Augmented ɛ-constraint-based matheuristic methodology for Bi-objective production scheduling problems

Matheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi-objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ-constraint, which transforms multi-objective problems into single-objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ-constraint-based matheuristic methodology (ɛ-MH) is proposed to apply the idea of ɛ-constraint to embedded MILP models, so that Pareto fronts obtained by meta-heuristics can be further improved by solving a set of MILP models. Afterwards, four speed-up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ-MH. Finally, several real-world bi-objective scheduling problems are discussed to present potential applications for the proposed methodology.