Stegmaier Bravo, Raul Enrique

A common practice in inventory systems with several customers requiring differentiated service levels is to group them into two or three classes, where a customer class is a group of customers with the same preset service level in terms of product availability. However, there is no evidence that grouping customers into two or three classes is optimal in terms of the ordering policy parameters. This paper studies the effect of the number of customer classes on the inventory level of a single-period inventory system with stochastic demand and individual service-level requirements from multiple customer classes. Using a Sample Average Approximation approach, we formulate computationally tractable multi-class service level models, under responsive and anticipative priority policies in cases of shortage, as mixed integer linear problems (MIPs). The effect of the number of classes on the inventory level is determined using a round-up aggregation scheme; i.e., given a sufficiently large initial number of classes, it is reduced by adding the lower service level classes to the next higher class. We analytically characterize the optimal inventory level under responsive and anticipative priority policies as a function of the initial number of classes and the number of classes grouped based on the round-up aggregation scheme. Under a responsive priority policy, we show that there is an optimal number of classes, while under an anticipative priority policy, the optimal number of classes is equal to the initial number of classes. The effect of free-riders resulting from the round-up aggregation scheme on the optimal inventory level is studied through numerical experiments.

Popular measures of product availability in inventory systems seek to control different aspects of stock shortages. However, none of them simultaneously control all aspects of shortages, because stock shortages in inventory systems are complex random events. This paper analyzes the performance ofαLservice measure, defined as the probability that stockouts do not occur during a replenishment cycle, to cover different aspects of stock shortages when used to design an optimal continuous review (Q,r) policy. We show that explicitly controlling the frequency of replenishment cycle stockouts, using theαLservice-level, allows to implicitly control the size of the stockouts at an arbitrary time, the size of accumulated backorders at an arbitrary time, and the duration of the replenishment cycle stockouts. However, the cost of controlling the frequency of replenishment cycle stockouts is greater than the cost of controlling the size of stockouts and the duration of the replenishment cycle stockouts.