A Hybrid Algorithm Based on Seeker Optimization Algorithm and Salp Swarm Algorithm for PID Parameters Optimization

doi:10.16182/j.issn1004731x.joss.20-1036

Abstract

Abstract:

Aiming at the premature convergence of seeker optimization algorithm(SOA) during optimizing the global problems, a new SOA-SSA hybrid algorithm based on seeker optimization algorithm and salp swarm algorithm (SSA) is proposed.The SOA-SSA algorithm is based on a double population evolution strategy, in which some individuals of the population are evolved by seeker optimization algorithm and the rest are evolved from salp swarm algorithm. The individuals in SOA and SSA both employ an information sharing mechanism to realize the coevolution. These strategies increase the diversity of the population and avoid the premature convergence. The experimental results show that the proposed algorithm can be used in both the high-dimensional cases and the PID control parameter optimization. Compared to the other eleven algorithms, the SOA-SSA has the higher, convergence speed, precision and robustness, and has a better optimization performance.

Key words: hybrid, seeker optimization algorithm, salp swarm algorithm, benchmark function optimization, PID control parameter

CLC Number:

TP391

Shaomi Duan, Huilong Luo, Haipeng Liu. A Hybrid Algorithm Based on Seeker Optimization Algorithm and Salp Swarm Algorithm for PID Parameters Optimization[J]. Journal of System Simulation, 2022, 34(6): 1230-1246.

Figures/Tables 10

Fig. 1

Table 1

Description of benchmark functions

测试函数	范围	最优值
$f 1 = ∑ i = 1 n x i 2$	[-100, 100]	0
$f 2 (x) = ∑ i = 1 n x i + ∏ i = 1 n x i$	[-10, 10]	0
$f 3 (x) = ∑ i = 1 n ∑ j = 1 j x j 2$	[-100, 100]	0
$f 4 (x) = m a x i {\| x i \|, ? 1 ≤ i ≤ n}$	[-100, 100]	0
$f 5 (x) = ∑ i = 1 n - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$	[-30, 30]	0
$f 6 (x) = ∑ i = 1 n (x i + 0.5) 2$	[-100, 100]	0
$f 7 (x) = ∑ i = 1 n i x i 4 + r a n d m [0,1)$	[-1.28, 1.28]	0
$f 8 (x) = ∑ i = 1 n i x i 2$	[-10, 10]	0
$f 9 (x) = ∑ i = 1 n x i i + 1$	[-1.28, 1.28]	0
$f 10 (x) = - ∑ i = 1 n x i s i n (x i)$	[-500, 500]	-418.982 9×1 000
$f 11 (x) = ∑ i = 1 n [x i 2 - 10 c o s (2 π x i) + 10]$	[-5.12, 5.12]	0
$f 12 (x) = 20 + e - 20 e - 0.2 1 n ∑ i = 1 n x i 2 - e 1 n ∑ i = 1 n c o s (2 π x i)$	[-32, 32]	0

Table 1

Table 2

Parameters set of algorithms

算法名称	参数和设定值
PSO^[31]	惯性常数: 0.9~0.4, 第一加速度系数: 1.496 2, 第二加速度系数: 1.496 2
SA_GA^[32]	选择概率: 0.6, 交叉概率: 0.7, 变异概率: 0.05, 初始温度: 100, 温度递减系数: 0.98
DE^[33]	变异概率: 0.6, 交叉概率: 0.09
DA^[1]	s 为分离权重: 0~0.2, a 为对齐权重: 0~0.2,c 为凝聚权重: 0~0.2, f 为食物因子: 0~2,e 为敌情因子: 0~0.1, $ω$ 为惯性权重: 0.9~0.4
BSO^[34]	集群数量: 10, 集群的选择概率: 0.8, 集群中心的选择替换概率: 0.2, 使用中心的概率: 0.4
MFO^[35]	定义对数螺旋形状的常数: b=1, 随机数: t = -1~1
GSA^[36]	在第一个宇宙量子区间的引力常数: G₀=100, alfa=20，个体与个体间的引力之比为2
SCA^[5]	随机数组: r₁=0~2, r₂=0~2π, r₃=0~2, r₄=0~1
SSA^[6]	随机数组: r₁=0~2, r₂=0~1, r₃=0~1
MVO^[7]	虫洞存在概率: WEP_max =1, WEP_min=0.2, 距离移动率: TDR=0~1, 随机数组: r₁=0~1,r₂=0~1, r₃=0~1
SOA^[8]	最大隶属度值: 0.95, 最小隶属度值: 0.011 1, 最大惯性权重值: 0.8, 最小惯性权重值: 0.2
SOA-SSA	最大隶属度值: 0.95, 最小隶属度值: 0.011 1, 最大惯性权重值: 0.9, 最小惯性权重值: 0.1.随机数组: r₁=0~2, r₂=0~1, r₃=0~1

Table 2

Table 3

Comparison of performance of algorithms in benchmark function of 30 independent runs in high dimension

函数	性能	PSO	SA_GA	DE	DA	BSO	MFO	GSA	SCA	MVO	SSA	SOA	SOA-SSA
f₁	平均值	9.7e+3	2.6e+6	1.3e+6	3.7e+5	1.5e+5	2.5e+6	9.8e+4	3.5e+5	3.6e+5	1.9e+5	8.4e+3	2.1e+3
	标准差	1.6e+3	5.7e+4	3.4e+4	1.8e+5	1.6e+4	5.6e+4	5.0e+3	1.4e+5	1.6e+4	9.6e+3	1.6e+3	2.9e+3
	最差适应度	1.4e+4	2.7e+6	1.4e+6	7.9e+5	1.9e+5	2.6e+6	1.1e+5	7.7e+5	4.0e+5	2.1e+5	1.3e+4	1.2e+4
	最佳适应度	7.0e+3	2.5e+6	1.2e+6	1.0e+5	1.2e+5	2.4e+6	8.7e+4	4.0e+4	3.3e+5	1.7e+5	4.8e+3	12.4
	排名	3	12	10	6	7	11	5	4	9	8	2	1
f₂	平均值	Inf	Inf	Inf	1.0e+3	N/A	Inf	1.8e+281	Inf	1.3e+273	1.1e+3	7.8e+278	78.9
	标准差	N/A	N/A	N/A	3.3e+2	N/A	N/A	Inf	N/A	Inf	20.2	Inf	6.4
	最差适应度	Inf	Inf	Inf	1.7e+3	N/A	Inf	3.8e+282	Inf	4.0e+274	1.1e+3	2.3e+280	95.9
	最佳适应度	2.9e+2	Inf	Inf	3.1e+2	N/A	Inf	4.6e+244	Inf	2.3e+209	1.0e+3	2.1e+3	67.6
	排名	2	8	8	3	12	8	7	8	6	4	5	1
f₃	平均值	3.3e+6	3.3e+7	6.2e+8	2.3e+7	1.9e+6	1.4e+7	2.0e+6	2.4e+7	6.2e+6	4.2e+6	2.4e+6	7.3e+5
	标准差	1.4e+6	1.1e+7	1.9e+7	6.0e+6	4.7e+5	2.3e+6	8.2e+5	4.7e+6	4.3e+5	1.9e+6	7.7e+5	3.5e+5
	最差适应度	8.0e+6	8.3e+7	6.5e+8	3.8e+7	3.2e+6	1.8e+7	4.8e+6	3.5e+7	7.4e+6	8.5e+6	4.0e+6	1.7e+6
	最佳适应度	1.7e+6	2.0e+7	5.7e+8	1.1e+7	1.2e+6	9.2e+6	9.6e+5	1.4e+7	5.5e+6	1.5e+6	2.0e+5	3.8e+3
	排名	6	10	12	9	4	8	3	11	7	5	2	1
f₄	平均值	18.3	99.5	95.1	74.0	43.0	99.4	28.7	99.5	97.4	43.9	37.6	30.5
	标准差	1.3	0.2	0.3	14.4	3.4	0.2	1.8	0.1	0.7	3.0	8.8	8.1
	最差适应度	21.5	99.8	95.6	88.6	50.5	99.8	33.7	99.7	98.8	53.5	44.9	39.5
	最佳适应度	16.4	99.0	94.3	19.5	36.6	99.0	25.9	99.2	95.66	39.5	12.1	2.5
	排名	3	10	8	4	6	11	5	12	9	7	2	1
f₅	平均值	2.7e+5	1.1e+10	4.0e+9	4.8e+8	8.6e+7	1.1e+10	1.7e+7	3.3e+9	6.8e+8	7.5e+7	1.0e+8	1.8e+5
	标准差	1.5e+5	3.3e+8	1.5e+8	3.9e+8	1.5e+7	3.6e+8	1.7e+6	7.0e+8	7.4e+7	8.9e+6	2.5e+7	1.9e+5
	最差适应度	7.5e+5	1.3e+10	4.2e+9	1.5e+9	1.2e+8	1.2e+10	2.2e+7	4.7e+9	8.4e+8	1.1e+8	1.5e+8	1.0e+6
	最佳适应度	1.2e+5	1.1e+10	3.5e+9	1.9e+7	5.7e+7	1.0e+10	1.5e+7	2.1e+9	5.5e+8	6.0e+7	6.3e+7	2.6e+4
	排名	2	12	10	4	5	11	3	9	8	6	7	1
f₆	平均值	1.0e+4	2.6e+6	1.3e+6	2.9e+5	1.6e+5	2.5e+6	9.9e+4	4.1e+5	3.6e+5	1.9e+5	8.0e+3	7.2e+2
	标准差	1.4e+3	4.1e+4	3.11e+4	1.2e+5	1.8e+4	4.5e+4	4.9e+3	1.3e+5	1.8e+4	1.0e+4	1.4e+3	1.2e+3
	最差适应度	1.4e+4	2.6e+6	1.4e+6	5.6e+5	2.0e+5	2.6e+6	1.1e+5	7.4e+5	4.0e+5	2.1e+5	1.2e+4	4.4e+3
	最佳适应度	6.2e+3	2.5e+6	1.2e+6	6.2e+4	1.2e+5	2.4e+6	9.1e+4	1.6e+5	3.2e+5	1.7e+5	4.8e+3	1.6e+2
	排名	3	12	10	4	6	11	5	7	9	8	2	1
f₇	平均值	1.1e+2	1.8e+5	5.2e+4	7.9e+3	6.6e+3	1.7e+5	5.3e+3	4.8e+4	8.7e+3	1.1e+3	2.0e+4	3.3e+2
	标准差	22.3	6.1e+3	2.9e+3	5.9e+3	2.1e+3	5.9e+3	5.8e+2	1.1e+4	771.6	129.1	2.7e+3	54.4
	最差适应度	1.7e+2	1.9e+5	5.9e+4	2.0e+4	1.3e+4	1.8e+5	6.9e+3	7.0e+4	1.1e+4	1.34e+3	2.6e+4	4.6e+2
	最佳适应度	81.0	1.7e+5	4.7e+4	667.2	3.2e+3	1.6e+5	4.4e+3	2.7e+4	7.3e+3	878.7	1.4e+4	2.1e+2
	排名	1	12	10	3	5	11	6	9	7	4	8	2
f₈	平均值	4.7e+4	1.3e+7	6.2e+6	1.9e+6	1.4e+6	1.2e+7	3.9e+5	1.8e+6	1.5e+6	8.6e+5	1.6e+6	6.4e+3
	标准差	6.6e+3	3.6e+5	1.9e+5	9.6e+5	1.6e+5	3.0e+5	2.2e+4	5.6e+5	6.5e+4	5.2e+4	1.1e+5	1.7e+3
	最差适应度	6.1e+4	1.3e+7	6.9e+6	3.8e+6	1.6e+6	1.26e+7	4.4e+5	2.7e+6	1.7e+6	9.9e+5	1.9e+6	1.1e+4
	最佳适应度	3.7e+4	1.1e+7	6.0e+6	6.5e+5	1.01e+6	1.1e+7	3.5e+5	4.5e+5	1.4e+6	7.9e+5	1.3e+6	3.2e+3
	排名	2	12	10	5	7	11	3	4	9	6	8	1
f₉	平均值	9.8e-9	1.6e+92	5.0e+55	4.8e+27	N/A	2.0e+90	7.7e-5	9.1e+83	1.0e+56	1.7e-6	1.4e+12	6.1e-8
	标准差	3.e-8	7.4e+92	1.0e+56	2.6e+28	N/A	1.1e+91	2. 6e-4	4.7e+84	5.5e+56	1.3e-6	5.5e+12	5.9e-8
	最差适应度	1.5e-7	4.0e+93	4.1e+56	1.4e+29	N/A	5.9e+91	1.3e-3	2.6e+85	3.0e+57	5.3e-6	2.6e+13	2.8e-7
	最佳适应度	5.5e-16	1.1e+7	1.5e+48	4.8e-4	N/A	2.3e+74	2.0e-9	9.3e+69	5.6e+38	4.1e-7	2.4e-2	7.7e-9
	排名	1	7	9	5	12	11	2	10	8	4	6	3
f₁₀	平均值	-1.7e+4	-5.8e+4	-6.0e+4	-3.2e+4	-6.6e+4	-1.1e+5	-1.5e+4	-2.3e+4	-1.3e+5	-1.2e+5	-1.3e+5	-1.6e+5
	标准差	2.6e+3	3.4e+3	1.7e+3	4.9e+3	4.3e+3	8.8e+3	2.3e+3	1.4e+3	6.0e+3	6.9e+3	2.7e+4	5.5e+4
	最差适应度	-1.0e+4	-5.3e+4	-5.7e+4	-2.4e+4	-5.5e+4	-9.4e+4	-1.1e+4	-2.0e+4	-1.2e+5	-1.18e+5	-9.1e+4	-1.1e+5
	最佳适应度	-2.2e+4	-6.7e+4	-6.5e+4	-4.2e+4	-7.4e+4	-1.3e+5	-1.8e+4	-2.7e+4	-1.5e+5	-1.4e+5	-2.0e+5	-3.7e+5
	排名	11	7	8	9	6	5	12	10	3	4	2	1
f₁₁	平均值	2.8e+3	1.6e+4	1.3e+4	94e+3	9.4e+3	1.5e+4	5.8e+3	1.8e+3	1.4e+4	6.3e+3	10.0e+3	1.3e+3
	标准差	1.7e+2	1.4e+2	84.5	1.2e+3	2.0e+2	2.4e+2	1.6e+2	8.7e+2	3.0e+2	1.6e+2	3.1e+2	3.8e+2
	最差适应度	3.3e+3	1.6e+4	1.3e+4	1.1e+4	9.8e+3	1.5e+4	6.0e+3	3.9e+3	1.4e+4	6.7e+3	1.1e+4	2.2e+3
	最佳适应度	2.5e+3	1.5e+4	1.3e+4	7.0e+3	9.1e+3	1.4e+4	5.4e+3	5.0e+2	1.3e+4	6.1e+3	9.4e+3	8.4e+2
	排名	3	12	9	6	7	11	4	1	10	5	8	2
f₁₂	平均值	4.5	20.8	19.8	15.5	14.4	20.2	10.3	18.9	20.9	14.4	10.4	4.6
	标准差	0.2	0.03	0.06	1.9	0.4	0.2	0.2	4.0	0.02	0.2	0.6	3.9
	最差适应度	5.1	20.9	19.9	18.7	15.2	20.5	10.6	20.8	21.0	14.8	11.1	11.5
	最佳适应度	4.2	20.7	19.7	11.4	13.5	20.0	9.9	8.5	20.9	14.0	9.2	0.8
	排名	2	11	9	6	7	10	5	3	12	8	4	1
平均排名		3.0	10.7	9.5	5.4	6.8	10.2	5.1	7.4	8.1	5.7	4.4	1.4
总体排名		2	12	10	6	7	11	5	8	9	4	3	1

Table 3

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Table 4

Fig. 6

References 37

1	Mirjalili Seyedali. Dragon fly Algorithm: A New Metaheuristic Optimization Technique for Solving Single-Objective, Discrete, and Multi-Objective Problems[J]. Neural Computing and Applications(S0941-0643), 2015, 29: 1-21.
2	Shi Y. An Optimization Algorithm Based on Brainstorming Process[J]. International Journal of Swarm Intelligence Research(S1947-9263), 2011, 2(4): 35-62.
3	Mirjalili Seyedali. Moth Flame Optimization Algorithm: A Novel Nature-inspired Heuristic Paradigm[J]. Knowledge-Based Systems(S0950-7051), 2015, 89: 228-249.
4	Rashedi E, Nezamabadi-Pour H, Saryazdi S. GSA: A Gravitational Search Algorithm[J]. Information Sciences(S0020-025), 2009, 179(13): 2232-2248.
5	Mirjalilia Seyedali. SCA: A Sine Cosine Algorithm for Solving Optimization Problems[J]. Knowledge-Based Systems(S0950-7051), 2016, 96: 120-133.
6	Mirjalili S, Mirjalili S M, Hatamlou A. Multi-Verse Optimizer: A Nature-Inspired Algorithm for Global Optimization[J]. Neural Computing and Applications(S0941-0643), 2016, 27: 495-513.
7	Mirjalili Seyedal, Gandomi Amir H, Seyedeh Zahra Mirjalili. Salp Swarm Algorithm: A Bio-inspired Optimizer for Engineering Design Problems[J]. Advances in Engineering Software(S0965-9978), 2017, 114: 163-191.
8	Tuba M, Brajevic I, Jovanovic R. Hybrid Seeker Optimization Algorithm for Global Optimization[J]. Applied Mathematics & Information Sciences(S1935-0090), 2013, 7(3): 867-875.
9	余玉丰, 李国, 徐晨. 基于分组的PSO与DE的混合算法[J]. 计算机应用研究, 2014, 31(8): 2294-2298.
	Yu Yufeng, Li Guo, Xu Chen. Hybrid Algorithm Based on Grouped PSO and DE[J]. Application Research of Computers, 2014, 31(8): 2294-2298.
10	白书华, 况明星. 基于PSO-GA-BP神经网络的林火预测设计与研究[J]. 系统仿真学报, 2018, 30(5): 1739-1748.
	Bai Shuhua, Kuang Mingxing. Design and Study of Forest Fire Forecasting Based on PSO-GA-BP Neural Network[J]. Journal of System Simulation, 2018, 30(5): 1739-1748.
11	王修岩, 刘艳敏, 张革文, 等. 基于PSO和CRO联合算法的飞机客舱能耗预测[J]. 系统仿真学报, 2018, 30(8): 3074-3081.
	Wang Xiuyan, Liu Yanmin, Zhang Gewen, et al. Prediction of Aircraft Cabin Energy Consumption Based on PSO and CRO Algorithms[J]. Journal of System Simulation, 2018, 30(8): 3074-3081.
12	卿东升, 张晓芳, 李建军, 等. 基于蜂群-粒子群算法的天然林空间结构优化[J]. 系统仿真学报, 2020, 32(3): 371-381.
	Dongsheng Qing, Zhang Xiaofang, Li Jianjun, et al. Spatial Structure Optimization of Natural Forest Based on Bee Colony-particle Swarm Algorithm[J]. Journal of System Simulation, 2020, 32(3): 371-381.
13	Pitakaso Rapeepan, Sethanan Kanchana, Jamrus Thitipong. Hybrid PSO and ALNS Algorithm for Software and Mobile Application for Transportation in Ice Manufacturing Industry 3.5[J]. Computers & Industrial Engineering (S0360-8352), 2020, 144: 1-13.
14	Dai C, Zhu Y, Chen W. Seeker Optimization Algorithm[C]// 2006 Inter. Conf. Computational Intelligence and Security. Guangzhou: IEEE Press, 2006: 225-229.
15	Dai C, Zhu Y, Chen W. Seeker Optimization Algorithm[J]. Lecture Notes in Computer Science (S0302-9743), 2007, 4456: 167-176.
16	Dai C, Chen W, Zhu Y, et al. Seeker Optimization Algorithm for Optimal Reactive Power Dispatch[J]. IEEE Transactions on Power Systems (S0885-8950), 2009, 24(3): 1218-1231.
17	Dai C, Chen W, Song Y, et al. Seeker Optimization Algorithm: A novel Stochastic Search Algorithm for Global Numerical Optimization[J]. Journal of Systems Engineering and Electronics(S1004-4132), 2010, 21(2): 300-311.
18	Dai C, Chen W, Zhu Y, Seeker Optimization Algorithm for Digital IIR Filter Design[J]. IEEE Transactions on Industrial Electronics(S0278-0046), 2010, 57(5): 1710-1718.
19	Dai C, Chen W, Zhu Y, et al. Seeker Optimization Algorithm for Tuning the Structure and Parameters of Neural Networks[J]. Neurocomputing (S0925-2312), 2011, 74(6): 876-883.
20	Dai C, Cheng Z, Li Q, et al. Seeker Optimization Algorithm for Global Optimization: A Case Study on Optimal Modelling of Proton Exchange Membrane Fuel Cell (PEMFC)[J]. International Journal of Electrical Power and Energy Systems (S0142-0615), 2011, 33(3): 369-376.
21	Dai C, Chen W, Ran L, et al. Human Group Optimizer with Local Search[J]. Lecture Notes in Computer Science(S0302-9743), 2011, 6728: 310-320.
22	Zhu Y, Dai C, Chen W. Seeker Optimization Algorithm for Several Practical Applications[J]. International Journal of Computational Intelligence Systems (S1875-6891), 2014, 7(2): 1-15.
23	杨博, 钟林恩, 朱德娜, 等. 部分遮蔽下改进樽海鞘群算法的光伏系统最大功率跟踪[J]. 控制理论与应用, 2019, 36(3): 339-352.
	Yang Bo, Zhong Linen, Zhu Dena, et al. Modified Salp Swarm Algorithm Based Maximum Power Point Tracking of Power-voltage System Under Partial Shading Condition[J]. Control Theory & Applications, 2019, 36(3): 339-352.
24	王梦秋, 王艳, 纪志成. 基于改进樽海鞘群算法的PMSM多参数辨识[J]. 系统仿真学报, 2018,30(11): 4284-4291, 4297.
	Wang Mengqiu, Wang Yan, Ji Zhicheng. Permanent Magnet Synchronous Motor Multi-parameter Identification Based on Improved Salp Swarm Algorithm[J]. Journal of System Simulation, 2018, 30(11): 4284-4291, 4297.
25	邢致恺, 贾鹤鸣, 宋文龙. 基于莱维飞行樽海鞘群优化算法的多阈值图像分割[J]. 自动化学报, 2021, 47(2): 363-377.
	Xing Zhikai, Jia Heming, Song Wenlong. Levy Flight Trajectory-Based Salp Swarm Algorithm for Multilevel Thresholding Image Segmentation[J]. Acta Automatica Sinica, 2021, 47(2): 363-377.
26	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 Tao, Wang Mengxin, Huang Xiangsong. Time Difference of Arrival Passive Location Based on Salp Swarm Algorithm[J]. Journal of Electronics & Information Technology, 2018, 40(7): 1591-1597.
27	Xia H, Wang Z, Zhou Y. Double-Population Differential Evolution Based on Logistic Model[J]. Journal of Information and Computational Science(S1548-7741), 2014, 11(15): 5549-5557.
28	Bao Z, Zhou Y, Li L. A Hybrid Global Optimization Algorithm Based on Wind Driven Optimization and Differential Evolution[J]. Mathematical Problems in Engineering(S1024-123X), 2015: No. 389630.
29	Li L, Zhou Y, Xie J. A Free Search Krill Herd Algorithm for Functions Optimization[J]. Mathematical Problems in Engineering(S1024-123X), 2014: No. 936374.
30	Li X, Zhang J, Yin M. Animal Migration Optimization: An Optimization Algorithm Inspired by Animal Migration Behavior[J]. Neural Computing and Applications (S0941-0643), 2014, 24(7/8): 1867-1877.
31	Kennedy J. Particle Swarm Optimization[M]// Encyclopedia of Machine Learning. New York: Springer, 2010: 760-766.
32	Yua H, Fang H, Yao P, et al. A Combined Genetic Algorithm/Simulated Annealing Algorithm for Large Scale System Energy Integration[J]. Computers & Chemical Engineering (S0098-1354), 2000, 24(8): 2023-2035.
33	Yildizdana Gülnur, Ömer Kaan Baykan. A novel Modified Bat Algorithm Hybridizing by Differential Evolution Algorithm[J]. Expert Systems with Applications(S0957-4174), 2019(141): No. 112949.
34	Li Chunquan, Luo Zu, Song Zhenshou, et a. An Enhanced Brain Storm Sine Cosine Algorithm for Global Optimization Problem [J]. IEEE Access(S2169-3536), 2019, 7: 28211-28229.
35	Li Chunquan, Niu Zheng, Song Zhenshou, et al. A Double Evolutionary Learning Moth-Flame Optimization for Real-Parameter Global Optimization Problems[J]. IEEE Access(S2169-3536), 2018, 6: 76700-76727.
36	Dowlatshahi M B, Nezamabadi-pour H, Mashinchi M. A Discrete Gravitational Search Algorithm for Solving Combinatorial Optimization Problems[J]. Information Sciences(S0020-0255), 2014, 258: 94-107.
37	Zhou Y, Bao Z, Luo Q, et al. A Complex-valued Encoding Wind Driven Optimization with Greedy Strategy for 0-1 Knapsack Problem[J]. Applied Intelligence(S0924-669X), 2017, 46: 684-702.

函数	性能	PSO	SA_GA	DE	DA	BSO	MFO	GSA	SCA	SSA	MVO	SOA	SOA-SSA
g₁	平均值	0.22	0.31	0.26	0.14	0.43	0.07	0.45	0.09	0.06	0.25	0.19	0.05
	标准差	0.08	0.06	0.09	0.08	0.14	0.02	0.15	0.02	0.05	0.05	0.11	0.00
	最差适应度	0.42	0.43	0.47	0.30	0.88	0.11	0.86	0.11	0.33	0.26	0.42	0.06
	最佳适应度	0.04	0.10	0.06	0.05	0.26	0.04	0.27	0.04	0.04	0.05	0.05	0.04
	排名	5	10	9	6	11	3	12	4	2	7	8	1
g₂	平均值	0.35	0.50	0.50	0.29	0.68	0.10	0.64	0.10	0.26	0.39	0.28	0.10
	标准差	0.20	0.10	0.09	0.21	0.15	4.7e-4	0.12	8.77e-4	0.21	0.19	0.18	0.00
	最差适应度	0.54	0.61	0.61	0.54	1.04	0.10	0.96	0.10	0.54	0.53	0.53	0.12
	最佳适应度	0.10	0.15	0.25	0.10	0.39	0.09	0.38	0.10	0.09	0.10	0.09	0.09
	排名	8	9	10	5	12	3	11	5	1	5	4	2
g₃	平均值	58.47	62.45	60.91	58.15	62.43	52.35	60.77	24.84	30.96	59.58	42.15	11.06
	标准差	7.75	0.12	4.01	9.55	0.80	16.91	5.30	21.52	29.63	7.65	27.90	18.68
	最差适应度	62.59	62.61	62.84	62.49	63.08	62.50	64.14	62.52	62.58	62.50	62.54	56.82
	最佳适应度	36.04	62.03	42.65	14.36	58.30	7.47	42.77	0.48	0.33	32.60	0.39	0.30
	排名	8	12	9	6	11	5	10	4	2	7	3	1
g₄	平均值	1.85e+2	2.72e+2	2.71e+2	1.35e+2	2.72e+2	39.95	2.77e+2	29.04	41.75	1.09e+2	2.62e+2	12.39
	标准差	59.64	0.62	0.37	71.50	5.5362	47.83	10.30	11.98	40.96	56.67	44.81	2.72
	最差适应度	3.15e+2	2.74e+2	2.72e+2	1.98e+2	2.80e+2	1.98e+2	3.16e+2	50.00	1.50e+2	1.96e+2	2.75e+2	18.36
	最佳适应度	32.54	2.71	2.70e+2	24.91	2.44e+2	13.17	2.71e+2	14.55	9.10	20.04	26.57	8.89
	排名	9	1	11	7	10	4	12	5	3	6	8	2
g₅	平均值	1.77e+2	55.35	93.24	81.65	41.44	35.07	2.34e+2	85.19	64.19	35.72	46.10	35.20
	标准差	4.22e+2	36.00	40.91	79.81	11.38	0.40	2.18e+2	1.01e+2	41.70	1.41	26.99	0.80
	最差适应度	2.13e+3	163.65	1.96e+2	3.65e+2	90.50	36.33	1.15e+3	4.20e+2	2.10e+2	41.28	1.85e+2	37.71
	最佳适应度	34.62	34.62	40.23	34.63	35.04	34.63	58.32	34.86	34.62	34.64	34.74	34.62
	排名	2	4	11	6	10	5	12	9	3	7	8	1
平均排名		6.4	7.2	10	6	10.8	4	11.4	5.4	2.2	6.4	6.2	1.4
总体排名		7	9	10	5	11	3	12	4	2	7	6	1

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