Optimal Control of a Brownian Production/Inventory System with Average Cost Criterion

Abstract

In this paper we study a stochastic production/inventory system with finite production capacity and random demand. The cumulative production and demand are modeled by a two-dimensional Brownian motion process. There is a setup cost for switching on the production and a convex holding and shortage cost, and our objective is to find the optimal production/inventory control that minimizes the average cost. Both lost-sales and backlog cases are studied. For the lost-sales model we show that, within a large class of policies, the optimal production strategy is either to produce according to an (s, S) policy, or never turn on the machine at all (thus it is optimal for the firm to not enter the business); whereas for the backlog model, we prove that the optimal production policy is always of the (s, S) type. Our approach first develops a lower bound for the average cost among a large class of nonanticipating policies and then shows that the value function of the desired policy reaches the lower bound. The results offer insights on the structure of the optimal control policies as well as the interplays between system parameters.