Optimal stocking policies for inventory systems with uncertain returns

The purpose of this paper is to enlarge the scope of existing EOQ-type models by incorporating uncertain product returns that may include time or quantity restrictions. In this inventory control problem, the item’s demand is deterministic and occurs at a known constant rate over time. Product returns arrive according to a Poisson process. The amount returned to the seller at each return epoch is a random or fixed fraction of the accumulated number of units (net of previous returns) that have been demanded so far within a stock cycle. The ordering policy is a continuous review policy that orders a fixed quantity every time an order is placed. The replenishment lead time is instantaneous, and shortages are not allowed. The objective is to find optimal order quantities by minimising the long-run average total costs per unit time. Costs include a fixed plus proportional cost per order, and a holding cost per unit held in inventory per unit time. A level-crossing approach is used to derive a closed-form expression for the expected ordering and inventory holding costs as a function of the order size and other relevant parameters.