An Evolutionary Algorithm with Clustering-Based Assisted Selection Strategy for Multimodal Multiobjective Optimization
Table 4
Results obtained by four other MMOEAs and MMOEA-CAS on IGDF.
Benchmark
Omnioptimizer
DN-NSGA-II
MO-Ring-PSO-SCD
TriMOEA-TA&R
MMOEA-CAS
SYM-PART simple
1.216e−02 (1.72e−03)+
1.230e−02 (1.78e−03)+
4.042e−02 (5.50e−03)−
3.414e−02 (4.99e−03)−
2.628e−02 (3.07e−03)
SYM-PART rotated
1.252e−02 (1.70e−03)+
1.539e−02 (2.12e−03)+
4.604e−02 (4.32e−03)−
2.843e−02 (3.88e−03)+
3.201e−02 (5.92e−03)
Omnitest
6.776e−03 (2.47e−04)+
7.911e−03 (5.10e−04)+
4.232e−02 (5.35e−03)−
1.841e−02 (4.18e−03)+
2.953e−02 (2.03e−03)
MMF1
3.788e−03 (5.27e−04)+
4.451e−03 (4.07e−04)+
3.707e−03 (1.58e−04)+
4.926e−03 (9.72e−04)∼
4.762e−03 (5.42e−04)
MMF1_z
3.231e−03 (4.30e−04)+
3.649e−03 (4.91e−04)+
3.603e−03 (1.57e−04)+
4.418e−03 (9.38e−04)+
5.427e−03 (1.06e−03)
MMF1_e
1.701e−02 (9.98e−03)+
2.171e−02 (1.52e−02)+
1.229e−02 (1.50e−03)+
6.981e−03 (1.42e−03)+
1.191e−01 (5.32e−02)
MMF2
1.801e−02 (9.68e−03)∼
3.691e−02 (3.43e−02)∼
1.994e−02 (4.34e−03)∼
2.521e−02 (8.18e−03)−
1.839e−02 (3.99e−03)
MMF3
2.203e−02 (2.40e−02)∼
2.320e−02 (1.42e−02)∼
1.614e−02 (3.22e−03)∼
5.876e−02 (7.78e−02)−
1.990e−02 (9.70e−03)
MMF4
2.818e−03 (2.63e−04)−
3.209e−03 (1.81e−04)−
3.652e−03 (3.93e−04)−
3.445e−02 (6.70e−02)−
2.694e−03 (4.49e−04)
MMF5
3.471e−03 (6.47e−04)+
3.793e−03 (3.10e−04)+
3.748e−03 (1.94e−04)+
4.220e−03 (6.03e−04)∼
4.284e−03 (5.54e−04)
MMF6
3.154e−03 (3.20e−04)+
3.747e−03 (2.97e−04)+
3.538e−03 (1.34e−04)+
4.602e−03 (2.06e−03)+
7.700e−03 (2.43e−03)
MMF7
3.162e−03 (2.96e−04)−
3.957e−03 (4.17e−04)−
3.830e−03 (3.43e−04)−
4.273e−03 (1.25e−03)−
2.937e−03 (6.51e−04)
MMF8
3.145e−03 (1.88e−04)+
3.983e−03 (5.03e−04)+
4.838e−03 (2.83e−04)∼
5.667e−03 (7.00e−03)−
5.583e−03 (1.69e−03)
MMF9
1.271e−02 (1.28e−03)∼
1.441e−02 (1.84e−03)−
1.562e−02 (1.42e−03)−
6.975e−02 (4.17e−03)−
1.332e−02 (1.42e−03)
MMF10
1.822e−01 (4.17e−02)−
1.938e−01 (4.11e−02)−
1.297e−01 (1.51e−02)−
2.285e−01 (5.05e−03)−
3.839e−02 (3.63e−02)
MMF11
9.565e−02 (9.39e−04)−
9.823e−02 (2.07e−03)−
8.508e−02 (6.16e−03)−
1.633e−01 (7.27e−03)−
2.827e−02 (3.10e−03)
MMF12
8.611e−02 (8.13e−03)−
8.404e−02 (2.68e−03)−
6.431e−02 (1.27e−02)−
8.601e−02 (1.47e−03)−
5.118e−03 (3.54e−04)
MMF13
1.472e−01 (2.08e−03)−
1.507e−01 (4.40e−03)−
9.329e−02 (1.73e−02)−
2.431e−01 (6.87e−03)−
4.139e−02 (9.33e−03)
MMF14
9.765e−02 (3.95e−03)−
1.080e−01 (8.20e−03)−
8.042e−02 (2.94e−03)−
8.616e−02 (1.12e−03)−
6.788e−02 (1.14e−03)
MMF14_a
1.049e−01 (4.57e−03)−
1.178e−01 (6.69e−03)−
7.818e−02 (2.50e−03)−
7.878e−02 (1.15e−03)−
7.506e−02 (1.69e−03)
MMF15
2.010e−01 (7.37e−03)−
2.162e−01 (1.14e−02)−
1.741e−01 (3.27e−03)−
2.067e−01 (7.10e−04)−
1.134e−01 (1.21e−02)
MMF15_a
2.074e−01 (8.92e−03)−
2.259e−01 (1.05e−02)−
1.744e−01 (3.72e−03)−
1.936e−01 (3.67e−03)−
1.483e−01 (3.59e−03)
best/all
9/22
0/22
2/22
1/22
10/22
+/–/∼
9/10/3
9/11/2
5/14/3
5/15/2
– –
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.
