Sales and Restocking Policies in a Single Item Inventory System

A single product system with linear costs is considered when customers are of two different types. Penalty costs for lost sales differ for the two types of customers and; therefore, optimal system control requires different shortage probabilities for the two classes. In one case, high penalty customers independently arrive while at the other extreme it is assumed that there is only one such customer. Single critical number policies are optimal in the simple situations. If the priority type customers become active and register displeasure by changing their demand pattern when their demands are not satisfied, simple convexity or Polya frequency function arguments are not sufficient to guarantee that the optimal policy remains one of the single critical number variety.