Scheduling in a flexible flow shop with unrelated parallel machines and machine-dependent process stages: Trade-off between Makespan and production costs

This paper aims to tackle bi-objective scheduling problem in a flexible flow shop containing unrelated parallel machines in the first stage. Due to the different technology levels of the parallel machines, their process speeds and production costs vary to each other. Therefore, minimizing the maximum completion time (Makespan) and the total production cost are considered as two objective functions. In addition, setup times are considered sequence-dependent and this system considers machine-dependent process steps and the process steps of an order depend on the assigned machine in the first stage. First, the problem is described and formulated as a bi-objective mathematical model. Since the problem is known to be strongly NP-hard, an approximate solution method is introduced based on the Non-dominated Sorting Genetic Algorithm (NSGA-II) to provide proper solutions for decision makers. The performance of the proposed solution method is investigated in comparison to another powerful multi-objective algorithm (SPEA 2) by solving different test problems. The computational results using various metrics such as Error Ratio (ER) and Generational Distance (GD) show the effectiveness of the proposed method in terms of optimality. The other indices such as Spacing (S), Diversification (D), and Mean Ideal Distance (MID) emphasize the superiority of the proposed algorithm compared to the rival algorithm in solving medium and large instances. In addition, supplementary analysis provided proper trade-off between two objectives for managers to select the best solution based on their preferences.