Continuous Inventory Models of Diffusion Type: Long-term Average Cost Criterion

Abstract

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.