Table 2.
Statistical results for IGD + values obtained by AB-MOEA, EDN-ARMOEA, CSEA, K-RVEA, CPS-MOEA, and TSDEA for 3 objectives with the same number of real FEs.
| Problem | D | ABSAEA | EDNARMOEA | CSEA | KRVEA | CPSMOEA | TSDEA |
|---|---|---|---|---|---|---|---|
| DTLZ1 | 20 | 2.8377e + 2 (4.18e + 1) − | 3.1773e + 2 (3.29e + 1) − | 1.9572e + 2 (3.09e + 1) + | 2.9757e + 2 (4.07e + 1) − | 2.5304e + 2 (3.40e + 1) = | 2.5177e + 2 (3.90e + 1) |
| 50 | 1.2029e + 3 (5.78e + 1) − | 1.1786e + 3 (6.02e + 1) − | 1.2007e + 3 (6.85e + 1) − | 1.2028e + 3 (5.17e + 1) − | 9.2159e + 2 (6.21e + 1) + | 1.0601e + 3 (7.56e + 1) | |
| 100 | 2.5830e + 3 (1.02e + 2) = | 2.6066e + 3 (8.88e + 1) = | 2.5773e + 3 (9.33e + 1) = | 2.5892e + 3 (9.36e + 1) = | 2.0772e + 3 (1.32e + 2) + | 2.6183e + 3 (1.52e + 2) | |
| DTLZ2 | 20 | 3.7368e−1 (6.11e−2) − | 7.8888e−1 (5.34e−2) − | 3.5342e−1 (7.05e−2) − | 6.7551e−1 (8.89e−2) − | 7.2468e−1 (9.03e−2) − | 1.0799e−1 (2.29e−2) |
| 50 | 2.7232e + 0 (1.26e−1) − | 2.7178e + 0 (1.49e−1) − | 2.7187e + 0 (1.06e−1) − | 2.7262e + 0 (1.00e−1) − | 1.8272e + 0 (2.90e−1) + | 2.2003e + 0 (3.02e−1) | |
| 100 | 6.1104e + 0 (1.75e−1) − | 6.0763e + 0 (2.14e−1) − | 6.0986e + 0 (1.69e−1) − | 6.0221e + 0 (2.40e−1) − | 3.7748e + 0 (6.32e−1) + | 5.5076e + 0 (3.72e−1) | |
| DTLZ3 | 20 | 8.1850e + 2 (9.30e + 1) = | 1.0593e + 3 (9.83e + 1) − | 5.4428e + 2 (9.12e + 1) + | 8.0779e + 2 (1.27e + 2) = | 7.5407e + 2 (8.48e + 1) = | 7.8757e + 2 (1.21e + 2) |
| 50 | 3.7581e + 3 (1.54e + 2) − | 3.7402e + 3 (1.49e + 2) − | 3.7792e + 3 (1.60e + 2) − | 3.6825e + 3 (1.81e + 2) − | 2.7113e + 3 (2.10e + 2) + | 3.2691e + 3 (2.82e + 2) | |
| 100 | 8.3160e + 3 (2.44e + 2) − | 8.3178e + 3 (2.26e + 2) − | 8.3788e + 3 (2.56e + 2) − | 8.4012e + 3 (1.70e + 2) − | 6.0293e + 3 (5.49e + 2) + | 7.5396e + 3 (4.07e + 2) | |
| DTLZ4 | 20 | 5.9489e−1 (1.56e−1) − | 5.4263e−1 (1.07e−1) − | 3.5993e−1 (6.91e−2) = | 8.0141e−1 (1.54e−1) − | 8.7691e−1 (1.07e−1) − | 3.9862e−1 (8.90e−2) |
| 50 | 3.0091e + 0 (1.33e−1) − | 3.0107e + 0 (1.46e−1) − | 2.9674e + 0 (1.65e−1) − | 2.9792e + 0 (1.44e−1) − | 2.0761e + 0 (2.28e−1) − | 1.1918e + 0 (4.75e−1) | |
| 100 | 6.2932e + 0 (2.51e−1) − | 6.3591e + 0 (2.13e−1) − | 6.2433e + 0 (3.10e−1) − | 6.3384e + 0 (2.00e−1) − | 4.1180e + 0 (4.97e−1) + | 4.7921e + 0 (7.66e−1) | |
| DTLZ5 | 20 | 2.8049e−1 (5.70e−2) − | 6.5502e−1 (8.01e−2) − | 3.5172e−1 (8.06e−2) − | 5.0959e−1 (1.12e−1) − | 6.1211e−1 (7.01e−2) − | 3.6663e−2 (8.70e−3) |
| 50 | 2.6478e + 0 (1.26e−1) − | 2.6128e + 0 (1.65e−1) − | 2.6151e + 0 (1.75e−1) − | 2.6487e + 0 (1.46e−1) − | 1.8445e + 0 (2.87e−1) + | 2.0239e + 0 (3.62e−1) | |
| 100 | 6.0182e + 0 (1.66e−1) − | 6.0197e + 0 (2.16e−1) − | 6.0197e + 0 (2.08e−1) − | 6.0220e + 0 (2.01e−1) − | 3.8176e + 0 (4.58e−1) + | 5.6228e + 0 (3.05e−1) | |
| DTLZ6 | 20 | 1.0193e + 1 (6.66e−1) = | 1.4021e + 1 (5.08e−1) − | 1.3653e + 1 (9.14e−1) − | 8.7982e + 0 (6.45e−1) + | 1.0100e + 1 (8.54e−1) = | 1.0106e + 1 (8.07e−1) |
| 50 | 4.1888e + 1 (2.54e−1) − | 4.1857e + 1 (2.92e−1) − | 4.1934e + 1 (2.04e−1) − | 4.1936e + 1 (2.07e−1) − | 3.0712e + 1 (1.61e + 0) + | 3.9430e + 1 (1.05e + 0) | |
| 100 | 8.6484e + 1 (2.20e−1) − | 8.6423e + 1 (3.17e−1) − | 8.6414e + 1 (3.16e−1) − | 8.6402e + 1 (2.84e−1) − | 6.7582e + 1 (2.25e + 0) + | 8.3949e + 1 (9.41e−1) | |
| DTLZ7 | 20 | 2.9149e−1 (2.57e−1) + | 1.8748e + 0 (5.79e−1) − | 2.6080e + 0 (8.35e−1) − | 8.0765e−2 (7.12e−3) + | 6.8441e + 0 (6.46e−1) − | 3.9655e−1 (1.16e−1) |
| 50 | 9.3695e + 0 (4.80e−1) − | 9.3282e + 0 (5.33e−1) − | 9.3144e + 0 (4.46e−1) − | 9.2600e + 0 (5.87e−1) − | 8.8788e + 0 (6.28e−1) − | 1.6948e + 0 (4.08e−1) | |
| 100 | 9.9101e + 0 (3.66e−1) − | 9.9868e + 0 (3.17e−1) − | 9.8956e + 0 (3.93e−1) − | 9.9129e + 0 (3.82e−1) − | 9.9689e + 0 (3.44e−1) − | 4.0322e + 0 (6.86e−1) | |
| WFG1 | 20 | 1.7682e + 0 (1.41e−1) = | 1.8841e + 0 (6.68e−2) − | 1.5608e + 0 (4.97e−2) + | 1.6873e + 0 (1.06e−1) = | 2.1749e + 0 (6.89e−2) − | 1.7178e + 0 (6.60e−2) |
| 50 | 2.1798e + 0 (3.92e−2) − | 2.1800e + 0 (3.99e−2) − | 2.1668e + 0 (3.74e−2) − | 2.1866e + 0 (3.62e−2) − | 2.1726e + 0 (4.08e−2) − | 1.7517e + 0 (8.83e−2) | |
| 100 | 2.1399e + 0 (4.32e−2) − | 2.1398e + 0 (3.14e−2) − | 2.1380e + 0 (3.82e−2) − | 2.1297e + 0 (6.43e−2) − | 2.1890e + 0 (4.28e−2) − | 1.7368e + 0 (7.76e−2) | |
| WFG2 | 20 | 4.6832e−1 (7.01e−2) + | 6.7783e−1 (3.65e−2) − | 4.7347e−1 (4.88e−2) + | 5.2556e−1 (8.44e−2) + | 7.3697e−1 (3.62e−2) − | 6.2701e−1 (6.09e−2) |
| 50 | 8.2622e−1 (2.26e−2) − | 8.4181e−1 (3.07e−2) − | 8.2902e−1 (3.17e−2) − | 8.3482e−1 (3.28e−2) − | 8.1111e−1 (3.23e−2) − | 7.1664e−1 (3.00e−2) | |
| 100 | 7.9937e−1 (1.24e−2) − | 8.0253e−1 (1.21e−2) − | 8.0586e−1 (1.01e−2) − | 8.0315e−1 (1.17e−2) − | 8.2161e−1 (3.24e−2) − | 7.7195e−1 (2.38e−2) | |
| WFG3 | 20 | 5.3483e−1 (5.74e−2) = | 7.1054e−1 (2.06e−2) − | 5.9672e−1 (3.96e−2) = | 6.8531e−1 (2.83e−2) − | 7.1392e−1 (3.54e−2) − | 5.7025e−1 (5.98e−2) |
| 50 | 7.7479e−1 (1.27e−2) = | 7.7204e−1 (1.03e−2) = | 7.7281e−1 (1.15e−2) = | 7.7620e−1 (1.05e−2) = | 8.2662e−1 (3.64e−2) − | 7.7108e−1 (2.07e−2) | |
| 100 | 8.0149e−1 (5.94e−3) + | 8.0014e−1 (6.80e−3) + | 7.9855e−1 (5.02e−3) + | 7.9998e−1 (6.88e−3) + | 8.5612e−1 (3.47e−2) − | 8.3122e−1 (1.22e−2) | |
| WFG4 | 20 | 3.7885e−1 (1.41e−2) − | 4.7765e−1 (8.91e−3) − | 3.8485e−1 (2.47e−2) − | 4.6788e−1 (1.67e−2) − | 4.7811e−1 (1.62e−2) − | 3.5559e−1 (3.83e−2) |
| 50 | 5.9425e−1 (3.70e−2) − | 5.9784e−1 (2.28e−2) − | 5.8958e−1 (2.61e−2) − | 5.9419e−1 (2.59e−2) − | 5.4109e−1 (1.23e−2) − | 4.6740e−1 (2.37e−2) | |
| 100 | 5.6382e−1 (2.21e−2) − | 5.6800e−1 (2.09e−2) − | 5.6628e−1 (2.48e−2) − | 5.6533e−1 (2.22e−2) − | 5.6517e−1 (1.26e−2) − | 5.1529e−1 (1.90e−2) | |
| WFG5 | 20 | 5.0294e−1 (4.89e−2) − | 6.0558e−1 (2.00e−2) − | 4.1618e−1 (3.18e−2) − | 3.5924e−1 (3.29e−2) + | 5.3751e−1 (2.26e−2) − | 3.8537e−1 (4.36e−2) |
| 50 | 7.3912e−1 (7.68e−3) − | 7.4212e−1 (8.50e−3) − | 7.4348e−1 (7.34e−3) − | 7.4152e−1 (7.88e−3) − | 6.1829e−1 (2.08e−2) − | 5.8332e−1 (3.79e−2) | |
| 100 | 7.4456e−1 (4.59e−3) − | 7.4316e−1 (3.81e−3) − | 7.4498e−1 (4.71e−3) − | 7.4388e−1 (5.05e−3) − | 6.5238e−1 (1.58e−2) + | 6.8311e−1 (2.63e−2) | |
| WFG6 | 20 | 7.4754e−1 (3.87e−2) − | 8.1620e−1 (1.74e−2) − | 5.9865e−1 (4.18e−2) = | 6.7313e−1 (4.14e−2) − | 8.5811e−1 (2.78e−2) − | 5.8559e−1 (6.24e−2) |
| 50 | 9.1586e−1 (1.18e−2) − | 9.1389e−1 (1.03e−2) − | 9.1750e−1 (1.20e−2) − | 9.1677e−1 (1.06e−2) − | 9.6917e−1 (1.68e−2) − | 8.0509e−1 (5.25e−2) | |
| 100 | 9.2375e−1 (5.81e−3) −− | 9.2435e−1 (5.91e−3) − | 9.2394e−1 (6.97e−3) − | 9.2452e−1 (5.96e−3) − | 9.9976e−1 (1.85e−2) − | 8.9024e−1 (2.72e−2) | |
| WFG7 | 20 | 4.9000e−1 (2.68e−2) = | 6.2300e−1 (1.16e−2) − | 5.1177e−1 (3.32e−2) − | 6.3967e−1 (1.24e−2) − | 6.3042e−1 (1.77e−2) − | 4.6640e−1 (4.72e−2) |
| 50 | 6.8819e−1 (8.02e−3) − | 6.9093e−1 (1.30e−2) − | 6.9258e−1 (8.65e−3) − | 6.9117e−1 (8.11e−3) − | 7.0440e−1 (1.71e−2) − | 6.1482e−1 (3.28e−2) | |
| 100 | 6.8773e−1 (4.28e−3) − | 6.8943e−1 (5.35e−3) − | 6.8872e−1 (5.83e−3) − | 6.8955e−1 (4.95e−3) − | 7.4123e−1 (1.79e−2) − | 6.8255e−1 (1.08e−2) | |
| WFG8 | 20 | 5.4647e−1 (3.79e−2) + | 6.9437e−1 (1.80e−2) − | 5.9470e−1 (6.37e−2) = | 6.6032e−1 (1.96e−2) − | 7.5424e−1 (2.52e−2) − | 5.8752e−1 (4.63e−2) |
| 50 | 7.8780e−1 (1.69e−2) − | 7.8458e−1 (1.71e−2) − | 7.9211e−1 (1.63e−2) − | 7.9064e−1 (1.31e−2) − | 7.9562e−1 (2.21e−2) − | 6.7104e−1 (2.73e−2) | |
| 100 | 7.5193e−1 (9.07e−3) − | 7.5356e−1 (8.34e−3) − | 7.5323e−1 (1.02e−2) − | 7.4997e−1 (8.69e−3) − | 8.0820e−1 (1.48e−2) − | 7.0777e−1 (2.16e−2) | |
| WFG9 | 20 | 6.5404e−1 (6.78e−2) − | 8.0876e−1 (4.35e−2) − | 6.6440e−1 (8.02e−2) − | 7.5853e−1 (5.95e−2) − | 7.3172e−1 (4.92e−2) − | 5.6977e−1 (1.10e−1) |
| 50 | 9.4366e−1 (2.57e−2) − | 9.3203e−1 (2.20e−2) − | 9.4855e−1 (2.58e−2) − | 9.4803e−1 (2.51e−2) − | 8.5468e−1 (5.66e−2) − | 8.0587e−1 (7.22e−2) | |
| 100 | 9.5590e−1 (9.97e−3) − | 9.5152e−1 (1.06e−2) − | 9.5291e−1 (9.51e−3) − | 9.5801e−1 (1.45e−2) − | 9.3799e−1 (5.04e−2) − | 8.9854e−1 (5.57e−2) | |
| + /−/ = | 4/37/7 | 1/45/2 | 5/37/6 | 5/39/4 | 12/33/3 | ||
The best results are highlighted in bold.