Random Review Production/Inventory Systems with Compound Poisson Demands and Arbitrary Processing Times

A production/inventory system is considered in which a single production facility produces items of a given type. The demand for the item is assumed to arrive according to a compound Poisson process. The time required to produce an item is assumed to follow an arbitrary distribution. An (s, S) policy is considered in which production stops at the instant that the inventory level is raised to S and production resumes at an inspection point when the inventory level is observed to have dropped to or below s for the first time. The time interval between two successive inspection points during a nonproduction period is a random variable which follows an arbitrary distribution. Under a cost structure which includes a set-up cost, a linear holding cost and a linear backorder cost, an expression for the expected cost per unit time is obtained for given control values. The operating cost during a cycle is shown to be convex in S for a given value of S-s. For the continuous review simple Poisson demand system, as well as some other special cases of the system considered here, it is shown that the expected cost per unit time is unimodal. Based on these properties, a procedure to find the optimal (s, S) policy is presented.