{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:07Z","timestamp":1760060107879,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T00:00:00Z","timestamp":1754352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["72171054","51805152","2023J06015","XJ2024001902"],"award-info":[{"award-number":["72171054","51805152","2023J06015","XJ2024001902"]}]},{"name":"Natural Science Foundation of Fujian","award":["72171054","51805152","2023J06015","XJ2024001902"],"award-info":[{"award-number":["72171054","51805152","2023J06015","XJ2024001902"]}]},{"name":"Doctoral Research Start-up Foundation of Hubei University of Technology","award":["72171054","51805152","2023J06015","XJ2024001902"],"award-info":[{"award-number":["72171054","51805152","2023J06015","XJ2024001902"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In lace textile manufacturing, the dyeing process in parallel machine environments faces challenges from sequence-dependent setup times due to color family transitions, machine eligibility constraints based on weight capacities, and probabilistic re-dyeing operations arising from quality inspection failures, which often lead to increased tardiness. To tackle this multi-constrained problem, a stochastic integer programming model is formulated to minimize total estimated tardiness. A novel symmetry-driven two-population collaborative differential evolution (TCDE) algorithm is then proposed. It features two symmetrically complementary subpopulations that achieve a balance between global exploration and local exploitation. One subpopulation employs chaotic parameter adaptation through a logistic map for symmetrically enhanced exploration, while the other adjusts parameters based on population diversity and convergence speed to facilitate symmetry-aware exploitation. Moreover, it also incorporates a symmetrical collaborative mechanism that includes the periodic migration of top individuals between subpopulations, along with elite-set guidance, to enhance both population diversity and convergence efficiency. Extensive computational experiments were conducted on 21 small-scale (optimally validated via CVX) and 15 large-scale synthetic datasets, as well as 21 small-scale (similarly validated) and 20 large-scale industrial datasets. These experiments demonstrate that TCDE significantly outperforms state-of-the-art comparative methods. Ablation studies also further verify the critical role of its symmetry-based components, with computational results confirming its superiority in solving the considered problem.<\/jats:p>","DOI":"10.3390\/sym17081243","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T10:50:21Z","timestamp":1754391021000},"page":"1243","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Symmetry-Driven Two-Population Collaborative Differential Evolution for Parallel Machine Scheduling in Lace Dyeing with Probabilistic Re-Dyeing Operations"],"prefix":"10.3390","volume":"17","author":[{"given":"Jing","family":"Wang","sequence":"first","affiliation":[{"name":"Hubei Yangtze River Shipping Development Research Center, Wuhan 430014, China"},{"name":"Hubei Key Laboratory of Modern Manufacturing and Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China"}]},{"given":"Jingsheng","family":"Lian","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Modern Manufacturing and Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China"}]},{"given":"Youpeng","family":"Deng","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Modern Manufacturing and Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China"}]},{"given":"Lang","family":"Pan","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Modern Manufacturing and Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3446-659X","authenticated-orcid":false,"given":"Huan","family":"Xue","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Modern Manufacturing and Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China"}]},{"given":"Yanming","family":"Chen","sequence":"additional","affiliation":[{"name":"Hubei Standardization and Quality Institute, Wuhan 430000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4077-3107","authenticated-orcid":false,"given":"Debiao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Fuzhou University, Fuzhou 350108, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5796-3479","authenticated-orcid":false,"given":"Xixing","family":"Li","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Modern Manufacturing and Quality Engineering, School of Mechanical Engineering, Hubei University of Technology, Wuhan 430068, China"}]},{"given":"Deming","family":"Lei","sequence":"additional","affiliation":[{"name":"School of Automation, Wuhan University of Technology, Wuhan 430070, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,5]]},"reference":[{"key":"ref_1","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_2","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_3","doi-asserted-by":"crossref","first-page":"5198","DOI":"10.15376\/biores.18.3.5198-5208","article-title":"Optimization of panel furniture plates rework based on intelligent manufacturing","volume":"18","author":"Luo","year":"2023","journal-title":"Bioresources"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wang, Y.S., Shi, J.W., Wang, W.J., and Li, C. (2024). Re-entrant green scheduling problem of bearing production shops considering job reworking. Machines, 12.","DOI":"10.3390\/machines12040281"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"142916","DOI":"10.1016\/j.jclepro.2024.142916","article-title":"A multi-objective integrated scheduling of remanufacturing system considering time window constrained outsourcing option","volume":"468","author":"Guo","year":"2024","journal-title":"J. Clean. Prod."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"145008","DOI":"10.1016\/j.jclepro.2025.145008","article-title":"A sustainable circular economic supply chain model with green production, delays in payment, and carbon tax regulation","volume":"495","author":"Jauhari","year":"2025","journal-title":"J. Clean. Prod."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1108\/AA-09-2020-0132","article-title":"Modeling and algorithm for resource-constrained multi-project scheduling problem based on detection and rework","volume":"41","author":"Zhu","year":"2021","journal-title":"Assem. Autom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1080\/00207543.2021.2005267","article-title":"Stochastic modelling of process scheduling for reduced rework cost and scrap","volume":"61","author":"Mahmoud","year":"2023","journal-title":"Int. J. Prod. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"62744","DOI":"10.1007\/s11356-023-26219-7","article-title":"A new uncertain remanufacturing scheduling model with rework risk using hybrid optimization algorithm","volume":"30","author":"Zhang","year":"2023","journal-title":"Environ. Sci. Pollut. Res."},{"key":"ref_10","first-page":"871","article-title":"Energy-efficient scheduling for a flexible job shop problem considering rework processes and new job arrival","volume":"15","author":"Albayrak","year":"2024","journal-title":"Int. J. Ind. Eng. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109848","DOI":"10.1016\/j.cie.2023.109848","article-title":"Multi-objective evolutionary scheduling based on collaborative virtual workflow model and adaptive rules for flexible production process with operation reworking","volume":"187","author":"Quan","year":"2024","journal-title":"Comput. Ind. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111062","DOI":"10.1016\/j.cie.2025.111062","article-title":"A genetic programming based reinforcement learning algorithm for dynamic hybrid flow shop scheduling with reworks under general queue time limits","volume":"203","author":"Kim","year":"2025","journal-title":"Comput. Ind. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Peng, J.L., Su, Z., and Liu, X. (2025). Multi skill project scheduling optimization based on quality transmission and rework network reconstruction. Sci. Rep., 15.","DOI":"10.1038\/s41598-025-92342-9"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.cie.2018.04.053","article-title":"Impact of integrating equipment health in production scheduling for semiconductor fabrication","volume":"120","author":"Kao","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106505","DOI":"10.1016\/j.cie.2020.106505","article-title":"Meta-heuristics for unrelated parallel machines scheduling with random rework to minimize expected total weighted tardiness","volume":"145","author":"Wang","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s43069-023-00233-4","article-title":"Sensitivity analysis of the unrelated parallel machine scheduling problem with rework processes and machine eligibility restrictions","volume":"4","author":"Rezaeian","year":"2023","journal-title":"Oper. Res. Forum"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"106968","DOI":"10.1016\/j.cor.2024.106968","article-title":"Unrelated parallel machine scheduling with random rework and limited preemption","volume":"177","author":"Wang","year":"2025","journal-title":"Comput. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.cor.2018.10.010","article-title":"A mathematical model and heuristic algorithms for an unrelated parallel machine scheduling problem with sequence-dependent setup times, machine eligibility restrictions and a common server","volume":"103","author":"Bektur","year":"2019","journal-title":"Comput. Oper. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"104886","DOI":"10.1016\/j.cor.2020.104886","article-title":"Analysis of variable neighborhood descent as a local search operator for total weighted tardiness problem on unrelated parallel machines","volume":"117","year":"2020","journal-title":"Comput. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1007\/s10479-025-06468-0","article-title":"Scheduling problems on parallel machines with machine-dependent generalized due-dates","volume":"347","author":"Mor","year":"2025","journal-title":"Ann. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Li, M.B., Xiong, H., and Lei, D.M. (2022). An Artificial Bee Colony with Adaptive Competition for the Unrelated Parallel Machine Scheduling Problem with Additional Resources and Maintenance. Symmetry, 14.","DOI":"10.3390\/sym14071380"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"106859","DOI":"10.1016\/j.cor.2024.106859","article-title":"Novel mathematical formulations for parallel-batching processing machine scheduling problems","volume":"173","author":"Zheng","year":"2025","journal-title":"Comput. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s00500-024-10395-w","article-title":"Benders decomposition for the multi-agent location and scheduling problem on unrelated parallel machines","volume":"29","author":"Liu","year":"2025","journal-title":"Soft Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.ejor.2024.09.032","article-title":"Minimizing total completion time and makespan for a multi-scenario bi-criteria parallel machine scheduling problem","volume":"321","author":"Zhang","year":"2025","journal-title":"Eur. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"110828","DOI":"10.1016\/j.cie.2024.110828","article-title":"The integration of shared renewable resources considering setup times for the parallel machine scheduling problem","volume":"200","author":"Shafiee","year":"2025","journal-title":"Comput. Ind. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"107110","DOI":"10.1016\/j.cor.2025.107110","article-title":"Designing state-of-the-art metaheuristics: What have we learned from the parallel-machine scheduling problem with setups","volume":"182","author":"Rolim","year":"2025","journal-title":"Comput. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential eevolution\u2013a simple and efficient heuristic for global optimization over continuous spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Fakhouri, H.N., Ishtaiwi, A., Makhadmeh, S.N., Al-Betar, M.A., and Alkhalaileh, M. (2024). Novel hybrid crayfish optimization algorithm and self-adaptive differential evolution for solving complex optimization problems. Symmetry, 16.","DOI":"10.3390\/sym16070927"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"109709","DOI":"10.1016\/j.engappai.2024.109709","article-title":"An improved reinforcement learning-based differential evolution algorithm for combined economic and emission dispatch problems","volume":"140","author":"Wang","year":"2025","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4226","DOI":"10.1080\/00207543.2023.2254858","article-title":"A multi-objective discrete differential evolution algorithm for energy-efficient distributed blocking flow shop scheduling problem","volume":"62","author":"Zhao","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1016\/j.jmsy.2025.04.001","article-title":"A Q-learning improved differential evolution algorithm for human-gentric dynamic distributed flexible job shop scheduling problem","volume":"80","author":"Li","year":"2025","journal-title":"J. Manuf. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"101783","DOI":"10.1016\/j.swevo.2024.101783","article-title":"A multi-strategy self-adaptive differential evolution algorithm for assembly hybrid flowshop lot-streaming scheduling with component sharing","volume":"92","author":"Lu","year":"2025","journal-title":"Swarm Evol. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"109878","DOI":"10.1016\/j.engappai.2024.109878","article-title":"Long-term scheduling strategy of hydro-wind-solar complementary system based on chaotic elite selection differential evolution algorithm with death penalty function","volume":"142","author":"He","year":"2025","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Wang, C.Y., and Liu, H.Y. (2025). RADE: A symmetry-inspired resource-adaptive differential evolution for lightweight dendritic learning in classification tasks. Symmetry, 17.","DOI":"10.3390\/sym17060891"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2722","DOI":"10.1080\/00207543.2020.1740341","article-title":"A hybrid differential evolution algorithm for parallel machine scheduling of lace dyeing considering colour families, sequence-dependent setup and machine eligibility","volume":"59","author":"Li","year":"2020","journal-title":"Int. J. Prod. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00207543.2024.2323065","article-title":"A genetic algorithm for fuzzy identical parallel machine scheduling of minimizing total weighted tardiness under resource constraint","volume":"62","author":"Li","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_37","unstructured":"Montgomery, D.C. (2019). Design and Analysis of Experiments, John Wiley & Sons. [10th ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1243\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:23:29Z","timestamp":1760034209000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1243"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,5]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081243"],"URL":"https:\/\/doi.org\/10.3390\/sym17081243","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,8,5]]}}}