极值个体引导的人工蜂群算法

doi:10.3778/j.issn.1673-9418.2104105

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (11): 2628-2641.DOI: 10.3778/j.issn.1673-9418.2104105

极值个体引导的人工蜂群算法

陈兰¹, 王联国²^,⁺()

1.甘肃农业大学机电工程学院，兰州 730070
2.甘肃农业大学信息科学技术学院，兰州 730070

收稿日期:2021-04-29 修回日期:2021-07-02 出版日期:2022-11-01 发布日期:2021-07-15
通讯作者: + E-mail: wanglg@gsau.edu.cn
作者简介:陈兰（1995—），女，甘肃临洮人，硕士研究生，主要研究方向为智能信息处理。
王联国（1968—），男，甘肃临夏人，博士，教授，主要研究方向为计算智能、智能信息处理、农业信息技术。
基金资助:
国家自然科学基金(61751313);甘肃省教育信息化建设专项任务项目(2011-02);甘肃省重点研发计划项目(21YF5GA088)

Extreme Individual Guided Artificial Bee Colony Algorithm

CHEN Lan¹, WANG Lianguo²^,⁺()

1. College of Mechanical and Electrical Engineering, Gansu Agricultural University, Lanzhou 730070, China
2. College of Information Science and Technology, Gansu Agricultural University, Lanzhou 730070, China

Received:2021-04-29 Revised:2021-07-02 Online:2022-11-01 Published:2021-07-15
About author:CHEN Lan, born in 1995, M.S. candidate. Her re-search interest is intelligent information processing.
WANG Lianguo, born in 1968, Ph.D., professor. His research interests include computational in-telligence, intelligent information processing and agricultural information technology.
Supported by:
National Natural Science Foundation of China(61751313);Special Task Project of Education Informationization Construction in Gansu Province(2011-02);Key Research and Development Program of Gansu Province(21YF5GA088)

摘要/Abstract

摘要：

针对目前人工蜂群算法（ABC）在求解函数优化问题时存在开发能力差、易陷入局部最优、收敛速度慢等问题，提出了一种极值个体引导的人工蜂群算法（EABC）。首先，该算法在雇佣蜂和跟随蜂的搜索中利用全局极值个体和邻域极值个体引导搜索，全局极值个体引导搜索有利于种群中优良个体的保留和发展，使算法跳出局部极值，避免早熟收敛。邻域极值个体引导搜索有利于增强搜索精度，提高算法的收敛速度，并通过随机数 $r$ 平衡两种搜索机制。其次，在搜索过程中引入小概率变异算子，对蜜蜂个体的各维度以较小的概率进行变异，克服算法陷入局部极值并出现早熟收敛的现象。最后，采用基于目标函数值的贪婪选择策略，提高算法的优化性能；采用28个测试函数进行仿真实验，并与其他几种算法进行比较，实验结果表明改进算法具有较高的优化性能和较快的收敛速度。

关键词: 人工蜂群算法（ABC）, 极值个体引导, 小概率变异, 目标函数值

Abstract:

To overcome the drawbacks of poor development ability, easy to fall into local optimum, slow conver-gence speed of artificial bee colony (ABC) algorithm in solving function optimization problems, an extreme indi-vidual guided artificial bee colony (EABC) algorithm is proposed. Firstly, global extremum and neighborhood ex-tremum individuals are used to guide search for employed bees and following bees. The global extremum individual guided search is good for the retention and development of excellent individuals in the population, so that the algorithm jumps out of local extremum and avoids premature convergence. The neighborhood extremum individual guided search is good for enhancing the search accuracy and improving the convergence speed of the algorithm, and the random number $r$ is used to balance two search mechanisms. Secondly, the small probability mutation operator is introduced into search process, and each dimension of bee individual is mutated with a small probability to overcome local extremum and premature convergence of the algorithm. Finally, the greedy selection strategy based on the value of objective function is adopted to improve the optimization performance of the algorithm. Simulation experiments are carried out with 28 test functions and the algorithm proposed in this paper is compared with other algorithms. Experimental results show that the improved algorithm has higher optimization performance and faster convergence speed.

Key words: artificial bee colony (ABC) algorithm, extreme individual guidance, small probability mutation, ob-jective function value

中图分类号:

TP301.6

陈兰, 王联国. 极值个体引导的人工蜂群算法[J]. 计算机科学与探索, 2022, 16(11): 2628-2641.

CHEN Lan, WANG Lianguo. Extreme Individual Guided Artificial Bee Colony Algorithm[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(11): 2628-2641.

图/表 7

表1 28个标准测试函数参数

Table 1 Parameters of 28 standard test functions

Function	Function name		D	C	Searching space	Function expression
$f 1$	Sphere		30	US	[-100,100]	$f 1 (x) = ∑ i = 1 n x i 2$
$f 2$	Elliptic		30	UN	[-100,100]	$f 2 (x) = ∑ i = 1 n (106) (i - 1) / (n - 1) x i 2$
$f 3$	SumSquares		30	US	[-10,10]	$f 3 (x) = ∑ i = 1 n i x i 2$
$f 4$	SumPower		30	MS	[-10,10]	$f 4 (x) = ∑ i = 1 n \| x i \| (i + 1)$
$f 5$	Schwefel2.22		30	UN	[-10,10]	$f 5 (x) = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$
$f 6$	Schwefel2.21		30	UN	[-100,100]	$f 6 (x) = m a x i {\| x i \|, 1 ≤ i ≤ n}$
$f 7$	Step		30	US	[-100,100]	$f 7 (x) = ∑ i = 1 n x i + 0.5 2$
$f 8$	Quartic		30	US	[-1.28,1.28]	$f 8 (x) = ∑ i = 1 n i x i 4$
$f 9$	QuarticWN		30	US	[-1.28,1.28]	$f 9 (x) = ∑ i = 1 n i x i 4 + r a n d o m [0,1)$
$f 10$	Rosebrock		30	UN	[-10,10]	$f 10 (x) = ∑ i = 1 n - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$
$f 11$	Rastrigin		30	MS	[-5.12,5.12]	$f 11 (x) = ∑ i = 1 n - 1 [x i 2 - 10 c o s (2 π x i) + 10]$
$f 12$	Non_Continuous Rastrigin		30	MS	[-5.12,5.12]	$f 12 (x) = ∑ i = 1 n - 1 [y i 2 - 10 c o s (2 π y i) + 10]$ , $y i = x i, \| x i \| < 12 r o u n d (2 x i) 2, \| x i \| ≥ 12$
$f 13$	Griewank		30	MN	[-600,600]	$f 13 (x) = 1 4 000 ∑ i = 1 n - 1 x i 2 - ∏ i = 1 n c o s x i i + 1$
$f 14$	Schwefel2.26		30	UN	[-500,500]	$f 14 (x) = 418.98 × n - ∑ i = 1 n x i s i n \| x i \|$
$f 15$	Ackley		30	MN	[-32,32]	$f 15 (x) = - 20 e x p - 0.2 1 n ∑ i = 1 n x i 2 - e x p 1 n c o s (2 π x i) + 20 + e$
$f 16$	Penalized1		30	MN	[-50,50]	$f 16 (x) = π n 10 s i n 2 (π y 1) + ∑ i = 1 n - 1 (y i - 1) 2 [1 + 10 s i n 2 (π y i + 1)] + (y n - 1) 2 + ∑ u (x i, 10,100,4)$ $y i = 1 + 14 (x i + 1)$ , $u x i, a, k, m = k (x i - a) m, x i > a 0, - a ≤ x i ≤ a k (x i - a) m, x i < a$
$f 17$	Penalized2		30	MN	[-50,50]	$f 17 (x) = 110 s i n 2 (π x 1) + ∑ i = 1 n - 1 (x i - 1) 2 [1 + s i n 2 (3 π x i + 1)] + (x n - 1) 2 [1 + s i n 2 (2 π x i + 1)] + ∑ i = 1 n u (x i, 5,100,4)$
$f 18$	Alpine		30	MS	[-10,10]	$f 18 (x) = ∑ i = 1 n \| x i s i n x i + 0.1 × x i \|$
$f 19$	Levy		30	MN	[-10,10]	$f 19 (x) = ∑ i = 1 n - 1 (x i - 1) 2 [1 + s i n 2 (3 π x i + 1)] + s i n 2 (3 π x 1) + \| x n - 1 \| [1 + s i n 2 (3 π x n)]$
$f 20$	Weierstrass		30	MN	[-0.5,0.5]	$f 20 (x) = ∑ i = 1 D ∑ k = 0 k m a x [a k + c o s (2 π b k (x i + 0.5))] - D ∑ k = 0 k m a x [a k c o s (2 π b k 0.5)], a = 0.5, b = 3, k m a x = 20$
$f 21$	Schaffer		30	MN	[-100,100]	$f 21 (x) = 0.5 + s i n ∑ i = 1 n x i 2 - 0.5 1 + 0.001 × ∑ i = 1 n x i 2 2$
$f 22$	Himmelblau		30	MS	[-5,5]	$f 22 (x) = 1 n ∑ i = 1 n - 1 (x i 4 - 16 x i 2 + 5 x i 2)$
$f 23$	Michalewicz		30	MS	[0,π]	$f 23 (x) = - ∑ i = 1 n - 1 s i n x i s i n 20 i × x i 2 π$
$f 24$	Shifted_Sphere		30	US	[-100,100]	$f 24 (x) = ∑ i = 1 n z i 2, z = x - o$
Function		Function name	D	C	Searching space	Function expression
$f 25$		Shifted_Rastrigin	30	MS	[-5.12,5.12]	$f 25 (x) = ∑ i = 1 n [z i 2 - 10 c o s (2 π z i) + 10], z = x - o$
$f 26$		Shifted_Griewank	30	MN	[-600,600]	$f 26 (x) = 1 4 000 ∑ i = 1 n - 1 z i 2 - ∏ i = 1 n c o s z i i + 1, z = x - o$
$f 27$		Shifted_Ackley	30	MN	[-32,32]	$f 27 (x) = - 20 e x p - 0.2 1 n ∑ i = 1 n z i 2 - e x p 1 n c o s (2 π z i), z = x - o$
$f 28$		Shifted_Alpine	30	MN	[-10,10]	$f 28 (x) = ∑ i = 1 n \| z i × s i n z i + 0.1 × z i \|, z = x - o$

表1 28个标准测试函数参数

Table 1 Parameters of 28 standard test functions

Function	Function name		D	C	Searching space	Function expression
$f 1$	Sphere		30	US	[-100,100]	$f 1 (x) = ∑ i = 1 n x i 2$
$f 2$	Elliptic		30	UN	[-100,100]	$f 2 (x) = ∑ i = 1 n (106) (i - 1) / (n - 1) x i 2$
$f 3$	SumSquares		30	US	[-10,10]	$f 3 (x) = ∑ i = 1 n i x i 2$
$f 4$	SumPower		30	MS	[-10,10]	$f 4 (x) = ∑ i = 1 n \| x i \| (i + 1)$
$f 5$	Schwefel2.22		30	UN	[-10,10]	$f 5 (x) = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$
$f 6$	Schwefel2.21		30	UN	[-100,100]	$f 6 (x) = m a x i {\| x i \|, 1 ≤ i ≤ n}$
$f 7$	Step		30	US	[-100,100]	$f 7 (x) = ∑ i = 1 n x i + 0.5 2$
$f 8$	Quartic		30	US	[-1.28,1.28]	$f 8 (x) = ∑ i = 1 n i x i 4$
$f 9$	QuarticWN		30	US	[-1.28,1.28]	$f 9 (x) = ∑ i = 1 n i x i 4 + r a n d o m [0,1)$
$f 10$	Rosebrock		30	UN	[-10,10]	$f 10 (x) = ∑ i = 1 n - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$
$f 11$	Rastrigin		30	MS	[-5.12,5.12]	$f 11 (x) = ∑ i = 1 n - 1 [x i 2 - 10 c o s (2 π x i) + 10]$
$f 12$	Non_Continuous Rastrigin		30	MS	[-5.12,5.12]	$f 12 (x) = ∑ i = 1 n - 1 [y i 2 - 10 c o s (2 π y i) + 10]$ , $y i = x i, \| x i \| < 12 r o u n d (2 x i) 2, \| x i \| ≥ 12$
$f 13$	Griewank		30	MN	[-600,600]	$f 13 (x) = 1 4 000 ∑ i = 1 n - 1 x i 2 - ∏ i = 1 n c o s x i i + 1$
$f 14$	Schwefel2.26		30	UN	[-500,500]	$f 14 (x) = 418.98 × n - ∑ i = 1 n x i s i n \| x i \|$
$f 15$	Ackley		30	MN	[-32,32]	$f 15 (x) = - 20 e x p - 0.2 1 n ∑ i = 1 n x i 2 - e x p 1 n c o s (2 π x i) + 20 + e$
$f 16$	Penalized1		30	MN	[-50,50]	$f 16 (x) = π n 10 s i n 2 (π y 1) + ∑ i = 1 n - 1 (y i - 1) 2 [1 + 10 s i n 2 (π y i + 1)] + (y n - 1) 2 + ∑ u (x i, 10,100,4)$ $y i = 1 + 14 (x i + 1)$ , $u x i, a, k, m = k (x i - a) m, x i > a 0, - a ≤ x i ≤ a k (x i - a) m, x i < a$
$f 17$	Penalized2		30	MN	[-50,50]	$f 17 (x) = 110 s i n 2 (π x 1) + ∑ i = 1 n - 1 (x i - 1) 2 [1 + s i n 2 (3 π x i + 1)] + (x n - 1) 2 [1 + s i n 2 (2 π x i + 1)] + ∑ i = 1 n u (x i, 5,100,4)$
$f 18$	Alpine		30	MS	[-10,10]	$f 18 (x) = ∑ i = 1 n \| x i s i n x i + 0.1 × x i \|$
$f 19$	Levy		30	MN	[-10,10]	$f 19 (x) = ∑ i = 1 n - 1 (x i - 1) 2 [1 + s i n 2 (3 π x i + 1)] + s i n 2 (3 π x 1) + \| x n - 1 \| [1 + s i n 2 (3 π x n)]$
$f 20$	Weierstrass		30	MN	[-0.5,0.5]	$f 20 (x) = ∑ i = 1 D ∑ k = 0 k m a x [a k + c o s (2 π b k (x i + 0.5))] - D ∑ k = 0 k m a x [a k c o s (2 π b k 0.5)], a = 0.5, b = 3, k m a x = 20$
$f 21$	Schaffer		30	MN	[-100,100]	$f 21 (x) = 0.5 + s i n ∑ i = 1 n x i 2 - 0.5 1 + 0.001 × ∑ i = 1 n x i 2 2$
$f 22$	Himmelblau		30	MS	[-5,5]	$f 22 (x) = 1 n ∑ i = 1 n - 1 (x i 4 - 16 x i 2 + 5 x i 2)$
$f 23$	Michalewicz		30	MS	[0,π]	$f 23 (x) = - ∑ i = 1 n - 1 s i n x i s i n 20 i × x i 2 π$
$f 24$	Shifted_Sphere		30	US	[-100,100]	$f 24 (x) = ∑ i = 1 n z i 2, z = x - o$
Function		Function name	D	C	Searching space	Function expression
$f 25$		Shifted_Rastrigin	30	MS	[-5.12,5.12]	$f 25 (x) = ∑ i = 1 n [z i 2 - 10 c o s (2 π z i) + 10], z = x - o$
$f 26$		Shifted_Griewank	30	MN	[-600,600]	$f 26 (x) = 1 4 000 ∑ i = 1 n - 1 z i 2 - ∏ i = 1 n c o s z i i + 1, z = x - o$
$f 27$		Shifted_Ackley	30	MN	[-32,32]	$f 27 (x) = - 20 e x p - 0.2 1 n ∑ i = 1 n z i 2 - e x p 1 n c o s (2 π z i), z = x - o$
$f 28$		Shifted_Alpine	30	MN	[-10,10]	$f 28 (x) = ∑ i = 1 n \| z i × s i n z i + 0.1 × z i \|, z = x - o$

图1 两种贪婪选择策略收敛曲线

Fig.1 Convergence curves of two greedy selection strategies

表2 两种算法的实验结果

Table 2 Experimental results of two optimization algorithms

Function	Algorithm	The worst value	Optimal value	Average optimal value	Standard deviation	Average running time/s
$f 1$	ABC	1.251 4E-016	1.251 4E-017	8.710 8E-017	2.806 4E-017	0.028 7
$f 1$	EABC	4.107 4E-118	2.752 4E-121	6.086 2E-119	9.813 2E-119	0.021 3
$f 2$	ABC	1.405 4E-016	4.605 3E-017	9.196 0E-017	3.008 0E-017	0.328 0
$f 2$	EABC	1.237 7E-114	6.297 8E-118	9.041 5E-116	2.311 8E-115	0.331 1
$f 3$	ABC	1.332 5E-016	4.579 0E-017	8.331 8E-017	2.789 2E-017	0.028 9
$f 3$	EABC	7.500 9E-119	5.381 9E-122	7.332 3E-120	1.475 3E-119	0.032 4
$f 4$	ABC	9.285 9E-018	2.443 5E-018	5.656 9E-018	2.106 9E-018	0.369 1
$f 4$	EABC	2.637 4E-047	6.695 1E-079	8.791 4E-049	4.734 3E-048	0.384 6
$f 5$	ABC	3.448 9E-016	1.920 4E-016	2.497 6E-016	3.828 5E-017	0.187 4
$f 5$	EABC	2.402 5E-060	1.066 4E-062	2.920 6E-061	4.225 7E-061	0.160 7
$f 6$	ABC	1.089 4E-001	1.647 5E-002	5.440 0E-002	2.169 1E-002	0.085 0
$f 6$	EABC	7.262 9E+000	3.160 0E+000	5.011 3E+000	8.501 2E-001	0.093 4
$f 7$	ABC	1.336 9E-016	5.618 4E-017	9.417 8E-017	2.841 7E-017	0.028 6
$f 7$	EABC	0	0	0	0	0.039 4
$f 8$	ABC	2.332 1E-017	9.192 5E-018	1.481 3E-017	3.050 3E-018	0.030 2
$f 8$	EABC	9.110 7E-238	1.676 0E-244	6.633 0E-239	0	0.044 1
$f 9$	ABC	6.688 5E-002	2.047 8E-002	4.648 1E-002	1.180 2E-002	0.031 9
$f 9$	EABC	3.054 4E-002	7.575 2E-003	1.969 7E-002	5.576 0E-003	0.044 8
$f 10$	ABC	8.707 8E+000	2.227 4E-004	1.402 5E-001	2.357 1E-001	1.344 0
$f 10$	EABC	7.413 6E+000	1.077 9E-004	2.698 0E-001	1.327 4E+000	0.971 8
$f 11$	ABC	0	0	0	0	0.468 9
$f 11$	EABC	0	0	0	0	0.471 4
$f 12$	ABC	0	0	0	0	0.338 6
$f 12$	EABC	0	0	0	0	0.297 8
$f 13$	ABC	1.968 9E-010	0	2.216 3E-011	5.255 7E-010	0.257 7
$f 13$	EABC	1.722 4E-011	0	5.825 1E-013	3.090 7E-012	0.243 0
$f 14$	ABC	8.661 8E-10	8.661 8E-010	8.661 8E-010	6.183 4E-010	0.300 3
$f 14$	EABC	5.784 3E-012	8.789 7E-013	1.432 5E-012	5.457 0E-013	0.279 6
$f 15$	ABC	7.694 1E-015	4.141 4E-015	6.509 9E-014	1.674 8E-015	0.226 2
$f 15$	EABC	3.611 5E-014	1.124 6E-014	2.048 4E-014	5.549 5E-015	0.214 4
$f 16$	ABC	1.391 0E-016	4.020 5E-017	9.379 2E-017	2.843 3E-017	0.470 9
$f 16$	EABC	1.570 4E-032	1.570 4E-032	1.570 4E-032	2.736 9E-048	0.418 8
$f 17$	ABC	1.298 0E-016	4.404 4E-017	8.927 4E-017	2.871 6E-017	0.460 7
$f 17$	EABC	1.349 6E-032	1.349 6E-032	1.349 7E-032	5.473 8E-048	0.422 0
$f 18$	ABC	7.261 7E-007	6.644 4E-016	2.453 2E-008	1.302 9E-007	0.268 6
$f 18$	EABC	1.075 0E-014	5.200 3E-049	9.708 1E-016	2.236 1E-015	0.242 0
$f 19$	ABC	1.554 5E-016	8.695 5E-017	1.224 5E-016	1.615 1E-17	1.324 7
$f 19$	EABC	2.625 5E-032	2.416 8E-032	1.216 6E-032	1.763 8E-048	1.112 6
$f 20$	ABC	0	0	0	0	1.009 8
$f 20$	EABC	0	0	0	0	0.756 8
$f 21$	ABC	3.960 9E-001	1.269 9E-001	3.095 5E-001	5.028 9E-002	0.051 3
$f 21$	EABC	3.455 9E-001	1.782 2E-001	2.679 8E-001	5.536 2E-002	0.064 6
$f 22$	ABC	-7.738 9E+001	-7.833 2E+001	-7.830 1E+001	1.691 7E-001	0.656 7
$f 22$	EABC	-7.833 2E+001	-7.833 2E+001	-7.833 2E+001	1.797 5E-014	0.629 0
$f 23$	ABC	-2.859 8E+001	-2.863 0E+001	-2.861 8E+001	8.494 8E-003	0.875 7
$f 23$	EABC	-2.857 0E+001	-2.863 0E+001	-2.861 8E+002	1.728 0E-002	0.843 0
$f 24$	ABC	8.474 8E-016	4.825 6E-017	6.426 4E-016	8.348 0E-017	0.186 3
$f 24$	EABC	0	0	0	0	0.113 3
$f 25$	ABC	0	0	0	0	0.303 1
$f 25$	EABC	0	0	0	0	0.285 0
$f 26$	ABC	1.876 8E-009	0	2.535 5E-010	5.525 5E-010	0.365 3
$f 26$	EABC	1.765 4E-011	0	5.876 9E-012	3.523 5E-012	0.252 3
$f 27$	ABC	7.694 1E-014	4.141 4E-015	6.809 5E-014	5.674 8E-015	0.226 2
$f 27$	EABC	3.611 5E-014	1.124 6E-014	2.048 4E-014	5.549 5E-015	0.214 4
$f 28$	ABC	8.592 2E-009	5.443 6E-009	7.289 5E-009	7.691 0E-010	0.594 7
$f 28$	EABC	7.867 7E-015	5.384 5E-016	6.883 8E-017	6.553 6E-017	0.291 4

表2 两种算法的实验结果

Table 2 Experimental results of two optimization algorithms

Function	Algorithm	The worst value	Optimal value	Average optimal value	Standard deviation	Average running time/s
$f 1$	ABC	1.251 4E-016	1.251 4E-017	8.710 8E-017	2.806 4E-017	0.028 7
$f 1$	EABC	4.107 4E-118	2.752 4E-121	6.086 2E-119	9.813 2E-119	0.021 3
$f 2$	ABC	1.405 4E-016	4.605 3E-017	9.196 0E-017	3.008 0E-017	0.328 0
$f 2$	EABC	1.237 7E-114	6.297 8E-118	9.041 5E-116	2.311 8E-115	0.331 1
$f 3$	ABC	1.332 5E-016	4.579 0E-017	8.331 8E-017	2.789 2E-017	0.028 9
$f 3$	EABC	7.500 9E-119	5.381 9E-122	7.332 3E-120	1.475 3E-119	0.032 4
$f 4$	ABC	9.285 9E-018	2.443 5E-018	5.656 9E-018	2.106 9E-018	0.369 1
$f 4$	EABC	2.637 4E-047	6.695 1E-079	8.791 4E-049	4.734 3E-048	0.384 6
$f 5$	ABC	3.448 9E-016	1.920 4E-016	2.497 6E-016	3.828 5E-017	0.187 4
$f 5$	EABC	2.402 5E-060	1.066 4E-062	2.920 6E-061	4.225 7E-061	0.160 7
$f 6$	ABC	1.089 4E-001	1.647 5E-002	5.440 0E-002	2.169 1E-002	0.085 0
$f 6$	EABC	7.262 9E+000	3.160 0E+000	5.011 3E+000	8.501 2E-001	0.093 4
$f 7$	ABC	1.336 9E-016	5.618 4E-017	9.417 8E-017	2.841 7E-017	0.028 6
$f 7$	EABC	0	0	0	0	0.039 4
$f 8$	ABC	2.332 1E-017	9.192 5E-018	1.481 3E-017	3.050 3E-018	0.030 2
$f 8$	EABC	9.110 7E-238	1.676 0E-244	6.633 0E-239	0	0.044 1
$f 9$	ABC	6.688 5E-002	2.047 8E-002	4.648 1E-002	1.180 2E-002	0.031 9
$f 9$	EABC	3.054 4E-002	7.575 2E-003	1.969 7E-002	5.576 0E-003	0.044 8
$f 10$	ABC	8.707 8E+000	2.227 4E-004	1.402 5E-001	2.357 1E-001	1.344 0
$f 10$	EABC	7.413 6E+000	1.077 9E-004	2.698 0E-001	1.327 4E+000	0.971 8
$f 11$	ABC	0	0	0	0	0.468 9
$f 11$	EABC	0	0	0	0	0.471 4
$f 12$	ABC	0	0	0	0	0.338 6
$f 12$	EABC	0	0	0	0	0.297 8
$f 13$	ABC	1.968 9E-010	0	2.216 3E-011	5.255 7E-010	0.257 7
$f 13$	EABC	1.722 4E-011	0	5.825 1E-013	3.090 7E-012	0.243 0
$f 14$	ABC	8.661 8E-10	8.661 8E-010	8.661 8E-010	6.183 4E-010	0.300 3
$f 14$	EABC	5.784 3E-012	8.789 7E-013	1.432 5E-012	5.457 0E-013	0.279 6
$f 15$	ABC	7.694 1E-015	4.141 4E-015	6.509 9E-014	1.674 8E-015	0.226 2
$f 15$	EABC	3.611 5E-014	1.124 6E-014	2.048 4E-014	5.549 5E-015	0.214 4
$f 16$	ABC	1.391 0E-016	4.020 5E-017	9.379 2E-017	2.843 3E-017	0.470 9
$f 16$	EABC	1.570 4E-032	1.570 4E-032	1.570 4E-032	2.736 9E-048	0.418 8
$f 17$	ABC	1.298 0E-016	4.404 4E-017	8.927 4E-017	2.871 6E-017	0.460 7
$f 17$	EABC	1.349 6E-032	1.349 6E-032	1.349 7E-032	5.473 8E-048	0.422 0
$f 18$	ABC	7.261 7E-007	6.644 4E-016	2.453 2E-008	1.302 9E-007	0.268 6
$f 18$	EABC	1.075 0E-014	5.200 3E-049	9.708 1E-016	2.236 1E-015	0.242 0
$f 19$	ABC	1.554 5E-016	8.695 5E-017	1.224 5E-016	1.615 1E-17	1.324 7
$f 19$	EABC	2.625 5E-032	2.416 8E-032	1.216 6E-032	1.763 8E-048	1.112 6
$f 20$	ABC	0	0	0	0	1.009 8
$f 20$	EABC	0	0	0	0	0.756 8
$f 21$	ABC	3.960 9E-001	1.269 9E-001	3.095 5E-001	5.028 9E-002	0.051 3
$f 21$	EABC	3.455 9E-001	1.782 2E-001	2.679 8E-001	5.536 2E-002	0.064 6
$f 22$	ABC	-7.738 9E+001	-7.833 2E+001	-7.830 1E+001	1.691 7E-001	0.656 7
$f 22$	EABC	-7.833 2E+001	-7.833 2E+001	-7.833 2E+001	1.797 5E-014	0.629 0
$f 23$	ABC	-2.859 8E+001	-2.863 0E+001	-2.861 8E+001	8.494 8E-003	0.875 7
$f 23$	EABC	-2.857 0E+001	-2.863 0E+001	-2.861 8E+002	1.728 0E-002	0.843 0
$f 24$	ABC	8.474 8E-016	4.825 6E-017	6.426 4E-016	8.348 0E-017	0.186 3
$f 24$	EABC	0	0	0	0	0.113 3
$f 25$	ABC	0	0	0	0	0.303 1
$f 25$	EABC	0	0	0	0	0.285 0
$f 26$	ABC	1.876 8E-009	0	2.535 5E-010	5.525 5E-010	0.365 3
$f 26$	EABC	1.765 4E-011	0	5.876 9E-012	3.523 5E-012	0.252 3
$f 27$	ABC	7.694 1E-014	4.141 4E-015	6.809 5E-014	5.674 8E-015	0.226 2
$f 27$	EABC	3.611 5E-014	1.124 6E-014	2.048 4E-014	5.549 5E-015	0.214 4
$f 28$	ABC	8.592 2E-009	5.443 6E-009	7.289 5E-009	7.691 0E-010	0.594 7
$f 28$	EABC	7.867 7E-015	5.384 5E-016	6.883 8E-017	6.553 6E-017	0.291 4

表3 ABC和EABC的实验结果 ( D = 60 )

Table 3 Experimental results of ABC and EABC ( D = 60 )

Function	ABC			EABC
Function	Average optimal value	Standard deviation	Average running time/s	Average optimal value	Standard deviation	Average running time/s
$f 1$	3.351 8E-016	3.342 4E-017	0.039 3	3.983 8E-055	3.838 4E-055	0.043 3
$f 2$	3.724 2E-016	5.172 9E-017	0.608 3	7.480 3E-052	1.372 4E-052	0.555 5
$f 3$	3.371 8E-016	4.722 6E-017	0.039 5	2.029 2E-055	3.039 2E-055	0.044 2
$f 4$	3.516 1E-017	1.841 0E-017	0.645 8	6.108 8E-030	3.129 8E-029	0.592 7
$f 5$	1.463 8E-015	1.016 2E-016	0.262 4	1.239 7E-028	1.178 9E-028	0.211 9
$f 6$	1.169 8E+001	1.635 5E+000	0.138 0	1.532 1E+001	1.902 4E+000	0.137 9
$f 7$	3.371 6E-016	5.412 8E-017	0.039 1	0	0	0.024 6
$f 8$	1.204 4E-016	2.229 9E-017	0.042 0	1.009 7E-113	1.611 8E-113	0.051 1
$f 9$	1.894 1E-001	3.440 0E-002	0.046 5	9.453 8E-002	1.785 8E-002	0.052 0
$f 10$	2.468 1E-001	4.482 5E-001	1.821 2	1.041 8E+001	2.458 3E+001	2.024 9
$f 11$	2.486 9E-015	1.339 2E-014	1.028 5	6.633 1E-002	2.481 9E-001	1.024 7
$f 12$	0	0	0.617 0	0	0	0.591 3
$f 13$	2.465 4E-004	1.327 6E-003	0.477 3	2.465 4E-004	1.327 6E-003	0.400 8
$f 14$	3.149 2E+001	5.240 2E+001	0.572 7	1.642 9E-001	4.817 6E-002	0.545 0
$f 15$	4.724 8E-014	5.709 0E-015	0.424 6	3.587 9E-014	7.277 0E-015	0.412 1
$f 16$	3.148 0E-016	5.657 5E-017	0.871 8	7.852 2E-033	1.368 5E-048	0.845 1
$f 17$	3.393 1E-016	5.511 1E-017	1.008 5	1.349 7E-032	5.473 8E-048	0.965 0
$f 18$	2.093 1E-005	2.894 1E-005	0.498 0	2.723 7E-007	7.691 1E-007	0.424 1
$f 19$	2.640 6E+003	3.220 6E+002	2.442 7	2.613 0E+003	2.371 4E+002	2.176 6
$f 20$	0	0	1.194 1	0	0	0.900 7
$f 21$	4.877 7E-001	5.149 7E-003	0.065 2	4.702 7E-001	1.556 0E-002	0.056 7
$f 22$	-7.833 2E+001	1.397 2E-014	1.261 8	-7.833 2E+001	1.244 3E-014	1.176 5
$f 23$	-5.829 6E+001	7.890 6E-002	2.814 6	-5.849 7E+001	3.908 6E-002	1.824 7
$f 24$	1.496 6E-015	1.053 5E-016	0.131 3	1.486 7E-055	1.169 6E-054	0.128 8
$f 25$	0	0	0.594 1	0	0	0.500 7
$f 26$	1.548 9E-003	8.350 9E-004	0.700 0	1.503 3E-003	8.564 2E-005	0.643 8
$f 27$	1.994 8E-014	9.566 1E-014	0.498 5	1.906 0E-014	5.965 4E-015	0.520 9
$f 28$	1.583 7E-005	1.108 5E-005	0.553 1	1.565 9E-007	1.261 3E-008	0.533 0

表3 ABC和EABC的实验结果 ( D = 60 )

Table 3 Experimental results of ABC and EABC ( D = 60 )

Function	ABC			EABC
Function	Average optimal value	Standard deviation	Average running time/s	Average optimal value	Standard deviation	Average running time/s
$f 1$	3.351 8E-016	3.342 4E-017	0.039 3	3.983 8E-055	3.838 4E-055	0.043 3
$f 2$	3.724 2E-016	5.172 9E-017	0.608 3	7.480 3E-052	1.372 4E-052	0.555 5
$f 3$	3.371 8E-016	4.722 6E-017	0.039 5	2.029 2E-055	3.039 2E-055	0.044 2
$f 4$	3.516 1E-017	1.841 0E-017	0.645 8	6.108 8E-030	3.129 8E-029	0.592 7
$f 5$	1.463 8E-015	1.016 2E-016	0.262 4	1.239 7E-028	1.178 9E-028	0.211 9
$f 6$	1.169 8E+001	1.635 5E+000	0.138 0	1.532 1E+001	1.902 4E+000	0.137 9
$f 7$	3.371 6E-016	5.412 8E-017	0.039 1	0	0	0.024 6
$f 8$	1.204 4E-016	2.229 9E-017	0.042 0	1.009 7E-113	1.611 8E-113	0.051 1
$f 9$	1.894 1E-001	3.440 0E-002	0.046 5	9.453 8E-002	1.785 8E-002	0.052 0
$f 10$	2.468 1E-001	4.482 5E-001	1.821 2	1.041 8E+001	2.458 3E+001	2.024 9
$f 11$	2.486 9E-015	1.339 2E-014	1.028 5	6.633 1E-002	2.481 9E-001	1.024 7
$f 12$	0	0	0.617 0	0	0	0.591 3
$f 13$	2.465 4E-004	1.327 6E-003	0.477 3	2.465 4E-004	1.327 6E-003	0.400 8
$f 14$	3.149 2E+001	5.240 2E+001	0.572 7	1.642 9E-001	4.817 6E-002	0.545 0
$f 15$	4.724 8E-014	5.709 0E-015	0.424 6	3.587 9E-014	7.277 0E-015	0.412 1
$f 16$	3.148 0E-016	5.657 5E-017	0.871 8	7.852 2E-033	1.368 5E-048	0.845 1
$f 17$	3.393 1E-016	5.511 1E-017	1.008 5	1.349 7E-032	5.473 8E-048	0.965 0
$f 18$	2.093 1E-005	2.894 1E-005	0.498 0	2.723 7E-007	7.691 1E-007	0.424 1
$f 19$	2.640 6E+003	3.220 6E+002	2.442 7	2.613 0E+003	2.371 4E+002	2.176 6
$f 20$	0	0	1.194 1	0	0	0.900 7
$f 21$	4.877 7E-001	5.149 7E-003	0.065 2	4.702 7E-001	1.556 0E-002	0.056 7
$f 22$	-7.833 2E+001	1.397 2E-014	1.261 8	-7.833 2E+001	1.244 3E-014	1.176 5
$f 23$	-5.829 6E+001	7.890 6E-002	2.814 6	-5.849 7E+001	3.908 6E-002	1.824 7
$f 24$	1.496 6E-015	1.053 5E-016	0.131 3	1.486 7E-055	1.169 6E-054	0.128 8
$f 25$	0	0	0.594 1	0	0	0.500 7
$f 26$	1.548 9E-003	8.350 9E-004	0.700 0	1.503 3E-003	8.564 2E-005	0.643 8
$f 27$	1.994 8E-014	9.566 1E-014	0.498 5	1.906 0E-014	5.965 4E-015	0.520 9
$f 28$	1.583 7E-005	1.108 5E-005	0.553 1	1.565 9E-007	1.261 3E-008	0.533 0

表4 ABC和EABC的实验结果 ( D = 100 )

Table 4 Experimental results of ABC and EABC ( D = 100 )

Function	ABC			EABC
Function	Average optimal value	Standard deviation	Average running time/s	Average optimal value	Standard deviation	Average running time/s
$f 1$	1.737 2E-015	1.870 7E-016	0.056 9	5.109 5E-030	5.727 5E-030	0.074 1
$f 2$	3.589 0E-012	7.980 5E-012	1.043 2	1.829 4E-026	2.198 1E-026	1.005 5
$f 3$	1.576 3E-015	1.742 7E-016	0.059 7	2.528 0E-030	2.508 0E-030	0.070 0
$f 4$	1.250 6E-007	2.671 9E-007	1.207 3	5.621 0E-022	1.067 4E-021	1.073 0
$f 5$	4.441 6E-010	3.857 4E-010	0.682 9	6.158 2E-016	1.865 3E-016	0.421 0
$f 6$	4.055 6E+001	2.886 1E+000	0.231 4	3.010 8E+001	2.252 8E+000	0.227 6
$f 7$	1.694 6E-015	2.169 0E-016	0.059 4	0	0	0.070 7
$f 8$	2.952 0E-016	5.553 5E-017	0.063 4	3.552 2E-063	1.550 3E-062	0.073 9
$f 9$	4.675 8E-001	7.989 0E-002	0.073 0	2.673 4E-001	3.147 9E-002	0.081 6
$f 10$	5.271 1E+000	1.696 1E+001	3.012 6	3.854 0E+001	4.619 5E+001	3.103 4
$f 11$	1.994 4E-001	4.735 1E-001	1.581 3	2.695 3E-001	5.078 1E-001	1.547 7
$f 12$	0	0	0.965 0	0	0	0.946 1
$f 13$	1.134 1E-003	4.274 4E-003	0.747 3	5.682 8E-004	1.869 1E-003	0.701 3
$f 14$	7.434 5E+002	2.809 7E+002	0.829 7	1.344 2E+002	9.612 8E+001	0.847 7
$f 15$	7.384 4E-009	6.387 5E-009	0.618 7	1.329 9E-013	4.859 9E-014	0.637 9
$f 16$	1.204 1E-015	1.161 5E-016	1.356 0	6.359 6E-032	4.714 8E-032	1.332 0
$f 17$	1.759 2E-015	7.969 9E-016	1.339 6	2.235 9E-030	2.082 6E-030	1.321 8
$f 18$	5.720 8E-003	6.563 2E-003	0.743 1	3.559 5E-004	1.465 4E-003	0.740 5
$f 19$	4.582 2E+003	3.893 6E+002	3.553 3	4.687 2E+003	3.830 5E+002	3.534 3
$f 20$	0	0	1.294 1	0	0	1.103 2
$f 21$	4.996 1E-001	1.571 0E-004	0.074 5	4.983 0E-001	4.693 1E-004	0.090 7
$f 22$	-7.832 3E+001	5.075 2E-002	1.972 6	-7.833 2E+001	1.216 9E-014	1.958 3
$f 23$	-9.697 3E+001	2.763 5E-001	2.617 6	-9.754 4E+001	1.276 4E-001	2.604 5
$f 24$	2.694 0E-015	1.513 6E-014	0.153 1	2.685 0E-015	1.019 5E-014	0.167 8
$f 25$	0	0	1.088 7	0	0	1.290 5
$f 26$	3.262 0E-003	1.271 6E-003	0.969 2	3.216 9E-003	1.586 7E-004	0.006 9
$f 27$	1.995 2E-010	7.666 0E-010	0.721 2	1.996 3E-010	6.247 5E-011	0.722 1
$f 28$	8.050 6E-003	3.397 3E-003	1.367 4	9.582 0E-004	9.729 6E-005	1.392 6

表4 ABC和EABC的实验结果 ( D = 100 )

Table 4 Experimental results of ABC and EABC ( D = 100 )

Function	ABC			EABC
Function	Average optimal value	Standard deviation	Average running time/s	Average optimal value	Standard deviation	Average running time/s
$f 1$	1.737 2E-015	1.870 7E-016	0.056 9	5.109 5E-030	5.727 5E-030	0.074 1
$f 2$	3.589 0E-012	7.980 5E-012	1.043 2	1.829 4E-026	2.198 1E-026	1.005 5
$f 3$	1.576 3E-015	1.742 7E-016	0.059 7	2.528 0E-030	2.508 0E-030	0.070 0
$f 4$	1.250 6E-007	2.671 9E-007	1.207 3	5.621 0E-022	1.067 4E-021	1.073 0
$f 5$	4.441 6E-010	3.857 4E-010	0.682 9	6.158 2E-016	1.865 3E-016	0.421 0
$f 6$	4.055 6E+001	2.886 1E+000	0.231 4	3.010 8E+001	2.252 8E+000	0.227 6
$f 7$	1.694 6E-015	2.169 0E-016	0.059 4	0	0	0.070 7
$f 8$	2.952 0E-016	5.553 5E-017	0.063 4	3.552 2E-063	1.550 3E-062	0.073 9
$f 9$	4.675 8E-001	7.989 0E-002	0.073 0	2.673 4E-001	3.147 9E-002	0.081 6
$f 10$	5.271 1E+000	1.696 1E+001	3.012 6	3.854 0E+001	4.619 5E+001	3.103 4
$f 11$	1.994 4E-001	4.735 1E-001	1.581 3	2.695 3E-001	5.078 1E-001	1.547 7
$f 12$	0	0	0.965 0	0	0	0.946 1
$f 13$	1.134 1E-003	4.274 4E-003	0.747 3	5.682 8E-004	1.869 1E-003	0.701 3
$f 14$	7.434 5E+002	2.809 7E+002	0.829 7	1.344 2E+002	9.612 8E+001	0.847 7
$f 15$	7.384 4E-009	6.387 5E-009	0.618 7	1.329 9E-013	4.859 9E-014	0.637 9
$f 16$	1.204 1E-015	1.161 5E-016	1.356 0	6.359 6E-032	4.714 8E-032	1.332 0
$f 17$	1.759 2E-015	7.969 9E-016	1.339 6	2.235 9E-030	2.082 6E-030	1.321 8
$f 18$	5.720 8E-003	6.563 2E-003	0.743 1	3.559 5E-004	1.465 4E-003	0.740 5
$f 19$	4.582 2E+003	3.893 6E+002	3.553 3	4.687 2E+003	3.830 5E+002	3.534 3
$f 20$	0	0	1.294 1	0	0	1.103 2
$f 21$	4.996 1E-001	1.571 0E-004	0.074 5	4.983 0E-001	4.693 1E-004	0.090 7
$f 22$	-7.832 3E+001	5.075 2E-002	1.972 6	-7.833 2E+001	1.216 9E-014	1.958 3
$f 23$	-9.697 3E+001	2.763 5E-001	2.617 6	-9.754 4E+001	1.276 4E-001	2.604 5
$f 24$	2.694 0E-015	1.513 6E-014	0.153 1	2.685 0E-015	1.019 5E-014	0.167 8
$f 25$	0	0	1.088 7	0	0	1.290 5
$f 26$	3.262 0E-003	1.271 6E-003	0.969 2	3.216 9E-003	1.586 7E-004	0.006 9
$f 27$	1.995 2E-010	7.666 0E-010	0.721 2	1.996 3E-010	6.247 5E-011	0.722 1
$f 28$	8.050 6E-003	3.397 3E-003	1.367 4	9.582 0E-004	9.729 6E-005	1.392 6

表5 EABC与ABC改进算法的性能比较（Ave±Std）

Table 5 Performance comparison of EABC with improved algorithms for ABC（Ave±Std）

Function	ABC	GABC	ABCBest1	MABC	ABCVSS	MSSABC	EABC
$f 1$	3.88E-16±8.06E-17	4.34E-16±7.54E-17	3.76E-47±6.76E-47	9.43E-32±6.67E-32	1.53E-81±8.37E-81	7.92E-87±1.04E-86	9.78E-119±1.90E-118
$f 2$	3.32E-16±5.77E-17	3.12E-16±4.45E-17	1.30E-40±4.33E-41	3.66E-28±5.96E-28	4.82E-82±2.63E-81	3.40E-84±4.78E-84	4.93E-116±9.77E-116
$f 3$	3.45E-16±6.25E-17	4.66E-16±6.64E-17	6.50E-48±6.04E-48	2.10E-32±1.56E-32	3.19E-89±1.48E-88	1.07E-87±1.90E-87	2.21E-119±6.71E-119
$f 4$	2.77E-17±9.81E-18	1.34E-17±8.73E-18	1.77E-86±7.02E-86	2.70E-69±5.38E-69	5.55E-115±3.04E-114	3.37E-82±1.76E-81	1.03E-51±5.49E-51
$f 5$	1.09E-15±1.40E-16	1.03E-15±1.38E-16	2.44E-25±2.33E-26	2.40E-17±9.02E-18	7.89E-43±4.32E-42	5.65E-45±7.38E-45	3.64E-61±3.93E-61
$f 6$	1.76E-01±5.48E-01	1.33E-01±5.01E-02	4.43E+00±1.43E-01	1.02E+01±1.49E+00	4.08E-02±2.20E-02	1.77E-01±4.62E-02	5.25E+00±8.86E-01
$f 7$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 8$	9.52E-16±2.29E-17	9.88E-16±3.54E-17	2.53E-95±5.53E-98	1.45E-67±2.28E-67	3.25E-154±1.78E-153	1.94E-176±0.00	3.09E-238±0.00
$f 9$	4.15E-02±5.06E-02	2.42E-02±3.41E-03	1.26E-02±3.45E-03	3.71E-02±8.53E-03	1.81E-02±5.27E-03	2.54E-02±7.36E-03	1.05E-02±5136E-03
$f 10$	0.93±0.16	0.40±0.13	13.30±23.10	0.61±0.45	0.38±1.54	3.27±12.20	1.42±2.44
$f 11$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 12$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 13$	2.22E-11±4.44E-10	3.46E-15±5.66E-15	0.00±0.00	0.00±0.00	0.00±0.00	2.46E-04±1.32E-03	0.00±0.00
$f 14$	1.17E-10±3.55E-09	7.57E-01±5.66E-01	4.22E-10±8.12E-11	1.21E-13±4.53E-13	4.85E-13±8.18E-13	7.41E+00±2.79E+01	3.86E+00±2.12E+01
$f 15$	2.97E-14±2.32E-15	2.91E-14±3.36E-15	3.11E-14±2.21E-13	4.12E-14±2.17E-15	2.45E-14±4.00E-15	3.07E-14±2.71E-15	1.98E-14±4.55E-15
$f 16$	5.34E-16±6.43E-17	4.44E-16±8.53E-17	1.47E-32±4.13E-48	1.90E-32±3.70E-33	1.57E-32±5.57E-48	1.57E-32±2.73E-48	1.57E-32±2.73E-48
$f 17$	3.76E-16±8.58E-17	4.66E-16±7.21E-17	1.42E-32±5.98E-48	2.23E-31±1.46E-31	1.35E-32±5.57E-48	1.34E-32±5.47E-48	1.34E-32±5.47E-48
$f 18$	2.71E-08±9.78E-08	1.47E-09±1.43E-07	3.34E-16±8.43E-16	1.58E-16±2.48E-16	3.66E-44±1.93E-43	1.49E-06±3.02E-06	7.44E-14±3.99E-13
$f 19$	4.45E-14±8.76E-13	3.45E-15±5.63E-16	1.35E-31±6.68E-47	1.48E-31±2.30E-32	1.35E-31±6.68E-47	3.28E-31±1.76E-31	1.25E-32±2.30E-32
$f 20$	0.00±0.00	0.00±0.00	4.74E-16±1.80E-15	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 21$	2.50E-01±5.47E-02	2.47E-01±6.65E-02	2.52E-01±6.76E-02	2.95E-01±3.17E-03	2.84E-01±5.69E-02	2.65E-01±4.53E-02	2.59E-01±4.58E-02
$f 22$	-7.83E+01±1.56E-1	-7.83E+01±3.50E-14	-7.83E+01±6.45E-01	-7.83E+01±3.17E-07	-7.83E+01±3.02E-10	-7.83E+01±3.06E-14	-7.83E+01±1.51E-14
$f 23$	-3.45E+01±4.48E-01	-3.44E+01±4.59E-02	-2.45E+01±3.42E-01	-9.07E+01±5.03E-01	-9.94E+01±8.84E-02	-2.85E+01±7.94E-02	-2.86E+01±1.16E-02
$f 24$	4.34E-16±8.34E-17	3.91E-16±8.32E-17	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 25$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 26$	2.65E-13±4.90E-12	3.01E-15±5.42E-15	6.76E-16±8.80E-15	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 27$	3.90E-14±2.35E-15	8.43E-14±8.80E-15	3.84E-14±1.43E-15	4.89E-14±8.12E-15	2.41E-14±7.90E-15	2.41E-14±1.78E-14	1.00E-16±4.02E-17
$f 28$	1.89E-07±3.56E-07	6.34E-05±2.35E-05	5.89E-16±3.56E-16	9.38E-16±4.11E-16	7.67E-17±1.43E-16	5.45E-17±3.90E-16	1.02E-18±1.68E-14

表5 EABC与ABC改进算法的性能比较（Ave±Std）

Table 5 Performance comparison of EABC with improved algorithms for ABC（Ave±Std）

Function	ABC	GABC	ABCBest1	MABC	ABCVSS	MSSABC	EABC
$f 1$	3.88E-16±8.06E-17	4.34E-16±7.54E-17	3.76E-47±6.76E-47	9.43E-32±6.67E-32	1.53E-81±8.37E-81	7.92E-87±1.04E-86	9.78E-119±1.90E-118
$f 2$	3.32E-16±5.77E-17	3.12E-16±4.45E-17	1.30E-40±4.33E-41	3.66E-28±5.96E-28	4.82E-82±2.63E-81	3.40E-84±4.78E-84	4.93E-116±9.77E-116
$f 3$	3.45E-16±6.25E-17	4.66E-16±6.64E-17	6.50E-48±6.04E-48	2.10E-32±1.56E-32	3.19E-89±1.48E-88	1.07E-87±1.90E-87	2.21E-119±6.71E-119
$f 4$	2.77E-17±9.81E-18	1.34E-17±8.73E-18	1.77E-86±7.02E-86	2.70E-69±5.38E-69	5.55E-115±3.04E-114	3.37E-82±1.76E-81	1.03E-51±5.49E-51
$f 5$	1.09E-15±1.40E-16	1.03E-15±1.38E-16	2.44E-25±2.33E-26	2.40E-17±9.02E-18	7.89E-43±4.32E-42	5.65E-45±7.38E-45	3.64E-61±3.93E-61
$f 6$	1.76E-01±5.48E-01	1.33E-01±5.01E-02	4.43E+00±1.43E-01	1.02E+01±1.49E+00	4.08E-02±2.20E-02	1.77E-01±4.62E-02	5.25E+00±8.86E-01
$f 7$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 8$	9.52E-16±2.29E-17	9.88E-16±3.54E-17	2.53E-95±5.53E-98	1.45E-67±2.28E-67	3.25E-154±1.78E-153	1.94E-176±0.00	3.09E-238±0.00
$f 9$	4.15E-02±5.06E-02	2.42E-02±3.41E-03	1.26E-02±3.45E-03	3.71E-02±8.53E-03	1.81E-02±5.27E-03	2.54E-02±7.36E-03	1.05E-02±5136E-03
$f 10$	0.93±0.16	0.40±0.13	13.30±23.10	0.61±0.45	0.38±1.54	3.27±12.20	1.42±2.44
$f 11$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 12$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 13$	2.22E-11±4.44E-10	3.46E-15±5.66E-15	0.00±0.00	0.00±0.00	0.00±0.00	2.46E-04±1.32E-03	0.00±0.00
$f 14$	1.17E-10±3.55E-09	7.57E-01±5.66E-01	4.22E-10±8.12E-11	1.21E-13±4.53E-13	4.85E-13±8.18E-13	7.41E+00±2.79E+01	3.86E+00±2.12E+01
$f 15$	2.97E-14±2.32E-15	2.91E-14±3.36E-15	3.11E-14±2.21E-13	4.12E-14±2.17E-15	2.45E-14±4.00E-15	3.07E-14±2.71E-15	1.98E-14±4.55E-15
$f 16$	5.34E-16±6.43E-17	4.44E-16±8.53E-17	1.47E-32±4.13E-48	1.90E-32±3.70E-33	1.57E-32±5.57E-48	1.57E-32±2.73E-48	1.57E-32±2.73E-48
$f 17$	3.76E-16±8.58E-17	4.66E-16±7.21E-17	1.42E-32±5.98E-48	2.23E-31±1.46E-31	1.35E-32±5.57E-48	1.34E-32±5.47E-48	1.34E-32±5.47E-48
$f 18$	2.71E-08±9.78E-08	1.47E-09±1.43E-07	3.34E-16±8.43E-16	1.58E-16±2.48E-16	3.66E-44±1.93E-43	1.49E-06±3.02E-06	7.44E-14±3.99E-13
$f 19$	4.45E-14±8.76E-13	3.45E-15±5.63E-16	1.35E-31±6.68E-47	1.48E-31±2.30E-32	1.35E-31±6.68E-47	3.28E-31±1.76E-31	1.25E-32±2.30E-32
$f 20$	0.00±0.00	0.00±0.00	4.74E-16±1.80E-15	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 21$	2.50E-01±5.47E-02	2.47E-01±6.65E-02	2.52E-01±6.76E-02	2.95E-01±3.17E-03	2.84E-01±5.69E-02	2.65E-01±4.53E-02	2.59E-01±4.58E-02
$f 22$	-7.83E+01±1.56E-1	-7.83E+01±3.50E-14	-7.83E+01±6.45E-01	-7.83E+01±3.17E-07	-7.83E+01±3.02E-10	-7.83E+01±3.06E-14	-7.83E+01±1.51E-14
$f 23$	-3.45E+01±4.48E-01	-3.44E+01±4.59E-02	-2.45E+01±3.42E-01	-9.07E+01±5.03E-01	-9.94E+01±8.84E-02	-2.85E+01±7.94E-02	-2.86E+01±1.16E-02
$f 24$	4.34E-16±8.34E-17	3.91E-16±8.32E-17	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 25$	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 26$	2.65E-13±4.90E-12	3.01E-15±5.42E-15	6.76E-16±8.80E-15	0.00±0.00	0.00±0.00	0.00±0.00	0.00±0.00
$f 27$	3.90E-14±2.35E-15	8.43E-14±8.80E-15	3.84E-14±1.43E-15	4.89E-14±8.12E-15	2.41E-14±7.90E-15	2.41E-14±1.78E-14	1.00E-16±4.02E-17
$f 28$	1.89E-07±3.56E-07	6.34E-05±2.35E-05	5.89E-16±3.56E-16	9.38E-16±4.11E-16	7.67E-17±1.43E-16	5.45E-17±3.90E-16	1.02E-18±1.68E-14

图2 函数收敛曲线

Fig.2 Convergence curve of functions

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极值个体引导的人工蜂群算法

Extreme Individual Guided Artificial Bee Colony Algorithm

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