We consider a continuous-review inventory system in which the setup cost of each order is a general function of the order quantity and the demand process is modeled as a Brownian motion with a positive drift. Assuming the holding and shortage cost to be a convex function of the inventory level, we obtain the optimal ordering policy that minimizes the long-run average cost by a lower bound approach. To tackle some technical issues in the lower bound approach under the quantity-dependent setup cost assumption, we establish a comparison theorem that enables one to prove the global optimality of a policy by examining a tractable subset of admissible policies. Since the smooth pasting technique does not apply to our Brownian inventory model, we also propose a selection procedure for computing the optimal policy parameters when the setup cost is a step function.

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: discountoptimal 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. 1. Introduction. This paper concerns the optimal control of Brownian inventory models under a long-run average cost criterion. It serves two purposes. First, it provides a tutorial on the powerful lower-bound approach, known as the "the verification theorem", for proving the optimality of a control band policy among all feasible policies. The tutorial is rigorous and self-contained with the exception of the standard Itô's formula. Second, it contributes to the literature by proving the existence of a "smooth" solution to the free boundary problem with a general convex holding cost function. The existence proof leads naturally to a numerical algorithm to compute the optimal control band parameters. The companion paper [33] studies the optimal control of Brownian inventory models under a discounted cost criterion.

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: discountoptimal controls", Stochastic Systems, Vol. 3,No. 2, explains how to adapt the lower-bound approach to study a Brownian inventory model under a discounted cost criterion.