A Probabilistic Production and Inventory Problem

R. Howard (R. Howard.1960. Dynamic Programming and Markov Processes. John Wiley and Sons, Inc., New York.) and A. Manne (A. Manne. 1960. Linear programming and sequential decisions. Management Science, April.) have suggested two methods for optimizing average costs per unit time by means of a sequential decision rule. This rule gives order quantities as a function of initial stock levels, and is based upon the assumption of a Markov process which is both homogenous and discrete. The formulation does not include any restrictive hypothesis on the structure of costs.

This paper presents a synthesis of both dynamic and linear programming methods applicable to the same kind of model for the case of time discounting.