Optimality of Base Stock Policy under Unknown Demand Distributions: New Results and a New Method

Abstract

This paper advances significantly the literature on the optimality of the base stock policy by generalizing the demand distribution and prior belief to be updated as demands, assumed to be continuous i.i.d. random variables, are observed over time. As the value function depends on the belief, the functional Bellman equation is infinite-dimensional. More importantly, we derive a functional equation for the derivative of the value function that provides a direct approach to obtain the optimal base stock. We characterize the way the base stock depends on the belief and implement it in two important cases. In the case of conjugate probabilities, the infinite-dimensional state reduces to a finite-dimensional sufficient statistic allowing us to solve numerically an example of Weibull demand. The second case considers the demand to come from one of two possible distributions, but we do not know which. This gives a functional equation in one hyperparameter expressing the ratio of the weights assigned to the two distributions. We develop an approximation scheme to solve it, obtain the formulas for the base stock, and show numerically that the procedure converges.