A Hybrid Moth Flame Optimization Algorithm for Global Optimization

Abstract

The Moth Flame Optimization (MFO) algorithm shows decent performance results compared to other meta-heuristic algorithms for tackling non-linear constrained global optimization problems. However, it still suffers from obtaining quality solution and slow convergence speed. On the other hand, the Butterfly Optimization Algorithm (BOA) is a comparatively new algorithm which is gaining its popularity due to its simplicity, but it also suffers from poor exploitation ability. In this study, a novel hybrid algorithm, h-MFOBOA, is introduced, which integrates BOA with the MFO algorithm to overcome the shortcomings of both the algorithms and at the same time inherit their advantages. For performance evaluation, the proposed h-MFOBOA algorithm is applied on 23 classical benchmark functions with varied complexity. The tested results of the proposed algorithm are compared with some well-known traditional meta-heuristic algorithms as well as MFO variants. Friedman rank test and Wilcoxon signed rank test are employed to measure the performance of the newly introduced algorithm statistically. The computational complexity has been measured. Moreover, the proposed algorithm has been applied to solve one constrained and one unconstrained real-life problems to examine its problem-solving capability of both type of problems. The comparison results of benchmark functions, statistical analysis, real-world problems confirm that the proposed h-MFOBOA algorithm provides superior results compared to the other conventional optimization algorithms.

Appendices

Appendix 1

Formulation of twenty-three 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	$f\left(x\right)=\sum_{i=1}^{D}{{x}_{i}}^{2}\times i$	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	$\mathrm{f}\left(\mathrm{x}\right)=\sum_{\mathrm{j}=1}^{\mathrm{d}}{{\mathrm{x}}_{\mathrm{i}}}^{2}+{\left(0.5\sum_{\mathrm{j}=1}^{\mathrm{d}}{\mathrm{jx}}_{\mathrm{j}}\right)}^{2}+{\left(0.5\sum_{\mathrm{j}=1}^{\mathrm{d}}{\mathrm{jx}}_{\mathrm{j}}\right)}^{4}$	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	$f\left(x\right)={sin}^{2}\left(\pi {x}_{1}\right)+\sum_{i=1}^{D-1}{\left({x}_{i}-1\right)}^{2}\left[1+10{sin}^{2}\left(\pi {x}_{i}+1\right)\right]+{\left({x}_{D}-1\right)}^{2}\left[1+{sin}^{2}\left(2\pi {x}_{D}\right)\right]$ Where, ${x}_{i}=1+\frac{1}{4}$(${x}_{i}-1), i=\mathrm{1,2},\dots \dots \dots D$	30	0	[-10, 10]
F11	Michalewicz	$f\left(x\right)=-\sum_{i=1}^{D}\mathrm{sin}{(x}_{i}){sin}^{2m}(\frac{{i{x}_{i}}^{2}}{\pi })$, m = 10	10	-9.66015	[0, $\pi$]
F12	Alpine	$f\left(x\right)=\sum_{i=1}^{D}\left|{x}_{i}\mathrm{sin}{(x}_{i})+0.1{x}_{i}\right|$	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	$f\left(x\right)= \sum_{i=1}^{D}\left[{\left(\sum_{k=1}^{D}{{(x}_{k}}^{i})-{b}_{i}\right)}^{2}\right]$
F15	Penalized2	$f\left(x\right)=0.1\left\{ 10{sin}^{2}\left(\pi {x}_{i}\right)+\sum_{i=1}^{D-1}{\left({x}_{i}-1\right)}^{2}[1+10{sin}^{2}\left(3\pi {x}_{i+1}\right)+{\left({x}_{D}-1\right)}^{2}[1+{sin}^{2}\left(2\pi {x}_{D}\right)]]\right\}+\sum_{i=1}^{D}u\left({x}_{i},\mathrm{5,100,4}\right)$		0	[-50, 50]
F16	Kowalik	$\mathrm{f}\left(\mathrm{x}\right)=\sum_{\mathrm{j}=1}^{11}{\left[{\mathrm{a}}_{\mathrm{j}}-\frac{{\mathrm{x}}_{1}\left({{\mathrm{b}}_{\mathrm{j}}}^{2}+{\mathrm{b}}_{\mathrm{j}}{\mathrm{x}}_{2}\right)}{({{\mathrm{b}}_{\mathrm{j}}}^{2}-{\mathrm{b}}_{\mathrm{j}}{\mathrm{x}}_{3}-{\mathrm{x}}_{4}}\right]}^{2}$	4	0.0003075	[-5, 5]
F17	Foxholes	$\mathrm{f}\left(\mathrm{x}\right)={\left[\frac{1}{500}+ \sum_{\mathrm{j}=1}^{25}\frac{1}{\mathrm{j}}+\sum_{\mathrm{i}=1}^{\mathrm{D}}{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{ij}}\right)}^{6}\right]}^{-1}$	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	$f\left(x\right)=-\sum_{i=1}^{4}{\alpha }_{i}\mathrm{exp}(-\sum_{j=1}^{3}{a}_{ij}{\left({x}_{j-{b}_{ij}}\right)}^{2})$	3	-3.86	[0, 1]
F20	Hartmann6	$f\left(x\right)=-\sum_{i=1}^{4}{\alpha }_{i}\mathrm{exp}(-\sum_{j=1}^{6}{a}_{ij}{\left({x}_{j-{b}_{ij}}\right)}^{2})$	6	-3.32	[0, 1]
F21	Shekel 5	$\mathrm{f}\left(\mathrm{x}\right)=-\sum_{\mathrm{j}=1}^{5}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{j}}\right]}^{-1}$	4	-10.1499	[0, 10]
F22	Shekel-7	$\mathrm{f}\left(\mathrm{x}\right)=-\sum_{\mathrm{j}=1}^{7}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{j}}\right]}^{-1}$	4	-10.3999	[0, 10]
F23	Shekel-10	$\mathrm{f}\left(\mathrm{x}\right)=-\sum_{\mathrm{j}=1}^{10}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{j}}\right]}^{-1}$	4	-10.5319	[0, 10]

Appendix 2

Optimal Capacity of Gas Production Facilities

$$\mathrm{Min f}\left(\mathrm{x}\right)=61.8+5.72\times {\mathrm{x}}_{1}\times 0.2623\times {\left[\left(40-{\mathrm{x}}_{1}\right)\times \mathrm{ln}\frac{{\mathrm{x}}_{2}}{200}\right]}^{-0.85}+0.087\times \left(40-{\mathrm{x}}_{1}\right)\times \mathrm{ln}\frac{{\mathrm{x}}_{2}}{200}+700.23\times {\mathrm{x}}_{2}^{-0.75}$$

$${\mathrm{x}}_{1}\ge 17.5, {\mathrm{x}}_{2}\ge 200, 17.5\le {\mathrm{x}}_{1}\le 40, 300\le {\mathrm{x}}_{2}\le 600;$$

Appendix 3

Three-bar truss 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:

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

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

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

$$and P=2, L=100 \& \sigma =2$$