On the Structure of Optimal Ordering Policies for Stochastic Inventory Systems with Minimum Order Quantity

We study a single-product periodic-review inventory model in which the ordering quantity is either zero or at least a minimum order size+ The ordering cost is a linear function of the ordering quantity, and the demand in different time periods are independent random variables+ The objective is to characterize the inventory policies that minimize the total discounted ordering, holding, and backorder penalty costs over a finite time horizon+ We introduce the concept of an M-increasing function+ These functions allow us to characterize the optimal inventory policies everywhere in the state space outside of an interval for each time period+ Furthermore, we identify easily computable upper bounds and asymptotic lower bounds for these intervals+ Finally, examples are given to demonstrate the complex structure of the optimal inventory policies+