A comparison between the order and the volume fill rate for a base-stock inventory control system under a compound renewal demand process

TLDR: This work considers a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered, and develops exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process.

Abstract: The order fill rate (OFR) is sometimes suggested as an alternative to the volume fill rate (VFR) (most often just denoted fill rate) as a performance measure for inventory control systems. We consider a continuous review, base-stock policy, where replenishment orders have a constant lead time and unfilled demands are backordered. For this policy, we develop exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process. We also elaborate on when the OFR can be interpreted as the (extended) ready rate. For the case when customer orders are generated by a negative binomial distribution, we show that it is the size of the shape parameter of this distribution that determines the relative magnitude of the two fill rates. In particular, we show that when customer orders are generated by a geometric distribution, the OFR and the VFR are equal.