{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:44:26Z","timestamp":1760233466188,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,15]],"date-time":"2021-01-15T00:00:00Z","timestamp":1610668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["No.[2019YFB1404700]"],"award-info":[{"award-number":["No.[2019YFB1404700]"]}],"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 paper explores the combination of a classic mathematical function named \u201chyperbolic tangent\u201d with a metaheuristic algorithm, and proposes a novel hybrid genetic algorithm called NSGA-II-BnF for multi-objective decision making. Recently, many metaheuristic evolutionary algorithms have been proposed for tackling multi-objective optimization problems (MOPs). These algorithms demonstrate excellent capabilities and offer available solutions to decision makers. However, their convergence performance may be challenged by some MOPs with elaborate Pareto fronts such as CFs, WFGs, and UFs, primarily due to the neglect of diversity. We solve this problem by proposing an algorithm with elite exploitation strategy, which contains two parts: first, we design a biased elite allocation strategy, which allocates computation resources appropriately to elites of the population by crowding distance-based roulette. Second, we propose a self-guided fast individual exploitation approach, which guides elites to generate neighbors by a symmetry exploitation operator, which is based on mathematical hyperbolic tangent function. Furthermore, we designed a mechanism to emphasize the algorithm\u2019s applicability, which allows decision makers to adjust the exploitation intensity with their preferences. We compare our proposed NSGA-II-BnF with four other improved versions of NSGA-II (NSGA-IIconflict, rNSGA-II, RPDNSGA-II, and NSGA-II-SDR) and four competitive and widely-used algorithms (MOEA\/D-DE, dMOPSO, SPEA-II, and SMPSO) on 36 test problems (DTLZ1\u2013DTLZ7, WGF1\u2013WFG9, UF1\u2013UF10, and CF1\u2013CF10), and measured using two widely used indicators\u2014inverted generational distance (IGD) and hypervolume (HV). Experiment results demonstrate that NSGA-II-BnF exhibits superior performance to most of the algorithms on all test problems.<\/jats:p>","DOI":"10.3390\/sym13010136","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T02:36:05Z","timestamp":1611196565000},"page":"136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Elite Exploitation: A Combination of Mathematical Concept and EMO Approach for Multi-Objective Decision Making"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0608-0649","authenticated-orcid":false,"given":"Wenxiao","family":"Li","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9091-0459","authenticated-orcid":false,"given":"Yushui","family":"Geng","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4293-4309","authenticated-orcid":false,"given":"Jing","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7148-6106","authenticated-orcid":false,"given":"Kang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3048-6000","authenticated-orcid":false,"given":"Jianxin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Miettinen, K. (1999). Nonlinear Multiobjective Optimization, Kluwer.","DOI":"10.1007\/978-1-4615-5563-6"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ins.2019.03.016","article-title":"A region search evolutionary algorithm for many-objective optimization","volume":"488","author":"Liu","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.swevo.2019.05.011","article-title":"Improved NSGA-III with selection-and-elimination operator","volume":"49","author":"Cui","year":"2019","journal-title":"Swarm Evol. Comput."},{"key":"ref_4","unstructured":"Zitzler, E., Laumanns, M., and Thiele, L. (2020, December 30). Spea2: Improving the strength pareto evolutionary algorithm. TIK-Report Computer Engineering and Communication Networks Lab(TIK), Swiss Federal Institute of Technology (ETH) Zurich, ETH Zentrum, Gloriastrasse 35, CH-8092 Zurich, Switzerland, Sept. 2001, 103. Available online: https:\/\/doi.org\/10.3929\/ethz-a-004284029."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: Nsga-ii","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1109\/TEVC.2018.2866854","article-title":"A Strengthened Dominance Relation Considering Convergence and Diversity for Evolutionary Many-Objective Optimization","volume":"23","author":"Tian","year":"2018","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.ins.2019.08.065","article-title":"An Expanded Particle Swarm Optimization Based on Multi-Exemplar and Forgetting Ability","volume":"508","author":"Xia","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.swevo.2018.12.009","article-title":"Particle swarm optimization based on dimensional learning strategy","volume":"45","author":"Xu","year":"2019","journal-title":"Swarm Evol. Comput."},{"key":"ref_9","unstructured":"Nebro, A.J., Durillo, J.J., Garcia-Nieto, J., Coello, C.A.C., and Alba, E. (April, January 30). Smpso: A new pso-based metaheuristic for multi-objective optimization. Proceedings of the IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making, Nashville, TN, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.ins.2018.12.078","article-title":"An angle dominance criterion for evolutionary many-objective optimization","volume":"509","author":"Liu","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1109\/TEVC.2010.2041060","article-title":"The r-Dominance: A New Dominance Relation for Interactive Evolutionary Multicriteria Decision Making","volume":"14","author":"Said","year":"2010","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.eswa.2015.10.039","article-title":"Multiobjective grey wolf optimizer: A novel algorithm for multi-criterion optimization","volume":"47","author":"Mirjalili","year":"2016","journal-title":"Expert Syst. Appl."},{"key":"ref_13","first-page":"1","article-title":"An Indicator-Based Many-Objective Evolutionary Algorithm With Boundary Protection","volume":"99","author":"Liang","year":"2020","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1109\/TEVC.2018.2881153","article-title":"A scalable indicator-based evolutionary algorithm for large-scale multiobjective optimization","volume":"23","author":"Hong","year":"2018","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1109\/TEVC.2017.2749619","article-title":"An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility","volume":"22","author":"Tian","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1109\/TII.2013.2272945","article-title":"A generalized framework for optimal sizing of distributed energy resources in microgrids using an indicator-based swarm approach","volume":"10","author":"Silvestre","year":"2013","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.asoc.2018.02.048","article-title":"A two-stage r2 indicator based evolutionary algorithm for many-objective optimization","volume":"67","author":"Li","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1109\/TEVC.2018.2791283","article-title":"IGD Indicator-Based Evolutionary Algorithm for Many-Objective Optimization Problems","volume":"23","author":"Sun","year":"2018","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_19","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":"2018","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1109\/TEVC.2015.2424251","article-title":"Are all the subproblems equally important? resource allocation in decomposition-based multiobjective evolutionary algorithms","volume":"20","author":"Zhou","year":"2015","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.ins.2014.02.002","article-title":"Objective space partitioning using conflict information for solving manyobjective problems","volume":"268","author":"Jaimes","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2388","DOI":"10.1109\/TCYB.2017.2739185","article-title":"A diversity-enhanced resource allocation strategy for decomposition-based multiobjective evolutionary algorithm","volume":"48","author":"Lin","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_23","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":"2008","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1109\/TSMC.2017.2654301","article-title":"A new decomposition-based NSGA-II for many-objective optimization","volume":"48","author":"Elarbi","year":"2017","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1109\/TEVC.2017.2704118","article-title":"On tchebycheff decomposition approaches for multiobjective evolutionary optimization","volume":"22","author":"Ma","year":"2018","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ins.2019.03.062","article-title":"An adaptive decomposition-based evolutionary algorithm for many-objective optimization","volume":"491","author":"Han","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Liu, W., and Li, H. (2009, January 18\u201321). The performance of a new version of moea\/d on cec09 unconstrained mop test instances. Proceedings of the 2009 IEEE Congress on Evolutionary Computation, Trondheim, Norway.","DOI":"10.1109\/CEC.2009.4982949"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1109\/TEVC.2015.2457616","article-title":"Constrained subproblems in a decomposition-based multiobjective evolutionary algorithm","volume":"20","author":"Wang","year":"2015","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zhang, J., Zhou, A., and Zhang, G. (2015). A multiobjective evolutionary algorithm based on decomposition and preselection. Bio-Inspired Computing-Theories and Applications, Springer.","DOI":"10.1007\/978-3-662-49014-3_56"},{"key":"ref_30","unstructured":"Mart\u00ednez, S.Z., and Coello, C.A.C. (2011, January 12\u201316). A multi-objective particle swarm optimizer based on decomposition. Proceedings of the Conference on Genetic & Evolutionary, Dublin, Ireland."},{"key":"ref_31","first-page":"440","article-title":"A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition","volume":"21","author":"Trivedi","year":"2016","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Xu, Q., Xu, Z., and Ma, T. (2019, January 28\u201331). Short Survey and Challenges for Multiobjective Evolutionary Algorithms Based on Decomposition. Proceedings of the 2019 International Conference on Computer, Information and Telecommunication Systems (CITS), Beijing, China.","DOI":"10.1109\/CITS.2019.8862046"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2480741.2480752","article-title":"Exploration and exploitation in evolutionary algorithms: A survey","volume":"45","author":"Crepinsek","year":"2013","journal-title":"ACM Comput. Surv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.asoc.2011.08.047","article-title":"A hybrid evolutionary algorithm for tuning a cloth-simulation model","volume":"12","author":"Mongus","year":"2012","journal-title":"Appl. Soft Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1109\/MCI.2010.936309","article-title":"Memetic Computation - Past, Present & Future","volume":"5","author":"Ong","year":"2010","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_36","unstructured":"Deb, K., Thiele, L., Laumanns, M., and Zitzler, E. (2006). Scalable Test Problems for Evolutionary Multiobjective Optimization. Evolutionary Multiobjective Optimization, Springer."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Huband, S., Barone, L.C., While, L., and Hingston, P.F. (2005, January 9\u201311). A scalable multiobjective test problem toolkit. Proceedings of the International Conference on Evolutionary Multi-Criterion Optimization, Guanajuato, Mexico.","DOI":"10.1007\/978-3-540-31880-4_20"},{"key":"ref_38","unstructured":"Zhang, Q., Zhou, A., Zhao, S., Suganthan, P.N., Liu, W., and Tiwari, S. (2009). Multiobjective optimization test instances for the CEC 2009 special session and competition. Mechanical Engineering, American Society of Mechanical Engineers(ASME)."},{"key":"ref_39","first-page":"115","article-title":"Simulated binary crossover for continuous search space","volume":"9","author":"Agrawal","year":"1995","journal-title":"Complex Syst."},{"key":"ref_40","first-page":"109","article-title":"Simulated annealing algorithm based on cauchy and gaussian distributed state generator","volume":"40","author":"ling","year":"2000","journal-title":"J. Tsinghua Univ. (Sci. Technol.)"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0010-4655(91)90071-R","article-title":"A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver","volume":"66","author":"Yabe","year":"1991","journal-title":"Comput. Phys. Commun."},{"key":"ref_42","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. Evol. Comput."},{"key":"ref_43","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. Evol. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/136\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:11:34Z","timestamp":1760159494000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,15]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010136"],"URL":"https:\/\/doi.org\/10.3390\/sym13010136","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,1,15]]}}}