A Dynamic Inventory Model with Random Replenishment Quantities

A periodic-review, random-demand inventory model is analyzed under the assumption that replenishment quantities are random fractions of the amounts ordered. Results of a previous study of a single-period model are generalized to form an easily computed heuristic adaptation of the (s, S) policy for use in this environment. The heuristic is based on the simple practice of scaling down pipeline inventories to estimate the inventory position, and scaling up the order quantity in anticipation of an average replenishment yield. Simulation experiments are used to estimate the most cost-efficient (s, S) policies and to estimate the performance of heuristic policies in environments where replenishment randomness ranges from mild (0-20% defectives) to moderate (0-50% defectives). The heuristic is shown to perform quite well, with expected total costs typically within a few percent of the best (s, S) costs. The results tend to support common practice in industry which is similar to the approach studied here. Although the heuristic is naive in the sense that it ignores the degree of randomness in the replenishment quantity, the simulation results support the speculation that unless the target service level is extremely high, the replenishment process must be extremely random for its variability to be a significant explicit factor in the selection of a practical, cost-effective policy. Key words—inventory control, replenishment, random replenishment quantities Article: Consider a problem that might, for example, confront the production manager of an electronics manufacturing firm. The manager is responsible for producing a highly sophisticated component that must meet very stringent quality specifications. Each production run yields a batch having only a random portion of the components which pass inspection. The component is produced to stock in response to continuing demands of random size. The problem at hand is to specify the inventory replenishment policy for the component which will drive the production process. It is likely that an optimal policy will compensate by prescribing an order for more units than are actually desired. The problem is an example of a situation that is not addressed by traditional inventory management models. That is, one cannot prespecify the exact size of an inventory replenishment at the time of ordering. Rather, one is limited to choosing from a set of ordering levels that influence the probability distribution of a random replenishment quantity. While it is certainly true that all real inventory systems have this property, many systems have replenishment processes that are sufficiently predictable to assume that the quantities are deterministic. This is often the case when stocking a simple, standard item that is in abundant supply. On the other hand, it may not be appropriate to assume deterministic replenishment quantities in situations when quality requirements are unusually high or when the replenishment process itself is inherently random, such as in agriculture, blood donations and semiconductor chip production. Topics in the areas of random replenishment inventory systems and variable yield production systems have received considerable attention in recent years, yet little is known about optimal policies in a multi-period, random demand environment other than the fact that they do not possess many of the more elegant properties of their deterministic-replenishment counterparts. Silver [17], Sepheri et al. [15], Lee and Yano [10], and Shih [16] analyze models with deterministic demand. Shih [16], Karlin [9], Noori and Keller [13] and Ehrhardt and Taube [4] discuss single-order problems with random demand and various assumptions about the ordering process, cost structure and demand distributions. Ehrhardt and Taube [4] show that when the replenishment quantity is a random fraction of the amount ordered, an optimal single-period ordering policy can be found with a simple generalization of the traditional newsvendor result. They also show that a simple scaling-up heuristic is an effective approximation to optimal performance. The heuristic computes an order size by starting with the order size that would be optimal with deterministic replenishment, and dividing it by the expected value of the replenishment yield fraction. Moinzadeh and Lee [11] study a continuous review system under Poisson demand, and analyze ordering policies confined to the approximately-optimal (r, Q) form, where the reorder point r is based on the sum of inventory on hand plus that on order. They perform an exact Markov-chain analysis of the system and find that the best values of r and Q, while difficult to compute, are nicely approximated by an intuitively appealing adaptation of the heuristic of Hadley and Whitin [6]. The adaptation, similar in spirit to Ehrhardt and Taube's [4], consists of merely scaling up the mean of the Poisson process by dividing it by the expected value of the replenishment yield fraction. Gerchak et al. [5] analyze a multi-period model with linear ordering cost and random demand. They establish the existence of order points, and find that optimal policies are in general difficult to compute and non-myopic in character. A significant theoretical contribution was made by Henig and Gerchak [7], who discuss singleand multi-period models with more general assumptions about the random replenishment distribution and the cost structure. They show that for a single-period model there exists, under very general conditions, an optimal order point whose value is independent of replenishment randomness. They also analyze a zero-leadtime multi-period model having replenishment quantities that are random fractions of the order sizes, and linear order costs. They know that optimal policies are not of the order-up-to type, and that the optimal order quantity and reorder point both tend to increase with replenishment randomness, although the strength of the relationships are not characterized. They also establish convergence of value functions to their infinite horizon counterparts, and that infinite-horizon optimal order points are no smaller than when replenishment is deterministic. Recently Bollapragada and Morton [1] analyzed a multiperiod model with normally distributed demand, linear cost functions, and replenishment quantities that are random fractions of the quantities ordered. They show that the problem is nearly myopic under certain assumptions, and present several accurate heuristics. Their model is very effectively analyzed using myopic and near-myopic methods, but it does not, however, include an order setup cost or a delivery lead time. In the remainder of this paper, we analyze a multi-period model with independently distributed demands and an expected-cost optimality criterion. Specifically, in the next section we specify a dynamic model having linear holding and shortage costs, an ordering setup cost, and a rather simple mechanism for generating replenishment quantities. An Appendix is also provided to place the model in a more general context, and to discuss the theoretical reasons for difficulties in computing exactly optimal ordering policies. In the following section we introduce an easily-computed heuristic (s, S) policy that is inspired by results from a previously published single-period version of the model. Then we present the results of a simulation experiment comparing the performance of the heuristic policy with that of the best policy of the (s, S) form. Finally, we briefly draw