Complexity

Research Article

A Simplified Hypervolume-Based Evolutionary Algorithm for Many-Objective Optimization

Table 1

The mean and standard deviation values of IGD obtained by SHEA, NSGAIII, MOEA/DD, KnEA, and RVEA while “+,” “=,” “−” mean SHEA is better than, the same as, and worse than the compared algorithms.
ProblemSHEANSGAIIIMOEA/DDKnEARVEA
MaF1-51.1614e − 1 (1.32e − 3)2.0830e − 1 (1.00e − 2) +3.0206e − 1 (1.07e − 2) +1.3136e − 1 (1.91e − 3) =3.2908e − 1 (6.21e − 2) +
MaF2-59.6828e − 2 (2.82e − 3)1.3012e − 1 (2.56e − 3) =1.3665e − 1 (3.72e − 3) =1.3734e − 1 (3.62e − 3) =1.2736e − 1 (1.42e − 3) =
MaF3-56.5417e − 2 (2.81e − 3)9.7048e − 2 (1.52e − 3) =1.1693e − 1 (1.75e − 3) +1.6971e − 1 (9.55e − 2) +8.0906e − 2 (6.70e − 3) =
MaF4-51.8707e + 0 (4.09e − 2)3.2220e + 0 (5.56e − 1) +7.7080e + 0 (2.14e − 1) +2.9005e + 0 (2.52 − 1) +4.8145e + 0 (1.30e − 0) +
MaF5-51.8660e + 0 (3.63e − 2)2.5845e + 0 (1.15e + 0) +6.1823e + 0 (1.00e + 0) +2.6408e + 0 (8.04e − 2) +2.3218e + 0 (3.07e − 1) +
MaF6-53.9514e - 3 (4.97e - 4)4.9054e - 2 (9.12e − 3) +7.6075e − 2 (4.42e − 3) +8.0721e − 3 (2.28e − 3) =9.6534e − 2 (3.42e − 2) +
MaF7-52.7153e − 1 (9.43e − 3)3.4413e − 1 (1.20e − 2) +1.7891e + 0 (8.07e − 1) +3.2757e − 1 (8.09e − 3) +4.4844e − 1 (1.10e − 3) +
MaF8-58.9494e − 2 (2.04e − 3)2.1547e − 1 (2.19e − 2) +3.3063e − 1 (3.61e − 2) +2.9874e − 1 (8.77e − 2) +4.8799e − 1 (8.73e − 2) +
MaF9-59.3455e − 2 (3.24e − 3)6.5706e − 1 (1.32e − 1) +2.5294e − 1 (1.34e − 2) +5.6348e − 1 (1.82e − 1) +3.6742e − 1 (7.00e − 2) +
MaF10-55.9695e − 1 (6.92e − 2)4.2888e − 1 (3.81e − 3)  − 5.4591e − 1 (3.64e − 2)  − 5.1174e − 1 (7.85e − 3)  − 4.3054e − 1 (4.90e − 3)  − 
MaF11-59.5446e − 1 (1.30e − 1)4.6337e − 1 (1.69e − 3)  − 5.7805e − 1 (9.69e − 3)  − 5.7122e − 1 (2.10e − 2)  − 4.4446e − 1 (8.16e − 3) -
MaF12-51.0185e + 0 (1.27e − 2)1.1183e + 0 (4.00e − 3) +1.2858e + 0 (1.43e − 2) +1.2609e + 0 (1.70e − 2) +1.1224e + 0 (2.51e − 3) +
MaF13-51.0079e − 1 (3.64e − 3)2.9677e − 1 (5.22e − 2) +2.4087e − 1 (2.54e − 2) +2.2221e − 1 (2.01e − 2) +6.6957e − 1 (1.36e − 1) +
MaF14-55.7391e − 1 (1.04e − 1)7.9085e − 1 (3.74e − 1) +3.8193e − 1 (8.93e − 2)  − 7.6615e − 1 (2.77e − 1) +7.1001e − 1 (2.02e − 1) +
MaF15-58.3177e − 1 (3.31e − 2)1.0511e + 0 (4.49e − 2) +5.9102e − 1 (4.35e − 2)  − 3.4169e + 0 (1.84e + 0) +6.1117e − 1 (4.50e − 2) −
MaF1-103.0459e − 1 (1.30e − 2)3.1563e − 1 (7.02e − 3) =4.8879e − 1 (4.58e − 2) +2.4052e − 1 (2.33e − 3)  − 6.6423e − 1 (8.71e − 2) +
MaF2-101.9990e − 1 (4.87e − 3)2.3692e − 1 (2.48e − 2) +2.9196e − 1 (7.15e − 2) +1.6521e − 1 (7.85e − 3) =4.8292e − 1 (1.82e − 1) +
MaF3-108.3915e − 2 (2.62e − 3)9.3337e − 1 (3.12e + 0) +1.1204e − 1 (1.03e − 3) =1.1840e + 9 (5.06e + 9) +9.8006e − 2 (5.33e − 3) =
MaF4-109.3360e + 1 (3.79e + 1)1.2678e + 2 (6.34e + 0) +4.3124e + 2 (1.96e + 1) +7.1006e + 1 (6.77e + 0) −2.3329e + 2 (5.10e + 1) +
MaF5-106.0339e + 1 (6.65e + 0)1.1938e + 2 (2.75e − 1) +2.9174e + 2 (7.70e + 0) +8.1774e + 1 (5.26e + 0) +1.0827e + 2 (1.73e + 1) +
MaF6-103.9073e − 3 (3.35e − 4)2.1879e − 1 (4.79e − 2) +1.2060e − 1 (8.53e − 3) +8.4648e + 0 (7.78e + 0) +3.3999e − 1 (2.44e − 1) +
MaF7-109.6508e − 1 (1.66e − 2)1.1620e + 0 (7.18e − 2) +2.7026e + 0 (3.72e − 1) +9.4855e − 1 (1.01e − 2) =2.4904e + 0 (3.56e − 1) +
MaF8-101.3864e − 1 (3.50e − 3)4.6344e − 1 (6.16e − 2) +9.1389e − 1 (2.39e − 2) +2.6509e − 1 (3.86e − 2) +1.0244e + 0 (2.14e − 1) +
MaF9-108.9161e − 1 (1.92e − 1)9.5771e − 1 (3.22e − 1) +5.9743e − 1 (2.33e − 3) −8.5718e + 1 (6.92e + 1) +1.1161e + 0 (2.69e − 1) +
MaF10-101.2266e + 0 (7.28e − 2)1.0965e + 0 (4.85e − 2) −1.2608e + 0 (2.88e − 2) =1.2117e + 0 (5.05e − 2) =1.1948e + 0 (1.04e − 1) =
MaF11-104.2556e + 0 (6.29e − 1)1.3422e + 0 (1.50e − 1) -1.4289e + 0 (1.06e − 2)  − 1.3490e + 0 (4.51e − 2)  − 1.3839e + 0 (4.88e − 2) −
MaF12-104.3224e + 0 (5.40e − 2)5.1072e + 0 (1.42e − 1) +6.0813e + 0 (2.79e − 1) +5.2874e + 0 (6.12e − 2) +4.8913e + 0 (4.87e − 2) +
MaF13-101.6901e − 1 (1.21e − 2)4.1263e − 1 (6.56e − 2) +4.4820e − 1 (3.29e − 2) +2.1228e − 1 (2.37e − 2) +9.3878e − 1 (2.89e − 1) +
MaF14-108.7548e − 1 (1.52e − 1)1.4523e + 0 (5.34e − 1) +5.3428e − 1 (6.22e − 2)  − 1.7663e + 2 (2.00e + 2) +6.6978e − 1 (5.91e − 2)  − 
MaF15-102.0029e + 0 (1.32e − 1)3.0468e + 0 (4.38e + 0) +1.0133e + 0 (6.38e − 2)  − 1.9364e + 1 (8.54e + 0) +1.0594e + 0 (4.47e − 2)  − 
MaF1-154.1181e − 1 (1.89e − 2)3.3510e − 1 (7.36e − 3)  − 6.3920e − 1 (4.13e − 2) +3.3929e − 1 (3.83e − 2)  − 7.3558e − 1 (5.35e − 2) +
MaF2-152.2440e − 1 (1.08e − 2)2.4645e − 1 (2.39e − 2) =3.1534e − 1 (3.10e − 2) +1.9241e − 1 (5.26e − 3) =7.8173e − 1 (8.16e − 2) +
MaF3-151.0297e − 1 (2.22e − 2)1.7607e + 0 (4.79e + 0) +1.1719e − 1 (1.28e − 3) =2.2295e + 9 (4.30e + 9) +9.6650e − 2 (7.32e − 3) =
MaF4-151.8657e + 4 (6.70e + 4)4.5055e + 3 (4.36e + 2) −1.5432e + 4 (2.45e + 3) −1.7224e + 3 (2.01e + 2) −7.7257e + 3 (1.96e + 3) −
MaF5-152.2259e + 3 (4.00e + 2)3.1331e + 3 (3.77e + 1) +7.3038e + 3 (6.71e + 1) +2.0495e + 3 (8.e + 1)  − 3.2985e + 3 (2.82e + 2) +
MaF6-154.5840e − 3 (7.29e − 4)3.7141e − 1 (1.41e − 1) +1.6153e − 1 (3.43e − 3) +4.8429e + 1 (9.21e + 0) +1.9771e − 1 (1.17e − 1) +
MaF7-151.7351e + 0 (4.48e − 2)7.7037e + 0 (9.37e − 1) +3.3764e + 0 (7.80e − 2) +2.4545e + 0 (2.36e − 1) +4.3845e + 0 (1.58e + 0) +
MaF8-151.6721e − 1 (2.97e − 3)4.0978e − 1 (4.32e − 2) +1.5460e + 0 (2.82e − 2) +1.9336e − 1 (8.95e − 3)=1.1960e + 0 (1.92e − 1) +
MaF9-152.0922e − 1 (1.44e − 2)2.2160e + 0 (4.13e + 0) +1.3164e + 0 (2.41e + 0) +5.2669e − 1 (4.32e − 1) +1.6125e + 0 (3.85e − 1) +
MaF10-151.7186e + 0 (6.28e − 2)1.6348e + 0 (8.58e − 2)  − 1.9986e + 0 (4.06e − 2) +1.6232e + 0 (5.10e − 2)  − 1.7417e + 0 (9.08e − 2) +
MaF11-158.8230e + 0 (1.22e + 0)1.8586e + 0 (8.91e − 2) −2.1955e + 0 (6.49e − 3) −1.7816e + 0 (6.20e − 2) −1.9469e + 0 (8.30e − 2) −
MaF12-158.4176e + 0 (1.66e − 1)8.8410e + 0 (9.87e − 2) +1.1457e + 1 (3.64e − 1) +7.3382e + 0 (1.38e − 1)  − 9.1349e + 0 (6.05e − 2) +
MaF13-151.8693e − 1 (1.01e − 2)3.8155e − 1 (1.01e − 1) +6.1376e − 1 (1.12e − 1) +1.5808e − 1 (1.46e − 2) =1.3296e + 0 (4.18e − 1) +
MaF14-152.1776e + 0 (9.69e − 1)1.4120e + 0 (7.14e − 1) −4.4791e − 1 (7.87e − 2)  − 4.2956e + 1 (4.09e + 1) +7.8507e − 1 (2.14e − 1) −
MaF15-154.3682e + 0 (9.00e − 1)7.6508e + 0 (1.22e + 1) +1.1636e + 0 (4.10e − 2) −1.4233e + 2 (2.31e + 1) +1.1431e + 0 (3.90e − 2)  
+/−/=—32/9/429/12/425/11/930/10/5
The bold values indicate best performance.