Optimal order-limit policies for an (r, Q) inventory system

The reorder-point-reorder-quantity policies referred to as (r,Q) policies are widely used in industry and extensively studiedin the inventory literature. It should be noted, however, thatshortages often occur in practice even when the optimal reorderpoint and reorder quantity are achieved. In this paper, we presenta mathematical model to control a shortage period by an emergencyorder. The problem is the determination of the best timing todeliver items after a shortage occurs. By applying the conceptof repair-limit replacement policy in the context of maintenancetheory, the optimal order time limits based on three kinds ofcost criteria are derived in the framework of a simple (r, Q)inventory system. Finally, some examples of the inventory controlmodel with a stochastic lead time are given to explain the optimalorder-limit policies.