A Technical Note on Drift Control with Changeover Costs

In this note we present a numerical example related to the problem addressed in ”Drift control with changeover costs”. We also provide more detailed versions of some of the proofs. First we present a description of the problem and the results. We consider the problem of managing capacity in a build-to-order environment like the personal computer manufacturing operations of Dell. We model this problem as a Brownian control problem and seek a policy that minimizes the long-run average cost. The workload of orders awaiting processing incurs a delay cost or waiting cost per unit per unit time that reflects the opportunity cost of revenue waiting to be recognized or the customer service impacts of delaying delivery of orders. On the other hand, maintaining capacity at a higher production rate also incurs a higher cost for capacity per unit time. 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. We model the cost of changing the processing rate from u to v as a fixed cost K(u, v) > 0 and assume the cost function K satisfies the usual triangle inequality so that changing the processing rate from u to v in a single step is no more expensive than accomplishing the same change via a series of intermediate steps. Even with changes in the processing rate, the workload of waiting orders can grow without limit. To ensure that delivery commitments can be met, we introduce additional boundary controls. In particular, we impose a fixed maximum level on the workload of waiting orders and exert instantaneous controls, e.g., rejecting orders, to keep from exceeding this limit. Similarly, the workload of waiting orders can drain to 0 and to ensure that the process remains non-negative, i.e., that the manufacturer does not build product in advance of orders, we impose instantaneous controls at the lower boundary corresponding roughly to idling lines briefly until additional orders arrive. In solving this problem a restricted version called the S-restricted Brownian control problem, in which the controller may change the drift rate only when the workload of waiting orders reaches a value in a given finite set S, is considered. Choosing S to be the set of non-negative integers up to the maximum workload for example, imposes no limitation on the system in practice, but allows us to formulate a linear program approximating the problem by considering only a special class of policies called control band policies. We show that an optimal solution to our structured linear program can be found among the special class of deterministic non-overlapping control band policies and that an optimal solution to the linear program actually solves the S-restricted Brownian control problem, i.e., a deterministic non-overlapping control band policy is optimal among all nonanticipating policies for the S-restricted problem. We further provide a lower bound on the average cost of any non-anticipating policy (i.e., any policy whose decisions at time t depend only on the