Abstract
Moth flame optimization (MFO) algorithm is a relatively new nature-inspired optimization algorithm based on the moth’s movement towards the moon. Premature convergence and convergence to local optima are the main demerits of the algorithm. To avoid these drawbacks, a modified dynamic opposite learning-based MFO algorithm (m-DMFO) is presented in this paper, incorporating a modified dynamic opposite learning (DOL) strategy. To validate the performance of the proposed m-DMFO algorithm, it is tested via twenty-three benchmark functions, IEEE CEC’2014 test functions and compared with a wide range of optimization algorithms. Moreover, Friedman rank test, Wilcoxon rank test, convergence analysis, and diversity measurement have been conducted to measure the robustness of the proposed m-DMFO algorithm. The numerical results show that, the proposed m-DMFO algorithm achieved superior results in more than 90% occasions. The proposed m-DMFO achieves the best rank in Friedman rank test and Wilcoxon rank test respectively. In addition, four engineering design problems have been solved by the suggested m-DMFO algorithm. According to the results, it achieves extremely impressive results, which also illustrates that the algorithm is qualified in solving real-world problems. Analyses of numerical results, diversity measure, statistical tests and convergence results ensure the enhanced performance of the proposed m-DMFO algorithm.
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Appendices
Appendix 1: Formulation of 23 benchmark functions
Sl. no. | Functions | Formulation of objective functions | d | Fmin | Search space |
---|---|---|---|---|---|
Unimodal benchmark functions | |||||
F1 | Beale | \(\mathrm{f}\left(\mathrm{x}\right)={\left(1.5-{\mathrm{x}}_{1}+{{\mathrm{x}}_{1}\mathrm{x}}_{2}\right)}^{2}+{\left(2.25-{\mathrm{x}}_{1}+{\mathrm{x}}_{1}{x}_{2}^{2}\right)}^{2}+{ \left(2.625-{\mathrm{x}}_{1}+{\mathrm{x}}_{1}{x}_{2}^{3}\right)}^{2}\) | 2 | 0 | [− 100, 100] |
F2 | Booth | \(\mathrm{f}\left(\mathrm{x}\right)={\left(2{\mathrm{x}}_{1}+{\mathrm{x}}_{2}-5\right)}^{2}+{\left({\mathrm{x}}_{1}+2{\mathrm{x}}_{2}-7\right)}^{2}\) | 2 | 0 | [− 10, 10] |
F3 | Matyas | \(\mathrm{f}\left(\mathrm{x}\right)=0.26\left({{\mathrm{x}}_{1}}^{2}+{{\mathrm{x}}_{2}}^{2}\right)-0.48{\mathrm{x}}_{1}{\mathrm{x}}_{2}\) | 2 | 0 | [− 10, 10] |
F4 | Sumsquare | \(f\left(x\right)= \sum\limits_{i=1}^{D}{{x}_{i}}^{2}\times i\) | 30 | 0 | [− 10, 10] |
F5 | Zettl | \(\mathrm{f}\left(\mathrm{x}\right)={\left(\mathrm{x}-{1}^{2}+\mathrm{x}-{2}^{2}-2{\mathrm{x}}_{1}\right)}^{2}+0.25{\mathrm{x}}_{1}\) | 2 | − 0.00379 | [− 1, 5] |
F6 | Leon | \(\mathrm{f}\left(\mathrm{x}\right)= 100{\left({\mathrm{x}}_{2}-{{\mathrm{x}}_{1}}^{3}\right)}^{2}+{\left(1-{\mathrm{x}}_{1}\right)}^{2}\) | 2 | 0 | [− 1.2, 1.2] |
F7 | Zakhrov | \(\mathrm{f}\left(\mathrm{x}\right)=\sum\limits_{\mathrm{j}=1}^{\mathrm{d}}{{\mathrm{x}}_{\mathrm{i}}}^{2}+{\left(0.5\sum\limits_{\mathrm{j}=1}^{\mathrm{d}}{\mathrm{jx}}_{\mathrm{j}}\right)}^{2}+{\left(0.5\sum\limits_{\mathrm{j}=1}^{\mathrm{d}}{\mathrm{jx}}_{\mathrm{j}}\right)}^{4}\) | 2 | 0 | [− 5, 10] |
Multimodal benchmark functions | |||||
F8 | Bohachevsky | \(\mathrm{f}\left(\mathrm{x}\right)={{\mathrm{x}}_{1}}^{2}+2{{\mathrm{x}}_{2}}^{2}-0.3\mathrm{cos}\left(3\uppi {\mathrm{x}}_{1}\right)-0.3\) | 2 | 0 | [− 100, 100] |
F9 | Bohachevsky 3 | \(\mathrm{f}\left(\mathrm{x}\right)={{\mathrm{x}}_{1}}^{2}+2{{\mathrm{x}}_{2}}^{2}-0.3\mathrm{cos}\left(3\uppi {\mathrm{x}}_{1}\right)-0.3\) | 2 | 0 | [− 50, 50] |
F10 | Levy | \(f\left(x\right)={sin}^{2}\left(\pi {x}_{1}\right)+\sum\limits_{i=1}^{D-1}{\left({x}_{i}-1\right)}^{2}\left[1+10{sin}^{2}\left(\pi {x}_{i}+1\right)\right]+{\left({x}_{D}-1\right)}^{2}\left[1+{sin}^{2}\left(2\pi {x}_{D}\right)\right]\) Where, \({x}_{i}=1+\frac{1}{4}\)(\({x}_{i}-1), i=\mathrm{1,2},\dots \dots \dots D\) | 30 | 0 | [− 10, 10] |
F11 | Michalewicz | \(f\left(x\right)=-\sum\limits_{i=1}^{D}\mathrm{sin}{(x}_{i}){sin}^{2m}(\frac{{i{x}_{i}}^{2}}{\pi })\), m = 10 | 10 | − 9.66015 | [0, \(\pi\)] |
F12 | Alpine | \(f\left(x\right)=\sum\limits_{i=1}^{D}\left|{x}_{i}\mathrm{sin}{(x}_{i})+0.1{x}_{i}\right|\) | 30 | 0 | [− 10, 10] |
F13 | Schaffers | \(f\left(x\right)=0.5+\frac{{sin}^{2}\left({{x}_{1}}^{2}+{{x}_{2}}^{2}\right)-0.5}{{\left[1+0.001\left({{x}_{1}}^{2}+{{x}_{2}}^{2}\right)\right]}^{2}}\) | 2 | 0 | [− 100, 100] |
F14 | Powersum | \(f\left(x\right)= \sum\limits_{i=1}^{D}\left[{\left(\sum\limits_{k=1}^{D}{{(x}_{k}}^{i})-{b}_{i}\right)}^{2}\right]\) | 30 | 0 | [− 10, 10] |
F15 | Penalized2 | \(f\left(x\right)=0.1\left\{ 10{sin}^{2}\left(\pi {x}_{i}\right)+\sum\limits_{i=1}^{D-1}{\left({x}_{i}-1\right)}^{2}[1+10{sin}^{2}\left(3\pi {x}_{i+1}\right)+{\left({x}_{D}-1\right)}^{2}[1+{sin}^{2}\left(2\pi {x}_{D}\right)]]\right\}+\sum\limits_{i=1}^{D}u\left({x}_{i},\mathrm{5,100,4}\right)\) | 30 | 0 | [− 50, 50] |
F16 | Kowalik | \(\mathrm{f}\left(\mathrm{x}\right)=\sum\limits_{\mathrm{j}=1}^{11}{\left[{\mathrm{a}}_{\mathrm{j}}-\frac{{\mathrm{x}}_{1}\left({{\mathrm{b}}_{\mathrm{j}}}^{2}+{\mathrm{b}}_{\mathrm{j}}{\mathrm{x}}_{2}\right)}{({{\mathrm{b}}_{\mathrm{j}}}^{2}-{\mathrm{b}}_{\mathrm{j}}{\mathrm{x}}_{3}-{\mathrm{x}}_{4}}\right]}^{2}\) | 4 | 0.0003075 | [− 5, 5] |
F17 | Foxholes | \(\mathrm{f}\left(\mathrm{x}\right)={\left[\frac{1}{500}+ \sum\limits_{\mathrm{j}=1}^{25}\frac{1}{\mathrm{j}}+\sum\limits_{\mathrm{i}=1}^{\mathrm{D}}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{ij}}\right)}^{6}\right]}^{-1}\) | 2 | 3 | [− 65, 65] |
Fixed dimension multimodal benchmark functions | |||||
F18 | Goldstein and Price | \(f\left(x\right)=\left[1+{\left(1+{x}_{1}+{x}_{2}\right)}^{2}\left(10-14{x}_{1}-14{x}_{2}+6{x}_{1}{x}_{2}+3{{x}_{1}}^{2}+3{{x}_{2}}^{2}\right)\right]\times \left[30+\left(2{x}_{1}-3{{x}_{2}}^{2}\right)\left(18-32{x}_{1}+12{{x}_{1}}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{{x}_{2}}^{2}\right)\right]\) | 2 | 3 | [− 2, 2] |
F19 | Hartmann3 | \(f\left(x\right)=-\sum\limits_{i=1}^{4}{\alpha }_{i}\mathrm{exp}(-\sum\limits_{j=1}^{3}{a}_{ij}{\left({x}_{j-{b}_{ij}}\right)}^{2})\) | 3 | − 3.86 | [0, 1] |
F20 | Hartmann6 | \(f\left(x\right)=-\sum\limits_{i=1}^{4}{\alpha }_{i}\mathrm{exp}(-\sum\limits_{j=1}^{6}{a}_{ij}{\left({x}_{j-{b}_{ij}}\right)}^{2})\) | 6 | − 3.32 | [0, 1] |
F21 | Shekel 5 | \(\mathrm{f}\left(\mathrm{x}\right)=-\sum\limits_{\mathrm{j}=1}^{5}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{j}}\right]}^{-1}\) | 4 | − 10.1499 | [0, 10] |
F22 | Shekel-7 | \(\mathrm{f}\left(\mathrm{x}\right)=-\sum\limits_{\mathrm{j}=1}^{7}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{j}}\right]}^{-1}\) | 4 | − 10.3999 | [0, 10] |
F23 | Shekel-10 | \(\mathrm{f}\left(\mathrm{x}\right)=-\sum\limits_{\mathrm{j}=1}^{10}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{j}}\right]}^{-1}\) | 4 | − 10.5319 | [0, 10] |
Appendix 2: Three-bar truss design problem
Objective function:
Subject to:
where, \(0\le {k}_{1},{k}_{2}\le 1,and\) \(P=2, L=100 \, \& \, \sigma =2.\)
Appendix 3: Tension/compression spring design problem
Let us consider
\(\overrightarrow{{\varvec{x}}}=\left[{x}_{1} {x}_{2} {x}_{3}\right]\) = [d D N],
Minimize f (\(\overrightarrow{{\varvec{x}}}\)) =\({{x}_{1}}^{2}{x}_{2}\) (2 + \({x}_{3}\)),
Subjected to \({g}_{1}\left(\overrightarrow{{\varvec{x}}}\right)=\) 71,785 \({{x}_{1}}^{4}-{{x}_{2}}^{3}{x}_{3}\le 0\)
where, \(0.05\le {x}_{1}\le 2.00\), \(0.25\le {x}_{2}\le 1.30\), 2 \(\le {x}_{3}\le 15.0\)
Appendix 4: Cantilever beam design problem
Let us consider
\(\overrightarrow{{\varvec{x}}}=\left[{x}_{1} {x}_{2} {x}_{3} {x}_{4} {x}_{5}\right]\) = [d D N],
Minimize f (\(\overrightarrow{{\varvec{x}}}\)) = \(0.6224\left({x}_{1}+ {x}_{2}+{x}_{3}+{x}_{4}+ {x}_{5}\right)\), Subjected to g \(\left(\overrightarrow{{\varvec{x}}}\right)= \frac{61}{{{x}_{1}}^{3}}+ \frac{27}{{{x}_{2}}^{3}}+\frac{19}{{{x}_{3}}^{3}}+\frac{7}{{{x}_{4}}^{3}}+\frac{1}{{{x}_{5}}^{3}}\) 71,785 \({{x}_{1}}^{4}-{{x}_{2}}^{3}{x}_{3}\le 0\)
Where, 0.01 \(\le {x}_{1} , {x}_{2}, {x}_{3} {x}_{4}, {x}_{5} \le 100\)
Appendix 5: Carside crash design problem
Objective function:
Subject to:
where,
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Sahoo, S.K., Saha, A.K., Nama, S. et al. An improved moth flame optimization algorithm based on modified dynamic opposite learning strategy. Artif Intell Rev 56, 2811–2869 (2023). https://doi.org/10.1007/s10462-022-10218-0
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DOI: https://doi.org/10.1007/s10462-022-10218-0