《计算机应用》唯一官方网站 ›› 2022, Vol. 42 ›› Issue (2): 542-549.DOI: 10.11772/j.issn.1001-9081.2021020337
• 先进计算 • 上一篇
收稿日期:
2021-03-08
修回日期:
2021-04-25
接受日期:
2021-04-28
发布日期:
2021-05-11
出版日期:
2022-02-10
通讯作者:
史非凡
作者简介:
史非凡(1993—),男,辽宁沈阳人,硕士研究生,主要研究方向:系统建模与优化;基金资助:
Received:
2021-03-08
Revised:
2021-04-25
Accepted:
2021-04-28
Online:
2021-05-11
Published:
2022-02-10
Contact:
Feifan SHI
About author:
SHI Feifan, born in 1993, M. S. candidate. His research interests include system modeling and optimization.Supported by:
摘要:
针对目前用多目标进化算法(MOEA)处理约束多目标优化问题(CMOP)的研究通常以解决单一类型约束为主,而在面对不同种类的复杂约束时算法难以收敛或者种群分布性差的问题,以基于分解的多目标进化算法(MOEA/D)框架为基础,提出一种基于参考向量的自适应约束多目标进化算法(ARVCMOEA)。首先将参考向量分成主参考向量及辅助参考向量两部分,然后在算法起始阶段通过无约束的辅助参考向量指导种群快速跨越不可行区间,最后通过自适应地调整辅助参考向量的位置及弱化对其的分布性要求来提高算法分布性及搜索能力。实验在30个具有不同种类复杂约束的测试函数上进行了验证,结果表明所提算法面对不同种类的约束时均可以很好地收敛,在总体性能上均优于NSGA-II(Non-dominated Sorting Genetic Algorithm II)、C-MOEA/D(Constraint-MOEA/D)及MOEA/D-DAE,并且相较于目前性能优异的CCMO(Coevolutionary Constrained Multi-objective Optimization framework)在部分测试函数上可以得到更优异的结果。可见,所提算法在面对不同种类的CMOP时具有优异的性能。
中图分类号:
史非凡, 史旭华. 基于参考向量的自适应约束多目标进化算法[J]. 计算机应用, 2022, 42(2): 542-549.
Feifan SHI, Xuhua SHI. Adaptive reference vector based constrained multi-objective evolutionary algorithm[J]. Journal of Computer Applications, 2022, 42(2): 542-549.
函数 | 指标 | NSGA-II | C-MOEA/D | MOEA/D-DAE | CCMO | ARVCMOEAD |
---|---|---|---|---|---|---|
C1-DTLZ1 | IGD | 2.278 2E-3 (2.59E-4) | 1.870 8E-3 (1.30E-4) | 2.101 6E-3 (9.18E-5) | 1.863 9E-3 (3.11E-5) | 1.8434E-3 (7.11E-5) |
GD | 3.594 8E-5 (4.39E-5) | 3.311 8E-5 (3.12E-5) | 2.334 7E-5 (1.44E-5) | 2.173 8E-5 (1.37E-5) | 1.5665E-5 (1.89E-5) | |
C1-DTLZ3 | IGD | 3.002 4E+0 (2.58E+0) | 1.640 7E-2 (3.66E-2) | 5.126 9E-3 (2.57E-4) | 4.272 3E-3 (1.85E-4) | 4.1949E-3 (1.51E-4) |
GD | 3.002 2E-1 (2.58E-1) | 2.178 5E-3 (6.30E-3) | 6.803 5E-5 (3.39E-5) | 8.678 8E-5 (5.01E-5) | 7.3414E-5 (4.46E-5) | |
C2-DTLZ2 | IGD | 5.042 3E-3 (1.73E-4) | 3.966 0E-3 (7.89E-8) | 5.214 6E-3 (2.51E-4) | 4.181 6E-3 (6.27E-5) | 3.9660E-3 (1.45E-8) |
GD | 9.561 0E-5 (2.15E-5) | 1.9469E-6 (7.02E-7) | 8.221 3E-5 (2.36E-5) | 7.373 6E-5 (2.63E-5) | 2.321 3E-6 (2.14E-7) | |
C3-DTLZ4 | IGD | 1.513 4E-1 (4.47E-1) | 8.0812E-3 (1.40E-4) | 7.714 0E-1 (5.45E-1) | 1.000 2E-2 (5.38E-4) | 8.744 7E-3 (6.34E-4) |
GD | 3.107 8E-4 (1.18E-4) | 1.5538E-4 (4.56E-5) | 4.022 7E-2 (3.39E-2) | 3.172 5E-4 (4.51E-5) | 2.071 2E-4 (1.49E-4) | |
DC1-DTLZ1 | IGD | 7.498 1E-4 (2.55E-5) | 1.743 9E-3 (2.49E-5) | 7.179 1E-4 (3.42E-5) | 6.3100E-4 (1.76E-5) | 1.777 0E-3 (1.30E-5) |
GD | 5.851 3E-6 (4.06E-6) | 5.631 5E-6 (1.96E-6) | 9.939 7E-6 (9.97E-6) | 9.332 4E-6 (6.68E-6) | 4.1193E-6 (1.41E-6) | |
DC2-DTLZ1 | IGD | — | 1.186 0E-1 (1.01E-1) | 2.162 2E-3 (5.84E-5) | 1.849 5E-3 (1.68E-5) | 1.7877E-3 (3.77E-6) |
GD | — | 1.252 6E-2 (1.08E-2) | 8.783 8E-6 (7.75E-6) | 1.397 0E-5 (9.56E-6) | 4.4786E-6 (3.71E-6) | |
DC3-DTLZ1 | IGD | 1.707 3E-1 (1.41E-1) | 6.355 5E-2 (9.94E-2) | 6.646 9E-2 (9.76E-2) | 6.2681E-4 (2.30E-5) | 1.780 0E-3 (5.89E-6) |
GD | 1.783 6E-2 (1.49E-2) | 1.051 1E-2 (1.70E-2) | 8.115 2E-3 (1.06E-2) | 7.993 5E-6 (7.50E-6) | 4.0869E-6 (1.32E-6) | |
DC1-DTLZ3 | IGD | 1.131 1E-1 (1.43E-4) | 1.131 9E-1 (2.30E-5) | 1.131 0E-1 (7.43E-5) | 1.128 9E-1 (5.94E-5) | 1.1120E-1 (3.24E-6) |
GD | 1.223 3E-2 (2.48E-4) | 1.653 9E-2 (3.93E-4) | 1.226 9E-2 (1.74E-4) | 1.168 5E-2 (1.34E-4) | 9.5750E-3 (1.37E-3) | |
DC2-DTLZ3 | IGD | — | 5.557 0E-1 (6.48E-5) | 3.355 1E-1 (2.84E-1) | 4.182 4E-3 (2.01E-4) | 3.9773E-3 (2.85E-5) |
GD | — | 5.599 9E-2 (2.14E-4) | 3.335 5E-2 (2.87E-2) | 5.040 9E-5 (4.55E-5) | 9.6741E-6 (1.10E-5) | |
DC3-DTLZ3 | IGD | 1.508 0E+0 (5.59E-1) | 9.795 1E-1 (4.61E-1) | 5.892 8E-1 (7.77E-1) | 1.1301E-1 (3.52E-4) | 1.603 2E-1 (1.49E-1) |
GD | 1.502 4E-1 (5.60E-2) | 1.506 3E-1 (7.22E-2) | 5.994 6E-2 (7.67E-2) | 1.569 4E-2 (3.32E-4) | 1.4122E-2 (1.49E-2) | |
DASCMOP1 | IGD | 7.450 5E-1 (4.01E-2) | 7.199 3E-1 (3.64E-2) | 6.930 4E-1 (4.21E-2) | 6.901 7E-1 (5.14E-2) | 3.8640E-2 (9.67E-2) |
GD | 5.573 7E-2 (5.57E-4) | 1.203 5E-1 (1.59E-2) | 1.975 8E-1 (8.32E-2) | 4.480 1E-2 (2.36E-2) | 1.1658E-4 (4.75E-5) | |
DASCMOP2 | IGD | 3.017 1E-1 (3.01E-2) | 2.687 6E-1 (4.68E-2) | 2.301 3E-1 (3.30E-2) | 2.321 3E-1 (2.62E-2) | 8.0187E-3 (5.97E-4) |
GD | 2.2317E-5 (4.72E-6) | 4.985 8E-4 (8.43E-4) | 2.392 2E-4 (1.83E-4) | 7.214 9E-5 (5.24E-5) | 3.207 2E-4 (8.02E-5) | |
DASCMOP3 | IGD | 3.655 0E-1 (4.20E-2) | 3.746 3E-1 (4.87E-2) | 3.836 6E-1 (1.22E-1) | 3.455 5E-1 (2.40E-3) | 8.6837E-2 (1.38E-1) |
GD | 2.4035E-4 (1.03E-4) | 8.590 9E-4 (2.78E-4) | 6.144 0E-2 (8.54E-2) | 2.464 2E-4 (3.80E-5) | 1.031 9E-3 (1.09E-3) | |
DASCMOP4 | IGD | 9.442 1E-2 (1.32E-1) | 7.688 5E-2 (1.04E-1) | 3.033 9E-2 (8.49E-2) | 1.9854E-3 (1.26E-3) | 1.054 8E-2 (8.79E-3) |
GD | 1.520 9E-2 (2.42E-2) | 1.736 6E-3 (2.58E-3) | 5.813 0E-5 (3.19E-5) | 2.1810E-5 (8.52E-6) | 1.148 5E-4 (8.03E-5) | |
DASCMOP5 | IGD | 3.895 4E-1 (2.29E-1) | 5.358 4E-2 (6.33E-2) | 5.592 6E-3 (4.53E-4) | 2.8263E-3 (1.13E-4) | 5.552 7E-3 (1.27E-4) |
GD | 5.891 5E-2 (3.02E-2) | 5.033 2E-3 (8.26E-3) | 3.717 1E-4 (3.96E-5) | 2.4826E-4 (2.33E-5) | 3.053 1E-4 (5.13E-5) | |
DASCMOP6 | IGD | 5.467 5E-1 (1.02E-1) | 5.361 5E-1 (2.49E-1) | 4.326 2E-2 (3.63E-2) | 6.614 2E-2 (5.72E-2) | 2.3694E-2 (1.95E-2) |
GD | 6.107 4E-2 (2.72E-3) | 1.098 7E-1 (4.70E-2) | 1.987 1E-3 (1.72E-3) | 5.6096E-4 (1.26E-4) | 1.226 2E-3 (2.00E-3) | |
MW1 | IGD | 6.491 5E-2 (1.67E-1) | 6.444 4E-3 (2.84E-3) | 1.895 8E-2 (2.96E-2) | 1.6174E-3 (1.95E-5) | 3.121 2E-3 (4.83E-3) |
GD | 7.076 5E-5 (7.60E-5) | 3.024 3E-4 (1.02E-4) | 7.628 7E-4 (1.23E-3) | 1.7831E-5 (1.26E-5) | 1.207 4E-4 (1.32E-4) | |
MW2 | IGD | 3.820 5E-2 (2.32E-2) | 1.655 0E-2 (9.14E-3) | 8.241 9E-2 (3.89E-2) | 1.889 2E-2 (8.54E-3) | 1.3157E-2 (1.05E-2) |
GD | 3.317 6E-3 (2.53E-3) | 1.2653E-3 (1.08E-3) | 8.013 8E-3 (4.12E-3) | 1.281 0E-3 (8.33E-4) | 1.221 2E-2 (1.13E-2) | |
MW3 | IGD | 1.700 5E-2 (3.46E-2) | 5.710 2E-3 (6.06E-4) | 5.876 7E-3 (3.27E-4) | 4.928 2E-3 (1.76E-4) | 4.0370E-3 (2.71E-4) |
GD | 3.619 5E-4 (1.09E-4) | 2.681 2E-4 (9.42E-5) | 3.155 5E-4 (5.79E-5) | 2.443 1E-4 (3.92E-5) | 2.0654E-4 (5.51E-5) | |
MW4 | IGD | 5.534 6E-2 (2.46E-3) | 4.215 1E-2 (1.64E-3) | 4.265 4E-2 (2.71E-4) | 4.0941E-2 (4.17E-4) | 4.160 8E-2 (3.63E-3) |
GD | 9.079 2E-4 (2.17E-4) | 7.858 8E-4 (6.26E-4) | 7.877 3E-4 (2.80E-4) | 8.163 0E-4 (2.30E-4) | 5.0707E-4 (3.20E-4) | |
MW5 | IGD | 2.837 0E-1 (3.36E-1) | 8.019 8E-2 (2.34E-1) | 9.788 0E-2 (1.72E-1) | 9.0406E-4 (7.50E-4) | 2.330 4E-3 (9.80E-4) |
GD | 7.934 3E-4 (1.04E-4) | 1.604 5E-3 (3.85E-4) | 1.826 2E-3 (1.06E-3) | 8.130 0E-4 (1.48E-4) | 5.2198E-4 (1.69E-4) | |
MW6 | IGD | 1.164 1E-1 (1.96E-1) | 2.011 8E-2 (2.66E-2) | 4.545 7E-1 (3.31E-1) | 7.300 5E-2 (1.50E-1) | 1.3577E-2 (2.08E-2) |
GD | 4.207 4E-3 (4.19E-3) | 1.9039E-3 (2.48E-3) | 1.782 7E-2 (1.46E-2) | 3.388 1E-3 (3.85E-3) | 5.067 0E-3 (5.83E-3) | |
MW7 | IGD | 9.773 3E-3 (1.46E-2) | 4.843 5E-3 (1.92E-4) | 4.858 3E-3 (2.54E-4) | 5.033 9E-3 (4.63E-4) | 3.0354E-3 (2.39E-4) |
GD | 2.534 3E-4 (5.92E-5) | 1.938 4E-4 (2.95E-5) | 1.880 3E-4 (3.95E-5) | 1.4898E-4 (6.21E-5) | 1.511 1E-4 (3.07E-5) | |
MW8 | IGD | 5.855 8E-2 (6.62E-3) | 5.240 8E-2 (5.62E-3) | 9.100 7E-2 (4.67E-2) | 4.5941E-2 (1.90E-3) | 1.409 0E-1 (4.55E-3) |
GD | 2.354 1E-3 (1.23E-3) | 1.7774E-3 (1.27E-3) | 8.003 5E-3 (5.57E-3) | 1.888 8E-3 (4.36E-4) | 7.447 4E-3 (4.38E-3) | |
MW9 | IGD | 1.590 6E-1 (2.91E-1) | 1.302 0E-2 (5.44E-3) | 5.851 2E-3 (2.24E-3) | 4.6306E-3 (1.40E-4) | 7.094 7E-3 (2.07E-3) |
GD | 1.745 5E-2 (1.32E-2) | 9.808 4E-3 (1.95E-3) | 6.609 7E-3 (1.17E-3) | 6.4907E-3 (9.26E-4) | 6.655 9E-3 (1.07E-3) | |
MW10 | IGD | 2.176 7E-1 (2.39E-1) | 4.855 2E-2 (3.60E-2) | 2.778 6E-1 (2.45E-1) | 2.2215E-2 (1.24E-2) | 1.832 0E-1 (2.76E-1) |
GD | 3.139 5E-3 (1.57E-3) | 2.258 1E-3 (1.18E-3) | 3.174 6E-3 (1.08E-3) | 1.0306E-3 (4.34E-4) | 5.242 1E-3 (2.78E-3) | |
MW11 | IGD | 4.232 9E-1 (3.52E-1) | 8.114 0E-2 (2.16E-1) | 7.029 0E-3 (5.13E-4) | 6.025 5E-3 (2.07E-4) | 5.6965E-3 (4.15E-4) |
GD | 7.524 3E-3 (9.63E-3) | 1.865 1E-2 (6.53E-3) | 9.745 5E-3 (8.06E-4) | 1.690 0E-2 (1.21E-3) | 6.7058E-3 (3.68E-3) | |
MW12 | IGD | 8.230 4E-2 (2.43E-1) | 7.205 1E-2 (2.12E-1) | 1.431 8E-1 (2.76E-1) | 4.853 5E-3 (9.00E-5) | 4.1428E-3 (5.19E-4) |
GD | 1.047 7E-2 (1.44E-2) | 6.166 4E-3 (1.92E-2) | 1.454 0E-2 (1.92E-2) | 3.999 9E-3 (1.64E-6) | 3.0836E-3 (8.23E-4) | |
MW13 | IGD | 1.925 1E-1 (1.96E-1) | 8.5635E-2 (4.02E-2) | 2.143 9E-1 (8.56E-2) | 1.031 0E-1 (7.80E-2) | 3.681 8E-1 (3.56E-1) |
GD | 5.118 0E-3 (1.54E-3) | 4.2576E-3 (2.66E-3) | 1.679 6E-2 (9.72E-3) | 5.949 0E-3 (6.90E-3) | 4.600 3E-2 (7.36E-2) | |
MW14 | IGD | 1.240 2E-1 (4.49E-3) | 2.118 2E-1 (9.10E-4) | 1.257 5E-1 (5.09E-2) | 9.7902E-2 (2.00E-3) | 2.092 4E-1 (5.13E-4) |
GD | 4.091 3E-3 (8.98E-4) | 3.010 3E-3 (1.27E-3) | 2.7331E-3 (3.44E-4) | 3.429 0E-3 (3.92E-4) | 3.247 3E-3 (1.53E-3) |
表1 五种算法在DTLZ、MW及DASCMOP上的IGD及GD
Tab. 1 IGD and GD of five algorithms on DTLZ,MW and DASCMOP
函数 | 指标 | NSGA-II | C-MOEA/D | MOEA/D-DAE | CCMO | ARVCMOEAD |
---|---|---|---|---|---|---|
C1-DTLZ1 | IGD | 2.278 2E-3 (2.59E-4) | 1.870 8E-3 (1.30E-4) | 2.101 6E-3 (9.18E-5) | 1.863 9E-3 (3.11E-5) | 1.8434E-3 (7.11E-5) |
GD | 3.594 8E-5 (4.39E-5) | 3.311 8E-5 (3.12E-5) | 2.334 7E-5 (1.44E-5) | 2.173 8E-5 (1.37E-5) | 1.5665E-5 (1.89E-5) | |
C1-DTLZ3 | IGD | 3.002 4E+0 (2.58E+0) | 1.640 7E-2 (3.66E-2) | 5.126 9E-3 (2.57E-4) | 4.272 3E-3 (1.85E-4) | 4.1949E-3 (1.51E-4) |
GD | 3.002 2E-1 (2.58E-1) | 2.178 5E-3 (6.30E-3) | 6.803 5E-5 (3.39E-5) | 8.678 8E-5 (5.01E-5) | 7.3414E-5 (4.46E-5) | |
C2-DTLZ2 | IGD | 5.042 3E-3 (1.73E-4) | 3.966 0E-3 (7.89E-8) | 5.214 6E-3 (2.51E-4) | 4.181 6E-3 (6.27E-5) | 3.9660E-3 (1.45E-8) |
GD | 9.561 0E-5 (2.15E-5) | 1.9469E-6 (7.02E-7) | 8.221 3E-5 (2.36E-5) | 7.373 6E-5 (2.63E-5) | 2.321 3E-6 (2.14E-7) | |
C3-DTLZ4 | IGD | 1.513 4E-1 (4.47E-1) | 8.0812E-3 (1.40E-4) | 7.714 0E-1 (5.45E-1) | 1.000 2E-2 (5.38E-4) | 8.744 7E-3 (6.34E-4) |
GD | 3.107 8E-4 (1.18E-4) | 1.5538E-4 (4.56E-5) | 4.022 7E-2 (3.39E-2) | 3.172 5E-4 (4.51E-5) | 2.071 2E-4 (1.49E-4) | |
DC1-DTLZ1 | IGD | 7.498 1E-4 (2.55E-5) | 1.743 9E-3 (2.49E-5) | 7.179 1E-4 (3.42E-5) | 6.3100E-4 (1.76E-5) | 1.777 0E-3 (1.30E-5) |
GD | 5.851 3E-6 (4.06E-6) | 5.631 5E-6 (1.96E-6) | 9.939 7E-6 (9.97E-6) | 9.332 4E-6 (6.68E-6) | 4.1193E-6 (1.41E-6) | |
DC2-DTLZ1 | IGD | — | 1.186 0E-1 (1.01E-1) | 2.162 2E-3 (5.84E-5) | 1.849 5E-3 (1.68E-5) | 1.7877E-3 (3.77E-6) |
GD | — | 1.252 6E-2 (1.08E-2) | 8.783 8E-6 (7.75E-6) | 1.397 0E-5 (9.56E-6) | 4.4786E-6 (3.71E-6) | |
DC3-DTLZ1 | IGD | 1.707 3E-1 (1.41E-1) | 6.355 5E-2 (9.94E-2) | 6.646 9E-2 (9.76E-2) | 6.2681E-4 (2.30E-5) | 1.780 0E-3 (5.89E-6) |
GD | 1.783 6E-2 (1.49E-2) | 1.051 1E-2 (1.70E-2) | 8.115 2E-3 (1.06E-2) | 7.993 5E-6 (7.50E-6) | 4.0869E-6 (1.32E-6) | |
DC1-DTLZ3 | IGD | 1.131 1E-1 (1.43E-4) | 1.131 9E-1 (2.30E-5) | 1.131 0E-1 (7.43E-5) | 1.128 9E-1 (5.94E-5) | 1.1120E-1 (3.24E-6) |
GD | 1.223 3E-2 (2.48E-4) | 1.653 9E-2 (3.93E-4) | 1.226 9E-2 (1.74E-4) | 1.168 5E-2 (1.34E-4) | 9.5750E-3 (1.37E-3) | |
DC2-DTLZ3 | IGD | — | 5.557 0E-1 (6.48E-5) | 3.355 1E-1 (2.84E-1) | 4.182 4E-3 (2.01E-4) | 3.9773E-3 (2.85E-5) |
GD | — | 5.599 9E-2 (2.14E-4) | 3.335 5E-2 (2.87E-2) | 5.040 9E-5 (4.55E-5) | 9.6741E-6 (1.10E-5) | |
DC3-DTLZ3 | IGD | 1.508 0E+0 (5.59E-1) | 9.795 1E-1 (4.61E-1) | 5.892 8E-1 (7.77E-1) | 1.1301E-1 (3.52E-4) | 1.603 2E-1 (1.49E-1) |
GD | 1.502 4E-1 (5.60E-2) | 1.506 3E-1 (7.22E-2) | 5.994 6E-2 (7.67E-2) | 1.569 4E-2 (3.32E-4) | 1.4122E-2 (1.49E-2) | |
DASCMOP1 | IGD | 7.450 5E-1 (4.01E-2) | 7.199 3E-1 (3.64E-2) | 6.930 4E-1 (4.21E-2) | 6.901 7E-1 (5.14E-2) | 3.8640E-2 (9.67E-2) |
GD | 5.573 7E-2 (5.57E-4) | 1.203 5E-1 (1.59E-2) | 1.975 8E-1 (8.32E-2) | 4.480 1E-2 (2.36E-2) | 1.1658E-4 (4.75E-5) | |
DASCMOP2 | IGD | 3.017 1E-1 (3.01E-2) | 2.687 6E-1 (4.68E-2) | 2.301 3E-1 (3.30E-2) | 2.321 3E-1 (2.62E-2) | 8.0187E-3 (5.97E-4) |
GD | 2.2317E-5 (4.72E-6) | 4.985 8E-4 (8.43E-4) | 2.392 2E-4 (1.83E-4) | 7.214 9E-5 (5.24E-5) | 3.207 2E-4 (8.02E-5) | |
DASCMOP3 | IGD | 3.655 0E-1 (4.20E-2) | 3.746 3E-1 (4.87E-2) | 3.836 6E-1 (1.22E-1) | 3.455 5E-1 (2.40E-3) | 8.6837E-2 (1.38E-1) |
GD | 2.4035E-4 (1.03E-4) | 8.590 9E-4 (2.78E-4) | 6.144 0E-2 (8.54E-2) | 2.464 2E-4 (3.80E-5) | 1.031 9E-3 (1.09E-3) | |
DASCMOP4 | IGD | 9.442 1E-2 (1.32E-1) | 7.688 5E-2 (1.04E-1) | 3.033 9E-2 (8.49E-2) | 1.9854E-3 (1.26E-3) | 1.054 8E-2 (8.79E-3) |
GD | 1.520 9E-2 (2.42E-2) | 1.736 6E-3 (2.58E-3) | 5.813 0E-5 (3.19E-5) | 2.1810E-5 (8.52E-6) | 1.148 5E-4 (8.03E-5) | |
DASCMOP5 | IGD | 3.895 4E-1 (2.29E-1) | 5.358 4E-2 (6.33E-2) | 5.592 6E-3 (4.53E-4) | 2.8263E-3 (1.13E-4) | 5.552 7E-3 (1.27E-4) |
GD | 5.891 5E-2 (3.02E-2) | 5.033 2E-3 (8.26E-3) | 3.717 1E-4 (3.96E-5) | 2.4826E-4 (2.33E-5) | 3.053 1E-4 (5.13E-5) | |
DASCMOP6 | IGD | 5.467 5E-1 (1.02E-1) | 5.361 5E-1 (2.49E-1) | 4.326 2E-2 (3.63E-2) | 6.614 2E-2 (5.72E-2) | 2.3694E-2 (1.95E-2) |
GD | 6.107 4E-2 (2.72E-3) | 1.098 7E-1 (4.70E-2) | 1.987 1E-3 (1.72E-3) | 5.6096E-4 (1.26E-4) | 1.226 2E-3 (2.00E-3) | |
MW1 | IGD | 6.491 5E-2 (1.67E-1) | 6.444 4E-3 (2.84E-3) | 1.895 8E-2 (2.96E-2) | 1.6174E-3 (1.95E-5) | 3.121 2E-3 (4.83E-3) |
GD | 7.076 5E-5 (7.60E-5) | 3.024 3E-4 (1.02E-4) | 7.628 7E-4 (1.23E-3) | 1.7831E-5 (1.26E-5) | 1.207 4E-4 (1.32E-4) | |
MW2 | IGD | 3.820 5E-2 (2.32E-2) | 1.655 0E-2 (9.14E-3) | 8.241 9E-2 (3.89E-2) | 1.889 2E-2 (8.54E-3) | 1.3157E-2 (1.05E-2) |
GD | 3.317 6E-3 (2.53E-3) | 1.2653E-3 (1.08E-3) | 8.013 8E-3 (4.12E-3) | 1.281 0E-3 (8.33E-4) | 1.221 2E-2 (1.13E-2) | |
MW3 | IGD | 1.700 5E-2 (3.46E-2) | 5.710 2E-3 (6.06E-4) | 5.876 7E-3 (3.27E-4) | 4.928 2E-3 (1.76E-4) | 4.0370E-3 (2.71E-4) |
GD | 3.619 5E-4 (1.09E-4) | 2.681 2E-4 (9.42E-5) | 3.155 5E-4 (5.79E-5) | 2.443 1E-4 (3.92E-5) | 2.0654E-4 (5.51E-5) | |
MW4 | IGD | 5.534 6E-2 (2.46E-3) | 4.215 1E-2 (1.64E-3) | 4.265 4E-2 (2.71E-4) | 4.0941E-2 (4.17E-4) | 4.160 8E-2 (3.63E-3) |
GD | 9.079 2E-4 (2.17E-4) | 7.858 8E-4 (6.26E-4) | 7.877 3E-4 (2.80E-4) | 8.163 0E-4 (2.30E-4) | 5.0707E-4 (3.20E-4) | |
MW5 | IGD | 2.837 0E-1 (3.36E-1) | 8.019 8E-2 (2.34E-1) | 9.788 0E-2 (1.72E-1) | 9.0406E-4 (7.50E-4) | 2.330 4E-3 (9.80E-4) |
GD | 7.934 3E-4 (1.04E-4) | 1.604 5E-3 (3.85E-4) | 1.826 2E-3 (1.06E-3) | 8.130 0E-4 (1.48E-4) | 5.2198E-4 (1.69E-4) | |
MW6 | IGD | 1.164 1E-1 (1.96E-1) | 2.011 8E-2 (2.66E-2) | 4.545 7E-1 (3.31E-1) | 7.300 5E-2 (1.50E-1) | 1.3577E-2 (2.08E-2) |
GD | 4.207 4E-3 (4.19E-3) | 1.9039E-3 (2.48E-3) | 1.782 7E-2 (1.46E-2) | 3.388 1E-3 (3.85E-3) | 5.067 0E-3 (5.83E-3) | |
MW7 | IGD | 9.773 3E-3 (1.46E-2) | 4.843 5E-3 (1.92E-4) | 4.858 3E-3 (2.54E-4) | 5.033 9E-3 (4.63E-4) | 3.0354E-3 (2.39E-4) |
GD | 2.534 3E-4 (5.92E-5) | 1.938 4E-4 (2.95E-5) | 1.880 3E-4 (3.95E-5) | 1.4898E-4 (6.21E-5) | 1.511 1E-4 (3.07E-5) | |
MW8 | IGD | 5.855 8E-2 (6.62E-3) | 5.240 8E-2 (5.62E-3) | 9.100 7E-2 (4.67E-2) | 4.5941E-2 (1.90E-3) | 1.409 0E-1 (4.55E-3) |
GD | 2.354 1E-3 (1.23E-3) | 1.7774E-3 (1.27E-3) | 8.003 5E-3 (5.57E-3) | 1.888 8E-3 (4.36E-4) | 7.447 4E-3 (4.38E-3) | |
MW9 | IGD | 1.590 6E-1 (2.91E-1) | 1.302 0E-2 (5.44E-3) | 5.851 2E-3 (2.24E-3) | 4.6306E-3 (1.40E-4) | 7.094 7E-3 (2.07E-3) |
GD | 1.745 5E-2 (1.32E-2) | 9.808 4E-3 (1.95E-3) | 6.609 7E-3 (1.17E-3) | 6.4907E-3 (9.26E-4) | 6.655 9E-3 (1.07E-3) | |
MW10 | IGD | 2.176 7E-1 (2.39E-1) | 4.855 2E-2 (3.60E-2) | 2.778 6E-1 (2.45E-1) | 2.2215E-2 (1.24E-2) | 1.832 0E-1 (2.76E-1) |
GD | 3.139 5E-3 (1.57E-3) | 2.258 1E-3 (1.18E-3) | 3.174 6E-3 (1.08E-3) | 1.0306E-3 (4.34E-4) | 5.242 1E-3 (2.78E-3) | |
MW11 | IGD | 4.232 9E-1 (3.52E-1) | 8.114 0E-2 (2.16E-1) | 7.029 0E-3 (5.13E-4) | 6.025 5E-3 (2.07E-4) | 5.6965E-3 (4.15E-4) |
GD | 7.524 3E-3 (9.63E-3) | 1.865 1E-2 (6.53E-3) | 9.745 5E-3 (8.06E-4) | 1.690 0E-2 (1.21E-3) | 6.7058E-3 (3.68E-3) | |
MW12 | IGD | 8.230 4E-2 (2.43E-1) | 7.205 1E-2 (2.12E-1) | 1.431 8E-1 (2.76E-1) | 4.853 5E-3 (9.00E-5) | 4.1428E-3 (5.19E-4) |
GD | 1.047 7E-2 (1.44E-2) | 6.166 4E-3 (1.92E-2) | 1.454 0E-2 (1.92E-2) | 3.999 9E-3 (1.64E-6) | 3.0836E-3 (8.23E-4) | |
MW13 | IGD | 1.925 1E-1 (1.96E-1) | 8.5635E-2 (4.02E-2) | 2.143 9E-1 (8.56E-2) | 1.031 0E-1 (7.80E-2) | 3.681 8E-1 (3.56E-1) |
GD | 5.118 0E-3 (1.54E-3) | 4.2576E-3 (2.66E-3) | 1.679 6E-2 (9.72E-3) | 5.949 0E-3 (6.90E-3) | 4.600 3E-2 (7.36E-2) | |
MW14 | IGD | 1.240 2E-1 (4.49E-3) | 2.118 2E-1 (9.10E-4) | 1.257 5E-1 (5.09E-2) | 9.7902E-2 (2.00E-3) | 2.092 4E-1 (5.13E-4) |
GD | 4.091 3E-3 (8.98E-4) | 3.010 3E-3 (1.27E-3) | 2.7331E-3 (3.44E-4) | 3.429 0E-3 (3.92E-4) | 3.247 3E-3 (1.53E-3) |
图3 五种算法在DC2-DTLZ3、DASCMOP6及MW8上的20次运算得到最优IGD的非支配解分布比较
Fig. 3 Non-dominated solution distribution comparison of optimal IGDs obtained by five algorithms after 20 runs on DC2-DTLZ3,DASCMOP6 and MW8
测试函数种类 | NSGA-II | C-MOEA/D | MOEA/D-DAE | CCMO | ARVCMOEA |
---|---|---|---|---|---|
MW | 3.3849 | 37.028 0 | 131.330 0 | 64.406 0 | 49.184 0 |
DTLZ | 1.6804 | 24.580 0 | 70.264 0 | 34.346 0 | 24.635 0 |
DASCMOP | 8.3568 | 28.006 0 | 92.320 0 | 29.321 0 | 54.791 0 |
表2 五种算法在三个系列测试函数上的平均计算时间 (s)
Tab. 2 Average computing times of five algorithms on three series of test functions
测试函数种类 | NSGA-II | C-MOEA/D | MOEA/D-DAE | CCMO | ARVCMOEA |
---|---|---|---|---|---|
MW | 3.3849 | 37.028 0 | 131.330 0 | 64.406 0 | 49.184 0 |
DTLZ | 1.6804 | 24.580 0 | 70.264 0 | 34.346 0 | 24.635 0 |
DASCMOP | 8.3568 | 28.006 0 | 92.320 0 | 29.321 0 | 54.791 0 |
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