Towards a Robust Scheduling on Unrelated Parallel Machines: A Scenarios Based Approach

Abstract

The article deals with scheduling on unrelated parallel machines under uncertainty of processing times. The performance measure is to minimize the makespan C _max when the splitting is tolerated. We show through an example that an optimal schedule of nominal data instance prior to its execution may be suboptimal due to minor or strong data uncertainty. Thus, we propose an artificial scenarios based approach to construct and identify a robust schedule to hedge against the uncertainty of processing times. To achieve this, we structure the uncertainty of processing time by mean of discrete scenarios, and evaluate the robustness according to the worst case strategy. We generate a family of artificial scenarios from the list of the potential realizations (processing time scenarios) to provide a family of feasible solutions using Lawler and Labetoulle’s linear formulation [10]. Then, we identify a robust solution by evaluating the maximal cost or the maximal regret of each solution when applied to the potential realizations. Extensive empirical tests have been carried out. We have determined that the artificial scenario of maximal processing times Smax and the artificial scenario average processing times Savg are able to generate the robust schedules regarding the potential realizations.