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An efficient salp swarm algorithm based on scale-free informed followers with self-adaption weight

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Abstract

Meta-heuristic algorithms are often leveraged to solve complicated engineering optimization and scientific problems. Salp swarm algorithm is one of the most useful meta-heuristic algorithms in recent years. To alleviate the slow convergence speed of the salp swarm algorithm, as well as the tendency to fall into local minima, we have proposed an efficient salp swarm algorithm called E-SSA, which combines the effective evolutionary strategies of basic salp swarm algorithm and two efficient mechanisms named self-adaption weight and scale-free network. These two mechanisms have been integrated into the follower evolution process of the algorithm to achieve the balance of exploration and exploitation. The performance of the E-SSA is benchmarked against a suit of CEC’2019 series functions and 23 commonly used international benchmarks. The algorithm is further validated via three engineering application problems. The experimental results indicate that the improved algorithm has clear advantages in optimization performance compared with other existing heuristic algorithms.

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Acknowledgments

This work was supported by the Program for Synergy Innovation in the Anhui Higher Education Institutions of China (Grant No. GXXT-2021-044), Scientific Research Foundation of Education Department of Anhui Province, China (Grant No. KJ2021A0506), Natural Science Foundation of Anhui Province, China(Grant No. 2108085MG237), Open Fund of Key Laboratory of Anhui Higher Education Institutes, China(Grant No.CS2021-02), Science and Technology Planning Project of Wuhu City, Anhui Province, China(Grant No. 2021jc1-2) and Research Start-Up Fund for Introducing Talents from Anhui Polytechnic University(Grant No. 2021YQQ066).

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Correspondence to Lu Wang or Kang Hao Cheong.

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Appendices

Appendix A

Table 22 Unimodal test functions
Table 23 Multimodal test functions
Table 24 Complex multimodal test function in fixed dimensions
Table 25 CEC2019 multimodal test function
Table 26 CEC’2019 multimodal test function of policy effectiveness results statistics table

Appendix B

I-The tension/compression spring design problem

The optimization objective is

$$ \text{Minimize}\qquad f(x) = (P + 2)\textit{D}\textit{d}^{2} $$
(B.1)

The corresponding constraints are

$$ g_{1} (x)=1-\frac{\textit{D}^{3} P}{71785\textit{d}^{4}} \leq 0 $$
(B.2a)
$$ g_{2} (x)=\frac{4\textit{D}^{2} - \textit{d} \textit{D}}{12566(\textit{D} \textit{d}^{3} - \textit{d}^{4})} + \frac{1}{5108\textit{d}^{2}} - 1 \leq 0 $$
(B.2b)
$$ g_{3} (x)=1 - \frac{140.45\textit{d}}{\textit{D}^{2} P} \leq 0 $$
(B.2c)
$$ g_{4} (x)=\frac{\textit{d} + \textit{D}}{1.5} - 1 \leq 0\ $$
(B.2d)

The range of variables is as follows

$$ 0.05 \leq \textit{d} \leq 2.00 $$
(B.3a)
$$ 0.25 \leq \textit{D} \leq 1.30 $$
(B.3b)
$$ 2.00 \leq P \leq 15.00 $$
(B.3c)

II-The cantilever beam design problem

The optimization objective is

$$ \text{Minimize}\qquad f(x_{1},x_{2},x_{3},x_{4},x_{5})=0.0624(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}) $$
(B.4)

The corresponding constraint is

$$ h_{1}=\frac{61}{{x_{1}^{3}}}+\frac{37}{{x_{2}^{3}}}+\frac{19}{{x_{3}^{3}}}+\frac{7}{{x_{4}^{3}}}+\frac{1}{{x_{5}^{3}}}-1\le 0 $$
(B.5)

III-The three-bar truss design problem

The optimization objective is

$$ \text{Minimize}\qquad f(\textit{A}_{1} , \textit{A}_{2}) = (2\sqrt{2} \textit{A}_{1} + \textit{A}_{2}) \times \textit{H} $$
(B.6)

The corresponding constraint is

$$ t_{1} =\frac{\sqrt{2} \textit{A}_{1} + \textit{A}_{2}}{\sqrt{2} \textit{A}_{1}^{2} + 2 \textit{A}_{1} \textit{A}_{2}} p - \sigma \leq 0 $$
(B.7a)
$$ t_{2} =\frac{\textit{A}_{2}}{\sqrt{2} \textit{A}_{1}^{2} + 2 \textit{A}_{1} \textit{A}_{2}} p - \sigma \leq 0 $$
(B.7b)
$$ t_{3} =\frac{1}{\textit{A}_{1} + \sqrt{2} \textit{A}_{2}} p - \sigma \leq 0 $$
(B.7c)

The range of variables is as follows

$$ 0 \leq \textit{A}_{1} \leq 1 $$
(B.8a)
$$ 0 \leq \textit{A}_{2} \leq 1 $$
(B.8b)

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Wang, C., Xu, Rq., Ma, L. et al. An efficient salp swarm algorithm based on scale-free informed followers with self-adaption weight. Appl Intell 53, 1759–1791 (2023). https://doi.org/10.1007/s10489-022-03438-y

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