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Description
1. Introduction
The primary motivation behind this research is our experience with a leading capital equipment manufacturer. This company owns research, development, and manufacturing facilities in the United States, Europe, and the Far East and distributes its systems across the globe. The company is at the top of the supply chain for many high technology products.
The systems that the company manufactures are very expensive investments and are critical to the operations of its customers. It is very costly to have unused capacity at a customer's manufacturing facility caused by equipment failure. The company has an extensive spare parts network to provide spare parts and service to customers to repair equipment failures and perform scheduled maintenance operations. The network consists of more than 70 company-owned distribution centers and depots across the globe. The company also has agreements with its leading customers to manage their stock rooms. Three regional distribution centers in North America, Asia, and Europe constitute the backbone of the network and are primarily responsible for procuring and distributing spare parts to depots and customer locations. The depot locations are such that they can provide a 4-hour service to those customers who do not have stock rooms operated by the company in the event of equipment failures ("down orders"). The regional distribution centers may also be used as a primary source for down orders for certain customers. The regional distribution centers also provide a second level of support for down orders that cannot be satisfied from the local depots. Customers also demand spare parts to be used in their scheduled maintenance activities ("lead time orders"). The regional distribution centers are the primary source to meet these demands, but local depots can also be used for certain customers. Even though the maintenance activities are scheduled and known in advance by the customers, the capital equipment manufacturer we study does not have access to these schedules. Since each location supports many customers and a large installation base, the capital equipment manufacturer perceives these orders as random.
Both types of customer orders (down and lead time) go through an order fulfillment engine that searches for available inventory in different locations according to a search sequence specific to each customer. Down orders need to be satisfied immediately (their request date is the date of order creation), whereas lead time orders need to be satisfied at a future date. A depot may be facing down and lead time demand from a variety of customers, while a regional distribution center may be facing down and lead time demand from external customers in addition to the "replenishment orders" requested by internal customers: the depots and stock rooms managed by the company. The operation of this complex network is further complicated by a vast number of parts, both consumable and non-consumable (more than 50 000 active parts need to be managed) and varying service level requirements for different customers.
Providing an implementable and "good" solution for the whole spares network is a proven challenge; we, however, focus on an important issue where improvements can provide immediate and significant benefits. At present, the company uses a regular base stock inventory system with one service level for all types of demand and does not account for demand lead time differences. Obviously, this approach is inefficient. We suggest an inventory model that recognizes both demand lead times and multiple demand classes, and allows for providing differentiated service levels through rationing.
Multiple demand classes occur naturally in many inventory systems. Examples include a distribution center facing demand from retailers as well as directly from end customers; a spare part that is used in equipment of varying criticality; or an item that is sold to many customers of different criticality. The reader is referred to Kleijn and Dekker (2000) for a comprehensive study illustrating various examples in which multiple demand classes arise.
Given a system with multiple demand classes, the easiest policy would be to use different stockpiles for each demand class. Inventory for each class could be managed separately to meet a different service level requirement. While this policy is practical and very appealing, the drawback is that no advantage could be gained from risk pooling and more safety stock would be needed. On the other hand, one could simply use the same pool of inventory to satisfy demand from various customer classes without differentiation. In this case, the total stock needed would be determined by the highest service level requirement. The drawback here is that the highest service level is offered to all demand classes, leading to increased inventory costs.
Rationing, or the so-called critical-level policy, lies between these two extremes. Rationing has proven effective for handling different demand classes with different stock-out costs or service levels. We will explain rationing assuming that there are two demand classes but the extension to several demand classes is straightforward. A part of the stock is reserved for high-priority demand: this is called the critical level. Once the inventory level drops to this level, demand from the lower priority demand class is no longer satisfied. If unsatisfied demand is backordered, one also has to decide how to handle arriving replenishment orders. Obviously, if there is a backorder for a high-priority customer upon the arrival of a replenishment order, an arriving replenishment order would be used to satisfy this backorder. In addition, if there is a backorder for a low-priority customer when a replenishment order arrives and the inventory level is at or above the critical level, one should use this replenishment order to satisfy this backorder. However, in the case of a low-priority backorder and an inventory level below the critical level, one can either satisfy this backorder or increase the inventory level. The latter option is referred to as the priority clearing mechanism and has been proven to be optimal under specific conditions. Under general conditions, however, determining which one of these is optimal depends on the problem settings. For example, if the backorder penalty is non-linear in the backorder length, it may be better to clear a low-priority backorder even though the inventory level is below the critical level. Note that the service level for the low-priority class is not affected by the way replenishment orders are handled.
Except for very specific cases, a simple critical-level policy with a static critical level will not be optimal. For example, if the inventory level is below the critical level, but it is known that a replenishment order will arrive within a short period of time, not satisfying a non-critical customer demand may not be optimal, especially if the probability of a critical demand arrival within this time is very small. Therefore, an optimal policy should take into account the remaining lead times of outstanding replenishment orders. However, there are two difficulties in employing a dynamic rationing policy. First, rationing problems are theoretically difficult. In fact, the exact expressions for the service level and the inventory on hand cannot be derived even for the seemingly simple static rationing policy with two demand classes with Poisson arrivals, deterministic lead time and backordering. Therefore, the existing literature and most of the ongoing research on rationing are limited to static policies. The only exception in the literature on backordering is Teunter and Haneveld (1996), which uses a heuristic under a very restrictive assumption. Even if theoretical results were readily available, employing such a dynamic rationing policy would be extremely difficult from a practical point of view. In fact, the fulfillment engine (a commercial software) that is used in the capital equipment manufacturer we study is not capable of promising orders based on the status of replenishment orders. Thus, we prefer to focus on a static rationing policy where the critical level does not change over time.
While the specific industrial application in this study requires a higher service level for the demand class that has no demand lead time, it is possible that other applications require a lower service level for this demand class. Consider, for example, a multi-channel retailer that sells its goods online as well as through a bricks-and-mortar store. Online customers submit their orders in advance and a commitment is made upon the acceptance of these orders. However, no prior commitment is made to the customers in the demand class without a demand lead time, who ask for inventory upon their arrival into the store. Obviously, the service level requirement for online customers would be higher than customers purchasing through the store.
We therefore study a more general model where each demand class is identified by two characteristics, namely its demand lead time requirement and its service level requirement. A demand class is either critical or non-critical (i.e., its service level requirement is either more or less than the other class) and its Demand Lead Time (DLT) is either zero or T. The four possible cases are illustrated in Fig. 1.
Fig. 1. The four possible cases. Critical class DLT = DLT = T DLT = [check] Non-critical class DLT = T [check]
When both DLTs are zero, the problem is the classical rationing problem for which we give an overview of the existing literature in Section 2. When both DLTs are T, the problem can again be reduced to the classical rationing problem in which the replenishment lead time is reduced by the common DLT of T (see Hariharan and Zipkin, 1995). The two cases of interest in this paper are represented by the check marks in Fig. 1. For these cases, without loss of generality, we assume that class 1 has a DLT of zero, and class 2 has a DLT of T. Our analysis is general for the two cases: (i) service level requirement for class 1 is higher than class 2; (ii) service level requirement for class 2 is higher than class 1.
We model the system as a single-location system facing Poisson demand in two classes with rates [[lambda].sub.1] and [[lambda].sub.2], respectively. The spare parts inventory is replenished according to a (S - 1, S) policy, S being the order-up-to level. For simplicity, we consider a deterministic replenishment lead time, L. The service level we consider will be the type I service level, i.e., the probability of no stock-out. Under these circumstances the policy works as follows: once a critical order comes, it is either satisfied (at its due date) or backlogged if there is no inventory. On the other hand, a non-critical order is satisfied only if the inventory level is above a critical level, [S.sub.c], otherwise it is backlogged. We assume that class 2 orders are always accepted and a delivery commitment is made for them at their due date. the objective is to find the optimum S and [S.sub.c] such that the given service level requirements [bar.[beta]].sub.1] and [bar.[beta]].sub.2] are satisfied.
The remainder of the paper is organized as follows. In Section 2, we review the literature on related inventory systems. In Section 3, we derive an exact expression for the non-critical customer class service level and an approximate expression for the critical customer class service level. We also show analytically that the approximate expression for the critical customer class service level is a lower bound for the actual service level. In addition, we present a service level optimization model to find the optimal base stock and critical levels that satisfy service level requirements. In Section 4, we present the results of our simulation study; these indicate that our approximation for the service level of the critical class works quite well for high service levels. In addition, we present the results of the optimization study which determines the settings where the rationing is most useful. These settings are when the non-critical demands are dominant in the arrival mix, when the service level requirements are significantly different and when the DLT is present for the critical class. Also in Section 4, we present our results on a case study using 64 parts from the capital equipment manufacturer that we described earlier. We conclude the paper in Section 5.
2. Literature review
We will review the literature on inventory systems with a DLT before elaborating on the literature about rationing. We will first focus on the periodic-review models and then proceed to the continuous-review models.
The concept of a DLT was first introduced by Simpson (1958), using the term "service time" for base stock, multi-stage production systems. Hariharan and Zipkin (1995) then coined the term "DLT" to describe inventory-distribution systems where customers do not require immediate delivery thus allowing a fixed delay. The key observation in both papers is that the DLT works just as the opposite of the supply lead time, reducing the inventory held for achieving the required service level. This fact also applies to our system, but the existence of the two service classes complicates the model. Moinzadeh and Aggarwal (1997) considered a two-echelon system with two modes of inventory replenishment. In their model all orders are satisfied on a first-come first-served basis and the two order classes differ only in their transportation lead times. We, however, consider a system where orders are satisfied on a first-due first-serve basis. Wang et al. (2002) analyzed a similar system in order to derive the transient and steady-state performance metrics of the system. This work is actually the most relevant to ours since it involves two classes of service differentiated by a DLT. Therefore, we will explore their work in detail.
Wang et al. (2002) first studied a single-location system and derived expressions for the inventory level distribution and random customer delay. They made a crucial observation: the service level for customers with positive DLTs is higher than for customers with a zero DLT as long as there is a positive probability that the replenishment order corresponding to a customer with a positive DLT arrives before its demand due date. After deriving the steady-state performance metrics for the single-location system, the model was extended to a two-echelon system. By following an approach similar to the well-known METRIC, the multi-echelon network was decomposed into single-location subsystems. Analysis of the two-echelon setting showed that the system with two service classes results in significant inventory cost savings.
The literature about rationing begins with Veinott (1965), who was the first to consider the problem of several demand classes in inventory systems. He analyzed a periodic-review inventory model with n demand classes and zero lead time with limited ordering, and introduced the critical-level policy. Topkis (1968) proved the optimality of this policy both for the backordering and lost sales cases, and showed that the critical-levels generally decrease with the remaining time until the next ordering opportunity. Evans (1968) and Kaplan (1969) independently derived the same results for two demand classes. Nahmias and Demmy (1981) derived expressions for the expected backorder levels for a multi-period model with zero lead times and an (s, S) inventory policy when a static critical... |
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