{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T14:44:05Z","timestamp":1782398645054,"version":"3.54.5"},"reference-count":48,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T00:00:00Z","timestamp":1754956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52305533"],"award-info":[{"award-number":["52305533"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022ZDZX0003"],"award-info":[{"award-number":["2022ZDZX0003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023YFB3307900"],"award-info":[{"award-number":["2023YFB3307900"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Sichuan Province of China","award":["52305533"],"award-info":[{"award-number":["52305533"]}]},{"name":"Sichuan Province of China","award":["2022ZDZX0003"],"award-info":[{"award-number":["2022ZDZX0003"]}]},{"name":"Sichuan Province of China","award":["2023YFB3307900"],"award-info":[{"award-number":["2023YFB3307900"]}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["52305533"],"award-info":[{"award-number":["52305533"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2022ZDZX0003"],"award-info":[{"award-number":["2022ZDZX0003"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2023YFB3307900"],"award-info":[{"award-number":["2023YFB3307900"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study tackles scheduling challenges in multi-product assembly within distributed manufacturing, where components are produced simultaneously at dedicated factories (single capacity per site) and assembled centrally upon completion. To minimize makespan and maximum tardiness, we design a symmetry-exploiting enhanced Non-dominated Sorting Genetic Algorithm II (NSGA-II) integrated with Q-learning. Our approach systematically explores the solution space using dual symmetric variable neighborhood search (VNS) strategies and two novel crossover operators that enhance solution-space symmetry and genetic diversity. An \u03b5-greedy policy leveraging maximum Q-values guides the symmetry-aware search toward optimality while enabling strategic exploration. We validate an MILP model (Gurobi-implemented) and present our symmetry-refined algorithm against six heuristics. Multi-scale experiments confirm superiority, with Friedman tests demonstrating statistically significant gains over benchmarks, providing actionable insights for efficient distributed manufacturing scheduling.<\/jats:p>","DOI":"10.3390\/sym17081306","type":"journal-article","created":{"date-parts":[[2025,8,12]],"date-time":"2025-08-12T15:51:02Z","timestamp":1755013862000},"page":"1306","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An Improved NSGA-II for Three-Stage Distributed Heterogeneous Hybrid Flowshop Scheduling with Flexible Assembly and Discrete Transportation"],"prefix":"10.3390","volume":"17","author":[{"given":"Zhiyuan","family":"Shi","sequence":"first","affiliation":[{"name":"Dongfang Electric Academy of Science and Technology Co., Ltd., Chengdu 610063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haojie","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fuqian","family":"Yan","sequence":"additional","affiliation":[{"name":"Dongfang Electric Academy of Science and Technology Co., Ltd., Chengdu 610063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xutao","family":"Deng","sequence":"additional","affiliation":[{"name":"Dongfang Electric Academy of Science and Technology Co., Ltd., Chengdu 610063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haiqiang","family":"Hao","sequence":"additional","affiliation":[{"name":"School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jialei","family":"Zhang","sequence":"additional","affiliation":[{"name":"Dongfang Electric Academy of Science and Technology Co., Ltd., Chengdu 610063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qingwen","family":"Yin","sequence":"additional","affiliation":[{"name":"Dongfang Electric Academy of Science and Technology Co., Ltd., Chengdu 610063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s00170-013-5027-5","article-title":"Training and assignment of multi-skilled workers for implementing seru production systems","volume":"69","author":"Liu","year":"2013","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"96115","DOI":"10.1109\/ACCESS.2020.2996305","article-title":"Memetic Algorithm with Meta-Lamarckian Learning and Simplex Search for Distributed Flexible Assembly Permutation Flowshop Scheduling Problem","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108830","DOI":"10.1016\/j.cie.2022.108830","article-title":"Research on assembly scheduling problem with nested operations","volume":"175","author":"Hao","year":"2023","journal-title":"Comput. Ind. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s12293-018-00278-7","article-title":"An improved differential evolution algorithm for solving a distributed assembly flexible job shop scheduling problem","volume":"11","author":"Wu","year":"2019","journal-title":"Memetic Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1080\/00207543.2018.1442948","article-title":"Scheduling in production, supply chain and Industry 4.0 systems by optimal control: Fundamentals, state-of-the-art and applications","volume":"57","author":"Dolgui","year":"2019","journal-title":"Int. J. Prod. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"116484","DOI":"10.1016\/j.eswa.2021.116484","article-title":"A matrix cube-based estimation of distribution algorithm for the energy-efficient distributed assembly permutation flow-shop scheduling problem","volume":"194","author":"Zhang","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2018","DOI":"10.1108\/K-11-2021-1112","article-title":"Distributed assembly permutation flow-shop scheduling problem with non-identical factories and considering budget constraints","volume":"52","author":"Hosseini","year":"2022","journal-title":"Kybernetes"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zheng, J., and Wang, Y. (2021). A Hybrid Bat Algorithm for Solving the Three-Stage Distributed Assembly Permutation Flowshop Scheduling Problem. Appl. Sci., 11.","DOI":"10.3390\/app112110102"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4053","DOI":"10.1080\/00207543.2020.1757174","article-title":"The distributed assembly permutation flowshop scheduling problem with flexible assembly and batch delivery","volume":"59","author":"Yang","year":"2021","journal-title":"Int. J. Prod. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5292","DOI":"10.1080\/00207543.2013.807955","article-title":"The Distributed Assembly Permutation Flowshop Scheduling Problem","volume":"51","author":"Hatami","year":"2013","journal-title":"Int. J. Prod. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"13","DOI":"10.4995\/ijpme.2015.3345","article-title":"The Distributed Assembly Parallel Machine Scheduling Problem with eligibility constraints","volume":"3","author":"Hatami","year":"2015","journal-title":"Int. J. Prod. Manag. Eng."},{"key":"ref_12","first-page":"1829","article-title":"Adaptive hybrid estimation of distribution algorithm for solving a certain kind of three-stage assembly flowshop scheduling problem","volume":"21","author":"Li","year":"2015","journal-title":"Comput. Integr. Manuf. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1080\/00207543.2014.962112","article-title":"Scheduling algorithms for remanufacturing systems with parallel flow-shop-type reprocessing lines","volume":"53","author":"Kim","year":"2015","journal-title":"Int. J. Prod. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"101128","DOI":"10.1016\/j.swevo.2022.101128","article-title":"A two-phase evolutionary algorithm for multi-objective distributed assembly permutation flowshop scheduling problem","volume":"74","author":"Huang","year":"2022","journal-title":"Swarm Evol. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2018","DOI":"10.1080\/0305215X.2023.2296538","article-title":"Integrated remanufacturing scheduling of disassembly, reprocessing and reassembly considering energy efficiency and stochasticity through group teaching optimization and simulation approaches","volume":"56","author":"Fu","year":"2024","journal-title":"Eng. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.simpat.2017.09.001","article-title":"A biased-randomized simheuristic for the distributed assembly permutation flowshop problem with stochastic processing times","volume":"79","author":"Ferone","year":"2017","journal-title":"Simul. Model. Pract. Theory"},{"key":"ref_17","first-page":"1705","article-title":"Integrated optimization of production planning and scheduling in uncertain re-entrance environment for fixed-position assembly workshops","volume":"42","author":"Jiang","year":"2022","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4713","DOI":"10.1080\/00207540600621029","article-title":"Evolutionary heuristics and an algorithm for the two-stage assembly scheduling problem to minimize makespan with setup times","volume":"44","author":"Allahverdi","year":"2006","journal-title":"Int. J. Prod. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"106223","DOI":"10.1016\/j.cie.2019.106223","article-title":"New efficient constructive heuristics for the two-stage multi-machine assembly scheduling problem","volume":"140","author":"Talens","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"125288","DOI":"10.1016\/j.eswa.2024.125288","article-title":"A multi-objective Immune Balancing Algorithm for Distributed Heterogeneous Batching-integrated Assembly Hybrid Flowshop Scheduling","volume":"259","author":"Hao","year":"2025","journal-title":"Expert Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1109\/TASE.2013.2274517","article-title":"Multiobjective Flexible Job Shop Scheduling Using Memetic Algorithms","volume":"12","author":"Yuan","year":"2015","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"122995","DOI":"10.1109\/ACCESS.2021.3110242","article-title":"Dynamic Jobshop Scheduling Algorithm Based on Deep Q Network","volume":"9","author":"Zhao","year":"2021","journal-title":"IEEE Access"},{"key":"ref_23","first-page":"979","article-title":"A comprehensive study on integrated optimization of flexible manufacturing system layout and scheduling for nylon components production","volume":"29","author":"Zhu","year":"2022","journal-title":"Int. J. Ind. Eng. Theory Appl. Pract."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7684","DOI":"10.1109\/LRA.2022.3184795","article-title":"Multi-Agent Reinforcement Learning for Real-Time Dynamic Production Scheduling in a Robot Assembly Cell","volume":"7","author":"Johnson","year":"2022","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Zhou, T., Luo, L., He, Y., Fan, Z., and Ji, S. (2023). Solving Panel Block Assembly Line Scheduling Problem via a Novel Deep Reinforcement Learning Approach. Appl. Sci., 13.","DOI":"10.3390\/app13148483"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.cie.2018.07.023","article-title":"Multi objective two-stage assembly flow shop with release time","volume":"124","author":"Sheikh","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Li, X., Chehade, H., Yalaoui, F., and Amodeo, L. (2011, January 18\u201322). A new method coupling simulation and a hybrid metaheuristic to solve a multiobjective hybrid flowshop scheduling problem. Proceedings of the EUSFLAT Conference, Aix-les-Bains, France.","DOI":"10.2991\/eusflat.2011.33"},{"key":"ref_28","unstructured":"Campos, S.C., and Arroyo, J.E.C. (2014, January 12\u201316). NSGA-II with iterated greedy for a bi-objective three-stage assembly flowshop scheduling problem. Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, Vancouver, BC, Canada."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Rahimi, I., Gandomi, A.H., Deb, K., Chen, F., and Nikoo, M.R. (2022). Scheduling by NSGA-II: Review and Bibliometric Analysis. Processes, 10.","DOI":"10.3390\/pr10010098"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.cirp.2024.04.010","article-title":"Bi-objective scheduling for energy-efficient distributed assembly blocking flow shop","volume":"73","author":"Du","year":"2024","journal-title":"CIRP Ann."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1080\/00207543.2022.2031331","article-title":"A novel shuffled frog-leaping algorithm with reinforcement learning for distributed assembly hybrid flow shop scheduling","volume":"61","author":"Cai","year":"2023","journal-title":"Int. J. Prod. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"102786","DOI":"10.1016\/j.rcim.2024.102786","article-title":"Digital twin-driven dynamic scheduling for the assembly workshop of complex products with workers allocation","volume":"89","author":"Gao","year":"2024","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1080\/00207543.2014.933273","article-title":"A Pareto block-based estimation and distribution algorithm for multi-objective permutation flow shop scheduling problem","volume":"53","author":"Tiwari","year":"2015","journal-title":"Int. J. Prod. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"68686","DOI":"10.1109\/ACCESS.2018.2879600","article-title":"Multi-Objective Parallel Variable Neighborhood Search for Energy Consumption Scheduling in Blocking Flow Shops","volume":"6","author":"Wang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.jmsy.2021.08.003","article-title":"An integrated job shop scheduling and assembly sequence planning approach for discrete manufacturing","volume":"61","author":"Wang","year":"2021","journal-title":"J. Manuf. Syst."},{"key":"ref_36","first-page":"256","article-title":"A fuzzy mathematical model for multi-objective flexible job-shop scheduling problem with new job insertion and earliness\/tardiness penalty","volume":"28","author":"Seyyedi","year":"2021","journal-title":"Int. J. Ind. Eng.-Theory Appl. Pract."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"107489","DOI":"10.1016\/j.cie.2021.107489","article-title":"Dynamic multi-objective scheduling for flexible job shop by deep reinforcement learning","volume":"159","author":"Luo","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3020","DOI":"10.1109\/TASE.2021.3104716","article-title":"Real-Time Scheduling for Dynamic Partial-No-Wait Multiobjective Flexible Job Shop by Deep Reinforcement Learning","volume":"19","author":"Luo","year":"2021","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4292","DOI":"10.1109\/TCYB.2022.3165074","article-title":"Multistep Multiagent Reinforcement Learning for Optimal Energy Schedule Strategy of Charging Stations in Smart Grid","volume":"53","author":"Zhang","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"107082","DOI":"10.1016\/j.cie.2020.107082","article-title":"A cooperative water wave optimization algorithm with reinforcement learning for the distributed assembly no-idle flowshop scheduling problem","volume":"153","author":"Zhao","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"108371","DOI":"10.1016\/j.asoc.2021.108371","article-title":"An adaptive artificial bee colony with reinforcement learning for distributed three-stage assembly scheduling with maintenance","volume":"117","author":"Wang","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"110427","DOI":"10.1016\/j.cie.2024.110427","article-title":"A self-learning particle swarm optimization for bi-level assembly scheduling of material-sensitive orders","volume":"195","author":"Hao","year":"2024","journal-title":"Comput. Ind. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"608","DOI":"10.14716\/ijtech.v11i3.3555","article-title":"A Batch Scheduling Model for a Three-stage Hybrid Flowshop Producing Products with Hierarchical Assembly Structures","volume":"11","author":"Maulidya","year":"2020","journal-title":"Int. J. Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"108385","DOI":"10.1016\/j.cie.2022.108385","article-title":"A multi-objective decomposition evolutionary algorithm based on the double-faced mirror boundary for a milk-run material feeding scheduling optimization problem","volume":"171","author":"Zhou","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.cor.2014.02.005","article-title":"Minimizing the total completion time in a distributed two stage assembly system with setup times","volume":"47","author":"Xiong","year":"2014","journal-title":"Comput. Oper. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.ijpe.2015.07.027","article-title":"Heuristics and metaheuristics for the distributed assembly permutation flowshop scheduling problem with sequence dependent setup times","volume":"169","author":"Hatami","year":"2015","journal-title":"Int. J. Prod. Econ."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"108126","DOI":"10.1016\/j.cie.2022.108126","article-title":"A cooperative memetic algorithm with feedback for the energy-aware distributed flow-shops with flexible assembly scheduling","volume":"168","author":"Wang","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"106432","DOI":"10.1016\/j.cor.2023.106432","article-title":"Automatic design of constructive heuristics for a reconfigurable distributed flowshop group scheduling problem","volume":"161","author":"Zhang","year":"2024","journal-title":"Comput. Oper. Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1306\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:25:55Z","timestamp":1760034355000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1306"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,12]]},"references-count":48,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081306"],"URL":"https:\/\/doi.org\/10.3390\/sym17081306","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,12]]}}}