Optimal Control Policy for Stochastic Inventory Models with Two Substitutable Products

In this paper, we consider a stochastic inventory system of two products where a fi xed proportion of the unsatisfied ed demand for one product purchases the other product as a substitute. We formulate the optimal inventory control problem as a dynamic program; and by developing two key technical results, we derive several important structural properties (including joint concavity and submodularity) of the value functions. Based on these properties, we show that the optimal inventory policy for each product in each period is a base-stock policy where the base-stock level is a decreasing function of the maximum of the initial inventory level of the other product and a critical control number. We further prove that, as the initial inventory level of the other product increases, the base-stock level of each product converges to the optimal base-stock level of an auxiliary single-product lost-sales inventory system. We also provide intuitive interpretations and a numerical example to facilitate the understanding of our analytical results