Optimal Drift Rate Control and Impulse Control for a Stochastic Inventory/Production System

In this paper, we consider joint drift rate control and impulse control for a stochastic inventory system under long-run average cost criterion. Assuming the inventory level must be nonnegative, we prove that a $\{(0,q^{\star},Q^{\star},S^{\star}),\{μ^{\star}(x): x\in[0, S^{\star}]\}\}$ policy is an optimal joint control policy, where the impulse control follows the control band policy $(0,q^{\star},Q^{\star},S^{\star})$, that brings the inventory level up to $q^{\star}$ once it drops to $0$ and brings it down to $Q^{\star}$ once it rises to $S^{\star}$, and the drift rate only depends on the current inventory level and is given by function $μ^{\star}(x)$ for the inventory level $x\in[0,S^{\star}]$. The optimality of the $\{(0,q^{\star},Q^{\star},S^{\star}),\{μ^{\star}(x): x\in[0,S^{\star}]\}\}$ policy is proven by using a lower bound approach, in which a critical step is to prove the existence and uniqueness of optimal policy parameters. To prove the existence and uniqueness, we develop a novel analytical method to solve a free boundary problem consisting of an ordinary differential equation (ODE) and several free boundary conditions. Furthermore, we find that the optimal drift rate $μ^{\star}(x)$ is firstly increasing and then decreasing as $x$ increases from $0$ to $S^{\star}$ with a turnover point between $Q^{\star}$ and $S^{\star}$.