Optimal inventory policies for a two-dimensional stochastic inventory model: A numerical investigation

This paper treats a discrete-time, two-item, stationary, infinite-horizon, stochastic inventory model. The system is reviewed at the beginning of each period where the inventory manager has the option of not ordering or ordering item 1 or item 2 or ordering jointly both items. If an order is made, then a set-up cost that depends on the ordered item(s) is accordingly incurred. The demand of each item in each period follows a known random distribution. In addition, the demands of the items are independent of each other and of the period. Inventory holding or backlogging costs are incurred depending on the stock level. This paper proposes and implements an exact stochastic dynamic program that determines the optimal steady state inventory policy. It exemplifies the form of the optimal policy for a number of demand distributions. This form turns out to be an extended (s, S) policy, where the state R² of possible inventory levels is naturally partitioned into two non overlapping sets: a stopping set and a continuation set. The stopping set consists of three non-overlapping subsets. The first corresponds to ordering item 1, the second to ordering item 2, and the third to jointly ordering items 1 and 2. In the continuation set, no action is required. The paper further illustrates the behavior of this policy for some interesting limiting cases, and discusses its sensitivity to the problem's parameters.