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Article Excerpt
We consider an assemble-to-order system with a high volume of prospective customers arriving per unit time. Our objective is to maximize expected infinite-horizon discounted profit by choosing product prices, component production capacities, and a dynamic policy for sequencing customer orders for assembly. We prove that a myopic discrete-review sequencing policy, which allocates scarce components among orders for different products to minimize instantaneous physical and financial holding costs, is asymptotically optimal. Furthermore, we prove that optimal prices and production capacity nearly balance the supply and demand for components (i.e., it is economically optimal to operate the system in heavy traffic), so system performance is characterized by a diffusion approximation. The diffusion approximation exhibits state-space collapse: Its dimension equals the number of components (rather than the number of components plus the number of products). These results complement the existing assemble-to-order literature, which focuses on managing component inventory and assumes FIFO sequencing of orders for assembly.
Key words: assemble-to-order systems; functional limit theorems; diffusion limits; Brownian motion; state-space collapse; discrete-review policy; control policy
MSC2000 subject classification: Primary: 90B05, 60F17; secondary: 90B22, 90B35, 60J70
OR/MS subject classification: Primary: queues, diffusion models, limit theorems, inventory/production, policies, review/lead times; secondary: probability, stochastic model applications
History: Received January 16, 2003; revised December 15, 2003, January 15, 2005, and October 5, 2005.
1. Introduction. Consumers are heterogeneous, and demand a variety of products. Manufacturers have difficulty in forecasting consumer choice, and therefore incur high costs in carrying finished-goods inventory in the wrong amounts, for the wrong products. For modular products like computers, an efficient alternative is to assemble to order; i.e., to hold inventories of components that can be rapidly assembled into a wide variety of finished products in response to customer orders.
As the Internet and new information technologies have enabled manufacturers to sell directly to customers rather than through retail outlets, assemble-to-order manufacturing has become increasingly prevalent. Dell, a leader in direct sales and assemble-to-order (ATO) manufacturing, has grown by 40% per year in recent years, although the PC industry as a whole has grown by less than 20% per year (Economist [14]). In addition to Dell, companies as diverse as General Electric, American Standard (Bylinsky [10]), BMW (Economist [14]), Toyota (Economist [15], Forbes [17]), Timbuk2 (Plambeck [30]), and National Bicycle (Agrawal and Cohen [2]) either have adopted or are considering adopting an ATO approach.
In an assemble-to-order system, pricing, capacity management, and dynamic execution are very challenging. To maximize profit, the first-order challenge is to choose product prices (which determine revenue and the demand for each component) and to choose the production capacity (or supply contract) for each component. However, due to stochastic fluctuations in supply and demand, component shortages will sometimes occur. Then, the manufacturer must dynamically ration scarce components among customer orders for various products, and/or pay to expedite component production. In practice, most firms adopt simple static rules to sequence customer orders for assembly, such as FIFO or proportional allocation (Agrawal and Cohen [2]) and expedite production on an ad hoc basis (Perman [29]). Even Dell has only recently begun to sequence orders for assembly dynamically, using real-time information about component availability (Perman [29]). Dynamic control is complex because Dell offers thousands of different products (computer configurations) from hundreds of different components. Therefore, theory is needed to guide business practice.
This paper and its sequel undertake a holistic analysis of ATO manufacturing. Specifically, we show how to set product prices and component production capacity, and then dynamically sequence customer orders for assembly and manage component inventory, in order to maximize expected infinite- horizon discounted profit. In this paper, the component production capacity equals the component production rate. However, in our sequel paper (Plambeck and Ward [31]) we allow for expediting extra components and salvaging excess components, so that the effective production rate of components differs from the component production capacity.
Our analysis begins with the deterministic analog of the system, ignoring stochastic variability in supply and demand for components. We solve a static planning problem: Set product prices and component production capacities to maximize profit in the deterministic system. At the optimal solution to the static planning problem, the long-run average production rate for each component is equal to the long-run average rate of demand. Hence, with stochastic variability, customers will experience delays and components will be held in inventory. Discounted expected profit is reduced from the optimal objective value in the static planning problem because components are purchased before they are needed, and customers will not pay until their products are assembled. However, assuming a high demand rate, we prove that the optimal prices and component production capacities are close to the solution of the static planning problem. That is, heavy traffic is the optimal operating regime.
Then, we can employ a diffusion approximation to the dynamic control problem. First, we characterize a simple, near-optimal rule for allocating scarce components to outstanding customer orders (dynamic sequencing). Next, we slightly perturb the static planning problem solution to increase expected infinite-horizon discounted profit.
The key assumption in our analysis is that a high volume of potential customers arrives per unit of time. Under this assumption, because the optimal production capacity for each component approximately equals the optimal demand rate, the functional central limit theorem dictates that the inventory position of each component (the number in stock minus the number required to assemble outstanding orders) changes gradually. However, the system manager can quickly adjust the customer order queue lengths. Consider Dell computer as a concrete example. Its OptiPlex assembly plant assembles more than 20,000 computers per day (an order for a computer arrives every 4.3 seconds), but the queue for a specific configuration of computer will typically be less than 100. By prioritizing that specific configuration for assembly, Dell could eliminate the queue in minutes. However, queues for other products would increase.
We propose to review the system and release orders for assembly at short intervals of time. If one or more components are in shortage, products are prioritized for assembly according to price and component requirements, so that the resulting configuration of queues minimizes the instantaneous financial "holding cost" for delaying assembly (and hence payment). Because the inventory position changes slowly, queue lengths will almost continuously track the minimum cost configuration. This discrete-review approach to dynamic control follows that of Harrison [21] and Maglaras [24, 25], except that the decision at each review time point pairs available components with outstanding orders instead of allocating server time amongst customer classes.
The following summarizes the main contributions of this paper.
(i) We prove that for a high-volume assemble-to-order system, optimal prices and component production capacity are close to the solution of the static planning problem; see Theorem 5.1. In particular, as in the systems studied in Maglaras [26], Maglaras and Zeevi [28], and Plambeck [30], heavy traffic is the optimal operating regime, meaning that diffusion approximations are relevant.
(ii) With dynamic assembly sequencing, the system exhibits state-space collapse; see Proposition 4.1. Hence, in managing component production and inventory, one need not track the customer order queue length for every product, only the component inventory positions. This reduction in problem dimensionality is very important, because a typical assemble-to-order system is designed to support thousands of different products from tens of different components.
(iii) We provide an asymptotically optimal policy for statically setting prices and component production capacity, and dynamically sequencing orders for assembly; see Theorem 5.1.
The remainder of this paper is organized as follows. We first review some relevant literature. Next, in [section]2, we discuss the model formulation. Section 3 sets up the static planning problem and introduces our assembly policy. Section 4 analyzes the asymptotic behavior of the system under the "first-attempt" policy specified in [section]3. Finally, we show that nearly balanced systems are optimal, and provide the appropriate "stochastic adjustment" factor to prices and component production capacity in [section]5. Appendix A contains all of our lemma proofs, and Appendix B provides a table of notation.
1.1. Literature review. Song and Zipkin [36] provide an excellent survey of the literature on managing ATO systems. This literature is focused on the management of component inventory, taking the demand process and distribution of component lead times as given, and assuming that customer orders must be filled FIFO. The subset of this literature that adopts a discrete-time formulation must further specify how to allocate scarce components among orders that arrive simultaneously, i.e., in the same period. Each of the following three papers assumes a different rule. In Agrawal and Cohen [2], scarce components are allocated "fairly" so that in each period, the fraction of demand that is satisfied is the same for every product. In Zhang [41], orders that arrive in the same period are prioritized according to price. Akcay and Xu [1] optimize the allocation of components (among orders that arrive in the same period) to maximize the fraction of orders assembled within the quoted maximum delay. All three papers assume independent base-stock control of component inventory, and optimize the base-stock levels. They find very different optimal base-stock levels, which suggests that the optimal inventory policy is very sensitive to the rule for allocating scarce components among orders for various products. Akqay and Xu [1] recommend that "inventory replenishment and component allocation optimizations must be made jointly" (p. 110). Clearly, these decisions are also sensitive to product prices and component lead times. This motivates our model formulation, which incorporates product prices, component production capacity, and the dynamic sequencing of orders for assembly as decision variables. We find that the optimal allocation of components to products (sequencing) collapses the state space, so the joint optimization recommended by Akqay and Xu [1] becomes more tractable than inventory optimization in isolation.
In an ATO system with only one component in shortage, the assembly-sequencing decision is similar to the decision of which class to serve in a multiclass, single-server queue. Much of the research in this area combines sequencing with dynamic lead-time quotation. Two representative publications are Wein [39] and Duenyas [13], and each of these papers provides an extensive review of related literature. When customers must be served within a class-specific maximum delay but production cannot be expedited (our follow-up paper Plambeck and Ward [31] has delay constraints and expediting), then some customers must be tumed away. Maglaras and Van Mieghem [27] and Plambeck et al. [32] consider dynamic sequencing and admission control in this setting.
In practice, many firms accept all customer orders, but set a lower bound on the fill rate (fraction of customer orders assembled within the quoted maximum delay). Researchers have provided methods for optimizing the base-stock level for each component in order to minimize long-run average inventory holding costs, subject to the lower bound on the fill rate. For example, when component lead times...
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