Joint replenishment policy for inventory system with demand substitution

In this article we study the optimal joint replenishment policy of single-period two-product inventory model with stochastic demands and downward substitution. The downward substitution means that using the stocks of products of higher class can satisfy demands for product of a lower class. Then we construct an optimization model and further prove that the objective function of this model is convex with respect to the inventory levels of these two products after their orders have arrived. Finally we obtain the optimal joint replenishment policy of this inventory system.