{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:08:37Z","timestamp":1760058517921,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T00:00:00Z","timestamp":1744156800000},"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":["42202125","2024NSFSC0828","GFZX02040205.3"],"award-info":[{"award-number":["42202125","2024NSFSC0828","GFZX02040205.3"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018542","name":"Natural Science Foundation of Sichuan Province","doi-asserted-by":"publisher","award":["42202125","2024NSFSC0828","GFZX02040205.3"],"award-info":[{"award-number":["42202125","2024NSFSC0828","GFZX02040205.3"]}],"id":[{"id":"10.13039\/501100018542","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013076","name":"National Major Science and Technology Projects of China","doi-asserted-by":"publisher","award":["42202125","2024NSFSC0828","GFZX02040205.3"],"award-info":[{"award-number":["42202125","2024NSFSC0828","GFZX02040205.3"]}],"id":[{"id":"10.13039\/501100013076","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Multi-objective optimization is a cornerstone of modern engineering and management, tackling challenges in complex system design, resource allocation, and financial portfolio optimization. Effective multi-objective optimization algorithms must strike a balance between convergence and diversity, a process that inherently reflects the symmetry in objectives and their trade-offs. However, real-world complexities introduce significant hurdles: the exponential increase in Pareto optimal solutions diminishes the effectiveness of dominance-based selection, while escalating problem complexity heightens the tension between convergence and diversity. To address these issues, we propose a two-archive evolutionary algorithm that integrates reference vectors and a novel hypervolume contribution strategy. This approach employs two complementary archives\u2014convergence and diversity\u2014for parallel optimization. Within the diversity archive, candidate solutions are first filtered using angular relationships to maintain uniform distribution. A novel hypervolume contribution evaluation strategy (HVindex) then determines whether updating solutions can improve the overall quality of the neighborhood population. For the convergence archive, we first select all the non-dominated solutions through non-dominated sorting. These solutions are further refined using reference vectors, and the final archive is completed by adding some opposite characteristic solutions based on distance measurements. The experimental results demonstrate that the proposed algorithm outperforms existing methods in multi-objective optimization.<\/jats:p>","DOI":"10.3390\/sym17040572","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T07:41:51Z","timestamp":1744270911000},"page":"572","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["TwoArchRH: Enhanced Two-Archive Algorithm for Many-Objective Optimization"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-5572-1143","authenticated-orcid":false,"given":"Jiang","family":"Quan","sequence":"first","affiliation":[{"name":"College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Caihua","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruoyu","family":"Hu","sequence":"additional","affiliation":[{"name":"Elmore Family School of Electrical and Computer Engineering, Purdue University, 465 North Western Ave, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Zeng","sequence":"additional","affiliation":[{"name":"College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9257-2820","authenticated-orcid":false,"given":"Honghui","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guangle","family":"Yao","sequence":"additional","affiliation":[{"name":"College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Huang, S., Wang, C., and Bian, W. (2024). A Hybrid Food Recommendation System Based on MOEA\/D Focusing on the Problem of Food Nutritional Balance and Symmetry. Symmetry, 16.","DOI":"10.3390\/sym16121698"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1502242","DOI":"10.1080\/23311916.2018.1502242","article-title":"A Review of Multi-Objective Optimization: Methods and Its Applications","volume":"5","author":"Gunantara","year":"2018","journal-title":"Cogent Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5605","DOI":"10.1007\/s11831-022-09778-9","article-title":"A Comprehensive Review on Multi-Objective Optimization Techniques: Past, Present and Future","volume":"29","author":"Sharma","year":"2022","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2285","DOI":"10.1007\/s11831-021-09663-x","article-title":"A Review of Multi-Objective Optimization: Methods and Algorithms in Mechanical Engineering Problems","volume":"29","author":"Pereira","year":"2022","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"997","DOI":"10.5755\/j01.itc.53.4.37410","article-title":"Flexible Job Shop Scheduling Optimization with Machine and AGV Integration Based on Improved NSGA-II","volume":"53","author":"Liu","year":"2024","journal-title":"Inf. Technol. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"105598","DOI":"10.1016\/j.autcon.2024.105598","article-title":"Deep Reinforcement Learning for Multi-Objective Optimization in BIM-Based Green Building Design","volume":"166","author":"Pan","year":"2024","journal-title":"Autom. Constr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"102233","DOI":"10.1016\/j.is.2023.102233","article-title":"Multi-Objective Optimization with Recommender Systems: A Systematic Review","volume":"117","author":"Zaizi","year":"2023","journal-title":"Inf. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1007\/s00366-021-01369-9","article-title":"A Multi-Objective Optimization Algorithm for Feature Selection Problems","volume":"38","author":"Abdollahzadeh","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_9","first-page":"259","article-title":"SPEA2: Improving the strength Pareto evolutionary algorithm","volume":"103","author":"Zitzler","year":"2001","journal-title":"TIK Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1109\/TEVC.2015.2420112","article-title":"A New Dominance Relation-Based Evolutionary Algorithm for Many-Objective Optimization","volume":"20","author":"Yuan","year":"2016","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1109\/TEVC.2016.2587808","article-title":"A Vector Angle-Based Evolutionary Algorithm for Unconstrained Many-Objective Optimization","volume":"21","author":"Xiang","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1109\/TEVC.2014.2373386","article-title":"An Evolutionary Many-Objective Optimization Algorithm Based on Dominance and Decomposition","volume":"19","author":"Li","year":"2015","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zitzler, E., and K\u00fcnzli, S. (2004, January 13\u201317). Indicator-Based Selection in Multiobjective Search. Proceedings of the Parallel Problem Solving from Nature\u2013PPSN VIII, International Conference on Parallel Problem Solving from Nature, Birmingham, UK.","DOI":"10.1007\/978-3-540-30217-9_84"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1162\/EVCO_a_00009","article-title":"HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization","volume":"19","author":"Bader","year":"2011","journal-title":"Evol. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.ins.2021.03.008","article-title":"On the Estimation of Pareto Front and Dimensional Similarity in Many-Objective Evolutionary Algorithm","volume":"563","author":"Li","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1109\/TEVC.2022.3190401","article-title":"Domination-Based Selection and Shift-Based Density Estimation for Constrained Multiobjective Optimization","volume":"27","author":"Zhou","year":"2023","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_17","first-page":"1426","article-title":"A Fast Nondominated Sorting-Based MOEA with Convergence and Diversity Adjusted Adaptively","volume":"7","author":"Gao","year":"2023","journal-title":"J. Supercomput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s00500-021-06467-w","article-title":"An Improved Multi-Objective Bacterial Colony Chemotaxis Algorithm Based on Pareto Dominance","volume":"26","author":"Lu","year":"2022","journal-title":"Soft Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1007\/s00500-021-06223-0","article-title":"A Novel Solver for Multi-Objective Optimization: Dynamic Non-Dominated Sorting Genetic Algorithm (DNSGA)","volume":"26","author":"Long","year":"2022","journal-title":"Soft Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"101082","DOI":"10.1016\/j.swevo.2022.101082","article-title":"A Twin-Archive Guided Decomposition Based Multi\/Many-Objective Evolutionary Algorithm","volume":"71","author":"Mallipeddi","year":"2022","journal-title":"Swarm Evol. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3495","DOI":"10.1109\/TCYB.2020.3008697","article-title":"A Fuzzy Decomposition-Based Multi\/Many-Objective Evolutionary Algorithm","volume":"52","author":"Liu","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1109\/TCYB.2018.2872803","article-title":"Evolutionary Many-Objective Optimization Based on Adversarial Decomposition","volume":"50","author":"Wu","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.swevo.2016.12.002","article-title":"A Fast Hypervolume Driven Selection Mechanism for Many-Objective Optimisation Problems","volume":"34","author":"Rostami","year":"2017","journal-title":"Swarm Evol. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1109\/TEVC.2021.3096669","article-title":"Analyzing Dominance Move (MIP-DoM) Indicator for Multiobjective and Many-Objective Optimization","volume":"26","author":"Lopes","year":"2022","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1109\/TEVC.2018.2848921","article-title":"ISDE+\u2014An Indicator for Multi and Many-Objective Optimization","volume":"23","author":"Pamulapati","year":"2019","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1109\/TEVC.2010.2058117","article-title":"Diversity Management in Evolutionary Many-Objective Optimization","volume":"15","author":"Adra","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1109\/TEVC.2007.892759","article-title":"MOEA\/D: A Multiobjective Evolutionary Algorithm Based on Decomposition","volume":"11","author":"Zhang","year":"2007","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1109\/TEVC.2008.925798","article-title":"Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA\/D and NSGA-II","volume":"13","author":"Li","year":"2009","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Khare, V., Yao, X., and Deb, K. (2003). Performance Scaling of Multi-Objective Evolutionary Algorithms. Lecture Notes in Computer Science, Volume 2632, Proceedings of the Evolutionary Multi-Criterion Optimization-EMO, Faro, Portugal, 8\u201311 April 2003, Springer.","DOI":"10.1007\/3-540-36970-8_27"},{"key":"ref_30","unstructured":"Praditwong, K., and Yao, X. (2007, January 25\u201328). How Well Do Multi-Objective Evolutionary Algorithms Scale to Large Problems. Proceedings of the IEEE Congress on Evolutionary Computation, Singapore."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1162\/EVCO_a_00053","article-title":"Diagnostic Assessment of Search Controls and Failure Modes in Many-Objective Evolutionary Optimization","volume":"20","author":"Hadka","year":"2012","journal-title":"Evol. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wagner, T., Beume, N., and Naujoks, B. (2007). Pareto-, Aggregation-, and Indicator-Based Methods in Many-Objective Optimization. Lecture Notes in Computer Science, Volume 4403, Proceedings of the Evolutionary Multi-Criterion Optimization, Matsushima, Japan, 5\u20138 March 2007, Springer.","DOI":"10.1007\/978-3-540-70928-2_56"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1109\/TEVC.2012.2227145","article-title":"A Grid-Based Evolutionary Algorithm for Many-Objective Optimization","volume":"17","author":"Yang","year":"2013","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Praditwong, K., and Yao, X. (2006, January 3\u20136). A New Multi-Objective Evolutionary Optimization Algorithm: The Two-Archive Algorithm. Proceedings of the International Conference on Computational Intelligence and Security, Guangzhou, China.","DOI":"10.1109\/ICCIAS.2006.294139"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Li, B., Li, J., Tang, K., and Yao, X. (2014, January 6\u201311). An Improved Two Archive Algorithm for Many-Objective Optimization. Proceedings of the 2014 IEEE Congress on Evolutionary Computation (CEC), Beijing, China.","DOI":"10.1109\/CEC.2014.6900491"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1109\/TEVC.2018.2855411","article-title":"Two-Archive Evolutionary Algorithm for Constrained Multiobjective Optimization","volume":"23","author":"Li","year":"2019","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1109\/TEVC.2014.2350987","article-title":"Two-Arch2: An Improved Two-Archive Algorithm for Many-Objective Optimization","volume":"19","author":"Wang","year":"2015","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Emmerich, M., Beume, N., and Naujoks, B. (2005). An EMO Algorithm Using the Hypervolume Measure as Selection Criterion. Lecture Notes in Computer Science, Volume 3410, Proceedings of the Evolutionary Multi-Criteria Optimization, Guanajuato, Mexico, 9\u201311 March 2005, Springer.","DOI":"10.1007\/978-3-540-31880-4_5"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3832","DOI":"10.1109\/TITS.2020.3048844","article-title":"Resource Allocation in 5G IoV Architecture Based on SDN and Fog-Cloud Computing","volume":"22","author":"Cao","year":"2021","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1109\/MNET.011.1900536","article-title":"A Many-Objective Optimization Model of Industrial Internet of Things Based on Private Blockchain","volume":"34","author":"Cao","year":"2020","journal-title":"IEEE Netw."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.eswa.2017.06.009","article-title":"Optimal Reactive Power Dispatch Problem Using a Two-Archive Multi-Objective Grey Wolf Optimizer","volume":"87","author":"NNuaekaew","year":"2017","journal-title":"Expert. Syst. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Naujoks, B., Beume, N., and Emmerich, M. (2005, January 2\u20135). Multi-Objective Optimisation Using S-Metric Selection: Application to Three-Dimensional Solution Spaces. Proceedings of the 2005 IEEE Congress on Evolutionary Computation, Edinburgh, UK.","DOI":"10.1109\/CEC.2005.1554838"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1137\/S1052623496307510","article-title":"Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems","volume":"8","author":"Das","year":"1998","journal-title":"SIAM J. Optim."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1109\/MCI.2017.2742868","article-title":"PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum]","volume":"12","author":"Tian","year":"2017","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1162\/106365600568202","article-title":"Comparison of Multiobjective Evolutionary Algorithms: Empirical Results","volume":"8","author":"Zitzler","year":"2000","journal-title":"Evol. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1109\/TCYB.2015.2507366","article-title":"Biased Multiobjective Optimization and Decomposition Algorithm","volume":"47","author":"Li","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/TEVC.2005.861417","article-title":"A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit","volume":"10","author":"Huband","year":"2006","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_48","unstructured":"Deb, K., Thiele, L., Laumanns, M., and Zitzler, E. (2002, January 12\u201317). Scalable Multiobjective Optimization Test Problems. Proceedings of the Congress on Evolutionary Computation (CEC), Honolulu, HI, USA."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1109\/4235.797969","article-title":"Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach","volume":"3","author":"Zitzler","year":"1999","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/TEVC.2003.810758","article-title":"Performance Assessment of Multiobjective Optimizers: An Analysis and Review","volume":"7","author":"Zitzler","year":"2003","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1653","DOI":"10.1016\/j.ejor.2006.08.008","article-title":"SMS-EMOA: Multiobjective Selection Based on Dominated Hypervolume","volume":"181","author":"Beume","year":"2007","journal-title":"Eur. J. Oper. 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