In (Q, r) inventory systems, when a shortage occurs, incoming demands are handled either by emergency orders (or lost sales) or backorders. However, the backorder costs are usually time-dependent, hence it is costly to backorder early in the lead time. On the other hand, it is obviously expensive to fill the shortages with emergency orders alone. In this paper, we propose a hybrid inventory control system using both backorders and emergency orders to handle shortages. In the early stage of the lead time, shortages are covered by emergency orders, and as the

Inventory systems with a pure backorder policy or a pure lost sales (emergency order) policy have been extensively discussed in the literature. Although it is mathematically easier to deal with these two extreme systems, they may not be the best from the cost perspective. Furthermore, these pure systems may not represent many real situations where a mixture of emergency orders and backorders is used (Silver and Peterson, 1985). In some cases customers are not willing to wait for the next replenishment. In other cases firms may set a time or quantity limit (rain check, for example) on backorders. Consequently, a number of hybrid (partial backorder) systems have been proposed.

Montgomery et al. (1973) propose a continuous review inventory system where a fraction of the unfilled demand is backordered and the remaining fraction is lost. Both the cases of deterministic and stochastic demands are considered, but the stochastic demand case is treated heuristically. In their paper, only the time-independent backorder cost is employed. Rosenberg (1979) reformulates the above model by introducing a "fictitious demand rate" that simplifies the analysis of the partial backorder policy and gives an economic interpretation of the circumstances under which this policy is optimal.

Kim and Park (1985) extend the Montgomery et al. (1973) stochastic demand model to one in which the cost of a backorder is assumed to be proportional to the length of time for which the backorder exists. Assuming at most one order outstanding at any point in time and an arbitrary continuous density function of lead time demand, they derive the equations from which the optimal order quantity and the reorder point can be iteratively computed. Assuming Poisson demand and an exponential lead time, Woo and Sphicas (1991) formulate a partial backorder model that allows a finite number of orders to be outstanding.