Table 7.
HV values for 5 MOEA and ASDMOPSO.
| Problem | M | D | DGEA | MOEADD | SPEAR | NSGAIII | IDBEA | ASDMOPSO | |
|---|---|---|---|---|---|---|---|---|---|
| ZDT1 | 2 | 30 |
Mean Std |
8.5274e−3 (1.83e−2)− | 6.0276e−1 (2.12e−2)− | 4.8568e−1 (3.46e−2)− | 6.1393e−1 (1.69e−2)− | 1.4767e−1 (5.05e−2)− |
7.2129e−01 (6.03e−05) |
| ZDT2 | 2 | 30 |
Mean Std |
2.1144e−2 (7.07e−2)− | 3.0344e−1 (2.33e−2)− | 7.0276e−2 (6.03e−2)− | 2.5351e−1 (3.78e−2)− | 0.0000e + 0 (0.00e + 0)− |
4.4657e−01 (2.05e−05) |
| ZDT3 | 2 | 30 |
Mean Std |
3.2427e−2 (6.18e−2)− | 4.9551e−1 (3.21e−2)− | 5.0081e−1 (1.28e−2)− | 5.5474e−1 (8.86e−3)− | 3.8083e−1 (2.14e−2)− |
6.0005e−01 (1.37e−04) |
| ZDT4 | 2 | 10 |
Mean Std |
2.0966e−3 (1.15e−2)− | 8.6915e−2 (8.59e−2)− | 4.2997e−3 (1.66e−2)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− |
6.9846e−01 (1.33e−01) |
| ZDT6 | 2 | 10 |
Mean Std |
3.6287e−1 (8.54e−2)≈ | 6.8634e−2 (4.79e−2)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− |
3.9015e−01 (1.31e−05) |
| UF1 | 2 | 30 |
Mean Std |
8.6414e−2 (6.86e−2)− | 5.3896e−1 (2.61e−2)− | 5.1022e−1 (3.30e−2)≈ | 5.4146e−1 (3.51e−2) + | 2.2314e−1 (6.73e−2)− |
4.9805e−01 (1.61e−02) |
| UF2 | 2 | 30 |
Mean Std |
5.1284e−1 (2.83e−2)− | 6.3799e−1 (6.69e−3)≈ | 6.3009e−1 (6.94e−3)− | 6.2399e−1 (7.30e−3)− | 5.1343e−1 (1.79e−2)− |
6.4061e−01 (4.58e−03) |
| UF3 | 2 | 30 |
Mean Std |
1.2226e−1 (2.17e−2)− | 2.0905e−1 (1.14e−2)− | 2.2358e−1 (1.13e−2)− | 1.9518e−1 (1.09e−2)− | 9.2357e−2 (1.40e−2)− |
3.4111e−01 (3.21e−02) |
| UF4 | 2 | 30 |
Mean Std |
2.8234e−1 (8.99e−3)− | 3.2372e−1 (4.38e−3)− | 3.2442e−1 (2.79e−3) + | 3.2202e−1 (4.20e−3)− | 2.9545e−1 (4.01e−3)− |
1.2726e−01 (3.36e−03) |
| UF5 | 2 | 30 |
Mean Std |
0.0000e + 0 (0.00e + 0)≈ | 2.4987e−4 (1.26e−3)≈ | 4.2485e−4 (2.33e−3)≈ | 2.2105e−3 (8.42e−3) + | 0.0000e + 0 (0.00e + 0)≈ |
0.0000e + 00 (1.75e−01) |
| UF6 | 2 | 30 |
Mean Std |
0.0000e + 0 (0.00e + 0)− | 1.4259e−2 (1.73e−2)− | 1.1077e−2 (1.56e−2)≈ | 3.0877e−2 (2.33e−2) + | 0.0000e + 0 (0.00e + 0)− |
7.1557e−03 (6.25e−02) |
| UF7 | 2 | 30 |
Mean Std |
2.4430e−2 (4.02e−2)− | 3.8790e−1 (5.65e−2)− | 3.3758e−1 (5.89e−2)− | 3.5981e−1 (7.82e−2)− | 1.1043e−1 (5.18e−2)− |
4.1627e−01 (1.26e−02) |
| UF8 | 3 | 30 |
Mean Std |
2.6973e−2 (2.99e−2)− | 1.9908e−1 (3.07e−2)− | 1.8468e−1 (3.08e−2)− | 2.7020e−1 (3.45e−2)− | 1.6876e−1 (4.45e−2)− |
3.4690e−01 (2.89e−02) |
| UF9 | 3 | 30 |
Mean Std |
7.4309e−2 (4.44e−2)− | 2.5124e−1 (5.40e−2)− | 2.5440e−1 (5.25e−2)− | 2.6808e−1 (3.87e−2)≈ | 2.0061e−1 (4.37e−2)− |
2.8639e−01 (2.11e−02) |
| UF10 | 3 | 30 |
Mean Std |
0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− |
2.7421e−01 (3.86e−02) |
| DTLZ1 | 3 | 7 |
Mean Std |
0.0000e + 0 (0.00e + 0)− | 7.6458e−3 (3.38e−2)≈ | 5.3905e−3 (2.44e−2)≈ | 3.6769e−2 (1.03e−1) + | 0.0000e + 0 (0.00e + 0)− |
5.9780e−04 (6.98e + 00) |
| DTLZ2 | 3 | 12 |
Mean Std |
4.1543e−1 (3.16e−2)− | 5.6361e−1 (1.31e−3) + | 5.5634e−1 (2.54e−3)− | 5.6283e−1 (1.08e−3)− | 5.5285e−1 (3.94e−3)− |
4.1042e−02 (8.54e−03) |
| DTLZ3 | 3 | 12 |
Mean Std |
0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− | 0.0000e + 0 (0.00e + 0)− |
4.1886e−03 (2.53e + 01) |
| DTLZ4 | 3 | 12 |
Mean Std |
1.9152e−1 (8.47e−2)− | 5.6283e−1 (2.15e−3) + | 5.5450e−1 (2.80e−3)− | 5.5407e−1 (3.98e−2)− | 4.9093e−1 (9.90e−2)− |
3.7090e−01 (8.65e−01) |
| DTLZ5 | 3 | 12 |
Mean Std |
1.4603e−1 (2.00e−2)− | 1.8308e−1 (2.45e−3)− | 1.8616e−1 (1.68e−3)− | 1.9679e−1 (6.86e−4) + | 1.9198e−1 (1.88e−3)− |
2.4535e−02 (1.29e + 00) |
| DTLZ6 | 3 | 12 |
Mean Std |
9.8346e−3 (3.90e−2)− | 1.1168e−1 (4.23e−2)− | 1.0343e−1 (4.56e−2)− | 1.8121e−1 (2.29e−2) + | 9.5555e−3 (2.92e−2)− |
1.3678e−02 (1.99e−04) |
| DTLZ7 | 3 | 22 |
Mean Std |
7.6737e−3 (3.15e−2)− | 2.1788e−1 (2.06e−2)− | 1.9161e−1 (1.47e−2)− | 2.2360e−1 (9.16e−3) + | 7.6151e−2 (2.70e−2)− |
1.6227e−01 (2.03e−02) |
| Best/all | 0/22 | 2/22 | 1/22 | 7/22 | 0/22 | 12/22 | |||
| ± /≈ | 0/20/2 | 2/17/3 | 1/17/4 | 7/14/1 | 0/21/1 | – |