Abstract: The fuzzy job shop scheduling problem with makespan minimisation has received considerable attention over the last decade. Different sets of benchmark instances have been made available, and many metaheuristic solutions and corresponding upper bounds of the optimal makespan have been given for these instances in different publications. However, unlike the deterministic case, very little work has been invested in proposing and solving mathematical models for the fuzzy problem. This has resulted both in a lack of a good characterisation of the hardness of existing benchmark instances and in the absence of reliable lower and upper bounds for the makespan. In consequence, it is difficult, if not impossible to properly assess and compare new proposals of exact or approximate solving methods, thus hindering progress in this field. In this work we intend to fill this gap by proposing and solving two mathematical models, a mixed integer linear programming model and a constraint programming model. A thorough analysis on the scalability of solving these mathematical models with commercial solvers is carried out. A state-of-the-art metaheuristic algorithm from the literature is also used as reference point for a better understanding of the results. Using solvers of different nature allows us to improve known upper and lower bounds for all existing instances, and certify optimality for many of them for the first time. It also enables us to structurally characterise the instances? hardness beyond their size.

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