# Optimal Inventory Policy with Multiple Set-Up Costs

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## Abstract

We consider a deterministic, single product, discrete review, finite time horizon inventory problem, called the multiple set-up cost problem. The holding cost in each period is a nondecreasing (and sometimes concave) function. The distinguishing feature of our model is the ordering cost function which is neither concave nor convex. The ordering cost is such that a natural interpretation of it consists in assuming that the order in period i is delivered in trucks with capacity M i and that the cost of delivery for each truck is a nondecreasing concave function of the amount delivered by that truck. We establish the existence of an optimal production schedule such that for each period (1) there are no partially filled trucks in period i if the inventory entering period i is positive and (2) the inventory at the end of period i is less than M i. Exploiting this information, an efficient algorithm is developed. In part II, we study the stationary, infinite horizon version of the multiple set-up cost problem. We single out a countable set S of schedules, each of which possess a periodic property in addition to properties (1) and (2) above, and we show that S contains a schedule with minimal cost per unit time. Moreover, if the ratio of demand per period to M i is rational, then S contains a schedule with minimal discounted cost.

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