We investigate the efficiency of a discretization procedure utilizing a time-indexed mathematical optimization model for finding accurate solutions to flexible job shop scheduling problems considering objectives comprising the makespan and the tardiness of jobs, respectively. The time-indexed model is used to find solutions to these problems by iteratively employing time steps of decreasing length. The solutions and computation times are compared with results from a known benchmark formulation and an alternative model, which is a modified and slightly enhanced version of the benchmark formulation. The proposed method finds significantly better solutions for the largest instances within the same time frame considering both objectives as compared to the other models, although there is a large difference in the performance of the models depending on which objective is considered. This implies that the evaluation of scheduling algorithms must be performed with respect to an objective that is suitable for the real application for which they are intended. The minimization of the makespan is no such objective, although it is the most widely used objective in research. We propose an objective incorporating tardiness. The iterative procedure for solving the time-indexed model outperforms the other models regarding the time to find the best feasible solution. We conclude that our iterative procedure with the time-indexed model is competitive in comparison with state-of-the-art mathematical optimization models. Since the proposed procedure quickly finds solutions of good quality to large instances, our findings imply that the new procedure is beneficially utilized for scheduling real flexible job shops.
Dept of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-421 96 Göteborg, Sweden