Production, Manufacturing and Logistics
Game-theoretic analyses of decentralized assembly supply chains: Non-cooperative equilibria vs. coordination with cost-sharing contracts

https://doi.org/10.1016/j.ejor.2009.10.011Get rights and content

Abstract

This paper considers a multiple-supplier, single manufacturer assembly supply chain where the suppliers produce components of a short life-cycle product which is assembled by the manufacturer. In this single-period problem the suppliers determine their production quantities and the manufacturer chooses the retail price. We assume that the manufacturer faces a random price-dependent demand in either additive or multiplicative form. For each case, we analyze both simultaneous-move and leader–follower games to respectively determine the Nash and Stackelberg equilibria, and find the globally-optimal solution that maximizes the system-wide expected profit. Then, we introduce appropriate buy-back and lost-sales cost-sharing contracts to coordinate this assembly supply chain, so that when all the suppliers and the manufacturer adopt their equilibrium solutions, the system-wide expected profit is maximized.

Introduction

In many supply chains assembled products are composed of complementary components. It is well-known that a large number of firms in industry outsource the production of the components to external suppliers in order to reduce costs and increase production flexibility. For example, in the US, Toyota outsources the production of car components to many suppliers who then deliver the components to Toyota’s assembly plant in Kentucky [7]. In particular, more than 75% of the parts and 98% of the steel used in the production of vehicles at this assembly plant come from US suppliers. In 2005, the plant had 350 suppliers across the continental United States. The Toyota production system (one of the first successful examples of a Just-In-Time system) is “all about producing only what’s needed and transferring only what’s needed.” By adopting such an efficient system known as a “pull”-type supply chain, Toyota’s assembly plant uses the components (delivered by its suppliers) to assemble final products when the orders of its customers (e.g., dealers) arrive; see, for example, Chopra and Meindl [7] and Simchi-Levi et al. [17]. As reported by Reinhardt in [15], Nokia also recently implemented the pull strategy to make built-to-order phones each with a unique faceplate with the operator’s logo on it and special keypad buttons that take users directly to certain wireless services, etc. Another well-known example is Dell which adopts the pull-type strategy to assemble computers only when its customers’ orders arrive online.

As Cachon [4] and Granot and Yin [8] have discussed, in a pull-type assembly supply chain all suppliers of the assembly plant need to determine their production quantities of components while the assembly plant needs to choose its sale price of final product. Moreover, in order to quickly respond to its customers, the plant also aims to strengthen partnering relationship with its suppliers and establish coordination with the suppliers for system-wide improvement. As another real example of a manufacturer’s effort to induce supply chain coordination, in 2006, Motorola decided to spend $60 million in Singapore to centralize and streamline global supply chain operations with its suppliers and customers. As outsourcing is considered to be one of the strategies achieving supply chain integration, there is an extensive literature focusing on outsourcing strategies and vertical integration in supply chains; see, Cachon and Harker [5].

As the above examples illustrate, the coordination of a decentralized supply chain with assembled products appears to be an interesting and important problem worth investigating. Motivated by these examples, in this paper we consider the following natural question: what mechanism can be developed to coordinate all members in such a supply chain? As Cachon [3] indicated, supply chain coordination is achieved if and only if all firms in a decentralized supply chain can behave (that is, make decisions) as if they are operating in a centralized supply chain. More specifically, in a decentralized supply chain, all firms primarily aim at optimizing their own individual objectives rather than the chain-wide objective, thus their self-serving focus may result in a deterioration of the chain-wide performance. To improve the supply chain’s performance, a proper mechanism must be developed to coordinate all channel members so that both the individual supply chain members’ objectives and the chain-wide performance can be optimized; see, e.g., Leng and Zhu [10]. A common (and useful) mechanism for supply chain coordination is to develop a set of properly-designed contracts among all supply chain members [3]. With the successful use of this mechanism the last decade has witnessed a rapidly increasing interest in supply chain coordination with contracts.

In this paper, we restrict our attention to a multiple-supplier, one-manufacturer supply chain with complementary products. (Hereafter, such a supply chain with complementary products is called an assembly supply chain, as in Carr and Karmarkar [6].) In this assembly supply chain, multiple-suppliers produce their complementary components, and serve a common manufacturer who assembles the final products with short life cycles and satisfies a random demand.Our assumption of short product life cycles reflects the following fact: the last two decades have witnessed rapid technological innovation and a high level of competition in the marketplace. In response to these developments, many firms (e.g., manufacturers of personal computers, cell phones, cars, etc.) have implemented the philosophy of “life-cycle management” to reduce life cycles of their products. As in many publications (such as Linh and Hong [11] and Parlar and Weng [12]) concerned with the assembly supply chains with short product life cycles, we construct our model and perform our analysis using the single-period (newsboy) setting. Similar to the pull-type system discussed in Cachon [4] and Granot and Yin [8], the n(2) suppliers determine their production quantities independently of each other. Moreover, we assume that all members of this supply chain are risk-neutral and the demand is only sensitive to the retail price chosen by the manufacturer. Note that, in practice, consumers’ demands may also depend on some other factors (e.g., the quality of the product that the consumers buy). However, in this paper, we only focus on the manufacturer’s pricing decision and the suppliers’ quantity decisions, as in most of previous publications regarding assembly supply chains. Accordingly, we assume that the demand is only dependent of the manufacturer’s retail price, and thus use Petruzzi and Dada’s additive and multiplicative demand forms [14]—which have been commonly used to analyze assembly supply chains with price-dependent demand—to characterize the random demand.

We use buy-back and lost-sales cost-sharing contracts between the n suppliers and the manufacturer to coordinate the supply chain. With the buy-back contract, the manufacturer returns the unused components to the suppliers at the buy-back price. Because all unused components can be returned to the suppliers (albeit at some loss), the manufacturer does not concern himself with the optimal order quantities of the components. Instead, he attempts to choose the optimal price to maximize his expected profit. Buy-back contracts have been widely used to analyze supply chain coordination. As described in Cachon [3], a typical buy-back contract (also called return policies) has two parameters; the ith supplier’s wholesale price, wi, and the buy-back price, vi,i=1,,n. Under such a contract, supplier i charges the manufacturer wi per unit purchased at the beginning of the single period, and pays the manufacturer vi per unit remaining at the end of the period. In our paper, we model the buy-back contract as a vector (w,v), where w=(w1,,wn) and v=(v1,,vn). For an early application of buy-back contracts, see Pasternack [13].

With the lost-sales cost-sharing contract, when shortages arise, suppliers and the manufacturer share the shortage cost. Assuming that one unit of the final product needs one unit of each of the n components, the manufacturer needs equal number of components for assembly from each supplier. If the number of components received from the suppliers happens to be different, the number of the final product the manufacturer can assemble equals the minimum of these quantities. In our lost-sales cost-sharing contract, the shortage penalty cost incurred by the manufacturer at the end of the single period is shared among all members of this supply chain. Given a unit shortage (underage) cost u, we define the percentage of this cost absorbed by the ith supplier as ϕi[0,1] with ϕ=i=1nϕi[0,1]. Thus, all suppliers pay the shortage cost ϕu, and the manufacturer bears the cost of (1-ϕ)u. The lost-sales cost-sharing contract is characterized by the vector ϕ=(ϕ1,,ϕn). See Table 1 for a complete list of the notation used in this paper.

Our paper uses game theory to analyze non-cooperative and coordinated assembly supply chains. In practical applications the manufacturer and n suppliers may simultaneously or sequentially make optimal decisions to maximize their individual expected profits. Accordingly, we analyze both a simultaneous-move game (in which all members of the assembly supply chain make their decisions concurrently) and a leader–follower game (in which the manufacturer announces its pricing decision before n suppliers make their production decisions). We use the Nash and Stackelberg equilibria to characterize all supply chain members’ optimal decisions for the simultaneous-move and leader–follower games, respectively. Note that Wang [18] also considered the two games but only with the multiplicative demand form, and used different contracts to induce supply chain coordination.

Since the supply chain members make optimal decisions to maximize their individual profits, the system-wide expected profit is usually lower than the case when they coordinate their decisions. Thus, to improve the supply chain’s performance, we design buy-back and lost-sales cost-sharing contracts which achieve supply chain coordination. Under properly-designed contracts, all supply chain members choose their equilibrium solutions and the maximum system-wide profit is realized.

In a recent literature review [9], Leng and Parlar surveyed a large number of publications that focus on supply chain-related game problems with substitutable products and indicate that there are only a few papers concerned with game-theoretic models for complementary products. We now briefly review some of the important papers that used game-theoretic models in assembly supply chains. Wang [18] considered joint pricing-production decision problems in supply chains with complementary products for a single period. Wang adopted a multiplicative demand model which is sensitive to sale price, and incorporated the consignment-sales and revenue-sharing contracts into both simultaneous-move and leader–follower games. Since Wang [18] analyzed assembly supply chains with the multiplicative demand form, in this paper we focus our analysis on the additive demand case, and provide only the major results for the multiplicative demand case without specific discussion.

Wang and Gerchak [19] examined a pricing-capacity decision problem for an assembly supply chain with an assembler and multiple-suppliers. Assuming that the demand is random but it is independent of price, the authors considered two game settings: The first is one where the assembler sets the prices, and the second is for the suppliers to simultaneously select the prices each wants to charge for its component. Bernstein and DeCroix [1] considered an assembly supply chain where two components are used to assemble a single final product that is then sold by an assembler to meet the random, price-independent demand. The authors investigated the equilibrium base-stock levels for the assembler and two component suppliers, and described a payment scheme to coordinate the assembly supply chain. Granot and Yin [8] investigated competition and cooperation in a multiple-supplier, one-manufacturer supply chain with complementary products. For the pull and push systems, these authors considered two levels of problems: at the first level, they used the concepts of Nash equilibrium and farsighted stability to identify stable coalitional structures among suppliers; and at the second level they developed a Stackelberg game to examine the interactions between the manufacturer and suppliers.

The remainder of this paper is organized as follows: in Section 2, we assume that the random price-dependent demand is given in additive form, and design a set of buy-back and lost-sales cost-sharing contracts to coordinate the assembly supply chain for both simultaneous-move and leader–follower games. In Section 3, we assume that the price-dependent demand is also random but it assumes the multiplicative form. For this case, we show that the supply chain can always be coordinated by a set of properly-designed contracts for both simultaneous-move and leader–follower games. The paper ends with conclusions in Section 4.

Section snippets

Non-cooperative equilibria and supply chain coordination with price-dependent random demand: Additive form

In this section, the manufacturer assembles the suppliers’ components to satisfy random price-dependent demand in an additive form. We consider the following two games: (i) a “simultaneous-move” game where the manufacturer and the suppliers concurrently make their decisions without any communication; (ii) a “leader–follower” game in which the manufacturer, as the leader, first announces his pricing decision and the suppliers, as the followers, respond to the leader’s decision. For the former

Non-cooperative equilibria and supply chain coordination with price-dependent random demand: Multiplicative form

In this section, the manufacturer faces the random price-dependent demand in a multiplicative form. Note that the difference between our analyses in this section and Section 2 is the form of the demand function. Thus, in order not to be repetitive, we only present Nash and Stackelberg equilibria, globally-optimal solution, and contract design in what follows.

As Petruzzi and Dada [14] discussed, the multiplicative demand function in the newsvendor context is commonly formulated asD(p,ε)=y(p)ε,

Conclusions and recommendations for further research

In this paper we considered an assembly supply chain where multiple suppliers produce complementary components that are used by a manufacturer who assembles the final product and sells them directly to a market. The single period demand for the final product is random and depends on the retail price chosen by the manufacturer.

The suppliers and the manufacturer have an agreement whereby each supplier buys back the unsold components and absorbs a portion of the lost-sales cost. Accordingly, we

Acknowledgements

The authors wish to thank the editor and three anonymous referees for their useful comments that helped improve the paper.

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    Arguably, non-cooperative game theory has been used extensively to model competition and coordination among players of a supply chain. Given the increasing supply chain costs, many articles used concepts and methods from non-cooperative game theory to model and solve certain problems such as inventory management (Halat and Hafezalkotob, 2019; Arda and Hennet, 2008; Cachon and Zepkin, 1999) and supply chain planning/pricing (Zamarripa et al., 2012; Leng and Parlar, 2010). Nevertheless, the main concern regarding of the application of non-cooperative game theory to supply chain network is whether using Nash and Stackelberg equilibrium concepts would provide a solution that maximizes the total profit (Zamarripa et al., 2012).

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1

Research supported by the Research and Postgraduate Studies Committee of Lingnan University under Research Project No. DR07B8.

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Research supported by the Natural Sciences and Engineering Research Council of Canada.

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