An optimization method: hummingbirds optimization algorithm

doi:10.21629/JSEE.2018.02.19

Journal of Systems Engineering and Electronics ›› 2018, Vol. 29 ›› Issue (2): 386-404.doi: 10.21629/JSEE.2018.02.19

• Software Algorithm and Simulation • Previous Articles Next Articles

An optimization method: hummingbirds optimization algorithm

Zhuoran ZHANG¹(), Changqiang HUANG¹(), Hanqiao HUANG^2,*(), Shangqin TANG¹(), Kangsheng DONG¹()

¹ Aeronautics Engineering College, Air Force Engineering University, Xi'an 710038, China
² Unmanned System Research Institute, Northwestern Polytechnical University, Xi'an 710072, China

Received:2017-10-20 Online:2018-04-26 Published:2018-04-27
Contact: Hanqiao HUANG E-mail:zhuoran1009@163.com;hcqxian@163.com;cnxahhq@126.com;carnationtang2@163.com;kgddks@163.com
About author:ZHANG Zhuoran was born in 1990. He received his M.S. degree in unmanned aircraft combat system and technology from Air Force Engineering University in 2015. He is a Ph.D. candidate of Air Force Engineering University. His research interests include the optimization algorithm and its application in unmanned aerial vehicle combat system. E-mail: zhuoran1009@163.com|HUANG Changqiang was born in 1961. He received his Ph.D. degree in navigation, guidance and control from Northwestern Polytechnical University in 2006. He is a professor and doctoral tutor of Air Force Engineering University. He has worked in aerial weapon system and application engineering for more than 30 years. His current research interest is artificial intelligence including knowledge extraction and air combat simulation system. E-mail: hcqxian@163.com|HUANG Hanqiao was born in 1982. He received his Ph.D. degree in navigation, guidance and control from Northwestern Polytechnical University in 2010. He is now an associate professor of Northwestern Polytechnical University and doing postdoctoral research. His research interests include the optimization algorithm and its application in cooperative combat for unmanned combat aerial vehicles. E-mail: cnxahhq@126.com|TANG Shangqin was born in 1984. He received his Ph.D. degree in armament science and technology from Air Force Engineering University in 2012. He is now a lecturer of Air Force Engineering University. His research interest revolves autonomous air combat for unmanned combat aerial vehicles, situation assessment and maneuver decision. E-mail: carnationtang2@163.com|DONG Kangsheng was born in 1988. He received his M.S. degree in unmanned aircraft combat system and technology from Air Force Engineering University in 2013. He is now a Ph.D. candidate in the Aeronautics and Astronautics Engineering College of Air Force Engineering University. His research interests include optimization algorithms and autonomous air combat for unmanned combat aerial vehicles. E-mail: kgddks@163.com
Supported by:
the National Natural Science Foundation of China(61601505);This work was supported by the National Natural Science Foundation of China (61601505)

Abstract

Abstract:

This paper introduces an optimization algorithm, the hummingbirds optimization algorithm (HOA), which is inspired by the foraging process of hummingbirds. The proposed algorithm includes two phases: a self-searching phase and a guide-searching phase. With these two phases, the exploration and exploitation abilities of the algorithm can be balanced. Both the constrained and unconstrained benchmark functions are employed to test the performance of HOA. Ten classic benchmark functions are considered as unconstrained benchmark functions. Meanwhile, two engineering design optimization problems are employed as constrained benchmark functions. The results of these experiments demonstrate HOA is efficient and capable of global optimization.

Key words: population-based algorithm, global optimization, hummingbirds optimization algorithm (HOA), engineering design optimization

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.

Figures/Tables 22

Fig 1

Fig 2

Table 1

Table 2

Fig 3

Table 3

Fig 4

Table 4

Table 5

Statistical results of five unimodal functions for nine algorithms"

Function	Metric	CS	GSA	FA	GWO	MFO	CSA	SCA	SBO	HOA
${f_{01}}$	Best	3.1873E-03	2.2356E-18	3.2175E-04	3.3152E-87	1.0556E-35	4.6416E-04	3.4963E-44	7.2106E-04	0
	Worst	1.8546E-02	5.1571E-18	9.9751E-04	5.0603E-84	1.5940E-31	2.2371E-02	6.8963E-35	4.8623E-03	0
	Mean	8.8861E-03	3.8401E-18	5.5797E-04	3.3557E-85	1.1267E-32	5.1425E-03	3.9021E-36	1.8193E-03	0
	SD	3.6808E-03	7.6262E-19	1.7113E-04	9.5395E-85	2.9352E-32	5.5721E-03	1.3694E-35	9.5993E-04	0
	Rank	9	5	6	2	4	8	3	7	1
${f_{02}}$	Best	4.6496E-01	7.7224E-09	1.1523E+01	2.5968E-50	3.4803E-21	1.7892E-01	4.9622E-28	9.2507E-03	1.0777E-222
	Worst	3.0992E+00	1.3183E-08	1.2012E+02	7.2046E-49	4.2380E-19	1.6354E+00	3.6748E-23	1.8586E-02	5.7755E-197
	Mean	1.2720E+00	1.0131E-08	5.7526E+01	2.6832E-49	1.0203E-19	7.8482E-01	4.9188E-24	1.1883E-02	1.9367E-198
	SD	5.6264E-01	1.2349E-09	3.3051E+01	2.0623E-49	9.7286E-20	4.9658E-01	9.5048E-24	2.2940E-03	0
	Rank	8	5	9	2	4	7	3	6	1
${f_{03}}$	Best	2.2965E+02	5.9359E+01	2.8976E+03	2.1684E-31	4.8183E-15	3.1720E+00	5.6985E-25	2.7381E+01	0
	Worst	7.7555E+02	2.1078E+02	9.6717E+03	1.0079E-25	6.6667E+03	2.1382E+01	7.7601E-14	1.2536E+02	0
	Mean	4.8197E+02	1.0621E+02	5.5386E+03	6.9282E-27	2.2222E+02	8.8877E+00	2.5954E-15	5.7587E+01	0
	SD	1.3122E+02	3.7795E+01	1.7061E+03	2.0630E-26	1.2172E+03	5.0462E+00	1.4166E-14	2.1035E+01	0
	Rank	8	6	9	2	7	4	3	5	1
${f_{04}}$	Best	2.9232E+01	2.5830E+01	2.2550E+01	2.4873E+01	8.1081E-01	2.5915E+01	6.2365E+00	3.7705E+00	1.2702E-03
	Worst	7.0603E+01	9.3011E+01	2.8434E+04	2.7920E+01	3.0199E+03	1.2433E+02	7.3753E+00	3.6187E+02	1.0433E+00
	Mean	3.8122E+01	2.8263E+01	2.7620E+03	2.6117E+01	3.1340E+02	4.3232E+01	6.8795E+00	7.1427E+01	1.7314E-01
	SD	1.0049E+01	1.2230E+01	6.0837E+03	7.3272E-01	9.1759E+02	2.7732E+01	4.0504E-01	8.5066E+01	2.8961E-01
	Rank	5	4	9	3	8	6	2	7	1
${f_{05}}$	Best	1.8749E-02	2.5682E-03	5.7134E+01	7.0008E-05	3.5000E-04	5.0821E-03	2.3530E-05	1.4931E-02	1.7315E-05
	Worst	7.2482E-02	1.0297E-02	8.5993E+01	5.2054E-04	2.8998E-03	1.9171E-02	1.1278E-03	3.6658E-02	2.9103E-03
	Mean	3.5559E-02	5.3473E-03	8.5031E+01	2.2807E-04	1.5665E-03	1.0681E-02	2.9819E-04	2.6566E-02	7.1999E-04
	SD	1.1799E-02	1.8592E-03	5.2689E+00	1.1727E-04	6.9356E-04	3.7847E-03	2.8251E-04	5.1333E-03	6.5413E-04
	Rank	8	5	9	1	4	6	2	7	3
Average rank		7.6	5	8.4	2	5.4	6.2	2.6	6.4	1.4
Overall rank		8	4	9	2	5	6	3	7	1

Table 5

Fig 5

Table 6

Statistical results of five multimodal functions for nine algorithms"

Function	Algorithm	CS	GSA	FA	GWO	MFO	CSA	SCA	SBO	HOA
${f_{06}}$	Best	– 9.2290E+03	– 4.2079E+03	– 9.0932E+03	– 7.5366E+03	– 1.0904E+04	– 9.1925E+03	– 2.7944E+03	– 8.1080E+03	– 9.4367E+03
	Worst	– 8.2272E+03	– 2.4576E+03	– 6.4878E+03	– 3.9478E+03	– 7.5818E+03	– 5.3633E+03	– 2.0971E+03	– 4.6926E+03	– 7.0501E+03
	Mean	– 8.6647E+03	– 3.1353E+03	– 7.7504E+03	– 6.2036E+03	– 9.1032E+03	– 7.2524E+03	– 2.4232E+03	– 6.0269E+03	– 8.1011E+03
	SD	2.3800E+02	3.9793E+02	5.2527E+02	8.2229E+02	7.6905E+02	8.7286E+02	1.6963E+02	8.9981E+02	5.7228E+02
	Rank	2	8	4	6	1	5	9	7	3
${f_{07}}$	Best	6.0216E+01	2.9849E+00	2.8854E+01	0	1.9899E+00	2.9980E+00	0	2.6865E+01	0
	Worst	1.1102E+02	1.1940E+01	1.0447E+02	0	3.6884E+01	3.4827E+01	2.3922E+01	5.5719E+01	0
	Mean	8.3875E+01	7.5285E+00	6.5800E+01	0	1.4472E+01	1.7269E+01	7.9740E-01	4.3258E+01	0
	SD	1.1548E+01	2.1496E+00	1.7857E+01	0	1.0085E+01	7.5528E+00	4.3676E+00	7.5163E+00	0
	Rank	9	4	8	1	5	6	3	7	1
${f_{08}}$	Best	1.8952E+00	1.2627E-09	2.1203E+00	7.9936E-15	4.4409E-15	1.5019E+00	8.8818E-16	7.1657E-03	8.8818E-16
	Worst	9.3048E+00	1.8461E-09	1.1949E+01	1.5099E-14	4.4409E-15	3.8363E+00	4.4409E-15	1.6477E-02	8.8818E-16
	Mean	4.5589E+00	1.5474E-09	4.0139E+00	1.0007E-14	4.4409E-15	2.6001E+00	4.0856E-15	1.0522E-02	8.8818E-16
	SD	2.0045E+00	1.4669E-10	1.9598E+00	2.5861E-15	0	6.0866E-01	1.0840E-15	2.2961E-03	0
	Rank	9	5	8	4	3	7	2	6	1
${f_{09}}$	Best	4.0521E-02	1.1490E+00	2.9338E-03	0	3.4448E-02	9.2387E-03	0	1.9840E-03	0
	Worst	2.4749E-01	2.5729E+00	6.8436E-02	1.7903E-02	2.6566E-01	9.8607E-02	2.7968E-01	7.8375E-01	0
	Mean	9.7669E-02	1.6150E+00	2.2698E-02	5.9675E-04	1.4386E-01	5.0072E-02	1.1920E-02	1.3485E-01	0
	SD	4.8698E-02	3.6852E-01	1.3938E-02	3.2686E-03	7.4666E-02	2.5063E-02	5.2127E-02	1.8202E-01	0
	Rank	6	9	4	2	8	5	3	7	1
${f_{10}}$	Best	4.5344E-01	1.7571E-20	2.6879E+00	3.0032E-07	4.7116E-32	5.7406E-03	1.3029E-02	3.1810E-06	6.3640E-05
	Worst	1.5468E+00	1.0367E-01	2.6805E+01	3.2467E-02	1.6909E-31	2.8469E+00	9.7563E-02	4.2273E-03	1.4864E-02
	Mean	9.6076E-01	1.7104E-02	1.0422E+01	1.2261E-02	5.5095E-32	6.7057E-01	4.7943E-02	1.5976E-04	2.7611E-03
	SD	2.5509E-01	3.8910E-02	5.1881E+00	8.5137E-03	2.2900E-32	6.3986E-01	1.9819E-02	7.6901E-04	3.5082E-03
	Rank	8	5	9	4	1	7	6	2	3
Average rank		6.8	6.2	6.6	3.4	3.6	6	4.6	5.8	1.8
Overall rank		9	7	8	2	3	6	4	5	1

Table 6

Fig 6

Table 7

Results of a Wilcoxon signed ranks test based on the best solution for each function with 30 independent runs $(a = 0.05)$"

Function	CS vs HOA				GSA vs HOA				FA vs HOA				GWO vs HOA				MFO vs HOA				CSA vs HOA				SCA vs HOA				SBO vs HOA
Function	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win	$p - {\rm{Value}}$	R+	R-	Win
${f_{01}}$	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+
${f_{02}}$	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+
${f_{03}}$	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+
${f_{04}}$	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+
${f_{05}}$	1.9209E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	2.2248E-04	412	53	+	1.7344E-06	465	0	+	6.8359E-03	101	364	-	1.7344E-06	465	0	+
${f_{06}}$	1.8910E-04	51	414	-	1.7344E-06	465	0	+	3.3269E-02	336	129	+	1.2506E-04	46	419	-	5.7924E-05	37	428	-	2.6134E-06	410	55	+	1.7344E-06	465	0	+	3.1817E-06	459	6	+
${f_{07}}$	1.7344E-06	465	0	+	1.6416E-06	465	0	+	1.7344E-06	465	0	+	1.0000E+00	0	0	≈	1.7224E-06	465	0	+	1.7344E-06	465	0	+	1.7971E-01	3	0	≈	1.7344E-06	465	0	+
${f_{08}}$	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	9.2745E-07	465	0	+	4.3205E-08	465	0	+	1.7344E-06	465	0	+	2.0346E-07	378	0	+	1.7344E-06	465	0	+
${f_{09}}$	1.7344E-06	465	0	+	1.7344E-06	465	0	+	1.7344E-06	465	0	+	3.1731E-01	1	0	≈	1.7344E-06	465	0	+	1.7344E-06	465	0	+	2.7708E-02	21	0	+	3.1731E-01	1	0	≈
${f_{10}}$	1.7344E-06	465	0	+	5.7096E-02	140	325	-	1.7344E-06	465	0	+	2.3704E-05	438	27	+	1.7344E-06	0	465	-	2.3534E-06	462	3	+	1.7344E-06	465	0	+	1.7344E-06	0	465	-
+/≈/–	9/0/1				9/0/1				10/0/0				7/2/1				8/0/2				10/0/0				7/2/1				9/0/1

Table 7

Table 8

Fig 7

Table 9

Fig 8

Table 10

Table 11

Fig 9

Table 12

Table 13

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Test function	Name	Type	Dim	Range	Optimum
${f_{01}}(x) = \sum\limits_{i = 1}^D {x_i^2} $	Sphere	U	30	[–100, 100]	0
${f_{02}}(x) = \sum\limits_{i = 1}^D \| {x_i}\| + \prod\limits_{i - 1}^D \| {x_i}\|$	Schwefel 2.22	U	30	[–10, 10]	0
${f_{03}}(x) = \sum\limits_{i = 1}^D {{{(\sum\limits_{i = 1}^D {{x_i}} )}^2}} $	Schwefel 1.2	U	30	[–100, 100]	0
${f_{04}}(x) = \sum\limits_{i = 1}^D 1 00{(x_{i + 1}^2 - x_i^2)^2} + {({x_i} - 1)^2}$	Rosenbrock	U	30	[–30, 30]	0
${f_{05}}(x) = \sum\limits_{i = 1}^D i x_i^4 + {\rm{random}}[0, 1)$	Quartic	U	30	[–1.28, 1.28]	0
${f_{06}}(x) = \sum\limits_{i = 1}^D - {x_i}\sin (\sqrt {\|{x_i}\|} )$	Schwefel 2.26	M	30	[–500, 500]	– 418.982 9^*Dim
${f_{07}}(x) = \sum\limits_{i = 1}^D {(x_i^2 - 10\cos (} z\pi {x_i}) + 10)$	Rastrigin	M	30	[–5.12, 5.12]	0
${f_{08}}(x) = 20 + {\rm{e}} - 20\exp ( - 0.2\sqrt {\frac{1}{D}\sum\limits_{i = 1}^D {x_i^2} } ) - \exp (\frac{1}{D}\sum\limits_{i = 1}^D {\cos } (2\pi {x_i}))$	Ackley	M	30	[–32, 32]	8.8818E-16
${f_{09}}(x) = \frac{1}{{4\;000}}\sum\limits_{i = 1}^D {(x_i^2)} \left( {\prod\limits_{i = 1}^D {\cos } (\frac{{{x_i}}}{{\sqrt i }})} \right) + 1$	Griewank	M	30	[–600, 600]	0
${f_{10}}(x) = \frac{\pi }{D}\{ 10{\sin ^2}(\pi {y_i}) + \sum\limits_{i = 1}^{D - 1} {{{({y_i} - 1)}^2}} [1 + 10{\sin ^2}(\pi {y_{i + 1}})] + $ $({y_D} - 1)\} + \sum\limits_{i = 1}^D u ({x_i}, 10, 100, 4), {y_i} = 1 + \frac{{{x_i} + 1}}{4}$	Penalized	M	30	[– 50, 50]	0
$u({x_i}, a, k, m) = \left\{ \begin{array}{l}k{({x_i} - a)^m}, \;\;\;{x_i} > a\\0, \;\;\; - a < {x_i} < a\\k{( - {x_i} - a)^m}, \;\;\;{x_i} < a\end{array} \right\}$

Function	Metric	PSO	AFSA	ABC	HOA
${f_{01}}$	Best	4.4661E-09	5.6944E+00	6.0932E-11	0
	Worst	5.3809E-07	8.0334E+00	1.5289E-09	0
	Mean	1.0592E-07	6.9494E+00	4.3182E-10	0
	SD	1.3464E-07	7.0391E-01	4.0839E-10	0
	Rank	3	4	2	1
${f_{02}}$	Best	5.1065E-06	1.0288E+01	6.2532E-07	1.3252E-229
	Worst	7.4840E-05	1.6562E+01	3.1236E-06	2.4570E-199
	Mean	1.9743E-05	1.4761E+01	2.0404E-06	8.3443E-201
	SD	1.4384E-05	1.4411E+00	7.0388E-07	0
	Rank	3	4	2	1
${f_{03}}$	Best	3.6639E+01	2.6562E+01	5.4211E+03	0
	Worst	1.3955E+02	5.0529E+01	1.4716E+04	0
	Mean	7.9331E+01	3.8830E+01	1.0391E+04	0
	SD	2.6585E+01	5.9073E+00	2.0738E+03	0
	Rank	3	2	4	1
${f_{05}}$	Best	1.5490E+01	6.8621E+02	7.8929E-03	1.2122E-03
	Worst	2.4689E+02	1.7763E+03	7.0644E+00	1.7633E+00
	Mean	6.1495E+01	1.1777E+03	2.0521E+00	2.1932E-01
	SD	6.3004E+01	2.5344E+02	2.1603E+00	4.0761E-01
	Rank	3	4	2	1
${f_{05}}$	Best	2.0420E-02	1.2235E+01	8.0045E-02	1.0886E-05
	Worst	1.3401E-01	2.7191E+01	2.3204E-01	2.5065E-03
	Mean	5.5557E-02	2.0114E+01	1.5399E-01	7.0678E-04
	SD	2.3476E-02	3.4812E+00	3.2530E-02	6.2312E-04
	Rank	2	4	3	1
Average rank		2.8	3.6	2.6	1
Overall rank		3	4	2	1

Function	Metric	PSO	AFSA	ABC	HOA
${f_{06}}$	Best	– 8.1474E+03	– 9.0509E+03	– 1.2451E+04	– 9.1108E+03
	Worst	– 5.1071E+03	– 7.1022E+03	– 1.2080E+04	– 6.3858E+03
	Mean	– 6.6989E+03	– 7.8385E+03	– 1.2258E+04	– 8.2889E+03
	SD	7.4309E+02	4.9643E+02	1.0206E+02	6.4126E+02
	Rank	4	3	1	2
${f_{07}}$	Best	2.3879E+01	2.0079E+02	2.5841E-10	0
	Worst	6.9647E+01	2.4204E+02	9.9497E-01	0
	Mean	4.0728E+01	2.2493E+02	6.6331E-02	0
	SD	1.1002E+01	1.0023E+01	2.5243E-01	0
	Rank	3	4	2	1
${f_{08}}$	Best	2.1826E-06	3.3679E+00	2.8043E-06	0
	Worst	2.5074E-05	3.9042E+00	1.8588E-05	0
	Mean	1.0056E-05	3.6668E+00	1.1395E-05	0
	SD	5.5968E-06	1.4558E-01	4.4298E-06	0
	Rank	2	4	3	1
${f_{09}}$	Best	2.9294E-07	1.3534E+01	1.7291E-10	0
	Worst	5.4083E-02	5.2048E+01	2.8638E-06	0
	Mean	1.4776E-02	3.5190E+01	2.5593E-07	0
	SD	1.4790E-02	1.1422E+01	6.9536E-07	0
	Rank	3	4	2	1
${f_{10}}$	Best	5.6210E-01	7.5819E-01	1.1747E-12	7.2501E-05
	Worst	4.5167E+00	1.2770E+00	8.2437E-11	1.2187E-02
	Mean	2.3753E+00	1.0301E+00	1.5396E-11	2.7549E-03
	SD	1.0560E+00	1.5043E-01	1.7705E-11	2.9718E-03
	Rank	4	3	1	2
Average rank		3.2	3.6	1.8	1.4
Overall rank		3	4	2	1

Function	HOA-S		HOA-G		HOA
Function	Mean	SD	Mean	SD	Mean	SD
${f_{01}}$	5.3855E-05	1.4262E-04	8.2585E+02	8.1274E+01	0	0
${f_{02}}$	1.4329E-03	1.6914E-03	6.3969E+01	3.1304E+01	2.9230E-194	0
${f_{05}}$	2.0396E-03	1.4584E-03	1.0719E+00	1.3378E-01	7.1633E-04	0
${f_{07}}$	1.3832E-13	7.4694E-13	1.3077E+02	6.9087E+00	0	0
${f_{08}}$	5.6283E-04	8.1531E-04	1.9364E+01	1.4999E-01	8.8818E-16	0
${f_{09}}$	1.4114E-05	3.6145E-05	8.3534E+00	7.3072E-01	0	0

Algorithm	PSO	ABC	CS	GSA	MFO	CSA	HOA
$O(\cdot)$	$O(2TND)$	$O(TN^2+5TND)$	$O(2TND)$	$O(TN^2+6TND)$	$O(TN^2+TND)$	$O(TND)$	$O(TN^2+4TND)$
Rank	2	6	2	7	4	1	5

An optimization method: hummingbirds optimization algorithm

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Algorithm	Specification
CS	$\beta = 1.5$ and $p_a = 0.25$ as in [7]
GSA	$G_O = 100$ and $\alpha = 20$ as in [18]
FA	$\alpha = 0.2, \beta _0 = 1, \gamma = 1$ as in [8]
GWO	$a = 2-1\times (2\rm/MaxCycle)$ as in [10]
MFO	$b{\rm{ }} = {\rm{ }}1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{and}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{flame}}\;{\rm{no }} = $${\kern 1pt} {\rm{round}}[N - l \cdot (N - 1)/T]$ as in [11]
CSA	$AP = 0.1$ and $fl = 2$ as in [12]
SCA	$\alpha = 2, r_3 = 2\cdot {{rand}}$ as in [39]
SBO	$\alpha = 0.94, Z = 0.02$ and mutationprobability is 0.05 as in [16]

Variable	HEA-ACT [42]	DEDS [43]	PSO-DE [44]	CS [45]	MBA [46]	CSA [12]	HOA
${x_1}$	0.788 68	0.788 675	0.788 675 1	0.788 67	0.788 565 0	0.788 675 128 4	0.788 675 142 372 453
${x_2}$	0.408 23	0.408 248	0.408 248 2	0.409 02	0.408 559 7	0.408 248 308 0	0.408 248 268 465 375
${g_1}$	– 0.000 000	1.77E-08	– 5.29E-11	– 0.000 29	– 5.29E-11	– 1.687539e-14	0
${g_2}$	– 1.464 118	– 1.464 101	– 1.463 747 5	– 0.268 53	– 1.463 747 5	– 1.464 101 595 2	– 1.464 101 640 145 903
${g_3}$	– 0.535 898	– 0.535 898 36	– 0.536 252 4	– 0.731 76	– 0.536 252 4	– 0.535 898 404 8	– 0.535 898 359 854 097
$f(x)$	263.895 843	263.895 843 4	263.895 843	263.971 6	263.895 852 2	263.895 843 376 5	2.638958433764684E+02

Method	Worst	Mean	Best	SD	FEs
SC [41]	263.969 756	263.903 356	263.895 846	1.3E-02	17 610
HEA-ACT [42]	263.896 099	263.895 865	263.895 843	4.9E-05	15 000
DEDS [43]	263.895 849	263.895 843	263.895 843	9.7E-07	15 000
PSO-DE [44]	263.895 843	263.895 843	263.895 843	4.5E-10	17 600
CS [45]	NA	264.066 9	263.971 56	9.00E-05	15 000
MBA [46]	263.915 983	263.897 996	263.895 852	3.93E-03	13 280
CSA [12]	263.895 843 377 0	263.895 843 376 5	263.895 843 376 5	1.0122543402E-10	25 000
HOA	2.638958445046976E+02	2.638958434379900E+02	2.638958433764684E+02	2.191366322234361E-07	13 000
HOA	2.638958433767606E+02	2.638958433764794E+02	2.638958433764684E+02	5.319778017413415E-11	20 000

Variable	CPSO [47]	HPSO [48]	CDE [49]	MBA [46]	GA3 [51]	CAEP[52]	WCA [53]	HOA
${x_1}$	0.202 369	0.205 73	0.203 137	0.205 729	0.205 986	0.205 700	0.205 728	0.205 729 639 786 079
${x_2}$	3.544 214	3.470 489	3.542 998	3.470 493	3.471 328	3.470 500	3.470 522	3.470 488 665 628 005
${x_3}$	9.048 210	9.036 624	9.033 498	9.036 626	9.020 224	9.036 600	9.036 620	9.036 623 910 357 633
${x_4}$	0.205 723	0.205 73	0.206 179	0.205 729	0.206 480	0.205 700	0.205 729	0.205 729 639 786 080
${g_1}$	– 13.655 547	– 0.025 399	– 44.578 568	– 0.001 614	– 0.103 049	1.988 676	– 0.034128	– 3.637978807091713E-12
${g_2}$	– 78.814 077	– 0.053 122	– 44.663 534	– 0.016 911	– 0.231 747	4.481 548	– 3.49E-05	0
${g_3}$	– 3.35E-03	0	– 0.003 042	– 2.40E-07	– 5E-04	0.000 000	– 1.19E-06	– 1.387778780781446E-16
${g_4}$	– 3.424 572	– 3.432 981	– 3.423 726	– 3.432 982	– 3.430 044	– 3.433 213	– 3.432 980	– 3.432 983 785 362 248
${g_5}$	– 0.077 369	– 0.080 73	– 0.078 137	– 0.080 729	– 0.080 986	– 0.080 700	– 0.080 728	– 0.080 729 639 786 079
${g_6}$	– 0.235 595	– 0.235 540	– 0.235 557	– 0.235 540	– 0.235 514	– 0.235 538	– 0.235 540	– 0.235 540 322 584 754
${g_7}$	– 4.472 858	– 0.031 555	– 38.028 268	– 0.001 464	– 58.646 888	– 2.603 347	– 0.013 503	– 2.728484105318785E-12
$f(x)$	1.728 024	1.724 852	1.733 462	1.724 853	1.728 226	1.724 852	1.724 856	1.724 852 308 597 365

[1]	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.
[2]	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.
[3]	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.
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Method	Worst	Mean	Best	SD	FEs
CPSO [47]	1.782 143	1.748 831	1.728 024	1.29E-02	240 000
HPSO [48]	1.814 295	1.749 040	1.724 852	4.01E-02	81 000
CDE [49]	1.824 105	1.768 158	1.733 461	0.022 194	204 800
FA [50]	2.345 579 3	1.878 656 0	1.731 206 5	0.267 798 9	50 000
GA3 [51]	1.993 408	1.792 654	1.728 226	7.47E-02	80 000
CAEP [52]	3.179 709	1.971 809	1.724 852	4.43E-01	50 020
WCA [53]	1.744 697	1.726 427	1.724 856	4.29E-03	46 450
ABC [54]	NA	1.741 913	1.724 852	0.03100	30 000
MBA [46]	1.724 853	1.724 853	1.724 853	6.94E-19	47 340
ISA [24]	2.670 0	2.497 3	2.381 2	1.02E-1	30 000
IGPSO [55]	1.724 852	1.724 852	1.724 852	4.76378E-09	120 000
HOA	1.785 904 800 543 632	1.727 531 203 494 599	1.724 852 308 597 814	0.011 185 363 062 822	29 900
HOA	1.724 852 309 234 164	1.724 852 308 619 237	1.724 852 308 597 365	1.161694738829006E-10	100 000