{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T13:03:24Z","timestamp":1780664604972,"version":"3.54.1"},"reference-count":57,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T00:00:00Z","timestamp":1754524800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61573264"],"award-info":[{"award-number":["61573264"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Batch processing machines (BPMs) are extensively present in high energy-consuming manufacturing processes such as casting, and they show some symmetries on adjacent batches and jobs within each batch. Preventive maintenance (PM) is very important for the stable running and energy saving of BPMs; however, PM in a parallel BPM shop is seldom studied. In this study, the energy-efficient parallel BPM scheduling problem with PM is considered and an imperialist competitive algorithm with three empires (TEICA) is presented to minimize makespan and total energy consumption. To obtain high-quality solutions, the number of empires is not used as a parameter and fixed at 3, a new way is applied to construct three initial empires, each of which has a new structure like two imperialists, a new assimilation is given, and an adaptive imperialist competition is implemented based on historical competition data. A number of computational experiments are conducted on 108 instances. The computational results show that the new strategies of TEICA are effective; TEICA can provide better results than all comparative methods on more than 90% instances of the considered BPM scheduling problem, and TEICA may be an effective way to solve other BPM scheduling problem.<\/jats:p>","DOI":"10.3390\/sym17081256","type":"journal-article","created":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T08:33:06Z","timestamp":1754555586000},"page":"1256","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Imperialist Competitive Algorithm with Three Empires for Energy-Efficient Parallel Batch Processing Machine Scheduling with Preventive Maintenance"],"prefix":"10.3390","volume":"17","author":[{"given":"Mingbo","family":"Li","sequence":"first","affiliation":[{"name":"School of Automation, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Deming","family":"Lei","sequence":"additional","affiliation":[{"name":"School of Automation, Wuhan University of Technology, Wuhan 430063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4091","DOI":"10.1080\/00207540410001704041","article-title":"Scheduling parallel batch processing machines with arbitrary job sizes and incompatible job families","volume":"42","author":"Koh","year":"2004","journal-title":"Int. J. Prod. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1080\/0951192X.2011.562545","article-title":"Weighted nested partitions based on differential evolution (WNPDE) algorithm-based scheduling of parallel batching processing machines (BPM) with incompatible families and dynamic lot arrival","volume":"24","author":"Su","year":"2011","journal-title":"Int. J. Comput. Integ. Manuf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105899","DOI":"10.1016\/j.cor.2022.105899","article-title":"Parallel batch processing machines scheduling in cloud manufacturing for minimizing total service completion time","volume":"146","author":"Zhang","year":"2022","journal-title":"Comput. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5131","DOI":"10.1080\/00207543.2021.1951446","article-title":"Scheduling parallel serial-batch processing machines with incompatible job families, sequence-dependent setup times and arbitrary sizes","volume":"60","author":"Gahm","year":"2022","journal-title":"Int. J. Prod. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.1109\/TR.2022.3201333","article-title":"Parallel batch processing machine scheduling under two-dimensional bin-packing constraints","volume":"72","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Reliab."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1016\/j.ejor.2023.04.019","article-title":"Parallel-batch scheduling with rejection: Structural properties and approximation algorithms","volume":"310","author":"Ou","year":"2023","journal-title":"Eur. J. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"106122","DOI":"10.1016\/j.cor.2022.106122","article-title":"Learning-augmented heuristics for scheduling parallel serial-batch processing machines","volume":"151","author":"Uzunoglu","year":"2023","journal-title":"Comput. Oper. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1080\/00207543.2023.2233626","article-title":"A hybrid discrete differential evolution-genetic algorithm approach with a new batch formation mechanism for parallel batch scheduling considering batch delivery","volume":"62","author":"Kucukkoc","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1287\/opre.1090.0788","article-title":"Coordinated multistage scheduling of parallel batch-processing machines under multiresource constraints","volume":"58","author":"Jula","year":"2010","journal-title":"Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/0951192X.2012.731612","article-title":"Heuristics to schedule uniform parallel batch processing machines with dynamic job arrivals","volume":"26","author":"Li","year":"2013","journal-title":"Int. J. Comput. Integ. Manuf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/j.ijpe.2013.04.053","article-title":"A bi-objective scheduling problem on batch machines via a Pareto-based ant colony system","volume":"145","author":"Xu","year":"2013","journal-title":"Int. J. Prod. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1680","DOI":"10.1080\/00207543.2014.952795","article-title":"Bi-objective optimisation for scheduling the identical parallel batch-processing machines with arbitrary job sizes, unequal job release times and capacity limits","volume":"53","author":"Abedi","year":"2015","journal-title":"Int. J. Prod. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijpe.2016.05.014","article-title":"An effective discrete differential evolution algorithm for scheduling uniform parallel batch processing machines with non-identical capacities and arbitrary job sizes","volume":"179","author":"Zhou","year":"2016","journal-title":"Int. J. Prod. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/j.asoc.2018.11.027","article-title":"Ant colony optimization algorithm for scheduling jobs with fuzzy processing times on parallel batch machines with different capacities","volume":"75","author":"Jia","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1080\/0305215X.2019.1613388","article-title":"Integrated scheduling on parallel batch processing machines with non-identical capacities","volume":"52","author":"Jia","year":"2020","journal-title":"Eng. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108115","DOI":"10.1016\/j.cie.2022.108115","article-title":"Scheduling identical parallel batch processing machines involving incompatible families with different job sizes and capacity constraints","volume":"169","author":"Li","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"117487","DOI":"10.1016\/j.eswa.2022.117487","article-title":"A bi-objective evolutionary algorithm scheduled on uniform parallel batch processing machines","volume":"204","author":"Li","year":"2022","journal-title":"Exp. Syst. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.cor.2018.12.002","article-title":"Coke production scheduling problem: A parallel machine scheduling with batch preprocessings and location-dependent processing times","volume":"104","author":"Liu","year":"2019","journal-title":"Comput. Oper. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.cie.2017.09.037","article-title":"Scheduling non-identical parallel batch processing machines to minimize total weighted tardiness using particle swarm optimization","volume":"113","author":"Hulett","year":"2017","journal-title":"Comput. Ind. Eng."},{"key":"ref_20","first-page":"12111","article-title":"Scheduling parallel-batching processing machines problem with learning and deterioration effect in fuzzy environment","volume":"40","author":"Wang","year":"2021","journal-title":"J. Intel. Fuzzy Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.knosys.2016.10.026","article-title":"A multi-objective artificial bee colony algorithm for parallel batch-processing machine scheduling in fabric dyeing processes","volume":"116","author":"Zhang","year":"2017","journal-title":"Knowl.-Based Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7704","DOI":"10.1080\/00207543.2024.2324452","article-title":"An adaptive shuffled frog-leaping algorithm for parallel batch processing machines scheduling with machine eligibility in fabric dyeing process","volume":"62","author":"Lei","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"112413","DOI":"10.1016\/j.asoc.2024.112413","article-title":"A dynamical teaching-learning-based optimization algorithm for fuzzy energy-efficient parallel batch processing machines scheduling in fabric dyeing process","volume":"167","author":"Wang","year":"2024","journal-title":"Appl. Soft Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.cor.2017.06.019","article-title":"Solving a bi-objective unrelated parallel batch processing machines scheduling problem: A comparison study","volume":"88","author":"Rastgar","year":"2017","journal-title":"Comput. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.asoc.2018.02.018","article-title":"A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity","volume":"66","author":"Lu","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_26","first-page":"254","article-title":"A random-keys genetic algorithm for scheduling unrelated parallel batch processing machines with different capacities and arbitrary job sizes","volume":"334","author":"Zhou","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_27","first-page":"21","article-title":"A bi-objective model for a scheduling problem of unrelated parallel batch processing machines with fuzzy parameters by two fuzzy multi-objective meta-heuristics","volume":"16","author":"Sadati","year":"2019","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_28","first-page":"1501","article-title":"Efficient algorithms to minimize makespan of the unrelated parallel batch-processing machines scheduling problem with unequal job ready times","volume":"55","author":"Zarook","year":"2021","journal-title":"PAIRO Oper. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"17063","DOI":"10.1007\/s00500-023-08754-0","article-title":"Unrelated parallel batch processing machine scheduling for production systems under carbon reduction policies: NSGA-II and MOGWO metaheuristics","volume":"27","author":"Fallahi","year":"2023","journal-title":"Soft Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6449","DOI":"10.1080\/00207543.2023.2252932","article-title":"A novel carbon reduction engineering method-based deep Q-learning algorithm for energy-efficient scheduling on a single batch-processing machine in semiconductor manufacturing","volume":"62","author":"Kong","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s10696-023-09488-9","article-title":"A tabu-based adaptive large neighborhood search for scheduling unrelated parallel batch processing machines with non-identical job sizes and dynamic job arrivals","volume":"36","author":"Xiao","year":"2024","journal-title":"Flex. Serv. Manuf. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"106474","DOI":"10.1016\/j.cor.2023.106474","article-title":"Unrelated parallel batch processing machine scheduling with time requirements and two-dimensional packing constraints","volume":"162","author":"Hu","year":"2024","journal-title":"Comput. Oper. Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.cie.2019.03.033","article-title":"Multi-objective energy-aware batch scheduling using ant colony optimization algorithm","volume":"131","author":"Jia","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"114145","DOI":"10.1016\/j.eswa.2020.114145","article-title":"A multi-objective co-evolutionary algorithm of scheduling on parallel non-identical batch machines","volume":"167","author":"Wang","year":"2021","journal-title":"Exp. Syst. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.cor.2018.04.009","article-title":"A multi-objective differential evolution algorithm for parallel batch processing machine scheduling considering electricity consumption cost","volume":"96","author":"Zhou","year":"2018","journal-title":"Comput. Oper. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1631","DOI":"10.1109\/TASE.2024.3368617","article-title":"Scheduling unrelated parallel batch processing machines under time-of-use electricity prices","volume":"22","author":"Tian","year":"2025","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"108608","DOI":"10.1016\/j.cie.2022.108608","article-title":"A bi-objective evolutionary algorithm for minimizing maximum lateness and total pollution cost on non-identical parallel batch processing machines","volume":"172","author":"Li","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2340002","DOI":"10.1142\/S0129626423400029","article-title":"A production plan considering parallel machines and deteriorating effects: Minimizing the makespan in the section of steel box girder processing","volume":"33","author":"Yang","year":"2023","journal-title":"Neural Process. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"108977","DOI":"10.1016\/j.cie.2023.108977","article-title":"Modified benders decomposition and metaheuristics for multi-machine parallel-batch scheduling and resource allocation under deterioration effect","volume":"176","author":"Jiang","year":"2023","journal-title":"Comput. Ind. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/TSM.2023.3326865","article-title":"Learning priority indices for energy-aware scheduling of jobs on batch processing machines","volume":"37","author":"Schorn","year":"2024","journal-title":"IEEE Trans. Semicond. Manuf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"4502","DOI":"10.1109\/TEM.2022.3182380","article-title":"A metaheuristic framework for energy-intensive industries with batch processing machines","volume":"71","author":"Abedi","year":"2024","journal-title":"IEEE Trans. Eng. Manag."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2056","DOI":"10.1109\/LRA.2025.3526571","article-title":"Deterioration-Aware Collaborative Energy-Efficient Batch Scheduling and Maintenance for Unrelated Parallel Machines Based on Improved MOEA\/D","volume":"10","author":"Wang","year":"2025","journal-title":"IEEE Rob. Autom. Lett."},{"key":"ref_43","unstructured":"Rudek, R. A fast local search for the identical parallel machine scheduling problem with the position-based deteriorating effect and maintenance, Int. Trans. Oper. Res., in press."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"125722","DOI":"10.1016\/j.eswa.2024.125722","article-title":"Scheduling identical parallel machines involving flexible maintenance activities","volume":"263","author":"Li","year":"2025","journal-title":"Exp. Syst. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"946","DOI":"10.1016\/j.jmsy.2024.11.004","article-title":"Generative deep reinforcement learning method for dynamic parallel machines scheduling with adaptive maintenance activities","volume":"77","author":"Wang","year":"2024","journal-title":"J. Manuf. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1016\/j.asoc.2014.08.024","article-title":"A survey on the imperialist competitive algorithm metaheuristic: Implementation in engineering domain and directions for future research","volume":"24","author":"Hosseini","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"110086","DOI":"10.1016\/j.cie.2024.110086","article-title":"Imperialist competitive algorithm for unrelated parallel machine scheduling with sequence-and-machine-dependent setups and compatibility and workload constraints","volume":"190","author":"Elyasi","year":"2024","journal-title":"Comput. Ind. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"108641","DOI":"10.1016\/j.cie.2022.108641","article-title":"A hybrid imperialist competitive algorithm for energy-efficient flexible job shop scheduling problem with variable-size sublots","volume":"172","author":"Li","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"104307","DOI":"10.1016\/j.engappai.2021.104307","article-title":"An imperialist competitive algorithm with feedback for energy-efficient flexible job shop scheduling with transportation and sequence-dependent setup times","volume":"103","author":"Li","year":"2021","journal-title":"Eng. Appl. Arti. Intel."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s40747-020-00193-w","article-title":"Discrete imperialist competitive algorithm for the resource-constrained hybrid flowshop problem with energy consumption","volume":"7","author":"Tao","year":"2021","journal-title":"Complex Intel. Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.swevo.2019.05.006","article-title":"Two-level imperialist competitive algorithm for energy-efficient hybrid flow shop scheduling problem with relative importance of objectives","volume":"49","author":"Li","year":"2019","journal-title":"Swarm Evolu. Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1240","DOI":"10.1080\/00207543.2013.848490","article-title":"An efficient imperialist competitive algorithm for scheduling in the two-stage assembly flow shop problem","volume":"52","author":"Seidgar","year":"2014","journal-title":"Int. J. Prod. Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multi-objective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE. Trans. Evolut. Comput."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1109\/4235.797969","article-title":"Multi objective evolutionary algorithms: A comparative case study and the strength Pareto approach","volume":"3","author":"Zitzler","year":"1999","journal-title":"IEEE. Trans. Evolut. Comput."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s00170-007-0945-8","article-title":"Pareto archive particle swarm optimization for multi-objective fuzzy job shop scheduling problems","volume":"37","author":"Lei","year":"2008","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1109\/TEVC.2003.810761","article-title":"The balance between proximity and diversity in multiobjective evolutionary algorithms","volume":"7","author":"Bosman","year":"2003","journal-title":"IEEE. Trans. Evolut. Comput."},{"key":"ref_57","unstructured":"Taguchi, G. (1986). Introduction to Quality Engineering, Asian Productivity Organization, Asian Productivity Organization."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1256\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:25:03Z","timestamp":1760034303000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1256"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,7]]},"references-count":57,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081256"],"URL":"https:\/\/doi.org\/10.3390\/sym17081256","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,7]]}}}