Fuzzy bi-objective formulation for a parallel machine scheduling problem with machine eligibility restrictions and sequence-dependent setup times

Abstract

In this paper, a fuzzy bi-objective mixed-integer linear programming (FBOMILP) model is presented. FBOMILP encompasses the minimisation workload imbalance and total tardiness simultaneously as a bi-objective formulation for an unrelated parallel machine scheduling problem. To make the proposed model more practical, sequence-dependent setup times, machine eligibility restrictions and release dates are also considered. Moreover, the inherent uncertainty of processing times, release dates, setup times and due dates are taken into account and modelled by fuzzy numbers. In order to solve the model for small-scale problems, a two-stage fuzzy approach is proposed. Nevertheless, since the problem belongs to the class of NP-hard problems, the proposed model is solved by two meta-heuristic algorithms, namely fuzzy multi-objective particle swarm optimisation (FMOPSO) and fuzzy non-dominated sorting genetic algorithm (FNSGA-II) for solving large-scale instances. Subsequently, through setting up various numerical examples, the performances of the two mentioned algorithms are compared. When α = 0.5 (α is a level of risk-taking and when it increases the decision-maker’s risk-taking decreases), FNSGA-II is fairly more effective than FMOPSO and has better performance especially in solving large-sized problems. However, when α rises, it can be stated that FMOPSO moderately becomes more appropriate. Finally, directions for future studies are suggested and conclusion remarks are drawn.