Inventory Control Driven by Demand Data: Optimality and Computation of Base Stocks

Abstract

This paper advances significantly the literature on the optimality of the base stock policy by generalizing the demand distribution and beginning with a completely general belief prior to be updated as demands are observed over time. As the value function depends on the belief, the functional Bellman equation is infinite-dimensional. We use unnormalized probabilities to linearize it and derive a functional equation for the derivative of the value function. This provides a constructive approach to obtain the base stock as well as the value function. We completely characterize the way the base stock depends on the belief, and implement it in two important cases. In the case of conjugate probabilities, we show rigorously that the infinite-dimensional problem reduces to one in terms of a finite-dimensional sufficient statistic. We solve numerically an example of Weibull demand. The second case considers the demand to come from one of two possible distributions, but we don’t know which. This gives a functional equation in two hyperparameters. We develop two approximation schemes to solve it, obtain the formulas for the base stock, and show numerically that both procedures converge and provide nearly the same base stocks. Finally, our methodology can be used for examples such as the case of multiple possible demands and the case when a fixed ordering cost is present.