Table 1.
IGD values of EGC-CMOPSO and eight comparison algorithms. significant values are in bold.
| Problem | NSGAIIARSBX | CAMOEA | MOEADCMA | PREA | Two_Arch2 | GrEA | NMPSO | CMOPSO | EGCCMOPSO |
|---|---|---|---|---|---|---|---|---|---|
| ZDT1 | 4.6531e−3 (1.98e−4)− | 4.5335e−3 (1.36e−4)− | 3.8994e−3 (1.09e−5)= | 4.2888e−3 (1.57e−4)− | 3.8092e−3 (3.08e−5)+ | 7.3423e−3 (1.10e−3)− | 2.8706e−2 (1.06e−2)− | 4.1138e−3 (7.60e−5)− | 3.9120e−3 (4.89e−5) |
| ZDT2 | 4.7307e−3 (1.63e−4)− | 4.4199e−3 (1.14e−4)− | 3.8224e−3 (7.68e−6)+ | 4.4551e−3 (1.59e−4)− | 4.7349e−3 (1.87e−5)− | 7.9473e−3 (1.22e−4)− | 1.9972e−2 (4.17e−3)− | 4.0752e−3 (7.02e−5)− | 3.8787e−3 (3.86e−5) |
| ZDT3 | 5.2867e−3 (1.90e−4)− | 5.1109e−3 (1.66e−4)− | 1.0836e−2 (3.63e−5)− | 5.9553e−3 (5.36e−3)− | 6.3968e−3 (7.46e−3)− | 1.4024e−2 (9.93e−4)− | 1.0095e−1 (8.10e−4)− | 4.6589e−3 (6.74e−5)− | 4.5705e−3 (4.83e−5) |
| ZDT4 | 4.5486e−3 (1.97e−4)− | 4.3805e−3 (1.82e−4)− | 3.8983e−3 (7.43e−6)− | 4.3147e−3 (1.61e−4)− | 3.9319e−3 (6.45e−5)− | 3.6131e−1 (1.45e−1)− | 2.9867e−2 (1.58e−2)− | 3.8398e−3 (2.96e−5)+ | 3.8743e−3 (3.87e−5) |
| ZDT6 | 3.7233e−3 (1.36e−4)− | 3.4295e−3 (8.64e−5)− | 3.1044e−3 (6.37e−6)− | 3.4157e−3 (7.25e−5)− | 3.3587e−3 (3.61e−5)− | 6.1647e−3 (2.07e−5)− | 4.4733e−3 (4.83e−4)− | 3.7399e−3 (1.39e−4)− | 3.0709e−3 (1.75e−5) |
| DTLZ1 | 2.6243e−2 (1.18e−3)+ | 2.1397e−2 (3.89e−4)+ | 2.0687e−2 (1.18e−4)+ | 2.1866e−2 (5.35e−4)+ | 2.1096e−2 (2.97e−4)+ | 6.9568e+0 (2.06e+0)− | 2.2874e−2 (1.43e−3)+ | 3.2460e+0 (4.50e+0)= | 2.4716e+0 (3.46e+0) |
| DTLZ2 | 6.9491e−2 (3.32e−3)− | 5.7071e−2 (8.98e−4)− | 5.5529e−2 (4.34e−4)− | 5.7401e−2 (7.02e−4)− | 5.8928e−2 (1.61e−3)− | 5.4954e−2 (7.12e−4)= | 7.6973e−2 (3.41e−3)− | 6.7091e−2 (1.94e−3)− | 5.4808e−2 (5.68e−4) |
| DTLZ3 | 6.8843e−2 (3.69e−3)+ | 5.8967e−2 (3.07e−3)+ | 1.9424e+0 (6.49e+0)+ | 5.7279e−2 (8.72e−4)+ | 5.8048e−2 (1.84e−3)+ | 2.7332e+1 (6.55e+0)+ | 7.6408e−2 (2.17e−3)+ | 5.2557e+1 (3.13e+1)+ | 8.1506e+1 (2.91e+1) |
| DTLZ4 | 6.7679e−2 (2.45e−3)− | 5.7335e−2 (7.50e−4)− | 1.1531e−1 (8.04e−2)− | 3.5310e−1 (3.27e−1)− | 8.6455e−2 (1.62e−1)= | 7.1384e−2 (8.87e−2)− | 7.6722e−2 (2.62e−3)− | 1.2859e−1 (2.22e−1)− | 5.6342e−2 (8.29e−4) |
| DTLZ5 | 5.8597e−3 (3.44e−4)− | 5.1122e−3 (1.67e−4)− | 2.2768e−2 (6.95e−5)− | 4.6739e−3 (1.35e−4)− | 5.8903e−3 (3.47e−4)− | 6.8862e−3 (5.52e−4)− | 1.5234e−2 (2.87e−3)− | 6.1616e−3 (3.68e−4)− | 4.2846e−3 (6.85e−5) |
| DTLZ6 | 5.6115e−3 (3.29e−4)− | 4.6541e−3 (1.26e−4)− | 2.2885e−2 (2.37e−5)− | 4.6779e−3 (1.36e−4)− | 6.2492e−3 (1.46e−5)− | 1.1328e−1 (2.48e−2)− | 1.4716e−2 (3.48e−3)− | 5.3144e−3 (2.52e−4)− | 4.1797e−3 (4.54e−5) |
| DTLZ7 | 7.8262e−2 (3.97e−3)− | 8.0415e−2 (7.31e−2)− | 1.5133e−1 (5.19e−3)− | 2.2530e−1 (2.61e−1)= | 6.6276e−2 (5.26e−2)− | 9.3116e−2 (5.47e−2)− | 6.9417e−2 (4.10e−3)− | 1.3865e−1 (1.48e−1)− | 5.9279e−2 (1.11e−3) |
| WFG1 | 2.2995e−1 (1.26e−2)− | 1.6666e−1 (6.40e−3)= | 4.4221e−1 (1.03e−1)− | 1.6064e−1 (4.68e−3)+ | 1.3892e−1 (2.96e−3)+ | 1.7872e−1 (5.57e−3)− | 5.7320e−1 (2.23e−1)− | 1.1708e+0 (1.48e−1)− | 1.6903e−1 (2.21e−2) |
| WFG2 | 2.1489e−1 (1.02e−2)− | 1.8171e−1 (6.01e−3)− | 2.4414e−1 (1.80e−2)− | 1.7557e−1 (5.65e−3)= | 1.5490e−1 (3.28e−3)+ | 2.2835e−1 (1.25e−2)− | 4.5097e−1 (5.36e−2)− | 2.1323e−1 (1.15e−2)− | 1.7509e−1 (5.28e−3) |
| WFG3 | 8.8850e−2 (1.50e−2)+ | 1.4234e−1 (1.23e−2)− | 1.3753e−1 (3.23e−3)− | 8.1256e−2 (7.24e−3)+ | 7.8758e−2 (4.42e−3)+ | 9.7682e−2 (5.01e−3)+ | 4.2503e−2 (3.10e−3)+ | 1.0580e−1 (2.02e−2)+ | 1.2652e−1 (8.56e−3) |
| WFG4 | 2.9248e−1 (1.19e−2)− | 2.3211e−1 (3.85e−3)− | 3.3374e−1 (1.62e−2)− | 2.2678e−1 (3.61e−3)− | 2.2439e−1 (4.93e−3)− | 2.4697e−1 (3.18e−3)− | 3.0795e−1 (1.19e−2)− | 3.0258e−1 (9.03e−3)− | 2.2122e−1 (3.19e−3) |
| WFG5 | 2.8400e−1 (1.33e−2)− | 2.3928e−1 (2.68e−3)− | 2.9660e−1 (9.95e−3)− | 2.3454e−1 (3.64e−3)− | 2.3454e−1 (4.76e−3)− | 2.5519e−1 (2.63e−3)− | 2.9702e−1 (9.99e−3)− | 2.8219e−1 (1.00e−2)− | 2.2340e−1 (2.13e−3) |
| WFG6 | 3.0213e−1 (1.12e−2)− | 2.6058e−1 (9.51e−3)− | 3.1159e−1 (3.04e−2)− | 2.4541e−1 (9.68e−3)− | 2.4640e−1 (1.10e−2)− | 2.6526e−1 (7.94e−3)− | 4.0612e−1 (9.45e−3)− | 2.8598e−1 (1.47e−2)− | 2.2486e−1 (7.01e−3) |
| WFG7 | 2.8636e−1 (1.07e−2)− | 2.3433e−1 (5.30e−3)− | 2.7430e−1 (1.95e−2)− | 2.2897e−1 (3.55e−3)− | 2.2269e−1 (4.62e−3)− | 2.4860e−1 (1.50e−3)− | 3.0892e−1 (1.36e−2)− | 2.7030e−1 (8.77e−3)− | 2.1478e−1 (3.01e−3) |
| WFG8 | 3.8214e−1 (1.15e−2)− | 3.2555e−1 (6.62e−3)− | 3.6561e−1 (6.28e−3)− | 2.8273e−1 (4.85e−3)+ | 2.9884e−1 (6.46e−3)+ | 2.9239e−1 (5.76e−3)+ | 3.4821e−1 (1.12e−2)− | 3.8795e−1 (1.15e−2)− | 3.0771e−1 (4.47e−3) |
| WFG9 | 2.6774e−1 (9.56e−3)− | 2.3130e−1 (4.08e−3)− | 2.9791e−1 (5.76e−3)− | 2.2529e−1 (3.63e−3)− | 2.2141e−1 (5.01e−3)+ | 2.3981e−1 (4.09e−3)− | 3.3511e−1 (4.94e−2)− | 2.6060e−1 (8.10e−3)− | 2.2201e−1 (3.43e−2) |
| +/−/= | 3/18/0 | 2/18/1 | 3/17/1 | 5/14/2 | 8/12/1 | 3/17/1 | 3/18/0 | 3/17/1 |