symmetry S S Article A Novel Multi-Population Artificial Bee Colony Algorithm for Energy-Efficient Hybrid Flow Shop Scheduling Problem Yandi Zuo 1 , Zhun Fan 2, *, Tierui Zou 3 and Pan Wang 1, *   Citation: Zuo, Y.; Fan, Z.; Zou, T.; Wang, P. A Novel Multi-Population Artificial Bee Colony Algorithm for Energy-Efficient Hybrid Flow Shop Scheduling Problem. Symmetry 2021, 13, 2421. https://doi.org/10.3390/ sym13122421 Academic Editors: Ming Li and Deming Lei Received: 17 November 2021 Accepted: 6 December 2021 Published: 14 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 School of Automation, Wuhan University of Technology, Wuhan 430062, China; yandizuo@whut.edu.cn 2 Department of Electronic and Information Engineering, Shantou University, Shantou 515063, China 3 Department of Electrical & Computer Engineering, University of Florida, Gainesville, FL 32611, USA; tieruizou@ufl.edu * Correspondence: zfan@stu.edu.cn (Z.F.); wangpan@whut.edu.cn (P.W.) Abstract: Considering green scheduling and sustainable manufacturing, the energy-efficient hybrid flow shop scheduling problem (EHFSP) with a variable speed constraint is investigated, and a novel multi-population artificial bee colony algorithm (MPABC) is developed to minimize makespan, total tardiness and total energy consumption (TEC), simultaneously. It is necessary for manufacturers to fully understand the notion of symmetry in balancing economic and environmental indicators. To improve the search efficiency, the population was randomly categorized into a number of subpopu- lations, then several groups were constructed based on the quality of subpopulations. A different search strategy was executed in each group to maintain the population diversity. The historical optimization data were also used to enhance the quality of solutions. Finally, extensive experiments were conducted. The results demonstrate that MPABC can achieve an outstanding performance on three metrics DI R , c and nd for the considered EHFSP. Keywords: energy-efficient; hybrid flow shop scheduling; artificial bee colony; multi-population 1. Introduction Shop scheduling is an essential subject and an effective way to improve resource utilization [1]. With the increase of energy demand and the emergence of various envi- ronmental issues, reducing energy consumption has become particularly urgent for the manufacturing industry. Many scholars devote themselves to the research of energy- efficient scheduling problems, related to single machine [2], unrelated parallel machines [3], job shop [4], permutation flow shop [5] and hybrid flow shop [6]. In the typical flow shop scheduling problem, sequencing jobs are processed in prede- termined orders, and each stage contains only one machine, which has been proved to be NP-hard [7]. Regarding the energy-efficient flow shop scheduling problem, Ding et al. [8] investigated a carbon-efficient scheduling of flow shops. Foumani et al. [9] considered the impact of various carbon reduction policies. Mansouri et al. [10] addressed a two-machine green flow shop scheduling. Jiang et al. [11] studied an energy-efficient permutation flow shop scheduling problem. As an extension of flow shop, hybrid flow shop employs multiple parallel machines to enlarge the capacity of stage and eliminate the restriction of the bottleneck stage [12]. The hybrid flow shop scheduling problem (HFSP) extensively exists in various real industrial scenarios, such as steel [13], textile [14], glass [15], paper [16] and electronics [17]. EHFSP often consists of green constraints, green objectives and three sub-problems including job permutation, machine assignment and speed selection, which is more com- plicated than energy-efficient flow shop scheduling problems and apparently NP-hard. In recent years, EHFSP has attracted much attention from researchers and manufacturers. Bruzzone et al. [18] utilized a heuristic method to obtain an energy-aware scheduling in a sustainable manufacturing process. Dai et al. [19] designed a novel genetic-simulated Symmetry 2021, 13, 2421. https://doi.org/10.3390/sym13122421 https://www.mdpi.com/journal/symmetry