Impulse Control of Brownian Motion: The Constrained Average Cost Case

When a manufacturer places repeated orders with a supplier to meet changing production requirements, he faces the challenge of finding the right balance between holding costs and the operational costs involved in adjusting the shipment sizes. We consider an inventory whose content fluctuates as a Brownian motion in the absence of control. At any moment, a controller can adjust the inventory level by any positive or negative quantity, but incurs both a fixed cost and a cost proportional to the magnitude of the adjustment. The inventory level must be nonnegative at all times and continuously incurs a linear holding cost. The objective is to minimize long-run average cost. We show that control band policies are optimal for the average cost Brownian control problem and explicitly calculate the parameters of the optimal control band policy. This form of policy is described by three parameters {q,Q,S}, 0 <q ≤ Q <S. When the inventory falls to zero (rises to S), the controller expedites (curtails) shipments to return it to q (Q). Employing apparently new techniques based on methods of Lagrangian relaxation, we show that this type of policy is optimal even with constraints on the size of adjustments and on the maximum inventory level. We also extend these results to the discounted cost problem.