We consider an inventory system in which inventory level fluctuates as aBrownian motion in the absence of control. The inventory continuouslyaccumulates cost at a rate that is a general convex function of the inventorylevel, which can be negative when there is a backlog. At any time, theinventory level can be adjusted by a positive or negative amount, which incursa fixed positive cost and a proportional cost. The challenge is to find anadjustment policy that balances the inventory cost and adjustment cost tominimize the expected total discounted cost. We provide a tutorial on using athree-step lower-bound approach to solving the optimal control problem under adiscounted cost criterion. In addition, we prove that a four-parameter controlband policy is optimal among all feasible policies. A key step is theconstructive proof of the existence of a unique solution to the free boundaryproblem. The proof leads naturally to an algorithm to compute the fourparameters of the optimal control band policy.
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