We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed cost and a proportional cost. The challenge is to find an adjustment policy that balances the holding cost and adjustment cost to minimize the long-run average cost. When both upward and downward fixed costs are positive, our problem is an impulse control problem. When both fixed costs are zero, our problem is a singular or instantaneous control problem. For the impulse control problem, we prove that a four-parameter control band policy is optimal among all feasible policies. For the singular control problem, we prove that a two-parameter control band policy is optimal.
We use a lower-bound approach, widely known as the “verification theorem”, to prove the optimality of a control band policy for both the impulse and singular control problems. Our major contribution is to prove the existence of a “smooth” solution to the free boundary problem under some mild assumptions on the holding cost function. The existence proof leads naturally to a numerical algorithm to compute the optimal control band parameters. We demonstrate that the lower-bound approach also works for a Brownian inventory model which prohibits an inventory backlog. A companion paper (“Brownian inventory models with convex holding cost, part 2: discount-optimal controls”, Stochastic Systems, Vol. 3, No. 2, pp. 500–573) explains how to adapt the lower-bound approach to study a Brownian inventory model under a discounted cost criterion.
"Brownian inventory models with convex holding cost, Part 1: Average-optimal controls." Stoch. Syst. 3 (2) 442 - 499, 2013. https://doi.org/10.1214/11-SSY041