{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,11]],"date-time":"2026-09-11T05:06:23Z","timestamp":1789103183907,"version":"build-2803163510"},"reference-count":42,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T00:00:00Z","timestamp":1788739200000},"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":["Grant No. 51505344, 11872046"],"award-info":[{"award-number":["Grant No. 51505344, 11872046"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["Grant No. 51505344, 11872046"],"award-info":[{"award-number":["Grant No. 51505344, 11872046"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Hubei Key Laboratory of Digital Textile Equipment Open Subjects","award":["No. 2020001"],"award-info":[{"award-number":["No. 2020001"]}]},{"name":"the Project of Outstanding Young and Middle-aged Scientific and Technological Innovation Team in Universities of Hubei Province","award":["No. T2022015"],"award-info":[{"award-number":["No. T2022015"]}]},{"name":"the Shandong Province key research and development plan projects refer to major Science and Technology Innovation Engineering","award":["2024CXGC010215"],"award-info":[{"award-number":["2024CXGC010215"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Biomimetics"],"abstract":"<jats:p>Multi-objective optimization plays an important role in modern design and complex engineering applications. However, achieving an effective balance between the convergence and diversity of Pareto-optimal solutions remains challenging. This paper proposes a Sobol-driven Multi-objective Whale Migration Algorithm (SMOWMA), which extends the Whale Migration Algorithm within a non-dominated sorting and elite-selection framework. A maximin scrambled Sobol initialization scheme is first employed to improve the distribution of the initial population. An archive-guided adaptive Student-t flight mechanism is then incorporated into the leader-whale position update to dynamically balance global exploration and local exploitation. In addition, archive crowding information and archive-entry success feedback are jointly used to adjust the search behavior according to both environmental diversity and recent search performance. SMOWMA is evaluated on five widely used multi-objective benchmark suites, namely ZDT, DTLZ, WFG, UF, and CF, using four performance indicators: generational distance (GD), inverted generational distance (IGD), spacing (SP), and hypervolume (HV). The results, together with Friedman tests and Holm-adjusted Wilcoxon tests, demonstrate that SMOWMA achieves competitive overall performance in terms of convergence, diversity, and objective-space coverage, although its relative advantage remains problem-dependent. The practical applicability of SMOWMA is further examined using multi-objective welded-beam design formulations, a bi-objective four-bar truss design problem, and a five-objective car side-impact design problem. The engineering results show that SMOWMA can obtain competitive and stable approximation sets for constrained design problems with different numbers of objectives, supporting its effectiveness and applicability in multi-objective engineering optimization.<\/jats:p>","DOI":"10.3390\/biomimetics11090642","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T13:53:28Z","timestamp":1788789208000},"page":"642","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Sobol-Driven Multi-Objective Whale Migration Algorithm for Engineering Optimization"],"prefix":"10.3390","volume":"11","author":[{"given":"Lizhen","family":"Du","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dahongnian","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5671-6871","authenticated-orcid":false,"given":"Xiaoshuang","family":"Xiong","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8945-7856","authenticated-orcid":false,"given":"Min","family":"Shen","sequence":"additional","affiliation":[{"name":"Three Dimensional Textile Hubei Engineering Research Center, Wuhan Textile University, Wuhan 430200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5027-8316","authenticated-orcid":false,"given":"Hongtao","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Mechancal and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lianqing","family":"Yu","sequence":"additional","affiliation":[{"name":"Three Dimensional Textile Hubei Engineering Research Center, Wuhan Textile University, Wuhan 430200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1196-5847","authenticated-orcid":false,"given":"Fei","family":"Fan","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430200, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"100586","DOI":"10.1016\/j.rineng.2022.100586","article-title":"Multi objective short term hydro-thermal- CHP scheduling using social spider algorithm","volume":"16","author":"Adhvaryyu","year":"2022","journal-title":"Results Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.neucom.2020.01.114","article-title":"A survey of decomposition approaches in multiobjective evolutionary algorithms","volume":"408","author":"Wang","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_3","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. 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