An improved moth flame optimization algorithm based on modified dynamic opposite learning strategy

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.

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

$$\overrightarrow{{\varvec{k}}}=\left\{{k}_{1}, {k}_{2},\right\}$$

    Objective function:

        

$$Min. f\left(k\right)=L\left\{{k}_{2}+2\sqrt{2}{k}_{1}\right\}$$

    Subject to:

$${h}_{1}\left(k\right)=\frac{{k}_{2}}{2{k}_{2}{k}_{1}+\sqrt{2}{k}_{1}^{2}} P-\sigma \le 0,$$

$${h}_{2}\left(k\right)=\frac{{k}_{2}+\sqrt{2}{k}_{1}}{2{k}_{2}{k}_{1}+\sqrt{2}{k}_{1}^{2}} P-\sigma \le 0,$$

$${h}_{3}\left(k\right)=\frac{1}{{k}_{1}+\sqrt{2}{k}_{2}}P-\sigma \le 0,$$

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 ${{x}_{1}}^{4}-{{x}_{2}}^{3}{x}_{3}\le 0$

$${g}_{2}\left(\overrightarrow{{\varvec{x}}}\right)= \frac{4{{x}_{2}}^{2}-{{x}_{1}x}_{2}}{12566\left({x}_{2}{{x}_{1}}^{3}-{{x}_{1}}^{4}\right)}-\frac{1}{5108{{x}_{1}}^{2}}\le 0$$

$${g}_{3}\left(\overrightarrow{{\varvec{x}}}\right)={{x}_{2}}^{2}{x}_{3}-140.45{x}_{1}\le 0$$

$${g}_{4}\left(\overrightarrow{{\varvec{x}}}\right)={x}_{1}+{x}_{2}-1.5\le 0$$

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

$$\overrightarrow{{\varvec{a}}}=\left\{{a}_{1},\boldsymbol{ }{a}_{2},{a}_{3},\boldsymbol{ }{a}_{4},\boldsymbol{ }{a}_{5},\boldsymbol{ }{a}_{6},\boldsymbol{ }{a}_{7},{a}_{8},{a}_{9},\boldsymbol{ }{a}_{10},{a}_{11}\right\}$$

Objective function:

$$Min\, f\left(a\right)=1.98+4.90 {a}_{1}+6.67 {a}_{2}+6.98 {a}_{3}+4.01 {a}_{4}+1.78 {a}_{5}+2.73 {a}_{7},$$

Subject to:

$${h}_{1}\left(a\right)=1.16-0.3717 {a}_{2} {a}_{4}-0.00931 {a}_{2} {a}_{10}-0.484 {a}_{3} {a}_{9}+$$

$$0.01343 {a}_{6} {a}_{10 }\le 1,$$

$${h}_{2}\left(a\right)=0.261-0.0159 {a}_{1} {a}_{2}-0.188 {a}_{1} {a}_{8}-0.019 {a}_{2} {a}_{7} +0.0144 {a}_{3} {a}_{5} + 0.0008757 {a}_{5} {a}_{10}+0.080405 {a}_{6} {a}_{9}+0.00139 {a}_{8} {a}_{11} + 0.00001575 {a}_{10} {a}_{11}\le 0.32,$$

$${h}_{3}\left(a\right)=0.214+0.00817 {a}_{5}-0.131 {a}_{1}{a}_{8}-0.0704 {a}_{1}{a}_{9}+0.03099 {a}_{2}{ a}_{6}-0.018 {a}_{2}{ a}_{7}+0.0208 {a}_{3}{ a}_{8}+0.121 {a}_{3}{ a}_{9} -0.00364 {a}_{5}{ a}_{6}+ 0.0007715 {a}_{5}{ a}_{10}-0.0005354 {a}_{6}{ a}_{10} +0.00121 {a}_{8}{ a}_{11}\le 0.32,$$

$${h}_{4}\left(a\right)=0.074-0.061 {a}_{2}-0.163 {a}_{3} {a}_{8}+0.001232 {a}_{3} {a}_{10}-0.166 {a}_{7 }{a}_{9} +0.227{ a}_{2}^{2} \le 0.32,$$

$${h}_{5}\left(a\right)=28.98+3.818 {a}_{3}-4.2 {a}_{1} {a}_{2}+0.0207{ a}_{5} {a}_{10}+6.63 {a}_{6} {a}_{9}-$$

$$7.7 {a}_{7} {a}_{8}+0.32 {a}_{9} {a}_{10}\le 32,$$

$${h}_{6}\left(a\right)=33.86+2.95 {a}_{3}+0.1792 {a}_{10}-5.05 {a}_{1} {a}_{2}-11.0 {a}_{2} {a}_{8} - 0.0215 {a}_{5} {a}_{10}-9.98 {a}_{7} {a}_{8}+22.0 {a}_{8} {a}_{9}\le 32,$$

$${h}_{7}\left(a\right)=46.36-9.9 {a}_{2}-12.9 {a}_{1} {a}_{8}+0.1107 {a}_{3} {a}_{10}\le 32,$$

$${h}_{8}\left(a\right)=4.72-0.5 {a}_{4}-0.19 {a}_{2} {a}_{3}-0.0122 {a}_{4} {a}_{10}+0.009325 {a}_{6} {a}_{10} +$$

$$0.000191 {a}_{11}^{2} \le 4,$$

$${h}_{9}\left(a\right)=10.58-0.674 {a}_{1} {a}_{2}-1.95 {a}_{2} {a}_{8}+0.02054 {a}_{3} {a}_{10}-$$

$$0.0198 {a}_{4} {a}_{10}+0.028 a\left(6\right) a\left(10\right)\le 9.9,$$

$${h}_{10}\left(a\right)=16.45-0.489 {a}_{3} {a}_{7}-0.843 {a}_{5} {a}_{6}+0.0432 {a}_{9} {a}_{10}-0.0556 {a}_{9} {a}_{11} -0.000786 {a}_{11}^{2} \le 15.7,$$

where,

$$0.5\le {a}_{i}\le 1.5, i=1, 2, 3, 4, 5, 6, 7$$

$${a}_{8},{a}_{9}\in \left(0.192, 0.345\right),$$

$$-30\le {a}_{10}, {a}_{11}\le 30.$$