Real time decision rule for online retailer on partial backordering

In this paper we consider the inventory system of an online retailer. The retailer continuously reviews and replenishes inventory according to a (Q, R) policy. We assume Poisson demand and constant lead times. Normally the customers will accept a reasonable period of waiting time after they issued their orders because of the purchasing convenience of the online business. With this period of waiting time, the online retailer has a response time to the customer orders even when a shortage occurs, but without incurring any penalty cost. We call this period of time as retailer committed service time. However, after the committed service time, if the retailer is still in shortage, the customer demands will be filled by emergency orders or backordered with a time dependent backorder cost. Based on the real time information of the outstanding orders of the retailer and the waiting time of the customers, we give a decision rule to minimize the expected cost under the assumption that no further emergency orders will take place. Numerical examples to illustrate the model are presented together with discussions on the conditions under which the real time decision rule provides more cost savings over the traditional systems.