The reorder‐point–reorder quantity policies referred to as (r, Q) policies are widely used in industry and extensively studied in the inventory literature. It should be noted, however, that shortages often occur in practice even when the optimal reorder point and reorder quantity are achieved. In this paper, we present a mathematical model to control a shortage period by an emergency order. The problem is the determination of the best timing to deliver items after a shortage occurs. By applying the concept of repair-limit replacement policy in the context of maintenance theory, the optimal order time limits based on three kinds of cost criteria are derived in the framework of a simple (r, Q) inventory system. Finally, some examples of the inventory control model with a stochastic lead time are given to explain the optimal order‐limit policies.
