Approximate order quantities and reorder points for inventory systems where orders arrive in two shipments

In this paper, the authors consider an inventory system where orders may arrive in two shipments, that is, the first shipment of the order may contain only part of the items ordered, while the rest of the items arrive in a second shipment. However, the amount of the first shipment is random. Such situations arise when the supplier partially fills an order if it does not have sufficient stock to satisfy the amount demanded in an order, or when an arriving order contains defective items that can be returned to and replaced by the supplier at some later date. We present the operating characteristics and an approximate cost function for such a system. The properties of the approximate cost function are exploited to bound the search for the order quantity and reorder point that minimize it. Finally, the effectiveness of the approximate cost function in providing near-optimal solutions is evaluated by means of numerical examples.