Page 197 - 《软件学报》2026年第3期

1160 软件学报 2026 年第 37 卷第 3 期

150
0.4 1.5 80
Objective value 0.3 Objective value 1.0 Objective value 60 Objective value 100
0.2
40
50
0.5
0.1
0 20 0 0
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10
Objective number Objective number Objective number Objective number
(a) Two-Arch/IS (b) Two-Arch (c) Two_Arch2 (d) C-TAEA
0.5
20 6
Objective value 0.3 Objective value 15 Objective value 4 2
0.4
10
0.2
0.1
0 5 0 0
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10
Objective number Objective number Objective number
(e) TriMOEATAR (f) TwoArchM (g) TwoArchA
图 6 Two-Arch/IS 和不同类型的 Two-Arch 算法在 10 目标 DTLZ1 问题上的种群平行坐标图

表 3 Two-Arch/IS 和其他高维多目标进化算法在高维 DTLZ 问题上的 IGD 结果

测试问题 目标数目 Two-Arch/IS NSGA-III KnEA IBEA HEA TSNSGA-II MOEA/D SMS-EMOA
4.9339E–2 4.9557E–2 1.1673E–1 2.3432E–1 4.8849E–2 5.1405E–2 6.2153E–2 1.9058E–1
4
(1.96E–2) (3.70E–3) – (5.15E–2) – (2.90E–2) – (6.53E–4) + (5.94E–3) – (5.15E–2) – (2.22E–1) –
9.7828E–2 1.2540E–1 2.0587E–1 2.6687E–1 9.2448E–2 9.9527E–2 9.8877E–2 2.2403E+0
6
(3.44E–2) (5.94E–2) – (9.59E–2) – (2.47E–2) – (2.49E–2) + (4.78E–3) – (5.52E–2) – (2.22E+0) –
1.5939E–1 2.1276E–1 5.4815E+0 2.9228E–1 1.1615E–1 1.9086E–1 1.2975E–1 9.6447E+0
8
(7.08E–2) (1.45E–1) – (3.95E+0) – (3.08E–2) – (3.33E–2) + (5.12E–2) – (4.87E–2) + (8.93E+0) –
DTLZ1
1.8356E–1 4.8366E–1 8.2897E+0 3.2506E–1 3.4630E–1 2.6442E–1 1.2462E–1 1.0657E+1
10
(7.37E–2) (3.57E–1) – (6.08E+0) – (3.00E–2) – (2.95E–1) – (6.81E–2) – (2.08E–2) + (8.06E+0) –
1.3163E–1 1.8638E–1 6.3392E+0 3.1762E–1 1.9248E–1 3.3986E–1 1.7276E–1 3.5431E+0
15
(2.73E–3) (2.08E–2) – (8.47E+0) – (3.61E–2) – (2.75E–2) – (2.28E–2) – (1.84E–2) – (2.50E+0) –
1.7919E–1 3.5380E–1 1.3528E+1 4.0626E–1 2.0535E–1 6.8319E–1 2.8527E–1 4.3477E+0
20
(7.45E–3) (9.04E–2) – (1.07E+1) – (1.69E–2) – (6.37E–3) – (3.20E–1) – (1.67E–1) – (3.34E+0) –
1.5525E–1 1.4032E–1 1.5067E–1 1.5805E–1 1.4032E–1 1.4065E–1 1.4036E–1 1.6290E–1
4
(3.11E–3) (1.15E–5) + (4.34E–3) + (1.68E–3) – (8.85E–5) + (1.66E–4) + (2.12E–5) + (4.05E–3) =
2.9212E–1 2.7946E–1 3.1294E–1 2.9499E–1 2.8007E–1 2.8235E–1 2.7836E–1 3.0449E–1
6
(2.87E–3) (1.35E–4) + (1.28E–2) – (3.98E–3) – (5.42E–4) + (9.45E–4) + (4.69E–4) + (4.52E–3) –
3.9197E–1 4.2221E–1 4.1931E–1 4.0198E–1 3.8930E–1 3.9494E–1 3.8472E–1 4.7517E–1
8
(2.74E–3) (6.55E–2) – (1.03E–2) – (4.84E–3) – (1.14E–3) = (2.79E–3) = (4.57E–4) = (9.28E–2) –
DTLZ2
4.7686E–1 5.6781E–1 5.1911E–1 5.0820E–1 5.3775E–1 5.6162E–1 5.0086E–1 7.4192E–1
10
(2.90E–3) (5.50E–2) – (1.37E–2) – (5.01E–3) – (2.74E–3) – (1.31E–2) – (6.30E–3) – (4.48E–2) –
5.4409E–1 6.4502E–1 6.6806E–1 5.7838E–1 7.1403E–1 1.5624E+0 1.1118E+0 9.3585E–1
15
(4.84E–3) (1.76E–2) – (1.65E–1) – (6.64E–3) – (5.65E–3) – (4.09E–1) – (8.20E–2) – (5.25E–2) –
6.2367E–1 1.0125E+0 7.5457E–1 6.7514E–1 7.6454E–1 2.0314E+0 8.3060E–1 1.0440E+0
20
(2.29E–3) (3.49E–2) – (1.58E–1) = (1.80E–2) = (2.69E–3) – (6.41E–1) – (1.24E–2) – (2.85E–2) –
4.8838E+0 1.1456E+0 8.2960E–1 6.1068E–1 2.5000E–1 4.4704E–1 5.1470E+0 4.0480E+0
4
(2.05E+0) (1.30E+0) + (8.46E–1) + (1.91E–1) + (2.62E–1) + (4.68E–1) + (5.45E+0) – (2.90E+0) +
4.4899E+0 5.3930E+0 1.2224E+1 7.6390E–1 5.5485E–1 2.4777E+0 3.7847E+0 6.9844E+1
6
(1.70E+0) (2.78E+0) = (4.95E+0) – (3.66E–1) + (3.72E–1) + (1.34E+0) + (4.55E+0) + (3.16E+1) –
4.5326E+0 7.4970E+0 1.5586E+2 7.8764E–1 7.9166E–1 4.8627E+0 4.4043E+0 1.1608E+2
DTLZ3 8
(1.95E+0) (4.56E+0) – (4.82E+1) – (2.41E–1) + (4.63E–1) + (2.98E+0) – (3.25E+0) = (3.74E+1) –
5.7788E+0 1.3058E+1 3.5237E+2 9.8908E–1 1.0207E+0 1.1255E+1 1.8070E+0 1.0782E+2
10
(1.67E+0) (6.28E+0) – (9.49E+1) – (3.75E–1) + (7.30E–1) + (7.65E+0) – (9.99E–1) + (3.65E+1) –
1.5993E+0 5.5582E+0 7.2278E+2 8.1705E–1 9.4543E–1 3.6017E+2 2.3254E+0 8.4731E+1
15
(8.97E–1) (3.38E+0) – (1.17E+2) – (2.07E–2) + (9.45E–2) + (2.85E+2) – (1.49E+0) – (1.69E+1) –