(Q, S) models for inventory policy with random component procurement lead times

This paper considers inventory models of (Q, s) type with Q the order-quantity and s the order point. In general, an optimal choice of control parameters (Q and s) will depend on the characteristics of replenishment lead time and the demand process, as well as holding and shortage costs. Although many studies have treated lead time as constant, focusing only on demand variability, a number of authors have shown that a stochastic lead time is an issue that can have significant impact on inventory models and systems. This paper addresses the model when supply is assumed to have an exponential lead time distribution, and the “customer” itself is an exponential unreliable manufacturing plant, aiming for a constant mean production rate; thus both supply and demand have random characteristics. Under a so-called averaging principle approximation, we derive closed form expressions of the quantities of interest and illustrate their application in the optimization of the Q and s parameters.