CONTROLS∗

We consider an inventory system in which inventory level fluctu-ates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex func-tion of the inventory level, which can be negative when there is a back-log. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a dis-counted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy