Improvement of the Seagull Algorithm and Its Application in Engineering Design Optimization

doi:10.11871/jfdc.issn.2096-742X.2022.06.012

Abstract

Abstract:

[Objective] This paper addresses the problem that the seagull algorithm has the defects of slow convergence speed and easy to fall into local optimization. [Methods] Firstly, the seagull population is initialized by combining the Fuch chaotic mapping and elite reverse learning strategy to improve the population quality. Secondly, the characteristic parameter A of its behavior is improved according to the cosine function to make the linear search of Seagull algorithm nonlinear. Finally, by adding Levy flight mechanism to increase the randomness of seagull flight, the algorithm is further optimized. [Results] The performance of I-SOA is tested by nine benchmark functions and three engineering design optimization problems. The experimental results show that for the nine benchmark functions, the I-SOA algorithm is superior to the standard SOA, PSO, and GA algorithms in optimization accuracy and convergence speed. Especially when solving f7 and f9, the theoretical optimal solution 0 is obtained. For the three engineering design optimization problems, I-SOA algorithm has obvious advantages in optimization accuracy and convergence speed compared to the standard SOA algorithm, and is stronger in adaptability and stability compared with the optimal value of other swarm intelligence optimization algorithms. [Conclusions] I-SOA algorithm has a great performance in benchmark functions and engineering design optimization problems, which proves the effectiveness of the improvement of the seagull algorithm.

Key words: seagull algorithm, Fuch chaotic mapping, elite reverse learning strategy, cosine function, Levy flight

YANG Tao,DAI Jian. Improvement of the Seagull Algorithm and Its Application in Engineering Design Optimization[J]. Frontiers of Data and Computing, 2022, 4(6): 129-144, https://cstr.cn/32002.14.jfdc.CN10-1649/TP.2022.06.012.

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测试函数	维度	搜索空间	最优值	测试函数类型
${{f}_{1}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,x_{i}^{2}$	30	[-100,100]	0	单模态
${{f}_{2}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left\| {{x}_{i}} \right\|+\underset{i=1}{\overset{n}{\mathop \prod }}\,\left\| {{x}_{i}} \right\|$	30	[-10,0]	0	单模态
${{f}_{3}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \underset{j=1}{\overset{i}{\mathop \sum }}\,{{x}_{j}} \right)}^{2}}$	30	[-100,100]	0	单模态
${{f}_{4}}=max\left\{ \left\| {{x}_{i}} \right\|,1\le i\le n \right\}$	30	[-100,100]	0	单模态
${{f}_{5}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,ix_{i}^{4}+random\left[ 0,1 \right)$	30	[-1.28,1.28]	0	单模态
${{f}_{6}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,-{{x}_{i}}\sin \left( \sqrt{\left\| {{x}_{i}} \right\|} \right)$	30	[-500,500]	-418.98×Dimⁿ	多模态
${{f}_{7}}=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ x_{i}^{2}-10\cos \left( 2\pi {{x}_{i}} \right)+10 \right]$	30	[-5.12,5.12]	0	多模态
${{f}_{8}}=e+20-20exp\left( -0.2\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\mathop \sum }}\,x_{i}^{2}} \right)-exp\left( \frac{1}{n}\underset{i=1}{\overset{n}{\mathop \sum }}\,\cos 2\pi {{x}_{i}} \right)$	30	[-32,32]	0	多模态
${{f}_{9}}=\frac{1}{4000}\underset{i=1}{\overset{n}{\mathop \sum }}\,x_{i}^{2}-\underset{i=1}{\overset{n}{\mathop \prod }}\,\cos \left( \frac{{{x}_{i}}}{\sqrt{i}} \right)+1$	30	[-600,600]	0	多模态

测试函数	算法	最优值	最差值	平均值	标准差
f₁	I-SOA	1.59E-44	8.88E-41	7.58767E-42	1.78334E-41
	SOA	2.24E-11	4.05E-06	5.09355E-07	9.56157E-07
	PSO	4.44E-04	2.44E-02	0.005362221	0.004785459
	GA	1.82E+02	7.06E+03	1899.030632	1538.904477
f₂	I-SOA	1.37E-28	2.18E-26	5.53778E-27	5.71373E-27
	SOA	1.34E-08	2.10E-06	3.58324E-07	4.38396E-07
	PSO	2.27E-02	1.47E-01	0.065551862	0.031852805
	GA	1.47E+01	5.03E+01	27.71107462	9.175842817
f₃	I-SOA	1.32E-11	1.84E-07	1.58772E-08	3.61107E-08
	SOA	1.64E+00	1.86E+03	190.6544601	336.4194883
	PSO	2.64E+01	1.61E+02	81.75993426	29.12498642
	GA	1.38E+04	4.60E+04	27217.60822	7318.158103
f₄	I-SOA	9.68E-11	8.68E-09	1.46604E-09	2.10226E-09
	SOA	1.71E-02	4.58E+00	0.510128247	0.881138441
	PSO	7.50E-01	2.10E+00	1.258233835	0.271775249
	GA	3.25E+01	8.18E+01	54.37515244	10.53775311
f₅	I-SOA	5.97E-05	1.87E-03	0.000674945	0.000536371
	SOA	6.53E-03	1.32E-01	0.047905832	0.033455955
	PSO	7.58E-02	4.03E-01	0.202157664	0.086749957
	GA	3.21E-02	4.93E-01	0.123260057	0.087433248
f₆	I-SOA	-9.93E+03	-7.68E+03	-9100.77899	524.5781804
	SOA	-4.70E+03	-3.65E+03	-4081.974185	234.6872121
	PSO	-8.18E+03	-2.82E+03	-5943.56994	1543.342753
	GA	-3.33E+03	-1.25E+03	-2254.340174	500.2642696
f₇	I-SOA	0.00E+00	0.00E+00	0	0
	SOA	1.07E-09	3.35E-03	0.000112695	0.000601548
	PSO	2.83E+01	7.96E+01	53.67875334	12.60401362
	GA	1.41E+02	2.67E+02	186.6718345	29.41262777
f₈	I-SOA	8.88E-16	7.99E-15	4.44089E-15	9.17307E-16
	SOA	1.57E-06	6.48E-04	6.0513E-05	0.000117366
	PSO	1.61E-02	1.08E-01	0.047530746	0.026267322
	GA	1.82E+01	2.04E+01	19.41380797	0.482471364
f₉	I-SOA	0.00E+00	3.69E-02	0.004924319	0.008916789
	SOA	4.88E-09	4.42E-01	0.034136721	0.084645165
	PSO	7.34E-04	1.61E-01	0.021110466	0.029915834
	GA	2.32E+00	4.64E+01	16.72956884	11.74057554

算法	最优值	平均值	最差值	标准差
SOA	5897.8055	5982.6953	6309.6429	102.9297
I-SOA	5885.4245	5886.1436	5894.7539	1.61157

算法	T_S	T_h	R	L	最优值
Improved HS	1.125000	0.625000	58.290150	43.692680	7197.7300
GSA	1.125000	0.625000	55.988659	84.454202	8538.8359
PSO(He and Wang)	0.812500	0.437500	42.091266	76.746500	6061.0777
GA(Coello)	0.812500	0.434500	40.323900	200.000000	6288.7445
GA(Coello and Montes)	0.812500	0.437500	40.097398	176.654050	6059.9463
GA(Deb and Gene)	0.937500	0.500000	48.329000	48.329000	6410.3811
ES(Montes and Coello)	0.812500	0.437500	42.098087	176.640518	6059.7456
DE(Huang et al.)	0.812500	0.437500	42.098411	176.637690	6059.7340
ACO(Kaveh and Talataheri)	0.812500	0.437500	42.103624	176.572656	6059.0888
Lagrangian multiplier	1.125000	0.625000	58.291000	43.690000	7198.0428
Branch-bound (San dgren)	1.125000	0.625000	47.700000	117.701000	8129.1036
MGWO-III	0.778197	0.384675	40.315493	199.959320	5884.0616
MGWO-II	0.778403	0.384823	40.331283	199.749890	5884.0810
MGWO-I	0.778181	0.384821	40.318142	199.931940	5884.2166
GWO	0.778535	0.384960	40.337793	199.650290	5884.3689
SCA	0.817577	0.417932	41.749390	183.572700	6137.3724
MVO	0.845719	0.418564	43.816270	156.381640	6011.5148
PSO	0.778961	0.384683	40.320913	200.000000	5891.3879
ERDSSA	0.778206	0.384674	40.321534	199.976268	5885.4771
SOA	0.783485	0.387615	40.582386	196.403346	5897.8055
I-SOA	0.778189	0.384660	40.320368	199.990024	5885.4245

算法	最优值	平均值	最差值	标准差
SOA	263.896127	263.8995306	263.9137091	0.003751377
I-SOA	263.8958462	263.8959933	263.8964569	0.000142977