Unrelated parallel machine scheduling with cost constraint under a fuzzy environment

This article investigates an unrelated parallel machine scheduling problem in which jobs possess fuzzy processing and fuzzy release times, while each machine incurs a unit-time rental cost. The objective is to minimize the total weighted fuzzy completion time. To solve this problem, a reasonable total cost threshold is first established, and a fuzzy mixed-integer programming (FMIP) model is formulated. Subsequently, an improved fuzzy genetic algorithm (IFGA) is designed by integrating an elite genetic algorithm with variable neighbourhood search (VNS). The algorithm incorporates an elitism mechanism, a population perturbation operator, a local enhancement procedure, two infeasible solution repair strategies, and four effective neighbourhood structures. Experimental results demonstrate that IFGA generates solutions closest to the optimal ones on 270 small-scale numerical instances, with the maximum relative deviation within 5%. On 360 large-scale numerical instances, it significantly outperforms other metaheuristic algorithms, and its computational time remains stable within 90 seconds.