{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T14:15:34Z","timestamp":1768313734021,"version":"3.49.0"},"reference-count":52,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T00:00:00Z","timestamp":1767744000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers"],"abstract":"<jats:p>We introduce AHA\u2013SCA, a compact hybrid optimiser that alternates the wave-based exploration of the Sine\u2013Cosine Algorithm (SCA) with the exploitation skills of the Artificial Hummingbird Algorithm (AHA) within a single population. Even iterations perform SCA moves with a linearly decaying sinusoidal amplitude to explore widely around the current best solution, while odd iterations invoke guided and territorial hummingbird flights using axial, diagonal, and omnidirectional patterns to intensify the search in promising regions. This simple interleaving yields an explicit and tunable balance between exploration and exploitation and incurs negligible overhead beyond evaluating candidate solutions. The proposed approach is evaluated on the CEC2014, CEC2017, and CEC2022 benchmark suites and on several constrained engineering design problems, including welded beam, pressure vessel, tension\/compression spring, speed reducer, and cantilever beam designs. Across these diverse tasks, AHA\u2013SCA demonstrates competitive or superior performance relative to stand-alone SCA, AHA, and a broad panel of recent metaheuristics, delivering faster early-phase convergence and robust final solutions. Statistical analyses using non-parametric tests confirm that improvements are significant on many functions, and the method respects problem constraints without parameter tuning. The results suggest that alternating wave-driven exploration with hummingbird-inspired refinement is a promising general strategy for continuous engineering optimisation.<\/jats:p>","DOI":"10.3390\/computers15010035","type":"journal-article","created":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T09:34:12Z","timestamp":1767778452000},"page":"35","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hybrid Sine\u2013Cosine with Hummingbird Foraging Algorithm for Engineering Design Optimisation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9060-7188","authenticated-orcid":false,"given":"Jamal","family":"Zraqou","sequence":"first","affiliation":[{"name":"Computer Science Department, University of Petra, Amman 11196, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7222-2433","authenticated-orcid":false,"given":"Ahmad Sami","family":"Al-Shamayleh","sequence":"additional","affiliation":[{"name":"Department of Data Science and Artificial Intelligence, Faculty of Information Technology, Al-Ahliyya Amman University, Amman 19328, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Riyad","family":"Alrousan","sequence":"additional","affiliation":[{"name":"Design & Visual Communication Department, School of Architecture and Built Environment (SABE), German Jordanian University (GJU), Amman 11180, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9170-3291","authenticated-orcid":false,"given":"Hussam","family":"Fakhouri","sequence":"additional","affiliation":[{"name":"Artificial Intelligence Faculty, Al-Balqa Applied University, Amman 11196, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7548-0529","authenticated-orcid":false,"given":"Faten","family":"Hamad","sequence":"additional","affiliation":[{"name":"Information Studies Department, Sultan Qaboos University, Muscat P.O. Box 50, Oman"},{"name":"Information Sciences and Educational Technology Department, School of Educational Sciences, The University of Jordan, Amman 11196, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-5328-7812","authenticated-orcid":false,"given":"Niveen","family":"Halalsheh","sequence":"additional","affiliation":[{"name":"Department of Journalism, Media, and Digital Communication, School of Arts, The University of Jordan, Amman 11942, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,7]]},"reference":[{"key":"ref_1","unstructured":"Kochenderfer, M.J., and Wheeler, T.A. (2019). Algorithms for Optimization, Mit Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Diwekar, U.M. (2020). Introduction to Applied Optimization, Springer.","DOI":"10.1007\/978-3-030-55404-0"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3229","DOI":"10.1137\/17M1135086","article-title":"Stochastic methods for composite and weakly convex optimization problems","volume":"28","author":"Duchi","year":"2018","journal-title":"SIAM J. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"106040","DOI":"10.1016\/j.cie.2019.106040","article-title":"A survey on new generation metaheuristic algorithms","volume":"137","author":"Dokeroglu","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_5","first-page":"732","article-title":"Addax Optimization Algorithm: A Novel Nature-Inspired Optimizer for Solving Engineering Applications","volume":"17","author":"Hamadneh","year":"2024","journal-title":"Int. J. Intell. Eng. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Abualhaj, M.M., Al-Khatib, S.N., Abu-Shareha, A.A., Almomani, O., Al-Mimi, H., Al-Allawee, A., and Anbar, M. (2025). Spam Detection Boosted by Firefly-Based Feature Selection and Optimized Classifiers. Int. J. Adv. Soft Comput. Its Appl., 17.","DOI":"10.15849\/IJASCA.251130.02"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/j.future.2020.03.055","article-title":"Slime mould algorithm: A new method for stochastic optimization","volume":"111","author":"Li","year":"2020","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103541","DOI":"10.1016\/j.engappai.2020.103541","article-title":"Tunicate Swarm Algorithm: A new bio-inspired based metaheuristic paradigm for global optimization","volume":"90","author":"Kaur","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"115079","DOI":"10.1016\/j.eswa.2021.115079","article-title":"RUN beyond the metaphor: An efficient optimization algorithm based on Runge Kutta method","volume":"181","author":"Ahmadianfar","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"105190","DOI":"10.1016\/j.knosys.2019.105190","article-title":"Equilibrium optimizer: A novel optimization algorithm","volume":"191","author":"Faramarzi","year":"2020","journal-title":"Knowl.-Based Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103300","DOI":"10.1016\/j.engappai.2019.103300","article-title":"Manta ray foraging optimization: An effective bio-inspired optimizer for engineering applications","volume":"87","author":"Zhao","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Raidl, G.R., Puchinger, J., and Blum, C. (2019). Metaheuristic hybrids. Handbook of Metaheuristics, Springer.","DOI":"10.1007\/978-3-319-91086-4_12"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sorensen, K., Sevaux, M., and Glover, F. (2017). A history of metaheuristics. arXiv.","DOI":"10.1007\/978-3-319-07124-4_4"},{"key":"ref_14","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_15","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_16","first-page":"99","article-title":"Sine-Cosine Algorithm: A Novel Metaheuristic Optimization Algorithm","volume":"61","author":"Mirjalili","year":"2016","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_17","first-page":"107","article-title":"Hummingbird Foraging Algorithm: A Novel Metaheuristic Algorithm","volume":"110","author":"Zhao","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107250","DOI":"10.1016\/j.cie.2021.107250","article-title":"Aquila Optimizer: A novel meta-heuristic optimization algorithm","volume":"157","author":"Abualigah","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5887","DOI":"10.1002\/int.22535","article-title":"Artificial gorilla troops optimizer: A new nature-inspired metaheuristic algorithm for global optimization problems","volume":"36","author":"Abdollahzadeh","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_20","first-page":"103731","article-title":"Vultures Optimization Algorithm: A New Nature-Inspired Algorithm","volume":"94","author":"Abdollahzadeh","year":"2021","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"7665","DOI":"10.1007\/s00521-018-3592-0","article-title":"On the exploration and exploitation in popular swarm-based metaheuristic algorithms","volume":"31","author":"Hussain","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Abdel-Basset, M., Abdel-Fatah, L., and Sangaiah, A.K. (2018). Metaheuristic algorithms: A comprehensive review. Computational Intelligence for Multimedia Big Data on the Cloud with Engineering Applications, Academic Press.","DOI":"10.1016\/B978-0-12-813314-9.00010-4"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s10489-017-1019-8","article-title":"Grasshopper optimization algorithm for multi-objective optimization problems","volume":"48","author":"Mirjalili","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"00017","DOI":"10.1051\/e3sconf\/20186000017","article-title":"Improving the Process of Coal Extraction Based on the Parameter Optimization of Mining Equipment","volume":"60","author":"Hrinov","year":"2018","journal-title":"E3S Web Conf."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Siarry, P. (2016). Metaheuristics, Springer.","DOI":"10.1007\/978-3-319-45403-0"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"114194","DOI":"10.1016\/j.cma.2021.114194","article-title":"Artificial hummingbird algorithm: A new bio-inspired optimizer with its engineering applications","volume":"388","author":"Zhao","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mirjalili, S.M., Mirjalili, S.Z., Saremi, S., and Mirjalili, S. (2020). Sine cosine algorithm: Theory, literature review, and application in designing bend photonic crystal waveguides. Nature-Inspired Optimizers: Theories, Literature Reviews and Applications, Springer.","DOI":"10.1007\/978-3-030-12127-3_12"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Chopard, B., and Tomassini, M. (2018). An Introduction to Metaheuristics for Optimization, Springer.","DOI":"10.1007\/978-3-319-93073-2"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1007\/s10489-020-01893-z","article-title":"Archimedes optimization algorithm: A new metaheuristic algorithm for solving optimization problems","volume":"51","author":"Hashim","year":"2021","journal-title":"Appl. Intell."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"11833","DOI":"10.1007\/s10489-022-03994-3","article-title":"Sea-horse optimizer: A novel nature-inspired meta-heuristic for global optimization problems","volume":"53","author":"Zhao","year":"2023","journal-title":"Appl. Intell."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.knosys.2015.12.022","article-title":"SCA: A sine cosine algorithm for solving optimization problems","volume":"96","author":"Mirjalili","year":"2016","journal-title":"Knowl. Based Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s10462-023-10653-7","article-title":"A novel metaheuristic inspired by horned lizard defense tactics","volume":"57","author":"Sinha","year":"2024","journal-title":"Artif. Intell. Rev."},{"key":"ref_33","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_34","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_35","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s00521-015-1870-7","article-title":"Multi-Verse Optimizer: A nature-inspired algorithm for global optimization","volume":"27","author":"Mirjalili","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"116158","DOI":"10.1016\/j.eswa.2021.116158","article-title":"Reptile search algorithm (RSA): A nature-inspired meta-heuristic optimizer","volume":"191","author":"Abualigah","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10571","DOI":"10.1007\/s00500-023-08202-z","article-title":"GMO: Geometric mean optimizer for solving engineering problems","volume":"27","author":"Rezaei","year":"2023","journal-title":"Soft Comput."},{"key":"ref_38","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_39","doi-asserted-by":"crossref","first-page":"113609","DOI":"10.1016\/j.cma.2020.113609","article-title":"The arithmetic optimization algorithm","volume":"376","author":"Abualigah","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"116924","DOI":"10.1016\/j.eswa.2022.116924","article-title":"Golden jackal optimization: A novel nature-inspired optimizer for engineering applications","volume":"198","author":"Chopra","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"103282","DOI":"10.1016\/j.advengsoft.2022.103282","article-title":"Mountain gazelle optimizer: A new nature-inspired metaheuristic algorithm for global optimization problems","volume":"174","author":"Abdollahzadeh","year":"2022","journal-title":"Adv. Eng. Softw."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"107224","DOI":"10.1016\/j.cie.2021.107224","article-title":"Flow direction algorithm (FDA): A novel optimization approach for solving optimization problems","volume":"156","author":"Karami","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"121615","DOI":"10.1109\/ACCESS.2022.3223388","article-title":"Giant trevally optimizer (GTO): A novel metaheuristic algorithm for global optimization and challenging engineering problems","volume":"10","author":"Sadeeq","year":"2022","journal-title":"IEEE Access"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1115\/1.2912596","article-title":"Nonlinear Integer and Discrete Programming in Mechanical Design Optimization","volume":"112","author":"Sandgren","year":"1990","journal-title":"J. Mech. Des."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"100693","DOI":"10.1016\/j.swevo.2020.100693","article-title":"A test-suite of non-convex constrained optimization problems from the real-world and some baseline results","volume":"56","author":"Kumar","year":"2020","journal-title":"Swarm Evol. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Dehghani, M., Montazeri, Z., Dhiman, G., Malik, O., Morales-Menendez, R., Ramirez-Mendoza, R.A., Dehghani, A., Guerrero, J.M., and Parra-Arroyo, L. (2020). A spring search algorithm applied to engineering optimization problems. Appl. Sci., 10.","DOI":"10.3390\/app10186173"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4847","DOI":"10.1016\/j.matpr.2020.10.324","article-title":"Multi-objective optimization and numerical modelling of helical coil spring for automotive application","volume":"46","author":"Rahul","year":"2021","journal-title":"Mater. Today Proc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"419043","DOI":"10.1155\/2013\/419043","article-title":"Design optimization of a speed reducer using deterministic techniques","volume":"2013","author":"Lin","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_49","unstructured":"Tudose, L., Buiga, O., Jucan, D., and \u015etefanache, C. (2025, November 12). Optimal Design of Two-Stage Speed Reducer. Available online: https:\/\/www.semanticscholar.org\/paper\/Optimal-design-of-two-stage-speed-reducer-Tudose-Buiga\/66138b3cb8e6752f0c4f2619fc6cff35faa2702e."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/0094-114X(82)90049-0","article-title":"Weight minimization of a speed reducer by heuristic and decomposition techniques","volume":"17","author":"Datseris","year":"1982","journal-title":"Mech. Mach. Theory"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2623","DOI":"10.1061\/(ASCE)0733-9445(1994)120:9(2623)","article-title":"Design of cantilever steel beams: Refined approach","volume":"120","author":"Essa","year":"1994","journal-title":"J. Struct. Eng."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Canbaz, B., Yannou, B., and Yvars, P.A. (2011, January 28\u201331). A new framework for collaborative set-based design: Application to the design problem of a hollow cylindrical cantilever beam. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA.","DOI":"10.1115\/DETC2011-48153"}],"container-title":["Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-431X\/15\/1\/35\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:29:46Z","timestamp":1768292986000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-431X\/15\/1\/35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,7]]},"references-count":52,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["computers15010035"],"URL":"https:\/\/doi.org\/10.3390\/computers15010035","relation":{},"ISSN":["2073-431X"],"issn-type":[{"value":"2073-431X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,7]]}}}