种群分段变异学习和S型权重变色龙群算法

doi:10.16182/j.issn1004731x.joss.21-0968

摘要/Abstract

摘要：

探索寻优能力强、良好的可靠性和稳定性是智能算法应用到具体领域中的最佳选择。针对变色龙群算法存在求解不稳定、收敛精度低下和搜索开发之间不平衡等缺陷，提出一种种群多样性分段变异学习和S型权重的变色龙群算法(RMSCSA)。引入折射镜像学习(refraction mirror learning，RML)策略使变色龙更符合自然界中的观察，增强它的多样性；引入种群多样性分段变异使适应度较差的个体得到保留，并引导它向最优位置靠近；S型递减权重的引入让它进一步平衡算法的全局搜索和开发能力，通过收敛性分析得出S型递减权重因子的优势。利用经典函数集和CEC 2017函数集进行性能验证，结果表明3种策略比CSA具有更好寻优精度和效率。通过对独立运行30次的结果进行Wilcoxon秩和检验、Friedman’s以及Holm后续检验统计分析，结果表明引入的3种策略与CSA相比都有更好的寻优能力。

关键词: 变色龙群算法, 折射镜像学习, 多样性变异, S型递减权重, 统计分析

Abstract:

It is the best choice for intelligent algorithms to be applied to specific fields to explore strong searching ability, good reliability and stability.In this paper, aiming at the defects of chameleon swarm algorithm, such as unstable solution, low convergence accuracy and unbalanced search and development, a chameleon swarm algorithm (RMSCSA) based on population diversity segmental mutation learning and S-type weight is proposed. The refraction mirror learning strategy (RML) is introduced to make the chameleon more consistent with the observation in nature and enhance its diversity. The introduction of segmental variation of population diversity can keep the individuals with poor fitness and guide them to the optimal position. The introduction of S-type decreasing weight makes it further balance the global search and explore ability of the algorithm, and obtains the advantage of S-type decreasing weight factor through convergence analysis. The classical test function and CEC 2017 competition function are used for performance verification, and the results show that the three strategies have better optimization accuracy and efficiency for CSA. In order to compare the performance of different algorithms, the results of 30 independent runs are statistically analyzed by Wilcoxon rank-sum test, Friedman's test and Holm follow-up test. The analysis shows that the three strategies introduced have better optimization ability compared with CSA.

Key words: chameleon swarm algorithm(CSA), refraction mirror learning(RML), diversity variation, S-type decreasing weight, statistical analyzed

中图分类号:

TP301.6

张达敏, 王义, 张琳娜. 种群分段变异学习和S型权重变色龙群算法[J]. 系统仿真学报, 2023, 35(1): 11-26.

Damin Zhang, Yi Wang, Linna Zhang. Chameleon Swarm Algorithm for Segmental Variation Learning of Population and S-type Weight[J]. Journal of System Simulation, 2023, 35(1): 11-26.

图/表 17

表1

猎物搜索的参数说明

参数名称	说明
$x i j t$	变色龙 $i$ 在 $j$ 维空间中，第 $t 次迭代$ 的位置
$P i j t$	变色龙 $i$ 在 $j$ 维空间经 $t 次迭代$ 的最佳位置
$G j$	变色龙在 $t 次迭代$ 的全局最优位置
$p 1, p 2$	用于控制算法开发能力的正系数
$r 1, r 2, r 3, r$	取值范围为(0, 1)的随机数
$P p$	变色龙的感知概率
$μ$	随 $t$ 控制的搜索能力参数
$s g n (r a n d - 0.5)$	控制旋转方向

表1

图1

图2

图3

表2

各算法的主要参数设置

算法	主要参数
CSA	$P p = 0.1, α = 3.5,$ $p 1 = 0.25,$ $p 2 = 1.5, c 1 = 1.75, c 2 = 1.75$
RCSA	$n 1 = 1.0 ~ 1.5, k = 1, A P = 0.35$
MCSA	$n = 50, N$ 为种群大小
SCSA	$g e n = 51, ω m a x = 0.95, ω m i n = 0.55$
SSA	--
WOA	$a m a x = 2, a m i n = 0$

表2

表3

14个经典测试函数信息

函数名称	函数公式	特征	取值空间	最优取值
Sphere	$F 1 X = ∑ i = 1 n x i 2$	US	[-100, 100]	0
Schwefel 2.22	$F 2 X = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$	US	[-100, 100]	0
Schwefel 1.2	$F 3 X = ∑ i = 1 n (∑ j = 1 n x j) 2$	UN	[-100, 100]	0
Schwefel 2.21	$F 4 X = m a x 1 ≤ i ≤ n {\| x i \|}$	US	[-100, 100]	0
Rosenbrock	$F 5 X = ∑ i = 1 n 100 x i + 1 - x i 2) 2 + (1 - x i 2) 2]$	UN	[-30, 30]	0
Quartic	$F 6 X = ∑ i = 1 n i x i 4 + r a n d o m [0, 1)$	US	[-1.28, 1.28]	0
Rastrigin	$F 7 X = ∑ i = 1 n x i 2 - 10 c o s (2 π x i + 10$	MS	[-5.12, 5.12]	0
Ackley	$F 8 X = - 20 e x p - 0.2 ∑ i = 1 n x i 2 n - e x p ∑ i = 1 n c o s 2 π x i n + 20 + e$	MN	[-32, 32]	0
Griewank	$F 9 X = 1 4 000 ∑ i = 1 n x i 2 - ∏ i = 1 n c o s x i i + 1$	MN	[-600, 600]	0
Penalized 2	$F 10 X = 0.1 {s i n 2 3 π x 1 + ∑ i = 1 n - 1 (x i - 1) 2 [1 + s i n 2 3 π x i + 1]$ $+ (x n - 1) 2 + ∑ i = 1 n u (x i, 5,100, 4}$	MN	[-50, 50]	0
Foxholes	$F 11 X = [1500 + ∑ j = 1 25 1 j + ∑ j = 1 2 (x i - a i j) 6] - 1$	MN	[-65.53, 65.53]	$≈ 1$
Kowalik	$F 12 X = ∑ i = 1 11 [a i - x 1 (b i 2 + b i x 2) b i 2 + b i x 3 x 4] 2$	MN	[-5, 5]	3.075 $× 10 - 4$
Branin	$F 13 X = (x 2 - 5.1 4 π 2 x 12 + 5 π x 1 - 6) 2 + 10 1 - 1 8 π c o s x 1 + 10$	MN	$x 1$ ∈[5, 10] $x 2$ ∈[0, 15]	0.398
Hartman	$F 14 X = - ∑ i = 1 4 c i e x p [- ∑ j = 1 6 a i j (x j - p i j) 2]$	MS	[0,1]	-3.32

表3

表4

不同维度下各策略的测试函数对比

函数	算法名	$d = 10$			$d = 30$			$d = 50$
函数	算法名	平均值	标准差	耗时/s	平均值	标准差	耗时/s	平均值	标准差	耗时/s
$F 1$	CSA	5.71×10^-10	1.44×10^-10	1.66	3.37×10^-7	1.31×10^-7	6.60	3.28×10^-6	5.32×10^-7	10.67
	RCSA	2.97×10^-90	1.57×10^-89	1.82	3.71×10^-86	1.45×10^-85	6.89	6.03×10^-89	3.28×10^-88	12.46
	MCSA	2.2×10^-163	6.4×10^-162	1.65	4.3×10^-117	9.9×10^-117	6.71	1.2×10^-116	6.6×10^-116	13.55
	SCSA	4.48×10^-78	2.37×10^-77	1.71	1.31×10^-34	4.56×10^-32	7.01	3.97×10^-21	3.38×10^-20	11.62
	RMSCSA	0	0	1.88	0	0	7.51	0	0	11.65
$F 2$	CSA	1.51×10^-6	6.84×10^-6	1.67	2.47	3.13	6.58	1.31×10²	8.40×10	10.54
	RCSA	4.78×10^-44	2.57×10^-43	1.81	2.75×10^-43	1.49×10^-42	6.61	5.84×10^-45	3.18×10^-44	11.28
	MCSA	5.13×10^-84	2.32×10^-83	2.10	2.37×10^-61	6.22×10^-61	7.56	2.23×10^-65	1.09×10^-64	11.25
	SCSA	2.35×10^-35	6.32×10^-33	1.71	6.19×10^-18	4.48×10^-17	6.84	2.36×10^-13	1.84×10^-12	11.22
	RMSCSA	1.3×10^-¹⁹⁷	3.2×10^-¹⁹⁵	2.15	1.1×10^-188	1.1×10^-187	6.77	2.12×10^-⁹⁹	7.9×10^-⁹⁹	11.32
$F 3$	CSA	3.92×10^-7	9.31×10^-7	1.77	7.88	9.53	7.00	4.39×10³	1.09×10³	11.02
	RCSA	8.64×10^-91	3.11×10^-90	2.03	2.23×10^-83	1.22×10^-82	6.95	2.95×10^-82	1.16×10^-81	10.99
	MCSA	5.7×10^-113	4.5×10^-112	1.88	1.14×10^-75	5.39×10^-75	7.51	4.55×10^-77	2.49×10^-76	11.43
	SCSA	2.03×10^-79	3.72×10^-78	1.81	9.42×10^-31	1.69×10^-29	7.22	3.38×10^-37	2.94×10^-35	11.33
	RMSCSA	0	0	2.02	0	0	7.35	3.0×10^-²⁸¹	9.1×10^-²⁸¹	11.19
$F 4$	CSA	1.14×10^-5	7.20×10^-6	1.67	3.50×10^-3	6.50×10^-3	6.69	2.25×10^-2	2.14×10^-2	10.50
	RCSA	4.75×10^-45	2.54×10^-44	1.83	5.65×10^-43	3.09×10^-42	6.89	5.78×10^-45	2.28×10^-44	10.79
	MCSA	2.41×10^-80	8.67×10^-80	2.16	2.33×10^-37	1.27×10^-36	8.12	5.08×10^-51	5.77×10^-50	10.20
	SCSA	3.79×10^-44	1.39×10^-44	1.72	2.61×10^-17	2.63×10^-18	6.88	7.59×10^-15	3.36×10^-16	9.95
	RMSCSA	4.5×10^-¹⁷⁸	3.6×10^-¹⁷⁸	2.11	6.3×10^-¹⁸²	3.3×10^-¹⁸¹	6.91	5.7×10^-¹⁷⁸	2.9×10^-¹⁷⁶	10.07
$F 5$	CSA	1.95	4.68	1.71	2.46	2.59	6.69	2.51×10²	3.81×10²	10.59
	RCSA	4.06×10^-7	7.73×10^-6	2.01	1.81×10^-5	4.04×10^-5	7.03	5.29×10^-5	1.03×10^-4	11.83
	MCSA	8. 2×10^-4	7.54×10^-3	2.06	2.89×10^-2	1.70×10^-2	7.24	4.0×10^-7	3.31×10^-⁵	11.56
	SCSA	1.11	7.51×10^-1	1.72	2.86	2.14×10^-1	7.89	4.87	4.89	11.21
	RMSCSA	1.40×10^-⁹	1.62×10^-⁹	2.39	1.92×10^-⁹	3.21×10^-⁷	7.21	2.63×10^-⁷	8.61×10^-5	11.31
$F 6$	CSA	3.55	3.02	1.68	1.26×10²	2.21×10	6.62	5.45×10²	5.36×10	10.52
	RCSA	3.26×10^-5	4.31×10^-5	1.96	2.49×10^-5	2.28×10^-5	6.71	2.65×10^-5	3.95×10^-5	12.51
	MCSA	4.17×10^-5	2.27×10^-5	2.21	2.53×10^-5	2.14×10^-⁶	6.55	1.38×10^-7	1.03×10^-6	11.85
	SCSA	1.21×10^-2	4.10×10^-1	1.95	2.98×10^-2	5.58×10^-1	6.55	5.36×10^-2	1.17×10^-2	11.29
	RMSCSA	3.61×10^-⁷	1.25×10^-⁷	2.44	2.62×10^-⁶	1.77×10^-5	6.52	1.37×10^-⁷	1.00×10^-⁶	11.18
$F 7$	CSA	9.44×10^-1	1.15	1.67	1.39×10	2.75×10	6.05	3.11×10²	1.94×10	10.63
	RCSA	0	0	1.96	0	0	6.56	0	0	11.02
	MCSA	0	0	2.12	0	0	6.71	0	0	11.43
	SCSA	3.85×10^-98	2.95×10^-97	1.81	2.55×10^-97	1.78×10^-96	6.96	0	0	11.41
	RMSCSA	0	0	2.09	0	0	6.31	0	0	12.69
$F 8$	CSA	2.92×10^-5	2.32×10^-6	1.71	1.02	1.54	6.03	1.91	1.02×10^-1	10.64
	RCSA	8.88×10^-¹⁶	8.88×10^-¹⁶	2.00	8.88×10^-¹⁶	8.88×10^-¹⁶	6.21	8.88×10^-¹⁶	8.88×10^-¹⁶	12.08
	MCSA	8.88×10^-¹⁶	8.88×10^-¹⁶	2.05	8.88×10^-¹⁶	8.88×10^-¹⁶	6.05	8.88×10^-¹⁶	8.88×10^-¹⁶	10.66
	SCSA	8.88×10^-¹⁶	8.88×10^-¹⁶	1.77	2.04×10^-12	2.47×10^-12	6.22	7.26×10^-13	5.47×10^-12	11.03
	RMSCSA	8.88×10^-¹⁶	8.88×10^-¹⁶	2.07	8.88×10^-¹⁶	8.88×10^-¹⁶	6.55	8.88×10^-¹⁶	8.88×10^-¹⁶	10.68
$F 9$	CSA	5.65×10^-1	4.37×10^-1	1.72	1.33×10^-4	3.12×10^-4	6.15	2.79×10^-3	6.19×10^-3	10.63
	RCSA	0	0	1.89	0	0	6.04	0	0	11.00
	MCSA	0	1.6×10^-311	2.00	4.05×10^-211	3.87×10^-209	6.29	1.4×10^-266	9.5×10^-264	11.52
	SCSA	0	0	1.91	0	0	6.22	0	0	10.99
	RMSCSA	0	0	2.00	0	0	6.49	0	0	10.71
$F 10$	CSA	1.90×10^-11	8.75×10^-11	1.97	9.53×10^-8	3.60×10^-8	6.56	3.70×10^-4	2.01×10^-3	11.26
	RCSA	1.51×10^-8	1.44×10^-7	2.10	1.12×10^-¹⁰	1.74×10^-8	7.06	6.95×10^-8	1.17×10^-8	12.28
	MCSA	4.21×10^-23	2.20×10^-20	2.30	2.99×10^-8	5.83×10^-8	6.89	4.99×10^-8	1.66×10^-9	12.09
	SCSA	3.85×10^-10	7.06×10^-8	2.03	2.56×10^-8	1.31×10^-8	6.66	4.27×10^-5	4.98×10^-4	11.76
	RMSCSA	3.51×10^-²⁶	1.18×10^-²⁴	2.51	1.57×10^-10	1.18×10^-⁸	7.05	7.55×10^-¹³	1.43×10^-¹²	11.81

表4

表5

固定维度下的优化对比

函数	算法名称	平均值	标准差	耗时/s	维度
$F 11$	CSA	9.979×10^-1	2.23×10^-4	1.28	2
	RCSA	9.980×10^-1	5.90×10^-11	1.45	2
	MCSA	9.981×10^-1	6.20×10^-2	1.38	2
	SCSA	9.980×10^-1	2.55×10^-8	1.45	2
	RMSCSA	9.982×10^-¹	4.38×10^-¹⁶	1.55	2
$F 12$	CSA	5.110×10^-4	4.99×10^-3	1.27	4
	RCSA	3.117×10^-4	5.53×10^-3	1.36	4
	MCSA	3.083×10^-4	6.58×10^-3	1.92	4
	SCSA	3.100×10^-4	2.87×10^-4	1.37	4
	RMSCSA	3.079×10^-⁴	2.38×10^-⁶	1.81	4
$F 13$	CSA	3.974×10^-1	1.55×10^-2	0.22	2
	RCSA	3.978×10^-¹	1.20	0.21	2
	MCSA	3.986×10^-1	8.21×10^-2	0.29	2
	SCSA	3.978×10^-¹	3.62×10^-5	0.24	2
	RMSCSA	3.978×10^-¹	2.98×10^-6	0.36	2
$F 14$	CSA	-3.322	3.01×10^-2	1.37	6
	RCSA	-3.324	5.80×10^-⁹	1.54	6
	MCSA	-3.320	1.99×10^-1	1.79	6
	SCSA	-3.322	6.75×10^-2	1.97	6
	RMSCSA	-3.320	7.25×10^-8	1.94	6

表5

图4

表6

表7

CEC 2017测试下各种算法性能对比

函数名	评价指标	RMSCSA	CSA	RCSA	MCSA	SCSA	SSA	WOA
CEC04	平均值	4.705 2×10²	2.120 8×10³	7.353 7×10²	4.724 4×10²	7.489 6×10²	6.425 5×10²	6.890 7×10²
	标准差	7.298×10^-4	1.628×10	1.587×10^-4	1.698×10	1.093×10²	2.055×10	6.773×10
	耗时/s	4.08	6.32	2.69	3.13	2.01	1.25	0.83
CEC05	平均值	5.209 7×10²	1.006 5×10	6.885 6×10²	7.511 1×10²	5.385 6×10²	6.970 5×10²	8.454 6×10²
	标准差	1.625×10^-⁵	2.013×10	1.526×10	2.296×10	2.160×10	3.852×10	5.344×10
	耗时/s	3.88	6.44	2.67	3.14	2.01	1.61	0.75
CEC08	平均值	9.235 5×10²	1.237 0×10³	1.237 0×10³	1.005 9×10³	1.037 9×10³	9.958 0×10²	1.129 4×10³
	标准差	2.684×10^-⁵	6.200×10^-4	1.954×10^-3	2.192×10	1.573×10	4.910 2×10	6.337×10
	耗时/s	3.91	2.91	2.71	4.19	2.26	1.53	0.88
CEC16	平均值	3.258 6×10³	6.706 5×10³	3.749 8×10³	4.503 9×10³	3.450 6×10³	1.855 9×10⁴	4.921 2×10³
	标准差	3.105×10²	2.176×10³	2.112×10³	3.051×10²	3.220×10²	3.771 5×10³	4.472×10²
	耗时/s	3.96	3.73	2.66	4.26	2.84	1.32	0.77
CEC17	平均值	2.356 9×10³	2.499 6×10⁴	4.446 5×10³	3.004 0×10³	2.282 3×10³	2.966 2×10³	2.986 8×10³
	标准差	3.144×10	3.334×10⁴	2.781×10²	8.944×10	1.395×10²	2.836×10²	2.991×10²
	耗时/s	4.11	3.96	2.78	4.37	2.93	1.55	0.56
CEC20	平均值	2.630 2×10³	3.646 7×10³	3.410 6×10³	3.012 0×10³	2.758 8×10³	2.795 5×10³	3.127 9×10³
	标准差	1.361×10	1.592×10²	9.995×10	7.638×10	2.135×10²	5.541×10³	1.823×10²
	耗时/s	4.14	3.85	2.81	4.40	2.90	1.97	0.57
CEC21	平均值	2.610 0×10³	2.926 4×10³	2.849 0×10³	2.604 0×10³	2.526 2×10³	2.917 2×10³	2.831 9×10³
	标准差	6.886	5.267×10	3.944×10	2.114×10	1.941×10	5.541 0×10²	3.544×10
	耗时/s	4.26	3.82	3.23	4.38	3.07	1.97	0.42
CEC23	平均值	3.211 7×10³	4.012 8×10³	3.781 0×10³	3.257 7×10³	3.893 0×10³	3.698 8×10³	3.987 5×10³
	标准差	4.617×10	2.371×10²	1.298×10²	4.601×10	2.389×10²	4.629×10³	1.256×10²
	耗时/s	2.95	3.76	3.16	3.45	3.36	1.45	0.95
CEC27	平均值	3.060 0×10³	5.167 4×10³	3.508 0×10³	5.611 0×10³	3.302 8×10³	4.305 7×10³	4.558 1×10³
	标准差	2.015×10	3.219×10³	1.143×10²	5.111×10²	3.083×10²	2.609×10	8.075×10
	耗时/s	4.06	4.13	4.46	3.17	4.35	1.30	0.56
CEC29	平均值	3.652 9×10³	2.614 1×10⁴	8.485 1×10³	8.485 8×10³	4.419 8×10³	4.202 8×10³	5.034 0×10³
	标准差	3.505	2.067×10⁴	5.081×10	5.081×10³	2.094×10²	2.024×10²	5.363×10²
	耗时/s	5.70	4.09	3.07	3.07	4.12	1.28	0.42

表7

表8

与其他智能算法的平均值比较

算法	$F 1$	$F 2$	$F 3$	$F 4$	$F 5$	$F 6$	$F 7$	$F 8$	$F 9$	$F 10$
RMSCSA	0	2.21×10^-179	0	2.77×10^-¹⁸¹	1.92×10^-⁵	2.39×10^-⁶	0	8.88×10^-16	0	5.52×10^-9
MPEA^[20]	2.70×10^-11	2.77×10^-91	1.52×10^-20	1.04×10^-68	6.74×10^-12	2.35×10^-1	6.38×10^-6	2.84×10^-13	4.3×10^-7	2.27×10^-5
SSA^[21]	0	2.12×10^-259	0	5.72×10^-177	9.24×10^-4	1.34×10^-4	0	8.88×10^-16	0	7.17×10^-12
HCMOIWO^[22]	3.17×10^-55	3.66×10^-51	5.41×10^-10	9.34×10^-13	5.28×10^-1	8.41×10^-3	4.62×10^-1	1.86×10^-11	3.29×10^-2	1.35×10^-3
HCUGOA^[23]	0	0	0	3.23×10^-176	1.78×10^-3	4.96×10^-5	0	8.88×10^-16	0
GOBL-RNADE^[24]	1.63×10^-240	1.80×10^-126	7.15×10^-276	6.26×10^-115	5.94×10^-2	7.70×10^-2	0	0	0	2.28×10^-¹²
WOAEP^[25]	1.59×10^-96	1.27×10^-48	3.97×10^-96	1.46×10^-49	2.89×10	2.67×10^-4	0	8.88×10^-16	0	2.99
FO-FPA^[26]	1.51×10^-184	5.04×10^-93	1.23×10^-183	9.97×10^-93	2.89×10	1.13×10^-4	0	8.88×10^-16	0	9.65×10^-6
SHDNMRA^[18]	1.80×10^-60	1.80×10^-25	5.70×10^-61	8.20×10^-24	8.00×10^-2	6.40×10^-4	8.20×10^-10	8.20×10^-10	4.60×10^-17	3.02×10^-7
SCCSA^[27]	9.22×10^-69	8.24×10^-41	4.31×10^-13	2.15×10^-17	5.90	1.33×10^-3	5.46	8.88×10^-16	1.34×10^-2	3.00

表8

表9

经典测试函数下的秩和检验值

函数	$p 1$	$p 2$	$p 3$	$p 4$
$F 1$	3.00×10^-11	3.01×10^-11	3.00×10^-11	3.00×10^-11
$F 2$	3.02×10^-11	3.02×10^-11	3.02×10^-11	3.02×10^-11
$F 3$	2.99×10^-11	2.99×10^-11	2.99×10^-11	2.99×10^-11
$F 4$	3.02×10^-11	3.00×10^-11	3.02×10^-11	3.00×10^-11
$F 5$	2.14×10^-9	4.04×10^-11	2.06×10^-10	1.16×10^-10
$F 6$	0.282	0.365	5.66×10^-11	3.02×10^-11
$F 7$	NA	NA	NA	1.21×10^-12
$F 8$	5.28×10^-5	3.99×10^-3	NA	1.21×10^-12
$F 9$	NA	3.68×10^-3	NA	1.21×10^-12
$F 10$	6.84×10^-7	3.02×10^-11	1.17×10^-9	1.21×10^-12
$F 11$	0.4247	3.02×10^-11	1.17×10^-9	1.21×10^-12
$F 12$	3.02×10^-11	1.49×10^-4	8.15×10^-11	8.48×10^-9
$F 13$	0.595	1.23×10^-9	2.78×10^-11	1.09×10^-12
$F 14$	7.70×10^-8	5.57×10^-10	3.02×10^-11	3.02×10^-11
+/-/=	9/3/2	12/1/1	11/0/3	14/0/0

表9

表10

经典测试函数下算法平均排名

算法	$R 1$	$R 2$	$R 3$	$R ¯$	排名
RMSCSA	2.35	2.2	2.05	2.2	1
RCSA	4.7	2.2	2.45	3.116 666 7	3
MCSA	2.35	2.5	2.25	2.366 666 7	2
SCSA	2.85	3.3	3.35	3.166 666 7	4
CSA	2.75	4.8	4.9	4.15	5

表10

表11

CEC 2017函数平均排名

算法	$R$	排名
RMSCSA	1.8	1
CSA	6.75	7
RCSA	4.55	5
MCSA	3.9	4
SCSA	2.5	2
SSA	3.7	3
WOA	4.8	6

表11

表12

经典函数Holm后续校验结果

$i$	RMSCSA vs.	$z ¯$ 值	$p ¯$ 值	$α / i$	是否拒绝
4	CSA	3.677 0	2.36×10^-4	0.0125	是
3	SCSA	2.877 5	4.01×10^-3	0.016 666 7	是
2	RCSA	2.388 7	1.69×10^-2	0.025	是
1	MCSA	0.893 1	0.3725	0.05	否

表12

表13

CEC 2017函数Holm后续校验结果

$i$	RMSCSA vs.	$z$ 值	$p$ 值	$α / i$	是否拒绝
6	CSA	5.124	2.99×10^-7	0.008 333	是
5	WOA	3.105	1.903×10^-3	0.01	是
4	RCSA	2.847	4.413×10^-3	0.025	是
3	MCSA	2.617 4	8.86×10^-3	0.016 67	是
2	SSA	2.359 4	1.83×10^-2	0.025	是
1	SCSA	0.725	0.468 452	0.05	否

表13

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