A Minimax Ordering Policy for the Infinite Stage Dynamic Inventory Problem

This paper obtains closed form expressions for minimax ordering decisions in a dynamic inventory problem in which demands in successive periods are independent random variables that have known (finite) mean and variance, but whose distributions are otherwise arbitrary and may change from period to period. We assume zero setup cost, linear holding and backlogging costs, and immediate delivery of orders. The situation of a decision maker facing this unknown sequence of demand distributions can be viewed as a sequential zero-sum game against nature. A minimax ordering policy is one that minimizes the maximum expected discounted cost over the planning horizon, where the maximum is taken over all the probability distributions in the class described. A limiting base stock policy characterizes a minimax ordering policy for the infinite-stage model. We show that the limiting base stock policy is an optimal stationary policy for a finite horizon model if the terminal value function - cx is used. We also obtain conditions under which a myopic minimax ordering policy exists for an inventory model when the class of demand distributions is allowed to be nonstationary.