Production, Manufacturing and Logistics
A game-theoretical cooperative mechanism design for a two-echelon decentralized supply chain

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Abstract

The paper analyses non-cooperative behaviour in a two-echelon decentralized supply chain, composed of one supplier and n retailers. For sufficient supply from the supplier, we build the approximate decision model of their base stock level, in which the suppliers' reactions are not considered, and its non-cooperative behaviour is obtained. For insufficient supply from the supplier, much more complicated non-cooperative behaviour is obtained, and we find that competition will occur between all the retailers as well as the supplier. In order to guarantee optimal cooperation in the system, several Nash equilibrium contracts are designed in echelon inventory games and local inventory games.

Introduction

The literature of supply chain inventory management mostly assumes that policies are set by a central decision-maker to optimize total supply chain performance (Cohen and Lee, 1988; Ishii et al., 1988; Pyke and Cohen, 1994; van Houtum et al., 1996; Maloni and Benton, 1997; Chopra and Meindl, 2001; Velde and Meijer, 2002). In the multi-echelon inventory domain, early in 1960, Clark and Scarf (1960) gave the exact decomposition procedure to find the optimal control parameters of a serial supply chain. Recently, Diks and de Kok (1999) gave a method to solve the general n-echelon divergent supply chain problem.

However, the central optimization approach is only an approximation of the real supply chain system, since most supply chain systems are decentralized. The central optimization models have three drawbacks:

  • (1)

    Ignoring the independence of the supply chain members. Competitive behaviour between members may lower the supply chain efficiency.

  • (2)

    The cost of information processing may be expensive. The central decision maker must gather all the information from every supply chain member and finally issue instructions to members.

  • (3)

    The capacity of the central optimazation models. If the problem is fairly large and difficult, however, it may be impossible to be modelled and calculated.


Recently the decentralized supply chains have been studied by many researchers (Monahan, 1984; Goyal and Gupta, 1989; Lee and Billington, 1993; Cachon and Zipkin, 1999; Chen, 1999; Lee and Whang, 1999). The decentralized supply chains can be categorised as two kinds: the intra-organizational-coordination one and an inter-organizational-cooperation one. In an intra-supply chain, all members are less selfish within an organization/firm, and there may exist a central power who controls the whole system to some extent. This kind of supply chain optimization is in the distributed problem solving realm, and its main problem is to design all members' sub-goals or performance measurement schemes to mitigate the incentives problems of all members in the supply chain. Monahan (1984) first used the quantity discount as a measurment scheme in a two-echelon fixed demand system. Federgreun and Zipkin (1984) found that by constructing cost functions appropriately, a decentralized system can perform equally as a decentralized one for fixed demand two multi-echelon inventory with a single retailer, and can perform almost as well as a centralized one with multiple retailers. Lee and Whang (1999) gave a set of measurement schemes for a stochastic demand one–one supply chain inventory system.

An inter-organizational-cooperation supply chain is composed of several self-interested members/firms. In order to achieve supply chain efficiency, all members are willing to cooperate, but the cooperative mechanism, e.g. contracts, must be carefully negotiated and designed. Many studies show that competition between members lowers the system efficiency, and usually is not optimal. How we can coordinate the supply chain members? A contract-based coordination mechanism should be used to fulfill this work, and the contract must comply with the following rules:

  • (1)

    Ensure the members' profit is not lower than the profit level before supply chain cooperation and properly share the total cooperative revenue––Participative constraint.

  • (2)

    Diminish the incentives to deviate from the system optimal solution––Incentive compatible constraint.


Cooperation in fixed demand inventory models has been studied carefully in the inventory literature as well as in the microeconomics area. Goyal and Gupta (1989) analysed the quantity discount in a two-echelon fixed demand inventory model from the joint buyer–seller perspective. The information asymmetric problem also attracted some attention, even though not many researchers (Corbett and Groote, 2000) have studied the optimal coordination contract in two-echelon fixed demand inventory system under asymmetric holding costs.

In the stochastic inventory management area, however, the coordination mechanisms have fewer studies. Cachon and Zipkin (1999) analysed the non-cooperative game behaviour and the joint optimum of a two-echelon serial supply chain, which is composed of a supplier and a retailer, and given all supply members have full information. In this paper, we extend their model to a one-supplier and n-retailers situation. If there exist multiple retailers, the supply from a supplier might not satisfy the demand of multiple retailers. The problem is how to design the distributing scheme of the supplier, and this makes models of supply chain systems more complex. We separate the sufficient supply from the supplier and insufficient supply from the supplier in our model, which is not mentioned in Cachon and Zipkin's model (1999).

The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 considers the non-cooperative game model. Section 4 deals with the system optimal policy. Section 5 gives the cooperative mechanisms based on linear contracts. Section 6 shows a numerical example.

Section snippets

Model description

The model is similar to the model presented by Cachon and Zipkin (1999), but we consider a one-product inventory system with one supplier (denoted by suffix 0) and n retailers (denoted by suffix 1 to n) as shown in Fig. 1. Time is divided into an infinite number of discrete periods, and consumer demand at each retailer is stochastic, independent across periods and stationary. The following is the sequence of events during a period: (1) shipments arrive; (2) orders are submitted and shipments

Echelon inventory games and local inventory games

In a supply chain system, all members could track their stocks by echelon inventory policies or local inventory policies. A firm's local inventory is its on-hand inventory, and its echelon inventory is its local inventory plus all inventory held lower in the supply chain. Using an echelon base stock level, each period the firm orders a sufficient amount to raise its echelon inventory position plus outstanding orders to that level. A firm's local base stock level is similar, except the local

System optimal solution

In Section 3, we have analysed the non-cooperative solution. Because the supplier and retailers are selfish profit maximizing centers, the sum of their own maximum profits is not the optimal solution of the whole system. If the whole system has an optimal solution, which minimizes the total average cost per period, and this will be the objective of cooperation. Diks and de Kok (1999) demonstrated an echelon based stock policy (S1J,…,SiJ,…,SnJ,S0J) is optimal in this setting:hi−(hi+h0+p)1−αii(SiJ

Cooperative mechanism design

In order to achieve the optimal cooperative solution, supply chain members must sign some contracts to diminish the incentives of the selfish profit maximizing motivation of partners. Below we consider a kind of contract that is based on transferable subsidy.

  • In echelon inventory games, the members' decision functions under contract are:

  • For the retailers, Ui=Ui+Ti, where i=1,2,…,n; for the supplier, U0=U0+T0.

  • In local inventory games, the members' decision functions under contracts are:

  • For the

Numerical study

We consider a supply chain system of one supplier and two retailers. The supplier and two retailers review their inventory level by a period of one week. One period demand of each retailer is normally distributed with mean 100 units and standard deviation 30. The supplier's holding cost is h0=1 per week per unit, and h0+h1=3(h1=2) and h0+h2=4(h2=3) are the unit holding costs for retailer 1 and retailer 2 respectively. The backorder penalty is p=4 per week per unit, and α=0.2. The lead time of

Conclusion and suggestions for future research

In this paper, we have analysed the non-cooperative behaviour in a two-echelon decentralized supply chain. In a decentralized supply chain, all partners' decision must be carefully coordinated to achieve the system optimal solution. When all partners are independent profit centers, their behaviour must be carefully studied and formal contracts must be assigned between all partners to diminish the derivation from system optimal solution to selfish motivations. Therefore, the partners' decision

Acknowledgements

The research is jointly supported by the National Natural Science Foundation of China (no. 70171015), the Excellent Young Teachers Fund, Foundation for University Key Teacher by the Ministry of Education, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, PR China.

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