An Inventory Model with Full Load Ordering

Abstract

We consider a discrete review, single product, dynamic inventory model with stochastic demands. The ordering cost is assumed to consist of a linear purchase cost and a cost K for each vehicle used. We examine the simple policy: “If the stock level at the beginning of period n does not exceed some critical number Tn, order the smallest number of full vehicle loads which will raise the inventory level just above Tn. If the stock level is above Tn do not order.” Conditions ensuring that this full load ordering rule minimizes total expected cost are given. We prove that the sequence of critical numbers T1, T2, … is bounded and nondecreasing and we find a limiting value T that characterizes an optimal infinite horizon policy. The conditions include typical assumptions, such as the convexity and differentiability of the expected one period holding and shortage cost. The most interesting condition involves the variation in the demand distribution over intervals of length equal to the vehicle capacity. As one would expect, it is satisfied if capacity is small or K is large. The case of exponential demand and linear holding and shortage costs is treated. In this situation, the conditions are easy to check and T is found explicitly.