{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T12:23:51Z","timestamp":1780921431791,"version":"3.54.1"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T00:00:00Z","timestamp":1649376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Foundation of Guizhou University","award":["Guidateganghezi [2021]04"],"award-info":[{"award-number":["Guidateganghezi [2021]04"]}]},{"DOI":"10.13039\/501100001809","name":"NNSF of China","doi-asserted-by":"publisher","award":["No.61640014"],"award-info":[{"award-number":["No.61640014"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Industrial Project of Guizhou province","award":["No. Qiankehe Zhicheng [2022]017, [2019]2152"],"award-info":[{"award-number":["No. Qiankehe Zhicheng [2022]017, [2019]2152"]}]},{"name":"Innovation group of Guizhou Education Department under Grant Qianjiaohe","award":["No.KY[2021]012"],"award-info":[{"award-number":["No.KY[2021]012"]}]},{"name":"cience and Technology Fund of Guizhou Province under Grant Qiankehe","award":["No.[2020]1Y266), Qiankehejichu [No.ZK[2022]Yiban103"],"award-info":[{"award-number":["No.[2020]1Y266), Qiankehejichu [No.ZK[2022]Yiban103"]}]},{"name":"CASE Library of IOT","award":["KCALK201708"],"award-info":[{"award-number":["KCALK201708"]}]},{"name":"platform about IoT of Guiyang National High technology industry development zone","award":["No. 2015"],"award-info":[{"award-number":["No. 2015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Only the smell perception rule is considered in the butterfly optimization algorithm (BOA), which is prone to falling into a local optimum. Compared with the original BOA, an extra operator, i.e., color perception rule, is incorporated into the proposed hybrid-flash butterfly optimization algorithm (HFBOA), which makes it more in line with the actual foraging characteristics of butterflies in nature. Besides, updating the strategy of the control parameters by the logistic mapping is used in the HFBOA for enhancing the global optimal ability. The performance of the proposed method was verified by twelve benchmark functions, where the comparison experiment results show that the HFBOA converges quicker and has better stability for numerical optimization problems, which are compared with six state-of-the-art optimization methods. Additionally, the proposed HFBOA is successfully applied to six engineering constrained optimization problems (i.e., tubular column design, tension\/compression spring design, cantilever beam design, etc.). The simulation results reveal that the proposed approach demonstrates superior performance in solving complex real-world engineering constrained tasks.<\/jats:p>","DOI":"10.3390\/e24040525","type":"journal-article","created":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T12:11:14Z","timestamp":1649419874000},"page":"525","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["Hybrid-Flash Butterfly Optimization Algorithm with Logistic Mapping for Solving the Engineering Constrained Optimization Problems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8546-9972","authenticated-orcid":false,"given":"Mengjian","family":"Zhang","sequence":"first","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"},{"name":"School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4936-8773","authenticated-orcid":false,"given":"Deguang","family":"Wang","sequence":"additional","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6407-1276","authenticated-orcid":false,"given":"Jing","family":"Yang","sequence":"additional","affiliation":[{"name":"Electrical Engineering College, Guizhou University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1109\/TEVC.2005.857610","article-title":"Comprehensive learning particle swarm optimizer for global optimization of multimodal functions","volume":"10","author":"Liang","year":"2006","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.engappai.2006.03.003","article-title":"An effective co-evolutionary particle swarm optimization for constrained engineering design problems","volume":"20","author":"He","year":"2007","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1656","DOI":"10.1109\/TSMCB.2012.2227469","article-title":"Particle Swarm Optimization for Feature Selection in Classification: A Multi-Objective Approach","volume":"43","author":"Xue","year":"2013","journal-title":"IEEE Trans. Cybern."},{"key":"ref_4","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle swarm optimization. Proceedings of the ICNN\u201995-International Conference on Neural Networks, Perth, WA, Australia."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Shen, Y., Cai, W., Kang, H., Sun, X., Chen, Q., and Zhang, H. (2021). A Particle Swarm Algorithm Based on a Multi-Stage Search Strategy. Entropy, 23.","DOI":"10.3390\/e23091200"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1109\/3477.484436","article-title":"Ant system: Optimization by a colony of cooperating agents","volume":"26","author":"Dorigo","year":"1996","journal-title":"IEEE Trans. Syst. Man Cybern. Part B (Cybern.)"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Teodorovic, D., Lucic, P., Markovic, G., and Dell\u2019Orco, M. (2006, January 25\u201327). Bee Colony Optimization: Principles and Applications. Proceedings of the 2006 8th Seminar on Neural Network Applications in Electrical Engineering, Belgrade, Serbia.","DOI":"10.1109\/NEUREL.2006.341200"},{"key":"ref_8","unstructured":"Yang, X.S. (2010). Nature-Inspired Metaheuristic Algorithms, Luniver Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","author":"Mirjalili","year":"2014","journal-title":"Adv. Eng. Softw."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1109\/TSMC.1986.289288","article-title":"Optimization of Control Parameters for Genetic Algorithms","volume":"16","author":"Grefenstette","year":"1996","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential Evolution\u2014A Simple and Efficient Heuristic for global Optimization over Continuous Spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/s00500-018-3102-4","article-title":"Butterfly optimization algorithm: A novel approach for global optimization","volume":"23","author":"Arora","year":"2019","journal-title":"Soft Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"114766","DOI":"10.1016\/j.applthermaleng.2019.114766","article-title":"Improved butterfly optimization algorithm for CCHP driven by PEMFC","volume":"173","author":"Zhi","year":"2019","journal-title":"Appl. Therm. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4809","DOI":"10.1007\/s00500-019-04234-6","article-title":"m-MBOA: A novel butterfly optimization algorithm enhanced with mutualism scheme","volume":"24","author":"Sharma","year":"2019","journal-title":"Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhang, M., Long, D., Qin, T., and Yang, J. (2020). A chaotic hybrid butterfly optimization algorithm with particle swarm optimization for high-dimensional optimization problems. Symmetry, 12.","DOI":"10.3390\/sym12111800"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3325","DOI":"10.1007\/s13369-017-2471-9","article-title":"Node localization in wireless sensor networks using butterfly optimization algorithm","volume":"42","author":"Arora","year":"2017","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"106518","DOI":"10.1016\/j.asoc.2020.106518","article-title":"Wavelet neural networks based solutions for elliptic partial differential equations with improved butterfly optimization algorithm training","volume":"95","author":"Tan","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.eswa.2018.08.051","article-title":"Binary butterfly optimization approaches for feature selection","volume":"116","author":"Arora","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"e12786","DOI":"10.1111\/exsy.12786","article-title":"A hybrid feature selection model based on butterfly optimization algorithm: COVID-19 as a case study","volume":"39","author":"Elhoseny","year":"2022","journal-title":"Expert Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"An, J., Li, X., Zhang, Z., Zhang, G., Man, W., Hu, G., He, J., and Yu, D. (2022). A Novel Method for Inverse Kinematics Solutions of Space Modular Self-Reconfigurable Satellites with Self-Collision Avoidance. Aerospace, 9.","DOI":"10.3390\/aerospace9030123"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1007\/s00442-004-1761-6","article-title":"Priority of color over scent during flower visitation by adult Vanessa indica butterflies","volume":"142","author":"Honda","year":"2005","journal-title":"Oecologia"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/4235.585893","article-title":"No free lunch theorems for optimization","volume":"1","author":"Wolpert","year":"1997","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/261459a0","article-title":"Simple mathematical models with very complicated dynamics","volume":"261","author":"May","year":"1976","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1109\/TEVC.2008.927706","article-title":"Differential evolution algorithm with strategy adaptation for global numerical optimization","volume":"13","author":"Qin","year":"2009","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s00366-011-0241-y","article-title":"Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems","volume":"29","author":"Gandomi","year":"2013","journal-title":"Eng. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"113702","DOI":"10.1016\/j.eswa.2020.113702","article-title":"Heap-based optimizer inspired by corporate rank hierarchy for global optimization","volume":"161","author":"Askari","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BF02491474","article-title":"Competitors of the Wilcoxon signed rank test","volume":"39","author":"Maesono","year":"1987","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1111\/j.2044-8317.1980.tb00779.x","article-title":"Unified analysis of variance by ranks","volume":"33","author":"Meddis","year":"1980","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1002\/(SICI)1097-0207(19960315)39:5<829::AID-NME884>3.0.CO;2-U","article-title":"Structural optimization using a new local approximation method","volume":"39","author":"Chickermane","year":"1996","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"302","DOI":"10.3846\/13923730.2014.897986","article-title":"An introduction of krill herd algorithm for engineering optimization","volume":"22","author":"Gandomi","year":"2013","journal-title":"J. Civ. Eng. Manag."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1109\/4235.873238","article-title":"Stochastic ranking for constrained evolutionary optimization","volume":"4","author":"Runarsson","year":"2000","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_32","first-page":"1076","article-title":"Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms","volume":"187","author":"Tavazoei","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106734","DOI":"10.1016\/j.asoc.2020.106734","article-title":"Cooperation search algorithm: A novel metaheuristic evolutionary intelligence algorithm for numerical optimization and engineering optimization problems","volume":"98","author":"Feng","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2592","DOI":"10.1016\/j.asoc.2012.11.026","article-title":"Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems","volume":"13","author":"Sadollah","year":"2013","journal-title":"Appl. Soft Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/j.future.2019.02.028","article-title":"Harris hawks optimization: Algorithm and applications","volume":"97","author":"Heidari","year":"2019","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_36","unstructured":"Zhang, M., Wen, G., and Yang, J. (2021). Duck swarm algorithm: A novel swarm intelligence algorithm. arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The whale optimization algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.compstruc.2014.03.007","article-title":"Symbiotic organisms search: A new metaheuristic optimization algorithm","volume":"139","author":"Cheng","year":"2014","journal-title":"Comput. Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.knosys.2015.07.006","article-title":"Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm","volume":"89","author":"Mirjalili","year":"2015","journal-title":"Knowl.-Based Syst."},{"key":"ref_40","first-page":"1587","article-title":"Research on convergence of grey wolf optimization algorithm based on Markov chain","volume":"48","author":"Zhang","year":"2020","journal-title":"Acta Electron. Sin."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/525\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:50:27Z","timestamp":1760136627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/525"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,8]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["e24040525"],"URL":"https:\/\/doi.org\/10.3390\/e24040525","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,8]]}}}