Coordination in decentralized assembly systems with uncertain component yields

Abstract

The literature on assembly systems with random component yields has focused on centralized systems, where a single decision maker chooses all components’ production quantities and incurs all the costs. We consider a decentralized setting where the component suppliers choose their production quantities based solely on their own cost/reward structure, and the assembly firm makes ordering decisions based on its own cost/reward structure. When the suppliers control their inputs but the outputs exhibit random yields, coordination in such systems becomes quite complex. In such situations, incentive alignment control mechanisms are needed so that the suppliers will choose production quantities as in the centralized system case. One such mechanism is to penalize the supplier with the worse delivery performance. We analyze the conditions under which system coordination is achieved while respecting participation constraints. Further, we determine the optimal component ordering policy for the assembly firm and derive the optimal coordinating penalties.

Introduction

In many systems, overall performance is determined by the agent who produces or delivers the least amount. “Minimum effort” or “weakest link” scenarios in team performance are explored by Van Huyck et al., 1990, Knez and Camerer, 1994, Cachon and Camerer, 1996. In the domain of production management, the prime examples are assembly systems—the number of complete kits available depends on the component in shortest supply (e.g., Gerchak and Wang, 2004). When agents control the inputs or efforts of their own activities, but the outputs exhibit random yields, coordination in such assembly systems becomes quite complex.

To date, the literature on assembly systems with random component yields (Singh et al., 1990, Gerchak et al., 1994, Gurnani et al., 1996, Gurnani et al., 2000) has focused on centralized systems, where a single decision maker chooses all components’ production quantities and incurs all the costs. For example, Gerchak et al. (1994) considered such systems with a product whose demand is known or uncertain, perfect or imperfect assembly stage, and stochastically proportional yields. Due to the need for simultaneously selecting production quantities of inter-dependent components, that problem is quite difficult to solve except in some special circumstances. Conceptually, however, the setting is a natural extension of well-studied single-item random yield models (e.g., Anupindi and Akella, 1993, Bassok and Akella, 1991, Bollapragada and Morton, 1999, Henig and Gerchak, 1990, Yano and Lee, 1995).

There is an extensive literature concerning the choice of quality by suppliers, contracting over it, and inspection policy for the buyer (Reyniers and Tapiero, 1995, Baiman et al., 2000). These papers do not deal with assembly systems or with lot-sizing. Baiman et al., 2004, Baiman and Netessine, 2004 do deal with assembly systems, but consider product quality only and not lot-sizing. So while the literature focuses on quality choice and monitoring, we focus on lot-sizing for given yield distributions. We assume a single supplier for each component, and thus do not explore multiple-sourcing (as in Cachon and Zhang, 2004).

The basic scenario we envision here is as follows. An assembly firm receives an order for d units of some product that is assembled using two components in amounts scaled to be equal (later, in Section 5, we discuss the extension of the model to the case of n components with identical cost and yield structures). It has to then decide how many units to order from its components suppliers. There are many examples of products assembled using critical components—NASA estimated that in the Challenger, there were approximately 4686 critical parts, each one, potentially crucial to the success of the mission. Other examples include using critical components to assemble cars, VCRs, etc.

The production process of the component suppliers is assumed to be imperfect (as is the case, for example, in semiconductor and electronics industries), and hence, their yields are random. In the first part of the paper, we consider a centralized system and determine optimal production quantities for the components. The centralized system solution serves as a benchmark for the decentralized setting.

For the decentralized problem, we first consider a contract form (referred to as the Additional Penalty contract) whereby the supplier with the worse delivery performance pays an extra penalty (in addition to the regular shortage penalty) to the assembly firm. The component suppliers choose their production quantities based solely on their own cost/reward structure in order to maximize profits subject to a participation constraint. The assembly firm would place orders from the suppliers, which may differ from its customer’s order size, based on its own cost/reward structure. In this contract, we assume that since both components are required in equal quantity, the assembly firm orders equal number of units from both the suppliers. However, the firm may choose penalty costs that are different for each supplier. The situation here can be viewed as two strategic “nested games” with uncertainty. The first is a Stackelberg game where the leader (the assembly firm) selects an order size, and the followers (the component suppliers facing random production yields) respond by selecting their production quantities. But the multiple “followers” (component suppliers) make their choices simultaneously, and, since their costs are interrelated, they may be viewed as actually playing a Nash-game, which is “nested” within the Stackelberg game. Comparative statics for the equilibrium are provided.

We also consider a second contract form where the firm may order unequal order quantities from the two suppliers. In this contract, we consider the case when there are regular shortage penalty costs only, that is, the assembly firm does not charge an additional penalty to the worse supplier.¹

Another issue we explore is the ordering policy of the assembly firm. Since the suppliers make allowances for their own yield randomness (and, in case of the extra penalty contract, even for the other supplier’s yield randomness), need the firm do so as well? Using the optimal reaction function of the suppliers, we determine the optimal component ordering decisions for the assembly firm.

In order to achieve channel coordination, we show that by using the Additional Penalty contract, the firm can suitably place (equal) orders from the suppliers and set the regular and additional shortage penalty prices such that participation constraint for both the suppliers will be binding and thus the firm could retain all of the above-reservation channel profits. The Additional Penalty contract also offers the advantage in that the assembly firm, by suitably choosing the penalty prices, can order exactly equal to its demand and let the suppliers make their allowances for yield uncertainty. In the second contract with unequal order quantities, again we show that the assembly firm is able to achieve coordination and retain all above-reservation profits by suitable selection of regular shortage penalties. In this case, since the firm has the flexibility to order different quantities from the suppliers, it turns out that additional penalty costs are not necessary as regular shortage penalties are sufficient to achieve coordination. Interestingly, we also observe that if the suppliers are symmetric (that is, the components are identical), again we can just use regular shortage penalties in order to achieve coordination, that is, additional penalties are not needed when the suppliers are symmetric.

The recently emerging literature within operations management stressing the importance of coordination and control mechanisms in decentralized supply chains (e.g., Cachon, 2003, Gavirneni, 2001) did not consider assembly systems (but see Gerchak and Wang, 2004), and, by and large, also did not consider the implications of random yields. When yields are random, a direct link between an agent’s (component supplier’s) actions and their results, and thus impacts on the chain, does not exist, as they are separated by a layer of uncertainty.

In the OM literature on decentralized systems, “coordination” has to do with economic aspects—appropriate incentive systems are sought so that agents will behave as if system is centralized (as thus economically efficient), and/or in a manner most profitable to the leader. In an assembly setting, on the other hand, coordination is a “physical” concept (equal component quantities delivered in unison) as well as an economic one (these equal quantities are to be the “right” ones). Thus we mean more by “coordination” than the non-assembly literature.

The rest of the paper is organized as follows. In the next section, we discuss the model assumptions and formulate the expected profit function for the suppliers for the additional penalty contract, as well as the resulting participation constraints. In Section 3, we consider the firm’s ordering problem for given penalty parameters and determine the optimal ordering policy. In Section 4, we analyze the conditions required for channel coordination where the firm can use the penalty parameters as levers, and show that by using the additional penalty contract, the firm is able to retain all of the above-reservation channel profits. We also model the second contract with unequal order quantities and show that regular shortage penalties are sufficient to achieve channel coordination. In Section 5, we discuss the extension of the model to the case of n-symmetric suppliers. Finally in Section 6, we summarize the results and discuss extensions.