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 a long-run average cost criterion. Assuming the inventory level must be nonnegative, we prove that a f(0; q*;Q*; S*); {mu*(x) : x epsilon [0; S*]gg policy is an optimal joint control policy, where the impulse control follows the control band policy (0; q*;Q*; S*), which brings the inventory level up to q? once it drops to 0 and brings it down to Q* once it rises to S*, and the drift rate only depends on the current inventory level and is given by function mu*(x) for the inventory level x 2 [0; S*]. The optimality of the f(0; q*;Q*; S*); {mu*(x) : x 2 [0; S*]gg 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 and several free boundary conditions. Furthermore, we find that the optimal drift rate mu*(x) is first increasing and then decreasing as x increases from 0 to S* with a turnover point between Q* and S*.