Minimax regret makespan scheduling with interval release dates on unrelated machines

In this paper, we present a study of the robust (non-deterministic) job scheduling problem with interval release dates on unrelated machines. Robust optimization is a tractable alternative to stochastic optimization suited for problems in which parameter distributions are undetermined due to insufficient data. Our problem formulation involves minimizing the maximum regret, defined as the worst-case deviation from an optimal makespan, with the assumption that each release date belongs to a well-defined interval. Mathematical analysis is accomplished to investigate the properties of the robust scheduling problem examining the inapproximability, determination of the upper bound of a regret function, and selection of feasible scenarios. In order to solve the robust scheduling problem, we propose an algorithm that implements a greedy strategy and schedules jobs using only the deterministic criterion (the makespan) instead of assessing the worst-case regret value. We compare the quality of our schedules with the solutions obtained by the properly adapted and tuned simulated annealing algorithm. Additionally, the applicability of our robust approach (robust framework) in the Scheduling-Location problem (deterministic problem in which the parameters are precisely defined), which combines the facility location problem and job scheduling problem, is demonstrated. The numerical evaluation shows that including the uncertainty and regret-averse (the minimax regret criterion) can allow, in certain cases, to achieve optimal solutions for incomplete dataset.