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Article Excerpt
1. Introduction
The practice of rationing inventory (or capacity) among different customer classes is an increasingly important tool for balancing supply with demand in environments where requirements for service vary widely. The practice of issuing stock to some customers while refusing or delaying demand fulfillment for others is analogous to the highly successful yield management policies adopted by airlines and hotels. In this paper, we analyze a stock-rationing scheme that is useful for managing inventory in a continuous review (Q, r) environment with two customer demand classes defined by unique arrival rates and service costs. The scheme is characterized by a threshold inventory level, K, which signals when to reserve stock for higher-priority customers. The associated (Q, r, K) inventory policy serves all customers on a first-come-first-serve (FCFS) basis while on-hand inventory is above K, and cuts off service to low-priority customers when on-hand inventory falls below this threshold.
Our interest in this policy grew from an empirical study of the military's logistic system supporting service parts for military weapon systems (Cohen et al. 1998). The military recently moved the management of these parts from the individual military services (e.g., separate Army and Navy warehouses) to a central inventory-control point within the Defense Logistics Agency (DLA). While this change offers inventory-pooling benefits for common parts, it has led to some disagreement across the military services about the appropriate safety stock levels. The disagreement stems from the fact that the criticality of a part often differs significantly for each military service. DLA's current policy for managing these demand classes is to "round-up" each part's availability requirements across the various military services. For example, if the Army requires a service level of 85% while the Navy requires 95%, DLA stocks the part to meet an aggregate service level of 95%. Once stocked, inventory is allocated to custome rs on a FCFS basis. There are two obvious shortcomings to this approach. First, by rounding-up requirements, DLA may be investing too much inventory in noncritical items. Second, processing orders on a FCFS basis allows a low-priority customer to possibly preempt more critical customers. The military's previous strategy of managing separate pools of stock for each service avoided these problems, but did so at the cost of no inventory pooling. A threshold rationing policy, similar to the (Q, r, K) policy studied here, has been proposed as a way to avoid the problems inherent in the round-up policy, while still taking advantage of inventory pooling.
While such rationing policies have been implemented from time to time in the military, there is no methodology for determining how to select the parameters for these policies. Very little research exists on how to optimize policy parameters for multiple customer classes, particularly in cases with fixed setup costs, positive lead times, and backlogged customer demand (Kleijn and Dekker 1998). This environment is challenging because positive backorders and positive on-hand inventory can coexist, making it difficult to calculate backorder distributions from the inventory-level distribution. The goal of this paper is to develop a tractable and implementable solution to the stock-rationing problem and offer managerial insights into when our proposed policy is attractive relative to traditional "round-up" and "separate stock" policies. We do so by developing a methodology for selecting optimal control parameters for the (Q, r, K) policy, i.e., to select policy parameters to minimize inventory, delay, and backorder ing costs.
Optimization within this static (Q, r, K) policy depends on how customer backlogs are cleared once an order arrives. The optimal scheme is to always clear higher-priority customers first. However, this "priority clearing" scheme is intractable because it does not allow closed-form expressions for the stockout levels, and average number of demand in backlog, for each demand class. To overcome this problem, we introduce a tractable "threshold clearing" scheme to approximate the system dynamics and develop an efficient algorithm for calculating the optimal control parameters under this scheme. Our proposed solution is to use the control parameters chosen under the "threshold clearing" scheme, but implement these parameters using the "priority clearing" scheme. Numerical results show that this "hybrid" policy closely approximates the optimal "priority clearing" scheme. Numerical results also reveal that, compared to DLA's current round-up policy, our hybrid policy is most beneficial when the arrival rate for low criticality demand is significantly higher than that of the higher criticality class. Using a round-up policy in this case is wasteful because a large amount of inventory is used to support the higher service level for the low-criticality demand class.
While the hybrid policy proposed here is easy to implement and performs well compared to traditional policies, other nonstationary policies (i.e., where K may vary with the state of the system) could perform better. A secondary goal of this paper is to establish when our hybrid policy is a reasonable approximation to the optimal nonstationary rationing policy. This is accomplished by developing a lower bound over all possible policies. Numerical results suggest that the performance gap between our hybrid policy and the optimal nonstationary rationing policy is small for cost and demand parameters typical of military service parts and other environments where setup costs are extremely high (e.g., semiconductor equipment which is a high-technology, make-to-order, capital-intensive industry). It does not perform as well when both setup costs are small and penalty costs for the two demand classes are significantly different.
To summarize, in this paper we compare four different control policies for the inventory management/rationing problem.
1. Priority clearing: An optimal (Q, r, K) policy assuming a priority discipline for clearing the backlog.
2. Threshold clearing: An optimal (Q, r, K) policy assuming a threshold discipline for clearing the backlog.
3. Hybrid policy: Using the parameters of the threshold clearing policy but operating the priority discipline for backlog clearing.
4. Optimal rationing policy: An undefined policy that allows state-dependent ordering and rationing control parameters.
Our principle result is the development of a model and algorithm for the threshold clearing policy, to serve as input to our recommended hybrid policy. The hybrid policy dominates the other three policies in ease of implementation, allowing the tractable parameter search of the threshold clearing policy while using the more natural priority clearing scheme.
While our problem is motivated by the dynamics observed in the service parts division of the U.S. military, we expect our solution approach is applicable to a wide range of industry settings. Inventory systems with multiple demand classes having different priorities are common to a number of industries. For example, Cohen et al. (1998) study a service parts application in the computer industry where a retailer could place normal replenishment orders and emergency orders, in case of stockout, at the warehouse. Kleijn and Dekker (1998) provide an overview of inventory systems with several demand classes, including examples ranging from airlines to petrochemical companies.
The paper continues in [section]2, where we position our research with respect to previous literature. Section 3 introduces the threshold rationing model, backlog-clearing mechanisms, and cost function that drive our methodology Performance measures, structural results, and a solution algorithm are developed in [section]4, assuming a threshold clearing discipline. Section 5 compares these results with the priority clearing and hybrid policies. Section 6 reveals the benefits of our hybrid policy over DLA's current approaches and compares the policy's cost to a lower bound on the optimal nonstationary rationing policy. We also test the assumption of independent Poisson demand processes by comparing our results to a perfectly correlated demand case. Section 7 provides a formulation of our rationing policy for more than two demand classes and briefly discusses how the analysis would change to accommodate this more complex system. The paper concludes in ??8 with a discussion of possible extensions.
2. Literature Review
The task of dynamically allocating inventory to different demand classes lies at the heart of many yield management problems. These problems are typically characterized by limited capacity and perishable inventory (e.g., seats on an airplane, cars in a rental fleet, or rooms in a hotel) which is allocated to different classes of demand (e.g., first class, business class, or economy). Kimes (1989) provides an overview of research in this area. In this environment, the key decision variables are normally the prices charged to each demand class as well as the possible rationing levels (i.e., booking limits) to impose. Some examples include Belobaba (1989), who examines booking limits for airline seats with different price classes, and Bitran and Gilbert (1996), who develop heuristic rationing procedures for managing hotel reservations. In these problems, the capacity or inventory level is fixed so the decision of how much inventory to order and when to replenish are not relevant. The presence of obsolescence, ho wever, leads to nonstationary control policies that dynamically adjust as time to expiration approaches. Examples of dynamic allocation models for yield management include Lee and Hersh (1993), Bitran and Mondschein (1995), Subramanian et al. (1999), and Zhao and Zheng (2001). The major differences between our stock-rationing problem and traditional yield management problems are that we allow for multiple replenishment opportunities and assume inventory is not perishable. We also focus on static, rather than dynamic, policies.
Turning to the inventory literature, Veinott (1965) was one of the first to consider multiple demand classes in a multiperiod, single-product, nonstationary inventory environment. While he focuses on the question of how much to order and when to replenish, he does so in the context of a periodic review system without rationing levels. Topkis (1968) extends Veinott's work by considering how inventory should be allocated between demand classes within a single period of a periodic review model. Here each demand class is characterized by a different shortage cost. The analysis is facilitated by breaking each review interval into a finite number of subperiods. At the end of each subperiod, the decision maker allocates inventory to demand that has been realized thus far. The allocation is based on a trade-off between the benefit of filling demand for low-class items in the current subperiod and reserving inventory to fill higher-class items in subsequent subperiods. Within a single review interval, Topkis proves th ere exists optimal, nonnegative, rationing levels for each demand class which, under certain conditions, are decreasing in time.
Our rationing policy differs from Topkis' (1968) in three fundamental ways. First, we make the decision of whether to fill or delay an order at the moment the order arrives. Topkis delays this decision until the end of each subperiod. Making the decision up front reduces order delays. Second, our rationing level is stationary, which is consistent with our continuous review environment, where there are no defined time intervals for revising decisions. Third, our replenishment order cycles are based on inventory position, taking into account setup costs, lead times, and the possibility that multiple replenishment orders maybe in the pipeline. Models similar to Topkis' under different operating environments have been considered by Kaplan (1969) and Frank et al. (1999).
Nahmias and Demmy (1981) were the first to analyze a rationing policy in a (Q, r) environment. They consider a continuous review system with Poisson demand and two demand classes (as we do). However, they focus on evaluating fill rates for given rationing and reorder levels rather than on optimizing the policy parameters...
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