Optimal Control Policy for Stochastic Inventory Systems with Markovian Discount Opportunities

In this paper, we study a single-item continuous-review inventory system with Poisson demand. In addition to the standard cost structure of a fixed setup cost and a quasiconvex expected inventory holding and shortage cost, special opportunities for placing orders at a discounted setup cost occur according to a Poisson process that is independent of the demand process. This model has been studied as a subproblem of multi-item/location inventory systems where there are economies-of-scale in joint replenishment. For the single-item model, the literature proposes the (s, c, S) policy, under which an order is placed to increase the inventory position to S either when the inventory position drops to s, or when the inventory position is at or below c and a discount opportunity occurs. We prove that the (s, c, S) policy is optimal for the model, develop an efficient algorithm for computing optimal control parameters s*, c*, S*, and carry out a parametric analysis showing the effects of changes in problem parameters on the optimal control parameters and the minimum cost.