Fuzzy correlation entropy-based NSGA-II for energy-efficient hybrid flow-shop scheduling problem

Green scheduling of manufacturing industry with energy saving as the core has been paid more and more attention in academia and industry. As a classic scheduling problem, hybrid flow-shop scheduling problem (HFSP) has been receiving increasing research attention. However, most studies only focus on time-related metrics while neglecting energy consumption. In this article, we studied energy-efficient HFSP and assumed that machines can operate at different speeds. This is an assumption that has been rarely explored but can make the problem more relevant to real-world production. Moreover, the energy-efficient HFSP at a variable machine speed (EHFSP-VMS) was formulated as a multiobjective mathematical optimization model aiming to optimize make-span and total energy consumption simultaneously. As a landmark achievement in the field of multiobjective optimization, non-dominated sorting genetic algorithm-II (NSGA-II) is adopted and improved as the solver, which is called fuzzy correlation entropy (FCE)-based NSGA-II (FCENSGA-II). Firstly, FCE, a fusion of fuzzy mathematics and information theory, is used to describe the difference and the FCE-based crowding distance is proposed for the first time. Its time complexity is lower than the original crowding distance. In addition, a machine learning strategy, namely opposition-based learning (OBL), is used to learn from opposite regions of the search space and increase the exploratory ability of the algorithm and the diversity of solutions. Finally, a critical path knowledge-based energy saving strategy (ESC) is adopted to discover non-dominant solutions by reducing the speed of machines on non-critical paths. A large number of experiments are conducted to test the performance of FCENSGA-II. The results show that in all test instances, the average, best and worst values of the solution obtained by FCENSGA-II are better than the compared state-of-the-art algorithms, and even the worst values obtained by FCENSGA-II in 75% of test instances are better than the best values of compared algorithms, which strongly confirms that FCENSGA-II outperforms the compared state-of-the-art algorithms for solving EHFSP-VMS.