Mathematical modelling and a meta-heuristic for flexible job shop scheduling

Abstract

This study develops new solution methodologies for the flexible job shop scheduling problem (F-JSSP). As a first step towards dealing with this complex problem, mathematical modellings have been used; two novel effective position- and sequence-based mixed integer linear programming (MILP) models have been developed to fully characterise operations of the shop floor. The developed MILP models are capable of solving both partially and totally F-JSSPs. Size complexities, solution effectiveness and computational efficiencies of the developed MILPs are numerically explored and comprehensively compared vis-à-vis the makespan optimisation criterion. The acquired results demonstrate that the proposed MILPs, by virtue of its structural efficiencies, outperform the state-of-the-art MILPs in literature. The F-JSSP is strongly NP-hard; hence, it renders even the developed enhanced MILPs inefficient in generating schedules with the desired quality for industrial scale problems. Thus, a meta-heuristic that is a hybrid of Artificial Immune and Simulated Annealing (AISA) Algorithms has been proposed and developed for larger instances of the F-JSSP. Optimality gap is measured through comparison of AISA’s suboptimal solutions with its MILP exact optimal counterparts obtained for small- to medium-size benchmarks of F-JSSP. The AISA’s results were examined further by comparing them with seven of the best-performing meta-heuristics applied to the same benchmark. The performed comparative analysis demonstrated the superiority of the developed AISA algorithm. An industrial problem in a mould- and die-making shop was used for verification.

Keywords:

• scheduling
• flexible job shop
• mixed integer linear programming
• hybrid artificial immune algorithms
• simulated annealing
• size complexity
• optimality gap

Acknowledgements

This research was conducted in the Intelligent Manufacturing Systems Center at the University of Windsor, Canada. Research funding from the Canada Research Chairs program and the Natural Sciences and Engineering Council (NSERC) of Canada are gratefully acknowledged.