Optimal Policies for Inventory Systems with a Generalized Cost Model

We study a single-stage inventory system with a generalized shortage penalty cost that includes the following three components: (i) a cost that is an increasing function of the number of backordered units in a period, (ii) a fixed cost incurred for each period in which there is a backorder irrespective of how many units are backordered, and finally (iii) a cost that is an increasing function of the number of periods a customer is backordered. We show the problem can be transformed into one in which the backorder cost depends on the inventory position only. Then, we present two sets of conditions, therst one of which restricts attention to a special case of the generalized penalty cost model while the second restricts attention to stationary demand models with some distributional assumptions. Under the first (second) set of conditions, we show that the expected cost in a period can be expressed as a convex (quasi-convex) function of the after-ordering inventory position. We use this property to prove the optimality of order-up-to policies under both sets of conditions, and discuss extensions to the cases where either a fixed ordering cost or a batch ordering constraint is present.