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Article Excerpt
Fast service is clearly important. Less obvious is how to go about obtaining fast service from suppliers or service providers. One technique is to make servers compete by allocating business to them based on their performance, i.e., the faster server is rewarded with a greater share of demand. For example, Sun Microsystems maintains multiple memory chip suppliers and allocates demand with a scorecard system. A score that depends on a number of factors, delivery lead time among them is periodically assigned to each supplier, and a supplier's allocation of Sun's business increases as they improve their score relative to the other suppliers (Farlow et al. 1996). GE Lighting and Air Products and Chemicals also allocate demand towards better-performing suppliers (Pyke and Johnson 2003).
This paper studies, in the context of a stylized queueing model, the issue of how performance-based demand allocation can induce competition among suppliers to obtain faster service or delivery lead times. A precursor to this line of research is the extensive body of work on queue-joining behavior, pioneered by Naor (1969). That literature focuses on the behavior of strategic customers/jobs: e.g., whether or not to join a queue (e.g., Naor 1969), or which of several queues to join (Bell and Stidham 1983). It is generally found that the behavior of individual jobs creates externalities on other jobs (e.g., overcongestion of the faster server). (See Hassin and Haviv 2003 for a review of the queue-joining literature with strategic customers/jobs.) Those externalities do not occur in our setting because a single buyer controls all of the jobs. Instead, we have strategic servers--servers that can regulate how fast they work, and working faster is costly.
In our model the buyer pays a fixed amount for each job, so the buyer's task is to choose an allocation policy to minimize the average lead time to complete jobs. We study allocation policies that can be classified into two groups, state-dependent policies (the allocation of a job to a server depends on the servers' current workload) and state-independent policies (the allocation of a job does not depend on the number of jobs currently in the servers' queues).
With nonstrategic servers it is clear that a state-dependent policy can deliver faster lead times than a state-independent policy because, in part, a state-independent policy risks allocating jobs to busy servers while other servers remain idle, i.e., a state-dependent policy can do a better job of pooling the servers' capacities. (1) However, are state-dependent policies better when the servers are strategic? Suppose a state-independent policy induces servers to work more quickly than a state-dependent policy. Then the buyer may be better off with a state-independent policy even though the system's capacity is not as effectively utilized. In other words, incentives may trump efficiency. In fact, Gilbert and Weng (1998) arrive at that conclusion. Nevertheless, there are several reasons why this might not be the best conclusion. First, we show that there is an error in their equilibrium existence proof, so it is not always meaningful to compare their two allocation policies. Second, and more importantly, they do not compare optimal policies. We compare the buyer's best state-dependent policy with the buyer's best state-independent policy and find that the buyer is better off with the state-dependent policy, i.e., the buyer can have both incentives and efficiency. In general, we find that there is considerable variation in the performance of intuitively reasonable policies. For example, the buyer's optimal state-independent policy with nonstrategic servers is found to perform poorly in the presence of strategic servers, and proportional allocation, which is probably the most intuitive allocation policy, can be the worst performer of the policies we consider.
The next section describes our model in detail. Section 2 expands upon the related literature. Section 3 studies the buyer's allocation policy choice and the competition between servers under several different allocation policies. Section 4 discusses several extensions to the model. The final section concludes with a summary of our results.
1. The Model
A buyer procures a good (e.g., a make-to-order component, as in the Sun Microsystems example) or a service. For ease of exposition, we assume a service is procured. There are two servers. (Most of our results extend to more than two servers; see Zhang 2004 for details.) Demand for the service arrives according to a Poisson process with rate [lambda]. Each demand is referred to as a job and all jobs are eventually completed. Server i's average service rate is [[micro].sub.i] and service times are exponentially distributed. We refer to [[micro].sub.i] as server i's capacity and [micro] = ([[micro].sub.1], [[micro].sub.2]) denotes the capacity vector. A server with capacity [[micro].sub.i] incurs a capacity cost at rate c([[micro].sub.i]), no matter whether the capacity is utilized or idle, where c(0) = 0, c'(.) > 0, and c"(.) [greater than or equal to] are assumed. The servers' variable cost per job is normalized to zero.
We say that a job is allocated to a server when it is certain that server will complete the job. The buyer pays R per allocated job. We assume R > [r.sub.1], where
[r.sub.1] = c([lambda]/2)/([lambda]/2),
because it is the minimal requirement for the suppliers to earn a nonnegative profit and deliver finite lead times (see Zhang 2004). We assume R is exogenous: There could be an industry standard price that the buyer is unable to negotiate away from, or the price could be set via negotiations that involve issues beyond the scope of this model.
The buyer controls her allocation policy (i.e., how jobs are allocated to servers) and the servers choose their capacities. The buyer seeks to minimize the average delivery lead time over an infinite-horizon subject to the constraint that each server earns a nonnegative profit, and the servers seek to maximize their average profit:
[[pi].sub.i]([micro]) = R[[lambda].sub.i]([micro]) - c([micro].sub.i), (1)
where [[lambda].sub.i]([micro]) is the rate at which server i is allocated jobs. (2) Hence, we assume that the buyer and the servers do not discount future cash flows and that they expect a long-term relationship. We focus on equilibria in which the servers adopt open-loop strategies, i.e., strategies that are independent of the history of play. As a result, this infinite-horizon capacity game among servers can be analyzed as a single-decision capacity game. Previous research on strategic servers also restricts attention to open-loop strategies. In [section]3.3 we discuss lead time-based allocation rather than capacity-based allocation.
2. Literature Review
Kalai et al. (1992) were the first to study strategic servers, but they only consider a simple state-dependent policy in which jobs are allocated to idle servers with equal probability. Gilbert and Weng (1998) expand upon their model to include a state-independent allocation policy that allocates jobs to servers immediately upon arrival. They conclude that a state-independent policy can be better for the buyer than a state-dependent policy. Our results are different, as we explain in detail in the subsequent sections. Christ and Avi-Itzhak (2002) extend those models to include customer balking, but we do not have balking.
Ha et al. (2003) study the competition between two suppliers serving one buyer, in which delivery frequency is an element of the buyer's allocation decision. However, they study deterministic demand, so although they consider issues similar to ours, a direct comparison between their work and ours is not meaningful.
There are papers that compare sole sourcing versus dual sourcing, whereas we assume that a dual-sourcing strategy has been adopted: e.g., Anton and Yao (1989, 1992), Anupindi and Akella (1993), Benjaafar et al. (2007), Seshadri (1995), and Seshadri et al. (1991). See Minner (2003) and Elmaghraby (2000) for reviews of the literature on sourcing strategies.
There are papers that study a buyer's procurement policy when there are multiple suppliers with exogenously determined characteristics: e.g., Bonser and Wu (2001), Chen et al. (2001), Li and Kouvelis (1999), Martinez de Albeniz and Simchi-Levi (2003), Sedarage et al. (1999), and Talluri (2002). In our model the servers' lead times depend on their choices and the buyer's allocation policy.
Several papers study coordination and competition in supply chains with multiple suppliers: Bernstein and DeCroix (2004); Wang and Gerchak (2003); and Nagarajan and Bassok (2003). In these papers, limited capacity leads to demand truncation rather than slower delivery times. Bernstein and de Vericourt (2005) consider a market with multiple suppliers and multiple buyers. Their suppliers have fixed processing rates and compete by offering different lead times to buyers, which they obtain via holding inventory.
There are a number of papers that study server competition in which firms choose operational strategies to adjust their delivery times: e.g., Allon and Federgruen (2003), Cachon and Harker (2002), Chayet and Hopp (2002), Lederer and Li (1997), and So (2000). In those papers the structure of how firms compete is exogenous, whereas in our model it is determined by the buyer via her allocation policy.
There is literature on capacity allocation (e.g., Cachon and Lariviere 1999a, b, c; Deshpande and Schwarz 2002), in which a single manufacturer allocates scarce capacity among multiple buyers. Although allocation policies similar to ours are implemented, those models are analytically quite different.
Li (1992) and Armony and Plambeck (2005) study models in which a buyer submits duplicate orders to multiple suppliers. In our model, each job is allocated to a single server, but we briefly discuss order duplication in [section]4.
3. Allocation and the Servers' Capacity Game
Our model can be analyzed in two interdependent parts. The first part is the buyer's allocation policy choice--i.e., how will the buyer allocate jobs among the two...
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