  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.
