Managing Capacity by Drift Control

We model the problem of managing capacity in a build-to-order environment as a Brownian drift control problem and seek a policy that minimizes the long-term average cost. We assume the controller can, at some cost, shift the processing rate among a finite set of alternatives by, for example, adding or removing staff, increasing or reducing the number of shifts or opening or closing production lines. The controller incurs a cost for capacity per unit time and a delay cost that reflects the opportunity cost of revenue waiting to be recognized or the customer service impacts of delaying delivery of orders. Furthermore he incurs a cost per unit to reject orders or idle resources as necessary to keep the workload of waiting orders within a prescribed range. We introduce a practical restriction on this problem, called the $Ss$-restricted Brownian control problem, and show how to model it via a structured linear program. We demonstrate that an optimal solution to the $Ss$-restricted problem can be found among a special class of policies called deterministic non-overlapping control band policies. These results exploit apparently new relationships between complementary dual solutions and relative value functions that allow us to obtain a lower bound on the average cost of any non-anticipating policy for the problem even without the $Ss$ restriction. Under mild assumptions on the cost parameters, we show that our linear programming approach is asymptotically optimal for the unrestricted Brownian control problem in the sense that by appropriately selecting the $Ss$-restricted problem, we can ensure its solution is within an arbitrary finite tolerance of a lower bound on the average cost of any non-anticipating policy for the unrestricted Brownian control problem.