An Evolutionary Algorithm with Clustering-Based Assisted Selection Strategy for Multimodal Multiobjective Optimization
Table 3
Results obtained by four other MMOEAs and MMOEA-CAS on IGDX.
Benchmark
Omnioptimizer
DN-NSGA-II
MO-Ring-PSO-SCD
TriMOEA-TA&R
MMOEA-CAS
SYM-PART simple
5.558e + 00 (2.61e + 00)−
3.810e + 00 (9.39e−01)−
1.663e−01 (2.19e−02)−
2.181e−02 (3.01e−03) +
7.633e−02 (4.92e−03)
SYM-PART rotated
4.405e + 00 (1.22e + 00)−
4.613e + 00 (1.99e + 00)−
1.972e−01 (3.50e−02)+
1.674e + 00 (9.48e−01)−
4.240e−01 (6.09e−01)
Omnitest
1.501e + 00 (2.29e−01)−
1.410e + 00 (2.10e−01)−
4.057e−01 (8.74e−02)−
2.536e−01 (1.09e−01)−
1.092e−01 (3.23e−03)
MMF1
9.905e−02 (1.81e−02)−
9.963e−02 (1.70e−02)−
4.951e−02 (2.06e−03)+
7.290e−02 (8.54e−03)−
5.350e−02 (5.06e−03)
MMF1_z
7.949e−02 (1.57e−02)−
7.898e−02 (1.01e−02)−
3.576e−02 (2.13e−03)+
7.040e−02 (1.17e−02)−
5.593e−02 (6.27e−03)
MMF1_e
1.314e + 00 (7.26e−01)∼
1.055e + 00 (4.70e−01)∼
5.709e−01 (2.57e−01)+
1.940e + 00 (7.07e−01)−
1.074e + 00 (2.73e−01)
MMF2
1.266e−01 (1.01e−01)−
1.389e−01 (8.51e−02)−
3.802e−02 (1.72e−02)∼
7.150e−02 (2.82e−02)−
3.370e−02 (1.36e−02)
MMF3
9.213e−02 (4.50e−02)−
9.220e−02 (3.53e−02)−
3.164e−02 (1.19e−02)+
9.315e−02 (3.98e−02)−
4.730e−02 (1.46e−02)
MMF4
8.358e−02 (2.70e−02)−
8.963e−02 (2.55e−02)−
2.705e−02 (1.71e−03)−
1.008e−01 (1.43e−01)−
2.221e−02 (1.91e−03)
MMF5
1.763e−01 (2.39e−02)−
1.703e−01 (2.06e−02)−
8.615e−02 (5.66e−03)∼
1.114e−01 (8.42e−03)−
8.754e−02 (8.77e−03)
MMF6
1.441e−01 (1.68e−02)−
1.471e−01 (1.66e−02)−
7.260e−02 (3.87e−03)+
9.006e−02 (9.24e−03) +
1.136e−01 (2.14e−02)
MMF7
4.629e−02 (9.25e−03)−
5.681e−02 (1.40e−02)−
2.610e−02 (1.54e−03)−
4.468e−02 (2.13e−02)−
2.428e−02 (3.96e−03)
MMF8
3.286e−01 (1.65e−01)−
3.012e−01 (9.95e−02)−
6.719e−02 (5.03e−03)+
3.438e−01 (1.04e−01)−
1.448e−01 (3.29e−02)
MMF9
2.200e−02 (1.09e−02)−
2.529e−02 (1.32e−02)−
7.882e−03 (3.55e−04)+
3.124e−03 (9.15e−05) +
8.759e−03 (1.30e−03)
MMF10
1.646e−01 (3.31e−02)−
1.617e−01 (4.48e−02)−
1.004e−01 (3.69e−02)−
2.014e−01 (7.52e−05)−
2.755e−02 (4.78e−02)
MMF11
2.502e−01 (3.58e−04)−
2.505e−01 (4.85e−04)−
2.169e−01 (2.72e−02)−
2.524e−01 (5.79e−05)−
9.196e−03 (1.25e−03)
MMF12
2.396e−01 (1.68e−02)−
2.467e−01 (9.44e−04)−
1.875e−01 (4.53e−02)−
2.477e−01 (6.84e−04)−
3.120e−03 (1.84e−04)
MMF13
2.858e−01 (9.42e−03)−
2.895e−01 (1.54e−02)−
2.325e−01 (1.43e−02)−
2.714e−01 (6.55e−03)−
1.055e−01 (1.46e−02)
MMF14
8.873e−02 (6.29e−03)−
9.794e−02 (9.76e−03)−
5.329e−02 (1.86e−03)−
3.651e−02 (4.92e−04) +
4.651e−02 (1.11e−03)
MMF14_a
1.110e−01 (9.66e−03)−
1.196e−01 (5.83e−03)−
5.997e−02 (1.97e−03)+
5.604e−02 (1.59e−03) +
7.757e−02 (4.54e−03)
MMF15
2.471e−01 (2.10e−02)−
2.238e−01 (2.59e−02)−
1.562e−01 (1.63e−02)−
2.711e−01 (2.53e−04)−
5.256e−02 (7.43e−03)
MMF15_a
2.133e−01 (1.99e−02)−
2.086e−01 (1.46e−02)−
1.641e−01 (1.31e−02)−
2.194e−01 (2.83e−03)−
9.684e−02 (6.43e−03)
best/all
0/22
0/22
8/22
4/22
10/22
+/–/∼
0/21/1
0/21/1
9/11/2
5/17/0
– –
Wilcoxon’s rank-sum test () is used for experimental comparison where +, −, and = represent that the performance of the results obtained by competitors is better, worse, or similar.
