Shuffled Frog Leaping Algorithm Driven by Nuclear Center and Its Application

doi:10.3778/j.issn.1673-9418.2108067

Abstract

Abstract:

Aiming at the defects of slow evolution speed and easy to fall into local convergence caused by the inertia provided by the current position of individual frog and the jump step of shuffled frog leaping algorithm (SFLA), a shuffled frog leaping algorithm driven by nuclear center (NCSFLA) is proposed, in which the jump evolution behavior of individual frog is defined as quantum mechanical behavior. In the global optimization, the concentric circle centered on the nucleus is used as the electron orbit to form the frog population. In the local optimization, three different local search strategies are used to update the worst individual in the population, such as jumping to the local optimal individual with the transition step as the radius, jumping to the global optimal individual with the driving step as the radius, and randomly generating non repeated frog individual components. Taking the electron orbit center, that is, the local optimal individual, as the inertial guidance of the transition, makes the convergence in the population more conducive to finding the local optimal solution and improving the search ability. If it falls into local optimization, the inertial guidance driven by the nuclear center, that is, the global optimal individual, makes the frog individuals gather around the nuclear center as much as possible, so as to speed up the convergence speed. The algorithm is applied to solving the capacity-limited vehicle routing problem (CVRP), and a shuffled frog leaping algorithm driven by nuclear center for capacity-limited vehicle routing problem (NCSFLA-CVRP) is proposed. In the test of 20 benchmark functions such as single peak value, multi-peak function and composite function, the expe-rimental results show that the improved shuffled frog leaping algorithm driven by nuclear center has the charac-teristics of fast convergence and high accuracy compared with other five algorithms. The test results of Solomon example standard test data show that this method can effectively improve the optimization performance of capacity-limited vehicle routing problem.

Key words: shuffled frog leaping algorithm (SFLA), nuclear center, track center, driving strategy, capacity-limited vehicle routing problem (CVRP)

摘要：

针对混合蛙跳算法（SFLA）青蛙个体当前位置提供的惯性以及跳跃步长引起的进化速度慢,易陷入局部收敛的缺陷,将青蛙个体跳跃进化行为定义为量子力学行为,提出一种核中心驱动混合蛙跳算法（NCSFLA）。在全局寻优中,以原子核为中心的同心圆作为电子轨道构成青蛙族群;在局部寻优中,分别以跃迁步长为半径向局部最优个体跳跃,以驱动步长为半径向全局最优个体跳跃,随机产生不重复的青蛙个体分量等三种不同的局部搜索策略对族群内最差个体进行更新。以电子轨道中心即局部最优个体为跃迁的惯性指导,使得族群内的收敛更加有利于寻找局部最优解,提升搜索能力;如果陷入局部最优,则以原子核中心即全局最优个体为驱动的惯性指导,使得青蛙个体尽可能聚集在原子核中心周围,从而加快收敛速度。将该算法应用于解决容量限制车辆路径问题（CVRP）,提出一种核中心驱动混合蛙跳算法的容量限制车辆路径优化算法（NCSFLA-CVRP）。实验结果显示,在单峰值、多峰值函数以及复合函数等20个测试函数上,改进后的核中心驱动混合蛙跳算法相比其他五种算法具有收敛速度快、精度高的特点。Solomon算例标准测试数据测试结果表明该方法可有效提高容量限制车辆路径的优化性能。

关键词: 混合蛙跳算法（SFLA）, 核中心, 轨道中心, 驱动策略, 容量限制车辆路径问题（CVRP）

CLC Number:

TP301.6

LIU Liqun, GU Renyuan. Shuffled Frog Leaping Algorithm Driven by Nuclear Center and Its Application[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(5): 1169-1181.

刘立群, 顾任远. 核中心驱动混合蛙跳算法及其应用[J]. 计算机科学与探索, 2022, 16(5): 1169-1181.

Figures/Tables 28

Fig.1 Principle of shuffled frog leaping algorithm driven by nuclear center

Fig.2 Flow chart of improved capacity-limited vehicle routing problem

Table 1 Benchmark functions

测试函数	公式	维数D	范围	$f m i n$	峰值
Sphere^[27]	$f 1 (x) = ∑ i = 1 D x i 2$	30	$[- 100,100]$	0	单峰
Griewank^[27-28]	$f 2 (x) = 14000 ∑ i = 1 D x i 2 - ∏ i = 1 D c o s x i i + 1$	30	$[- 600,600]$	0	多峰
Bent Cigar^[28]	$f 3 (x) = x 12 + 10 6 ∑ i = 2 D x i 2$	30	$[- 10,10]$	0	单峰
Sumpower^[28]	$f 4 (x) = ∑ i = 1 D \| x i \| (i + 1)$	30	$[- 100,100]$	0	单峰
Zakharov^[28]	$f 5 (x) = ∑ i = 1 D x i 2 + ∑ i = 1 D 0.5 x i 2 + ∑ i = 1 D 0.5 x i 4$	30	$[- 5,10]$	0	单峰
Discus^[28]	$f 6 (x) = 10 6 x 12 + ∑ i = 2 D x i 2$	30	$[- 10,10]$	0	单峰
Weierstrass^[28]	$f 7 (x) = ∑ i = 1 D ∑ k = 0 k m a x [a k c o s (2 π b k (x i + 0.5))] - D ∑ k = 0 k m a x [a k c o s (2 π b k 0.5)]$	30	$[- 1,1]$	0	多峰
Katsuura^[28]	$f 8 (x) = 10 D 2 ∏ i = 1 D 1 + i ∑ j = 1 32 \| 2 j x i - r o u n d (2 j x i) \| 2 j 10 D 1.2 - 10 D 2$	30	$[- 5,5]$	0	多峰
HappyCat^[28]	$f 9 (x) = ∑ i = 1 D x i 2 - D 14 + 0.5 ∑ i = 1 D x i 2 + ∑ i = 1 D x i D + 0.5$	30	$[- 100,100]$	$-$ 1	多峰
HGBat^[28]	$f 10 (x) = ∑ i = 1 D x i 2 2 - ∑ i = 1 D x i 2 12 + 0.5 ∑ i = 1 D x i 2 + ∑ i = 1 D x i D + 0.5$	30	$[- 100,100]$	$-$ 1	多峰
Schwefel222^[27]	$f 11 (x) = ∑ i = 1 D \| x i \| + ∏ i = 1 D \| x i \|$	30	$[- 10,10]$	0	单峰
aa^[27]	$f 12 (x) = ∑ i = 1 D ∑ j = 1 i x j 2$	30	$[- 100,100]$	0	单峰
Schwefel221^[27]	$f 13 (x) = m a x {\| x i \|, 1 ≤ i ≤ D}$	30	$[- 100,100]$	0	单峰
Step^[27]	$f 14 (x) = ∑ i = 1 D (\| x i + 0.5 \|) 2$	30	$[- 100,100]$	0	单峰
dd^[27]	$f 15 (x) = π D 10 s i n (π y 1) + ∑ i = 1 D - 1 (y i - 1) 2 1 + 10 s i n 2 (π y i + 1) + (y D - 1) 2 + ∑ i = 1 D u (x i, 10,100,4)$	30	$[- 50,50]$	0	多峰
ee^[27]	$f 16 (x) = 0.1 s i n 2 3 π x 1 + ∑ i = 1 D (x i - 1) 2 [1 + s i n 2 (3 π x i + 1)] + (x D - 1) 2 [1 + s i n 2 (2 π x D)] + ∑ i = 1 D u (x i, 5,100,4)$	30	$[- 50,50]$	0	多峰
Rosenbrock^[27-28]	$f 17 (x) = ∑ i = 1 D - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$	30	$[- 10,10]$	0	多峰
Schwefel2_26^[28]	$f 18 (x) = 418.98288727243380 × D - ∑ i = 1 D x i s i n \| x i \|$	30	$[- 500,500]$	0	多峰
cf01^[28]	$f 19 (x) = ∑ i = 1 N {ω i × [λ i g i (x) + b i a s i]} + F *$ $g 1 (x) = ∑ i = 1 D - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$ $g 2 (x) = ∑ i = 1 D (106) i - 1 D - 1 x i 2$ $g 3 (x) = ∑ i = 1 D [x i 2 - 10 c o s (2 π x i) + 10]$	30	$[- 10,10] [- 20,20] [- 30,30]$	$[0,0, 0]$	复合
cf02^[28]	$f 20 (x) = ∑ i = 1 N {ω i × [λ i g i (x) + b i a s i]} + F *$ $g 1 (x) = ∑ i = 1 D [x i 2 - 10 c o s (2 π x i) + 10]$ $g 2 (x) = 14000 ∑ i = 1 D x i 2 - ∏ i = 1 D c o s x i i + 1$ $g 3 (x) = 418.98288727243380 × D - ∑ i = 1 D x i s i n \| x i \|$	30	$[- 10,10] [- 20,20] [- 30,30]$	$[0,0, 0]$	复合

Table 1 Benchmark functions

测试函数	公式	维数D	范围	$f m i n$	峰值
Sphere^[27]	$f 1 (x) = ∑ i = 1 D x i 2$	30	$[- 100,100]$	0	单峰
Griewank^[27-28]	$f 2 (x) = 14000 ∑ i = 1 D x i 2 - ∏ i = 1 D c o s x i i + 1$	30	$[- 600,600]$	0	多峰
Bent Cigar^[28]	$f 3 (x) = x 12 + 10 6 ∑ i = 2 D x i 2$	30	$[- 10,10]$	0	单峰
Sumpower^[28]	$f 4 (x) = ∑ i = 1 D \| x i \| (i + 1)$	30	$[- 100,100]$	0	单峰
Zakharov^[28]	$f 5 (x) = ∑ i = 1 D x i 2 + ∑ i = 1 D 0.5 x i 2 + ∑ i = 1 D 0.5 x i 4$	30	$[- 5,10]$	0	单峰
Discus^[28]	$f 6 (x) = 10 6 x 12 + ∑ i = 2 D x i 2$	30	$[- 10,10]$	0	单峰
Weierstrass^[28]	$f 7 (x) = ∑ i = 1 D ∑ k = 0 k m a x [a k c o s (2 π b k (x i + 0.5))] - D ∑ k = 0 k m a x [a k c o s (2 π b k 0.5)]$	30	$[- 1,1]$	0	多峰
Katsuura^[28]	$f 8 (x) = 10 D 2 ∏ i = 1 D 1 + i ∑ j = 1 32 \| 2 j x i - r o u n d (2 j x i) \| 2 j 10 D 1.2 - 10 D 2$	30	$[- 5,5]$	0	多峰
HappyCat^[28]	$f 9 (x) = ∑ i = 1 D x i 2 - D 14 + 0.5 ∑ i = 1 D x i 2 + ∑ i = 1 D x i D + 0.5$	30	$[- 100,100]$	$-$ 1	多峰
HGBat^[28]	$f 10 (x) = ∑ i = 1 D x i 2 2 - ∑ i = 1 D x i 2 12 + 0.5 ∑ i = 1 D x i 2 + ∑ i = 1 D x i D + 0.5$	30	$[- 100,100]$	$-$ 1	多峰
Schwefel222^[27]	$f 11 (x) = ∑ i = 1 D \| x i \| + ∏ i = 1 D \| x i \|$	30	$[- 10,10]$	0	单峰
aa^[27]	$f 12 (x) = ∑ i = 1 D ∑ j = 1 i x j 2$	30	$[- 100,100]$	0	单峰
Schwefel221^[27]	$f 13 (x) = m a x {\| x i \|, 1 ≤ i ≤ D}$	30	$[- 100,100]$	0	单峰
Step^[27]	$f 14 (x) = ∑ i = 1 D (\| x i + 0.5 \|) 2$	30	$[- 100,100]$	0	单峰
dd^[27]	$f 15 (x) = π D 10 s i n (π y 1) + ∑ i = 1 D - 1 (y i - 1) 2 1 + 10 s i n 2 (π y i + 1) + (y D - 1) 2 + ∑ i = 1 D u (x i, 10,100,4)$	30	$[- 50,50]$	0	多峰
ee^[27]	$f 16 (x) = 0.1 s i n 2 3 π x 1 + ∑ i = 1 D (x i - 1) 2 [1 + s i n 2 (3 π x i + 1)] + (x D - 1) 2 [1 + s i n 2 (2 π x D)] + ∑ i = 1 D u (x i, 5,100,4)$	30	$[- 50,50]$	0	多峰
Rosenbrock^[27-28]	$f 17 (x) = ∑ i = 1 D - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$	30	$[- 10,10]$	0	多峰
Schwefel2_26^[28]	$f 18 (x) = 418.98288727243380 × D - ∑ i = 1 D x i s i n \| x i \|$	30	$[- 500,500]$	0	多峰
cf01^[28]	$f 19 (x) = ∑ i = 1 N {ω i × [λ i g i (x) + b i a s i]} + F *$ $g 1 (x) = ∑ i = 1 D - 1 [100 (x i + 1 - x i 2) 2 + (x i - 1) 2]$ $g 2 (x) = ∑ i = 1 D (106) i - 1 D - 1 x i 2$ $g 3 (x) = ∑ i = 1 D [x i 2 - 10 c o s (2 π x i) + 10]$	30	$[- 10,10] [- 20,20] [- 30,30]$	$[0,0, 0]$	复合
cf02^[28]	$f 20 (x) = ∑ i = 1 N {ω i × [λ i g i (x) + b i a s i]} + F *$ $g 1 (x) = ∑ i = 1 D [x i 2 - 10 c o s (2 π x i) + 10]$ $g 2 (x) = 14000 ∑ i = 1 D x i 2 - ∏ i = 1 D c o s x i i + 1$ $g 3 (x) = 418.98288727243380 × D - ∑ i = 1 D x i s i n \| x i \|$	30	$[- 10,10] [- 20,20] [- 30,30]$	$[0,0, 0]$	复合

Table 2 Comparison of convergence precision and convergence speed

函数	算法	平均最优值	标准差	最大值	最小值	运行时间/s	函数	算法	平均最优值	标准差	最大值	最小值	运行时间/s
$f 1$	NCSFLA	1.34E+03	5.34E+03	4.68E+04	6.25E-05	1.51E-01	$f 11$	NCSFLA	4.91E+00	1.47E+01	1.18E+02	8.66E-05	1.98E-01
	MSFLA	1.34E+04	1.65E+04	4.68E+04	4.53E+02	1.85E-01		MSFLA	1.21E+01	1.55E+01	1.18E+02	1.98E+00	2.20E-01
	GSF²LA	6.37E+03	1.42E+04	4.68E+04	1.64E+01	1.76E-01		GSF²LA	4.57E+00	1.49E+01	1.18E+02	8.48E-01	1.78E-01
	SFLA	6.37E+03	1.41E+04	4.68E+04	4.24E+01	1.70E-01		SFLA	5.29E+00	1.29E+01	1.18E+02	1.56E+00	1.78E-01
	HSA	4.64E+04	3.01E+02	4.68E+04	4.54E+04	1.09E-01		HSA	1.07E+02	4.51E+00	1.18E+02	1.05E+02	1.17E-01
	ABCA	1.66E+04	2.01E+04	7.31E+04	1.45E+01	1.84E+00		ABCA	3.53E+08	3.92E+09	5.45E+10	6.03E-01	1.62E+00
$f 2$	NCSFLA	1.26E+01	4.82E+01	4.22E+02	2.97E-02	4.30E-01	$f 12$	NCSFLA	2.19E+03	8.82E+03	8.35E+04	6.18E+00	3.05E-01
	MSFLA	4.14E+02	1.92E+01	4.22E+02	3.45E+02	2.71E-01		MSFLA	6.00E+04	1.95E+04	8.35E+04	3.33E+04	2.41E-01
	GSF²LA	3.20E+02	1.15E+02	4.22E+02	9.77E+01	2.47E-01		GSF²LA	2.79E+04	3.49E+04	8.35E+04	2.29E+02	2.15E-01
	SFLA	3.21E+02	1.15E+02	4.22E+02	9.81E+01	2.50E-01		SFLA	2.62E+04	3.45E+04	8.35E+04	2.63E+02	2.18E-01
	HSA	4.19E+02	1.98E+00	4.22E+02	4.18E+02	1.89E-01		HSA	8.35E+04	0.00E+00	8.35E+04	8.35E+04	7.60E-02
	ABCA	1.45E+02	1.88E+02	5.87E+02	1.44E+00	2.20E+00		ABCA	3.62E+04	1.51E+04	9.13E+04	2.52E+04	1.87E+00
$f 3$	NCSFLA	1.30E+07	5.75E+07	4.59E+08	2.12E-01	2.97E-01	$f 13$	NCSFLA	5.38E-01	5.92E+00	8.20E+01	0.00E+00	2.28E-01
	MSFLA	1.83E+07	6.47E+07	4.59E+08	1.86E+05	2.76E-01		MSFLA	5.89E+01	6.94E+00	8.20E+01	5.40E+01	2.58E-01
	GSF²LA	8.61E+06	5.46E+07	4.59E+08	4.02E+04	2.10E-01		GSF²LA	8.07E+01	3.64E-01	8.20E+01	8.05E+01	1.85E-01
	SFLA	7.45E+06	4.87E+07	4.59E+08	1.20E+05	1.98E-01		SFLA	7.81E+01	3.70E+00	8.20E+01	7.20E+01	1.84E-01
	HSA	4.35E+08	1.53E+07	4.59E+08	4.25E+08	1.89E-01		HSA	8.20E+01	0.00E+00	8.20E+01	8.20E+01	1.50E-01
	ABCA	1.30E+08	2.10E+08	8.07E+08	4.88E+04	1.96E+00		ABCA	7.42E+01	9.04E+00	9.33E+01	6.07E+01	1.95E+00
$f 4$	NCSFLA	4.58E+43	6.45E+44	9.15E+45	5.20E+01	4.35E-01	$f 14$	NCSFLA	1.28E+03	5.29E+03	4.67E+04	6.79E-06	2.02E-01
	MSFLA	1.04E+45	2.87E+45	9.15E+45	1.18E+18	2.87E-01		MSFLA	1.37E+04	1.63E+04	4.67E+04	6.04E+02	2.07E-01
	GSF²LA	9.69E+44	2.80E+45	9.15E+45	1.00E+00	2.24E-01		GSF²LA	6.36E+03	1.41E+04	4.67E+04	4.06E+01	1.75E-01
	SFLA	9.81E+44	2.81E+45	9.15E+45	1.30E+01	2.22E-01		SFLA	6.37E+03	1.41E+04	4.67E+04	3.73E+01	1.80E-01
	HSA	7.52E+45	1.49E+45	9.15E+45	6.16E+45	1.90E-01		HSA	4.67E+04	0.00E+00	4.67E+04	4.67E+04	1.46E-01
	ABCA	1.32E+48	4.79E+48	2.50E+49	4.98E+06	1.25E+00		ABCA	1.18E+04	1.94E+04	7.32E+04	9.36E-01	1.48E+00
$f 5$	NCSFLA	2.84E+04	3.99E+05	5.65E+06	4.16E-01	4.25E-01	$f 15$	NCSFLA	5.07E+06	3.19E+07	3.19E+08	-9.21E-01	5.57E-01
	MSFLA	1.72E+05	9.64E+05	5.65E+06	1.91E+02	4.11E-01		MSFLA	2.03E+07	7.02E+07	3.19E+08	2.31E-01	4.07E-01
	GSF²LA	2.16E+05	1.07E+06	5.65E+06	1.59E+01	3.30E-01		GSF²LA	1.31E+07	5.77E+07	3.19E+08	-1.00E+00	2.47E-01
	SFLA	2.05E+05	1.02E+06	5.65E+06	1.10E+01	3.45E-01		SFLA	1.27E+07	5.68E+07	3.19E+08	-9.92E-01	2.48E-01
	HSA	1.33E+06	2.40E+06	5.65E+06	5.80E+02	1.48E-01		HSA	3.19E+08	5.36E-07	3.19E+08	3.19E+08	1.97E-01
	ABCA	2.58E+07	1.89E+08	2.08E+09	2.79E+02	1.26E+00		ABCA	3.58E+07	7.76E+07	4.59E+08	-9.07E-01	1.21E+00
$f 6$	NCSFLA	3.60E+01	1.33E+02	7.63E+02	4.61E-07	1.65E-01	$f 16$	NCSFLA	1.23E+07	6.72E+07	6.09E+08	2.10E-05	6.63E-01
	MSFLA	9.07E+01	1.56E+02	7.63E+02	6.67E+00	1.76E-01		MSFLA	4.54E+07	1.44E+08	6.09E+08	2.19E+01	4.24E-01
	GSF²LA	1.85E+01	1.02E+02	7.63E+02	7.98E-01	1.67E-01		GSF²LA	2.82E+07	1.16E+08	6.09E+08	1.64E-01	2.69E-01
	SFLA	2.21E+01	1.08E+02	7.63E+02	1.43E+00	1.71E-01		SFLA	2.76E+07	1.15E+08	6.09E+08	2.39E-01	2.81E-01
	HSA	7.63E+02	2.16E-12	7.63E+02	7.63E+02	1.01E-01		HSA	6.09E+08	2.03E-06	6.09E+08	6.09E+08	1.55E-01
	ABCA	1.61E+04	5.40E+04	3.67E+05	1.21E+03	1.39E+00		ABCA	1.47E+08	2.63E+08	9.27E+08	3.90E-02	1.13E+00
$f 7$	NCSFLA	1.09E+01	1.10E+01	4.91E+01	3.99E+00	2.53E+01	$f 17$	NCSFLA	2.18E+04	9.17E+04	8.17E+05	6.24E-01	8.17E-01
	MSFLA	2.32E+01	6.47E+00	4.91E+01	1.78E+01	3.54E+01		MSFLA	1.83E+04	8.60E+04	8.17E+05	1.36E+03	5.62E-01
	GSF²LA	8.67E+00	8.34E+00	4.91E+01	3.82E+00	1.36E+01		GSF²LA	1.75E+04	9.77E+04	8.17E+05	1.72E+03	2.94E-01
	SFLA	8.93E+00	8.17E+00	4.91E+01	2.99E+00	1.60E+01		SFLA	1.73E+04	8.45E+04	8.17E+05	3.80E+03	2.99E-01
	HSA	4.77E+01	1.40E+00	4.91E+01	4.55E+01	2.08E-01		HSA	8.17E+05	2.33E-10	8.17E+05	8.17E+05	1.56E-01
	ABCA	1.44E+01	1.28E+01	5.54E+01	2.61E+00	2.36E+00		ABCA	1.66E+05	3.60E+05	1.69E+06	1.72E+02	2.13E+00
$f 8$	NCSFLA	3.35E-01	2.47E-01	9.62E-01	9.74E-03	4.71E+01	$f 18$	NCSFLA	-2.69E+19	1.14E+19	9.46E+03	-3.75E+19	1.58E+00
	MSFLA	5.32E-01	1.35E-01	9.62E-01	4.53E-01	6.72E+01		MSFLA	9.26E+03	3.58E+01	9.46E+03	9.25E+03	4.48E-01
	GSF²LA	4.82E-01	2.33E-01	9.62E-01	1.90E-01	3.40E+01		GSF²LA	9.38E+03	1.50E+02	9.46E+03	9.08E+03	4.48E-01
	SFLA	4.16E-01	1.71E-01	9.62E-01	2.42E-01	5.74E+01		SFLA	9.39E+03	2.02E+01	9.46E+03	9.38E+03	4.20E-01
	HSA	7.67E-01	1.59E-01	9.62E-01	6.37E-01	1.78E-01		HSA	9.25E+03	6.89E+01	9.46E+03	9.03E+03	1.57E-01
	ABCA	3.55E-01	1.22E+00	9.20E+00	6.02E-03	2.02E+00		ABCA	5.30E+03	2.25E+03	1.08E+04	2.36E+03	1.44E+00
$f 9$	NCSFLA	5.40E+01	8.83E+01	7.95E+02	2.15E+01	1.82E-01	$f 19$	NCSFLA	1.30E+04	5.88E+03	7.64E+04	1.16E+04	2.32E+00
	MSFLA	7.72E+02	1.75E+01	7.95E+02	7.48E+02	1.92E-01		MSFLA	2.83E+04	1.73E+04	7.64E+04	1.29E+04	1.14E+00
	GSF²LA	5.65E+02	9.01E+01	7.95E+02	4.95E+02	1.80E-01		GSF²LA	3.43E+04	1.59E+04	7.64E+04	1.93E+04	1.79E+00
	SFLA	5.78E+02	8.68E+01	7.95E+02	5.07E+02	1.72E-01		SFLA	3.47E+04	1.35E+04	7.64E+04	1.75E+04	1.86E+00
	HSA	7.08E+02	5.50E+01	7.95E+02	6.73E+02	1.67E-01		HSA	6.58E+04	4.52E+03	7.64E+04	6.26E+04	1.54E-01
	ABCA	4.78E+02	2.75E+02	1.29E+03	2.28E+02	1.74E+00		ABCA	2.14E+04	1.53E+04	6.98E+04	1.19E+04	1.29E+00
$f 10$	NCSFLA	2.77E+03	5.60E+03	4.78E+04	9.31E+02	2.00E-01	$f 20$	NCSFLA	1.24E+02	1.46E+01	2.45E+02	1.18E+02	1.25E+00
	MSFLA	4.69E+04	9.22E+02	4.78E+04	4.57E+04	1.91E-01		MSFLA	1.68E+02	4.36E+01	2.45E+02	1.29E+02	9.06E-01
	GSF²LA	3.50E+04	5.28E+03	4.78E+04	3.13E+04	1.76E-01		GSF²LA	1.80E+02	2.94E+01	2.45E+02	1.40E+02	1.15E+00
	SFLA	3.49E+04	5.33E+03	4.78E+04	3.06E+04	1.77E-01		SFLA	1.68E+02	3.64E+01	2.45E+02	1.30E+02	9.91E-01
	HSA	4.78E+04	1.31E-10	4.78E+04	4.78E+04	1.57E-01		HSA	2.45E+02	4.83E-13	2.45E+02	2.45E+02	1.50E-01
	ABCA	2.27E+04	1.48E+04	7.93E+04	1.34E+04	1.51E+00		ABCA	1.57E+02	4.27E+01	2.90E+02	1.24E+02	3.01E+00

Table 2 Comparison of convergence precision and convergence speed

函数	算法	平均最优值	标准差	最大值	最小值	运行时间/s	函数	算法	平均最优值	标准差	最大值	最小值	运行时间/s
$f 1$	NCSFLA	1.34E+03	5.34E+03	4.68E+04	6.25E-05	1.51E-01	$f 11$	NCSFLA	4.91E+00	1.47E+01	1.18E+02	8.66E-05	1.98E-01
	MSFLA	1.34E+04	1.65E+04	4.68E+04	4.53E+02	1.85E-01		MSFLA	1.21E+01	1.55E+01	1.18E+02	1.98E+00	2.20E-01
	GSF²LA	6.37E+03	1.42E+04	4.68E+04	1.64E+01	1.76E-01		GSF²LA	4.57E+00	1.49E+01	1.18E+02	8.48E-01	1.78E-01
	SFLA	6.37E+03	1.41E+04	4.68E+04	4.24E+01	1.70E-01		SFLA	5.29E+00	1.29E+01	1.18E+02	1.56E+00	1.78E-01
	HSA	4.64E+04	3.01E+02	4.68E+04	4.54E+04	1.09E-01		HSA	1.07E+02	4.51E+00	1.18E+02	1.05E+02	1.17E-01
	ABCA	1.66E+04	2.01E+04	7.31E+04	1.45E+01	1.84E+00		ABCA	3.53E+08	3.92E+09	5.45E+10	6.03E-01	1.62E+00
$f 2$	NCSFLA	1.26E+01	4.82E+01	4.22E+02	2.97E-02	4.30E-01	$f 12$	NCSFLA	2.19E+03	8.82E+03	8.35E+04	6.18E+00	3.05E-01
	MSFLA	4.14E+02	1.92E+01	4.22E+02	3.45E+02	2.71E-01		MSFLA	6.00E+04	1.95E+04	8.35E+04	3.33E+04	2.41E-01
	GSF²LA	3.20E+02	1.15E+02	4.22E+02	9.77E+01	2.47E-01		GSF²LA	2.79E+04	3.49E+04	8.35E+04	2.29E+02	2.15E-01
	SFLA	3.21E+02	1.15E+02	4.22E+02	9.81E+01	2.50E-01		SFLA	2.62E+04	3.45E+04	8.35E+04	2.63E+02	2.18E-01
	HSA	4.19E+02	1.98E+00	4.22E+02	4.18E+02	1.89E-01		HSA	8.35E+04	0.00E+00	8.35E+04	8.35E+04	7.60E-02
	ABCA	1.45E+02	1.88E+02	5.87E+02	1.44E+00	2.20E+00		ABCA	3.62E+04	1.51E+04	9.13E+04	2.52E+04	1.87E+00
$f 3$	NCSFLA	1.30E+07	5.75E+07	4.59E+08	2.12E-01	2.97E-01	$f 13$	NCSFLA	5.38E-01	5.92E+00	8.20E+01	0.00E+00	2.28E-01
	MSFLA	1.83E+07	6.47E+07	4.59E+08	1.86E+05	2.76E-01		MSFLA	5.89E+01	6.94E+00	8.20E+01	5.40E+01	2.58E-01
	GSF²LA	8.61E+06	5.46E+07	4.59E+08	4.02E+04	2.10E-01		GSF²LA	8.07E+01	3.64E-01	8.20E+01	8.05E+01	1.85E-01
	SFLA	7.45E+06	4.87E+07	4.59E+08	1.20E+05	1.98E-01		SFLA	7.81E+01	3.70E+00	8.20E+01	7.20E+01	1.84E-01
	HSA	4.35E+08	1.53E+07	4.59E+08	4.25E+08	1.89E-01		HSA	8.20E+01	0.00E+00	8.20E+01	8.20E+01	1.50E-01
	ABCA	1.30E+08	2.10E+08	8.07E+08	4.88E+04	1.96E+00		ABCA	7.42E+01	9.04E+00	9.33E+01	6.07E+01	1.95E+00
$f 4$	NCSFLA	4.58E+43	6.45E+44	9.15E+45	5.20E+01	4.35E-01	$f 14$	NCSFLA	1.28E+03	5.29E+03	4.67E+04	6.79E-06	2.02E-01
	MSFLA	1.04E+45	2.87E+45	9.15E+45	1.18E+18	2.87E-01		MSFLA	1.37E+04	1.63E+04	4.67E+04	6.04E+02	2.07E-01
	GSF²LA	9.69E+44	2.80E+45	9.15E+45	1.00E+00	2.24E-01		GSF²LA	6.36E+03	1.41E+04	4.67E+04	4.06E+01	1.75E-01
	SFLA	9.81E+44	2.81E+45	9.15E+45	1.30E+01	2.22E-01		SFLA	6.37E+03	1.41E+04	4.67E+04	3.73E+01	1.80E-01
	HSA	7.52E+45	1.49E+45	9.15E+45	6.16E+45	1.90E-01		HSA	4.67E+04	0.00E+00	4.67E+04	4.67E+04	1.46E-01
	ABCA	1.32E+48	4.79E+48	2.50E+49	4.98E+06	1.25E+00		ABCA	1.18E+04	1.94E+04	7.32E+04	9.36E-01	1.48E+00
$f 5$	NCSFLA	2.84E+04	3.99E+05	5.65E+06	4.16E-01	4.25E-01	$f 15$	NCSFLA	5.07E+06	3.19E+07	3.19E+08	-9.21E-01	5.57E-01
	MSFLA	1.72E+05	9.64E+05	5.65E+06	1.91E+02	4.11E-01		MSFLA	2.03E+07	7.02E+07	3.19E+08	2.31E-01	4.07E-01
	GSF²LA	2.16E+05	1.07E+06	5.65E+06	1.59E+01	3.30E-01		GSF²LA	1.31E+07	5.77E+07	3.19E+08	-1.00E+00	2.47E-01
	SFLA	2.05E+05	1.02E+06	5.65E+06	1.10E+01	3.45E-01		SFLA	1.27E+07	5.68E+07	3.19E+08	-9.92E-01	2.48E-01
	HSA	1.33E+06	2.40E+06	5.65E+06	5.80E+02	1.48E-01		HSA	3.19E+08	5.36E-07	3.19E+08	3.19E+08	1.97E-01
	ABCA	2.58E+07	1.89E+08	2.08E+09	2.79E+02	1.26E+00		ABCA	3.58E+07	7.76E+07	4.59E+08	-9.07E-01	1.21E+00
$f 6$	NCSFLA	3.60E+01	1.33E+02	7.63E+02	4.61E-07	1.65E-01	$f 16$	NCSFLA	1.23E+07	6.72E+07	6.09E+08	2.10E-05	6.63E-01
	MSFLA	9.07E+01	1.56E+02	7.63E+02	6.67E+00	1.76E-01		MSFLA	4.54E+07	1.44E+08	6.09E+08	2.19E+01	4.24E-01
	GSF²LA	1.85E+01	1.02E+02	7.63E+02	7.98E-01	1.67E-01		GSF²LA	2.82E+07	1.16E+08	6.09E+08	1.64E-01	2.69E-01
	SFLA	2.21E+01	1.08E+02	7.63E+02	1.43E+00	1.71E-01		SFLA	2.76E+07	1.15E+08	6.09E+08	2.39E-01	2.81E-01
	HSA	7.63E+02	2.16E-12	7.63E+02	7.63E+02	1.01E-01		HSA	6.09E+08	2.03E-06	6.09E+08	6.09E+08	1.55E-01
	ABCA	1.61E+04	5.40E+04	3.67E+05	1.21E+03	1.39E+00		ABCA	1.47E+08	2.63E+08	9.27E+08	3.90E-02	1.13E+00
$f 7$	NCSFLA	1.09E+01	1.10E+01	4.91E+01	3.99E+00	2.53E+01	$f 17$	NCSFLA	2.18E+04	9.17E+04	8.17E+05	6.24E-01	8.17E-01
	MSFLA	2.32E+01	6.47E+00	4.91E+01	1.78E+01	3.54E+01		MSFLA	1.83E+04	8.60E+04	8.17E+05	1.36E+03	5.62E-01
	GSF²LA	8.67E+00	8.34E+00	4.91E+01	3.82E+00	1.36E+01		GSF²LA	1.75E+04	9.77E+04	8.17E+05	1.72E+03	2.94E-01
	SFLA	8.93E+00	8.17E+00	4.91E+01	2.99E+00	1.60E+01		SFLA	1.73E+04	8.45E+04	8.17E+05	3.80E+03	2.99E-01
	HSA	4.77E+01	1.40E+00	4.91E+01	4.55E+01	2.08E-01		HSA	8.17E+05	2.33E-10	8.17E+05	8.17E+05	1.56E-01
	ABCA	1.44E+01	1.28E+01	5.54E+01	2.61E+00	2.36E+00		ABCA	1.66E+05	3.60E+05	1.69E+06	1.72E+02	2.13E+00
$f 8$	NCSFLA	3.35E-01	2.47E-01	9.62E-01	9.74E-03	4.71E+01	$f 18$	NCSFLA	-2.69E+19	1.14E+19	9.46E+03	-3.75E+19	1.58E+00
	MSFLA	5.32E-01	1.35E-01	9.62E-01	4.53E-01	6.72E+01		MSFLA	9.26E+03	3.58E+01	9.46E+03	9.25E+03	4.48E-01
	GSF²LA	4.82E-01	2.33E-01	9.62E-01	1.90E-01	3.40E+01		GSF²LA	9.38E+03	1.50E+02	9.46E+03	9.08E+03	4.48E-01
	SFLA	4.16E-01	1.71E-01	9.62E-01	2.42E-01	5.74E+01		SFLA	9.39E+03	2.02E+01	9.46E+03	9.38E+03	4.20E-01
	HSA	7.67E-01	1.59E-01	9.62E-01	6.37E-01	1.78E-01		HSA	9.25E+03	6.89E+01	9.46E+03	9.03E+03	1.57E-01
	ABCA	3.55E-01	1.22E+00	9.20E+00	6.02E-03	2.02E+00		ABCA	5.30E+03	2.25E+03	1.08E+04	2.36E+03	1.44E+00
$f 9$	NCSFLA	5.40E+01	8.83E+01	7.95E+02	2.15E+01	1.82E-01	$f 19$	NCSFLA	1.30E+04	5.88E+03	7.64E+04	1.16E+04	2.32E+00
	MSFLA	7.72E+02	1.75E+01	7.95E+02	7.48E+02	1.92E-01		MSFLA	2.83E+04	1.73E+04	7.64E+04	1.29E+04	1.14E+00
	GSF²LA	5.65E+02	9.01E+01	7.95E+02	4.95E+02	1.80E-01		GSF²LA	3.43E+04	1.59E+04	7.64E+04	1.93E+04	1.79E+00
	SFLA	5.78E+02	8.68E+01	7.95E+02	5.07E+02	1.72E-01		SFLA	3.47E+04	1.35E+04	7.64E+04	1.75E+04	1.86E+00
	HSA	7.08E+02	5.50E+01	7.95E+02	6.73E+02	1.67E-01		HSA	6.58E+04	4.52E+03	7.64E+04	6.26E+04	1.54E-01
	ABCA	4.78E+02	2.75E+02	1.29E+03	2.28E+02	1.74E+00		ABCA	2.14E+04	1.53E+04	6.98E+04	1.19E+04	1.29E+00
$f 10$	NCSFLA	2.77E+03	5.60E+03	4.78E+04	9.31E+02	2.00E-01	$f 20$	NCSFLA	1.24E+02	1.46E+01	2.45E+02	1.18E+02	1.25E+00
	MSFLA	4.69E+04	9.22E+02	4.78E+04	4.57E+04	1.91E-01		MSFLA	1.68E+02	4.36E+01	2.45E+02	1.29E+02	9.06E-01
	GSF²LA	3.50E+04	5.28E+03	4.78E+04	3.13E+04	1.76E-01		GSF²LA	1.80E+02	2.94E+01	2.45E+02	1.40E+02	1.15E+00
	SFLA	3.49E+04	5.33E+03	4.78E+04	3.06E+04	1.77E-01		SFLA	1.68E+02	3.64E+01	2.45E+02	1.30E+02	9.91E-01
	HSA	4.78E+04	1.31E-10	4.78E+04	4.78E+04	1.57E-01		HSA	2.45E+02	4.83E-13	2.45E+02	2.45E+02	1.50E-01
	ABCA	2.27E+04	1.48E+04	7.93E+04	1.34E+04	1.51E+00		ABCA	1.57E+02	4.27E+01	2.90E+02	1.24E+02	3.01E+00

Fig.3 Evolution curves of f1

Fig.4 Evolution curves of f2

Fig.5 Evolution curves of f3

Fig.6 Evolution curves of f4

Fig.7 Evolution curves of f5

Fig.8 Evolution curves of f6

Fig.9 Evolution curves of f7

Fig.10 Evolution curves of f8

Fig.11 Evolution curves of f9

Fig.12 Evolution curves of f10

Fig.13 Evolution curves of f11

Fig.14 Evolution curves of f12

Fig.15 Evolution curves of f13

Fig.16 Evolution curves of f14

Fig.17 Evolution curves of f15

Fig.18 Evolution curves of f16

Fig.19 Evolution curves of f17

Fig.20 Evolution curves of f18

Fig.21 Evolution curves of f19

Fig.22 Evolution curves of f20

Table 3 Comparison of vehicle routing optimization algorithms for Solomon example data while K = 2

算法	最短路径值	车辆1优化路径	车辆2优化路径
NCSFLA-CVRP	554.08	5-81-4-46-22-90-6-79	19-11-59-51-65-34-50-85
MSFLA-CVRP	629.68	59-46-4-81-9-11-67-94	22-1-3-50-88-82-95-87
GSF²LA-CVRP	634.21	98-16-4-17-58-67-53-65-82	38-59-11-75-32-69-8-12
SFLA-CVRP	647.69	56-79-85-22-58-18-57-86-74	12-96-78-11-5-50-31-29

Table 4 Comparison of vehicle routing optimization algorithms for Solomon example data while K = 4

算法	最短路径值	车辆1优化路径	车辆2优化路径	车辆3优化路径	车辆4优化路径
NCSFLA-CVRP	479.75	69-40-41-49-11-5-46-12	71-4-76-96-24-83-61-13-86-98-60-64-47	33-65-9-38-55-77-44-19	34-63-22-20-17-10-91-66-78
MSFLA-CVRP	552.85	87-97-17-68-36-89-85-57-65	12-58-16-27-34-95-33-30-35-83	90-64-74-46-60-75-28-21-53-98-18-80	11-86-94-56-22-26-38
GSF²LA-CVRP	566.62	64-21-15-80-5-87-53-36-92-83-52	96-89-42-85-93-94-25-44-6-17	23-76-97-73-49-30-33-40-69-65	10-38-61-18-68-75-95
SFLA-CVRP	578.84	86-7-85-11-82-45-90-50-8-3	95-36-49-34-22-93-66	78-21-73-96-57-62-87-24-23-48	38-26-46-13-97-61-35-70-91-68-55

Fig.23 Evolution curves of each algorithm for Solomon example data while K = 2

Fig.24 Evolution curves of each algorithm for Solomon example data while K = 4

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