Optimal ordering policies for periodic-review systems with replenishment cycles

Abstract

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.

Introduction

Despite the attractiveness of continuous-review inventory systems, periodic-review models are still applied in many situations, especially for inventory systems where many different items are purchased from the same supplier and the coordination of ordering and transportation is important. See, e.g., Chiang and Gutierrez [5] and Silver et al. [16] for other reasons of adopting periodic-review systems. Often, periodic-review systems have the review periods that are one or few weeks (or months) long and the supply lead-time is shorter than a review period.

Studies on the periodic-review inventory models (see, e.g., Porteus [15] and references therein) often assume that the supply lead-time is a (integer) multiple of a review period, and develop optimal policies (for backorder models) that are of either the base-stock type or the (s, S) type (i.e., whenever the inventory is reviewed or drops to s at the beginning of a period, an order is placed to raise the inventory to a predetermined level S), depending on whether or not a fixed cost of ordering is present. Efficient procedures are also developed to find the optimal s and S (see, e.g., Zheng and Federgruen [18]) or near-optimal solutions for the lost-sales model with positive lead times (see, e.g., Morton [11], Nahmias [13], van Donselaar et al. [6], and Johansen and Hill [9]). As Chiang [1] and Chiang and Gutierrez [4], [5] point out in the two-supply-mode setting, such periodic models could be regarded as an approximation of continuous-review inventory systems, for the review periods are typically modeled as small as one day and the holding and shortage costs are computed based on the ending inventory of periods.

For periodic inventory systems where the review periods are one or few weeks long, it is appropriate to compute the holding and shortage costs based on respectively, the average period inventory and the duration of shortage (for the backorder model). Exact analysis of such systems assumes the (R, T) policy and derives the average annual cost expression (for specific cases only) that is difficult to compute, especially for the lost-sales model (see, e.g., Hadley and Whitin [7, Sections 5–6 and 5–13]). Consequently, the heuristic or approximate treatment of such systems is often used by standard textbooks (see, e.g., [7, Section 5–2] and [16, Section 7.9.4]) and research papers (see, e.g., Chiang [3] and Moses and Seshadri [12]) to obtain easy-to-implement solutions.

There are many shortcomings for the approximate treatment of periodic inventory systems with one-or-few-weeks-long review periods. First, the average on-hand inventory is derived by assuming that backorders or lost sales are incurred in very small quantities. This approximation is poor if backorders or lost sales are not an insignificant portion of demand, or if demand is highly volatile as discussed by Nahmias and Smith [14] for the lost-sales model. To overcome this shortcoming, van der Heijden and de Kok [17] propose an improved approximate method to estimate the mean physical stock given a target fill rate. Second, for the backorder model, the shortage cost is charged per unit of shortage irrespective of the duration of shortage [7, p. 238], while for the lost-sales model, the effect of lost sales occurring between the time an order is placed and the time it arrives is ignored [7, p. 241]. Third, the lost-sales model assumes the base-stock policy, which is in general not optimal, as demonstrated by Karlin and Scarf [10]. In this paper, we study periodic inventory systems with one-or-few-weeks-long review periods, and use dynamic programming to formulate both the backorder and lost-sales models.

To be somewhat consistent with the periodic-review literature, we will rename the one-or-few-weeks-long review periods as replenishment cycles (or simply cycles), and let a cycle consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle (i.e., at a review epoch), as in the (R, T) policy. This assumption is reasonable if the fixed cost of ordering is small or negligible, which is especially true if an order for a specific item is part of a joint order and the ordering cost for a joint order (which is incurred every time a joint order is placed) is irrelevant to individual items. We assume that the length of cycles is handled outside our models (e.g., determined by the need of coordinating replenishments of many different items), as in Chiang [1] and Chiang and Gutierrez [4], [5]. The holding and shortage costs will be computed based on the ending inventory of periods (rather than only on the ending inventory of cycles). As periods can be defined to be any time units for the purpose of an application, this approach actually computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and thus remedies the shortcomings mentioned above for the approximate treatment of periodic systems with replenishment cycles.

It should be noted that Chiang and Gutierrez [5] and Chiang [2] also consider periodic inventory systems with replenishment cycles that consist a number of periods. However, periods in their models are defined to be such that an emergency order can be placed at the beginning of them. Also, they develop only the backorder models.

We notice that the proposed dynamic programming approach to computing the holding and shortage costs has several economic implications. First, the proposed backorder and lost-sales models that minimize the expected discounted cost over a planning horizon consider the time value of money, while the approximate models [7, Section 5–2] that minimize the expected annual cost do not take it into account. Second, as we discuss above, periods can be defined to suit the needs of an application. For example, if customers consider the level of disservice in proportion to its time expressed in days (or even hours), periods are defined as days (or hours). This is often true of inventory systems for the service parts of equipment or cars and computer products. Customers usually escalate their unhappiness as their waiting for service parts to arrive continues. In such situations, the shortage cost should be charged in proportion to the duration of shortage (in addition to the amount of shortage). By contrast, the approximate backorder model computes the shortage cost irrespective of the duration of shortage and may not be applicable to these situations.

We will show that a base-stock policy is optimal for the backorder model. This agrees with the periodic-review inventory literature. We also develop a simple expression for computing the optimal base-stock level. Moreover, we show that the optimal order quantity is a function of the on-hand inventory for the lost-sales model. This generalizes Theorem 4 of Karlin and Scarf [10] for the one-period-lag inventory problem. We also derive the convergence conditions of stopping dynamic programming computation and obtaining the optimal operational parameters for the infinite-horizon lost-sales model. As computational results indicate that it takes only a few cycles for operational parameters to converge, we advocate that firms use the proposed method (of computing the holding and shortage costs) and implement the optimal policy.