Single-Item Continuous-Review Inventory Models with Random Supplies

Abstract: This paper analyzes single-item continuous-review inventory models with random supplies in which the inventory dynamic between orders is described by a diffusion process, and a long-term average cost criterion is used to evaluate decisions. The class of models have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. Random yield is described by a (probability) transition function which depends on the inventory-on-hand and the {\em nominal} amount ordered; it is assumed to be a distribution with support in the interval determined by the order-from and the nominal order-to locations of the stock level. Using weak convergence arguments involving average expected occupation and ordering measures, conditions are given for the optimality of an $(s,S)$ ordering policy in the general class of policies with finite expected cost. The characterization of the cost of an $(s,S)$-policy as a function of two variables naturally leads to a nonlinear optimization problem over the stock levels $s$ and $S$ and existence of an optimizing pair $(s^,S^)$ is established under weak conditions. Thus, optimal policies of inventory models with random supplies can be (easily) numerically computed. The range of applicability of the optimality result is illustrated on several inventory models with random yields.