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A mixed sine cosine butterfly optimization algorithm for global optimization and its application

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Abstract

This paper proposes a hybrid sine cosine butterfly optimization algorithm (m-SCBOA), in which a modified butterfly optimization algorithm is combined with sine cosine algorithm to achieve superior exploratory and exploitative search capabilities. The newly suggested m-SCBOA algorithm has been tested on 39 benchmark functions and compared with seven state-of-the-art algorithms. Moreover, it is tested with the CEC 2017 test suite, and the results are compared with seven more algorithms. Experimental results have demonstrated superior results of the proposed algorithm in nearly 75% of occasions on average whereas, similar results in nearly 20% of cases. Simulation results and convergence graphs show the supremacy of the m-SCBOA over the compared algorithms. The Friedman rank test validates that the m-SCBOA is statistically the best among the experiments. Additionally, four real-world engineering design problems and a priori multi-objective problem called parameter optimization of Al–4.5% Cu–TiC metal matrix composite are solved and the outcomes are compared to a wide range of algorithms. The results of these case studies establish the merits of m-SCBOA in solving challenging, real-world problems with unknown global optima.

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All data generated or analyzed during this study are included in this paper.

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The codes associated with this study are available upon reasonable request.

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Acknowledgements

The authors are extremely thankful to the editor and the reviewers for their valuable suggestions and comments, which helped improve the manuscript.

Funding

N/A.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Agartala, Agartala, Tripura, 799046, India

    Sushmita Sharma, Apu Kumar Saha & Susmita Roy

  2. Centre for Artificial Intelligence Research and Optimisation, Torrens University Australia, Fortitude Valley, Brisbane, QLD, 4006, Australia

    Seyedali Mirjalili

  3. Yonsei Frontier Lab, Yonsei University, Seoul, Korea

    Seyedali Mirjalili

  4. Department of Science & Humanities, Gomati District Polytechnic, Udaipur, Tripura, 799013, India

    Sukanta Nama

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  1. Sushmita Sharma
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  2. Apu Kumar Saha
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Correspondence to Apu Kumar Saha.

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Appendices

A Appendix

Thirty-nine classical benchmark functions used in this experiment (Fmin: Global minima, C: Convex, U: Unimodal, M: Multimodal, S: Separable, N: Non-separable, D: Dimension, Nc: Non convex, d: Differentiable, C: Continuous).

F_No

Function

Dimension

[Range]

F_min

Characteristics

 

Ackley

30

[− 32.768, 32.768]

0

M, N, C, d

 

Cross in tray

2

[− 10, 10]

 − 2.06261

M, Nc, Nd, N

 

Drop Wave

2

[− 5.12, 5.12]

 − 1

U, Nc

 

Gramacy & Lee

1

[0.5, 2]

 − 0.869011134989500

M, C, Nc

 

Griewank

30

[− 600, 600]

0

M, C, N, Nc

 

Langerman

2

[0, 10]

 

M, N

 

Levy

30

[− 10, 10]

0

M, S

 

Rastrigin

30

[− 5.12, 5.12]

0

M, S

 

Schaffer2

2

[− 100, 100]

0

M, N

 

Shubert

2

[− 10, 10]

 − 186.7309

M, S

 

Bohachevsky1

2

[− 100, 100]

0

M, S

 

Bohachevsky2

2

[− 100, 100]

0

M, N

 

Perm 0, d, Beta

30

[− 30, 30]

0

M, N

 

Rotated Hyper-Ellipsoid

30

[− 65.536, 65.536]

 

U

 

Sphere

30

[− 5.12, 5.12]

0

U, S

 

Modified sphere

30

[− 5.12, 5.12]

0

U, S

 

Sum of Different Powers

30

[− 1, 1]

0

U

 

Booth

2

[− 10, 10]

0

M, S

 

MATYAS

2

[− 10, 10]

0

U, N

 

Power Sum

4

[0, 4]

0

M, N

 

Three-Hump Camel

2

[− 5, 5]

0

M, N

 

Dixon-Price

30

[− 10, 10]

 

U, N

 

Rosenbrock

30

[− 5, 10]

0

U, N

 

Modified Rosenbrock

4

[− 5, 10]

0

U, N

 

De Jong5

2

[− 65.536, 65.536]

 

M

 

Easom

2

[− 100, 100]

 − 1

M, S

 

Michalewicz

10

[0, \(\pi\)]

 − 0.966015

M, S

 

Beale

2

[− 4.5, 4.5]

0

U, N

 

Colville

4

[− 10, 10]

0

U, N

 

Goldstein-Price

2

[− 2, 2]

3

M, N

 

Rescaled Goldstein-Price

2

[− 2, 2]

3

M, N

 

Hartmann 3

3

[0, 1]

 − 3.862782

M, N

 

Hartmann 4

4

[0, 1]

 

M, N

 

Hartmann 6

6

[0, 1]

 − 3.32236

M, N

 

RescaledHartmann 6

3

[0, 1]

 − 3.32237

M, N

 

Powell

30

[− 4, 5]

0

U, N

 

Schwefel 1.2

20

[− 100, 100]

0

U, N

 

Schwefel 2.21

30

[− 10, 10]

0

U, S

 

Schwefel2.22

30

[− 10, 10]

0

U, N

B Appendix

2.1 B.1 Multi-plate disc clutch brake problem

$$Minimize f\left(y\right)=\pi \left({y}_{2}^{2}-{y}_{1}^{2}\right){y}_{3}({y}_{5}+1)\rho$$

Subject to \({c}_{1}\left(y\right)={y}_{2}-{y}_{1}-\Delta R\ge 0\)

$${c}_{2}\left(y\right)={l}_{max}-({y}_{5}+1)({y}_{3}+\delta )\ge 0,$$
$${c}_{3}\left(y\right)={P}_{max}-{P}_{rz}\ge 0,$$
$${c}_{4}\left(y\right)={P}_{mx}{v}_{sr max}-{P}_{rz}{v}_{sr}\ge 0,$$
$${c}_{5}\left(y\right)={v}_{sr max}-{v}_{sr}\ge 0,$$
$${c}_{6}\left(y\right)={T}_{max}-T\ge 0,$$
$${c}_{7}\left(y\right)={M}_{h}-s{M}_{s}\ge 0,$$
$${c}_{8}\left(y\right)=T\ge 0,$$

where \({M}_{h}=\frac{2}{3}\mu {y}_{4}{y}_{5}\frac{{y}_{2}^{3}-{y}_{1}^{3}}{{y}_{2}^{2}-{y}_{1}^{2}}\)

$$w=\frac{\pi n}{30} \mathrm{rad}/\mathrm{s},$$
$$A=\pi ({y}_{2}^{2}-{y}_{1}^{2}){\mathrm{ mm}}^{2},$$
$${p}_{rz}=\frac{{y}_{4}}{A} \mathrm{N}/{\mathrm{mm}}^{2},$$
$${v}_{sr}=\frac{{\pi R}_{sr}n}{30} \mathrm{mm}/\mathrm{s},$$
$${R}_{sr}=\frac{2}{3}\frac{{y}_{2}^{3}-{y}_{1}^{3}}{{y}_{2}^{2}-{y}_{1}^{2}} \mathrm{mm},$$
$$T=\frac{{I}_{z}\pi n}{30\left({M}_{h}+{M}_{f}\right)} \mathrm{mm},$$
$$\Delta R=20 \mathrm{mm} {L}_{max}=30 \mathrm{mm}, \mu =0.6,$$
$${T}_{max}=15 \mathrm{s}, s= 1.5, {M}_{s}=40 \mathrm{Nm},{P}_{max}=1 \mathrm{MPa}, \rho =0.0000078\frac{\mathrm{kg}}{{\mathrm{mm}}^{3}},$$
$${v}_{sr max}=10 \frac{\mathrm{m}}{\mathrm{s}} , \delta =0.5 \mathrm{mm}, s=1.5 n=250 \mathrm{rpm}, {I}_{z}=55 \mathrm{kg} {\mathrm{m}}^{2}, {M}_{s}=40 \mathrm{Nm}, {M}_{f}=3 \mathrm{Nm},$$
$$60\le {y}_{1}\le 80, 90\le {y}_{2}\le 110, 1\le {y}_{3}\le 3,$$
$$60\le {y}_{4}\le 80, 90\le {y}_{5}\le 110, i=1, 2, 3, 4, 5.$$

2.2 B.2 Speed reducer problem

$$Minimize f\left(y\right)=0.785{y}_{1}{y}_{2}^{3}\left(3.333{y}_{3}^{2}+14.9334{y}_{3}-42.0934\right),$$
$${c}_{1}\left(y\right)=\frac{27}{{y}_{1}{y}_{2}^{2}{y}_{3}}-1\le 0,$$
$${c}_{2}\left(y\right)=\frac{397.5}{{y}_{1}{y}_{3}^{2}{y}_{2}}-1\le 0,$$
$${c}_{3}\left(y\right)=\frac{1.93{y}_{4}^{3}}{{y}_{1}{y}_{6}^{4}{y}_{3}}-1\le 0,$$
$${c}_{4}\left(y\right)=\frac{1.93{y}_{4}^{3}}{{y}_{1}{y}_{7}^{4}{y}_{3}}-1\le 0,$$
$${c}_{5}\left(y\right)=\frac{1}{110{y}_{6}^{3}}\sqrt{{\left(\frac{745{y}_{4}}{{y}_{2}{y}_{3}}\right)}^{2}+16.9*{10}^{6}}-1\le 0,$$
$${c}_{6}\left(y\right)=\frac{1}{85{y}_{7}^{3}}\sqrt{{\left(\frac{745{y}_{4}}{{y}_{2}{y}_{3}}\right)}^{2}+157.5*{10}^{6}}-1\le 0.$$

2.3 B.3 Car side impact design problem

$$Minimize f\left(u\right)=1.98+4.90 u\left(1\right)+6.67 u\left(2\right)+6.98 u\left(3\right)+4.01 u\left(4\right)+1.78 u\left(5\right)+2.73 u\left(7\right),$$

Subject to

$${c}_{1}\left(u\right)=1.16-0.3717 u\left(2\right) u\left(4\right)-0.00931 u\left(2\right) u\left(10\right)-0.484 u\left(3\right) u\left(9\right)+0.01343 u\left(6\right) u\left(10\right)\le 1,$$
$${c}_{2}\left(u\right)=0.261-0.0159 u\left(1\right) u\left(2\right)-0.188 u\left(1\right) u\left(8\right)-0.019 u\left(2\right) u\left(7\right)+0.0144 u\left(3\right) u\left(5\right) + 0.0008757 u\left(5\right) u\left(10\right)+0.080405 u\left(6\right) u\left(9\right)+0.00139 u\left(8\right)u\left(11\right)+ 0.00001575 u\left(10\right) u\left(11\right)\le 0.32,$$
$${c}_{3}\left(u\right)=0.214+0.00817 u\left(5\right)-0.131 u\left(1\right) u\left(8\right)-0.0704 u\left(1\right) u\left(9\right)+0.03099 u\left(2\right)u\left(6\right) -0.018 u\left(2\right) u\left(7\right)+0.0208 u\left(3\right) y\left(8\right)+0.121u\left(3\right) u\left(9\right)-0.00364 u\left(5\right) u\left(6\right)+ 0.0007715 u\left(5\right) u\left(10\right)-0.0005354 u\left(6\right) u\left(10\right)+0.00121 u\left(8\right) u\left(11\right)\le 0.32,$$
$${c}_{4}\left(u\right)=0.074-0.061 u\left(2\right)-0.163 u\left(3\right) u\left(8\right)+0.001232 u\left(3\right) u\left(10\right)-0.166 u\left(7\right)u\left(9\right) +0.227u{\left(2\right)}^{2} \le 0.32,$$
$${c}_{5}\left(u\right)=28.98+3.818 u\left(3\right)-4.2u\left(1\right) u\left(2\right)+0.0207u\left(5\right) u\left(10\right)+6.63u\left(6\right) u\left(9\right)-7.7 u\left(7\right) u\left(8\right)+0.32 u\left(9\right) u\left(10\right)\le 32,$$
$${c}_{6}\left(u\right)=33.86+2.95 u\left(3\right)+0.1792 u\left(10\right)-5.05 u\left(1\right) u\left(2\right)-11.0 u\left(2\right) u\left(8\right)- 0.0215 u\left(5\right) u\left(10\right)-9.98 u\left(7\right) u\left(8\right)+22.0 u\left(8\right) u\left(9\right)\le 32,$$
$${c}_{7}\left(u\right)=46.36-9.9 u\left(2\right)-12.9 u\left(1\right) u\left(8\right)+0.1107 u\left(3\right) u\left(10\right)\le 32,$$
$${c}_{8}\left(u\right)=4.72-0.5 u\left(4\right)-0.19 u\left(2\right) u\left(3\right)-0.0122 u\left(4\right) u\left(10\right)+0.009325 u\left(6\right) u\left(10\right) +0.000191 u{\left(11\right)}^{2}\le 4,$$
$${c}_{9}\left(u\right)=10.58-0.674 u\left(1\right) u\left(2\right)-1.95 u\left(2\right) u\left(8\right)+0.02054 u\left(3\right) u\left(10\right)-0.0198 u\left(4\right) u\left(10\right)+0.028 u\left(6\right) u\left(10\right)\le 9.9,$$
$${c}_{10}\left(u\right)=16.45-0.489 u\left(3\right) u\left(7\right)-0.843 u\left(5\right) u\left(6\right)+0.0432 u\left(9\right) u\left(10\right)-0.0556 u\left(9\right) u\left(11\right)-0.000786 u{\left(11\right)}^{2}\le 15.7,$$

where \(0.5\le u\left(1\right)-u\left(7\right)\le 1.5\)

$$u\left(8\right), u\left(9\right)\in \left(0.192, 0.345\right),$$
$$-30\le u\left(10\right),u\left(11\right)\le 30.$$

2.4 B.4 Cantilever beam design problem

$$Minimize f\left(y\right)=0.6224({y}_{1}+{y}_{2}+{y}_{3}+{y}_{4}+{y}_{5}$$

Subject to \(c\left(y\right)=\frac{61}{{y}_{1}^{3}}+\frac{37}{{y}_{2}^{3}}+\frac{19}{{y}_{3}^{3}}+\frac{7}{{y}_{4}^{3}}+\frac{1}{{y}_{5}^{3}}\le 1\)

Variables range 0.01 ≤ \({y}_{1}to {y}_{5}\le 100\)

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Sharma, S., Saha, A.K., Roy, S. et al. A mixed sine cosine butterfly optimization algorithm for global optimization and its application. Cluster Comput 25, 4573–4600 (2022). https://doi.org/10.1007/s10586-022-03649-5

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