Optimal Ordering Policy for TwoProduct Inventory Models with Fixed Ordering Costs

We study the problem of finding an optimal ordering policy that minimizes the present value of the costs incurred over the infinite horizon in a continuous-time two-products inventory model with fixed setup costs, unit production costs, and linear holding and backlog costs. We first consider a deterministic demand and then a stochastic demand involving Brownian motion. A two-dimensionalquasi-variational inequality (QVI) is developed for the value function of the problem. This results ina free-boundary problem that shows that the optimality of a (γ, Γ) policy, where γ and Γ are the ordering and order-up-to boundaries, respectively, such that we order only when the inventory is below γ and then order up to a specified point in Γ. We illustrate our results by constructing the policy boundaries using a finite difference method to solve the QVI