A Hybrid Inventory System with a Time Limit on Backorders

Traditionally, in ( Q , r ) inventory systems, when a shortage occurs, incoming demands are either filled by emergency orders or backordered. 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 to handle shortages. A cutoff time τis incorporated with the traditional ( Q , r ) systems. Poisson demands and a constant lead time are assumed. Shortages are covered by emergency orders before τand backordered after τ. The traditional backorder and emergency order systems are special cases of the hybrid system with τ= 0 and τ= lead time, respectively. The optimal control variables Q *, r *, and τ* are determined by minimizing the expected annual cost. Numerical examples to illustrate the model are presented together with discussions on the conditions under which the hybrid system provides cost savings over the two traditional systems.