Solving the drift control problem

Abstract

We model the problem of managing capacity in a build-to-order environment as a Brownian drift control problem. We formulate a structured linear program that models a practical discretization of the problem and exploit a strong relationship between relative value functions and dual solutions to develop a functional lower bound for the continuous problem from a dual solution to the discrete problem. Refining the discretization proves a functional strong duality for the continuous problem. The linear programming formulation is so badly scaled, however, that solving it is beyond the capabilities of standard solvers. By demonstrating the equivalence between strongly feasible bases and deterministic unichain policies, we combinatorialize the pivoting process and by exploiting the relationship between dual solutions and relative value functions, develop a mechanism for solving the LP without ever computing its coefficients. Finally, we exploit the relationship between relative value functions and dual solutions to develop a scheme analogous to column generation for refining the discretization so as to drive the gap between the discrete approximation and the continuous problem to zero quickly while keeping the LP small. Computational studies show our scheme is much faster than simply solving a regular discretization of the problem both in terms of finding a policy with a low average cost and in terms of providing a lower bound on the optimal average cost.