  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.
