使用高斯分布估计策略的改进樽海鞘群算法

doi:10.12305/j.issn.1001-506X.2022.07.19

Abstract

Abstract:

In order to address the shortcomings of the salp swarm algorithm (SSA) in solving complex optimization problems, such as reduced population diversity and easy to fall into local optimum, an improved SSA using elite pool strategy and Gaussian distribution estimation strategy (GDESSA) is proposed. Firstly, an elite pool strategy is proposed. When the leader position is updated at each time, an individual from the elite pool is randomly selected as a food source, which enhances the exploration ability of the leader and enriches the population diversity. Secondly, the follower formula is improved using a Gaussian distribution estimation strategy. By fitting the dominant population information, the evolutionary direction of the population is modified, and the algorithm's optimization ability is enhanced. The proposed algorithm is tested using the CEC2017 test suite and the performance of the improved algorithm is evaluated by statistical analysis, convergence analysis, stability analysis, Wilcoxon test, Friedman test, and Iman-Davenport test. The simulation results show that the improved strategy proposed can effectively improve the performance of the algorithm, and the proposed algorithm has faster convergence speed and better convergence accuracy compared with other algorithms.

Key words: salp swarm algorithm (SSA), Gaussian distribution estimation, elite pool, function optimization

CLC Number:

TP301.6

Andi TANG, Tong HAN, Dengwu XU, Huan ZHOU, lei XIE. An improved salp swarm algorithm using Gaussian distribution estimation strategy[J]. Systems Engineering and Electronics, 2022, 44(7): 2229-2240.

Figures/Tables 11

Fig.1

Fig.2

Table 1

Comparison results of different improvement strategies"

函数	SSA		SSA-1		SSA-2		GDESSA
函数	均值(排名)	耗时(排名)	均值(排名)	耗时(排名)	均值(排名)	耗时(排名)	均值(排名)	耗时(排名)
F1	1.85E+00(3)	6.49E+00(3)	1.12E+04(4)	6.69E+00(4)	6.84E-08(2)	2.64E+00(1)	4.32E-08(1)	2.77E+00(2)
F2	9.49E+01(3)	6.47E+00(3)	1.00E+02(4)	6.69E+00(4)	5.70E+01(1)	2.66E+00(1)	6.80E+01(2)	2.72E+00(2)
F3	1.18E+02(4)	6.65E+00(3)	3.27E+01(2)	6.91E+00(4)	3.95E+01(3)	2.90E+00(1)	2.23E+01(1)	3.01E+00(2)
F4	2.65E+01(4)	7.67E+00(3)	8.17E-01(3)	7.92E+00(4)	2.75E-04(2)	3.98E+00(1)	2.72E-05(1)	4.14E+00(2)
F5	1.50E+02(4)	6.73E+00(3)	5.56E+01(2)	6.99E+00(4)	6.73E+01(3)	2.96E+00(1)	4.32E+01(1)	3.07E+00(2)
F6	1.16E+02(4)	6.70E+00(3)	2.67E+01(2)	6.96E+00(4)	4.03E+01(3)	2.89E+00(1)	1.98E+01(1)	3.08E+00(2)
F7	1.93E+03(4)	6.77E+00(3)	6.34E+00(3)	7.00E+00(4)	5.52E-09(2)	2.93E+00(1)	2.51E-09(1)	3.04E+00(2)
F8	3.85E+03(4)	6.94E+00(3)	2.42E+03(1)	7.20E+00(4)	3.50E+03(3)	3.19E+00(1)	2.56E+03(2)	3.34E+00(2)
F9	2.11E+02(4)	6.54E+00(3)	1.31E+02(3)	6.81E+00(4)	2.24E+01(2)	2.73E+00(1)	1.81E+01(1)	2.89E+00(2)
F10	9.30E+06(4)	6.73E+00(3)	5.61E+06(3)	6.98E+00(4)	3.80E+02(2)	2.91E+00(1)	3.72E+02(1)	3.08E+00(2)
F11	1.46E+05(4)	6.60E+00(3)	6.09E+04(3)	6.85E+00(4)	5.06E+01(1)	2.78E+00(1)	5.76E+01(2)	2.93E+00(2)
F12	1.18E+04(4)	6.96E+00(3)	8.87E+03(3)	7.22E+00(4)	4.12E+01(2)	3.15E+00(1)	4.08E+01(1)	3.32E+00(2)
F13	5.78E+04(4)	6.55E+00(3)	2.86E+04(3)	6.83E+00(4)	2.18E+01(1)	2.72E+00(1)	2.34E+01(2)	2.86E+00(2)
F14	8.67E+02(4)	6.72E+00(3)	5.10E+02(3)	6.97E+00(4)	4.05E+02(2)	2.91E+00(1)	2.82E+02(1)	3.02E+00(2)
F15	3.82E+02(4)	7.52E+00(3)	2.30E+02(3)	7.73E+00(4)	1.05E+02(2)	3.70E+00(1)	7.76E+01(1)	3.87E+00(2)
F16	2.15E+05(4)	6.80E+00(3)	1.38E+05(3)	6.98E+00(4)	2.71E+01(1)	2.90E+00(1)	2.72E+01(2)	3.05E+00(2)
F17	7.59E+05(4)	1.17E+01(3)	5.21E+05(3)	1.19E+01(4)	2.06E+01(2)	7.91E+00(1)	2.03E+01(1)	8.35E+00(2)
F18	4.29E+02(4)	7.72E+00(3)	2.52E+02(3)	7.91E+00(4)	1.81E+02(2)	3.86E+00(1)	1.01E+02(1)	4.09E+00(2)
F19	2.98E+02(4)	8.19E+00(3)	2.34E+02(2)	8.38E+00(4)	2.42E+02(3)	4.37E+00(1)	2.20E+02(1)	4.61E+00(2)
F20	4.21E+02(2)	8.54E+00(3)	1.00E+02(1)	8.72E+00(4)	9.67E+02(4)	4.70E+00(1)	7.63E+02(3)	4.97E+00(2)
F21	4.48E+02(4)	9.00E+00(3)	3.79E+02(2)	9.19E+00(4)	3.82E+02(3)	5.18E+00(1)	3.60E+02(1)	5.50E+00(2)
F22	5.12E+02(4)	9.29E+00(3)	4.45E+02(2)	9.46E+00(4)	4.54E+02(3)	5.50E+00(1)	4.38E+02(1)	5.79E+00(2)
F23	4.02E+02(3)	8.94E+00(3)	4.09E+02(4)	9.15E+00(4)	3.87E+02(2)	5.11E+00(1)	3.87E+02(1)	5.40E+00(2)
F24	1.53E+03(4)	9.80E+00(3)	1.28E+03(3)	9.96E+00(4)	1.25E+03(2)	5.98E+00(1)	1.04E+03(1)	6.32E+00(2)
F25	5.24E+02(4)	1.04E+01(3)	5.10E+02(3)	1.06E+01(4)	4.99E+02(2)	6.64E+00(1)	4.95E+02(1)	7.06E+00(2)
F26	4.36E+02(4)	9.76E+00(3)	4.33E+02(3)	9.98E+00(4)	3.73E+02(2)	6.00E+00(1)	3.68E+02(1)	6.33E+00(2)
F27	9.01E+02(4)	8.85E+00(3)	7.05E+02(3)	9.05E+00(4)	5.69E+02(2)	5.07E+00(1)	5.26E+02(1)	5.34E+00(2)
F28	2.85E+06(4)	1.30E+01(3)	2.68E+06(3)	1.33E+01(4)	2.16E+03(2)	9.27E+00(1)	2.14E+03(1)	9.79E+00(2)
平均排名	3.82	3.00	2.75	4.00	2.18	1.00	1.25	2.00

Table 1

Fig.3

Table 2

Table 3

Results statistics of seven algorithms in the CEC2017 30D test"

函数	比较项	HFPSO	GEDGWO	VCS	MPA	SMA	HHO	GDESSA
F1	均值	2.21E+02	4.79E-07	7.32E+03	2.73E-01	4.94E-01	1.68E+03	4.32E-08
	标准差	1.35E+02	1.83E-07	5.58E+03	3.30E-01	3.18E-01	7.95E+02	2.36E-08
	排名	5	2	7	3	4	6	1
F2	均值	4.22E+01	4.25E+01	7.39E+01	8.68E+01	8.99E+01	1.23E+02	6.80E+01
	标准差	2.87E+01	2.64E+01	3.48E+01	6.25E+00	5.12E+00	3.33E+01	3.36E+01
	排名	1	2	4	5	6	7	3
F3	均值	7.42E+01	4.53E+01	1.09E+02	6.89E+01	8.17E+01	2.05E+02	2.23E+01
	标准差	1.82E+01	1.24E+01	2.56E+01	1.86E+01	1.99E+01	3.62E+01	1.01E+01
	排名	4	2	6	3	5	7	1
F4	均值	9.20E-01	7.22E-03	8.72E-01	1.92E+00	7.35E-01	5.62E+01	2.72E-05
	标准差	1.29E+00	1.47E-02	8.88E-01	1.10E+00	3.21E-01	5.92E+00	3.06E-06
	排名	5	2	4	6	3	7	1
F5	均值	8.74E+01	7.18E+01	1.79E+02	1.00E+02	1.18E+02	4.98E+02	4.32E+01
	标准差	2.05E+01	1.30E+01	1.20E+02	1.68E+01	2.40E+01	6.57E+01	7.56E+00
	排名	3	2	6	4	5	7	1
F6	均值	7.06E+01	4.50E+01	9.19E+01	7.05E+01	9.39E+01	1.40E+02	1.98E+01
	标准差	2.31E+01	1.17E+01	2.25E+01	1.44E+01	2.03E+01	2.13E+01	9.77E+00
	排名	4	2	5	3	6	7	1
F7	均值	8.99E+00	1.76E-03	5.22E+02	8.87E+01	9.95E+02	4.69E+03	2.51E-09
	标准差	1.11E+01	1.25E-02	1.44E+03	4.60E+01	1.22E+03	8.28E+02	5.04E-10
	排名	3	2	5	4	6	7	1
F8	均值	2.87E+03	3.01E+03	4.00E+03	2.50E+03	3.04E+03	4.35E+03	2.56E+03
	标准差	6.39E+02	4.30E+02	1.03E+03	4.10E+02	5.06E+02	7.25E+02	7.00E+02
	排名	3	4	6	1	5	7	2
F9	均值	9.43E+01	1.56E+01	1.25E+02	4.59E+01	1.16E+02	1.61E+02	1.81E+01
	标准差	3.12E+01	1.54E+01	2.42E+01	2.30E+01	4.33E+01	4.86E+01	1.60E+01
	排名	4	1	6	3	5	7	2
F10	均值	1.04E+05	7.75E+02	3.51E+06	9.82E+04	1.31E+06	7.61E+06	3.72E+02
	标准差	8.70E+04	5.82E+02	1.89E+06	7.77E+04	1.09E+06	4.21E+06	2.53E+02
	排名	4	2	6	3	5	7	1
F11	均值	1.83E+04	5.80E+01	9.00E+04	1.39E+03	2.71E+04	1.51E+05	5.76E+01
	标准差	2.99E+04	1.90E+01	4.08E+04	3.80E+02	2.64E+04	9.05E+04	2.82E+01
	排名	4	2	6	3	5	7	1
F12	均值	1.05E+04	3.66E+01	1.49E+04	4.57E+01	4.71E+04	3.82E+04	4.08E+01
	标准差	1.06E+04	1.32E+01	1.80E+04	1.08E+01	2.84E+04	4.25E+04	7.05E+00
	排名	4	1	5	3	7	6	2
F13	均值	9.39E+03	2.49E+01	1.59E+04	9.26E+01	1.99E+04	6.86E+04	2.34E+01
	标准差	1.12E+04	1.56E+01	1.82E+04	2.60E+01	1.57E+04	4.86E+04	6.31E+00
	排名	4	2	5	3	6	7	1
F14	均值	7.09E+02	5.90E+02	8.28E+02	3.40E+02	8.18E+02	1.55E+03	2.82E+02
	标准差	2.30E+02	2.89E+02	2.45E+02	1.84E+02	2.83E+02	3.56E+02	2.08E+02
	排名	4	3	6	2	5	7	1
F15	均值	2.11E+02	9.90E+01	2.56E+02	7.97E+01	4.34E+02	7.48E+02	7.76E+01
	标准差	8.26E+01	7.71E+01	1.22E+02	3.97E+01	1.64E+02	2.19E+02	3.09E+01
	排名	4	3	5	2	6	7	1
F16	均值	1.67E+05	3.31E+01	2.30E+05	1.11E+02	3.75E+05	6.90E+05	2.72E+01
	标准差	1.45E+05	7.76E+00	1.74E+05	2.67E+01	3.55E+05	8.77E+05	2.36E+00
	排名	4	2	5	3	6	7	1
F17	均值	6.60E+03	2.73E+01	1.49E+04	4.38E+01	3.00E+04	1.46E+05	2.03E+01
	标准差	8.00E+03	8.80E+00	1.87E+04	8.44E+00	2.11E+04	1.42E+05	2.80E+00
	排名	4	2	5	3	6	7	1
F18	均值	2.70E+02	1.60E+02	2.22E+02	1.11E+02	3.59E+02	6.71E+02	1.01E+02
	标准差	1.03E+02	6.87E+01	6.41E+01	5.76E+01	1.59E+02	2.01E+02	6.61E+01
	排名	5	3	4	2	6	7	1
F19	均值	2.69E+02	2.41E+02	2.78E+02	1.40E+02	2.93E+02	4.06E+02	2.20E+02
	标准差	1.94E+01	1.40E+01	2.98E+01	7.03E+01	2.17E+01	3.51E+01	8.73E+00
	排名	4	3	5	1	6	7	2
F20	均值	9.77E+02	1.00E+02	1.07E+02	1.04E+02	2.90E+03	2.39E+03	7.63E+02
	标准差	1.53E+03	6.36E-06	6.10E+00	4.14E+00	1.36E+03	2.37E+03	1.17E+03
	排名	5	1	3	2	7	6	4
F21	均值	4.52E+02	3.93E+02	4.62E+02	3.82E+02	4.35E+02	7.05E+02	3.60E+02
	标准差	2.90E+01	1.61E+01	2.35E+01	2.39E+01	1.95E+01	7.35E+01	1.32E+01
	排名	5	3	6	2	4	7	1
F22	均值	5.34E+02	4.58E+02	5.32E+02	4.83E+02	5.30E+02	8.26E+02	4.38E+02
	标准差	3.47E+01	1.28E+01	3.20E+01	1.79E+01	2.95E+01	7.42E+01	1.07E+01
	排名	6	2	5	3	4	7	1
F23	均值	3.79E+02	3.87E+02	3.78E+02	3.87E+02	3.88E+02	4.11E+02	3.87E+02
	标准差	4.73E+00	1.49E-02	3.35E+00	1.71E+00	1.69E+00	1.87E+01	7.58E-01
	排名	2	4	1	3	6	7	5
F24	均值	1.50E+03	1.25E+03	1.55E+03	3.00E+02	1.98E+03	3.94E+03	1.04E+03
	标准差	8.58E+02	3.15E+02	9.39E+02	5.84E-02	3.42E+02	1.10E+03	1.17E+02
	排名	4	3	5	1	6	7	2
F25	均值	4.45E+02	4.92E+02	5.00E+02	5.03E+02	5.11E+02	6.05E+02	4.95E+02
	标准差	1.01E+01	1.33E+01	1.51E-04	7.53E+00	1.17E+01	4.00E+01	1.52E+01
	排名	1	2	4	5	6	7	3
F26	均值	4.05E+02	3.13E+02	4.96E+02	4.08E+02	4.46E+02	4.62E+02	368.3704
	标准差	4.98E+01	3.48E+01	5.05E+00	6.99E+00	2.89E+01	2.60E+01	43.01458
	排名	3	1	7	4	5	6	2
F27	均值	5.53E+02	5.11E+02	5.69E+02	5.32E+02	7.75E+02	1.32E+03	5.26E+02
	标准差	1.13E+02	6.00E+01	1.43E+02	6.92E+01	1.83E+02	2.56E+02	7.14E+01
	排名	4	1	5	3	6	7	2
F28	均值	2.96E+03	1.98E+03	9.38E+03	4.46E+03	1.27E+04	1.01E+06	2.14E+03
	标准差	3.54E+03	2.30E+01	7.22E+03	1.42E+03	5.29E+03	6.08E+05	1.09E+02
	排名	3	1	5	4	6	7	2
	平均排名	3.79	2.14	5.07	3.00	5.46	6.86	1.68

Table 3

Fig.4

Fig.5

Table 4

Wilcoxon signed rank test results (α=0.05)"

函数	HFPSO				GEDGWO				VCS
函数	P-value	R+	R-	Win	P-value	R+	R-	Win	P-value	R+	R-	Win
F1	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F2	1.41E-04	257	1 069	-	8.58E-05	244	1 082	-	5.42E-01	728	598	=
F3	5.80E-10	1 324	2	+	4.17E-09	1 290	36	+	5.15E-10	1 326	0	+
F4	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F5	5.15E-10	1 326	0	+	1.05E-09	1 314	12	+	5.15E-10	1 326	0	+
F6	5.15E-10	1 326	0	+	7.35E-10	1 320	6	+	5.15E-10	1 326	0	+
F7	5.15E-10	1 326	0	+	9.66E-09	51	1 275	-	5.15E-10	1 326	0	+
F8	4.01E-02	882	444	+	1.04E-03	1 013	313	+	7.74E-09	1 279	47	+
F9	5.15E-10	1 326	0	+	1.39E-01	505	821	=	5.15E-10	1 326	0	+
F10	5.15E-10	1 326	0	+	2.57E-04	1 053	273	+	5.15E-10	1 326	0	+
F11	5.15E-10	1 326	0	+	5.30E-01	730	596	=	5.15E-10	1 326	0	+
F12	5.15E-10	1 326	0	+	5.69E-03	368	958	+	5.15E-10	1 326	0	-
F13	5.15E-10	1 326	0	+	4.76E-01	739	587	=	5.15E-10	1 326	0	+
F14	2.97E-09	1 296	30	+	1.52E-06	1 176	150	+	7.80E-10	1 319	7	+
F15	5.23E-09	1 286	40	+	3.73E-01	758	568	=	6.53E-10	1 322	4	+
F16	5.15E-10	1 326	0	+	3.43E-05	1 105	221	+	5.15E-10	1 326	0	+
F17	5.15E-10	1 326	0	+	1.49E-05	1 125	201	+	5.15E-10	1 326	0	+
F18	2.50E-09	1 299	27	+	2.76E-04	1 051	275	+	2.32E-08	1 259	67	+
F19	5.15E-10	1 326	0	+	4.17E-09	1 290	36	+	9.66E-09	1 275	51	+
F20	6.80E-01	619	707	=	5.15E-10	0	1 326	-	4.65E-01	741	585	=
F21	5.15E-10	1 326	0	+	2.23E-09	1 301	25	+	5.15E-10	1 326	0	+
F22	5.15E-10	1 326	0	+	1.58E-08	1 266	60	+	5.15E-10	1 326	0	+
F23	5.57E-07	129	1 197	+	1.45E-07	102	1 224	-	3.94E-09	35	1 291	-
F24	5.24E-04	1 033	293	+	1.31E-05	1 128	198	+	1.26E-04	1 072	254	+
F25	5.15E-10	0	1 326	+	3.88E-01	571	755	=	4.10E-02	881	445	+
F26	1.58E-04	1 066	260	+	5.15E-08	82	1244	-	5.15E-10	1 326	0	+
F27	1.34E-01	823	503	=	3.11E-01	555	771	=	1.20E-01	829	497	=
F28	9.78E-01	666	660	=	5.15E-10	0	1 326	-	5.15E-10	1 326	0	+
+/-/=	24/1/3				16/6/6				23/2/3

函数	MPA				SMA				HHO
函数	P-value	R+	R-	Win	P-value	R+	R-	Win	P-value	R+	R-	Win
F1	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F2	8.26E-05	1 083	243	+	8.49E-06	1 138	188	+	7.74E-09	1 279	47	+
F3	6.15E-10	1 323	3	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F4	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F5	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F6	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F7	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	=
F8	7.43E-01	628	698	=	2.86E-04	1 050	276	+	1.18E-09	1 312	14	+
F9	2.53E-07	1 213	113	+	5.80E-10	1 324	2	+	5.15E-10	1 326	0	+
F10	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F11	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F12	7.14E-03	950	376	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F13	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F14	2.19E-01	794	532	=	1.67E-09	1 306	20	+	5.15E-10	1 326	0	+
F15	7.22E-01	625	701	=	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F16	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F17	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F18	2.30E-01	791	535	=	1.99E-09	1303	23	+	5.15E-10	1 326	0	+
F19	4.17E-08	78	1 248	-	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F20	4.65E-01	741	585	=	8.69E-08	1 234	92	+	8.87E-06	1 137	189	+
F21	5.57E-07	1 197	129	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
F22	5.46E-10	1 325	1	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	-
F23	9.03E-01	676	650	=	1.00E-04	1 078	248	+	7.80E-10	1 319	7	+
F24	5.15E-10	0	1 326	-	7.35E-10	1 320	6	+	6.93E-10	1 321	5	+
F25	2.04E-03	992	334	+	4.37E-07	1 202	124	+	5.15E-10	1 326	0	-
F26	1.26E-06	1 180	146	+	5.80E-10	1324	2	+	5.15E-10	1 326	0	+
F27	6.87E-01	706	620	=	1.50E-08	1 267	59	+	5.15E-10	1 326	0	+
F28	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+	5.15E-10	1 326	0	+
+/-/=	19/2/7				28/0/0				27/2/1

Table 4

Table 5

Fig.6

References 28

1	HOLLAND J H . Genetic algorithms[J]. Scientific American, 1992, 267, 66- 73. doi: 10.1038/scientificamerican0792-66
2	STORN R , PRICE K . Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces[J]. Journal of Global Optimization, 1997, 11 (4): 341- 359. doi: 10.1023/A:1008202821328
3	HWANG C R . Simulated annealing: theory and applications[J]. Acta Applicandae Mathematica, 1988, 12 (1): 108- 111. doi: 10.1007/BF00047572
4	RASHEDI E , NEZAMABADI P H , SARYAZDI S . GSA: a gravitational search algorithm[J]. Information Sciences, 2009, 179 (13): 2232- 2248. doi: 10.1016/j.ins.2009.03.004
5	WEI Z , HUANG C Q , WANG X F , et al. Nuclear reaction optimization: a novel and powerful physics-based algorithm for global optimization[J]. IEEE Access, 2019, 7, 66084- 66109. doi: 10.1109/ACCESS.2019.2918406
6	MIRJALILI S , MIRJALILI S M , LEWIS A . Grey wolf optimizer[J]. Advances in Engineering Software, 2014, 69, 46- 61. doi: 10.1016/j.advengsoft.2013.12.007
7	KARABOGA D , AKAY B . A comparative study of artificial bee colony algorithm[J]. Applied Mathematics & Computation, 2009, 214 (1): 108- 132.
8	MIRJALILI S , LEWIS A . The whale optimization algorithm[J]. Advances in Engineering Software, 2016, 95, 51- 67. doi: 10.1016/j.advengsoft.2016.01.008
9	WOLPERT D H , MACREADY W G . No free lunch theorems for optimization[J]. IEEE Trans.on Evolutionary Computation, 1997, 1 (1): 67- 82. doi: 10.1109/4235.585893
10	MIRJALILI S , GANDOMI A H , MIRJALILI S Z , et al. Salp swarm algorithm: a bio-inspired optimizer for engineering design problems[J]. Advances in Engineering Software, 2017, 114, 163- 169. doi: 10.1016/j.advengsoft.2017.07.002
11	TOLBA M , REZK H , DIAB A , et al. A novel robust methodology based salp swarm algorithm for allocation and capacity of renewable distributed generators on distribution grids[J]. Energies, 2018, 11 (10): 1- 34.
12	陈涛, 王梦馨, 黄湘松. 基于樽海鞘群算法的无源时差定位[J]. 电子与信息学报, 2018, 40 (7): 1591- 1597.
	CHEN T , WANG M X , HUANG X S . Time difference of arrival passive location based on salp swarm algorithm[J]. Journal of Electronics & Information Technology, 2018, 40 (7): 1591- 1597.
13	GOUDOS S , ATHANASIADOU G E . Application of an ensemble method to uav power modeling for cellular communications[J]. IEEE Antennas and Wireless Propagation Letters, 2019, 18 (11): 2340- 2344. doi: 10.1109/LAWP.2019.2926784
14	SINGH N , SINGH S B , HOUSSEIN E H . Hybridizing salp swarm algorithm with particle swarm optimization algorithm for recent optimization functions[J]. Evolutionary Intelligence, 2020, 22, 468- 476.
15	ZHANG H L , WANG Z Y , CHEN W B , et al. Ensemble mutation-driven salp swarm algorithm with restart mechanism: framework and fundamental analysis[J]. Expert Systems with Applications, 2020, 165, 113897.
16	ZHANG Q , CHEN H , HEIDARI A A , et al. Chaos-induced and mutation-driven schemes boosting salp chains-inspired optimizers[J]. IEEE Access, 2019, 7, 31243- 31261. doi: 10.1109/ACCESS.2019.2902306
17	张铸, 张仕杰, 饶盛华, 等. 基于自适应正态云模型的引力樽海鞘群算法[J]. 控制与决策, 2022, 37 (2): 344- 352.
	ZHANG Z , ZHANG S J , RAO S H , et al. Gravity salp swarm algorithm based on adaptive normal cloud model[J]. Control and Decision, 2022, 37 (2): 344- 352.
18	白钰, 彭珍瑞. 基于自适应惯性权重的樽海鞘群算法[J]. 控制与决策, 2022, 37 (1): 237- 246.
	BAI Y , PENG Z R . Salp swarm algorithm based on adaptive inertia weight[J]. Control and Decision, 2022, 37 (1): 237- 246.
19	IBRAHIM B A . A hybrid firefly and particle swarm optimization algorithm for computationally expensive numerical problems[J]. Applied Soft Computing, 2018, 6, 232- 249.
20	WANG X F , ZHAO H , HAN T , et al. A grey wolf optimizer using Gaussian estimation of distribution and its application in the multi-UAV multi-target urban tracking problem[J]. Applied Soft Computing, 2019, 78, 240- 260. doi: 10.1016/j.asoc.2019.02.037
21	LI M D , ZHAO H , WENG X W , et al. A novel nature-inspired algorithm for optimization: virus colony search[J]. Advances in Engineering Software, 2016, 92, 65- 88. doi: 10.1016/j.advengsoft.2015.11.004
22	FARAMARZI A , HEIDARINEJAD M , MIRJALILI S , et al. Marine predators algorithm: a nature-inspired metaheuristic[J]. Expert Systems with Applications, 2020, 152, 113377. doi: 10.1016/j.eswa.2020.113377
23	LI S M , CHEN H L , WANG M J , et al. Slime mould algorithm: a new method for stochastic optimization[J]. Future Generation Computer Systems, 2020, 111, 300- 323. doi: 10.1016/j.future.2020.03.055
24	HEIDARI A A , MIRJALILI S , FARIS H , et al. Harris hawks optimization: algorithm and applications[J]. Future Generation Computer Systems, 2019, 97, 849- 872. doi: 10.1016/j.future.2019.02.028
25	SALVADOR G , MOLINA D , LOZANO M , et al. A study on the use of non-parametric tests for analyzing the evolutionary algorithms'behaviour: a case study on the CEC'2005 special session on real parameter optimization[J]. Journal of Heuristics, 2009, 15 (6): 617- 644. doi: 10.1007/s10732-008-9080-4
26	GARCÍA S , MOLINA D , LOZANO M , et al. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms[J]. Journal of Heuristics, 2009, 15 (6): 617- 644. doi: 10.1007/s10732-008-9080-4
27	DERRAC J , GARCIA S , MOLINA D , et al. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms[J]. Swarm and Evolutionary Computation, 2011, 1 (1): 3- 18. doi: 10.1016/j.swevo.2011.02.002
28	VECEK N , MERNIK M , CREPINSEK M . A chess rating system for evolutionary algorithms: a new method for the comparison and ranking of evolutionary algorithms[J]. Information Sciences, 2014, 277, 154- 177. doi: 10.1016/j.ins.2014.02.014

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An improved salp swarm algorithm using Gaussian distribution estimation strategy

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算法	参数设置
HFPSO	c₁=c₂=1.494 45, a=0.2, B₀=2γ=1, w_i=0.9, w_f=0.5
GEDGWO	r₃∈(0, 1), r₄∈(0, 1), r₅∈(0, 1),
VCS	λ=0.5, σ=0.3
MPA	FADs =0.2, P=0.5
SMA	z=0.03
HHO	α=0.01, β=1.5

测试项目	HFPSO	GEDGWO	VCS	MPA	SMA	HHO	GDESSA	检验结果
均值排名	3.79	2.14	5.07	3.00	5.46	6.86	1.68	Friedman测试的显著概率3.330 7E-25 卡方分布值128.04 6×168自由度的F分布值86.501 3 Iman-Davenport测试的显著概率2.220 4E-16
标准差排名	4.57	2.39	5.00	2.86	4.54	6.50	2.14	Friedman测试的显著概率1.699 0E-17 卡方分布值91.21 6×168自由度的F分布值32.069 6 Iman-Davenport测试的显著概率4.584 2E-11
时间排名	1.00	2.00	3.86	5.00	7.00	6.00	3.14	Friedman测试的显著概率2.441 9E-33 卡方分布值166.53 6×168自由度的F分布值3.05E+03 Iman-Davenport测试的显著概率0