A Novel Variant of Moth Flame Optimizer for Higher Dimensional Optimization Problems

Abstract

Moth Flame Optimization (MFO) is a nature-inspired optimization algorithm, based on the principle of navigation technique of moth toward moon. Due to less parameter and easy implementation, MFO is used in various field to solve optimization problems. Further, for the complex higher dimensional problems, MFO is unable to make a good trade-off between global and local search. To overcome these drawbacks of MFO, in this work, an enhanced MFO, namely WF-MFO, is introduced to solve higher dimensional optimization problems. For a more optimal balance between global and local search, the original MFO’s exploration ability is improved by an exploration operator, namely, Weibull flight distribution. In addition, the local optimal solutions have been avoided and the convergence speed has been increased using a Fibonacci search process-based technique that improves the quality of the solutions found. Twenty-nine benchmark functions of varying complexity with 1000 and 2000 dimensions have been utilized to verify the projected WF-MFO. Numerous popular algorithms and MFO versions have been compared to the achieved results. In addition, the robustness of the proposed WF-MFO method has been evaluated using the Friedman rank test, the Wilcoxon rank test, and convergence analysis. Compared to other methods, the proposed WF-MFO algorithm provides higher quality solutions and converges more quickly, as shown by the experiments. Furthermore, the proposed WF-MFO has been used to the solution of two engineering design issues, with striking success. The improved performance of the proposed WF-MFO algorithm for addressing larger dimensional optimization problems is guaranteed by analyses of numerical data, statistical tests, and convergence performance.

Appendix 1: Formulation of 29 benchmark functions

Sl. no	Functions	Formulation of objective functions	d	F min	Search space
		Unimodal benchmark functions
F1	Sphere	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {x_{j}^{2} }$	30	0	[− 100, 100]
F2	Matyas	$f\left( x \right) = 0.26\left( {x_{1}^{2} + x_{2}^{2} } \right) - 0.48x_{1} x_{2}$	2	0	[− 10, 10]
F3	Sumsquare	$f\left( x \right) = \sum\nolimits_{i = 1}^{D} {x_{i}^{2} \times i}$	30	0	[− 10, 10]
F4	Zettl	$f\left( x \right) = \left( {x - 1^{2} + x - 2^{2} - 2x_{1} } \right)^{2} + 0.25x_{1}$	2	− 0.00379	[− 1, 5]
F5	Zakhrov	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {x_{j}^{2} } + \left( {0.5\sum\nolimits_{j = 1}^{d} {jx_{j} } } \right)^{2} + \left( {0.5\sum\nolimits_{j = 1}^{d} {jx_{j} } } \right)^{4}$	2	0	[− 5, 10]
F6	High conditioned elliptic function	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {\left( {1000000^{{\left( {\frac{j - 1}{{D - 1}}} \right)}} \times x_{j}^{2} } \right)}$	30	0	[− 100, 100]
F7	Brown	$f\left( x \right) = \sum\nolimits_{j = 1}^{n - 1} {\left( {x_{j}^{2} } \right)^{{(x_{j + 1}^{2} + 1)}} + \left( {\left( {x_{j + 1}^{2} } \right)^{{(x_{j}^{2} + 1)}} } \right)}$	30	0	[− 1, 4]
F8	Cube	$f\left( x \right) = 100\left( {x_{2} - x_{1}^{3} } \right)^{2} + \left( {1 - x_{1} } \right)^{2}$	2	0	[− 10, 10]
F9	Rotated hyperellipsoid	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {\left( {d + 1 - j} \right)x_{j}^{2} }$	30	0	[− 100, 100]
F10	Schwefel 1.2	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {\sum\nolimits_{k = 1}^{j} {x_{j}^{2} } }$	30	0	[− 100, 100]
F11	Schwefel 2.21	$f\left( x \right) = \max \left\{ {\left| {x_{j} } \right|, 1 \le j \le d} \right\}$	30	0	[− 100, 100]
F12	Rosenbrock	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {[100\left( {x_{j + 1} - x_{j}^{2} } \right)^{2} + \left( {x_{j} - 1} \right)^{2} ]}$	30	0	[− 2.048, 2.048]
F13	Cigar	$f\left( x \right) = 10^{6} \sum\nolimits_{j = 1}^{d} {x_{j}^{2} }$	30	0	[− 100, 100]
F14	Step	$f\left( x \right) = \mathop \sum \limits_{j = 1}^{d} \left( {x_{j} + 0.5} \right)^{2}$	30	0	[− 100, 100]
Multimodal benchmark functions
F15	Bohachevsky	$f\left( x \right) = x_{1}^{2} + 2x_{2}^{2} - 0.3\cos \left( {3\pi x_{1} } \right) - 0.3$	2	0	[− 100, 100]
F16	Bohachevsky 3	$f\left( x \right) = x_{1}^{2} + 2x_{2}^{2} - 0.3\cos \left( {3\pi x_{1} } \right) - 0.3$	2	0	[− 50, 50]
F17	Levy	$f\left( x \right) = \sin^{2} \left( {\pi x_{1} } \right) + \sum\nolimits_{i = 1}^{D - 1} {\left( {x_{i} - 1} \right)^{2} \left[ {1 + 10\sin^{2} \left( {\pi x_{i} + 1} \right)} \right] + \left( {x_{D} - 1} \right)^{2} \left[ {1 + \sin^{2} \left( {2\pi x_{D} } \right)} \right]}$ where $x_{i} = 1 + \frac{1}{4}(x_{i} - 1), i = 1,2, \ldots D$	30	0	[− 10, 10]
F18	Alpine	$f\left( x \right) = \sum\nolimits_{i = 1}^{D} {\left| {x_{i} \sin (x_{i} ) + 0.1x_{i} } \right|}$	30	0	[− 10, 10]
F19	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]
F20	Salomon	$f\left( x \right) = 1 - \cos \left( {2\pi \sqrt {\sum\nolimits_{j = 1}^{d} {x_{j}^{2} } } } \right) + 0.1\sqrt {\sum\nolimits_{j = 1}^{d} {x_{j}^{2} } }$	30	0	[− 100, 100]
F21	Penalized 2	$f\left( x \right) = 0.1\left\{ { 10\sin^{2} \left( {\pi x_{i} } \right) + \sum\nolimits_{i = 1}^{D - 1} {\left( {x_{i} - 1} \right)^{2} [1 + 10\sin^{2} \left( {3\pi x_{i + 1} } \right) + \left( {x_{D} - 1} \right)^{2} [1 + \sin^{2} \left( {2\pi x_{D} } \right)]]} } \right\} + \sum\nolimits_{i = 1}^{D} {u\left( {x_{i} ,5,100,4} \right)}$	30	0	[− 50, 50]
F22	Inverted cosine mixture	$f\left( x \right) = 0.1 \times 30 - \left[ {0.1 \times \sum\nolimits_{j = 1}^{d} {5\pi x_{j} } - \sum\nolimits_{j = 1}^{d} {x_{j}^{2} } } \right]$	30	0	[− 1, 1]
F23	Modified sphere	$f\left( x \right) = \sum\nolimits_{j = 1}^{6} {\frac{{\left( {x_{j}^{2} \times 2^{j} } \right) - 1745}}{899}}$	30	0	[− 5.12, 5.12]
F24	Drop wave	$f\left( x \right) = 1 - \frac{{1 + \cos \left( {12\sqrt {x_{1}^{2} + x_{2}^{2} } } \right)}}{{0.5\left( {x_{1}^{2} + x_{2}^{2} } \right) + 2}}$	2	− 1	[− 5.12, 5.12]
F25	Egg-holder	$f\left( x \right) = \left[ { - \left( {47 + x_{i + 1} } \right)\sin \begin{array}{*{20}c} {\sqrt {\left| {x_{i + 1} + \frac{{x_{i} }}{2} + 47} \right|} } \\ { - x_{i} \sin \sqrt {x_{i} - \left( {x_{i + 1} + 47} \right)} } \\ \end{array} } \right]$	30	− 959.6407	[− 512, 512]
F26	Rastrigin	$f\left( x \right) = 10D + \sum\nolimits_{j = 1}^{d} {[x_{j}^{2} - 10\cos (2\pi x_{j} )]}$	30	0	[− 5.12, 5.12]
F27	Schwefel 2.26	$f\left( x \right) = - \sum\nolimits_{j = 1}^{d} {x_{j} \sin \left( {\sqrt {\left| {x_{j} } \right|} } \right)}$	30	− 418.982	[− 500, 500]
F28	Ackley	$f\left( x \right) = 20 + e - 20e^{{\frac{1}{d}\left( {\sqrt {\left( {\frac{1}{d}\sum\nolimits_{j = 1}^{d} {x_{j}^{2} } } \right)} } \right)}} - e^{{\frac{1}{d}\left( {\sum \cos \left( {2\pi x_{j} } \right)} \right)}}$	30	0	[− 32.768, 32.768]
F29	Griewank	$f\left( x \right) = \sum\nolimits_{j = 1}^{d} {\frac{{x_{j}^{2} }}{4000}} - \prod\nolimits_{j = 1}^{d} {\cos \left( {\frac{{x_{j} }}{\sqrt j }} \right) - 1}$	30	0	[− 600, 600]