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

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