  This paper establishes conditions for optimality of an $(s,S)$ ordering
policy for the minimization of the long-term average cost of one-dimensional
diffusion inventory models. The class of such models under consideration have
general drift and diffusion coefficients and boundary points that are
consistent with the notion that demand should tend to decrease the inventory
level. Characterization of the cost of a general $(s,S)$ policy as a function
$F$ of two variables naturally leads to a nonlinear optimization problem over
the ordering levels $s$ and $S$. Existence of an optimizing pair $(s_*,S_*)$ is
established for these models. Using the minimal value $F_*$ of $F$, along with
$(s_*,S_*)$, a function $G$ is identified which is proven to be a solution of a
quasi-variational inequality provided a simple condition holds. At this level
of generality, optimality of the $(s_*,S_*)$ ordering policy is established
within a large class of ordering policies such that local martingale and
transversality conditions involving $G$ hold. For specific models, optimality
of an $(s,S)$ policy in the general class of admissible policies can be
established using comparison results. This most general optimality result is
shown for the classical drifted Brownian motion inventory model with holding
and fixed plus proportional ordering costs and for a geometric Brownian motion
inventory model with fixed plus level-dependent ordering costs. However, for a
drifted Brownian motion process with reflection at $\{0\}$, a new class of
non-Markovian policies is introduced which have lower costs than the $(s,S)$
policies. In addition, interpreting reflection at $\{0\}$ as "just-in-time"
ordering, a necessary and sufficient condition is given that determines when
just-in-time ordering is better than traditional $(s,S)$ policies.

Optimization and Control