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Article Excerpt
We study a multi-item stochastic inventory system in which customers may order different but possibly overlapping subsets of items, such as a multiproduct assemble-to-order system. The goal is to determine the right base-stock level for each item and to identify the key driving factors. We formulate a cost-minimization model with order-based backorder costs and compare it with the standard single-item, newsvendor-type model with item-based backorder cost. We show that the solution of the former can be bounded by that of the latter with appropriately imputed parameters. Starting with this upper bound, the optimal base-stock levels of the order-based problem can be obtained in a greedy fashion. We also show that the optimal base-stock levels increase in replenishment lead times but may increase or decrease in lead-time variability and demand correlation. Finally, we devise closed-form approximations of the optimal base-stock levels to see more clearly their dependence on the system parameters.
Subject classifications: inventory/production: stochastic, multi-item, assemble-to-order, component commonality; probability; multivariate Poisson process, correlation; optimization; submodular function minimization.
Area of review: Manufacturing, Service, and Supply Chain Operations.
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1. Introduction
Managing multi-item inventory systems is hard, especially when customers order different but possibly overlapping subsets of items and customer satisfaction is based on the fulfillment of the entire order. Manufacturers and distributors of multiple products face this challenge, and so do mail-order and online retailers. A typical example is component inventory planning for assemble-to-order (ATO) production systems, in which final products are assembled only after customer orders are realized. This is commonly known as a problem with component commonality. Here, a stockout of one component would delay the delivery of, possibly, several different products. Thus, the determination of the stock level of one item should take into account the stock level of other items. However, any model that aims to find the joint optimal inventory levels would involve evaluation of multidimensional probabilities and optimization of nonseparable functions, which is computationally demanding, if not intractable. This partly explains why in practice most systems simply apply single-item inventory planning tools to each item, ignoring the connections of the items. We call this approach the item-based approach. A natural question to ask is, to what extent is the item-based approach good enough? If the item-based approach is not satisfactory, how much can one learn from the solutions of this approach to devise an effective policy for the multi-item system with correlated demand?
The objective of this paper is to shed light on these issues and to develop efficient solution techniques for the multi-item system. We assume that each demand requires at most one unit of each item. This unit-demand model is commonly seen in practice. Examples include e-tailing and mail-order businesses of apparel, books, CDs, and toys. It is also typical in ATO systems for electronics, such as personal computers. We shall discuss possible extensions to a more general demand model in the concluding remarks.
Because the form of the optimal policy for the general system is unknown, we assume that independent base-stock policies are used to control the item inventories. That is, there is a target base-stock level for each item, and the replenishment decision for this item is solely determined by its inventory position relative to the target level. Of course, because a customer demand may require several items, the inventory positions of different items may drop simultaneously upon a customer arrival, which, in turn, triggers simultaneous replenishment orders of these items. For each fixed item, this kind of policy is well known to be optimal if there are no economies of scale in replenishment. Such a policy is also shown to be optimal for multiple-product systems with zero lead times. See Song and Zipkin (2003) for a review. Due to its simplicity, this type of policy is widely adopted in industry.
We formulate a cost-minimization model to determine the joint optimal base-stock levels. The inventory costs are measured at the item level, while the backorder costs are measured at the customer-order level. We compare this order-based model with a single-item cost-minimization model with appropriately chosen item backorder costs (the item-based formulation).
The key findings of our study include: (1) The items in the multi-item system are complementary in the sense that increasing the stock level of one item can only increase the desirability of stocking more of the other items. (2) The optimal base-stock levels increase when lead times are stochastically longer, but may increase or decrease in lead-time variability and demand correlation. These properties are different from those observed from the traditional item-based models. (3) However, using certain imputed item backorder cost rates derived from the original order-based demand and cost parameters, which essentially transfers the demand correlation into the dependence of the item back-order cost rates, the item-based formulation can lead to useful solutions. For instance, (4) with one particular set of induced item backorder cost rates, the item-based formulation yields an upper bound on the optimal base-stock levels. This upper bound provides a very good initial point in searching for the optimal base-stock levels in a greedy fashion. (5) With two other ways of setting imputed item backorder cost rates, the item-based formulation generates lower and upper bounds on the cost function. This, in turn, leads to closed-form approximations for the optimal base-stock levels.
Research on optimal planning of multi-item inventory systems with component commonality started as early as the 1970s. Early works mostly focus on single-period models or multiperiod models with zero lead times; see, for example, Baker et al. (1986), Gerchak and Henig (1989), and the references therein. In the past few years, motivated by the needs of managing ATO systems, there has been growing interest in this area. As a result, substantial progress has been made in developing tools to analyze models with dynamic demand and positive lead times. We refer the reader to Agrawal and Cohen (2001) and Song and Zipkin (2003) for reviews of this literature.
To our knowledge, most of these recent works focus on developing approximate solution techniques to solve constrained optimization models, and rely on numerical examples to gain managerial insights. For example, Agrawal and Cohen (2001), Cheng et al. (2002), Wang (1999), and Zhang (1997) examine the problem of minimizing average inventory holding cost (or total inventory investment in base-stocks) subject to service-level (e.g., fill rate) constraints. Hausman et al. (1998) study a model that maximizes a lower bound on the aggregate order fill rate with an inventory budget constraint.
This paper differs from the previous studies in several aspects. First, we put more emphasis on developing analytical insights. For this purpose, we use an unconstrained cost-optimization formulation by introducing order-based backorder costs. This approach makes it easier to make connections to the standard single-item inventory theory and therefore facilitates our understanding of the multi-item inventory planning problem. It also allows us to see the effect of relative importance of different products (through the changes in backorder cost rates). Second, our formulation lends itself to the application of the state-of-the-art optimization techniques of discretely convex functions, which leads to an exact greedy-type algorithm that guarantees optimality. Third, the optimality conditions developed in this paper enable us to establish certain ordering properties of the optimal solution, which would be more difficult to achieve in a constrained optimization problem. Finally, employing a decoupling approach to reduce the original problem on dependent variables to problems related to independent variables, we are able to obtain closed-form approximations of the optimal base-stock levels, which help sharpen our intuition on the key determinants of the optimal policy.
It is worth mentioning that several authors have studied unconstrained cost-minimization models in single-product assembly systems. Assuming deterministic component lead times, Schmidt and Nahmias (1985) and Rosling (1989) characterize the form of optimal policy for multistage assembly systems. Gallien and Wein (2001) consider a two-stage assembly system with i.i.d. lead times for each component. They focus on a type of coordinated base-stock policy that is inspired by Rosling's theory on systems with deterministic lead times. Assuming order synchronization (that is, the replenishments of all components triggered by the same customer demand are later assembled into the same final product), and assuming that the lead times follow Gumbel distributions, they develop a closed-form solution that is approximately optimal. The model considered in this paper allows i.i.d. lead times and multiple products in an ATO setting. Because there is no theory equivalent to Rosling's in the multiproduct systems, we focus on the widely used independent base-stock policies. However, we do not assume order synchronization.
In the context of repairable spare-parts inventory planning, Miller (1971a) studies a one-period problem of minimizing the sum of total investment in spare-part base-stock levels and the expected backorder cost. Assuming complete cannibalization, he obtains an expression of the expected total number of backordered aircrafts (due to the shortage of spare parts), which constitutes the backorder cost component in the objective function. Miller develops a general algorithm to minimize nonseparable functions defined on the integers and applies the algorithm to this problem. Due to different application domain, our objective function is different from Miller's. We consider an infinite-horizon problem with dynamic demand and replenishment lead times. We do not assume cannibalization, and we calculate the expected number of backorders for each different product. Our exact algorithm is built on more advanced development in discrete convex analysis, which to some extent was inspired by Miller's work. In the same application context as in Miller (1971a), Miller (1971b) compares two continuous versions of the problem (i.e., treating the base-stock levels as continuous variables)--one minimizes the sum of expected item backorders subject to a budget constraint; the other minimizes the expected order-based backorders (assuming complete cannibalization) with the same constraint. He obtains conditions under which the second problem yields higher base-stock levels. Our comparison of item-based and order-based solutions is similar in spirit, although the detailed formulation and analysis are completely different.
The rest of this paper is organized as follows. Section 2 provides the model details and some preliminaries on performance evaluation. Section 3 formulates the item-based and order-based cost-minimization models. Sections 4 and 5 present the main results of the paper; [section]4 focuses on the exact analysis, while [section]5 discusses closed-form approximations. These results are illustrated using a two-component, three-product system in [section]6. Numerical results for larger systems are reported in [section]7. Finally, [section]8 concludes the paper. All proofs are in the appendix.
2. Model Description
Consider an inventory system of m different items indexed by I = {1, 2,..., m}. Customer orders arrive at the system following a Poisson process with rate [lambda]. Each order may require several items simultaneously. For any subset of items K [??] I, we say an order (or a demand) is of type K if it consists of one unit of each item in K and zero units in I
K. We assume that there is a fixed probability [q.sup.K] that an order is of type K, [[summation].sub.K][q.sup.K] = 1. Each order's type is independent of the other orders' types and of all other events. Thus, the type-K order stream forms a Poisson process with rate [[lambda].sup.K] = [q.sup.K][lambda].
For each item i, let [K.sub.i] denote the set of all order types that requires item i. Then, the demand process for item i forms a Poisson process with rate [[lambda].sub.i] = [[summation].sub.K[member of][K.sub.i]][[lambda].sup.K].
Throughout this paper, we use subscripts to indicate the item types and superscripts for the demand types. Also, when K is singleton, say K = {i}, we simply call the type-K demand the type-i demand and denote [[lambda].sup.i] = [[lambda].sup.{i}].
Demands are filled on a first-come-first-served (FCFS) basis. If there is enough on-hand inventory for all the items required by a demand upon its arrival, the demand is filled immediately. In the ATO setting, this is equivalent to assuming that the final assembly time is negligible, which is typically true in practice due to the relatively long component procurement times. We also assume complete backlogging for demands that cannot be filled immediately. When a demand arrives and some of its required items are in stock but others are not, we put aside the in-stock items as committed inventory. A demand is considered backlogged until it is satisfied completely. When there are backorders, they also are filled on a FCFS basis. (Although FCFS is easy to implement and hence widely used across industries, it may not be an optimal allocation rule. Alternative approaches are considered, for example, by Agrawal and Cohen 2001, Akcay and Xu 2002, and Zhang 1997.)
The inventory of each item is controlled by an independent base-stock policy with
[s.sub.i] := the base-stock level for item i.
That is, upon each demand arrival, if the inventory position (=inventory on hand + inventory on order - backorders) of item i is less than [s.sub.i], then order up to [s.sub.i]; otherwise, do not order. Note that the inventory on hand does not include the committed inventory. Assuming that the initial on-hand inventory of item i equals [s.sub.i], then the arrival of each demand requiring item i triggers a replenishment order of one unit for this item.
The replenishment lead times for item i are i.i.d. random variables with a common cumulative distribution [G.sub.i]. Let [L.sub.i] denote the generic random variable with distribution [G.sub.i] and mean E[[L.sub.i]] = [l.sub.i]. Denote [G.sub.i.sup.c] = 1 - [G.sub.i]. Assume that the lead times are independent across items; that is, [L.sub.i] is independent of [L.sub.j] for any j [not equal to] i. Also, [G.sub.i] can be different from [G.sub.j] for i [not equal to] j. Note that this model includes the deterministic lead times as a special case.
Let [X.sub.i](t) be the number of outstanding orders (i.e., orders placed but have not arrived) of item i at time t. Then, as a result of following a base-stock policy, the outstanding...
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