On continuous review stochastic (s, S) inventory systems with ordering delays

In this paper a continuous review (s, S) inventory system with ordering delays is considered. Demands for the item arrive according to a Poisson process. When the inventory level drops to s, the order is triggered. However, due to an unavoidable delay, the order is actually placed after a random amount of time. Once the order is placed, it is instantaneously delivered bringing the inventory level back to S. Under a cost structure which includes a setup cost, a holding cost and a penalty cost, an expression for the expected cost per unit time for given control values is obtained. Then some properties of the cost functions are developed to characterize the optimal policy. Based on these properties, an efficient search procedure to find the optimal (s, S) policy is presented.