Journal of Shandong University(Engineering Science) ›› 2020, Vol. 50 ›› Issue (2): 34-43.doi: 10.6040/j.issn.1672-3961.0.2019.294

• Machine Learning & Data Mining • Previous Articles     Next Articles

Improved bird swarm algorithms based on mixed decision making

Wei YAN(),Damin ZHANG*(),Huijuan ZHANG,Ziyun XI,Zhongyun CHEN   

  1. College of Big Data and Information Engineering, Guizhou University, Guiyang 550025, Guiyang, China
  • Received:2019-06-10 Online:2020-04-20 Published:2020-04-16
  • Contact: Damin ZHANG E-mail:349552812@qq.com;1203813362@qq.com
  • Supported by:
    贵州省自然科学基金资助项目(黔科合基础[2017]1047号)

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Abstract:

Aiming at the problems of low precision and easy to fall into local optimum in solving complex function problems of traditional bird swarm algorithm (BSA), an improved bird swarm algorithm based on mixed decision-making was proposed while retaining the simplicity of BSA. The centroid opposition-based learning was used to initialize the bird population and maintain the better spatial solution distribution of the bird flock. In order to balance the global search ability and local detection ability of the algorithm in the optimization process, the period time of the birds flying to another area was dynamically adjusted. The weighting strategy of adaptive cosine function and weighted averaging idea were introduced to improve the producer's foraging formula, so as to increase the ability of the algorithm to get rid of difficulties after falling into local optimum. The performance of improved bird swarm algorithm based on mixed decision-making, bird swarm algorithm and particle swarm optimization were compared on the basis of nine test functions. The results showed that the accuracy and speed of the improved algorithm were greatly improved in the test of single-peak and multi-peak functions.

Key words: bird swarm algorithm, centroid opposition-based learning, the weighting strategy of adaptive cosine function, mixed decision making, the centroid opposition-based learning

CLC Number: 

  • TP391

Fig.1

Function curve of S-shaped"

Fig.2

Function curve of transformed S-shaped"

Table 1

Experimental setting parameters"

算法 参数设置
IBSA c1=c2=1.5, a1=a2=1, FQ=[4, 10], ω=[4, 9], a=12
BSA c1=c2=1.5, a1=a2=1, FQ=10
PSO ω=0.729, c1=c2=1.494 45,

Table 2

Basic information on standard test functions"

函数名 表达式 范围 最优值
Schwefel 1.2 $F_{1}(x)=\sum\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{i} x_{j}^{2}\right)^{2}$ [-10, 10] 0
Tablet $F_{2}(x)=10^{6} x_{1}^{2}+\sum\limits_{i=2}^{n} x_{i}^{2}$ [-100, 100] 0
Schwefel 2.22 $F_{3}(x)=\sum\limits_{i=1}^{n}\left|x_{i}\right|+\prod\limits_{i=1}^{n}\left|x_{i}\right|$ [-10, 10] 0
Sphere $F_{4}(x)=\sum\limits_{i=1}^{n} x_{i}^{2}$ [-100, 100] 0
Alpine $F_{5}(x)=\sum\limits_{i=1}^{n}\left|x_{i} \sin \left(x_{i}\right)+0.1 x_{i}\right|$ [-10, 10] 0
Rastrigin $F_{6}(x)=\sum\limits_{i=1}^{n}\left(x_{i}^{2}-10 \cos \left(2 \pi x_{i}\right)+10\right)$ [-5.12, 5.12] 0
Griewank $F_{7}(x)=\sum\limits_{i=1}^{n} \frac{x_{i}^{2}}{4000}-\prod\limits_{i=1}^{n} \cos \left(\frac{x_{i}}{\sqrt{i}}\right)+1$ [-600, 600] 0
Powell $F_{8}(x)=\sum\limits_{i=1}^{n / 4}\left[\left(x_{4 i-3}+10 x_{4 i-2}\right)^{2}+5\left(x_{4 i-1}-x_{4 i}\right)^{2}+\left(x_{4 i-2}-2 x_{4 i-1}\right)^{4}+10\left(x_{4 i-3}-x_{4 i}\right)^{4}\right]$ [-4, 5] 0
Zakharov $F_{9}(x)=\sum\limits_{i=1}^{n} x_{i}^{2}+\left(\sum\limits_{i=1}^{n} 0.5 i x_{i}\right)^{2}+\left(\sum\limits_{i=1}^{n} 0.5 i x_{i}\right)^{4}$ [-5, 10] 0

Table 3

Performance comparison between different algorithms"

函数 维数 算法 最优值 最差值 平均值 标准差 成功率/% 平均耗时/s
IBSA 0 0 0 0 100 1.464 4
10 BSA 0 0 0 0 100 1.865 5
PSO 1.670×10-134 6.565×10-45 6.565×10-47 6.565×10-46 97 1.187 2
IBSA 0 0 0 0 100 1.440 7
F1 20 BSA 0 0 0 0 100 1.826 9
PSO 5.522×10-7 3.037×10-3 2.129×10-4 4.340×10-4 0 1.217 4
IBSA 0 0 0 0 100 1.510 2
50 BSA 0 0 0 0 100 1.914 6
PSO 0.841 2.251×101 6.463 4.061 0 1.302 9
IBSA 0 0 0 0 100 1.829 9
10 BSA 2.579×10-252 5.409×10-187 7.623×10-189 0 100 2.230 9
PSO 7.388×10-53 1.621×10-4 4.465×10-6 2.167×10-5 0 1.596 4
IBSA 0 0 0 0 100 2.263 6
F2 20 BSA 3.363×10-247 3.795×10-165 4.579×10-167 0 100 2.712 6
PSO 1.359×10-2 2.687 3.255 3.694 0 2.081 7
IBSA 0 0 0 0 100 3.615 9
50 BSA 1.157×10-251 1.243×10-170 1.243×10-172 0 100 3.995 4
PSO 2.576 2.461 7.731 3.557 0 3.313 6
IBSA 4.303×10-221 7.367×10-195 1.539×10-196 0 100 1.833 9
10 BSA 7.813×10-112 2.020×10-77 2.102×10-79 2.020×10-78 20 2.299 6
PSO 7.269×10-5 1.331×10-1 1.326×10-2 2.201×10-2 0 1.617 4
IBSA 1.892×10-223 2.538×10-179 2.538×10-181 0 100 1.897 8
F3 20 BSA 7.098×10-110 6.962×10-74 6.962×10-76 6.962×10-75 15 2.317 0
PSO 2.199×10-1 3.028 1.029 0.597 0 1.581 6
IBSA 5.854×10-220 4.740×10-187 4.740×10-189 0 100 2.022 0
50 BSA 1.339×10-111 1.445×10-77 1.704×10-79 1.466×10-78 14 2.353 3
PSO 6.468 1.965×101 1.309×101 3.204 0 1.689 1
IBSA 0 0 0 0 100 1.453 1
10 BSA 5.489×10-247 7.667×10-182 7.669×10-184 0 100 1.822 3
PSO 1.066×10-56 6.109×10-5 6.113×10-7 6.109×10-6 0 1.160 9
IBSA 0 0 0 0 100 1.495 3
F4 20 BSA 4.011×10-247 1.652×10-183 1.808×10-185 0 100 1.858 1
PSO 0.108×10-1 0.762 0.161 0.150 0 1.235 4
IBSA 0 0 0 0 100 1.553 3
50 BSA 3.162×10-249 4.535×10-191 4.538×10-193 0 100 1.905 1
PSO 2.018 1.120×101 5.751 1.798 0 1.287 4
IBSA 1.067×10-216 5.371×10-194 5.510×10-196 0 100 1.940 5
10 BSA 1.331×10-106 8.476×10-84 1.415×10-85 9.703×10-85 24 2.368 1
PSO 4.454×10-7 0.331 0.017 0.042 0 1.722 3
IBSA 2.531×10-225 1.234×10-190 1.234×10-192 0 100 2.323 3
F5 20 BSA 1.036×10-105 7.532×10-74 7.532×10-76 7.532×10-75 21 2.812 9
PSO 9.482×10-3 2.922 0.408 0.551 0 2.064 9
IBSA 1.103×10-222 7.117×10-186 7.117×10-188 0 100 3.563 3
50 BSA 8.997×10-110 8.052×10-82 9.490×10-84 8.125×10-83 19 3.994 9
PSO 1.985 1.801×101 7.424 2.804 0 3.412 2
IBSA 0 0 0 0 100 1.847 6
10 BSA 0 0 0 0 100 2.264 0
PSO 1.990 2.288×101 8.467 3.793 0 1.613 9
IBSA 0 0 0 0 100 2.284 7
F6 20 BSA 0 0 0 0 100 2.688 4
PSO 6.236 4.507×101 1.817×101 5.829 0 2.073 9
IBSA 0 0 0 0 100 3.611 0
50 BSA 0 0 0 0 100 4.151 3
PSO 7.070×101 1.611×102 1.108×102 1.833×101 0 3.481 4
IBSA 0 0 0 0 100 2.261 9
10 BSA 0 0 0 0 100 2.673 1
PSO 0.832 5.868 2.435 0.899 0 2.102 7
IBSA 0 0 0 0 100 2.6604
F7 20 BSA 0 0 0 0 100 3.064 9
PSO 5.983 1.495×101 9.993 2.020 0 2.553 7
IBSA 0 0 0 0 100 3.882 9
50 BSA 0 0 0 0 100 4.294 4
PSO 2.518×101 5.273×101 3.723×101 4.637 0 3.970 0
IBSA 0 0 0 0 100 1.330 2
10 BSA 0 2.020×10-189 2.020×10-191 0 100 1.771 2
PSO 1.765×10-32 1.690×10-6 1.929×10-8 1.693×10-7 0 1.045 5
IBSA 0 0 0 0 100 1.462 0
F8 20 BSA 3.849×10-255 2.245×10-181 2.286×10-183 0 100 1.887 2
PSO 6.275×102 1.441×101 1.555 2.132 0 1.222 7
IBSA 0 0 0 0 100 1.903 9
50 BSA 8.628×10-250 4.011×10-189 4.011×10-191 0 100 2.340 6
PSO 1.284×101 8.807×101 4.002×101 1.65 0 1.672 1
IBSA 0 0 0 0 100 1.329 4
10 BSA 1.186×10-247 2.661×10-171 2.661×10-173 0 100 1.755 7
PSO 5.982×10-67 9.092×10-6 1.244×10-7 9.287×10-07 0 0.994 7
IBSA 0 0 0 0 100 1.327 4
F9 20 BSA 7.775×10-247 5.183×10-187 5.183×10-189 0 100 1.786 3
PSO 0.076 9 2.947×101 1.267 4.044 0 1.081 5
IBSA 0 0 0 0 100 1.705 5
50 BSA 2.081×10-250 6.203 9×10-168 6.203 9×10-170 0 100 2.080 3
PSO 1.158×101 4.111×102 6.729×101 5.663×101 0 1.407 0

Fig.3

Average convergence curve of test functions"

Table 4

Comparisons of the performance of the algorithm in reference[7] and the improved algorithm in this paper"

文献[7]中的函数 算法
 最优值 最差值 平均值 方差
F2 IBSA 0 0 0 0
AIBSO 1.051 6×10-24 1.006 4×102 1.634 4 1.052 7×101
F3 IBSA 0 0 0 0
AIBSO 1.509 7×10-18 1.463 1×10-1 5.072 8×10-3 2.187 9×10-2
F7 IBSA 0 0 0 0
AIBSO 0 1.400 4×10-8 3.545 1×10-10 2.030 6×10-9

Table 5

Comparisons of the performance of the algorithm in reference[9] and the improved algorithm in this paper"

文献[9]中的函数 算法 最优值 最差值 平均值 标准差
F1 IBSA 0 0 0 0
LFSABSA 3.47×10-200 8.44×10-152 8.44×10-153 2.67×10-152
F2 IBSA 0 0 0 0
LFSABSA 1.27×10-176 3.55×10-155 2.97×10-156 0
F3 IBSA 0 0 0 0
LFSABSA 7.92×10-12 1.19×10-7 1.42×10-8 3.72×10-8

Table 6

Comparisons of the performance of algorithm in reference[10] and the improved algorithm in this paper"

文献[10]中的函数
算法		 最优值 最差值 平均值 标准差
F1 IBSA 0 0 0 0
MMSBSA 2.629 4×10-278 1.867 5×10-214 6.225 0×10-216 4.314 7×10-215
F3 IBSA 3.349 1×10-232 2.146 7×10-190 7.155 6×10-192 0
MMSBSA 1.081 5×10-10 1.523 5×10-5 1.281 9×10-6 3.004 1×10-6
F4 IBSA 1.781 0×10-226 2.521 5×10-191 8.404 9×10-193 0
MMSBSA 3.818 3×10-4 0.133 6 0.013 5 0.034 0
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