A two-scale scheme for finite horizon switching problems with delays

Abstract

We consider the problem where an operator of n production units acts by turning off and on the units to track a stochastic demand. We investigate the situation when the running cost depends on the time that each unit has been in operation, covering, for example, the case when there are delays in the control response. In this setting, standard methods for numerical solution quickly become intractable as n increases. To resolve this we propose a numerical scheme where delay states are discretized with a different step size than that of the time line. We first show that the value function converges to the optimum as the step size goes to zero. We then show, by a counter-example, that the corresponding control does not necessarily converge to an optimal control for the original problem and propose a perturbation that resolves this issue.