In this paper we consider an inventory management problem motivated by a specific practical context that concerns the commonly used periodic review, reorder point, order-up-to-level control system. In particular, the reorder point and order-up-to level have to be selected so as to satisfy two managementspecified constraints. First, from a marketing (or, more generally, a business) perspective, the actual fill rate achieved must be no lower than a prescribed value. Second, for production or supply purposes, the average time between consecutive replenishments must be as close as possible to a target value. The demand distribution considered is the negative binomial. An efficient procedure is developed with illustrations.