OPTIMAL ORDERING POLICIES FOR BROWNIAN INVENTORY MODELS WITH GENERAL SETUP COSTS

Abstract

We consider continuous-review inventory models with general quantity-dependent setup costs. The demand processes of the inventory models are modeled as Brownian motions with a positive drift. The inventory level can be adjusted by a positive amount at any time and the lead time of each order is zero. Each order incurs a proportional cost and a setup cost that is a step function of the order quantity. We further assume that the holding cost is a general convex function of the inventory level. By a lower bound approach, we obtain optimal ordering policies for three continuous-review inventory models: (a) an inventory model without backlogs under the long-run average cost criterion; (b) an inventory model without backlogs under the discounted cost criterion; (c) an inventory model with backlogs under the discounted cost criterion. Since the smooth pasting technique does not apply when the setup cost is quantity-dependent, we propose a four-step procedure to obtain optimal policy parameters for the inventory models. To cope with the quantity-dependent setup cost and upward adjustments, we provide a comparison theorem under the discounted cost criterion. With this comparison theorem, we can prove the global optimality within a tractable subset of admissible policies.