A Dynamic Inventory Model with the Right of Refusal

Abstract

We consider a dynamic inventory (production) model with general order (production) costs and excess demand that can be backordered or refused by the firm. A unit backordered incurs a backorder cost, a unit refused incurs a lost sales charge. Endogenizing the sales decision is necessary in the presence of general convex order costs so that the firm is not forced to backorder a unit whose subsequent procurement would reduce total profits. In each period, the firm must determine the optimal order/production and sales strategy. We show that the optimal policy is characterized by an optimal buy up to level that increases with the initial inventory level and an order quantity that decreases with the initial inventory level. More importantly, we show the optimal sales strategy is characterized by a critical threshold, a backorder limit, that dictates when to stop selling. This threshold is independent of the initial inventory level and the amount purchased. We investigate various properties of this new policy. As demand stochastically increases, the amount purchased increases but the amount backordered decreases, reflecting a shift in the way excess demand is managed. We develop two regularity conditions, one that ensures some backorders are allowed in each period, and another that ensures the amount backordered is nondecreasing in the length of the planning horizon. We bound the buy up to levels in our model using buy up to levels from the pure lost sales and pure backlogging models. We illustrate our findings and results using several numerical examples.

Keywords: dynamic programming, inventory and production, convex order costs, backorder limit