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Article Excerpt
1. Introduction and Literature Review
In many inventory settings, a supply firm wishes to provide different levels of service to different customers. For instance, in a service-parts network, a customer can choose amongst different contracts, each with a different cost and level of service. A "gold contract" might provide a 99% fill rate within 24 hours, while a "bronze contract" promises an 85% fill rate within two days. In other settings, a supplier segments its customers based on the delivery channel or the price they pay; the supplier recognizes some customers as deserving higher priority over other customers. In other cases, a supplier provides price discounts for delivery flexibility, and then allows a customer to choose the delivery time when placing an order.
A common approach to such scenarios is to categorize the customers into a finite number of demand classes. Customers within a demand class receive the same level of service. The inventory challenge is then to determine how to meet the service-level expectations for each demand class with the least amount of inventory.
In this paper, we consider a single-item inventory system with stochastic demand and multiple demand classes. The key assumptions are Poisson demand, a deterministic lead time, a continuous-review (Q, R) replenishment policy, and demand backordering. As is common in the literature, we assume a critical-level policy for rationing the inventory across the demand classes.
We have organized this paper into eight sections. In the remainder of this section, we discuss the relevant literature. In the following section, we present our assumptions and a general framework to describe how we manage the inventory with a stationary critical-level policy. In particular, we introduce our allocation policy for clearing backorders. In [section]3, we show how to map this inventory system into a serial inventory system. In [section]4, we use this mapping to develop a model for cost evaluation and optimization under the assumptions of Poisson demand, a deterministic replenishment lead time, and a continuous-review (Q, R) policy with rationing. In [section]5, we pose the service-level problem (SLP), in which we find the critical-level policy that meets a specified fill-rate target for each demand class with the least inventory. Furthermore, we provide a heuristic solution approach for the SLP. We provide some justification for our allocation policy in [section]6. In [section]7, we provide a numerical experiment both to compare our proposed heuristic with the optimal solution and to show the value from rationing. We show with both bounds and a numerical experiment that this heuristic is quite robust and near optimal. In the final section, we discuss possible extensions and directions for future research. We show how to extend the model to permit service times, whereby different demand classes have different service times by which their demand is to be met. Finally, we describe how to use the single-item inventory system to characterize the inventories and backorders in a multi-echelon distribution system.
Kleijn and Dekker (1998) give an overview of inventory systems with multiple demand classes and provide examples of managing inventory with multiple demand classes, ranging from airline service companies to petrochemical companies. In Table 1, we provide a high-level categorization of the literature. Like much of the stochastic demand inventory literature, we can categorize the research by the control policy, periodic or continuous review, and by the treatment of shortages, lost sales or backorders. In addition, some of the key developments are restricted to or primarily focused on two demand classes, whereas other work is not.
Veinott (1965) analyzes an inventory model with several demand classes for a single product. He proposes to use critical inventory levels to ration the on-hand inventory among demand classes. Topkis (1968) subsequently analyzes the proposed critical-level policy for a periodic-review single-product inventory model with multiple demand classes.
Kaplan (1969) and Evans (1968) study periodic-review models with only two demand classes, similar to Topkis (1968). Recently, Atkins and Katircioglu (1996) and Frank et al. (2003) analyze periodic-review inventory systems with multiple stochastic demand classes. Atkins and Katircioglu (1996) require an associated service level for each demand class, which had not been analyzed in the previous literature. However, their model allows negative inventory allocations that are hard to explain and implement. Frank et al. (2003) apply rationing to avoid incurring high-fixed-ordering costs rather than saving inventory for high-priority demand.
Nahmias and Demmy (1981) study a continuous-review inventory policy with two demand classes. They assume a (Q, R) inventory replenishment policy, a critical-level policy, and at most one outstanding order at any time. This last assumption implies that whenever a reorder quantity is received, the inventory level and inventory position become identical. This allows them to calculate approximate expressions for expected backorders for both demand classes. Moon and Kang (1998) later extend this model to account for compound Poisson demand processes.
Deshpande et al. (2003) analyze the same (Q, R) inventory rationing model with two demand classes as in Nahmias and Demmy (1981), but without the restriction on the number of outstanding orders. They introduce the threshold clearing mechanism to fill backorders, which permits them to derive expressions for the expected number of backorders for both classes. Based on these expressions, they develop algorithms to calculate the optimal ordering and rationing parameters. They demonstrate numerically the effectiveness of their model, by comparison to a priority-based backlog clearing mechanism, where high-priority backorders are filled before low-priority backorders.
Deshpande and Cohen (2005) extend the analysis in Deshpande et al. (2003) to multiple demand classes under the assumption of the latter paper's threshold clearing policy. They derive expressions for computing the performance measures and state a series of structural results on these performance measures.
Fadiloglu and Bulut (2005) study the inventory rationing problem with two demand classes in Deshpande et al. (2003) using an embedded Markov chain approach. They assume a one-for-one inventory replenishment policy and they clear backorders using a priority clearing mechanism in which they clear class-2 backorders only after restoring the entire reserve stock for class 1. They derive a recursive procedure to calculate the transition probabilities of the corresponding Markov chain; based on this Markov chain, they develop an algorithm for finding the steady-state distribution of the inventory level in the system.
Melchiors et al. (2000) also analyze a (Q, R) inventory model with two demand classes. Unlike Nahmias and Demmy (1981) and Deshpande et al. (2003), they consider a lost sales environment so that demands from the low-priority class are rejected whenever the inventory level drops to the critical level. Melchiors (2001) extends the model in Melchiors et al. (2000) to multiple Poisson demand classes with stochastic replenishment lead times. Moreover, he considers a nonstationary critical-level policy that provides a benchmark to evaluate the stationary critical-level policy employed by Nahmias and Demmy (1981), Melchiors et al. (2000), and Deshpande et al. (2003).
Dekker et al. (1998) study an inventory model with two demand classes and a one-for-one replenishment policy. The model is similar to the one in Nahmias and Demmy (1981). They assume Poisson demand processes, a deterministic replenishment lead time, backordering of unfilled demands, and a critical-level policy to ration the inventory. Dekker et al. (1998) explore how best to handle and allocate incoming replenishment orders, which remains an open question in the literature.
Dekker et al. (2002) extend the model in Dekker et al. (1998) to multiple demand classes with stochastic replenishment lead times by switching to a lost-sales environment rather than by allowing backorders. They assume a one-for-one replenishment policy and a critical-level policy to ration inventory among demand classes. In a lost-sales environment, each incoming replenishment order simply replenishes the inventory. They develop efficient numerical solution methods to calculate the optimal base-stock level and critical levels with or without service-level constraints.
Ha (1997a, b) considers a make-to-stock production system with a single production facility and multiple demand classes for the end product. He assumes exponentially distributed production time, Poisson demand for each demand class, and either lost sales or backorders. He shows that a stationary critical-level policy is optimal. de Vericourt et al. (2000, 2002) consider the multiple-demand class extension of the two-demand class study in Ha (1997a). They characterize the optimal policy for the backorders case with zero setup costs and exponential lead times.
2. General Framework
Our work is most closely related to that of Nahmias and Demmy (1981) and Deshpande et al. (2003). However, whereas their work considers two demand classes, we have no restriction on the number of demand classes. We also develop the model in what we believe is a more transparent and natural way. Indeed, as will be seen, this allows us to extend the model to permit service times and to analyze a multi-echelon system with multiple demand classes.
We consider a facility that carries inventory for a single product to serve N customer classes. We differentiate customer classes based on their relative service-level requirements or shortage costs. For our analysis, we require the following standard inventory assumptions:
(i) We have a fixed replenishment lead time L > 0;
(ii) The demand from class i, [D.sub.i], i [member of] {|1, N|}, follows a stationary Poisson process with rate [[lambda].sub.i] that is independent of the demand from the other demand classes;
(iii) We replenish inventory with a continuous-review (Q, R) policy; and
(iv) We backorder any demand that is not immediately met from on-hand inventory.
In addition to these assumptions, we need to describe how we ration inventory across the demand classes. We number the demand classes according to their relative priority, where class 1 has the highest priority. As suggested by Veinott (1965), we use a critical-level policy given by c = {[c.sub.1], [c.sub.2],..., [c.sub.N-1] | [c.sub.i] [member of] [Z.sup.+] [union] {0} and [c.sub.i-1] [less than or equal to] [c.sub.i]}. We stop serving demand class i + 1 once the on-hand inventory reaches or falls below the class-i critical stock level [c.sub.i]; by assumption, we then backorder all demand for class i + 1 until the on-hand inventory is raised above [c.sub.i]. For class 1, we fill class-1 demand until the on-hand inventory is completely depleted, at which point we backorder any subsequent class-1 demand.
We also need an assumption on how we allocate the inventory replenishment when it is received. The primary issue is how to allocate the replenishment between backorders of different classes and between filling backorders versus restoring the inventory for higher-priority demand classes. To illustrate the challenge, consider a three-class system with order quantity Q = 4, and critical levels given by [c.sub.1] = 2, [c.sub.2] = 3, R = 5. Suppose that just before the order quantity arrives, we have on-hand inventory of one unit, no backorders for class 1, two backorders for class 2, and three backorders for class 3. We need to decide how to allocate the replenishment quantity, as it is insufficient to clear all of the backorders and restore the on-hand inventory above the class-2 critical level [c.sub.2] = 3. For instance, should the four units be used exclusively to clear backorders, and if so, which of the five backorders should be filled? Should we fill a class-2 backorder before a class-3 backorder, even if the class-3 backorder is older? Alternatively, we might want to hold some units in anticipation of class-1 demand. For instance, if we regard class-1 customers as extremely sensitive to service relative to the other classes, we might want to hold one unit to raise the on-hand inventory to the critical level for class 1 and then use the remaining three units to clear backorders. Another option is to "ignore" class 3 and use the four units to clear the class-2 backorders and rebuild the on-hand inventory to...
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