Chaos-enhanced moth-flame optimization algorithm for global optimization

doi:10.21629/JSEE.2019.06.10

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (6): 1144-1159.doi: 10.21629/JSEE.2019.06.10

• Systems Engineering • Previous Articles Next Articles

Chaos-enhanced moth-flame optimization algorithm for global optimization

Hongwei LI¹(), Jianyong LIU¹(), Liang CHEN^1,^2,*(), Jingbo BAI¹(), Yangyang SUN³(), Kai LU¹()

¹ College of Field Engineering, Army Engineering University of the PLA, Nanjing 210001, China
² Automobile Non-Commissioned Officer Academy, Army Military Transportation University, Bengbu 233011, China
³ College of National Defense Engineering, Army Engineering University of the PLA, Bengbu 233011, China

Received:2018-10-09 Online:2019-12-20 Published:2019-12-25
Contact: Liang CHEN E-mail:727802081@qq.com;jianyong1212@126.com;chenbb0708@163.com;baijingbo1982@163.com;bryant8011@163.com;xikaikaixi@outlook.com
About author:LI Hongwei was born in 1978. He received his M.S. degree from University of Science and Technology of the PLA in 2002. He is an associate professor in College of Field Engineering, Army Engineering University of the PLA. His current research interests are military operations research and intelligent unmanned technology. E-mail: 727802081@qq.com|LIU Jianyong was born in 1961. He received his Ph.D. degree from University of Science and Technology of the PLA in 2004. He is a professor in College of Field Engineering, Army Engineering University of the PLA. His current research interests are military operations research and intelligent unmanned technology. E-mail: jianyong1212@126.com|CHEN Liang was born in 1981. He received his B.S. degree and M.S. degree from University of Science and Technology of the PLA in 2004 and 2009, respectively. He is a Ph.D. candidate at College of Field Engineering, Army Engineering University of the PLA. He is a lecturer in Army Military Transportation University. His main research interests include operations research, swarm intelligence, multi-objective optimization and intelligent unmanned technology. E-mail: chenbb0708@163.com|BAI Jingbo was born in 1982. He received his B.S. degree and M.S. degree from University of Science and Technology of the PLA in 2005 and 2009, respectively. He is a lecturer and a Ph.D. candidate in College of Field Engineering, Army Engineering University of the PLA. His current research interest includes mission planning of military problems. E-mail: baijingbo1982@163.com|SUN Yangyang was born in 1983. He received his M.S. degree from University of Science and Technology of the PLA in 2009. He is a lecturer and a Ph.D. candidate in College of National Defense Engineering, Army Engineering University of the PLA. His current research interests include optical fiber sensing and system integration. E-mail: bryant8011@163.com|LU Kai was born in 1991. He received his B.S. degree and M.S. degree from Ordnance Engineering College in 2014 and 2017, respectively. He is a Ph.D. candidate in College of Field Engineering, Army Engineering University of the PLA. His research interest include bridge design theory and key technologies. E-mail: xikaikaixi@outlook.com
Supported by:
the Military Science Project of the National Social Science Foundation of China(15GJ003-141);This work was supported by the Military Science Project of the National Social Science Foundation of China (15GJ003-141)

Abstract

Abstract:

Moth-flame optimization (MFO) is a novel metaheuristic algorithm inspired by the characteristics of a moth's navigation method in nature called transverse orientation. Like other metaheuristic algorithms, it is easy to fall into local optimum and leads to slow convergence speed. The chaotic map is one of the best methods to improve exploration and exploitation of the metaheuristic algorithms. In the present study, we propose a chaos-enhanced MFO (CMFO) by incorporating chaos maps into the MFO algorithm to enhance its performance. The chaotic map is utilized to initialize the moths' population, handle the boundary overstepping, and tune the distance parameter. The CMFO is benchmarked on three groups of benchmark functions to find out the most efficient one. The performance of the CMFO is also verified by using two real engineering problems. The statistical results clearly demonstrate that the appropriate chaotic map (singer map) embedded in the appropriate component of MFO can significantly improve the performance of MFO.

Key words: moth-flame optimization (MFO), chaotic map, metaheuristic, global optimization

Hongwei LI, Jianyong LIU, Liang CHEN, Jingbo BAI, Yangyang SUN, Kai LU. Chaos-enhanced moth-flame optimization algorithm for global optimization[J]. Journal of Systems Engineering and Electronics, 2019, 30(6): 1144-1159.

Figures/Tables 22

Table 1

Table 2

Table 3

Table 4

Composite benchmark functions"

Formulation	Search range
$F_{13}({\rm{CF}}1): f_1, f_2, f_3, f_4, f_5, f_6, f_7, f_8, f_9, f_{10}=$ Sphere Function $[\sigma_1, \sigma_2, \ldots, \sigma_{10}]=[1, 1, \ldots, 1]$ $[\lambda_1, \lambda_2, \ldots, \lambda_{10}]=[5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100]$	$[-5, 5]$
$F_{14}({\rm{CF}}2): f_1, f_2, f_3, f_4, f_5, f_6, f_7, f_8, f_9, f_{10}=$ Griewank Function $[\sigma_1, \sigma_2, \ldots, \sigma_{10}]=[1, 1, \ldots, 1]$ $[\lambda_1, \lambda_2, \ldots, \lambda_{10}]=[5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100, 5/100]$	$[-5, 5]$
$F_{15}({\rm{CF}}3): f_1, f_2, f_3, f_4, f_5, f_6, f_7, f_8, f_9, f_{10}=$ Griewank Function $[\sigma_1, \sigma_2, \ldots, \sigma_{10}]=[1, 1, \ldots, 1]$ $[\lambda_1, \lambda_2, \ldots, \lambda_{10}]=[1, 1, \ldots, 1]$	$[-5, 5]$
$F_{16}({\rm{CF}}4): f_1, f_2=$ Ackley Function, $f_3, f_4=$ Rastrigin Function, $f_5, f_6=$ Weierstrass Function, $f_7, f_8=$ Griewank Function, $f_9, f_{10}=$ Sphere Function $[\sigma_1, \sigma_2, \ldots, \sigma_{10}]=[1, 1, \ldots, 1]$ $ [\lambda_1, \lambda_2, \ldots, \lambda_{10}]=[5/32, 5/32, 1, 1, 5/0.5, 5/0.5, 5/100, 5/100, 5/100, 5/100]$	$[-5, 5]$
$F_{17}({\rm{CF}}5): f_1, f_2=$ Rastrigin Function, $f_3, f_4=$ Weierstrass Function, $f_5, f_6=$ Griewank Function, $f_7, f_8=$ Ackley Function, $f_9, f_{10}=$ Sphere Function $[\sigma_1, \sigma_2, \ldots, \sigma_{10}]=[1, 1, \ldots, 1]$ $[\lambda_1, \lambda_2, \ldots, \lambda_{10}]=[1/5, 1/5, 5/0.5, 5/0.5, 5/100, 5/100, 5/32, 5/32, 5/100, 5/100]$	$[-5, 5]$
$F_{18}({\rm{CF}}6): f_1, f_2=$ Rastrigin Function, $f_3, f_4=$ Weierstrass Function, $f_5, f_6=$ Griewank Function, $f_7, f_8=$ Ackley Function, $f_9, f_{10}=$ Sphere Function $[\sigma_1, \sigma_2, \ldots, \sigma_{10}]=[0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]$ $[\lambda_1, \lambda_2, \ldots, \lambda_{10}]=[0.1\times1/5, 0.2\times1/5, 0.3\times5/0.5, 0.4\times5/0.5, 0.5\times5/100, 0.6\times5/100, 0.7\times5/32, 0.8\times5/32, $ $0.9\times5/100, 1]$	$[-5, 5]$

Table 4

Table 5

Table 6

Results of unimodal benchmark functions for chaotic initialization population"

Algorithm	$F_1$			$F_{2}$			$F_{3}$			$F_4$			$F_{5}$			$F_{6}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	1.58E+00	7.84E-01	$\underline {3.51E-01}$	2.37E+06	1.18E+06	$\underline {1.40E-01}$	1.44E+04	3.37E+03	$\underline {2.39E-01}$	$\textbf{1.67E+01}$	$\textbf{3.65E+00}$	$\textbf{N/A}$	$\textbf{3.11E+03}$	$\textbf{9.55E+02}$	$\textbf{N/A}$	3.97E+03	1.29E+03	$\underline{1.08E-01}$
CMFO1	1.43E+00	1.01E+00	$\underline {4.73E-01}$	1.93E+06	9.89E+05	$\underline {7.97E-01}$	1.57E+04	4.17E+03	4.99E-02	1.82E+01	4.60E+00	$\underline {2.98E-01}$	3.47E+03	7.80E+02	$\underline {1.72E-01}$	3.82E+03	7.71E+02	$\underline {8.10E-02}$
CMFO2	1.39E+00	5.36E-01	$\underline {8.18E-01}$	2.06E+06	1.23E+06	$\underline {6.55E-01}$	1.62E+04	4.52E+03	$\underline {5.31E-02}$	2.00E+01	3.49E+00	7.71E-03	3.30E+03	7.55E+02	$\underline{3.51E-01}$	3.48E+03	9.04E+02	$\underline {5.08E-01}$
CMFO3	5.98E+02	1.60E-02	6.80E-08	9.46E+08	6.41E+01	6.80E-08	5.92E+06	1.05E+04	6.80E-08	2.27E+02	2.67E+02	6.80E-08	1.31E+05	1.08E+00	6.80E-08	1.27E+05	9.50E-01	6.80E-08
CMFO4	1.42E+00	4.23E-01	$\underline {5.98E-01}$	2.36E+06	1.47E+06	$\underline {1.72E-01}$	1.58E+04	4.35E+03	$\underline {6.39E-02}$	9.16E+01	3.25E+01	6.92E-07	3.74E+03	9.26E+02	3.60E-02	3.65E+03	1.20E+03	$\underline {3.37E-01}$
CMFO5	$\textbf{1.31E+00}$	$\textbf{4.11E-01}$	$\textbf{N/A}$	2.32E+06	1.33E+06	$\underline {2.18E-01}$	1.68E+04	3.26E+03	1.95E-03	1.90E+01	4.21E+00	$\underline {1.08E-01}$	3.52E+03	7.22E+02	$\underline {1.20E-01}$	3.43E+03	1.17E+03	$\underline {8.18E-01}$
CMFO6	1.56E+00	5.05E-01	$\underline {5.65E-02}$	2.17E+06	1.37E+06	$\underline {5.25E-01}$	1.49E+04	3.69E+03	$\underline {1.26E-01}$	1.87E+01	4.33E+00	$\underline {1.56E-01}$	3.67E+03	1.05E+03	$\underline {8.59E-02}$	3.66E+03	9.74E+02	$\underline {3.94E-01}$
CMFO7	1.50E+00	6.09E-01	$\underline {4.73E-01}$	2.72E+06	1.46E+06	4.11E-02	1.66E+04	3.65E+03	5.56E-03	1.88E+01	2.95E+00	3.15E-02	3.65E+03	1.10E+03	$\underline{1.14E-01}$	3.36E+03	1.07E+03	$\underline {8.60E-01}$
CMFO8	1.63E+00	4.34E-01	$\underline {2.56E-02}$	3.13E+06	1.34E+06	6.22E-04	1.71E+04	4.24E+03	4.70E-03	1.87E+01	2.88E+00	3.85E-02	3.88E+03	1.16E+03	4.68E-02	3.56E+03	8.50E+02	$\underline {3.10E-01}$
CMFO9	1.69E+00	7.26E-01	$\underline {1.26E-01}$	2.79E+06	1.82E+06	$\underline {1.08E-01}$	$\textbf{1.33E+04}$	$\textbf{2.98E+03}$	$\textbf{N/A}$	1.91E+01	4.78E+00	$\underline {6.79E-02}$	3.73E+03	1.12E+03	$\underline {6.79E-02}$	$\textbf{3.29E+03}$	$\textbf{9.79E+02}$	$\textbf{N/A}$
CMFO10	1.38E+00	5.93E-01	$\underline {7.56E-01}$	$\textbf{1.73E+06}$	$\textbf{6.83E+05}$	$\textbf{N/A}$	1.70E+04	4.15E+03	6.04E-03	1.84E+01	3.36E+00	$\underline {9.09E-02}$	3.16E+03	8.82E+02	$\underline {8.18E-01}$	3.53E+03	1.10E+03	$\underline {5.25E-01}$

Table 6

Table 7

Results of multimodal benchmark functions for chaotic initialization population"

Algorithm	$F_{7}$			$F_{8}$			$F_{9}$			$F_{10}$			$F_{11}$			$F_{12}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	1.18E+01	1.01E+00	$\underline {9.03E-01}$	2.92E+01	1.01E+01	$\underline {5.98E-01}$	7.65E+05	1.02E+06	$\underline {1.90E-01}$	4.33E+06	3.27E+06	$\underline {2.62E-01}$	$\textbf{1.18E+02}$	$\textbf{1.76E+01}$	$\textbf{N/A}$	5.51E+03	4.96E+02	5.63E-04
CMFO1	1.18E+01	1.35E+00	$\underline {9.46E-01}$	3.28E+01	1.04E+01	$\underline {8.59E-02}$	$\textbf{3.98E+05}$	$\textbf{3.94E+05}$	$\textbf{N/A}$	$\textbf{2.78E+06}$	$\textbf{1.42E+06}$	$\textbf{N/A}$	1.27E+02	2.01E+01	$\underline {1.72E-01}$	5.34E+03	5.50E+02	7.71E-03
CMFO2	1.19E+01	1.04E+00	$\underline {4.09E-01}$	3.32E+01	1.04E+01	4.99E-02	9.36E+05	2.18E+06	$\underline {1.64E-01}$	4.55E+06	3.96E+06	$\underline {1.14E-01}$	1.24E+02	1.73E+01	$\underline {3.23E-01}$	5.64E+03	6.12E+02	4.16E-04
CMFO3	2.04E+01	3.83E-02	6.80E-08	1.16E+03	1.77E-02	6.80E-08	2.72E+09	2.85E-01	6.80E-08	4.47E+09	1.35E+00	6.80E-08	5.58E+02	3.21E+00	6.80E-08	1.06E+04	1.37E+02	6.80E-08
CMFO4	1.18E+01	7.33E-01	$\underline {8.82E-01}$	3.30E+01	9.07E+00	4.11E-02	8.87E+05	1.12E+06	$\underline {3.10E-01}$	5.02E+06	5.89E+06	$\underline {3.65E-01}$	1.28E+02	1.75E+01	$\underline {1.14E-01}$	5.40E+03	4.82E+02	3.34E-03
CMFO5	$\textbf{1.17E+01}$	$\textbf{1.23E+00}$	$\textbf{N/A}$	3.24E+01	5.27E+00	1.67E-02	5.01E+05	4.22E+05	$\underline {3.51E-01}$	5.86E+06	4.09E+06	3.97E-03	1.29E+02	1.53E+01	$\underline{7.20E-02}$	5.46E+03	5.03E+02	1.95E-03
CMFO6	1.20E+01	1.10E+00	$\underline {4.09E-01}$	3.04E+01	8.46E+00	$\underline {2.85E-01}$	9.13E+05	8.67E+05	4.68E-02	3.75E+06	4.20E+06	$\underline {7.35E-01}$	1.33E+02	2.17E+01	4.11E-02	5.82E+03	5.74E+02	4.17E-05
CMFO7	1.21E+01	1.21E+00	$\underline {2.18E-01}$	3.12E+01	7.58E+00	$\underline {1.02E-01}$	7.22E+05	8.10E+05	$\underline {3.65E-01}$	4.99E+06	3.77E+06	1.44E-02	1.25E+02	2.18E+01	$\underline{4.25E-01}$	5.15E+03	5.87E+02	$\underline {1.40E-01}$
CMFO8	1.17E+01	1.24E+00	$\underline {9.89E-01}$	3.25E+01	1.18E+01	$\underline {1.40E-01}$	2.86E+06	3.47E+06	1.10E-05	6.02E+06	4.52E+06	1.79E-04	1.30E+02	1.72E+01	4.99E-02	$\textbf{4.88E+03}$	$\textbf{5.30E+02}$	$\textbf{N/A}$
CMFO9	1.18E+01	1.07E+00	$\underline {9.68E-01}$	3.17E+01	8.96E+00	$\underline {1.48E-01}$	1.16E+06	9.33E+05	6.56E-03	6.41E+06	8.58E+06	$\underline {7.64E-02}$	1.34E+02	1.51E+01	6.56E-03	5.20E+03	4.20E+02	2.56E-02
CMFO10	1.22E+01	1.12E+00	$\underline {4.25E-01}$	$\textbf{2.74E+01}$	$\textbf{7.79E+00}$	$\textbf{N/A}$	6.53E+05	5.17E+05	$\underline {1.08E-01}$	3.78E+06	3.39E+06	$\underline {5.25E-01}$	1.41E+02	2.12E+01	6.87E-04	5.73E+03	4.36E+02	2.60E-05

Table 7

Table 8

Results of composite benchmark functions for chaotic initialization population"

Algorithm	$F_{13}$			$F_{14}$			$F_{15}$			$F_{16}$			$F_{17}$			$F_{18}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	8.36E+01	4.63E+01	$\underline {6.39E-02}$	9.27E+01	1.91E+01	$\underline {8.39E-01}$	9.77E+02	2.61E+02	$\underline {1.02E-01}$	7.50E+02	5.17E+01	1.79E-02	1.02E+02	2.40E+01	$\underline {7.15E-01}$	$\textbf{9.32E+02}$	$\textbf{7.19E+00}$	$\textbf{N/A}$
CMFO1	7.02E+01	2.16E+01	$\underline {4.73E-01}$	1.03E+02	4.97E+01	$\underline {5.98E-01}$	9.47E+02	2.08E+02	$\underline {9.09E-02}$	7.23E+02	5.36E+01	$\underline {4.57E-01}$	1.01E+02	1.86E+01	$\underline {4.57E-01}$	9.36E+02	9.46E+00	$\underline {2.39E-01}$
CMFO2	8.05E+01	4.70E+01	$\underline {5.65E-02}$	1.21E+02	8.93E+01	$\underline {4.57E-01}$	9.85E+02	1.59E+02	1.14E-02	7.30E+02	5.27E+01	$\underline {2.08E-01}$	1.02E+02	1.63E+01	$\underline {3.94E-01}$	9.34E+02	1.19E+01	$\underline {7.15E-01}$
CMFO3	1.48E+03	1.38E-01	6.80E-08	1.47E+03	4.18E+01	6.80E-08	2.08E+03	2.54E+00	6.80E-08	1.43E+03	3.87E+00	6.80E-08	1.52E+03	6.78E+01	6.80E-08	1.88E+03	1.42E-02	6.80E-08
CMFO4	6.85E+01	2.55E+01	$\underline {7.35E-01}$	9.13E+01	1.48E+01	$\underline {8.39E-01}$	8.84E+02	1.57E+02	$\underline {3.94E-01}$	7.80E+02	5.81E+01	2.00E-04	1.13E+02	3.46E+01	$\underline{1.33E-01}$	9.34E+02	1.10E+01	$\underline {8.18E-01}$
CMFO5	6.63E+01	1.49E+01	$\underline {3.51E-01}$	9.60E+01	1.63E+01	$\underline {3.10E-01}$	8.36E+02	2.30E+02	$\underline {9.89E-01}$	$\textbf{7.08E+02}$	$\textbf{5.03E+01}$	$\textbf{N/A}$	1.01E+02	2.87E+01	$\underline {6.36E-01}$	9.37E+02	9.67E+00	$\underline {8.59E-02}$
CMFO6	1.41E+02	4.36E+01	1.10E-05	1.74E+02	3.86E+01	1.20E-06	9.58E+02	2.02E+02	$\underline {7.64E-02}$	7.58E+02	4.35E+01	3.34E-03	1.95E+02	1.56E+01	6.80E-08	9.35E+02	1.09E+01	$\underline {5.61E-01}$
CMFO7	7.86E+01	6.57E+01	$\underline {3.79E-01}$	9.40E+01	1.66E+01	$\underline {5.79E-01}$	$\textbf{8.34E+02}$	$\textbf{1.85E+02}$	$\textbf{N/A}$	7.44E+02	5.29E+01	1.33E-02	1.13E+02	2.49E+01	3.37E-02	9.35E+02	1.06E+01	$\underline {4.73E-01}$
CMFO8	6.98E+01	1.51E+01	$\underline {2.29E-01}$	1.01E+02	5.14E+01	$\underline {7.76E-01}$	9.79E+02	1.68E+02	1.23E-02	7.55E+02	5.07E+01	5.12E-03	1.04E+02	1.48E+01	$\underline{1.08E-01}$	9.34E+02	1.07E+01	$\underline {9.03E-01}$
CMFO9	$\textbf{6.27E+01}$	$\textbf{1.41E+01}$	$\textbf{N/A}$	$\textbf{9.12E+01}$	$\textbf{1.53E+01}$	$\textbf{N/A}$	8.98E+02	2.03E+02	$\underline{2.98E-01}$	7.66E+02	8.19E+01	1.44E-02	1.03E+02	1.60E+01	$\underline{2.62E-01}$	9.34E+02	1.01E+01	$\underline {5.61E-01}$
CMFO10	1.04E+02	7.83E+01	1.79E-02	1.05E+02	6.65E+01	$\underline {7.97E-01}$	9.14E+02	2.17E+02	$\underline {3.51E-01}$	7.70E+02	3.67E+01	4.60E-04	$\textbf{9.73E+01}$	$\textbf{1.72E+01}$	$\textbf{N/A}$	9.36E+02	1.14E+01	$\underline {2.29E-01}$

Table 8

Table 9

Table 10

Results of unimodal benchmark functions for chaotic boundary handing"

Algorithm	$F_1$			$F_{2}$			$F_{3}$			$F_4$			$F_{5}$			$F_{6}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	1.58E+00	7.84E-01	$\underline {1.14E-01}$	2.37E+06	1.18E+06	$\underline {6.01E-02}$	$\textbf{1.44E+04}$	$\textbf{3.37E+03}$	$\textbf{N/A}$	$\textbf{1.67E+01}$	$\textbf{3.65E+00}$	$\textbf{N/A}$	$\textbf{3.11E+03}$	$\textbf{9.55E+02}$	$\textbf{N/A}$	3.97E+03	1.29E+03	2.34E-03
CMFO11	3.97E+00	2.42E+00	1.38E-06	6.48E+06	4.57E+06	1.81E-05	2.66E+04	7.57E+03	2.69E-06	2.39E+01	4.39E+00	1.41E-05	6.74E+03	1.49E+03	2.22E-07	6.41E+03	1.34E+03	6.80E-08
CMFO12	1.37E+00	6.84E-01	$\underline {5.79E-01}$	2.04E+06	1.17E+06	$\underline {5.61E-01}$	1.69E+04	3.55E+03	4.39E-02	1.97E+01	3.37E+00	1.06E-02	3.19E+03	1.02E+03	$\underline{7.15E-01}$	$\textbf{2.78E+03}$	$\textbf{5.88E+02}$	$\textbf{N/A}$
CMFO13	8.55E+00	7.23E+00	1.80E-06	5.60E+06	2.84E+06	1.38E-06	2.58E+04	6.77E+03	1.20E-06	4.77E+01	1.45E+01	6.80E-08	7.33E+03	2.50E+03	4.54E-07	7.47E+03	3.97E+03	2.96E-07
CMFO14	1.74E+00	7.01E-01	1.23E-02	2.16E+06	1.10E+06	$\underline{3.23E-01}$	1.80E+04	4.66E+03	1.79E-02	2.10E+01	4.94E+00	4.70E-03	3.68E+03	1.10E+03	$\underline{1.02E-01}$	3.77E+03	1.16E+03	2.80E-03
CMFO15	2.46E+00	9.06E-01	1.10E-05	2.73E+06	1.21E+06	7.71E-03	2.00E+04	5.33E+03	3.05E-04	2.17E+01	4.41E+00	8.36E-04	4.49E+03	1.34E+03	4.60E-04	4.21E+03	1.47E+03	4.16E-04
CMFO16	1.53E+00	5.77E-01	$\underline {6.39E-02}$	2.03E+06	9.69E+05	$\underline {5.08E-01}$	1.67E+04	5.05E+03	$\underline {1.64E-01}$	2.10E+01	4.47E+00	3.64E-03	3.91E+03	1.11E+03	2.75E-02	3.48E+03	8.14E+02	6.04E-03
CMFO17	1.66E+00	6.04E-01	1.44E-02	3.54E+06	2.37E+06	5.12E-03	1.97E+04	4.98E+03	9.21E-04	2.23E+01	3.91E+00	1.04E-04	4.25E+03	1.00E+03	1.01E-03	4.49E+03	1.25E+03	1.60E-05
CMFO18	1.55E+00	5.08E-01	3.15E-02	2.88E+06	2.08E+06	$\underline{5.65E-02}$	1.83E+04	5.45E+03	1.67E-02	1.96E+01	3.90E+00	2.39E-02	3.78E+03	1.26E+03	$\underline{1.02E-01}$	4.57E+03	1.31E+03	2.06E-06
CMFO19	$\textbf{1.19E+00}$	$\textbf{4.42E-01}$	$\textbf{N/A}$	2.42E+06	1.42E+06	$\underline {8.59E-02}$	1.69E+04	4.60E+03	$\underline {7.64E-02}$	1.90E+01	3.31E+00	4.99E-02	3.15E+03	9.35E+02	$\underline {9.46E-01}$	3.45E+03	1.02E+03	1.55E-02
CMFO20	1.37E+00	3.68E-01	$\underline {1.26E-01}$	$\textbf{1.73E+06}$	$\textbf{6.83E+05}$	$\textbf{N/A}$	1.58E+04	5.17E+03	$\underline {4.73E-01}$	2.01E+01	3.49E+00	3.97E-03	3.79E+03	1.30E+03	$\underline {6.79E-02}$	2.95E+03	7.72E+02	$\underline {4.09E-01}$

Table 10

Table 11

Results of multimodal benchmark functions for chaotic boundary handing"

Algorithm	$F_7$			$F_{8}$			$F_{9}$			$F_{10}$			$F_{11}$			$F_{12}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	1.18E+01	1.01E+00	$\underline {9.09E-02}$	2.92E+01	1.01E+01	$\underline {7.15E-01}$	7.65E+05	1.02E+06	$\underline {1.81E-01}$	4.33E+06	3.27E+06	$\underline {4.90E-01}$	$\textbf{1.18E+02}$	$\textbf{1.76E+01}$	$\textbf{N/A}$	5.51E+03	4.96E+02	2.36E-06
CMFO11	1.43E+01	1.49E+00	3.94E-07	5.53E+01	1.52E+01	2.06E-06	4.83E+06	4.60E+06	1.20E-06	1.35E+07	5.84E+06	3.50E-06	1.47E+02	1.58E+01	1.81E-05	5.13E+03	6.85E+02	1.44E-04
CMFO12	1.15E+01	9.23E-01	$\underline {4.41E-01}$	$\textbf{2.73E+01}$	$\textbf{7.23E+00}$	$\textbf{N/A}$	$\textbf{5.60E+05}$	$\textbf{9.72E+05}$	$\textbf{N/A}$	$\textbf{3.30E+06}$	$\textbf{2.02E+06}$	$\textbf{N/A}$	1.21E+02	2.20E+01	$\underline {8.60E-01}$	5.82E+03	4.13E+02	6.92E-07
CMFO13	1.56E+01	2.25E+00	2.56E-07	6.23E+01	2.85E+01	1.20E-06	3.58E+06	2.27E+06	2.06E-06	1.76E+07	9.50E+06	2.56E-07	1.91E+02	2.98E+01	2.22E-07	6.41E+03	4.99E+02	2.22E-07
CMFO14	1.22E+01	9.62E-01	2.14E-03	3.57E+01	9.21E+00	6.56E-03	5.74E+05	7.89E+05	$\underline {9.89E-01}$	5.80E+06	4.25E+06	$\underline {6.79E-02}$	1.31E+02	2.04E+01	3.15E-02	5.22E+03	6.41E+02	1.79E-04
CMFO15	1.26E+01	1.21E+00	4.60E-04	3.78E+01	1.17E+01	5.56E-03	1.25E+06	1.13E+06	4.32E-03	8.34E+06	6.46E+06	1.95E-03	1.41E+02	2.82E+01	3.97E-03	5.09E+03	6.07E+02	6.22E-04
CMFO16	1.19E+01	9.71E-01	4.68E-02	3.34E+01	9.71E+00	3.15E-02	6.71E+05	5.97E+05	$\underline {9.09E-02}$	5.25E+06	3.83E+06	$\underline {1.02E-01}$	1.32E+02	2.06E+01	2.39E-02	5.34E+03	4.71E+02	1.81E-05
CMFO17	1.25E+01	1.25E+00	7.58E-04	3.67E+01	7.97E+00	6.87E-04	1.57E+06	1.68E+06	2.34E-03	8.04E+06	7.99E+06	1.67E-02	1.40E+02	2.29E+01	2.34E-03	5.04E+03	2.82E+02	2.30E-05
CMFO18	1.23E+01	1.26E+00	5.12E-03	3.17E+01	9.49E+00	$\underline{1.08E-01}$	1.60E+06	1.71E+06	1.12E-03	6.67E+06	4.37E+06	6.56E-03	1.41E+02	2.25E+01	2.80E-03	4.69E+03	5.00E+02	1.67E-02
CMFO19	1.19E+01	1.10E+00	$\underline {6.39E-02}$	2.90E+01	8.21E+00	$\underline {7.97E-01}$	5.94E+05	4.99E+05	$\underline {2.50E-01}$	5.61E+06	5.55E+06	$\underline {1.26E-01}$	1.28E+02	2.48E+01	$\underline {1.81E-01}$	$\textbf{4.31E+03}$	$\textbf{6.92E+02}$	$\textbf{N/A}$
CMFO20	$\textbf{1.12E+01}$	$\textbf{9.69E-01}$	$\textbf{N/A}$	3.26E+01	8.36E+00	2.94E-02	5.64E+05	5.82E+05	$\underline {6.17E-01}$	4.00E+06	2.49E+06	$\underline {4.90E-01}$	1.31E+02	1.39E+01	2.23E-02	5.52E+03	5.86E+02	5.17E-06

Table 11

Table 12

Results of composite benchmark functions for chaotic boundary handing"

Algorithm	$F_{13}$			$F_{14}$			$F_{15}$			$F_{16}$			$F_{17}$			$F_{18}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	8.36E+01	4.63E+01	$\underline {2.85E-01}$	9.27E+01	1.91E+01	$\underline {1.48E-01}$	9.77E+02	2.61E+02	$\underline {2.29E-01}$	7.50E+02	5.17E+01	$\underline {6.36E-01}$	1.02E+02	2.40E+01	$\underline {5.98E-01}$	$\textbf{9.32E+02}$	$\textbf{7.19E+00}$	$\textbf{N/A}$
CMFO11	1.69E+02	1.15E+02	6.92E-07	2.22E+02	1.28E+02	6.80E-08	1.52E+03	1.91E+02	1.43E-07	9.40E+02	1.25E+02	3.07E-06	2.29E+02	1.04E+02	6.80E-08	9.77E+02	2.24E+01	7.90E-08
CMFO12	1.16E+02	8.27E+01	2.14E-03	1.13E+02	1.83E+01	1.05E-06	1.00E+03	2.14E+02	$\underline {8.10E-02}$	7.59E+02	4.97E+01	$\underline {1.90E-01}$	1.23E+02	3.66E+01	1.78E-03	9.33E+02	8.83E+00	$\underline {9.89E-01}$
CMFO13	2.69E+02	1.18E+02	6.80E-08	2.60E+02	1.03E+02	6.80E-08	1.63E+03	9.72E+01	6.80E-08	8.22E+02	7.85E+01	1.29E-04	3.04E+02	1.42E+02	6.80E-08	9.80E+02	2.47E+01	1.92E-07
CMFO14	6.71E+01	1.15E+01	$\underline {8.39E-01}$	9.35E+01	1.53E+01	4.68E-02	9.44E+02	2.57E+02	$\underline {3.79E-01}$	7.35E+02	5.32E+01	$\underline {8.82E-01}$	1.03E+02	2.24E+01	$\underline {3.65E-01}$	9.34E+02	8.78E+00	$\underline {8.60E-01}$
CMFO15	6.92E+01	2.14E+01	$\underline {8.39E-01}$	9.31E+01	1.78E+01	$\underline {6.01E-02}$	9.92E+02	2.12E+02	$\underline {1.14E-01}$	$\textbf{7.34E+02}$	$\textbf{5.42E+01}$	$\textbf{N/A}$	1.03E+02	2.14E+01	$\underline {3.10E-01}$	9.35E+02	1.13E+01	$\underline {6.17E-01}$
CMFO16	8.85E+01	4.91E+01	$\underline {9.62E-02}$	1.25E+02	9.29E+01	1.44E-02	9.66E+02	2.56E+02	$\underline {2.85E-01}$	7.63E+02	7.11E+01	$\underline {3.79E-01}$	1.21E+02	4.33E+01	4.39E-02	9.34E+02	1.01E+01	$\underline {6.55E-01}$
CMFO17	6.84E+01	1.64E+01	$\underline {7.56E-01}$	1.10E+02	4.94E+01	6.56E-03	9.81E+02	2.14E+02	$\underline {1.20E-01}$	7.40E+02	5.92E+01	$\underline {9.89E-01}$	1.05E+02	3.39E+01	$\underline {5.43E-01}$	9.34E+02	1.33E+01	$\underline {6.75E-01}$
CMFO18	6.96E+01	1.61E+01	$\underline {5.08E-01}$	$\textbf{8.34E+01}$	$\textbf{1.08E+01}$	$\textbf{N/A}$	8.89E+02	2.08E+02	$\underline {9.68E-01}$	7.46E+02	5.69E+01	$\underline {3.23E-01}$	$\textbf{9.47E+01}$	$\textbf{1.23E+01}$	$\textbf{N/A}$	9.39E+02	1.03E+01	2.94E-02
CMFO19	$\textbf{6.56E+01}$	$\textbf{1.71E+01}$	$\textbf{N/A}$	9.10E+01	1.42E+01	$\underline {1.64E-01}$	9.47E+02	1.71E+02	$\underline {2.98E-01}$	7.40E+02	4.10E+01	$\underline {2.50E-01}$	9.63E+01	1.62E+01	$\underline {8.60E-01}$	9.45E+02	8.33E+00	2.04E-05
CMFO20	8.57E+01	6.92E+01	$\underline {6.95E-01}$	9.29E+01	1.70E+01	3.15E-02	$\textbf{8.80E+02}$	$\textbf{1.97E+02}$	$\textbf{N/A}$	7.42E+02	6.10E+01	$\underline {7.15E-01}$	1.05E+02	2.59E+01	$\underline {2.39E-01}$	9.36E+02	9.50E+00	$\underline {1.20E-01}$

Table 12

Table 13

Table 14

Results of unimodal benchmark functions for tuning distance parameter $r$"

Algorithm	$F_1$			$F_{2}$			$F_{3}$			$F_4$			$F_{5}$			$F_{6}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	1.58E+00	7.84E-01	3.60E-02	2.37E+06	1.18E+06	1.79E-04	$\textbf{1.44E+04}$	$\textbf{3.37E+03}$	$\textbf{N/A}$	1.67E+01	3.65E+00	6.22E-04	3.11E+03	9.55E+02	$\underline {5.98E-01}$	3.97E+03	1.29E+03	7.58E-06
CMFO21	2.04E+01	5.20E+00	6.80E-08	4.75E+07	7.79E+06	6.80E-08	4.24E+04	6.87E+03	6.80E-08	1.51E+02	2.01E+02	6.80E-08	2.59E+04	3.07E+03	6.80E-08	2.50E+04	2.70E+03	6.80E-08
CMFO22	3.31E+00	1.24E+00	3.42E-07	8.32E+06	3.78E+06	6.80E-08	2.20E+04	5.15E+03	1.81E-05	3.37E+01	5.36E+00	6.80E-08	8.21E+03	1.99E+03	1.78E-03	8.34E+03	2.19E+03	6.80E-08
CMFO23	3.76E+01	7.99E+00	6.80E-08	9.13E+07	1.19E+07	6.80E-08	5.39E+04	6.96E+03	6.80E-08	1.69E+03	3.14E+03	6.80E-08	3.59E+04	3.40E+03	6.80E-08	3.63E+04	3.25E+03	6.80E-08
CMFO24	1.23E+00	5.55E-01	$\underline {4.41E-01}$	2.07E+06	7.70E+05	7.41E-05	1.65E+04	3.75E+03	$\underline {1.08E-01}$	2.06E+01	3.46E+00	6.01E-07	3.23E+03	7.25E+02	$\underline{3.65E-01}$	3.92E+03	1.01E+03	1.80E-06
CMFO25	2.49E+00	1.21E+00	7.58E-06	5.36E+06	2.30E+06	6.80E-08	1.55E+04	4.44E+03	$\underline {4.09E-01}$	2.93E+01	4.77E+00	7.90E-08	5.35E+03	1.44E+03	$\underline {3.10E-01}$	6.13E+03	1.77E+03	7.90E-08
CMFO26	2.42E+00	1.01E+00	2.60E-05	5.27E+06	1.98E+06	7.90E-08	1.81E+04	3.07E+03	1.95E-03	2.61E+01	3.44E+00	7.90E-08	6.34E+03	2.22E+03	4.39E-02	5.99E+03	1.96E+03	2.56E-07
CMFO27	3.20E+00	1.43E+00	7.95E-07	5.61E+06	2.45E+06	1.06E-07	2.03E+04	4.03E+03	6.61E-05	2.91E+01	5.11E+00	7.90E-08	6.26E+03	1.55E+03	$\underline{5.43E-01}$	7.83E+03	2.01E+03	6.80E-08
CMFO28	$\textbf{1.11E+00}$	$\textbf{5.20E-01}$	$\textbf{N/A}$	$\textbf{1.11E+06}$	$\textbf{5.16E+05}$	$\textbf{N/A}$	1.58E+04	5.77E+03	$\underline {7.56E-01}$	$\textbf{1.30E+01}$	$\textbf{2.51E+00}$	$\textbf{N/A}$	$\textbf{2.25E+03}$	$\textbf{6.79E+02}$	$\textbf{N/A}$	$\textbf{2.11E+03}$	$\textbf{5.79E+02}$	$\textbf{N/A}$
CMFO29	1.82E+00	1.05E+00	4.70E-03	3.22E+06	1.89E+06	1.29E-04	2.00E+04	4.34E+03	8.29E-05	1.79E+01	4.67E+00	9.28E-05	3.55E+03	1.35E+03	$\underline {8.60E-01}$	3.36E+03	1.46E+03	2.34E-03
CMFO30	1.52E+00	3.86E-01	9.21E-04	2.20E+06	1.39E+06	2.34E-03	1.45E+04	3.28E+03	$\underline {9.25E-01}$	1.88E+01	3.56E+00	2.30E-05	3.04E+03	8.77E+02	$\underline {2.39E-01}$	3.72E+03	9.56E+02	1.80E-06

Table 14

Table 15

Results of multimodal benchmark functions for tuning distance parameter $r$"

Algorithm	$F_7$			$F_{8}$			$F_{9}$			$F_{10}$			$F_{11}$			$F_{12}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	1.18E+01	1.01E+00	1.35E-03	2.92E+01	1.01E+01	9.05E-03	7.65E+05	1.02E+06	$\underline {2.62E-01}$	4.33E+06	3.27E+06	2.23E-02	1.18E+02	1.76E+01	$\underline {2.39E-01}$	5.51E+03	4.96E+02	$\underline {5.08E-01}$
CMFO21	1.87E+01	4.32E-01	6.80E-08	2.25E+02	2.30E+01	6.80E-08	5.73E+07	2.71E+07	6.80E-08	1.64E+08	5.31E+07	6.80E-08	2.90E+02	1.85E+01	6.80E-08	7.88E+03	5.03E+02	6.80E-08
CMFO22	1.50E+01	6.98E-01	6.80E-08	8.25E+01	1.80E+01	6.80E-08	5.03E+06	4.02E+06	2.56E-07	1.80E+07	8.20E+06	6.80E-08	1.82E+02	2.29E+01	1.06E-07	5.87E+03	5.56E+02	1.23E-02
CMFO23	1.94E+01	2.08E-01	6.80E-08	3.30E+02	2.77E+01	6.80E-08	1.42E+08	4.75E+07	6.80E-08	3.24E+08	7.36E+07	6.80E-08	3.23E+02	2.18E+01	6.80E-08	7.97E+03	4.19E+02	6.80E-08
CMFO24	1.20E+01	1.05E+00	3.05E-04	3.50E+01	9.77E+00	4.68E-05	$\textbf{4.36E+05}$	$\textbf{4.33E+05}$	$\textbf{N/A}$	5.02E+06	5.54E+06	3.60E-02	1.22E+02	1.91E+01	$\underline{8.10E-02}$	5.58E+03	4.61E+02	$\underline {1.99E-01}$
CMFO25	1.36E+01	9.40E-01	7.90E-08	5.13E+01	1.65E+01	1.23E-07	1.44E+06	1.07E+06	1.12E-03	1.19E+07	6.68E+06	1.80E-06	1.47E+02	2.01E+01	1.10E-05	5.53E+03	5.63E+02	$\underline {4.09E-01}$
CMFO26	1.40E+01	1.12E+00	7.90E-08	5.33E+01	1.35E+01	4.54E-07	1.20E+06	8.19E+05	2.14E-03	1.36E+07	7.04E+06	1.06E-07	1.64E+02	1.71E+01	3.94E-07	5.91E+03	5.54E+02	2.80E-03
CMFO27	1.46E+01	7.20E-01	6.80E-08	6.02E+01	1.74E+01	6.80E-08	6.08E+06	6.06E+06	9.13E-07	1.37E+07	7.79E+06	1.23E-07	1.61E+02	2.66E+01	2.69E-06	5.58E+03	8.93E+02	$\underline {4.09E-01}$
CMFO28	$\textbf{1.04E+01}$	$\textbf{1.21E+00}$	$\textbf{N/A}$	$\textbf{2.14E+01}$	$\textbf{5.73E+00}$	$\textbf{N/A}$	4.82E+05	6.02E+05	$\underline {6.75E-01}$	$\textbf{2.29E+06}$	$\textbf{1.76E+06}$	$\textbf{N/A}$	$\textbf{1.11E+02}$	$\textbf{2.14E+01}$	$\textbf{N/A}$	$\textbf{5.39E+03}$	$\textbf{4.57E+02}$	$\textbf{N/A}$
CMFO29	1.19E+01	1.41E+00	1.95E-03	3.05E+01	1.02E+01	1.95E-03	8.39E+05	8.56E+05	$\underline {1.56E-01}$	5.62E+06	4.18E+06	3.97E-03	1.55E+02	2.53E+01	7.58E-06	6.57E+03	5.69E+02	2.36E-06
CMFO30	1.15E+01	8.73E-01	1.78E-03	2.98E+01	6.77E+00	3.75E-04	8.89E+05	1.67E+06	$\underline {1.21E-01}$	4.49E+06	3.31E+06	1.23E-02	1.33E+02	2.11E+01	3.06E-03	5.71E+03	6.05E+02	$\underline {1.08E-01}$

Table 15

Table 16

Results of composite benchmark functions for tuning distance parameter $r$"

Algorithm	$F_{13}$			$F_{14}$			$F_{15}$			$F_{16}$			$F_{17}$			$F_{18}$
Algorithm	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value	Mean	SD	$p$ value
MFO	8.36E+01	4.63E+01	$\underline {2.85E-01}$	9.27E+01	1.91E+01	$\underline {9.62E-02}$	9.77E+02	2.61E+02	3.34E-03	7.50E+02	5.17E+01	$\underline {9.46E-01}$	1.02E+02	2.40E+01	2.22E-04	9.32E+02	7.19E+00	3.34E-03
CMFO21	3.34E+02	5.06E+01	7.90E-08	3.57E+02	5.20E+01	9.17E-08	1.74E+03	2.17E+01	6.80E-08	9.73E+02	6.97E+01	6.80E-08	3.85E+02	5.86E+01	6.80E-08	1.11E+03	3.37E+01	6.80E-08
CMFO22	1.33E+02	2.84E+01	4.54E-06	1.54E+02	3.72E+01	1.80E-06	1.40E+03	1.93E+02	1.92E-07	7.69E+02	5.32E+01	2.23E-02	1.64E+02	3.09E+01	7.90E-08	9.69E+02	1.82E+01	1.66E-07
CMFO23	4.39E+02	7.87E+01	6.80E-08	4.81E+02	6.69E+01	6.80E-08	1.78E+03	2.00E+01	6.80E-08	1.15E+03	6.60E+01	6.80E-08	5.07E+02	6.58E+01	6.80E-08	1.28E+03	3.51E+01	6.80E-08
CMFO24	9.81E+01	9.60E+01	$\underline {9.46E-01}$	1.09E+02	4.52E+01	2.34E-03	1.04E+03	2.03E+02	2.47E-04	$\textbf{7.36E+02}$	$\textbf{5.68E+01}$	$\textbf{N/A}$	1.15E+02	4.28E+01	4.17E-05	9.36E+02	1.49E+01	3.34E-03
CMFO25	9.13E+01	2.03E+01	2.56E-03	1.17E+02	2.57E+01	6.61E-05	1.16E+03	2.13E+02	7.58E-06	7.52E+02	5.01E+01	$\underline {9.09E-02}$	1.26E+02	1.74E+01	2.56E-07	9.58E+02	1.74E+01	2.22E-07
CMFO26	1.27E+02	9.10E+01	3.06E-03	1.23E+02	2.04E+01	4.54E-06	1.27E+03	1.97E+02	6.92E-07	7.57E+02	6.14E+01	$\underline {8.59E-02}$	1.39E+02	3.23E+01	1.23E-07	9.61E+02	1.46E+01	7.90E-08
CMFO27	1.16E+02	6.47E+01	1.44E-04	1.35E+02	2.55E+01	1.58E-06	1.34E+03	1.66E+02	1.23E-07	7.70E+02	5.39E+01	3.37E-02	1.47E+02	3.67E+01	1.66E-07	9.61E+02	1.87E+01	1.23E-07
CMFO28	8.53E+01	8.74E+01	$\underline {5.25E-01}$	9.22E+01	4.96E+01	$\underline {8.39E-01}$	$\textbf{7.26E+02}$	$\textbf{2.28E+02}$	$\textbf{N/A}$	7.46E+02	4.38E+01	$\underline {7.35E-01}$	$\textbf{8.29E+01}$	$\textbf{1.17E+01}$	$\textbf{N/A}$	$\textbf{9.25E+02}$	$\textbf{5.67E+00}$	$\textbf{N/A}$
CMFO29	1.30E+02	7.44E+01	2.30E-05	1.46E+02	3.10E+01	2.36E-06	1.20E+03	2.88E+02	2.30E-05	7.95E+02	5.39E+01	8.36E-04	1.38E+02	3.46E+01	4.54E-06	9.51E+02	1.52E+01	2.56E-07
CMFO30	$\textbf{7.56E+01}$	$\textbf{4.27E+01}$	$\textbf{N/A}$	$\textbf{9.08E+01}$	$\textbf{5.02E+01}$	$\textbf{N/A}$	9.05E+02	2.43E+02	3.37E-02	7.49E+02	5.34E+01	$\underline {9.25E-01}$	9.76E+01	1.56E+01	1.95E-03	9.31E+02	9.25E+00	$\underline {5.31E-02}$

Table 16

Table 17

Fig 1

Fig 2

Table 18

Fig 3

Table 19

References 48

1	GANDOMI A H, YANG X S, ALAVI A H. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Engineering with Computers, 2013, 29 (1): 17- 35. doi: 10.1007/s00366-011-0241-y
2	MAFARJA M M, MIRJALILI S. Hybrid whale optimization algorithm with simulated annealing for feature selection. Neurocomputing, 2017, 260, 302- 312. doi: 10.1016/j.neucom.2017.04.053
3	BHANDARI A K, SINGH V K, KUMAR A, et al. Cuckoo search algorithm and wind driven optimization based study of satellite image segmentation for multilevel thresholding using Kapur's entropy. Expert Systems with Applications, 2014, 41 (7): 3538- 3560. doi: 10.1016/j.eswa.2013.10.059
4	WANG H, WANG W J, SUN H, et al. A new cuckoo search algorithm with hybrid strategies for flow shop scheduling problems. Soft Computing, 2017, 21 (15): 4297- 4307. doi: 10.1007/s00500-016-2062-9
5	ZHANG X M. Parameter estimation of shallow wave equation via cuckoo search. Neural Computing & Applications, 2017, 28 (12): 4047- 4059.
6	SAAD A, KHAN S A, MAHMOOD A. A multi-objective evolutionary artificial bee colony algorithm for optimizing network topology design. Swarm and Evolutionary Computation, 2018, 38, 187- 201. doi: 10.1016/j.swevo.2017.07.010
7	HOLLAND J H. Adaptation in natural and artificial systems. Massachusetts: MIT Press, 1992.
8	EBERHART R, KENNEDY J. A new optimizer using particle swarm theory. Proc. of the 6th International Symposium on Micro Machine and Human Science, 1995, 39- 43.
9	DORIGO M, MANIEZZO V, COLORNI A. Ant system: optimization by a colony of cooperating agents. IEEE Trans. on Systems Man & Cybernetics-Part B, 1996, 26 (1): 29- 41.
10	STORN R, PRICE K. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 1997, 11 (4): 341- 359. doi: 10.1023/A:1008202821328
11	KARABOGA D, BASTURK B. An artificial bee colony (ABC) algorithm for numeric function optimization. Proc. of the IEEE Swarm Intelligence Symposium, 2006, 181- 184.
12	YANG X S, DEB S. Cuckoo search via levy flights. Mathematics, 2010, 210- 214.
13	ZONG W G, KIM J H, LOGANATHAN G V. A new heuristic optimization algorithm: harmony search. Simulation Transactions of the Society for Modeling & Simulation International, 2001, 76 (2): 60- 68.
14	YANG X S. A new metaheuristic bat-inspired algorithm. Computer Knowledge & Technology, 2010, 284 (12): 65- 74.
15	MENG X, LIU Y, GAO X, et al. A new bio-inspired algorithm: chicken swarm optimization. Proc. of the International Conference in Swarm Intelligence, 2014, 86- 94.
16	MIRJALILI S, MIRJALILI S M, LEWIS A. Grey wolf optimizer. Advances in Engineering Software, 2014, 69 (3): 46- 61.
17	YANG X S. Nature-inspired metaheuristic algorithms. Frome: Luniver Press, 2008.
18	MIRJALILI S, LEWIS A. The whale optimization algorithm. Advances in Engineering Software, 2016, 95, 51- 67. doi: 10.1016/j.advengsoft.2016.01.008
19	MIRJALILI S. The ant lion optimizer. Advances in Engineering Software, 2015, 83 (C): 80- 98.
20	MIRJALILI S. Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Knowledge-Based Systems, 2015, 89, 228- 249. doi: 10.1016/j.knosys.2015.07.006
21	JANGIR N, TRIVEDI I N, PANDYA M H, et al. Moth-flame optimization algorithm for solving real challenging constrained engineering optimization problems. Proc. of the IEEE Students' Conference on Electrical, Electronics and Computer Science, 2016, 1- 5.
22	SAVSANI V, TAWHID M A. Non-dominated sorting moth flame optimization (NS-MFO) for multi-objective problems. Engineering Applications of Artificial Intelligence, 2017, 63, 20- 32. doi: 10.1016/j.engappai.2017.04.018
23	MUANGKOTE N, SUNAT K, CHIEWCHANWATTANA S, et al. Multilevel thresholding for satellite image segmentation with moth-flame based optimization. Proc. of the 13th International Joint Conference on Computer Science and Software Engineering, 2016, 460- 465.
24	ABD EL AZIZ M, EWEESC A A, HASSANIEN A E. Whale optimization algorithm and moth-flame optimization for multilevel thresholding image segmentation. Expert Systems with Applications, 2017, 83, 242- 256. doi: 10.1016/j.eswa.2017.04.023
25	WANG M J, CHEN H L, YANG B, et al. Toward an optimal kernel extreme learning machine using a chaotic moth-flame optimization strategy with applications in medical diagnoses. Neurocomputing, 2017, 267, 69- 84. doi: 10.1016/j.neucom.2017.04.060
26	HASSANIEN A E, GABER T, MOKHTAR U, et al. An improved moth flame optimization algorithm based on rough sets for tomato diseases detection. Computers and Electronics in Agriculture, 2017, 136, 86- 96. doi: 10.1016/j.compag.2017.02.026
27	BUCH H, TRIVEDI I N, JANGIR P. Moth flame optimization to solve optimal power flow with non-parametric statistical evaluation validation. Cogent Engineering, 2017, 4 (1): 1286731.
28	SINGH P, PRAKASH S. Optical network unit placement in fiber-wireless (FiWi) access network by moth-flame optimization algorithm. Optical Fiber Technology, 2017, 36, 403- 411. doi: 10.1016/j.yofte.2017.05.018
29	ALLAM D, YOUSRI D A, ETEIBA M B. Parameters extraction of the three diode model for the multi-crystalline solar cell/module using moth-flame optimization algorithm. Energy Conversion and Management, 2016, 123, 535- 548. doi: 10.1016/j.enconman.2016.06.052
30	YAMANY W, FAWZY M, THARWAT A, et al. Moth-flame optimization for training multi-layer perceptrons. Proc. of the 11th International Computer Engineering Conference, 2015, 267- 272.
31	VIKAS, NANDA S J. Multi-objective moth flame optimization. Proc. of the International Conference on Advances in Computing, Communications and Informatics, 2016, 2470- 2476.
32	DUBEY H M, PANDIT M, PANIGRAHI B K. An overview and comparative analysis of recent bio-inspired optimization techniques for wind integrated multi-objective power dispatch. Swarm and Evolutionary Computation, 2018, 38, 12- 34. doi: 10.1016/j.swevo.2017.07.012
33	YANG L J, CHEN T L. Application of chaos in genetic algorithms. Communications in Theoretical Physics, 2002, 38 (2): 168- 172. doi: 10.1088/0253-6102/38/2/168
34	GUO Z, CHENG B, YE M, et al. Self-adaptive chaos differential evolution. Proc. of the International Conference on Natural Computation, 2006, 972- 975.
35	GANDOMI A H, YUN G J, YANG X S, et al. Chaos-enhanced accelerated particle swarm optimization. Communications in Nonlinear Science and Numerical Simulation, 2013, 18 (2): 327- 340. doi: 10.1016/j.cnsns.2012.07.017
36	GANDOMI A H, YANG X S, TALATAHARI S, et al. Firefly algorithm with chaos. Communications in Nonlinear Science and Numerical Simulation, 2013, 18 (1): 89- 98. doi: 10.1016/j.cnsns.2012.06.009
37	SAREMI S, MIRJALILI S, LEWIS A. Biogeography-based optimisation with chaos. Neural Computing & Applications, 2014, 25 (5): 1077- 1097.
38	HUANG L, DING S, YU S H, et al. Chaos-enhanced cuckoo search optimization algorithms for global optimization. Applied Mathematical Modelling, 2016, 40 (5/6): 3860- 3875.
39	WANG G G, DEB S, GANDOMI A H, et al. Chaotic cuckoo search. Soft Computing, 2016, 20 (9): 3349- 3362. doi: 10.1007/s00500-015-1726-1
40	WU W M, LI Z X, LIN Z Y, et al. Moth-flame optimization algorithm based on chaotic crisscross operator. Computer Engineering & Applications, 2018, 54 (3): 136- 141.
41	GASTON K J, BENNIE J, DAVIES T W, et al. The ecological impacts of nighttime light pollution: a mechanistic appraisal. Biological Reviews, 2013, 88 (4): 912- 927. doi: 10.1111/brv.12036
42	PADHYE N, DEB K, MITTAL P. Boundary handling approaches in particle swarm optimization. Advances in Intelligent Systems & Computing, 2013, 201, 287- 298.
43	LIANG J J, SUGANTHAN P N, DEB K. Novel composition test functions for numerical global optimization. Proc. of the Swarm Intelligence Symposium, 2005, 68- 75.
44	HE Q, WANG L. An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Engineering Applications of Artificial Intelligence, 2007, 20 (1): 89- 99.
45	SECTION I. Composite differential evolution with modified oracle penalty method for constrained optimization problems. Mathematical Problems in Engineering, 2014, 1, 1- 15.
46	COELLO C A C. Use of a self-adaptive penalty approach for engineering optimization problems. Computers in Industry, 2000, 41 (2): 113- 127. doi: 10.1016/S0166-3615(99)00046-9
47	COELLO C A C, MONTES E M. Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Advanced Engineering Informatics, 2002, 16 (3): 193- 203. doi: 10.1016/S1474-0346(02)00011-3
48	HUANG F Z, WANG L, HE Q. An effective co-evolutionary differential evolution for constrained optimization. Applied Mathematics and Computation, 2007 186 (1): 340- 356. doi: 10.1016/j.amc.2006.07.105

Number	Name	Chaotic map
1	Chebyshev	$x_{k+1}=\cos(k\arccos x_k)$
2	Circle map	$x_{k+1}=x_k+b-(a/2{\rm{\pi }})\sin(2{\rm{\pi }} x_k)\cdot \\ \bmod(1), ~~~a=0.5~{\rm{and}}~b=0.2$
3	Gaussian map	$x_{k+1}=\left\{\!\!\begin{array}{l} 0, ~~~x_k=0 \\ 1/\bmod(x_k, 1), ~~~{\rm otherwise}\end{array}\!\!\right.$
4	Iterative map	$x_{k+1}=\sin(a{\rm{\pi }}/x_k), ~~~a=0.7$
5	Logistic map	$x_{k+1}=ax_k(1-x_k), ~~~a=4$
6	Piecewise map	$x_{k+1}=$ $\left\{\!\!\begin{array}{l} \dfrac{x_k}{P}, ~~~0\le x_k < P\\[3mm] \dfrac{x_k-P}{0.5-P}, ~~~P\le x_k < 0.5\\[3mm] \dfrac{1{-}P{-}x_k}{0.5-P}, ~~~0.5\le x_k\le1-P\\[3mm] %\qquad P=0.4\\ \dfrac{1-x_k}{P}, ~~~1-P\le x_k\le1\\ \end{array}\!\!\right.$
7	Sine map	$x_{k+1}=\dfrac{a}{4}\sin({\rm{\pi }} x_k), ~~~a=4$
8	Singer map	$x_{k+1}=\mu(7.86x_k{-}23.31x_k^2{+}28.75x^3_k- \\ 13.302~875x^4_k), ~~~u=1.07$
9	Sinusoidal map	$x_{k+1}=ax^2_k\sin({\rm{\pi }} x_k), ~~~a=2.3$
10	Tent map	$x_{k+1}=\left\{\!\!\begin{array}{l} \dfrac{x_k}{0.7}, ~~~x_k < 0.7 \\ \dfrac{10}{3}(1-x_k), ~~~x_k\ge0.7\end{array}\!\!\right.$

Formulation	Search range
$F_1(\mathit{\boldsymbol{X}})=\sum\limits^n_{i=1}ix^4_i+{\rm random}[0, 1]$	$x_i\in[-1.28, 1.28]$
$F_2(\mathit{\boldsymbol{X}})=\sum\limits^{n-1}_{i=1}(100(x_{i+1}{-}x^2_i)^2{+}(1{-}x_i)^2)$	$x_i\in[-30, 30]$
$F_3(\mathit{\boldsymbol{X}})=\sum\limits^n_{i=1}\left(\sum\limits^i_{j-1}x_j\right)^2$	$x_i\in[-100, 100]$
$F_4(\mathit{\boldsymbol{X}})=\sum\limits^n_{i=1}\|x_i\|+\prod \limits^n_{i=1}\|x_i\|$	$x_i\in[-10, 10]$
$F_5(\mathit{\boldsymbol{X}})=\sum\limits^n_{i=1}x^2_i$	$x_i\in[-100, 100]$
$F_6(\mathit{\boldsymbol{X}})=\sum\limits^n_{i=1}(\lfloor x_i+0.5\rfloor)^2$	$x_i\in[-100, 100]$

Formulation	Search range
$F_7(\mathit{\boldsymbol{X}})=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum\limits^n_{i=1}x^2_i}\right)- \exp\left(\frac{1}{n}\sum\limits^n_{i=1}\cos2{\rm{\pi }} x^2_i\right)+20+e$	$x_i\in[-32, 32]$
$F_8(\mathit{\boldsymbol{X}})=\frac{1}{4~000}\sum\limits^n_{i=1}x^2_i-\prod \limits^n_{i=1}\frac{\cos x_i}{\sqrt{i}}+1$	$x_i\in[-600, 600]$
$F_9(\mathit{\boldsymbol{X}})=\frac{{\rm{\pi }}}{n}\Bigg\{10\sin^2({\rm{\pi }} y_1)+\sum\limits^n_{i=1}(y_i-1)^2[1+10\sin^2({\rm{\pi }} y_{i+1})]+ (y_n-1)^2\Bigg\}+$
$\sum\limits^n_{i=1}u(x_i, 10, 100, 4), y_i=1+(x_i+1)/4$	$x_i\in[-50, 50]$
$F_{10}(\mathit{\boldsymbol{X}})=0.1\Bigg\{\sin^3(3{\rm{\pi }} x_1)+\sum\limits^{n-1}_{i=1}(x_i-1)^2[1+\sin^2(3{\rm{\pi }} x_{i+1})]+ (x_n-1)^2[1+\sin^2(2{\rm{\pi }} x_n)]\Bigg\}+$
$\sum\limits^n_{i=1}u(x_i, 5, 100, 4)$	$x_i\in[-50, 50]$
$F_{11}(\mathit{\boldsymbol{X}})=\sum\limits^n_{i=1}(x^2_i-10\cos(2{\rm{\pi }} x_i)+10)$	$x_i\in[-5.12, 5.12]$
$F_{12}(\mathit{\boldsymbol{X}})=418.982~9n-\sum\limits^n_{i=1}(x_i\sin\sqrt{\|x_i\|})$	$x_i\in[-500, 500]$

Function	MFO			IMFO
Function	Mean	SD	$p$ value	Mean	SD	$p$ value
$F_{1}$	$\textbf{1.44E+00}$	$\textbf{6.71E-01}$	$\textbf{N/A}$	9.46E+00	6.03E+00	4.08E-33
$F_{2}$	$\textbf{2.19E+06}$	$\textbf{1.01E+06}$	$\textbf{N/A}$	1.60E+07	1.22E+07	4.08E-33
$F_{3}$	$\textbf{1.56E+04}$	$\textbf{3.67E+03}$	$\textbf{N/A}$	3.85E+04	9.44E+03	6.88E-34
$F_{4}$	$\textbf{1.91E+01}$	$\textbf{3.43E+00}$	$\textbf{N/A}$	6.48E+01	1.70E+01	5.93E-34
$F_{5}$	$\textbf{3.41E+03}$	$\textbf{1.07E+03}$	$\textbf{N/A}$	1.10E+04	4.92E+03	6.11E-34
$F_{6}$	$\textbf{3.35E+03}$	$\textbf{9.79E+02}$	$\textbf{N/A}$	1.06E+04	4.32E+03	5.35E-32
$F_{7}$	$\textbf{1.15E+01}$	$\textbf{1.22E+00}$	$\textbf{N/A}$	1.85E+01	1.66E+00	1.29E-33
$F_{8}$	$\textbf{3.19E+01}$	$\textbf{8.61E+00}$	$\textbf{N/A}$	9.77E+01	3.37E+01	3.78E-34
$F_{9}$	$\textbf{7.71E+05}$	$\textbf{9.13E+05}$	$\textbf{N/A}$	1.88E+07	3.19E+07	9.00E-34
$F_{10}$	$\textbf{4.08E+06}$	$\textbf{3.52E+06}$	$\textbf{N/A}$	4.80E+07	3.83E+07	2.99E-32
$F_{11}$	$\textbf{1.27E+02}$	$\textbf{2.15E+01}$	$\textbf{N/A}$	2.23E+02	2.72E+01	7.99E-34
$F_{12}$	5.44E+03	5.68E+02	8.34E-08	$\textbf{4.95E+03}$	$\textbf{6.48E+02}$	$\textbf{N/A}$
$F_{13}$	$\textbf{7.43E+01}$	$\textbf{4.99E+01}$	$\textbf{N/A}$	9.77E+01	5.56E+01	2.38E-08
$F_{14}$	$\textbf{1.01E+02}$	$\textbf{5.04E+01}$	N/A	1.18E+02	3.89E+01	8.69E-08
$F_{15}$	$\textbf{9.04E+02}$	$\textbf{1.97E+02}$	$\textbf{N/A}$	1.33E+03	2.58E+02	4.62E-23
$F_{16}$	$\textbf{7.39E+02}$	$\textbf{5.51E+01}$	$\textbf{N/A}$	7.95E+02	7.32E+01	1.11E-09
$F_{17}$	$\textbf{1.00E+02}$	$\textbf{2.36E+01}$	$\textbf{N/A}$	1.29E+02	3.86E+01	9.53E-10
$F_{18}$	$\textbf{9.31E+02}$	$\textbf{9.37E+00}$	$\textbf{N/A}$	9.78E+02	2.78E+01	4.33E-33

Group	Number	MFO	CMFO1	CMFO2	CMFO3	CMFO4	CMFO5	CMFO6	CMFO7	CMFO8	CMFO9	CMFO10
Group1	$num_1$	2	0	0	0	0	1	0	0	0	2	1
Group1	$num_2$	4	5	5	0	4	4	6	3	2	4	4
Group2	$num_1$	1	2	0	0	0	1	0	0	1	0	1
Group2	$num_2$	4	3	5	0	4	2	3	5	2	3	3
Group3	$num_1$	1	0	0	0	0	1	0	1	0	2	1
Group3	$num_2$	4	6	5	0	5	5	2	3	4	3	3

Chaos-enhanced moth-flame optimization algorithm for global optimization

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

Share this article

Figures/Tables 22

References 48

Related Articles 7

Recommended Articles

Metrics

Comments

Design variable	GA [46]	Coello& Montes [47]	CPSO [44]	CDE [48]	MoCoDE [45]	MFO28
$d(x_1)$	0.051 480	0.051 989	0.051 728	0.051 609	0.051 718	0.051 705
$D(x_2)$	0.351 661	0.363 965	0.357 644	0.354 714	0.357 418	0.357 103
$N(x_3)$	11.632 201	10.890 522	11.244 543	11.410 831	11.248 015	11.266 387
$f(\mathit{\boldsymbol{x}})$	0.012 705	0.012 681	0.0126 747	0.0126 702	0.012 665	0.012 665

Design variable	GA [46]	Coello & Montes [47]	CPSO [44]	CDE [48]	MoCoDE [45]	MFO28
$h(x_1)$	0.208 800	0.205 986	0.202 369	0.203 137	0.205 730	0.205 730
$l(x_2)$	3.420 500	3.471 328	3.544 214	3.542 998	3.470 489	3.470 489
$t(x_3)$	8.997 500	9.020 224	9.048 210	9.033 498	9.036 624	0.205 730
$b(x_4)$	0.210 000	0.206 480	0.205 723	0.206 179	0.205 730	0.205 7302
$f(\mathit{\boldsymbol{x}})$	1.748 309	1.728 226	1.728 024	1.733 462	1.724 852	1.724 852

[1]	Baiquan LU, Chenlong NI, Zhongwei ZHENG, Tingzhang LIU. A global optimization algorithm based on multi-loop neural network control [J]. Journal of Systems Engineering and Electronics, 2019, 30(5): 1007-1024.
[2]	Zhuoran ZHANG, Changqiang HUANG, Hanqiao HUANG, Shangqin TANG, Kangsheng DONG. An optimization method: hummingbirds optimization algorithm [J]. Journal of Systems Engineering and Electronics, 2018, 29(2): 386-404.
[3]	Ye Tian and Zhimao Lu. S-box: six-dimensional compound hyperchaotic map and artificial bee colony algorithm [J]. Systems Engineering and Electronics, 2016, 27(1): 232-.
[4]	Aijun Zhu, Chuanpei Xu, Zhi Li, JunWu, and Zhenbing Liu. Hybridizing grey wolf optimization with differential evolution for global optimization and test scheduling for 3D stacked SoC [J]. Journal of Systems Engineering and Electronics, 2015, 26(2): 317-328.
[5]	Yu Xue,Yi Zhuang, Tianquan Ni, Siru Ni, and Xuezhi Wen. Self-adaptive learning based discrete differential evolution algorithm for solving CJWTA problem [J]. Journal of Systems Engineering and Electronics, 2014, 25(1): 59-68.
[6]	Yong Liu and Liang Ma. Improved gravitational search algorithm based on free search differential evolution [J]. Journal of Systems Engineering and Electronics, 2013, 24(4): 690-698.
[7]	Chaohua Dai1,Weirong Chen1,Yonghua Song2,and Yunfang Zhu3. Seeker optimization algorithm:a novel stochastic search algorithm for global numerical optimization [J]. Journal of Systems Engineering and Electronics, 2010, 21(2): 300-311.