Optimal control of a multi-product inventory system with substitution

We study a single-period, multi-product inventory model that allows substitution among the products. The main decision is the replenishment (order) quantity for each product at the beginning of the period, before demands are realized. This decision, however, has to take into account the substitution rule, which will be carried out, also in an optimal fashion, at the end of the period when demands are realized. We first derive an optimal subsitution rule, and show that under this rule, the objective function is both concave and submodular in the order quantities - decision variables. Hence, the optimal replenishment decision can be effectively generated through solving a concave program.