Introduction
Related work
The proposed algorithm for many-objective optimization
The general framework of the proposed algorithm
The knee point-driven environmental selection
Knee point
Knee-oriented dominance relationship
Processs of environmental selection
Novel velocity update equation
Archive update
Computational complexity
Test function | Number of objectives (\(M\)) | Number of decision variables (\(N\)) | Parameters in \(k\left( x \right)\left( {A,B,s,p,l} \right)\) | Number of convex knees | |
---|---|---|---|---|---|
PMOP 1 | 3,5,8,10 | \(m + 9\) | (4, 1, -1, 1) | \(\left[ {A/2} \right]^{m - 1}\) | |
PMOP 2,3,8,11,12 | 3,5,8,10 | \(m + 9\) | (4, 1, 2, 1) | \(\left[ {A/2} \right]^{m - 1}\) | |
PMOP 5 | 3,5,8,10 | \(m + 9\) | (1, 1, 2, 1, 12) | \(\left[ {2 * A} \right]^{m - 1}\) | |
PMOP 6,9 | 3,5,8,10 | \(m + 9\) | (2,1,2,1) | \(\left[ {A - 1} \right]^{m - 1}\) | |
PMOP 13 | 3,5,8,10 | \(m + 9\) | (2,1,-2,1) | \(\infty\) | |
PMOP 14 | 3,5,8,10 | \(m + 9\) | (2,1,-1,1) | \(\infty\) |
Problem | M | D | KnEA | LA_MOEA | LBD_MOEA | MAPIO | KnMAPIO |
---|---|---|---|---|---|---|---|
PMOP1 | 3 | 12 | 1.8469e−3 (2.44e−4) = | 1.4322e−3 (1.35e−4) + | 1.4016e−3 (1.19e−4) + | 1.9981e−3 (3.36e−4) = | 1.9213e−3 (2.06e−4) |
5 | 14 | 2.2393e−2 (2.79e−3) + | 1.6326e−2 (1.38e−3) + | 1.6990e−2 (1.33e−3) + | 4.4279e−2 (7.95e−3) − | 3.3608e−2 (3.59e−3) | |
8 | 17 | 7.7401e−2 (2.00e−2) + | 5.3710e−2 (4.16e−3) + | 5.5873e−2 (4.30e−3) + | 1.8405e−1 (4.14e−2) = | 1.8334e−1 (1.78e−2) | |
10 | 19 | 9.6444e−2 (6.09e−2) + | 6.1521e−2 (5.47e−3) + | 6.2354e−2 (6.76e−3) + | 1.6543e−1 (8.11e−2) + | 2.3085e−1 (3.96e−2) | |
PMOP2 | 3 | 12 | 1.9565e−3 (2.90e−3) − | 3.1685e−3 (9.44e−4) − | 2.9804e−3 (8.07e−4) − | 7.0534e−4 (4.91e−5) + | 7.6479e−4 (3.33e−5) |
5 | 14 | 1.8891e−2 (9.17e−3) − | 5.7350e−3 (5.81e−4) − | 6.1427e−3 (7.61e−4) − | 1.1061e−1 (4.73e−1) − | 5.2127e−3 (1.35e−4) | |
8 | 17 | 2.8814e+0 (1.26e+0) − | 1.2864e−2 (2.02e−3) − | 1.2421e−2 (2.69e−3) = | 1.1515e−2 (1.62e−3) = | 1.1769e−2 (9.64e−4) | |
10 | 19 | 3.2156e+0 (2.12e+0) − | 2.0847e−2 (9.18e−3) − | 1.8909e−2 (5.68e−3) − | 1.3402e−2 (1.77e−3) = | 1.2415e−2 (1.81e−3) | |
PMOP3 | 3 | 12 | 9.3679e−2 (1.39e−1) − | 1.4563e−2 (1.49e−2) − | 1.3153e−2 (1.07e−2) − | 2.8556e−3 (5.07e−3) − | 2.0614e−3 (2.36e−3) |
5 | 14 | 3.0432e+0 (1.07e+0) − | 1.5511e−2 (1.05e−2) − | 1.6246e−2 (8.44e−3) − | 6.3091e−2 (2.58e−1) − | 6.3105e−3 (1.41e−3) | |
8 | 17 | 7.1034e+0 (1.90e+0) − | 1.3408e−2 (7.58e−3) = | 1.2494e−2 (4.78e−3) = | 1.3903e+0 (2.96e+0) − | 1.1617e−2 (3.31e−3) | |
10 | 19 | 5.0461e+0 (2.71e+0) − | 1.0155e−1 (4.52e−2) − | 1.0363e−1 (4.98e−2) − | 1.3361e+0 (2.10e+0) − | 7.0070e−2 (1.30e−1) | |
PMOP5 | 3 | 12 | 3.3007e+0 (8.58e+0) = | 7.2112e−1 (9.60e−1) = | 1.0358e+0 (8.92e−1) = | 3.8080e+3 (1.41e+4) = | 1.1629e+0 (1.90e+0) |
5 | 14 | 1.3326e+1 (2.28e+1) − | 9.3691e−1 (5.21e−1) = | 9.2124e−1 (9.94e−1) = | 1.7550e+4 (3.47e+4) = | 1.2684e+0 (2.09e+0) | |
8 | 17 | 1.4201e+3 (1.43e+3) − | 2.1125e+0 (1.93e+0) − | 1.5023e+0 (1.23e+0) = | 1.6264e+4 (5.02e+4) − | 9.1905e−1 (4.46e−2) | |
10 | 19 | 3.2018e+4 (9.13e+4) − | 2.2011e+1 (1.15e+1) − | 2.4439e+1 (1.10e+1) − | 5.8693e+4 (6.72e+4) − | 1.7880e+0 (3.44e+0) | |
PMOP6 | 3 | 12 | 1.3213e−1 (1.82e−1) − | 2.3207e−2 (4.64e−2) = | 4.3035e−3 (1.21e−2) = | 1.3016e−3 (2.73e−4) = | 1.5043e−3 (2.97e−4) |
5 | 14 | 3.5210e+0 (2.35e+0) − | 1.1492e−1 (1.38e−1) − | 5.9528e−2 (7.52e−2) − | 5.5949e−1 (2.46e+0) − | 1.0083e−2 (8.70e−4) | |
8 | 17 | 5.5865e+2 (2.08e+2) − | 1.0850e+0 (1.31e+0) − | 5.9012e−1 (4.69e−1) = | 3.7760e+1 (5.93e+1) = | 3.6419e−1 (8.06e−2) | |
10 | 19 | 8.2870e+3 (8.01e+3) − | 2.3691e+2 (3.48e+2) + | 2.7826e+2 (3.94e+2) + | 1.8033e+3 (1.95e+3) = | 1.6698e+3 (1.23e+3) | |
PMOP8 | 3 | 12 | 2.7807e−3 (3.44e−3) − | 1.1518e−3 (1.57e−4) − | 1.1338e−3 (2.18e−4) − | 5.6264e−4 (4.57e−5) = | 5.8972e−4 (2.91e−5) |
5 | 14 | 2.6941e−2 (7.77e−3) − | 3.1981e−3 (3.56e−4) + | 3.2131e−3 (3.49e−4) + | 3.0957e−3 (4.65e−4) + | 3.6720e−3 (1.36e−4) | |
8 | 17 | 1.9409e−1 (4.40e−2) − | 3.2794e−3 (7.56e−4) + | 3.1164e−3 (4.23e−4) + | 6.3988e−3 (1.28e−3) + | 8.3614e−3 (3.57e−4) | |
10 | 19 | 1.4787e−1 (2.45e−2) − | 2.1042e−2 (8.72e−3) − | 2.0419e−2 (7.14e−3) − | 1.2530e−2 (6.13e−3) = | 9.6205e−3 (1.59e−3) | |
PMOP9 | 3 | 12 | 1.9194e−2 (8.43e−3) − | 1.3688e−2 (8.53e−3) − | 1.1580e−2 (6.50e−3) − | 4.5841e−2 (6.16e−2) − | 9.9630e−4 (7.80e−4) |
5 | 14 | 2.8807e−1 (7.58e−2) − | 1.5171e−2 (8.83e−3) − | 1.6958e−2 (7.94e−3) − | 2.4622e−2 (2.68e−2) − | 7.3974e−3 (5.26e−3) | |
8 | 17 | 9.5797e−1 (2.50e−1) − | 2.6375e−2 (9.77e−3) + | 2.8023e−2 (1.66e−2) + | 8.7855e−2 (1.35e−1) − | 6.1833e−2 (3.97e−2) | |
10 | 19 | 1.5176e+0 (2.89e−1) − | 8.3021e−2 (3.72e−2) + | 7.6761e−2 (2.57e−2) + | 2.8071e−1 (2.81e−1) + | 5.3258e−1 (3.03e−1) | |
PMOP11 | 3 | 12 | 4.0225e−3 (4.92e−3) − | 6.6403e−3 (6.89e−3) − | 1.2664e−2 (1.33e−2) − | 7.3591e−4 (2.46e−4) = | 8.1330e−4 (5.06e−5) |
5 | 14 | 6.8024e−2 (3.37e−2) − | 1.3109e−1 (1.03e−1) − | 2.8282e−1 (3.11e−1) − | 1.1223e−2 (2.35e−3) + | 1.3067e−2 (4.65e−4) | |
8 | 17 | 9.0247e+0 (3.39e+0) − | 3.5381e−1 (1.68e−1) − | 3.5049e−1 (1.50e−1) − | 4.5320e−2 (6.83e−3) = | 4.4666e−2 (1.31e−3) | |
10 | 19 | 8.3257e+0 (4.67e+0) − | 8.7800e−1 (4.76e−1) − | 7.0604e−1 (3.59e−1) − | 7.5371e−2 (1.76e−2) − | 5.0304e−2 (3.50e−3) | |
PMOP12 | 3 | 12 | 1.7102e−2 (2.89e−2) − | 1.9103e−4 (1.03e−4) = | 7.7803e−4 (1.94e−3) = | 1.3622e−4 (4.20e−5) = | 1.7267e−4 (6.93e−5) |
5 | 14 | 4.8386e−1 (1.13e−1) − | 3.2192e−3 (6.61e−3) = | 2.5359e−3 (4.66e−3) − | 1.1588e−1 (3.80e−1) = | 3.5954e−4 (1.35e−4) | |
8 | 17 | 4.7259e−1 (1.93e−1) − | 2.8734e−4 (1.37e−4) + | 3.6082e−4 (1.85e−4) + | 3.6956e−1 (5.34e−1) = | 1.3624e−3 (3.13e−3) | |
10 | 19 | 1.7012e−1 (1.93e−1) − | 1.3372e−3 (4.90e−4) + | 1.3964e−3 (4.79e−4) + | 1.2453e−1 (2.73e−1) = | 4.1743e−2 (4.13e−2) | |
PMOP13 | 3 | 12 | 4.2588e−2 (2.72e−2) − | 2.3506e−2 (1.23e−2) − | 2.7068e−2 (1.83e−2) − | 4.9823e−2 (1.06e−1) − | 1.7025e−3 (3.19e−4) |
5 | 14 | 1.3874e+0 (4.57e−1) − | 3.0477e−1 (2.06e−1) − | 2.9364e−1 (2.21e−1) − | 4.3872e−1 (5.27e−1) − | 9.7955e−2 (9.70e−3) | |
8 | 17 | 4.4765e+1 (3.14e+1) − | 1.9912e+0 (1.93e+0) + | 2.1635e+0 (1.45e+0) + | 2.5893e+1 (1.89e+1) − | 1.0305e+1 (1.03e+1) | |
10 | 19 | 1.0412e+2 (1.61e+2) + | 1.0088e+1 (4.96e+0) + | 8.9263e+0 (5.66e+0) + | 8.7474e+1 (6.42e+1) + | 2.3184e+2 (1.38e+2) | |
PMOP14 | 3 | 12 | 2.6551e−2 (6.55e−2) − | 3.4955e−2 (5.01e−2) − | 1.7731e−2 (2.12e−2) − | 1.0437e−3 (3.77e−4) = | 1.0194e−3 (2.84e−4) |
5 | 14 | 2.2893e+0 (1.51e+0) − | 1.8049e−1 (1.46e−1) − | 1.6238e−1 (1.92e−1) − | 1.4398e+0 (6.31e+0) − | 2.4340e−2 (3.11e−3) | |
8 | 17 | 4.2772e+1 (2.48e+1) − | 2.1939e−1 (1.11e−1) + | 1.9790e−1 (1.64e−1) + | 6.8881e+1 (5.90e+1) − | 4.1384e−1 (7.81e−2) | |
10 | 19 | 4.6106e+1 (5.45e+1) = | 9.5097e−1 (6.28e−1) + | 8.8581e−1 (4.01e−1) + | 1.2108e+2 (1.28e+2) = | 2.4970e+1 (2.31e+1) | |
± / = | 4/37/3 | 15/23/6 | 15/21/8 | 7/18/19 |
Problem | M | D | KnEA | LA_MOEA | LBD_MOEA | MAPIO | KnMAPIO |
---|---|---|---|---|---|---|---|
PMOP1 | 3 | 12 | 2.6853e−1 (1.18e−1) = | 5.9505e−1 (1.26e−1) − | 5.4260e−1 (2.34e−2) − | 6.0434e−1 (1.83e−1) − | 2.7254e−1 (6.58e−2) |
5 | 14 | 1.0157e+0 (3.23e−1) = | 1.1116e+0 (7.93e−2) − | 1.2159e+0 (1.90e−1) − | 1.8858e+0 (5.11e−1) − | 1.0083e+0 (2.01e−1) | |
8 | 17 | 2.6883e+0 (5.57e−1) = | 2.0252e+0 (2.47e−1) + | 2.0805e+0 (2.47e−1) + | 3.8210e+0 (1.05e+0) − | 2.5159e+0 (5.98e−1) | |
10 | 19 | 3.3227e+0 (8.43e−1) = | 2.4065e+0 (5.82e−1) + | 2.4333e+0 (6.86e−1) + | 3.8954e+0 (1.49e+0) − | 3.1598e+0 (5.70e−1) | |
PMOP2 | 3 | 12 | 8.2361e−2 (3.27e−2) − | 1.9643e−1 (3.91e−2) − | 1.8918e−1 (2.89e−2) − | 1.2218e−1 (8.63e−2) − | 5.1144e−2 (2.77e−3) |
5 | 14 | 2.1464e−1 (3.66e−2) − | 2.5161e−1 (1.95e−2) − | 2.6696e−1 (9.84e−3) − | 2.7517e−1 (6.38e−2) − | 1.2601e−1 (3.83e−3) | |
8 | 17 | 1.0408e+0 (5.95e−1) − | 2.9526e−1 (1.57e−2) − | 2.8723e−1 (2.72e−2) − | 2.5802e−1 (4.53e−2) − | 1.7112e−1 (8.86e−3) | |
10 | 19 | 4.2602e+0 (1.36e+1) − | 2.5368e−1 (6.42e−2) − | 2.3400e−1 (3.19e−2) − | 2.4022e−1 (3.94e−2) − | 1.6852e−1 (1.82e−2) | |
PMOP3 | 3 | 12 | 2.5274e−1 (1.05e−1) + | 5.8548e−1 (1.85e−1) − | 5.8268e−1 (1.48e−1) − | 8.7264e−1 (4.08e−1) − | 3.6911e−1 (9.86e−2) |
5 | 14 | 3.7703e+0 (1.77e+0) − | 4.1051e−1 (7.52e−2) = | 4.1513e−1 (5.60e−2) = | 7.1453e−1 (3.25e−1) − | 4.5518e−1 (1.38e−1) | |
8 | 17 | 5.2670e+0 (2.46e+0) − | 8.0183e−1 (1.05e−1) − | 7.6525e−1 (1.22e−1) − | 9.3674e−1 (2.51e−1) − | 6.3808e−1 (1.92e−1) | |
10 | 19 | 2.2139e+0 (6.92e−1) − | 7.3272e−1 (1.70e−1) − | 7.5130e−1 (1.58e−1) − | 9.2550e−1 (3.80e−1) − | 4.6871e−1 (1.52e−1) | |
PMOP5 | 3 | 12 | 2.8170e+0 (2.13e+0) + | 3.3324e+0 (4.19e+0) + | 4.7727e+0 (3.75e+0) + | 4.2699e+0 (2.29e+0) = | 9.5969e+0 (1.49e+1) |
5 | 14 | 1.4829e+1 (1.11e+1) − | 4.3223e+0 (2.12e+0) + | 4.6777e+0 (4.05e+0) + | 7.0438e+0 (3.45e+0) = | 1.0108e+1 (1.56e+1) | |
8 | 17 | 3.3853e+2 (3.25e+2) − | 1.5657e+1 (1.85e+1) − | 8.5184e+0 (7.50e+0) = | 3.9376e+1 (4.42e+1) − | 7.4689e+0 (4.63e−1) | |
10 | 19 | 6.8345e+2 (1.12e+3) − | 8.8091e+1 (4.85e+1) − | 1.0208e+2 (5.66e+1) − | 9.3403e+1 (1.19e+2) − | 1.0832e+1 (1.14e+0) | |
PMOP6 | 3 | 12 | 1.7601e−1 (1.60e−1) + | 5.4568e−1 (1.47e−1) − | 4.6391e−1 (1.20e−1) − | 6.2970e−1 (1.49e−1) − | 2.7046e−1 (5.56e−2) |
5 | 14 | 4.3831e+0 (3.47e+0) − | 7.6308e−1 (2.75e−1) + | 7.3838e−1 (9.21e−2) + | 1.1036e+0 (4.67e−1) − | 8.8021e−1 (1.21e−1) | |
8 | 17 | 7.5155e+1 (3.37e+1) − | 1.7853e+1 (1.73e+0) + | 1.8490e+1 (1.59e+0) + | 2.1558e+1 (1.77e+0) = | 2.1015e+1 (2.21e+0) | |
10 | 19 | 2.5178e+3 (6.14e+2) − | 4.4665e+3 (8.82e+1) − | 4.4723e+3 (9.92e+1) − | 4.6534e+3 (1.26e+2) − | 1.9794e+3 (3.05e+2) | |
PMOP8 | 3 | 12 | 7.0425e−2 (1.13e−2) − | 1.4075e−1 (1.58e−2) − | 1.4191e−1 (1.35e−2) − | 8.2930e−2 (4.10e−2) − | 4.8745e−2 (3.21e−3) |
5 | 14 | 1.3441e−1 (4.09e−2) − | 1.0892e−1 (1.08e−2) − | 1.1311e−1 (1.21e−2) − | 1.2163e−1 (2.07e−2) − | 8.2575e−2 (4.88e−3) | |
8 | 17 | 8.8452e−1 (2.18e−1) − | 9.0989e−2 (7.80e−3) + | 8.9956e−2 (8.29e−3) + | 1.0620e−1 (9.73e−3) = | 1.0402e−1 (6.56e−3) | |
10 | 19 | 7.6039e−1 (1.50e−1) − | 2.4150e−1 (9.75e−2) − | 2.4283e−1 (7.85e−2) − | 1.4406e−1 (4.79e−2) − | 9.2146e−2 (7.88e−3) | |
PMOP9 | 3 | 12 | 1.3906e−1 (4.50e−2) + | 2.0702e−1 (1.61e−2) − | 2.2717e−1 (3.69e−2) − | 2.2026e−1 (8.45e−2) − | 1.6904e−1 (3.37e−2) |
5 | 14 | 5.8132e−1 (1.71e−1) − | 3.1323e−1 (3.14e−2) + | 3.0794e−1 (3.15e−2) + | 3.6916e−1 (1.07e−1) = | 3.8989e−1 (7.01e−2) | |
8 | 17 | 1.9717e+0 (5.31e−1) − | 1.2528e+0 (1.56e−1) − | 1.1900e+0 (1.28e−1) − | 1.2311e+0 (4.38e−1) = | 1.0381e+0 (1.91e−1) | |
10 | 19 | 2.7666e+0 (7.77e−1) = | 2.1324e+0 (8.83e−1) + | 2.2090e+0 (8.58e−1) + | 2.5009e+0 (8.30e−1) = | 3.0383e+0 (8.21e−1) | |
PMOP11 | 3 | 12 | 1.3908e−1 (8.77e−2) − | 4.3339e−1 (2.27e−1) − | 4.3479e−1 (1.63e−1) − | 2.8026e−1 (1.97e−1) − | 7.7811e−2 (5.72e−3) |
5 | 14 | 5.2547e−1 (1.04e−1) − | 1.1296e+0 (3.85e−1) − | 1.4137e+0 (5.62e−1) − | 5.1416e−1 (1.38e−1) − | 3.1333e−1 (8.19e−3) | |
8 | 17 | 3.9618e+0 (9.52e−1) − | 2.5680e+0 (6.86e−1) − | 2.2941e+0 (6.27e−1) − | 7.8853e−1 (1.37e−1) − | 5.4073e−1 (1.93e−2) | |
10 | 19 | 4.7537e+0 (2.20e+0) − | 3.0703e+0 (1.01e+0) − | 3.0707e+0 (1.07e+0) − | 1.0975e+0 (2.18e−1) − | 5.7539e−1 (4.82e−2) | |
PMOP12 | 3 | 12 | 2.4202e−2 (3.05e−2) + | 7.5334e−2 (1.25e−2) − | 7.1418e−2 (9.85e−3) − | 6.6945e−2 (2.24e−2) − | 3.9892e−2 (6.91e−3) |
5 | 14 | 1.9901e−1 (6.55e−2) − | 2.2920e−2 (6.16e−3) = | 2.3061e−2 (4.56e−3) − | 4.4371e−2 (2.65e−2) − | 1.9186e−2 (4.65e−3) | |
8 | 17 | 1.1159e−1 (3.93e−2) − | 1.4485e−2 (2.07e−3) = | 1.3576e−2 (1.71e−3) = | 1.8497e−2 (2.18e−2) = | 1.4013e−2 (3.24e−3) | |
10 | 19 | 3.7249e−2 (1.58e−2) − | 1.0771e−2 (2.31e−3) = | 1.0078e−2 (2.02e−3) = | 2.5190e−2 (9.69e−3) − | 1.4129e−2 (9.13e−3) | |
PMOP13 | 3 | 12 | 3.0682e−1 (1.41e−1) + | 5.7992e−1 (6.52e−2) − | 6.0788e−1 (8.69e−2) − | 5.6996e−1 (1.19e−1) − | 4.1210e−1 (1.29e−1) |
5 | 14 | 3.0814e+0 (9.15e−1) − | 2.1149e+0 (2.81e−1) = | 2.0965e+0 (3.49e−1) = | 2.7325e+0 (9.71e−1) = | 2.1606e+0 (3.61e−1) | |
8 | 17 | 2.8672e+1 (9.63e+0) = | 2.3196e+1 (1.97e+0) = | 2.2053e+1 (2.06e+0) = | 2.4960e+1 (6.15e+0) = | 2.4559e+1 (7.42e+0) | |
10 | 19 | 8.9153e+1 (3.43e+1) + | 7.4262e+1 (1.87e+1) + | 7.3543e+1 (1.53e+1) + | 9.6421e+1 (1.48e+1) + | 1.1515e+2 (3.09e+1) | |
PMOP14 | 3 | 12 | 1.7942e−1 (3.12e−1) + | 4.1954e−1 (9.96e−2) − | 4.0260e−1 (6.51e−2) − | 3.6961e−1 (9.20e−2) − | 2.6477e−1 (7.99e−2) |
5 | 14 | 2.5583e+0 (1.75e+0) − | 5.2732e−1 (1.64e−1) + | 4.9098e−1 (1.28e−1) + | 7.5403e−1 (5.47e−1) = | 5.5255e−1 (6.78e−2) | |
8 | 17 | 1.2453e+1 (6.48e+0) − | 1.5843e+0 (3.46e−1) = | 1.3726e+0 (2.94e−1) + | 1.8290e+0 (7.60e−1) = | 1.4863e+0 (1.95e−1) | |
10 | 19 | 2.1500e+1 (1.49e+1) − | 5.7148e+0 (1.23e+0) − | 5.9720e+0 (1.45e+0) − | 4.6215e+0 (3.41e+0) − | 2.2220e+0 (7.37e−1) | |
± / = | 8/30/6 | 11/26/7 | 12/26/6 | 1/31/12 |
Experimental results
Test function | Number of decision variables (\(n\)) | Parameters | Maxgen |
---|---|---|---|
DTLZ 1 | \(M - 1 + k\) | \(k = 5\) | 700 |
DTLZ 2 | \(M - 1 + k\) | \(k = 10\) | 250 |
DTLZ 3 | \(M - 1 + k\) | \(k = 10\) | 1000 |
DTLZ 4-DTLZ 6 | \(M - 1 + k\) | \(k = 10\) | 250 |
DTLZ 7 | \(M - 1 + k\) | \(k = 20\) | 250 |
WFG 1 | \(k + l\) | \(k = M - 1,l = 20\) | 1000 |
WFG 2 | \(k + l\) | \(k = M - 1,l = 20\) | 700 |
WFG3–WFG9 | \(k + l\) | \(k = M - 1,l = 20\) | 250 |
Verification of the performance of KnMAPIO to identify knee points
Benchmark problems and experimental settings
Problem | M | D | GrEA | MOEAD | NSGAIII | RVEA | VaEA | KnMAPIO |
---|---|---|---|---|---|---|---|---|
DTLZ1 | 4 | 8 | 4.5879e−1 (2.47e−1) − | 7.3624e−1 (3.82e−1) − | 4.2250e−1 (2.24e−1) − | 5.8777e−1 (3.10e−1) − | 5.9782e−1 (9.49e−2) − | 4.0408e−1 (7.60e−2) |
6 | 10 | 1.0480e−1 (6.87e−2) + | 4.4028e−1 (4.51e−1) = | 2.9848e−1 (1.78e−1) + | 6.2092e−1 (3.24e−1) − | 2.9040e−1 (7.16e−2) + | 4.5758e−1 (9.46e−2) | |
8 | 12 | 8.1838e−2 (8.28e−2) + | 2.7124e−1 (4.47e−1) + | 2.1132e−1 (1.52e−1) + | 6.0749e−1 (3.15e−1) − | 1.2778e−1 (3.20e−2) + | 2.8715e−1 (7.22e−2) | |
10 | 14 | 2.2461e−1 (1.20e−1) − | 4.6631e−1 (5.06e−1) = | 1.8267e−1 (1.03e−1) − | 5.4869e−1 (2.85e−1) − | 1.7188e−1 (2.77e−2) − | 1.0364e−1 (3.77e−2) | |
DTLZ2 | 4 | 13 | 2.7778e−2 (1.96e−2) = | 1.4866e−1 (8.81e−2) − | 1.2389e−1 (7.35e−2) − | 1.3906e−1 (8.42e−2) − | 3.0722e−1 (8.93e−2) − | 4.0556e−2 (2.11e−2) |
6 | 15 | 2.3737e−2 (1.80e−2) − | 7.6515e−2 (5.05e−2) − | 7.9293e−2 (5.58e−2) − | 4.4275e−2 (2.91e−2) − | 2.0505e−1 (5.50e−2) − | 7.5758e−3 (8.44e−3) | |
8 | 17 | 2.7244e−1 (1.61e−1) − | 5.4060e−2 (3.86e−2) − | 1.5000e−1 (9.74e−2) − | 3.9978e−2 (2.94e−2) − | 2.0385e−1 (6.50e−2) − | 8.9744e−3 (8.52e−3) | |
10 | 19 | 1.2412e−1 (6.98e−2) − | 2.6935e−2 (2.17e−2) − | 2.1867e−1 (1.59e−1) − | 1.6254e−1 (8.95e−2) − | 3.4267e−1 (4.24e−2) − | 1.5152e−2 (1.04e−2) | |
DTLZ3 | 4 | 13 | 1.9675e−1 (1.05e−1) = | 6.9837e−1 (3.45e−1) − | 1.0463e−1 (6.17e−2) + | 4.7460e−1 (2.56e−1) − | 1.3375e−1 (6.81e−2) + | 2.3634e−1 (1.27e−1) |
6 | 15 | 2.1212e−2 (2.42e−2) + | 6.0294e−1 (3.76e−1) − | 1.2980e−1 (7.83e−2) = | 4.7319e−1 (2.54e−1) − | 1.2828e−1 (4.84e−2) + | 1.7407e−1 (8.36e−2) | |
8 | 17 | 4.1239e−2 (3.22e−2) + | 4.4444e−1 (4.53e−1) = | 9.1026e−2 (7.93e−2) = | 4.6503e−1 (2.48e−1) − | 6.3034e−2 (3.51e−2) = | 7.9915e−2 (4.77e−2) | |
10 | 19 | 3.4873e−1 (1.85e−1) − | 3.5065e−1 (4.16e−1) − | 9.9394e−2 (7.29e−2) − | 4.6986e−1 (2.04e−1) − | 1.2703e−1 (2.98e−2) − | 1.0667e−2 (1.02e−2) | |
DTLZ4 | 4 | 13 | 2.0556e−2 (1.97e−2) = | 3.9313e−3 (1.56e−2) + | 7.7500e−2 (5.89e−2) − | 1.1776e−1 (8.04e−2) − | 2.7000e−1 (9.80e−2) − | 2.8333e−2 (1.88e−2) |
6 | 15 | 5.5556e−3 (6.87e−3) − | 0.0000e+0 (0.00e+0) = | 3.7879e−3 (4.77e−3) − | 8.9628e−3 (9.70e−3) − | 1.0480e−1 (4.90e−2) − | 0.0000e+0 (0.00e+0) | |
8 | 17 | 9.8291e−3 (9.33e−3) − | 0.0000e+0 (0.00e+0) = | 1.4957e−3 (2.76e−3) − | 5.8110e−3 (6.64e−3) − | 1.6453e−2 (1.38e−2) − | 0.0000e+0 (0.00e+0) | |
10 | 19 | 1.2121e−4 (6.64e−4) = | 0.0000e+0 (0.00e+0) = | 0.0000e+0 (0.00e+0) = | 2.4287e−4 (9.24e−4) = | 1.9394e−3 (2.82e−3) − | 0.0000e+0 (0.00e+0) | |
DTLZ5 | 4 | 13 | 2.2583e−1 (3.47e−2) − | 3.4251e−1 (1.13e−1) − | 5.5667e−1 (1.20e−1) − | 5.8905e−1 (8.79e−2) − | 2.6556e−1 (2.88e−2) − | 1.9111e−1 (3.38e−2) |
6 | 15 | 1.3359e−1 (4.08e−2) − | 3.8518e−1 (9.84e−2) − | 3.6869e−1 (8.50e−2) − | 3.1238e−1 (1.00e−1) − | 2.7778e−1 (3.54e−2) − | 1.9444e−2 (1.03e−2) | |
8 | 17 | 1.6346e−1 (5.65e−2) − | 4.4867e−1 (9.84e−2) − | 3.2650e−1 (9.61e−2) − | 4.9464e−1 (1.52e−1) − | 4.1303e−1 (5.79e−2) − | 1.3889e−2 (8.76e−3) | |
10 | 19 | 2.0279e−1 (7.03e−2) − | 5.1906e−1 (1.03e−1) − | 3.7382e−1 (7.12e−2) − | 5.6948e−1 (1.39e−1) − | 5.1479e−1 (8.46e−2) − | 4.7273e−3 (2.17e−3) | |
DTLZ6 | 4 | 13 | 1.6361e−1 (8.29e−2) − | 2.2552e−1 (2.69e−1) − | 3.8056e−1 (2.08e−1) − | 4.6218e−1 (1.91e−1) − | 3.8667e−1 (1.67e−1) − | 3.5278e−2 (2.49e−2) |
6 | 15 | 3.1061e−1 (1.40e−1) − | 4.6105e−1 (2.73e−1) − | 7.2020e−1 (2.42e−1) − | 4.0089e−1 (1.45e−1) − | 8.9672e−1 (2.11e−2) − | 1.3889e−2 (8.92e−3) | |
8 | 17 | 7.7885e−1 (2.31e−1) − | 5.1588e−1 (2.58e−1) − | 6.2329e−1 (1.55e−1) − | 6.0568e−1 (1.26e−1) − | 8.7073e−1 (1.50e−2) − | 6.6239e−3 (1.17e−3) | |
10 | 19 | 7.8085e−1 (1.28e−1) − | 5.4339e−1 (2.16e−1) − | 5.6812e−1 (1.27e−1) − | 6.8867e−1 (1.44e−1) − | 9.1576e−1 (1.62e−2) − | 3.6364e−3 (1.32e−18) | |
DTLZ7 | 4 | 23 | 7.3917e−1 (3.14e−1) − | 1.0312e−1 (6.12e−2) + | 9.5750e−1 (9.92e−2) − | 9.3947e−1 (1.41e−1) − | 1.0000e+0 (0.00e+0) − | 1.4083e−1 (5.86e−2) |
6 | 25 | 6.5227e−1 (2.72e−1) − | 4.6063e−3 (4.78e−3) + | 9.2626e−1 (1.71e−1) − | 8.5340e−1 (3.10e−1) − | 1.0000e+0 (0.00e+0) − | 1.2500e−1 (7.00e−2) | |
8 | 27 | 6.3718e−1 (3.02e−1) − | 4.2735e−4 (1.63e−3) + | 8.7842e−1 (2.78e−1) − | 8.3217e−1 (2.87e−1) − | 1.0000e+0 (0.00e+0) − | 1.2158e−1 (4.92e−2) | |
10 | 29 | 8.3709e−1 (3.71e−1) − | 0.0000e+0 (0.00e+0) + | 8.7455e−1 (2.86e−1) − | 8.3392e−1 (2.92e−1) − | 1.0000e+0 (0.00e+0) − | 3.7188e−1 (1.74e−1) | |
± / = | 4/20/4 | 6/16/6 | 3/22/3 | 0/27/1 | 4/23/1 |
Problem | M | D | CSEA | hpaEA | MaPSO−MC | KnMAPIO |
---|---|---|---|---|---|---|
DTLZ1 | 4 | 8 | 2.0738e−1 (8.19e−2) = | 5.6677e−1 (9.87e−2) = | 1.9483e−1 (4.16e−2) + | 3.5385e−1 (1.55e−1) |
6 | 10 | 1.2194e−1 (9.48e−2) + | 2.6597e−1 (5.13e−2) + | 2.2651e−1 (8.42e−2) + | 4.3182e−1 (1.13e−1) | |
8 | 12 | 1.3484e−1 (9.03e−2) + | 3.4003e−1 (4.40e−2) = | 6.0640e−1 (2.34e−1) = | 2.7949e−1 (7.91e−2) | |
10 | 14 | 1.2299e−1 (8.45e−2) = | 3.2379e−1 (5.84e−2) − | 7.3257e−1 (6.57e−2) − | 1.0691e−1 (1.88e−2) | |
DTLZ2 | 4 | 13 | 4.3457e−2 (4.55e−3) = | 2.3866e−1 (4.99e−2) − | 2.1864e−2 (1.14e−2) = | 3.3333e−2 (8.33e−3) |
6 | 15 | 1.4294e−2 (2.44e−3) = | 1.4643e−1 (2.05e−2) − | 0.0000e+0 (0.00e+0) = | 9.0909e−3 (8.30e−3) | |
8 | 17 | 6.9210e−3 (2.48e−3) = | 3.3982e−1 (4.71e−2) − | 2.5641e−2 (1.76e−2) = | 6.4103e−3 (6.41e−3) | |
10 | 19 | 6.2743e−3 (3.62e−3) = | 4.9282e−1 (1.03e−1) − | 6.6406e−2 (5.61e−3) − | 1.7455e−2 (1.13e−2) | |
DTLZ3 | 4 | 13 | 1.6000e−1 (1.02e−1) = | 1.1421e−1 (1.21e−1) = | 1.1682e−1 (6.00e−2) + | 2.3337e−1 (5.68e−2) |
6 | 15 | 7.7102e−2 (2.25e−2) = | 6.5152e−2 (6.78e−3) = | 2.0237e−2 (1.50e−2) = | 1.3333e−1 (7.64e−2) | |
8 | 17 | 3.0030e−2 (1.76e−2) = | 1.6785e−1 (2.52e−2) = | 3.7703e−1 (1.09e−1) − | 8.0769e−2 (6.50e−2) | |
10 | 19 | 3.6201e−2 (2.04e−2) = | 2.7883e−1 (7.72e−2) − | 4.6611e−1 (9.24e−2) − | 2.0364e−2 (1.38e−2) | |
DTLZ4 | 4 | 13 | 4.0185e−2 (6.46e−3) = | 7.6634e−2 (7.43e−2) = | 1.8333e−2 (2.24e−2) = | 1.5000e−2 (2.07e−2) |
6 | 15 | 4.5612e−3 (3.49e−3) − | 1.8417e−2 (1.95e−2) − | 0.0000e+0 (0.00e+0) = | 0.0000e+0 (0.00e+0) | |
8 | 17 | 8.9403e−4 (8.68e−4) − | 7.8836e−3 (7.69e−3) = | 0.0000e+0 (0.00e+0) = | 0.0000e+0 (0.00e+0) | |
10 | 19 | 1.2099e−4 (1.49e−4) = | 0.0000e+0 (0.00e+0) = | 0.0000e+0 (0.00e+0) = | 0.0000e+0 (0.00e+0) | |
DTLZ5 | 4 | 13 | 1.5942e−1 (3.69e−2) = | 3.0333e−1 (4.51e−2) − | 4.1731e−1 (3.99e−2) − | 1.8667e−1 (3.75e−2) |
6 | 15 | 1.4225e−1 (3.11e−2) − | 3.5606e−1 (4.25e−2) − | 4.2605e−1 (6.48e−2) − | 2.1212e−2 (6.34e−3) | |
8 | 17 | 1.3577e−1 (3.13e−2) − | 3.1923e−1 (2.85e−2) − | 2.6525e−1 (1.11e−1) − | 8.9744e−3 (5.73e−3) | |
10 | 19 | 1.3757e−1 (2.50e−2) − | 3.4473e−1 (3.81e−2) − | 3.4531e−1 (1.16e−1) − | 4.3636e−3 (1.63e−3) | |
DTLZ6 | 4 | 13 | 2.1035e−1 (2.81e−2) − | 4.3833e−1 (1.85e−1) − | 3.2251e−1 (8.11e−2) − | 4.3333e−2 (4.10e−2) |
6 | 15 | 2.8056e−1 (3.47e−2) − | 7.5748e−1 (2.16e−2) − | 5.5960e−1 (1.25e−1) − | 1.0606e−2 (4.15e−3) | |
8 | 17 | 1.5116e−1 (1.84e−2) − | 7.6537e−1 (4.40e−2) − | 5.6244e−1 (1.60e−1) − | 6.4103e−3 (0.00e+0) | |
10 | 19 | 1.3233e−1 (2.91e−2) − | 8.3200e−1 (2.92e−2) − | 7.0163e−1 (7.34e−2) − | 3.6364e−3 (0.00e+0) | |
DTLZ7 | 4 | 23 | 4.1363e−1 (6.31e−2) − | 9.8833e−1 (1.83e−2) − | 9.6988e−1 (4.70e−2) − | 1.2000e−1 (3.89e−2) |
6 | 25 | 2.6336e−1 (6.63e−2) = | 9.9394e−1 (9.88e−3) − | 9.7871e−1 (4.26e−2) − | 2.3485e−1 (1.16e−1) | |
8 | 27 | 2.4612e−1 (5.60e−2) − | 1.0000e+0 (0.00e+0) − | 9.7183e−1 (5.55e−2) − | 1.0256e−1 (2.94e−2) | |
10 | 29 | 4.4943e−1 (1.51e−1) = | 1.0000e+0 (0.00e+0) − | 1.0000e+0 (0.00e+0) − | 4.4873e−1 (8.48e−2) | |
± / = | 2/11/15 | 1/19/8 | 3/16/9 |
Performance measures
Experimental results and analysis
Approximation degree of KnMAPIO to the true Pareto front
Benchmark problems and experimental settings
Performance measures
Experimental results and analysis
Problem | M | D | GrEA | MOEAD | RVEA | NSGAIII | VaEA | KnMAPIO |
---|---|---|---|---|---|---|---|---|
WFG1 | 4 | 13 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) |
6 | 15 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
8 | 17 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
10 | 19 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
WFG2 | 4 | 13 | 6.6375e−1 (1.48e−1) + | 9.5170e−1 (8.27e−2) = | 8.6233e−1 (8.80e−2) + | 9.0833e−1 (6.43e−2) + | 9.2458e−1 (4.67e−2) + | 9.6625e−1 (1.80e−2) |
6 | 15 | 2.0833e−1 (1.08e−1) - | 3.9986e−1 (3.53e−1) - | 3.6283e−1 (1.37e−1) - | 3.4508e−1 (1.34e−1) - | 7.0492e−1 (1.12e−1) - | 1.6288e−1 (1.45e−1) | |
8 | 17 | 1.4103e−2 (1.56e−2) = | 7.0152e−2 (1.59e−1) = | 8.3573e−2 (1.06e−1) - | 7.2756e−2 (5.60e−2) - | 4.2051e−1 (1.05e−1) - | 1.6346e−2 (2.52e−2) | |
10 | 19 | 2.2982e−1 (8.50e−2) + | 7.9488e−2 (2.22e−1) + | 2.8001e−1 (1.21e−1) + | 2.9600e−1 (9.08e−2) + | 5.4618e−1 (1.34e−1) - | 3.6727e−1 (1.13e−1) | |
WFG3 | 4 | 13 | 2.0833e−1 (4.42e−2) - | 9.7711e−2 (3.76e−2) = | 2.9012e−1 (4.78e−2) - | 2.1667e−1 (3.05e−2) - | 1.7708e−1 (3.05e−2) - | 8.0833e−2 (2.21e−2) |
6 | 15 | 6.4773e−2 (1.52e−2) - | 5.6818e−3 (5.43e−3) + | 5.9463e−2 (2.44e−2) - | 5.7576e−2 (6.22e−2) - | 8.5227e−2 (3.18e−2) - | 2.0076e−2 (7.88e−3) | |
8 | 17 | 6.2500e−2 (1.50e−2) - | 0.0000e+0 (0.00e+0) + | 2.7750e−2 (1.95e−2) - | 7.7244e−2 (3.38e−2) - | 5.9615e−2 (2.37e−2) - | 1.2500e−2 (5.29e−3) | |
10 | 19 | 1.9636e−2 (4.78e−3) - | 0.0000e+0 (0.00e+0) + | 1.8235e−2 (9.73e−3) - | 6.5636e−2 (2.36e−2) - | 3.7636e−2 (1.15e−2) - | 8.0000e−3 (5.73e−3) | |
WFG4 | 4 | 13 | 6.4500e−1 (5.67e−2) = | 9.8609e−1 (2.66e−2) - | 9.4731e−1 (2.79e−2) - | 9.3625e−1 (3.26e−2) - | 9.5708e−1 (2.25e−2) - | 6.3625e−1 (6.78e−2) |
6 | 15 | 3.0720e−1 (3.87e−2) - | 4.8399e−1 (8.44e−2) - | 4.6930e−1 (6.09e−2) - | 4.7652e−1 (1.24e−1) - | 4.4659e−1 (6.51e−2) - | 2.0341e−1 (4.36e−2) | |
8 | 17 | 1.7051e−1 (3.23e−2) - | 6.2188e−1 (1.00e−1) - | 3.3488e−1 (6.29e−2) - | 2.7692e−1 (6.80e−2) - | 1.9712e−1 (2.93e−2) - | 1.2853e−1 (2.18e−2) | |
10 | 19 | 4.5636e−2 (2.06e−2) + | 7.2709e−1 (1.34e−1) - | 3.1210e−1 (7.07e−2) - | 9.1273e−2 (3.61e−2) = | 1.1309e−1 (2.20e−2) = | 1.0345e−1 (2.01e−2) | |
WFG5 | 4 | 13 | 8.4042e−1 (3.20e−2) - | 9.8041e−1 (1.90e−2) - | 9.7353e−1 (1.53e−2) - | 9.7833e−1 (1.56e−2) - | 9.7750e−1 (1.51e−2) - | 7.3167e−1 (3.18e−2) |
6 | 15 | 2.9924e−1 (5.21e−2) - | 8.0567e−1 (5.30e−2) - | 5.0910e−1 (6.64e−2) - | 4.7727e−1 (6.26e−2) - | 4.8485e−1 (4.66e−2) - | 1.3712e−1 (2.30e−2) | |
8 | 17 | 1.7821e−1 (2.95e−2) - | 9.0509e−1 (2.39e−2) - | 4.5119e−1 (8.53e−2) - | 3.3718e−1 (5.29e−2) - | 2.4519e−1 (2.40e−2) - | 7.3397e−2 (1.11e−2) | |
10 | 19 | 1.3327e−1 (1.75e−2) - | 9.1205e−1 (3.68e−2) - | 5.1249e−1 (9.85e−2) - | 2.9145e−1 (6.95e−2) - | 1.9418e−1 (2.17e−2) - | 3.9636e−2 (1.07e−2) | |
WFG6 | 4 | 13 | 9.7458e−1 (3.90e−2) + | 9.9680e−1 (7.04e−3) + | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 9.9917e−1 (3.73e−3) = | 1.0000e+0 (0.00e+0) |
6 | 15 | 5.9280e−1 (7.74e−2) = | 9.6534e−1 (4.40e−2) - | 6.8979e−1 (1.21e−1) - | 7.2538e−1 (7.50e−2) - | 7.0909e−1 (7.81e−2) - | 5.5530e−1 (3.62e−2) | |
8 | 17 | 3.5449e−1 (5.43e−2) - | 9.6251e−1 (3.44e−2) - | 6.5698e−1 (1.19e−1) - | 5.5994e−1 (5.22e−2) - | 3.7115e−1 (5.57e−2) - | 2.4231e−1 (3.04e−2) | |
10 | 19 | 2.2491e−1 (3.50e−2) - | 9.7176e−1 (2.97e−2) - | 6.0188e−1 (1.02e−1) - | 4.3164e−1 (8.20e−2) - | 3.2073e−1 (3.76e−2) - | 1.1436e−1 (1.52e−2) | |
WFG7 | 4 | 13 | 3.4333e−1 (3.40e−2) + | 9.5430e−1 (5.36e−2) - | 9.3964e−1 (3.31e−2) - | 9.5125e−1 (3.26e−2) - | 9.6292e−1 (2.03e−2) - | 3.9292e−1 (7.26e−2) |
6 | 15 | 2.2462e−1 (4.44e−2) - | 5.2022e−1 (1.24e−1) - | 4.5206e−1 (7.56e−2) - | 5.8030e−1 (6.29e−2) - | 4.9697e−1 (8.59e−2) - | 1.6856e−1 (2.84e−2) | |
8 | 17 | 1.9359e−1 (3.13e−2) - | 5.9153e−1 (1.35e−1) - | 3.7343e−1 (9.09e−2) - | 3.4135e−1 (1.02e−1) - | 2.7821e−1 (5.89e−2) - | 1.2436e−1 (2.37e−2) | |
10 | 19 | 4.4000e−2 (7.82e−3) + | 7.0603e−1 (1.21e−1) - | 3.9081e−1 (7.28e−2) - | 1.7545e−1 (6.85e−2) = | 1.8218e−1 (2.98e−2) - | 1.5636e−1 (3.83e−2) | |
WFG8 | 4 | 13 | 9.6208e−1 (1.68e−2) = | 9.9957e−1 (1.93e−3) - | 9.9640e−1 (6.79e−3) - | 9.9500e−1 (7.84e−3) - | 9.9625e−1 (5.04e−3) - | 9.7042e−1 (1.52e−2) |
6 | 15 | 8.3068e−1 (3.65e−2) - | 9.9955e−1 (2.01e−3) - | 9.3790e−1 (3.45e−2) - | 8.6250e−1 (7.71e−2) - | 8.6894e−1 (3.36e−2) - | 6.2652e−1 (5.08e−2) | |
8 | 17 | 7.4712e−1 (3.23e−2) - | 1.0000e+0 (0.00e+0) - | 8.8322e−1 (1.10e−1) - | 5.5481e−1 (2.11e−1) - | 7.0737e−1 (4.17e−2) - | 3.6218e−1 (4.56e−2) | |
10 | 19 | 6.5455e−2 (4.27e−2) + | 1.0000e+0 (0.00e+0) - | 7.6137e−1 (6.90e−2) - | 4.8055e−1 (1.02e−1) - | 6.3436e−1 (3.65e−2) - | 3.9364e−1 (4.09e−2) | |
WFG9 | 4 | 13 | 6.4167e−1 (6.78e−2) = | 9.9246e−1 (1.50e−2) - | 9.8242e−1 (1.87e−2) - | 9.8917e−1 (1.51e−2) - | 9.7250e−1 (2.89e−2) - | 7.4917e−1 (1.86e−1) |
6 | 15 | 3.3068e−1 (1.03e−1) - | 9.7253e−1 (4.65e−2) - | 7.0578e−1 (1.22e−1) - | 6.7727e−1 (1.27e−1) - | 7.3712e−1 (1.11e−1) - | 2.3295e−1 (1.53e−1) | |
8 | 17 | 2.0801e−1 (6.03e−2) - | 9.7050e−1 (4.67e−2) - | 5.0030e−1 (1.40e−1) - | 4.3590e−1 (1.63e−1) - | 3.7340e−1 (1.05e−1) - | 8.7500e−2 (3.12e−2) | |
10 | 19 | 5.3091e−2 (2.20e−2) - | 9.8128e−1 (3.52e−2) - | 3.6511e−1 (1.03e−1) - | 2.3382e−1 (1.00e−1) - | 2.1491e−1 (3.55e−2) - | 3.1818e−2 (1.33e−2) | |
± / = | 7/20/9 | 5/24/7 | 2/29/5 | 2/27/7 | 1/29/6 |
Problem | M | D | CSEA | hpaEA | MaPSO-MC | KnMAPIO |
---|---|---|---|---|---|---|
WFG1 | 4 | 13 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) |
6 | 15 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
8 | 17 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
10 | 19 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
WFG2 | 4 | 13 | 7.9324e−1 (9.24e−2) + | 9.3517e−1 (4.13e−2) = | 5.9935e−1 (2.09e−1) + | 9.6141e−1 (3.50e−2) |
6 | 15 | 3.8777e−1 (1.90e−1) - | 4.9106e−1 (1.18e−1) - | 8.8876e−2 (4.53e−2) = | 1.0303e−1 (1.17e−1) | |
8 | 17 | 1.1402e−1 (4.52e−2) - | 1.6848e−1 (6.10e−2) - | 4.0831e−3 (6.14e−3) = | 1.1538e−2 (1.60e−2) | |
10 | 19 | 1.6548e−1 (1.03e−1) + | 4.4655e−1 (5.25e−2) = | 4.3490e−2 (2.57e−2) + | 3.9055e−1 (1.30e−1) | |
WFG3 | 4 | 13 | 8.7882e−2 (3.44e−2) = | 2.4833e−1 (2.66e−2) - | 4.1569e−1 (8.80e−2) - | 7.8333e−2 (1.73e−2) |
6 | 15 | 1.3177e−1 (7.58e−2) - | 1.3030e−1 (9.88e−3) - | 2.9846e−1 (5.95e−2) - | 1.8182e−2 (6.78e−3) | |
8 | 17 | 7.3760e−2 (2.48e−2) - | 6.9231e−2 (3.64e−2) - | 3.4138e−1 (1.17e−1) - | 8.9744e−3 (7.31e−3) | |
10 | 19 | 5.8935e−2 (3.18e−2) - | 5.0909e−2 (1.23e−2) - | 7.0664e−2 (3.82e−2) - | 1.3091e−2 (7.54e−3) | |
WFG4 | 4 | 13 | 7.9583e−1 (2.39e−2) - | 9.7651e−1 (1.16e−2) - | 1.6537e−1 (5.07e−2) + | 6.6000e−1 (6.70e−2) |
6 | 15 | 3.4200e−1 (6.39e−2) - | 3.4739e−1 (7.20e−2) - | 1.4041e−1 (4.03e−2) + | 2.3030e−1 (4.72e−2) | |
8 | 17 | 8.1938e−2 (1.02e−2) + | 9.1026e−2 (4.09e−2) + | 7.7381e−2 (2.13e−2) + | 1.3974e−1 (1.39e−2) | |
10 | 19 | 3.8910e−2 (9.01e−3) + | 2.0364e−2 (1.35e−2) + | 6.8109e−2 (2.21e−2) + | 1.2218e−1 (2.43e−2) | |
WFG5 | 4 | 13 | 8.0637e−1 (1.64e−2) - | 9.8632e−1 (5.51e−3) - | 7.9901e−1 (5.26e−2) - | 7.1667e−1 (5.03e−2) |
6 | 15 | 4.4579e−1 (3.09e−2) - | 5.1614e−1 (7.01e−2) - | 2.8720e−1 (3.12e−2) - | 1.5455e−1 (2.05e−2) | |
8 | 17 | 2.7664e−1 (2.88e−2) - | 2.5960e−1 (5.32e−2) - | 1.3378e−1 (2.73e−2) - | 6.6667e−2 (9.72e−3) | |
10 | 19 | 1.6444e−1 (3.20e−2) - | 1.1055e−1 (1.51e−2) - | 8.8968e−2 (2.37e−2) - | 3.7818e−2 (7.54e−3) | |
WFG6 | 4 | 13 | 9.9621e−1 (7.27e−3) = | 1.0000e+0 (0.00e+0) = | 9.8457e−1 (2.15e−2) = | 9.9667e−1 (7.45e−3) |
6 | 15 | 6.3115e−1 (7.14e−2) = | 8.2101e−1 (4.09e−2) - | 3.2693e−1 (6.52e−2) + | 5.7121e−1 (2.76e−2) | |
8 | 17 | 2.6822e−1 (3.38e−2) = | 4.3341e−1 (9.76e−2) - | 2.9338e−1 (6.54e−2) = | 2.7308e−1 (4.29e−2) | |
10 | 19 | 1.7479e−1 (4.73e−2) - | 2.2764e−1 (5.20e−2) - | 2.1954e−1 (4.41e−2) - | 1.0764e−1 (1.83e−2) | |
WFG7 | 4 | 13 | 7.0001e−1 (2.78e−2) - | 9.4556e−1 (3.67e−2) - | 1.1432e−1 (2.58e−2) + | 3.7000e−1 (3.61e−2) |
6 | 15 | 3.0822e−1 (2.74e−2) - | 5.0667e−1 (3.09e−2) - | 9.6447e−2 (2.91e−2) + | 1.6818e−1 (3.00e−2) | |
8 | 17 | 9.0444e−2 (1.62e−2) = | 2.1282e−1 (4.91e−2) - | 1.0548e−1 (2.95e−2) = | 1.2564e−1 (3.41e−2) | |
10 | 19 | 3.5486e−2 (6.21e−3) + | 7.2000e−2 (2.47e−2) + | 5.4875e−2 (2.00e−2) + | 1.4691e−1 (1.11e−2) | |
WFG8 | 4 | 13 | 9.8938e−1 (4.95e−3) = | 9.9708e−1 (6.53e−3) = | 8.8770e−1 (2.27e−2) + | 9.9333e−1 (6.97e−3) |
6 | 15 | 8.1679e−1 (1.41e−1) - | 8.1250e−1 (3.22e−2) - | 5.5898e−1 (2.22e−2) = | 5.4091e−1 (5.07e−2) | |
8 | 17 | 4.0045e−1 (8.99e−2) = | 5.6923e−1 (7.26e−2) - | 3.9748e−1 (6.40e−2) = | 4.1795e−1 (3.75e−2) | |
10 | 19 | 1.4196e−1 (5.10e−2) + | 3.7745e−1 (6.53e−2) = | 2.8114e−1 (4.52e−2) = | 3.3964e−1 (3.87e−2) | |
WFG9 | 4 | 13 | 1.0000e+0 (0.00e+0) - | 9.3195e−1 (5.65e−2) = | 5.9301e−1 (8.81e−2) = | 6.4667e−1 (2.14e−1) |
6 | 15 | 6.3146e−1 (1.88e−1) - | 6.7048e−1 (6.09e−2) - | 2.3974e−1 (2.80e−2) = | 1.9242e−1 (5.45e−2) | |
8 | 17 | 4.5997e−1 (5.55e−2) - | 3.5734e−1 (8.59e−2) - | 2.3556e−1 (1.66e−1) - | 6.2821e−2 (2.10e−2) | |
10 | 19 | 2.0897e−1 (1.41e−1) - | 1.2145e−1 (2.94e−2) - | 1.0091e−1 (2.41e−2) - | 2.7636e−2 (7.97e−3) | |
± / = | 6/19/11 | 3/23/10 | 11/11/14 |
Problem | M | MAPIO | MAPIOC | MAPIOG | MAPIOLC | MAPIOGC | KnMAPIO |
---|---|---|---|---|---|---|---|
DTLZ1 | 4 | 5.4567e−1 (1.50e−1) = | 4.5275e−1 (2.63e−1) = | 5.8506e−1 (1.92e−1) = | 3.9092e−1 (1.61e−1) = | 5.7206e−1 (1.74e−1) = | 4.5156e−1 (1.06e−1) |
6 | 5.6970e−1 (6.63e−2) = | 5.6970e−1 (5.75e−2) = | 6.6818e−1 (1.10e−1) = | 7.0455e−1 (1.54e−1) = | 5.9394e−1 (1.45e−1) = | 6.3030e−1 (1.78e−1) | |
8 | 6.0333e−1 (9.14e−2) = | 6.3333e−1 (1.20e−1) = | 5.9103e−1 (2.00e−1) = | 4.8718e−1 (1.88e−1) = | 5.1410e−1 (1.40e−1) = | 5.2607e−1 (6.96e−2) | |
10 | 6.2109e−1 (4.86e−2) - | 4.5309e−1 (1.17e−1) = | 4.5382e−1 (5.85e−2) = | 4.6327e−1 (6.92e−2) = | 4.6182e−1 (1.04e−1) = | 3.7382e−1 (8.74e−2) | |
DTLZ2 | 4 | 9.5000e−2 (3.21e−2) - | 3.1667e−2 (1.09e−2) = | 3.6667e−2 (2.25e−2) = | 4.5000e−2 (2.54e−2) = | 3.3333e−2 (8.33e−3) = | 3.1667e−2 (2.07e−2) |
6 | 4.6970e−2 (2.10e−2) - | 1.9697e−2 (8.64e−3) = | 1.0606e−2 (4.15e−3) = | 7.5758e−3 (7.58e−3) = | 3.0303e−3 (4.15e−3) = | 1.0606e−2 (8.64e−3) | |
8 | 3.3333e−2 (9.51e−3) = | 2.3077e−2 (1.85e−2) = | 2.5641e−2 (1.63e−2) = | 1.9231e−2 (9.07e−3) = | 2.0513e−2 (1.05e−2) = | 1.9231e−2 (1.20e−2) | |
10 | 6.9818e−2 (2.11e−2) = | 8.1455e−2 (1.58e−2) - | 7.6364e−2 (2.37e−2) - | 6.2545e−2 (2.46e−2) = | 5.0182e−2 (2.20e−2) = | 4.6545e−2 (1.90e−2) | |
DTLZ3 | 4 | 2.0199e−1 (6.44e−2) = | 8.3043e−2 (6.64e−2) = | 1.1681e−1 (8.69e−2) = | 1.2603e−1 (7.89e−2) = | 7.2601e−2 (4.89e−2) = | 1.2543e−1 (1.18e−1) |
6 | 4.0222e−1 (9.77e−2) = | 3.8485e−1 (1.14e−1) = | 3.4848e−1 (1.31e−1) = | 4.2121e−1 (1.25e−1) = | 2.7917e−1 (1.12e−1) = | 3.1667e−1 (8.66e−2) | |
8 | 3.6597e−1 (9.46e−2) = | 4.0000e−1 (1.18e−1) = | 3.8462e−1 (8.64e−2) = | 3.9159e−1 (1.86e−1) = | 3.9872e−1 (1.31e−1) = | 4.0000e−1 (1.10e−1) | |
10 | 4.2691e−1 (8.06e−2) - | 4.2182e−1 (1.00e−1) - | 3.8182e−1 (8.88e−2) - | 3.5636e−1 (7.56e−2) - | 4.0727e−1 (7.47e−2) - | 1.9636e−1 (9.58e−2) | |
DTLZ4 | 4 | 7.1667e−2 (2.80e−2) - | 4.0000e−2 (3.25e−2) = | 1.1667e−2 (2.61e−2) = | 3.3333e−2 (1.02e−2) = | 2.5000e−2 (3.73e−2) = | 2.8333e−2 (1.73e−2) |
6 | 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) = | 0.0000e+0 (0.00e+0) | |
8 | 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) = | 0.0000e+0 (0.00e+0) | |
10 | 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) = | 0.0000e+0 (0.00e+0) | |
DTLZ5 | 4 | 2.6667e−1 (8.25e−2) = | 3.1833e−1 (1.16e−1) = | 3.6167e−1 (3.36e−2) = | 3.7833e−1 (4.66e−2) = | 3.0167e−1 (6.91e−2) = | 3.4000e−1 (1.13e−1) |
6 | 6.9697e−2 (3.14e−2) = | 1.3333e−1 (3.92e−2) - | 1.7424e−1 (6.15e−2) - | 1.1364e−1 (7.74e−2) = | 1.2273e−1 (8.81e−2) = | 7.4242e−2 (4.09e−2) | |
8 | 1.9231e−2 (4.53e−3) = | 2.1795e−2 (1.16e−2) = | 1.4103e−2 (8.36e−3) = | 4.2308e−2 (4.36e−2) = | 3.0769e−2 (1.05e−2) = | 2.6923e−2 (2.19e−2) | |
10 | 1.0182e−2 (3.98e−3) = | 4.3636e−3 (1.63e−3) + | 1.6727e−2 (7.09e−3) = | 1.2364e−2 (4.15e−3) = | 1.3818e−2 (5.39e−3) = | 1.5273e−2 (1.22e−2) | |
DTLZ6 | 4 | 2.7333e−1 (1.49e−2) - | 2.1167e−1 (1.54e−1) = | 2.3667e−1 (1.37e−1) - | 3.8167e−1 (3.54e−1) = | 2.5333e−1 (2.30e−1) = | 6.1667e−2 (4.74e−2) |
6 | 3.7424e−1 (1.85e−1) - | 3.6364e−2 (3.32e−2) = | 1.1970e−1 (1.34e−1) = | 1.6970e−1 (1.50e−1) = | 4.6970e−2 (3.45e−2) = | 9.8485e−2 (7.93e−2) | |
8 | 5.4744e−1 (1.66e−1) - | 2.0000e−1 (1.13e−1) - | 1.8718e−1 (1.35e−1) - | 4.0128e−1 (1.07e−1) - | 3.6282e−1 (2.66e−1) - | 3.2051e−2 (4.44e−2) | |
10 | 7.0255e−1 (9.28e−2) - | 8.1091e−1 (6.69e−2) - | 7.8109e−1 (2.96e−2) - | 7.2655e−1 (8.72e−2) - | 8.1891e−1 (7.34e−2) - | 2.3564e−1 (7.25e−2) | |
DTLZ7 | 4 | 9.8500e−1 (3.35e−2) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 9.9667e−1 (7.45e−3) |
6 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
8 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
10 | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) = | 1.0000e+0 (0.00e+0) | |
± / = | 0/9/19 | 1/5/22 | 0/6/22 | 0/3/25 | 0/3/25 |