Combining two-stage stochastic programming and recoverable robustness to minimize the number of late jobs in the case of uncertain processing times

Abstract

Minimizing the number of late jobs on a single machine is a classic scheduling problem, which can be used to model the situation that from a set of potential customers, we have to select as many as possible whom we want to serve, while selling no to the other ones. This problem can be solved by Moore–Hodgson’s algorithm, provided that all data are deterministic. We consider a stochastic variant of this problem, where we assume that there is a small probability that the processing times differ from their standard values as a result of some kind of disturbance. When such a disturbance occurs, then we must apply some recovery action to make the solution feasible again. This leads us to the area of recoverable robustness, which handles this uncertainty by modeling each possible disturbance as a scenario; in each scenario, the initial solution must then be made feasible by applying a given, simple recovery algorithm to it. Since we cannot accept previously rejected customers, our only option is to reject customers that would have been served in the undisturbed case. Our problem therefore becomes to find a solution for the undisturbed case together with a feasible recovery to every possible disturbance. Our goal hereby is to maximize the expected number of served customers; we assume here that we know the probability that a given scenario occurs. In this respect, our problem falls outside the area of the ‘standard’ recoverable robustness, which contains the worst-case recovery cost as a component of the objective. Therefore, we consider our approach as a combination of two-stage stochastic programming and recoverable robustness. We show that this problem is mathcal{NP}-hard in the ordinary sense even if there is only one scenario, and we present some sufficient conditions that allow us to find a part of the optimal solution in polynomial time. We further evaluate several solution methods to find an optimal solution, among which are dynamic programming, branch-and-bound, and branch-and-price.