well also. Its performance is not sensitive to changes in holding cost and setup costs but was somewhat sensitive to the difference between production and demand rates. But even in this case, the average deviation from the optimal solution is around 1.5%. The GC and GV methods based on the geometric policy provided solutions within an average of 1.57% of the optimal solution. The equal-shipment policy E had the maximum deviation from the optimal solution, around 6% on average, which is a significant deviation.

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Table 5 Frequency of m for 10,000 randomly generated problems m

0

1

2

3

P4

Frequency % Cumulative

702 7.02 7.02

2585 25.85 32.87

2282 22.82 55.69

1253 12.53 68.22

3178 31.78 100

The SLC and SLV policies are two simple policies that performed well on the problems solved above. In these policies, we have a single small lot followed by larger equal-size lots ðm ¼ 2Þ, as mentioned before. To see if such simple policies perform well in general on a more diversified problem domain, a set of random problems was generated as follows: The production and demand rates, P and D, are generated uniformly in the interval [100, 100,000], the holding costs, hv and hb, are generated uniformly in the interval [1, 100], and the setup and ordering costs, Av and Ab, are generated uniformly in the interval [10, 5000]. Ten thousand problems were solved using the optimal solution procedure. Table 5 provides the frequency of optimal solution with various values of m. For example, 32.38% of the problems have an optimal value m ¼ 1, and 55.69% of the problem have optimal values of m less than or equal to 2. This is in line with the results obtained earlier and shows that SLC policy provides good solutions. The comparison of the JELP models considered in this section provided additional insights into these models that cannot be captured directly from their formulations. In fact, some simple policies provide solutions that are very close to the optimal solution. These findings can be very useful when the model is extended to more complex situations such as three-layer supply chains or problems where other practical situations are taken into consideration such as quality, maintenance or setup cost reduction. Such extensions have already been considered as discussed in Section 2. However, they were developed using equal-sized shipments policies. The insights gained from this section can be used to explore other shipment policies in the context of the complex models of various JELP extensions. 6. Conclusion and directions for future research JELP received a great deal of attention in recent years as it is the building block for the wider supply chain. In this paper, we provided an up-to-date review for the deterministic integrated single-vendor single-buyer problem together with a mathematical description of the basic JELP models. We also provided a general formulation of the problem based on the structure of the optimal solution. Most policies suggested for this problem are special cases of the proposed generalized formulation. Furthermore, a comparative empirical study amongst the main policies proposed for JELP was conducted. The geometric-then-equal policies (GEC and GEV), in particular, provided solutions very close to the optimal one. Extensions to JELP are evolving in two main directions. The first line of research deals with extensions aimed at relaxing assumptions that may not be realistic in certain practical situations such as the assumptions of deterministic demand, perfect product quality, and completely reliable production systems. Research efforts in this direction are likely to continue in the future. A more recent trend involves looking at the integration of more than two stages. Many papers involving three stages appeared recently and this trend is likely to continue. Other promising areas of research include • Consideration of more complex supply chains networks involving many stages and many entities at each stage. • Explicit consideration of transportation in these models in addition to inventory and production decisions. The authors extended JELP to include transportation considerations (Ertogral et al., 2007) and are in the process of incorporating additional aspects to the extended model such as maintenance, quality, and stochastic demand. Some work is already underway to extend JELP to three-layer supply chain and investigate various shipment policies between the different stages.

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