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Article Excerpt
This paper examines a two-tier assemble-to-order system. Customer orders for various products must be filled within the product-specific target lead time, or become lost sales. A product can be assembled instantaneously if it required components are in stock at the assembly facility. The production facility for each component is geographically distant from the assemble cost per unit. The system manager must initially commit to the production capacity for each component. Then, in response to customer orders, he must dynamically manage production (expediting and salvaging) and shipping for each component, and the sequence of customer orders for assembly (how scarce components are allocated to outstanding orders). The objective is to minimize expected discounted costs for lost sales, production, and shipping. This discounted formulation accounts for financial inventory holding costs but not physical inventory holding costs. The main result is that as the order arrival rate for each product becomes large and the discount rate becomes small, a simple threshold policy with independent control of each component is asymptotically optimal. The policy is parameterized by five numbers for each component. Expressions for these parameters, the expected discounted cost, and the long-run average rates of salvaging and expediting are obtained by solving an approximating Brownian control problem. In a numerical example from the computer industry, the Brownian approximation is remarkably accurate.
Subject classifications: inventory/production: assemble-to-order: stochastic optimal control.
Area of review: Stochastic Models.
History: Received July 2006; revisions received December 2006. May 2007. July 2007; accepted August 2007. Published online in Articles in Advance September 17, 2008.
1. Introduction
An assemble-to-order manufacturer such as Dell offers a wide variety of products without tying up capital in finished-goods inventory. However, to fill customer orders in a timely manner requires inventory of components that can be rapidly assembled into final products. In the event of a component shortage, the manufacturer must dynamically allocate the scarce component among customer orders for various products, and/or pay to expedite component production. Dynamic control of an assemble-to-order system is challenging because the state space (outstanding orders and their due dates, and the inventory and production status for each component) is large.
This paper proves that a simple policy with independent threshold control of each component is near optimal for an assemble-to-order (ATO) system with the following characteristics:
* Customer orders must be filled within a product-specific target lead time or become "lost sales."
* Component production facilities are geographically distant from the assembly facility, and each shipment of components incurs a fixed cost in addition to the variable cost per unit.
* Expediting component production (above nominal capacity) incurs a high per-unit cost; excess components can be salvaged for a low per-unit cost.
* The objective is to minimize expected discounted costs of production, shipping, and lost sales.
* The discount rate is small and the order arrival rate is large.
The proposed policy is characterized by just five numbers for each component: [mu], I, [bar.I], r, and Q. The capacity investment (nominal rate of component production) is [mu]. Production is expedited when the inventory position (for the entire system) falls below [I.sub.m] and salvaging occurs when the inventory position reaches [bar.[I.sub.m]]. A shipment occurs when the inventory position for the assembly facility falls below the reorder point [r.sub.m]. The shipment brings the inventory position for the assembly facility up to [r.sub.m] + [Q*.sub.m] or as close as possible given limited inventory at the component production facility. (Federgruen and Zipkin 1986 conjectured that this shipping policy is near optimal for a single-item capacitated system with a fixed cost per shipment.)
The remainder of this paper is organized as follows. Section 2 reviews related literature. Section 3 formulates the model. Section 4 employs a Brownian approximation to derive expressions for [mu] I [bar.I], r, and Q and system performance, and to prove that the proposed policy is asymptotically optimal as the order arrival rate grows large and the discount rate becomes small. For a personal computer (PC) assemble-to-order system, [section]5 implements the proposed policy and shows that the Brownian approximation is accurate. Concluding remarks are in [section]6. All proofs are in the online appendix that can be found at http://or.journal.informs.org/.
2. ATO Literature Review
Many papers have addressed the question: How much inventory of each component should be held at the assembly facility? For certain assembly systems with a single product, the optimal policy has a simple structure. For example, when each component has a deterministic lead time, independent base-stock control of each component is optimal (Rosling 1986, Chen and Zheng 1994); and with capacitated production of each component (by an exponential single server), the optimal base-stock level for each component increases with the inventory of other components (Benjaafar and ElHafsi 2006). However, for assemble-to-order systems with more than one product, the structure of an optimal policy is not known. Researchers have assumed FIFO order fulfillment and independent base-stock control for the inventory of each component, and then proceeded to characterize system performance and optimize the base-stock levels. For example, with capacitated component production, Song et al. (1999) derive an expression for the fill rate (fraction of customer orders filled within the target delivery lead time); Glasserman and Wang (1998) and Wang (1999) constrain the fill rate to be close to one, establish a linear relationship between the target delivery lead time and component base-stock levels, and provide expressions for near-optimal base-stock levels. With i.i.d. component transportation lead times, Song (1998) and Lu et al. (2003) show how to calculate the fill rate, and Cheng et al. (2002) propose an algorithm to optimize base-stock levels, subject to a lower bound on the fill rate. Song and Zipkin (2003) provide an excellent survey of the assemble-to-order literature.
A few recent papers investigate alternatives to independent base-stock control of component inventory and FIFO order fulfillment. Song (2000) assumes that component m is shipped in a fixed batch size [Q.sub.m] (e.g., a container or truck-load). Specifically, whenever the inventory position for component m falls below the level [r.sub.m], one orders i[Q.sub.m] units, where i is the smallest integer that brings the inventory position above [r.sub.m]. Assuming Poisson demand and deterministic component lead times, Song (2000) provides expressions for the fill rate and the average number of backorders. Zhao and Simchi-Levi (2006) incorporate stochastic sequential component lead times into Song's model, and propose an efficient method for estimating the fill rate and average waiting time. Based on a multidimensional Brownian approximation, Kushner (1999) and Plambeck and Ward (2005, 2006) propose policies for integrated control of component inventories and non-FIFO order fulfillment. Plambeck and Ward (2007) also allow for non-FIFO order fulfillment, but characterize conditions under which independent control of each component is optimal.
The two-tier model in this paper is motivated by the observation that some prominent assemble-to-order manufacturers such as Dell rely on component suppliers to hold inventory and maintain relatively little inventory in their own assembly facilities, and the component production often is geographically distant from assembly (Perman 2001). Whereas other assemble-to-order researchers have focused on the assembly facility, this paper is the first to account for inventory at component production facilities, in transit, and at the assembly facility.
Derivation of the asymptotically optimal policy follows the general approach proposed by Harrison (1988): solve an approximating Brownian control problem, translate the solution into a policy for the original system, and verify that the policy is asymptotically optimal. This approach has generated important insights for a wide variety of non-ATO stochastic systems (Harrison and Wein 1990, Kumar 2000, Bell and Williams 2001, Maglaras and Zeevi 2003, Ata and Kumar 2005, Mandelbaum and Stolyar 2004, Armony and Maglaras 2004). In those papers, the control problem admits a solution that is optimal on every sample path. In contrast, one must solve a Hamilton-Jacobi-Bellman equation to characterize the optimal solution to the Brownian problem that approximates the assemble-to-order system. One other paper (Ward and Kumar 2008) provides a proof of asymptotic optimality where the solution to the approximating control problem is not pathwise optimal.
3. Model Formulation
Consider the assemble-to-order system with J products and M components in Figure 1. Orders for product j arrive according to a renewal process with rate [[lambda].sub.j]: specifically, [D.sub.j] (t) denotes the cumulative number of orders for product j up to time t [greater than or equal to] 0, where
[FIGURE 1 OMITTED]
[D.sub.j](t)[equivalent to]max{I[greater than or equal to]0:[I.summation over (i = 1)][u.sub.j](i)[less than or equal to][[lambda].sub.j]t} (1)
and {[u.sub.j] (i), i = 1, 2, ...} are i.i.d random variables with mean one and variance [[phi].sub.j.sup.2], and
E[|[u.sub.j](1)|.sup.2 + 2[[xi]] < [infinity] (2)
for some [xi] > 0, for j = 1, ..., J. For t <...
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