An enhanced whale migration algorithm for its application in engineering problems

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper,we proposed an enhanced whale migration algorithm (EWMA) that integrates two novel strategies: Normal cloud model mutation (NCMM) and Fast Random Opposition- Based Learning. NCMM enables adaptive uncertainty management through expectation-entropy- hyperentropy mechanisms to balance exploration and exploitation. FROBL improves population diversity and convergence speed via oscillatory perturbations and nonlinear scaling. EWMA outperformed eight competing algorithms when evaluated on 23 benchmark functions and the CEC 2019, achieving optimal results on 18 of the 23 benchmark functions and all 10 CEC 2019 functions. It ranked first overall, significantly surpassing the other algorithms. Statistical analysis confirmed notable improvements in solution accuracy, convergence speed, and stability, with standard deviations 2–4 orders of magnitude lower than those of competitors. Engineering applications, including pressure vessel design, cantilever beams, and reinforced concrete beams, further demonstrated EWMA’s practical effectiveness, yielding optimal designs with improved constraint handling. EWMA offers a robust optimization tool for complex engineering problems requiring global search capability and precise local refinement.

Keywords

whale migration algorithm / normal cloud model / opposition-based learning / engineering optimization / metaheuristic algorithms

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] X. Yang, Nature-inspired optimization algorithms: Challenges and open problems, J. Comput. Sci., 46 (2020), 101104. https://doi.org/10.1016/j.jocs.2020.101104

[2] H. Ma, S. Shen, M. Yu, Z. Yang, M. Fei, H. Zhou, Multi-population techniques in nature inspired optimization algorithms: A comprehensive survey, Swarm Evol. Comput., 44 (2019), 365-387. https://doi.org/10.1016/j.swevo.2018.04.011

[3] W. Gao, Y. Wang, L. Liu, L. Huang, A gradient-based search method for multi-objective optimization problems, Inf. Sci., 578 (2021), 129-146. https://doi.org/10.1016/j.ins.2021.07.051

[4] L. Jiao, J. Zhao, C. Wang, X. Liu, F. Liu, L. Li, et al., Nature-inspired intelligent computing: A comprehensive survey, Research, 7 (2024), 0442. https://doi.org/10.34133/research.0442

[5] T. A. Le, X. Yang, Generalized firefly algorithm for optimal transmit beamforming, IEEE Trans. Wirel. Commun., 23 (2024), 5863-5877. https://doi.org/10.1109/TWC.2023.3328713

[6] G. R. Harik, F. G. Lobo, D. E. Goldberg, The compact genetic algorithm, IEEE Trans. Evol. Comput., 3 (1999), 287-297. https://doi.org/10.1109/4235.797971

[7] T. Blackwell, J. Kennedy, Impact of communication topology in particle swarm optimization, IEEE Trans. Evol. Comput., 23 (2019), 689-702. https://doi.org/10.1109/TEVC.2018.2880894

[8] X. Chen, C. Ye, Y. Zhang, Strengthened grey wolf optimization algorithms for numerical optimization tasks and AutoML, Swarm Evol. Comput., 94 (2025), 101891. https://doi.org/10.1016/j.swevo.2025.101891

[9] R. Storn, K. Price, Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim., 11 (1997), 341-359. https://doi.org/10.1023/A:1008202821328

[10] L. Ye, H. Ding, H. Xu, B. Xiang, Y. Wu, M. Gong, MFWOA: Multifactorial Whale Optimization Algorithm, Swarm Evol. Comput., 91 (2024), 101768. https://doi.org/10.1016/j.swevo.2024.101768

[11] J. Xue, B. Shen, Dung beetle optimizer: A new meta-heuristic algorithm for global optimization, J. Supercomput., 79 (2023), 7305-7336. https://doi.org/10.1007/s11227-022-04959-6

[12] R. Zhang, S. Mao, S. Zhao, C. Liu, An arithmetic optimization algorithm with balanced diversity and convergence for multimodal multiobjective optimization, Swarm Evol. Comput., 91 (2024), 101724. https://doi.org/10.1016/j.swevo.2024.101724

[13] Y. Zhu, H. Huang, J. Wei, J. Yi, J. Liu, M. Li, ISO: An improved snake optimizer with multi- strategy enhancement for engineering optimization, Expert Syst. Appl., 281 (2025), 127660. https://doi.org/10.1016/j.eswa.2025.127660

[14] X. Zhu, C. Jia, J. Zhao, C. Xia, W. Peng, J. Huang, et al., An enhanced artificial lemming algorithm and its application in UAV path planning, Biomimetics, 10 (2025), 377. https://doi.org/10.3390/biomimetics10060377

[15] Y. Xiao, H. Cui, A. G. Hussien, F. A. Hashim, MSAO: A multi-strategy boosted snow ablation optimizer for global optimization and real-world engineering applications, Adv. Eng. Inf., 61 (2024), 102464. https://doi.org/10.1016/j.aei.2024.102464

[16] Q. Fu, Q. Li, X. Li, H. Wang, J. Xie, Q. Wang, MOFS-REPLS: A large-scale multi-objective feature selection algorithm based on real-valued encoding and preference leadership strategy, Inf. Sci., 667 (2024), 120483. https://doi.org/10.1016/j.ins.2024.120483

[17] R. Wang, S. Zhang, G. Zou, An improved multi-strategy crayfish optimization algorithm for solving numerical optimization problems, Biomimetics, 9 (2024), 361. https://doi.org/10.3390/biomimetics9060361

[18] M. Chen, G. Zeng, K. Lu, A many-objective population extremal optimization algorithm with an adaptive hybrid mutation operation, Inf. Sci., 498 (2019), 62-90. https://doi.org/10.1016/j.ins.2019.05.048

[19] M. Zhou, M. Cui, D. Xu, S. Zhu, Z. Zhao, A. Abusorrah, Evolutionary optimization methods for high-dimensional expensive problems: A survey, IEEE/CAA J. Autom. Sin., 11 (2024), 1092-1105. https://doi.org/10.1109/JAS.2024.124320

[20] J. Zhao, D. Chen, R. Xiao, Z. Cui, H. Wang, I. Lee, Multi-strategy ensemble firefly algorithm with equilibrium of convergence and diversity, Appl. Soft Comput., 123 (2022), 108938. https://doi.org/10.1016/j.asoc.2022.108938

[21] M. Li, Y. Wang, J. Geng, W. Hong, Chaos cloud quantum bat hybrid optimization algorithm, Nonlinear Dyn., 103 (2021), 1167-1193. https://doi.org/10.1007/s11071-020-06111-6

[22] X. Yu, W. Xu, C. Li, Opposition-based learning grey wolf optimizer for global optimization, Knowl.-Based Syst., 226 (2021), 107139. https://doi.org/10.1016/j.knosys.2021.107139

[23] Z. Zhan, L. Shi, K. C. Tan, J. Zhang, A survey on evolutionary computation for complex continuous optimization, Artif. Intell. Rev., 55 (2022), 59-110. https://doi.org/10.1007/s10462-021-10042-y

[24] M. Ghasemi, M. Deriche, P. Trojovský, Z. Mansor, M. Zare, E. Trojovská, et al., An efficient bio- inspired algorithm based on humpback whale migration for constrained engineering optimization, Results Eng., 25 (2025), 104215. https://doi.org/10.1016/j.rineng.2025.104215

[25] S. Mohapatra, P. Mohapatra, Fast random opposition-based learning Golden Jackal Optimization algorithm, Knowl.-Based Syst., 275 (2023), 110679. https://doi.org/10.1016/j.knosys.2023.110679

[26] G. Hu, J. Zhong, C. Zhao, G. Wei, C. Chang, LCAHA: A hybrid artificial hummingbird algorithm with multi-strategy for engineering applications, Comput. Methods Appl. Mech. Eng., 415 (2023), 116238. https://doi.org/10.1016/j.cma.2023.116238

[27] X. Yang, R. Wang, D. Zhao, F. Yu, C. Huang, A. A. Heidari, et al., An adaptive quadratic interpolation and rounding mechanism sine cosine algorithm with application to constrained engineering optimization problems, Expert Syst. Appl., 213 (2023), 119041. https://doi.org/10.1016/j.eswa.2022.119041

[28] F. Gao, G. Liu, X. Wu, W. Liao, Optimization algorithm-based approach for modeling large deflection of cantilever beam subject to tip load, Mech. Mach. Theory, 167 (2022), 104522. https://doi.org/10.1016/j.mechmachtheory.2021.104522

[29] A Tian, F. Liu, H. Lv, Snow Geese Algorithm: A novel migration-inspired meta-heuristic algorithm for constrained engineering optimization problems, Appl. Math. Model., 126 (2024), 327-347. https://doi.org/10.1016/j.apm.2023.10.045

[30] A. Kumar, G. Wu, M. Z. Ali, R. Mallipeddi, P. N. Suganthan, S. Das, A test-suite of non-convex constrained optimization problems from the real-world and some baseline results, Swarm Evol. Comput., 56 (2020), 100693. https://doi.org/10.1016/j.swevo.2020.100693