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Enhancing the Whale Optimisation Algorithm with sub-population and hybrid techniques for single- and multi-objective optimisation

  • Optimization
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Abstract

The Whale Optimisation Algorithm (WOA) is a meta-heuristic model inspired by the hunting behaviours of humpback whales. Similar to many other meta-heuristic models, e.g., Particle Swarm Optimisation (PSO), Artificial Bee Colony (ABC), and Ant Colony Optimisation (ACO), the WOA is susceptible to the issues of slow convergence and local optima. In this study, we address these short comings by first proposing an Enhanced WOA (EWOA) model for tackling single-objective optimisation. Specifically, EWOA integrates the WOA and Differential Evolution (DE). DE is a population-based algorithm that generates new candidate solutions by combining the existing ones, employing a simple yet robust formula. This amalgamation aids in generating diverse solutions during the exploration stage by utilising a non-linear coefficient vector, adaptive weight, and sub-population strategies. Furthermore, fast non-dominated sorting and crowding distance techniques from the Non-dominated Sorting Genetic Algorithm II (NSGA-II) are incorporated into EWOA, resulting in a multi-objective EWOA (MOEWOA) model. We evaluate both EWOA and MOEWOA with a broad spectrum of benchmark functions. The results from 51 single-objective optimisation problems indicate the usefulness of EWOA in terms of a fast convergence rate and with increased performance. On the other hand, MOEWOA demonstrates a better convergence rate and an effective balance between convergence and diversity in 12 multi-objective optimisation problems. In addition, MOEWOA successfully solves 21 complex multi-objective constrained mechanical design problems, outperforming other compared algorithms at the 95% confidence level. The empirical outcomes of our study indicate the potential of EWOA and MOEWOA for undertaking complex, real-world optimisation problems.

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  1. Yit Hong Choo, Vu Le, Chee Peng Lim and Mingyu Liao have contributed equally to this work.

Authors and Affiliations

  1. Institute for Intelligent Systems Research and Innovation, Deakin University, Geelong, Australia

    Zheng Cai, Yit Hong Choo, Vu Le & Chee Peng Lim

  2. Department of International Business, National Kaohsiung University of Science and Technology, Kaohsiung, Taiwan, R.O.C.

    Mingyu Liao

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  1. Zheng Cai
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Appendices

Appendix A: Pseudocodes of WOA and MOEWOA

Algorithm 1
figure a

Whale Optimisation Algorithm (WOA)

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Algorithm 2
figure b

Multi-objective EWOA

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Table 14 Description of uni-modal benchmark functions in Yao et al. (1999); Molga and Smutnicki (2005); Digalakis and Margaritis (2001); Yang (2010)
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Table 15 Description of multi-modal benchmark functions in Yao et al. (1999); Molga and Smutnicki (2005); Digalakis and Margaritis (2001); Yang (2010)
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Table 16 Description of fixed-dimension multi-modal benchmark functions in Yao et al. (1999); Molga and Smutnicki (2005); Digalakis and Margaritis (2001); Yang (2010)
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Table 17 Description of benchmark functions (F1–F28) in CEC 2013
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Appendix B: Single-objective benchmark functions

See Tables 14, 15, 16 and 17.

Table 18 Results of benchmarks functions (F1–F23) in Yao et al. (1999); Molga and Smutnicki (2005); Digalakis and Margaritis (2001); Yang (2010) for EWOA vs its competitors in Mirjalili and Lewis (2016)
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Table 19 Mean values and ranking of EWOA and several popular algorithms in Wang et al. (2022) for benchmark functions (F1–F28) in CEC 2013 competition
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Appendix C: Quantitative results for single-objective functions

See Tables 18, 19,

Appendix D: Quantitative results for multi-objective functions

See Tables 20, 21, 22 and 23

Table 20 Mean values, standard deviation, and ranking of MOEWOA and other well-known algorithms in Liu et al. (2020) in term of \(\gamma \) for all test problems
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Table 21 Mean values, standard deviation, and ranking of MOEWOA and other well-known algorithms in Liu et al. (2020) in term of \(\Delta \) for all test problems
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Table 22 Mean values, standard deviation, and ranking of MOEWOA and other well-known algorithms in Liu et al. (2020) in term of GD for all test problems
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Table 23 Mean values, standard deviation, and ranking of MOEWOA and other well-known algorithms in Liu et al. (2020) in term of IGD for all test problems
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Table 24 Description of 21 real-world constrained multi-objective mechanical design problems (RCM01–RCM21) in Kumar et al. (2021)
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Table 25 Mean values and ranking of MOEWOA and other well-known algorithms in Kumar et al. (2021); Premkumar et al. (2021) in term of HV for all mechanical design problems in Kumar et al. (2021)
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Appendix E: Real-world case study

See Tables 24, 25

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Cai, Z., Choo, Y.H., Le, V. et al. Enhancing the Whale Optimisation Algorithm with sub-population and hybrid techniques for single- and multi-objective optimisation. Soft Comput 28, 3941–3971 (2024). https://doi.org/10.1007/s00500-023-09351-x

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