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Analyzing the evolutionary stability of the vendor-managed inventory supply chains
Computers & Industrial Engineering 56 (2009) 274–282
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Analyzing the evolutionary stability of the vendor-managed inventory supply chains Haisheng Yu a, Amy Z. Zeng b,*, Lindu Zhao a a b
Institute of Systems Engineering, Southeast University Nanjing, Jiangsu 210096, PR China Department of Management, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA
a r t i c l e
i n f o
Article history: Received 3 December 2007 Received in revised form 30 May 2008 Accepted 31 May 2008 Available online 8 June 2008 Keywords: Supply chain coordination Vendor-managed inventory Game theory Evolutionary stability Replicator dynamic
a b s t r a c t One of the widely used strategies for achieving system integration in the arena of supply chain management is the vendor-managed inventory (VMI) approach. Although there exists a large amount of literature that examines various aspects of VMI, little looks at how this model evolves as the implementation progresses and matures. Therefore, this paper shows how to analyze the intrinsic evolutionary mechanism of the VMI supply chains by applying the evolutionary game theories. It is found that during the early stage of the VMI implementation, the upstream supplier will have some profit loss; however, as the transaction quantity increases in the long run, which will eventually benefit the entire chain, it is necessary for the downstream buyer to share profit with the upstream supplier to cover the supplier’s initial loss in order to exploit and sustain the benefits of the VMI. Additionally, the impact factors for VMI to become an evolutionary stable strategy are examined. All the results identify the conditions under which the VMI model is favorable over the traditional chain structure and shed lights on when and why collaboration is critical for a successful, long-term implementation of VMI. 2008 Elsevier Ltd. All rights reserved.
1. Introduction A great deal of evidence has shown that vendor-managed inventory (VMI) approach can improve supply chain performance by decreasing inventory-related costs and increasing customer service; as such, the industrial applications of VMI have grown constantly over the past decade. Unlike a traditional supply chain wherein each member manages its own inventories and makes individual stocking decision, VMI is a collaborative initiative where a downstream buyer shifts the ownership of inventories to its immediate upstream supplier and allows the supplier to access its demand information in return. In particular, a VMI process involves the following two steps: (1) a downstream buyer provides demand information to its immediate upstream supplier and leaves the stocking decisions to that supplier; and (2) the upstream supplier has the ownership of the inventories till the inventories are shipped to the buyer and bears the risk of demand uncertainty. It is not difficult to see that the VMI structure promotes collaborations between suppliers and buyers through information sharing and business process reengineering. The potential benefits of VMI are very compelling and the main benefits include substantial reduction of inventory-related costs for both suppliers and buyers and noticeable improvement of customer service levels as indicated by shorter order cycle times and higher fill rates (Achabal, * Corresponding author. Tel.: +1 508 831 6117; fax: +1 508 831 5720. E-mail addresses: [email protected] (H. Yu), [email protected] (A.Z. Zeng), [email protected] (L. Zhao). 0360-8352/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.05.016
Mclntyre, Smith, & Kalyanam, 2000). The success of VMI has been experienced by many well-known retailers and their first-tier suppliers, most notable is Wal-Mart and its key suppliers like Proctor and Gamble (Cetinkaya & Lee, 2000). However, a supply chain is a complex system consisting of numerous interrelated channel entities, such as suppliers, intermediaries, third-party service providers, and final customers. These firms are connected by material, financial and information flows that travel both directions along the supply chain. As such, despite of VMI’s growing popularity, implementing this approach can be more challenging and difficult in practice than what is perceived in theories, especially when the trust between supply chain members and their actual commitments are under question. The underlying cause that may destroy a trusting relationship is the bounded rationality, which, as pointed out by Sterman (1989), is one of the reasons responsible for the occurrence of the Bullwhip Effect. Therefore, from the standpoint of bounded rationality and in comparison with the classic game theories, the evolutionary games may provide an additional avenue to study the effectiveness of VMI. As such, analyzing the evolutionary stability of the VMI supply chains is the subject of this paper. Although there exists a large amount of literature that examines various aspects of VMI, little looks at how this model evolves as the implementation progresses and matures. The aim of this paper is to investigate the intrinsic evolutionary process of the VMI supply chains by obtaining the evolutionary stable strategy (ESS) through the help of the evolutionary game theories. The other contribution of the paper can be seen from the following two perspectives. First,
H. Yu et al. / Computers & Industrial Engineering 56 (2009) 274–282
this paper takes a few steps further from the existing research efforts and provides an in-depth analysis of the VMI supply chains using evolutionary game theories. Second, the results of this paper not only identify the conditions under which the VMI model is favorable over the traditional chain structure, but also explicitly demonstrates why it is necessary for the involving parties to share benefits during the early stage of VMI implementation in order for the benefits of this strategy to increase and sustain in a long run. The remainder of the paper is organized as follows: Section 2 provides a brief review of the previous works that are closely related to our research. In Section 3, we describe the details of a two-stage VMI supply chain model and obtain the profit functions of the upstream and downstream members. In Section 4, we apply Replicator dynamics to analyze the evolutionary stability of the VMI supply chain based on each channel member’s profit function, from which the short-term and long-term evolutionary mechanisms are obtained. In Section 5, we analyze the factors that affect the stability of the VMI approach. Finally, we conclude the paper with a summary and discussion in Section 6.
2. Literature review In recent years, the issues related to supply chain channel coordination in different forms and contexts have been addressed extensively in the literature. The VMI model is a channel coordination strategy between downstream and upstream players. In this model, based on their common interests and objectives, the two players reach an agreement that the upstream supplier will manage the downstream member’s inventory stock decisions and that both will monitor and modify their agreed-upon terms (Kisperska, 1999) in the course of VMI implementation. Since its adoption in the arena of supply chain management in early 1980s, the VMI concept has received a great deal of attention, as the model differs significantly from the traditional practice in that the chain members shift their relationships from an arm’s length to a strategic partnership. Moreover, the VMI method emphasizes integration and collaboration, which have been becoming the core theme of the contemporary supply chain management philosophy. In this section, we provide a brief review of the existing studies pertinent to our work, which are classified into four streams. The first stream focuses on the ‘‘why-VMI” question, that is, the perceived importance and benefits of this model. For example, the studies by Vergin and Barr (1999) and Lee et al. (2005) conclude that VMI is becoming an effective approach for implementing the channel coordination initiative, which is critical and imperative to improve the entire chain’s financial performance. Waller, Johnson, and Davis (1999) indicate further that the VMI method can improve inventory turnover and customer service levels at every stage of a supply chain. In a more in-depth analysis, Disney and Towill (2002) find that the kernel goal of VMI chains, which is minimizing the channel cost while simultaneously satisfying some degree of customer service levels, is achieved primarily by sharing demand and inventory information. The second research stream addresses the ‘‘how-to” question. Specifically, this stream builds upon the belief that the success of the VMI method requires extensive technology enabling, especially from the support of information technologies. The representative study of this stream is given by Xu, Dong, and Evers (2001), in which they examine the impacts of EDI and Internet-based technologies on the practice of VMI. They also find that the application of VMI will improve the demand and inventory information sharing between the upstream and downstream members. In contrast, a survey conducted by Tyan and Wee (2003) points out that aside from the computer technologies, the key of implementing VMI lies in the abilities of the related chain members to cooperate and to
275
understand the flows and processes concerning their products or services delivery. Although in theory, the VMI model is one of the most efficient coordination strategies for managing supply chains, it possesses numerous challenges and problems in practice that may lead to either a success or a failure like in a competitive game. Therefore, in the third research stream, many scholars reply on game theories to study VMI supply chains with specific structures and configurations. For example, a study by Huang and Li (2001) investigates a two-stage game mechanism, called upstream dominance and downstream dominance, in cooperative advertising models for a manufacturer–retailer supply chain. Wang, Guo, and Efstathiou (2004) analyze the non-cooperative behavior in a decentralized two-echelon supply chain consisting of one supplier and multiple retailers. They find that competition occurs among all the retailers as well as between the supplier and each retailer. In order to achieve the optimal cooperation in the system, the authors have designed several Nash equilibrium contracts based on the echelon-inventory games and local inventory games. The last research stream also employs the game theories but from a social and economic standpoint. In social and economic systems, the behavior of a member in a system often affects a few other members, and there exist numerous examples of situations under which the so-called free riders or defectors take advantage of others for a common good. In the context of channel coordination, a game-theoretic approach can be applied based on the Prisoners’ Dilemma game – a situation that captures the temptation to act in a selfish way to gain a higher reward instead of sharing a reward by cooperation (e.g., Kuhn, 2003). An example of such research effort is given by Eriksson and Lindgren (2005), in which they investigate how cooperation is driven by mutations in multi-person Prisoners’ Dilemma and analyze the impacts of the payoff parameters, the initial population, the mutation rate, and the group size on the efficiency of the channel coordination. The literature generated from the last two research streams mainly relies on the classic games. In understanding the bounded rationality of channel members of a supply chain, we think that the evolutionary game theory can provide another way to study the channel coordination mechanism. Evolutionary game theories originate as an application of the mathematical theory of games to biological problems based on the realization that frequencydependent fitness introduces a strategic aspect to evolution. Recently, there has been an increased interest in using this theory by economists, sociologists, and anthropologists that are social scientists in general as well as philosophers (e.g., Christina, 2007; Michael, 2008; Mizuho & Toshio, 2007; Xiao & Yu, 2006). The evolutionary game theory consists of two main approaches. The first approach derives from the work of Maynard-Smith and Price (1973) and employs the concept of an evolutionarily stable strategy as the principal tool of analysis. The second approach constructs an explicit model of the evolution process in which the frequency of strategies changes with the population sizes, and studies the properties of the evolutionary dynamics based on that model. Although there exists one study by Wang and Meng (2004) that develops an evolutionary game model for supply chain partnerships from an evolutionary game’s viewpoint, only a simple theoretical framework is presented, and the effects of the contract and the penalty mechanisms on the effectiveness of the channel coordination are not considered. Therefore, in this paper, we take a few steps further from the existing research efforts and provide an in-depth analysis of the VMI supply chains using evolutionary game theories. We will first present the profit functions of the members in a two-stage VMI chain, based on which we examine the channel’s evolutionary stability trend from a short-term and long-term perspective, respectively. We then analyze the impact
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factors for adopting the VMI model by taking the penalty mechanisms and other sensitive parameters into consideration. Customers
3. The VMI supply chain model To understand how the VMI supply chains work, we provide some fundamental results in this section. Specifically, we first present the principles of the VMI supply chain by comparing them with those chains with a traditional structure, followed by the development of the profit functions of the chain members in both circumstances. We then move on to create a payoff matrix to be used for subsequent analyses. 3.1. A comparison of the traditional and VMI supply chains In a traditional supply chain, each channel member operates individually with the interactions between them limited to the feed-forward flow of physical products and the feedback flow of information in the forms of purchase orders and cash (Disney, Potter, & Gardner, 2003). A typical schematic of a three-echelon supply chain comprising customers, retailers, distributors and manufacturers is illustrated in Fig. 1. When the order information is received at each echelon, the channel member manages its own stocking position and places orders to its adjacent upstream suppliers based on the quantity of incoming demand, inventory levels, goods shipped and received, and backorders. Consequently, orders placed to the upstream suppliers include not only the actual incoming demand, but also quantities covering the echelon’s own desired inventory position, customer service level, and cost requirements (Disney et al., 2003). It can be observed that under this operating structure, the traditional supply chain often suffers long lead-times, multiple decision points, unclear or redundant information, and minimal synchronization (Childerhouse & Towill, 2000). The inception of the VMI concept can be traced back to 1958 when Professor Magee studied the issue of who should have the authority to control inventories (Holmstro}m, Småros, Disney, & Towill, 2003). Although the concept emerged half a century ago, the interest in VMI started to grow only in the 1990s, during which organizations were seeking ways to improve their supply chain wide performance to create and maintain competitive advantages. In recent years, a drastic decline in telecommunication costs and a quick improvement in firms’ ability to provide accurate information have made it economically plausible to exchange sophisticated information quickly and accurately. As this is a critical requirement for a VMI supply chain to work effectively, it is also a driving force for the increasing applications of VMI. An example of VMI supply chains is shown in Fig. 2, where the triangle denotes inventory information. Under this circumstance, the upstream member (which is often a manufacturer, but can be a distributor) manages the adjacent downstream member’s inventories so as to ensure that a predetermined customer service level is maintained. In such a channel relationship, the upstream member is now responsible for making replenishment decisions for the downstream member and for scheduling the shipments of products at a quantity that can be either fixed (so as to maximize the
Customers
Retailers
Distributors
Manufacturers
Fig. 1. The structure of a traditional three-echelon supply chain.
Retailers
Distributors
Manufacturers
Fig. 2. The structure of a three-echelon VMI supply chain.
production or transportation efficiency) or variable (Waller et al., 1999). Replenishment occurs only when the stock level at the downstream echelon reaches a specified level based on both the average demand during the replenishment lead-time and a safety-stock requirement. Since the ownership of inventories in a VMI structure is shifted to upstream, in return, the downstream member is required to provide real-time demand information to the upstream supplier by allowing the supplier to access its demand databases. It is not difficult to see that a critical prerequisite for the VMI model to succeed is that a large amount of information must be transferred freely and timely between the channel members. The VMI approach generates numerous benefits for all channel members in a supply chain. First of all, the possibility of demand information amplification is dampened as the manufacturer now has a direct view at the end-consumer demand patterns, which will improve the forecasting accuracy. As a result, the manufacturer will be able to utilize the production facilities more efficiently, and the production outputs will not be ramped up and down frequently, which are often seen in a traditional supply chain with predicted large swings in demand. Second, there is only one inventory control point in the chain, which helps reduce the total buffer stocks along the chain. Third, the service levels can be improved as the product availability is increased (Waller et al., 1999). Finally, VMI can, in the long run, increase the profitability of both downstream and upstream members (Dong & Xu, 2002), because the downstream member can afford a price reduction as a result of lowered inventory costs, which in return may increase sales volumes for price-sensitive demands, thereby increasing the resulted profitability. 3.2. The profit functions of the channel members This section establishes the profit functions of the channel members in both traditional and VMI supply chains. Without loss of generality, we consider a two-stage supply chain where there is only one supplier and one downstream buyer, and we will examine their inventory management practices before and after the implementation of VMI. In this paper, we consider the situation under which the buying company dominates the supply chain and appears to be the ‘‘leader”. This decision scenario is inspired by our recently completed case study of a Chinese auto maker’s procurement method for one of its strategic MRO commodities (SEU Report, 2005). Because of the high risk and high cost of this strategic MRO item, this Chinese auto manufacturer wanted to make careful purchasing decisions and to take a leading role in the transaction. Specifically, the Chinese buyer first presented a price to its German MRO supplier, who then determined the supply quantity accordingly. Based on each individual profit, the two parties wanted to determine the feasibility of a VMI implementation. To proceed, we make several assumptions that are found common in the literature pertinent to inventory management (e.g., Silver, Pyke, & Peterson, 1998). First of all, the buyer’s inventory
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control policy is described by an economic order quantity (EOQ) rule based on deterministic demand and lead-time, and no stockouts. Second, both the unit order processing or setup cost and the unit cost of holding inventory are known and constant. Third, the buying firm determines the wholesale prices for certain quantities of products provided by the supplier, and then the supplier will choose a sales quantity to maximize its profit according the given wholesale price. Finally, the demand is inversely affected by the price; that is, the higher the price, the lower the demand; and hence, we use p(y) to represent the price and p0 (y) < 0. Additionally, a set of symbols and notation is used throughout the paper and is explained as follows. For simplicity and clarity, we use the subscript, ‘‘1”, to denote all the terms for the upstream supplier, and ‘‘2” for the downstream buyer. y: the annual purchase quantity of the downstream buyer p(y) the inverse demand function of the final product and is monotonically decreasing with respect to y, that is, p0 (y) < 0 w the wholesale price s the unit order/setup cost, in $/order the order quantity of the downstream buyer, and Q2 Q2 = (2ys2/I2Cv)1/2 Cv the unit product value, in $/unit I the unit inventory carrying cost, in %/year. With the above assumptions and the notation list, the time sequence of the one-shot game of the channel members is given as follows: (i) the downstream buyer brings forward the wholesale price, w, and then, the upstream supplier chooses a quantity, y, to maximize its profit based the wholesale price, w. (ii) And then, the two parties individually determines whether the VMI or a traditional strategy should be used based on their respective payoff. Thus, we can give the downstream buyer’s annual profit funcQ tion in a traditional supply chain, 02 ðy; wÞ, as follows 0 Y 2
ys2 I2 C v ðy; wÞ ¼ pðyÞy wy þ Q2 ; Q2 2
ð3:1Þ
where ys2/Q2 is the annual ordering cost, and I2Cv Q2/2 is the annual cycle inventory holding cost. It is reasonable to assume that the supplier’s total cost, which are associated with inventory, production and distribution, is an increasing convex function with respect to the quantity, y. As such, if the total cost function is represented by c(y), then c0 (y) > 0 and Q c00 (y) > 0, and the supplier’s profit function, 01 ðy; wÞ, can be given as: 0 Y 1
s1 y I 1 C v þ Q2 Q2 2 1=2 I2 C v s2 y s1 I 1 ¼ wy cðyÞ þ : 2 s2 I 2
ðy; wÞ ¼ wy cðyÞ
ð3:2Þ
Note that Q2 in (3.2) is replaced by the expression given in the notation list. Thus, for a given wholesale price, w, by the downstream buyer, the upstream supplier will choose a quantity, y, to maximize its profit. This optimal quantity can be determined from the following first-order condition obtained from the function in (3.2):
w ¼ c0 ðyÞ þ
1=2 1 I 2 C v s2 s1 I1 : þ 2 2y s2 I2
ð3:3Þ
Furthermore, at this price, which replaces w in (3.1), the downstream buyer then maximizes its profit by choosing an optimal purchase quantity, y*, such that:
p0 ðy Þy þ pðy Þ c0 ðy Þ c00 ðy Þy
1=2 1 2I2 C v s2 ¼ 0: 2 y
1=2 1 I2 C v s2 s1 I 1 þ 4 2y s2 I 2 ð3:4Þ
If the two members decide to adopt the VMI approach, then the downstream buyer transfers the ownership of inventories to the upstream supplier, who will make decisions regarding the inventory levels, order quantities, and replenishment frequencies on behalf of the buyer. As a result, the upstream member now has combined inventories with the new order setup cost (s1 + s2) and the new carrying cost (I1 + I2)Cv. Let wVMI represent the new wholesale price in Q and PVMI represent the new profit the presence of VMI, and VMII 1 2 function of the downstream and the upstream members, respectively. Then, VMI Y
ðy; wVMI Þ ¼ y wVMI cðyÞ ½2ðs1 þ s2 ÞðI1 þ I2 ÞC v y1=2
ð3:5Þ
ðy; wVMI Þ ¼ y½pðyÞ wVMI :
ð3:6Þ
1 VMI Y 2
When the upstream supplier with the VMI agreement in place aims to maximize its profit, the following relationship between the contractual wholesale price and quantity can be obtained from the first-order condition of the upstream member’s profit function in (3.5):
wVMI ¼ c0 ðyVMI Þ þ
1=2 1 2ðs1 þ s2 ÞðI1 þ I2 ÞC v : 2 yVMI
ð3:7Þ
The downstream buyer’s optimal purchase quantity can be determined by the first-order condition of its profit function in (3.5) with wVMI being replaced by (3.7). As such, we obtain the following relationship in a long-term use of the VMI:
yVMI p0 ðyVMI Þ þ pðyVMI Þ c0 ðyVMI Þ yVMI c00 ðyVMI Þ 1=2 1 2ðs1 þ s2 ÞðI1 þ I2 ÞC v ¼ 0: 4 yVMI
ð3:8Þ
3.3. A numerical illustration of the benefits in VMI supply chains To put the theoretical results into perspectives, we use the explicit functions for p(y) and c(y) to derive the benefits resulted from a long-term implementation of VMI. Assume that the relationship between the sales price of a product and the demand can be represented by a simple linear function: p(y) = a by, " a,b > 0, and that the total cost function of the supplier is given by: c(y) = dy + 0.8hy2, " d,h > 0. Additionally, the demand and cost parameters are fixed at the following values:
a ¼ 200; b ¼ 0:01; h ¼ 0:006; s2 ¼ 600; C v ¼ 100:
d ¼ 60;
I1 ¼ I2 ¼ 20%;
If s1 is set to 1800, we can calculate the buyer’s optimal purchase quantity, y*, in a traditional supply chain using (3.4) and in a long-term VMI chain using (3.8). As shown in Fig. 3, the purchase quantity increases in a long-term VMI supply chain, which confirms the result reported in the literature (Dong & Xu, 2002) that a matured VMI supply chain eventually leads to an increased purchase quantity of the buyer, which will increase the total chain profit. Although the results presented in Sections 3.2 and 3.3 are obtained from a standard sequential optimization method for finding the optimal decisions for the buyer and seller, they provide a foundation for our following analysis of the VMI chain’s evolution using the evolutionary game theory.
Purchase Quantity
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H. Yu et al. / Computers & Industrial Engineering 56 (2009) 274–282
3500 3490 3480 3470 3460 3450 3440 3430 3420 3410 3400
The idea here is that the ESS state, s, resists all possible ‘‘mutations”, x, either because (i) they are less fit or (ii) they are equally fit at the current state (where they are rare) but less fit when they are prevalent. Moreover, the ruled Replicator dynamic equation named by Taylor and Jonker (1978) is seen below:
3490
dxi ¼ ½f ðs; xÞ f ðx; xÞ xi : dt
3435
Traditional Chain
VMI Chain
Fig. 3. Changes of the optimal purchase quantities.
Table 1 The payoff matrix of the two supply chain strategies Downstream
Upstream VMI Traditional
VMI
Traditional
QVMI QVMI 1 ; 2 Q0 1 p1 ; 2 n
Q0
Q0
1 m; Q0 Q0 1; 2
Q0
2
p2
3.4. The payoff matrix of the two supply chain strategies Based on the results obtained in the previous section, we present the payoff matrix of the VMI and traditional supply chains in Table 1. It is seen that the payoff matrix is similar to that of the Prisoner’s Dilemma, a well-known simple game theory model used to demonstrate the complexity of cooperation in any system (see for example, Kuhn, 2003). In the payoff matrix, the parameters, (m, n), are the respective investment made by the upstream and downstream buyers to adopt the VMI strategy; for example, the costs spent on installing or upgrading the software and hardware for information sharing. The terms, (p1, p2), represent the two members’ respective financial penalties, such as the goodwill loss or opportunity loss, when they breach the channel contract.
Consider two populations from the upstream suppliers’ network and downstream buyers’ network that are interactive to each other. For the sake of simplicity, two populations are assumed to be of equal size. In each period t, member i in the supplier’s network plays a game with member j in the buyer’s network by choosing either VMI or a traditional supply chain strategy. Every member uses Replicator dynamic equation to adjust its strategy at every game stage. If we assume that the probability of an upstream supplier selecting the VMI strategy at the initial state is aVMI = a (0 6 a 6 1). (The probability can be equal to the percentage of the supplier population selecting the VMI strategy), then, with a0 = 1 a, it will use the traditional supply chain strategy at that state. Similarly, with the probability, bVMI = b (0 6 b 6 1), a downstream buyer will adopt the VMI strategy at the original state, and withb0 = 1 b, it will not do so. Based on these probabilities, we can obtain a series of equations regarding the expected values of a number of terms. Specifically, the average payoff of the upstream supplier if it selects to adopt the VMI strategy can be expressed as: VMI Y
As the numerical example in the preceding section suggests, the channel profit increases in a long-term VMI supply chain; but the profit change for each individual member is unclear. Evolutionary game theory can shed more insights on the intrinsic evolutionary mechanism of the VMI chains and examine the trend of the channel members’ evolutionary stability from a time-based perspective. We use the evolutionary game theory to study the evolutionary stability of the VMI strategy. Evolutionary game theory combines the static feature of an evolutionary stable strategy (ESS) with the dynamic nature of the Replicator dynamic initiated by Taylor and Jonker (1978). The concept of ESS, initially proposed by Maynard Smith (Maynard-Smith & Price, 1973; Maynard-Smith, 1974, 1982), has had perhaps more influence than any other theory has in the field of evolutionary games. The original definition of ESS for a single population is given in a linear fitness function, f(r, s) = rT As for some given N N matrix A. It is as follows. Definition 4.1. A state s 2 S is an ESS if for all other states x 2 S either
ðiÞ f ðs; sÞ > f ðx; sÞ; or ðiiÞf ðs; sÞ ¼ f ðx; sÞ and f ðs; xÞ > f ðx; xÞ:
¼b
1
VMI Y
þð1 bÞ
0 Y
1
! m ;
ð4:2Þ
1
Q Q where VMI and 01 are given in (3.5) and (3.2), respectively. Fur1 thermore, the supplier’s average payoff if choosing the traditional strategy is given by 0 Y
¼b
0 Y
1
! p1
þ ð1 bÞ
1
0 Y
:
ð4:3Þ
1
It is now straightforward to obtain the expected payoff of the upstream supplier as follows:
Y 1
4. Analysis of the evolutionary stability of the VMI supply chains
ð4:1Þ
¼a
VMI Y 1
þ ð1 aÞ
0 Y :
ð4:4Þ
1
As a result, the Replicator dynamic equation of the upstream supplier who selects the VMI strategy based on Definition 4.1 and Eq. (4.1) can be given as
! " ! # VMI Y VMI 0 Y Y Y da ¼ að1 aÞ b þp1 þ m m : ¼a dt 1 1 1 1
ð4:5Þ
Similarly, the Replicator dynamic equation of the downstream buyer if it chooses to adopt the VMI strategy can be obtained as:
! " ! # VMI Y VMI 0 Y Y Y db ¼ bð1 bÞ a þp2 þ n n : ¼b dt 2 2 2 2
ð4:6Þ
Q where 2 denotes the expected payoff of the downstream buyer, and its expression can be derived in a similar fashion to the series from (4.2),(4.3),(4.4). The differential equations in (4.5) and (4.6) describe the group evolutionary dynamics of the two-stage supply chain. They indicate that the trend of each channel member’s profit with VMI implementation depends on the probabilities of other channel members choosing to use VMI, the profit functions before and after the implementation of VMI, the original investments, and the financial penalties.
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According to Ellison and Fudenburg (2000), the stable state of the Replicator dynamic equation is the Nash equilibrium (NE), which is often referred to as the evolutionary equilibrium da dt
(EE). When
¼ 0; db dt ¼ 0, the
EE
of (4.5) and (4.6) are n QVMI Q 0
E1 = (0, 0), E2 = (0, 1), E3 = (1, 0), E4 = (1, 1), and E5 ¼
2
! m QVMI Q 0
1
þp1 þm 1
2
þp2 þn
;
: . Note that the two terms in the bracket of E5 are
both fractional. Based on the approach suggested by Friedman (1991), the stability of EE can be analyzed by the Jacobi matrix, which is given here as follows: by ddta and db dt 3 VMI 0 VMI 0 Q Q Q Q ð1 2 a Þ b þp þ m m a ð1 a Þ þp þ m 1 1 7 6 1 1 1 1 7 6 J¼6 VMI 0 VMI 0 7: 5 4 Q Q Q Q þp2 þ n ð1 2bÞ a þp2 þ n n bð1 bÞ 2
2
2
2
2
The two terms, detJ and trJ, associated with the above Jacobi matrix, are derived below:
(
VMI Y
det J ¼ ð1 2aÞ b
1
(
a
VMI Y
0 Y
2
0 Y
VMI Y
1
)
þp1 þ m m
1
!
VMI Y
þp1 þ m bð1 bÞ
1
2
and
trJ ¼ ð1 2aÞ b
VMI Y
1
þ ð1 2bÞ a
0 Y
!
2
0 Y
!
0 Y
! þp2 þ n ;
2
! !
þp2 þ n n :
2
Proposition 4.1. With a short-term application of VMI, when the channel penalty costs do not exist, i.e., both p1 and p2 are zero, then the Replicator dynamic equations in (4.5) and (4.6) have four EE, (E1, E2, E3, E4), out of which only E1 = (0, 0) is the ESS, implying that no channel member chooses to use the VMI strategy. Proof. In a short-term use of the VMI, the change of the upstream supplier’s profit can be calculated by
1
So, if s1 = s2 and I2 > I1/7, then VMI Y
1
0 Y
¼
1
1=2 h i2 1 I2 s2 y 2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 2 2
< 0: ð4:8Þ Similarly, the profit change of the downstream member can be found as VMI Y
2
0 Y
¼
2
1=2 h i2 1 I 2 s2 y > 0: 2 þ ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 2 2 ð4:9Þ
¼ y wVMI cðyÞ ½2ðs1 þ s2 ÞðI1 þ I2 ÞC v y
1
þp1 þm
R ½0; 1, i.e., E5 is not the EE of the Replicator dy-
1=2
1=2 # I 2 C v s2 y s1 I 1 : wy cðyÞ þ 2 s2 I 2
p1 >
h i2 1 : ðI2 s2 y=2Þ1=2 2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 2
Proof. Based on Table 2, E4 is the ESS if and only if the following is Q Q Q Q 01 þ p1 Þð VMI 02 þ p2 Þ > 0. Since Eq. (4.9) shows true: ð VMI 1 2 QVMI Q0 QVMI Q0 2 > 0, then 1 þ p1 > 0. Based on (4.8), that if 2 1
During the early stage of the VMI implementation, the sales and purchase quantities stay relatively unchanged due to market constraints and other contractual or public agreements between the channel members. For this reason, the short-term effects will be evaluated at the extent to which the annual purchase quantity, y, is the same as that in a tradition supply chain. The wholesale price, however, should be adjusted relatively quickly as a new VMI contract is negotiated and determined. Based on the results obtained in the preceding section, the next two propositions are presented for a supply chain with a short-term application of VMI.
1
1
1=2 h i2 1 I2 s2 y 2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 : 2 2
Proposition 4.2. With a short-term utilization, the VMI strategy will be accepted by the channel members on the conditions that E4 is the ESS, and if and only if the upstream member’s penalty cost, p1, satisfies the following requirement:
4.1. The ESS with a short-term VMI implementation
0 Y
¼
namic equations in (4.5) and (4.6). Furthermore, based on the Jacobi matrix, we report the results of the local stability analysis in Table 2. It is seen from Table 2 that E1 is the only ESS of Replicator dynamic equations. This means the related channel members will refute the VMI strategy if it is used only in a short-term and the channel penalty costs are not present. h
In the following section, we will use the Jacobi matrix to analyze the ESS with short-term and long-term VMI supply chains, respectively.
VMI Y
1
0 Y
1
þp1 þ m m
1 VMI Y
Thus, QVMI Q0
ð1 2bÞ
að1 aÞ
!
VMI Y
m
)
þp2 þ n n
2 0 Y
!
Based on Eqs. (3.3) and (3.7) and Q2 = EOQ2 = (2s2y/I2)1/2, we can simplify the upstream member’s profit change in (4.7) and obtain the following relationship:
"
ð4:7Þ
the following expression can be obtained, p1 > 12 ðI2 s2 y=2Þ1=2 h i2 , which proves the proposi2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 tion.
h
Interestingly but reasonably, the first proposition indicates the fact that because of the advantageous position, the downstream member is able to take away the majority of the cost savings from a VMI adoption. The upstream member, on the other hand, as indicated in the second proposition, not only receives nothing from this cost reduction, but has to accept some penalty in order for the VMI method to outperform the traditional approach. It implies that the profit increase of the downstream buyer equals the profit reduction of the upstream supplier after VMI is imple-
Table 2 Analysis of the local stability of EE with a short-term after VMI EE
det J
E1 = (0, 0) E2 = (0, 1) E3 = (1, 0) E4 = (1, 1)
mn Q ð VMI Q1 ð VMI Q2 ð VMI Q1 ð VMI 2
tr J Q 01 Þn Q 02 Þm Q 01 Þ Q 02 Þ
+
m n QVMI Q0 þn Q1VMI Q10 þm 2Q 2Q VMI ð 1 01 Þ QVMI Q0 ð 2 2Þ
Local stability
ESS Saddle-point Saddle-point Unsteady
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Replicator dynamic equations in (4.5) and (4.6) have five values of EE, (E1, E2, E3, E4 and E5), and that only E1 and E4 are the ESS. h
mented, because the downstream buyer’s inventory cost has been transferred to upstream supplier with VMI in place. Thus, from a short-term perspective, the downstream buyer has the desire to implement VMI but the upstream supplier has a disadvantage to adopt it. However, it should be stressed that this series of results is obtained based on the assumption that the downstream member keeps the same annual purchase quantity as used in a traditional chain. This condition limits the upstream member’s ability to increase its optimal profit in a long-term. The next section will discuss the ESS of the long-term utilization of the VMI strategy.
The result obtained herein may explain why the upstream member is skeptical about the VMI approach at the beginning of the implementation, but will gradually accept it as the opportunities for reducing other costs and for increasing the long-term profit exist. To achieve better coordination, the downstream buyer should compensate the upstream supplier for the loss by sharing revenues so that the upstream supplier will have sufficient financial incentive to accept VMI during the early stage of implementation. As the implementation of VMI progresses, the purchase quantity in a VMI supply chain will grow higher than that in a traditional chain, and as a result, the total supply chain profit will increase. For VMI to reach the ESS and to eventually benefit the entire chain, the supply chain members should be aware that the purchase quantity will increase in the long run, and thus, it may be necessary for the downstream buyer to share profit with the upstream supplier to cover the supplier’s loss during the early stage of VMI.
4.2. The ESS with a long-term application of the VMI It has been reported that supply chain members adopting the VMI approach have experienced increases in sales eventually, which will bring up the purchase quantity (as indicated in Fig. 3). In a long-term use of the VMI, the downstream buyer’s profit will increase, while the upstream supplier’s profit may increase under some specified conditions. Proposition 4.3 given below indicates such conditions.
5. Analysis of the impact factors on ESS
Proposition 4.3. In a long-term adoption of the VMI and without the channel penalty costs, the Replicator dynamic equations in (4.5) and (4.6) have five EE, (E1, E2, E3, E4, E5), out of which E1 and E4 are ESS.
We illustrate the dynamic evolutionary process of the channel members with a long-term utilization of VMI in Fig. 4, which is created by using the simulation program called ‘‘Dynamo” (Sandholm and Dokumaci, 2007). The input values of the diagram are (y, p1, p2, m and n), and the illustration depicts how a VMI supply chain evolves from its initial state. The curves of E2, E3 and E5 are the critical lines of the dynamic evolutionary games, along which they
Proof. Let
f ðyÞ ¼ p0 ðyÞy þ pðyÞ c0 ðyÞ c00 ðyÞy
1=2 1 I 2 C v s2 ðs1 =s2 þ I1 =I2 Þ: 4 2y
Using Eqs. (3.4) and (3.7), we obtain the following two functions: 1=2 2 C v s2 f ðyVMI Þ ¼ p0 ðyVMI ÞyVMI þ pðyVMI Þ c0 ðyVMI Þ c00 ðyVMI ÞyVMI 14 I2y VMI
ðs1 =s2 þ I1 =I2 Þ ¼ 0
E2
and
f ðy Þ ¼ p0 ðy Þy þ pðy Þ c0 ðy Þ c00 ðy Þy
E4
1=2 1 I 2 C v s2 4 2y
ðs1 =s2 þ I1 =I2 Þ P p0 ðy Þy þ pðy Þ c0 ðy Þ c00 ðy Þy 1=2 1=2 1 I2 C v s2 1 2I2 C v s2 ðs =s þ I =I Þ ¼ 0: 1 2 1 2 4 2y 2 y Thus, f(yVMI*) = 0 and f(y*) > 0, and f(y) is a strictly monotonic digressive function with respect to y. Furthermore, it can be shown that yVMI > y , or the purchase quantity in a VMI supply chain is higher than that in a traditional chain. (This is proven true numerically in Section 3.4.) In summary, a long run application of the VMI will increase the upstream Q Q0 member’s profit, which means that VMI 1 ðyVMI Þ 1 ðyÞ > 0. Similar to n m Proposition 4.1, since QVMI Q0 2 ½0; 1; QVMI Q 2 ½0; 1, E5 is 0 2
2
þp2 þn
1
1
E5
E1
þp1 þm
an EE. The results of the local stability analysis for a long-term application of VMI are summarized in Table 3, which clearly indicates that the
E3
Fig. 4. The process of the dynamic evolutionary game.
Table 3 Analysis of the local stability of EE with a long-term after VMI EE
det J
E1 = (0, 0) E2 = (0, 1) E3 = (1, 0)
mn Q Q ð VMI 01 þ p1 Þn Q1 Q0 ð VMI 2 2 þ p2 Þm QVMI Q0 ð 1 1 þ p1 Þ Q Q 02 þ p2 Þ ð VMI 2 QVMI Q0 mnð 2 2 þp2 Þ
QVMI Q0 þp þn
Q2 VMI Q2 0 2 þp
QVMI1 Q0 1 1
E4 = (1, 1) n E5 ¼ QVMI Q 0 2
2
þp2 þn
m ; QVMI Q 0 1
1
þp1 þm
1
tr J
1
þp1 þm
+ +
m n QVMI Q0 þ p1 þ n Q1VMI Q10 2 þ p2 þ m 2 Q Q ð VMI 0 þ p1 Þ Q 1 Q0 1 ð VMI 2 þ p2 Þ 2 0
Local stability + +
ESS Saddle-point Saddle-point
ESS
Saddle-point
H. Yu et al. / Computers & Industrial Engineering 56 (2009) 274–282
gradually evolve into E1 or E4 to become ESS. On the right upper corner of the region in which the evolution moves toward E4, the supply chain members will adopt the VMI model completely. In contrast, if the evolution moves toward the lower left corner, then the channel members tend to constringe to E1, which implies that they will prefer the traditional strategy over VMI. Note that the dynamic evolutionary game process can be long and may allow coexistence of the VMI and traditional strategies. It can be seen from Fig. 4 that whether the dynamic evolutionary game of the supply chain members converges to E1 or E4 is influenced by the initial state of the game. Moreover, E5is the pivotal point that affects final converging state, and therefore, it is the impact factor of the ESS; in other words, the impact factor of E1 or E in the capacity of ESS is determined by the following terms: Q4VMI Q0 QVMI Q0 2 , 1 1 , p1, p2, m and n. 2 To obtain the intrinsic evolutionary mechanism and improve the efficiency of the VMI supply chain, it would be interesting to examine the effects of the main parameters on the ESS. However, since it is difficult to obtain closed-form expressions and clearly defined results, we discuss the impact factor analysis of the ESS by simply observing the changes of the following six terms: QVMI Q0 QVMI Q0 2 , 1 1 , p1, p2, m and n, and using Fig. 4 as a guide2 line. First of all, under the long-term employment of the VMI, the downstream and upstream members’ profits will increase with Q Q Q Q the purchase quantity. As a result, VMI 02 and VMI 01 will 2 1 gradually increase, moving the location of E5 gradually close to the origin, which makes E4 the ESS with higher possibility. Second, the pair, (p1, p2), positively increases the likelihood that E4 will become the ESS, whereas the other pair, (m, n) decreases it. Finally, as the purchase quantity will increase in the long-term utilization of the VMI and the values of both m and n are fixed, the impact of (m, n) can be negligible. As a result, the absolute value of the profit change will approach to zero. From this point of view, E4 will remain the only ESS, that is, the supply chain members will fully adopt VMI. 6. Conclusions and discussions This study has focused on the evolutionary trend analysis of the VMI supply chains. The evolutionary game theory is used to analyze the evolutionary stable strategy (ESS) of the VMI supply chain to examine the intrinsic evolutionary mechanism of the VMI method. With the consideration of the bounded rationality of the supply chain channel members, we have analyzed the trend of the evolutionary stability of the VMI implementation from both short- and long-term points of view in order to identify the conditions under which the VMI method will outperform the traditional approach. In a short-term utilization of the VMI, if the chain members choose to adopt the VMI strategy, then the upstream supplier will have some profit loss as well as a penalty cost consisting of goodwill and opportunity loss at the beginning of implementation. However, as the implementation matures, increases in purchase quantity will be observed, which will increase both downstream and upstream members’ profits, and as such, the two chain members will eventually adopt VMI completely. This research provides practitioners with a clear understanding of how to make decisions with respect to VMI. As our results indicate, a downstream buyer does obtain certain inventory cost reductions at the beginning stage of VMI, but the upstream supplier cannot achieve inventory cost reductions; rather, it will experience reduced profit. However, with the implementation progresses, an increased transaction quantity between the chain members will be resulted from more information sharing, better visibility and availability of information, and more accurate forecast. As a result, there are opportunities for all involving members
281
to increase profits. For a more effective implementation of VMI and to exploit the benefits of the VMI, the downstream buyer can share some revenue with the upstream supplier to compensate the supplier’s partial loss during the early stage of VMI implementation. In contrast to most research efforts devoted to understanding the complexity of the VMI adoption and implementation, this paper takes a different approach – using the evolutionary game theory to analyze how VMI may evolve over a short or long period of time after it is implemented. Our research sheds insights on when and why the supply chain partners should coordinate during VMI implementations. However, our current analysis has a number of limitations, which suggest the following future research directions. (1) It will be interesting to quantify and analyze the Bullwhip Effect of a VMI supply chain by applying evolutionary games based on the bounded rationality; (2) when the service levels required at downstream and upstream stages are different, how will the trend of the evolutionary stability affect the adoption of the VMI model? And (3) Sensitivity analysis of the impacts of the parameters, including p1, p2, m and n, on the evolution of VMI to an ESS needs to be conducted. Acknowledgements This research is supported by the National Natural Science Foundation of China (706711021) for the project titled ‘‘Project Operation and Simulation of Emergency Response Logistics Network in the System of Anti-Bioterrorism”. We are also grateful to the anonymous reviewers for their constructive comments, which have improved the quality and presentation of this paper. References Achabal, D., Mclntyre, S., Smith, S., & Kalyanam, K. (2000). A decision support system for vendor managed inventory. Journal of Retailing, 76(4), 430–454. Cetinkaya, S., & Lee, C. Y. (2000). Stock replenishment and shipment scheduling for vendor-managed inventory systems. Management Science, 46(2), 217–232. Childerhouse, P., & Towill, D. R. (2000). Engineering supply chain to match customer requirements. Logistics Information Management, 13(6), 337–345. Christina, P. (2007). Finite populations choose an optimal language. Journal of Theoretical Biology, 249(3), 606–616. Disney, S. M., Potter, A. T., & Gardner, B. M. (2003). The impact of vendor managed inventory on transport operations. Transportation Research Part E, 39(5), 363–380. Disney, S. M., & Towill, D. R. (2002). A procedure for the optimization of the dynamic response of a vendor managed inventory system. Computers & Industrial Engineering, 43(1–2), 27–58. Dong, Y., & Xu, K. (2002). A supply chain model of vendor managed inventory. Transportation Research Part E, 38(2), 75–95. Ellison, G., & Fudenburg, D. (2000). Learning purified mixed equilibria. Journal of Economic Theory, 90(1), 84–115. Eriksson, A., & Lindgren, K. (2005). Cooperation driven by mutations in multi-person Prisoner’s Dilemma. Journal of Theoretical Biology, 232, 399–409. Friedman, D. (1991). Evolutionary games in economics. Econometrica, 59(3), 637–666. Holmstro}m, J., Småros, J., Disney, S. M., & Towill, D. R. (2003). Collaborative supply chain configurations: The implications for supplier performance in production and inventory control. In Presentation at the 8th symposium on logistics, July 6–8, 2003, Seville, Spain. Huang, Z., & Li, S. X. (2001). Co-op advertising models in manufactures-retailer supply chain: A game theory approach. European Journal of Operational Research, 135(3), 527–544. Kisperska, M. D. (1999). Warehousing conditions for holding inventory in polish supply chain. International Journal of Production Economics, 59, 123–128. Kuhn, S. (2003). Prisoner’s Dilemma, The Stanford Encyclopedia of Philosophy (Fall 2003 Edition). Available from: <
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Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Analyzing the evolutionary stability of the vendor-managed inventory supply chains Haisheng Yu a, Amy Z. Zeng b,*, Lindu Zhao a a b
Institute of Systems Engineering, Southeast University Nanjing, Jiangsu 210096, PR China Department of Management, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA
a r t i c l e
i n f o
Article history: Received 3 December 2007 Received in revised form 30 May 2008 Accepted 31 May 2008 Available online 8 June 2008 Keywords: Supply chain coordination Vendor-managed inventory Game theory Evolutionary stability Replicator dynamic
a b s t r a c t One of the widely used strategies for achieving system integration in the arena of supply chain management is the vendor-managed inventory (VMI) approach. Although there exists a large amount of literature that examines various aspects of VMI, little looks at how this model evolves as the implementation progresses and matures. Therefore, this paper shows how to analyze the intrinsic evolutionary mechanism of the VMI supply chains by applying the evolutionary game theories. It is found that during the early stage of the VMI implementation, the upstream supplier will have some profit loss; however, as the transaction quantity increases in the long run, which will eventually benefit the entire chain, it is necessary for the downstream buyer to share profit with the upstream supplier to cover the supplier’s initial loss in order to exploit and sustain the benefits of the VMI. Additionally, the impact factors for VMI to become an evolutionary stable strategy are examined. All the results identify the conditions under which the VMI model is favorable over the traditional chain structure and shed lights on when and why collaboration is critical for a successful, long-term implementation of VMI. 2008 Elsevier Ltd. All rights reserved.
1. Introduction A great deal of evidence has shown that vendor-managed inventory (VMI) approach can improve supply chain performance by decreasing inventory-related costs and increasing customer service; as such, the industrial applications of VMI have grown constantly over the past decade. Unlike a traditional supply chain wherein each member manages its own inventories and makes individual stocking decision, VMI is a collaborative initiative where a downstream buyer shifts the ownership of inventories to its immediate upstream supplier and allows the supplier to access its demand information in return. In particular, a VMI process involves the following two steps: (1) a downstream buyer provides demand information to its immediate upstream supplier and leaves the stocking decisions to that supplier; and (2) the upstream supplier has the ownership of the inventories till the inventories are shipped to the buyer and bears the risk of demand uncertainty. It is not difficult to see that the VMI structure promotes collaborations between suppliers and buyers through information sharing and business process reengineering. The potential benefits of VMI are very compelling and the main benefits include substantial reduction of inventory-related costs for both suppliers and buyers and noticeable improvement of customer service levels as indicated by shorter order cycle times and higher fill rates (Achabal, * Corresponding author. Tel.: +1 508 831 6117; fax: +1 508 831 5720. E-mail addresses: [email protected] (H. Yu), [email protected] (A.Z. Zeng), [email protected] (L. Zhao). 0360-8352/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.05.016
Mclntyre, Smith, & Kalyanam, 2000). The success of VMI has been experienced by many well-known retailers and their first-tier suppliers, most notable is Wal-Mart and its key suppliers like Proctor and Gamble (Cetinkaya & Lee, 2000). However, a supply chain is a complex system consisting of numerous interrelated channel entities, such as suppliers, intermediaries, third-party service providers, and final customers. These firms are connected by material, financial and information flows that travel both directions along the supply chain. As such, despite of VMI’s growing popularity, implementing this approach can be more challenging and difficult in practice than what is perceived in theories, especially when the trust between supply chain members and their actual commitments are under question. The underlying cause that may destroy a trusting relationship is the bounded rationality, which, as pointed out by Sterman (1989), is one of the reasons responsible for the occurrence of the Bullwhip Effect. Therefore, from the standpoint of bounded rationality and in comparison with the classic game theories, the evolutionary games may provide an additional avenue to study the effectiveness of VMI. As such, analyzing the evolutionary stability of the VMI supply chains is the subject of this paper. Although there exists a large amount of literature that examines various aspects of VMI, little looks at how this model evolves as the implementation progresses and matures. The aim of this paper is to investigate the intrinsic evolutionary process of the VMI supply chains by obtaining the evolutionary stable strategy (ESS) through the help of the evolutionary game theories. The other contribution of the paper can be seen from the following two perspectives. First,
H. Yu et al. / Computers & Industrial Engineering 56 (2009) 274–282
this paper takes a few steps further from the existing research efforts and provides an in-depth analysis of the VMI supply chains using evolutionary game theories. Second, the results of this paper not only identify the conditions under which the VMI model is favorable over the traditional chain structure, but also explicitly demonstrates why it is necessary for the involving parties to share benefits during the early stage of VMI implementation in order for the benefits of this strategy to increase and sustain in a long run. The remainder of the paper is organized as follows: Section 2 provides a brief review of the previous works that are closely related to our research. In Section 3, we describe the details of a two-stage VMI supply chain model and obtain the profit functions of the upstream and downstream members. In Section 4, we apply Replicator dynamics to analyze the evolutionary stability of the VMI supply chain based on each channel member’s profit function, from which the short-term and long-term evolutionary mechanisms are obtained. In Section 5, we analyze the factors that affect the stability of the VMI approach. Finally, we conclude the paper with a summary and discussion in Section 6.
2. Literature review In recent years, the issues related to supply chain channel coordination in different forms and contexts have been addressed extensively in the literature. The VMI model is a channel coordination strategy between downstream and upstream players. In this model, based on their common interests and objectives, the two players reach an agreement that the upstream supplier will manage the downstream member’s inventory stock decisions and that both will monitor and modify their agreed-upon terms (Kisperska, 1999) in the course of VMI implementation. Since its adoption in the arena of supply chain management in early 1980s, the VMI concept has received a great deal of attention, as the model differs significantly from the traditional practice in that the chain members shift their relationships from an arm’s length to a strategic partnership. Moreover, the VMI method emphasizes integration and collaboration, which have been becoming the core theme of the contemporary supply chain management philosophy. In this section, we provide a brief review of the existing studies pertinent to our work, which are classified into four streams. The first stream focuses on the ‘‘why-VMI” question, that is, the perceived importance and benefits of this model. For example, the studies by Vergin and Barr (1999) and Lee et al. (2005) conclude that VMI is becoming an effective approach for implementing the channel coordination initiative, which is critical and imperative to improve the entire chain’s financial performance. Waller, Johnson, and Davis (1999) indicate further that the VMI method can improve inventory turnover and customer service levels at every stage of a supply chain. In a more in-depth analysis, Disney and Towill (2002) find that the kernel goal of VMI chains, which is minimizing the channel cost while simultaneously satisfying some degree of customer service levels, is achieved primarily by sharing demand and inventory information. The second research stream addresses the ‘‘how-to” question. Specifically, this stream builds upon the belief that the success of the VMI method requires extensive technology enabling, especially from the support of information technologies. The representative study of this stream is given by Xu, Dong, and Evers (2001), in which they examine the impacts of EDI and Internet-based technologies on the practice of VMI. They also find that the application of VMI will improve the demand and inventory information sharing between the upstream and downstream members. In contrast, a survey conducted by Tyan and Wee (2003) points out that aside from the computer technologies, the key of implementing VMI lies in the abilities of the related chain members to cooperate and to
275
understand the flows and processes concerning their products or services delivery. Although in theory, the VMI model is one of the most efficient coordination strategies for managing supply chains, it possesses numerous challenges and problems in practice that may lead to either a success or a failure like in a competitive game. Therefore, in the third research stream, many scholars reply on game theories to study VMI supply chains with specific structures and configurations. For example, a study by Huang and Li (2001) investigates a two-stage game mechanism, called upstream dominance and downstream dominance, in cooperative advertising models for a manufacturer–retailer supply chain. Wang, Guo, and Efstathiou (2004) analyze the non-cooperative behavior in a decentralized two-echelon supply chain consisting of one supplier and multiple retailers. They find that competition occurs among all the retailers as well as between the supplier and each retailer. In order to achieve the optimal cooperation in the system, the authors have designed several Nash equilibrium contracts based on the echelon-inventory games and local inventory games. The last research stream also employs the game theories but from a social and economic standpoint. In social and economic systems, the behavior of a member in a system often affects a few other members, and there exist numerous examples of situations under which the so-called free riders or defectors take advantage of others for a common good. In the context of channel coordination, a game-theoretic approach can be applied based on the Prisoners’ Dilemma game – a situation that captures the temptation to act in a selfish way to gain a higher reward instead of sharing a reward by cooperation (e.g., Kuhn, 2003). An example of such research effort is given by Eriksson and Lindgren (2005), in which they investigate how cooperation is driven by mutations in multi-person Prisoners’ Dilemma and analyze the impacts of the payoff parameters, the initial population, the mutation rate, and the group size on the efficiency of the channel coordination. The literature generated from the last two research streams mainly relies on the classic games. In understanding the bounded rationality of channel members of a supply chain, we think that the evolutionary game theory can provide another way to study the channel coordination mechanism. Evolutionary game theories originate as an application of the mathematical theory of games to biological problems based on the realization that frequencydependent fitness introduces a strategic aspect to evolution. Recently, there has been an increased interest in using this theory by economists, sociologists, and anthropologists that are social scientists in general as well as philosophers (e.g., Christina, 2007; Michael, 2008; Mizuho & Toshio, 2007; Xiao & Yu, 2006). The evolutionary game theory consists of two main approaches. The first approach derives from the work of Maynard-Smith and Price (1973) and employs the concept of an evolutionarily stable strategy as the principal tool of analysis. The second approach constructs an explicit model of the evolution process in which the frequency of strategies changes with the population sizes, and studies the properties of the evolutionary dynamics based on that model. Although there exists one study by Wang and Meng (2004) that develops an evolutionary game model for supply chain partnerships from an evolutionary game’s viewpoint, only a simple theoretical framework is presented, and the effects of the contract and the penalty mechanisms on the effectiveness of the channel coordination are not considered. Therefore, in this paper, we take a few steps further from the existing research efforts and provide an in-depth analysis of the VMI supply chains using evolutionary game theories. We will first present the profit functions of the members in a two-stage VMI chain, based on which we examine the channel’s evolutionary stability trend from a short-term and long-term perspective, respectively. We then analyze the impact
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factors for adopting the VMI model by taking the penalty mechanisms and other sensitive parameters into consideration. Customers
3. The VMI supply chain model To understand how the VMI supply chains work, we provide some fundamental results in this section. Specifically, we first present the principles of the VMI supply chain by comparing them with those chains with a traditional structure, followed by the development of the profit functions of the chain members in both circumstances. We then move on to create a payoff matrix to be used for subsequent analyses. 3.1. A comparison of the traditional and VMI supply chains In a traditional supply chain, each channel member operates individually with the interactions between them limited to the feed-forward flow of physical products and the feedback flow of information in the forms of purchase orders and cash (Disney, Potter, & Gardner, 2003). A typical schematic of a three-echelon supply chain comprising customers, retailers, distributors and manufacturers is illustrated in Fig. 1. When the order information is received at each echelon, the channel member manages its own stocking position and places orders to its adjacent upstream suppliers based on the quantity of incoming demand, inventory levels, goods shipped and received, and backorders. Consequently, orders placed to the upstream suppliers include not only the actual incoming demand, but also quantities covering the echelon’s own desired inventory position, customer service level, and cost requirements (Disney et al., 2003). It can be observed that under this operating structure, the traditional supply chain often suffers long lead-times, multiple decision points, unclear or redundant information, and minimal synchronization (Childerhouse & Towill, 2000). The inception of the VMI concept can be traced back to 1958 when Professor Magee studied the issue of who should have the authority to control inventories (Holmstro}m, Småros, Disney, & Towill, 2003). Although the concept emerged half a century ago, the interest in VMI started to grow only in the 1990s, during which organizations were seeking ways to improve their supply chain wide performance to create and maintain competitive advantages. In recent years, a drastic decline in telecommunication costs and a quick improvement in firms’ ability to provide accurate information have made it economically plausible to exchange sophisticated information quickly and accurately. As this is a critical requirement for a VMI supply chain to work effectively, it is also a driving force for the increasing applications of VMI. An example of VMI supply chains is shown in Fig. 2, where the triangle denotes inventory information. Under this circumstance, the upstream member (which is often a manufacturer, but can be a distributor) manages the adjacent downstream member’s inventories so as to ensure that a predetermined customer service level is maintained. In such a channel relationship, the upstream member is now responsible for making replenishment decisions for the downstream member and for scheduling the shipments of products at a quantity that can be either fixed (so as to maximize the
Customers
Retailers
Distributors
Manufacturers
Fig. 1. The structure of a traditional three-echelon supply chain.
Retailers
Distributors
Manufacturers
Fig. 2. The structure of a three-echelon VMI supply chain.
production or transportation efficiency) or variable (Waller et al., 1999). Replenishment occurs only when the stock level at the downstream echelon reaches a specified level based on both the average demand during the replenishment lead-time and a safety-stock requirement. Since the ownership of inventories in a VMI structure is shifted to upstream, in return, the downstream member is required to provide real-time demand information to the upstream supplier by allowing the supplier to access its demand databases. It is not difficult to see that a critical prerequisite for the VMI model to succeed is that a large amount of information must be transferred freely and timely between the channel members. The VMI approach generates numerous benefits for all channel members in a supply chain. First of all, the possibility of demand information amplification is dampened as the manufacturer now has a direct view at the end-consumer demand patterns, which will improve the forecasting accuracy. As a result, the manufacturer will be able to utilize the production facilities more efficiently, and the production outputs will not be ramped up and down frequently, which are often seen in a traditional supply chain with predicted large swings in demand. Second, there is only one inventory control point in the chain, which helps reduce the total buffer stocks along the chain. Third, the service levels can be improved as the product availability is increased (Waller et al., 1999). Finally, VMI can, in the long run, increase the profitability of both downstream and upstream members (Dong & Xu, 2002), because the downstream member can afford a price reduction as a result of lowered inventory costs, which in return may increase sales volumes for price-sensitive demands, thereby increasing the resulted profitability. 3.2. The profit functions of the channel members This section establishes the profit functions of the channel members in both traditional and VMI supply chains. Without loss of generality, we consider a two-stage supply chain where there is only one supplier and one downstream buyer, and we will examine their inventory management practices before and after the implementation of VMI. In this paper, we consider the situation under which the buying company dominates the supply chain and appears to be the ‘‘leader”. This decision scenario is inspired by our recently completed case study of a Chinese auto maker’s procurement method for one of its strategic MRO commodities (SEU Report, 2005). Because of the high risk and high cost of this strategic MRO item, this Chinese auto manufacturer wanted to make careful purchasing decisions and to take a leading role in the transaction. Specifically, the Chinese buyer first presented a price to its German MRO supplier, who then determined the supply quantity accordingly. Based on each individual profit, the two parties wanted to determine the feasibility of a VMI implementation. To proceed, we make several assumptions that are found common in the literature pertinent to inventory management (e.g., Silver, Pyke, & Peterson, 1998). First of all, the buyer’s inventory
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control policy is described by an economic order quantity (EOQ) rule based on deterministic demand and lead-time, and no stockouts. Second, both the unit order processing or setup cost and the unit cost of holding inventory are known and constant. Third, the buying firm determines the wholesale prices for certain quantities of products provided by the supplier, and then the supplier will choose a sales quantity to maximize its profit according the given wholesale price. Finally, the demand is inversely affected by the price; that is, the higher the price, the lower the demand; and hence, we use p(y) to represent the price and p0 (y) < 0. Additionally, a set of symbols and notation is used throughout the paper and is explained as follows. For simplicity and clarity, we use the subscript, ‘‘1”, to denote all the terms for the upstream supplier, and ‘‘2” for the downstream buyer. y: the annual purchase quantity of the downstream buyer p(y) the inverse demand function of the final product and is monotonically decreasing with respect to y, that is, p0 (y) < 0 w the wholesale price s the unit order/setup cost, in $/order the order quantity of the downstream buyer, and Q2 Q2 = (2ys2/I2Cv)1/2 Cv the unit product value, in $/unit I the unit inventory carrying cost, in %/year. With the above assumptions and the notation list, the time sequence of the one-shot game of the channel members is given as follows: (i) the downstream buyer brings forward the wholesale price, w, and then, the upstream supplier chooses a quantity, y, to maximize its profit based the wholesale price, w. (ii) And then, the two parties individually determines whether the VMI or a traditional strategy should be used based on their respective payoff. Thus, we can give the downstream buyer’s annual profit funcQ tion in a traditional supply chain, 02 ðy; wÞ, as follows 0 Y 2
ys2 I2 C v ðy; wÞ ¼ pðyÞy wy þ Q2 ; Q2 2
ð3:1Þ
where ys2/Q2 is the annual ordering cost, and I2Cv Q2/2 is the annual cycle inventory holding cost. It is reasonable to assume that the supplier’s total cost, which are associated with inventory, production and distribution, is an increasing convex function with respect to the quantity, y. As such, if the total cost function is represented by c(y), then c0 (y) > 0 and Q c00 (y) > 0, and the supplier’s profit function, 01 ðy; wÞ, can be given as: 0 Y 1
s1 y I 1 C v þ Q2 Q2 2 1=2 I2 C v s2 y s1 I 1 ¼ wy cðyÞ þ : 2 s2 I 2
ðy; wÞ ¼ wy cðyÞ
ð3:2Þ
Note that Q2 in (3.2) is replaced by the expression given in the notation list. Thus, for a given wholesale price, w, by the downstream buyer, the upstream supplier will choose a quantity, y, to maximize its profit. This optimal quantity can be determined from the following first-order condition obtained from the function in (3.2):
w ¼ c0 ðyÞ þ
1=2 1 I 2 C v s2 s1 I1 : þ 2 2y s2 I2
ð3:3Þ
Furthermore, at this price, which replaces w in (3.1), the downstream buyer then maximizes its profit by choosing an optimal purchase quantity, y*, such that:
p0 ðy Þy þ pðy Þ c0 ðy Þ c00 ðy Þy
1=2 1 2I2 C v s2 ¼ 0: 2 y
1=2 1 I2 C v s2 s1 I 1 þ 4 2y s2 I 2 ð3:4Þ
If the two members decide to adopt the VMI approach, then the downstream buyer transfers the ownership of inventories to the upstream supplier, who will make decisions regarding the inventory levels, order quantities, and replenishment frequencies on behalf of the buyer. As a result, the upstream member now has combined inventories with the new order setup cost (s1 + s2) and the new carrying cost (I1 + I2)Cv. Let wVMI represent the new wholesale price in Q and PVMI represent the new profit the presence of VMI, and VMII 1 2 function of the downstream and the upstream members, respectively. Then, VMI Y
ðy; wVMI Þ ¼ y wVMI cðyÞ ½2ðs1 þ s2 ÞðI1 þ I2 ÞC v y1=2
ð3:5Þ
ðy; wVMI Þ ¼ y½pðyÞ wVMI :
ð3:6Þ
1 VMI Y 2
When the upstream supplier with the VMI agreement in place aims to maximize its profit, the following relationship between the contractual wholesale price and quantity can be obtained from the first-order condition of the upstream member’s profit function in (3.5):
wVMI ¼ c0 ðyVMI Þ þ
1=2 1 2ðs1 þ s2 ÞðI1 þ I2 ÞC v : 2 yVMI
ð3:7Þ
The downstream buyer’s optimal purchase quantity can be determined by the first-order condition of its profit function in (3.5) with wVMI being replaced by (3.7). As such, we obtain the following relationship in a long-term use of the VMI:
yVMI p0 ðyVMI Þ þ pðyVMI Þ c0 ðyVMI Þ yVMI c00 ðyVMI Þ 1=2 1 2ðs1 þ s2 ÞðI1 þ I2 ÞC v ¼ 0: 4 yVMI
ð3:8Þ
3.3. A numerical illustration of the benefits in VMI supply chains To put the theoretical results into perspectives, we use the explicit functions for p(y) and c(y) to derive the benefits resulted from a long-term implementation of VMI. Assume that the relationship between the sales price of a product and the demand can be represented by a simple linear function: p(y) = a by, " a,b > 0, and that the total cost function of the supplier is given by: c(y) = dy + 0.8hy2, " d,h > 0. Additionally, the demand and cost parameters are fixed at the following values:
a ¼ 200; b ¼ 0:01; h ¼ 0:006; s2 ¼ 600; C v ¼ 100:
d ¼ 60;
I1 ¼ I2 ¼ 20%;
If s1 is set to 1800, we can calculate the buyer’s optimal purchase quantity, y*, in a traditional supply chain using (3.4) and in a long-term VMI chain using (3.8). As shown in Fig. 3, the purchase quantity increases in a long-term VMI supply chain, which confirms the result reported in the literature (Dong & Xu, 2002) that a matured VMI supply chain eventually leads to an increased purchase quantity of the buyer, which will increase the total chain profit. Although the results presented in Sections 3.2 and 3.3 are obtained from a standard sequential optimization method for finding the optimal decisions for the buyer and seller, they provide a foundation for our following analysis of the VMI chain’s evolution using the evolutionary game theory.
Purchase Quantity
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3500 3490 3480 3470 3460 3450 3440 3430 3420 3410 3400
The idea here is that the ESS state, s, resists all possible ‘‘mutations”, x, either because (i) they are less fit or (ii) they are equally fit at the current state (where they are rare) but less fit when they are prevalent. Moreover, the ruled Replicator dynamic equation named by Taylor and Jonker (1978) is seen below:
3490
dxi ¼ ½f ðs; xÞ f ðx; xÞ xi : dt
3435
Traditional Chain
VMI Chain
Fig. 3. Changes of the optimal purchase quantities.
Table 1 The payoff matrix of the two supply chain strategies Downstream
Upstream VMI Traditional
VMI
Traditional
QVMI QVMI 1 ; 2 Q0 1 p1 ; 2 n
Q0
Q0
1 m; Q0 Q0 1; 2
Q0
2
p2
3.4. The payoff matrix of the two supply chain strategies Based on the results obtained in the previous section, we present the payoff matrix of the VMI and traditional supply chains in Table 1. It is seen that the payoff matrix is similar to that of the Prisoner’s Dilemma, a well-known simple game theory model used to demonstrate the complexity of cooperation in any system (see for example, Kuhn, 2003). In the payoff matrix, the parameters, (m, n), are the respective investment made by the upstream and downstream buyers to adopt the VMI strategy; for example, the costs spent on installing or upgrading the software and hardware for information sharing. The terms, (p1, p2), represent the two members’ respective financial penalties, such as the goodwill loss or opportunity loss, when they breach the channel contract.
Consider two populations from the upstream suppliers’ network and downstream buyers’ network that are interactive to each other. For the sake of simplicity, two populations are assumed to be of equal size. In each period t, member i in the supplier’s network plays a game with member j in the buyer’s network by choosing either VMI or a traditional supply chain strategy. Every member uses Replicator dynamic equation to adjust its strategy at every game stage. If we assume that the probability of an upstream supplier selecting the VMI strategy at the initial state is aVMI = a (0 6 a 6 1). (The probability can be equal to the percentage of the supplier population selecting the VMI strategy), then, with a0 = 1 a, it will use the traditional supply chain strategy at that state. Similarly, with the probability, bVMI = b (0 6 b 6 1), a downstream buyer will adopt the VMI strategy at the original state, and withb0 = 1 b, it will not do so. Based on these probabilities, we can obtain a series of equations regarding the expected values of a number of terms. Specifically, the average payoff of the upstream supplier if it selects to adopt the VMI strategy can be expressed as: VMI Y
As the numerical example in the preceding section suggests, the channel profit increases in a long-term VMI supply chain; but the profit change for each individual member is unclear. Evolutionary game theory can shed more insights on the intrinsic evolutionary mechanism of the VMI chains and examine the trend of the channel members’ evolutionary stability from a time-based perspective. We use the evolutionary game theory to study the evolutionary stability of the VMI strategy. Evolutionary game theory combines the static feature of an evolutionary stable strategy (ESS) with the dynamic nature of the Replicator dynamic initiated by Taylor and Jonker (1978). The concept of ESS, initially proposed by Maynard Smith (Maynard-Smith & Price, 1973; Maynard-Smith, 1974, 1982), has had perhaps more influence than any other theory has in the field of evolutionary games. The original definition of ESS for a single population is given in a linear fitness function, f(r, s) = rT As for some given N N matrix A. It is as follows. Definition 4.1. A state s 2 S is an ESS if for all other states x 2 S either
ðiÞ f ðs; sÞ > f ðx; sÞ; or ðiiÞf ðs; sÞ ¼ f ðx; sÞ and f ðs; xÞ > f ðx; xÞ:
¼b
1
VMI Y
þð1 bÞ
0 Y
1
! m ;
ð4:2Þ
1
Q Q where VMI and 01 are given in (3.5) and (3.2), respectively. Fur1 thermore, the supplier’s average payoff if choosing the traditional strategy is given by 0 Y
¼b
0 Y
1
! p1
þ ð1 bÞ
1
0 Y
:
ð4:3Þ
1
It is now straightforward to obtain the expected payoff of the upstream supplier as follows:
Y 1
4. Analysis of the evolutionary stability of the VMI supply chains
ð4:1Þ
¼a
VMI Y 1
þ ð1 aÞ
0 Y :
ð4:4Þ
1
As a result, the Replicator dynamic equation of the upstream supplier who selects the VMI strategy based on Definition 4.1 and Eq. (4.1) can be given as
! " ! # VMI Y VMI 0 Y Y Y da ¼ að1 aÞ b þp1 þ m m : ¼a dt 1 1 1 1
ð4:5Þ
Similarly, the Replicator dynamic equation of the downstream buyer if it chooses to adopt the VMI strategy can be obtained as:
! " ! # VMI Y VMI 0 Y Y Y db ¼ bð1 bÞ a þp2 þ n n : ¼b dt 2 2 2 2
ð4:6Þ
Q where 2 denotes the expected payoff of the downstream buyer, and its expression can be derived in a similar fashion to the series from (4.2),(4.3),(4.4). The differential equations in (4.5) and (4.6) describe the group evolutionary dynamics of the two-stage supply chain. They indicate that the trend of each channel member’s profit with VMI implementation depends on the probabilities of other channel members choosing to use VMI, the profit functions before and after the implementation of VMI, the original investments, and the financial penalties.
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According to Ellison and Fudenburg (2000), the stable state of the Replicator dynamic equation is the Nash equilibrium (NE), which is often referred to as the evolutionary equilibrium da dt
(EE). When
¼ 0; db dt ¼ 0, the
EE
of (4.5) and (4.6) are n QVMI Q 0
E1 = (0, 0), E2 = (0, 1), E3 = (1, 0), E4 = (1, 1), and E5 ¼
2
! m QVMI Q 0
1
þp1 þm 1
2
þp2 þn
;
: . Note that the two terms in the bracket of E5 are
both fractional. Based on the approach suggested by Friedman (1991), the stability of EE can be analyzed by the Jacobi matrix, which is given here as follows: by ddta and db dt 3 VMI 0 VMI 0 Q Q Q Q ð1 2 a Þ b þp þ m m a ð1 a Þ þp þ m 1 1 7 6 1 1 1 1 7 6 J¼6 VMI 0 VMI 0 7: 5 4 Q Q Q Q þp2 þ n ð1 2bÞ a þp2 þ n n bð1 bÞ 2
2
2
2
2
The two terms, detJ and trJ, associated with the above Jacobi matrix, are derived below:
(
VMI Y
det J ¼ ð1 2aÞ b
1
(
a
VMI Y
0 Y
2
0 Y
VMI Y
1
)
þp1 þ m m
1
!
VMI Y
þp1 þ m bð1 bÞ
1
2
and
trJ ¼ ð1 2aÞ b
VMI Y
1
þ ð1 2bÞ a
0 Y
!
2
0 Y
!
0 Y
! þp2 þ n ;
2
! !
þp2 þ n n :
2
Proposition 4.1. With a short-term application of VMI, when the channel penalty costs do not exist, i.e., both p1 and p2 are zero, then the Replicator dynamic equations in (4.5) and (4.6) have four EE, (E1, E2, E3, E4), out of which only E1 = (0, 0) is the ESS, implying that no channel member chooses to use the VMI strategy. Proof. In a short-term use of the VMI, the change of the upstream supplier’s profit can be calculated by
1
So, if s1 = s2 and I2 > I1/7, then VMI Y
1
0 Y
¼
1
1=2 h i2 1 I2 s2 y 2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 2 2
< 0: ð4:8Þ Similarly, the profit change of the downstream member can be found as VMI Y
2
0 Y
¼
2
1=2 h i2 1 I 2 s2 y > 0: 2 þ ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 2 2 ð4:9Þ
¼ y wVMI cðyÞ ½2ðs1 þ s2 ÞðI1 þ I2 ÞC v y
1
þp1 þm
R ½0; 1, i.e., E5 is not the EE of the Replicator dy-
1=2
1=2 # I 2 C v s2 y s1 I 1 : wy cðyÞ þ 2 s2 I 2
p1 >
h i2 1 : ðI2 s2 y=2Þ1=2 2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 2
Proof. Based on Table 2, E4 is the ESS if and only if the following is Q Q Q Q 01 þ p1 Þð VMI 02 þ p2 Þ > 0. Since Eq. (4.9) shows true: ð VMI 1 2 QVMI Q0 QVMI Q0 2 > 0, then 1 þ p1 > 0. Based on (4.8), that if 2 1
During the early stage of the VMI implementation, the sales and purchase quantities stay relatively unchanged due to market constraints and other contractual or public agreements between the channel members. For this reason, the short-term effects will be evaluated at the extent to which the annual purchase quantity, y, is the same as that in a tradition supply chain. The wholesale price, however, should be adjusted relatively quickly as a new VMI contract is negotiated and determined. Based on the results obtained in the preceding section, the next two propositions are presented for a supply chain with a short-term application of VMI.
1
1
1=2 h i2 1 I2 s2 y 2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 : 2 2
Proposition 4.2. With a short-term utilization, the VMI strategy will be accepted by the channel members on the conditions that E4 is the ESS, and if and only if the upstream member’s penalty cost, p1, satisfies the following requirement:
4.1. The ESS with a short-term VMI implementation
0 Y
¼
namic equations in (4.5) and (4.6). Furthermore, based on the Jacobi matrix, we report the results of the local stability analysis in Table 2. It is seen from Table 2 that E1 is the only ESS of Replicator dynamic equations. This means the related channel members will refute the VMI strategy if it is used only in a short-term and the channel penalty costs are not present. h
In the following section, we will use the Jacobi matrix to analyze the ESS with short-term and long-term VMI supply chains, respectively.
VMI Y
1
0 Y
1
þp1 þ m m
1 VMI Y
Thus, QVMI Q0
ð1 2bÞ
að1 aÞ
!
VMI Y
m
)
þp2 þ n n
2 0 Y
!
Based on Eqs. (3.3) and (3.7) and Q2 = EOQ2 = (2s2y/I2)1/2, we can simplify the upstream member’s profit change in (4.7) and obtain the following relationship:
"
ð4:7Þ
the following expression can be obtained, p1 > 12 ðI2 s2 y=2Þ1=2 h i2 , which proves the proposi2 ð1 þ s1 =s2 Þ1=2 ð1 þ I1 =I2 Þ1=2 tion.
h
Interestingly but reasonably, the first proposition indicates the fact that because of the advantageous position, the downstream member is able to take away the majority of the cost savings from a VMI adoption. The upstream member, on the other hand, as indicated in the second proposition, not only receives nothing from this cost reduction, but has to accept some penalty in order for the VMI method to outperform the traditional approach. It implies that the profit increase of the downstream buyer equals the profit reduction of the upstream supplier after VMI is imple-
Table 2 Analysis of the local stability of EE with a short-term after VMI EE
det J
E1 = (0, 0) E2 = (0, 1) E3 = (1, 0) E4 = (1, 1)
mn Q ð VMI Q1 ð VMI Q2 ð VMI Q1 ð VMI 2
tr J Q 01 Þn Q 02 Þm Q 01 Þ Q 02 Þ
+
m n QVMI Q0 þn Q1VMI Q10 þm 2Q 2Q VMI ð 1 01 Þ QVMI Q0 ð 2 2Þ
Local stability
ESS Saddle-point Saddle-point Unsteady
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Replicator dynamic equations in (4.5) and (4.6) have five values of EE, (E1, E2, E3, E4 and E5), and that only E1 and E4 are the ESS. h
mented, because the downstream buyer’s inventory cost has been transferred to upstream supplier with VMI in place. Thus, from a short-term perspective, the downstream buyer has the desire to implement VMI but the upstream supplier has a disadvantage to adopt it. However, it should be stressed that this series of results is obtained based on the assumption that the downstream member keeps the same annual purchase quantity as used in a traditional chain. This condition limits the upstream member’s ability to increase its optimal profit in a long-term. The next section will discuss the ESS of the long-term utilization of the VMI strategy.
The result obtained herein may explain why the upstream member is skeptical about the VMI approach at the beginning of the implementation, but will gradually accept it as the opportunities for reducing other costs and for increasing the long-term profit exist. To achieve better coordination, the downstream buyer should compensate the upstream supplier for the loss by sharing revenues so that the upstream supplier will have sufficient financial incentive to accept VMI during the early stage of implementation. As the implementation of VMI progresses, the purchase quantity in a VMI supply chain will grow higher than that in a traditional chain, and as a result, the total supply chain profit will increase. For VMI to reach the ESS and to eventually benefit the entire chain, the supply chain members should be aware that the purchase quantity will increase in the long run, and thus, it may be necessary for the downstream buyer to share profit with the upstream supplier to cover the supplier’s loss during the early stage of VMI.
4.2. The ESS with a long-term application of the VMI It has been reported that supply chain members adopting the VMI approach have experienced increases in sales eventually, which will bring up the purchase quantity (as indicated in Fig. 3). In a long-term use of the VMI, the downstream buyer’s profit will increase, while the upstream supplier’s profit may increase under some specified conditions. Proposition 4.3 given below indicates such conditions.
5. Analysis of the impact factors on ESS
Proposition 4.3. In a long-term adoption of the VMI and without the channel penalty costs, the Replicator dynamic equations in (4.5) and (4.6) have five EE, (E1, E2, E3, E4, E5), out of which E1 and E4 are ESS.
We illustrate the dynamic evolutionary process of the channel members with a long-term utilization of VMI in Fig. 4, which is created by using the simulation program called ‘‘Dynamo” (Sandholm and Dokumaci, 2007). The input values of the diagram are (y, p1, p2, m and n), and the illustration depicts how a VMI supply chain evolves from its initial state. The curves of E2, E3 and E5 are the critical lines of the dynamic evolutionary games, along which they
Proof. Let
f ðyÞ ¼ p0 ðyÞy þ pðyÞ c0 ðyÞ c00 ðyÞy
1=2 1 I 2 C v s2 ðs1 =s2 þ I1 =I2 Þ: 4 2y
Using Eqs. (3.4) and (3.7), we obtain the following two functions: 1=2 2 C v s2 f ðyVMI Þ ¼ p0 ðyVMI ÞyVMI þ pðyVMI Þ c0 ðyVMI Þ c00 ðyVMI ÞyVMI 14 I2y VMI
ðs1 =s2 þ I1 =I2 Þ ¼ 0
E2
and
f ðy Þ ¼ p0 ðy Þy þ pðy Þ c0 ðy Þ c00 ðy Þy
E4
1=2 1 I 2 C v s2 4 2y
ðs1 =s2 þ I1 =I2 Þ P p0 ðy Þy þ pðy Þ c0 ðy Þ c00 ðy Þy 1=2 1=2 1 I2 C v s2 1 2I2 C v s2 ðs =s þ I =I Þ ¼ 0: 1 2 1 2 4 2y 2 y Thus, f(yVMI*) = 0 and f(y*) > 0, and f(y) is a strictly monotonic digressive function with respect to y. Furthermore, it can be shown that yVMI > y , or the purchase quantity in a VMI supply chain is higher than that in a traditional chain. (This is proven true numerically in Section 3.4.) In summary, a long run application of the VMI will increase the upstream Q Q0 member’s profit, which means that VMI 1 ðyVMI Þ 1 ðyÞ > 0. Similar to n m Proposition 4.1, since QVMI Q0 2 ½0; 1; QVMI Q 2 ½0; 1, E5 is 0 2
2
þp2 þn
1
1
E5
E1
þp1 þm
an EE. The results of the local stability analysis for a long-term application of VMI are summarized in Table 3, which clearly indicates that the
E3
Fig. 4. The process of the dynamic evolutionary game.
Table 3 Analysis of the local stability of EE with a long-term after VMI EE
det J
E1 = (0, 0) E2 = (0, 1) E3 = (1, 0)
mn Q Q ð VMI 01 þ p1 Þn Q1 Q0 ð VMI 2 2 þ p2 Þm QVMI Q0 ð 1 1 þ p1 Þ Q Q 02 þ p2 Þ ð VMI 2 QVMI Q0 mnð 2 2 þp2 Þ
QVMI Q0 þp þn
Q2 VMI Q2 0 2 þp
QVMI1 Q0 1 1
E4 = (1, 1) n E5 ¼ QVMI Q 0 2
2
þp2 þn
m ; QVMI Q 0 1
1
þp1 þm
1
tr J
1
þp1 þm
+ +
m n QVMI Q0 þ p1 þ n Q1VMI Q10 2 þ p2 þ m 2 Q Q ð VMI 0 þ p1 Þ Q 1 Q0 1 ð VMI 2 þ p2 Þ 2 0
Local stability + +
ESS Saddle-point Saddle-point
ESS
Saddle-point
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gradually evolve into E1 or E4 to become ESS. On the right upper corner of the region in which the evolution moves toward E4, the supply chain members will adopt the VMI model completely. In contrast, if the evolution moves toward the lower left corner, then the channel members tend to constringe to E1, which implies that they will prefer the traditional strategy over VMI. Note that the dynamic evolutionary game process can be long and may allow coexistence of the VMI and traditional strategies. It can be seen from Fig. 4 that whether the dynamic evolutionary game of the supply chain members converges to E1 or E4 is influenced by the initial state of the game. Moreover, E5is the pivotal point that affects final converging state, and therefore, it is the impact factor of the ESS; in other words, the impact factor of E1 or E in the capacity of ESS is determined by the following terms: Q4VMI Q0 QVMI Q0 2 , 1 1 , p1, p2, m and n. 2 To obtain the intrinsic evolutionary mechanism and improve the efficiency of the VMI supply chain, it would be interesting to examine the effects of the main parameters on the ESS. However, since it is difficult to obtain closed-form expressions and clearly defined results, we discuss the impact factor analysis of the ESS by simply observing the changes of the following six terms: QVMI Q0 QVMI Q0 2 , 1 1 , p1, p2, m and n, and using Fig. 4 as a guide2 line. First of all, under the long-term employment of the VMI, the downstream and upstream members’ profits will increase with Q Q Q Q the purchase quantity. As a result, VMI 02 and VMI 01 will 2 1 gradually increase, moving the location of E5 gradually close to the origin, which makes E4 the ESS with higher possibility. Second, the pair, (p1, p2), positively increases the likelihood that E4 will become the ESS, whereas the other pair, (m, n) decreases it. Finally, as the purchase quantity will increase in the long-term utilization of the VMI and the values of both m and n are fixed, the impact of (m, n) can be negligible. As a result, the absolute value of the profit change will approach to zero. From this point of view, E4 will remain the only ESS, that is, the supply chain members will fully adopt VMI. 6. Conclusions and discussions This study has focused on the evolutionary trend analysis of the VMI supply chains. The evolutionary game theory is used to analyze the evolutionary stable strategy (ESS) of the VMI supply chain to examine the intrinsic evolutionary mechanism of the VMI method. With the consideration of the bounded rationality of the supply chain channel members, we have analyzed the trend of the evolutionary stability of the VMI implementation from both short- and long-term points of view in order to identify the conditions under which the VMI method will outperform the traditional approach. In a short-term utilization of the VMI, if the chain members choose to adopt the VMI strategy, then the upstream supplier will have some profit loss as well as a penalty cost consisting of goodwill and opportunity loss at the beginning of implementation. However, as the implementation matures, increases in purchase quantity will be observed, which will increase both downstream and upstream members’ profits, and as such, the two chain members will eventually adopt VMI completely. This research provides practitioners with a clear understanding of how to make decisions with respect to VMI. As our results indicate, a downstream buyer does obtain certain inventory cost reductions at the beginning stage of VMI, but the upstream supplier cannot achieve inventory cost reductions; rather, it will experience reduced profit. However, with the implementation progresses, an increased transaction quantity between the chain members will be resulted from more information sharing, better visibility and availability of information, and more accurate forecast. As a result, there are opportunities for all involving members
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to increase profits. For a more effective implementation of VMI and to exploit the benefits of the VMI, the downstream buyer can share some revenue with the upstream supplier to compensate the supplier’s partial loss during the early stage of VMI implementation. In contrast to most research efforts devoted to understanding the complexity of the VMI adoption and implementation, this paper takes a different approach – using the evolutionary game theory to analyze how VMI may evolve over a short or long period of time after it is implemented. Our research sheds insights on when and why the supply chain partners should coordinate during VMI implementations. However, our current analysis has a number of limitations, which suggest the following future research directions. (1) It will be interesting to quantify and analyze the Bullwhip Effect of a VMI supply chain by applying evolutionary games based on the bounded rationality; (2) when the service levels required at downstream and upstream stages are different, how will the trend of the evolutionary stability affect the adoption of the VMI model? And (3) Sensitivity analysis of the impacts of the parameters, including p1, p2, m and n, on the evolution of VMI to an ESS needs to be conducted. Acknowledgements This research is supported by the National Natural Science Foundation of China (706711021) for the project titled ‘‘Project Operation and Simulation of Emergency Response Logistics Network in the System of Anti-Bioterrorism”. We are also grateful to the anonymous reviewers for their constructive comments, which have improved the quality and presentation of this paper. References Achabal, D., Mclntyre, S., Smith, S., & Kalyanam, K. (2000). A decision support system for vendor managed inventory. Journal of Retailing, 76(4), 430–454. Cetinkaya, S., & Lee, C. Y. (2000). Stock replenishment and shipment scheduling for vendor-managed inventory systems. Management Science, 46(2), 217–232. Childerhouse, P., & Towill, D. R. (2000). Engineering supply chain to match customer requirements. Logistics Information Management, 13(6), 337–345. Christina, P. (2007). Finite populations choose an optimal language. Journal of Theoretical Biology, 249(3), 606–616. Disney, S. M., Potter, A. T., & Gardner, B. M. (2003). The impact of vendor managed inventory on transport operations. Transportation Research Part E, 39(5), 363–380. Disney, S. M., & Towill, D. R. (2002). A procedure for the optimization of the dynamic response of a vendor managed inventory system. Computers & Industrial Engineering, 43(1–2), 27–58. Dong, Y., & Xu, K. (2002). A supply chain model of vendor managed inventory. Transportation Research Part E, 38(2), 75–95. Ellison, G., & Fudenburg, D. (2000). Learning purified mixed equilibria. Journal of Economic Theory, 90(1), 84–115. Eriksson, A., & Lindgren, K. (2005). Cooperation driven by mutations in multi-person Prisoner’s Dilemma. Journal of Theoretical Biology, 232, 399–409. Friedman, D. (1991). Evolutionary games in economics. Econometrica, 59(3), 637–666. Holmstro}m, J., Småros, J., Disney, S. M., & Towill, D. R. (2003). Collaborative supply chain configurations: The implications for supplier performance in production and inventory control. In Presentation at the 8th symposium on logistics, July 6–8, 2003, Seville, Spain. Huang, Z., & Li, S. X. (2001). Co-op advertising models in manufactures-retailer supply chain: A game theory approach. European Journal of Operational Research, 135(3), 527–544. Kisperska, M. D. (1999). Warehousing conditions for holding inventory in polish supply chain. International Journal of Production Economics, 59, 123–128. Kuhn, S. (2003). Prisoner’s Dilemma, The Stanford Encyclopedia of Philosophy (Fall 2003 Edition). Available from: <
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