Robust preemptive scheduling on unrelated parallel machines under uncertain processing times

In this paper, we focus on the makespan minimization on restricted preemptive unrelated parallel machines scheduling under uncertain processing times. We propose to investigate the problem under discrete scenario representation of uncertain processing times and to use min-max objective to generate a schedule minimizing the worst-case makespan over all the scenarios. We show that the robust assignment counterpart of the deterministic formulation provides a lower bound of the worst-case makespan and we propose a Mixed Integer Linear Program that computes the robust schedule minimizing the worst-case makespan under the set of discrete scenarios with restricted number of preemptions. Moreover, we prove that this robust schedule is equivalent to the optimal schedule of the artificial worst-case scenario. Thus, it is too conservative. To reduce the conservativeness, we propose an improved robust formulation under which the sequence is scenario dependent.