Computational Intelligence and Neuroscience

Research Article

An Opposition-Based Evolutionary Algorithm for Many-Objective Optimization with Adaptive Clustering Mechanism

Table 8

The statistical results (mean and standard deviation) of the IGD values obtained by RVEA, NSGA-III, SPEAR, MOEA/DD, Two_Arch2, MOEA/DVA, and OBEA on WFG test suits. The best results are italicized.
ProblemMRVEANSGA-IIISPEARMOEA/DDTwo_Arch2MOEA/DVAOBEA
WFG131.353e + 0 (5.32e − 2) −1.351e + 0 (3.94e − 2) −1.943e + 0 (3.57e − 2) −1.505e + 0 (5.41e − 2) −1.158e + 0 (4.94e − 2) −1.358e + 0 (2.63e − 2) −5.083e − 1 (7.94e − 2)
51.924e + 0 (3.36e − 2) −1.992e + 0 (3.41e − 2) −2.387e + 0 (4.86e − 2) −2.201e + 0 (1.03e − 1) −1.627e + 0 (7.58e − 2) −1.733e + 0 (3.00e − 2) −9.814e − 1 (1.56e − 1)
82.564e + 0 (5.35e − 2) −2.686e + 0 (4.65e − 2) =3.072e + 0 (3.53e − 1) −2.790e + 0 (3.35e − 2) −2.436e + 0 (5.48e − 2) −2.780e + 0 (6.81e − 2) −1.959e + 0 (3.48e − 1)
102.924e + 0 (5.26e − 2) −3.154e + 0 (5.28e − 2) −3.327e + 0 (8.51e − 2) −3.108e + 0 (3.10e − 2) =2.799e + 0 (3.64e − 2) −3.252e + 0 (5.35e − 2) −2.088e + 0 (4.77e − 2)
153.518e + 0 (2.28e − 1) =4.022e + 0 (1.91e − 1) −3.997e + 0 (2.70e − 1) =4.272e + 0 (7.06e − 2) =3.790e + 0 (3.90e − 2) −4.230e + 0 (7.38e − 2) −3.115e + 0 (2.48e − 1)
WFG233.055e − 1 (2.62e − 2) −2.262e − 1 (1.78e − 2) −7.566e − 1 (2.46e − 1) −4.168e − 1 (1.08e − 1) −1.210e − 1 (5.90e − 3) +4.002e − 1 (2.45e − 2) −2.747e − 1 (6.59e − 2)
51.190e + 0 (4.01e − 1) =6.904e − 1 (6.05e − 2) −4.437e + 0 (1.64e + 0) −2.753e + 0 (4.53e − 1) −6.203e − 1 (1.06e − 1) +1.372e + 0 (1.77e − 1) =1.502e + 0 (5.72e − 1)
84.110e + 0 (9.56e − 1) −1.084e + 0 (1.17e − 1) +5.338e + 0 (1.98e + 0) −4.883e + 0 (8.37e − 1) −1.331e + 0 (4.07e − 1) =4.234e + 0 (3.15e − 1) −3.321e + 0 (8.42e − 1)
106.072e + 0 (2.26e + 0) =1.379e + 0 (1.47e − 1) +5.298e + 0 (1.63e + 0) =8.855e + 0 (1.82e + 0) −2.141e + 0 (9.47e − 2) +8.325e + 0 (3.81e + 0) −4.661e + 0 (1.41e + 0)
151.213e + 1 (3.43e + 0) =4.156e + 0 (1.46e + 0) −2.602e + 0 (6.92e − 1) +1.776e + 1 (2.05e + 0) =2.201e + 0 (7.24e − 1) +2.078e + 1 (4.53e + 0) −1.632e + 1 (3.19e + 0)
WFG333.070e − 1 (2.98e − 2) −2.471e − 1 (3.06e − 3) −4.032e − 1 (5.37e − 2) −3.944e − 1 (9.74e − 2) −1.153e − 1 (1.33e − 2) =8.484e − 2 (7.19e − 3) +1.730e − 1 (1.72e − 2)
57.938e − 1 (1.39e − 1) −1.005e + 0 (7.46e − 3) −8.296e − 1 (1.26e − 1) −7.444e − 1 (5.32e − 2) −3.525e − 1 (1.92e − 2) =1.225e + 0 (1.48e − 1) −5.355e − 1 (7.74e − 2)
81.895e + 0 (3.00e − 1) −3.039e + 0 (8.46e − 2) −5.639e + 0 (4.82e − 1) −2.037e + 0 (1.45e − 1) =9.309e − 1 (5.24e − 2) =3.879e + 0 (1.91e − 1) −1.208e + 0 (2.60e − 1)
102.323e + 0 (2.08e − 1) −4.409e + 0 (1.46e − 1) −5.276e + 0 (1.03e + 0) −2.588e + 0 (1.18e − 1) =1.228e + 0 (1.65e − 1) =5.244e + 0 (2.45e − 1) −1.464e + 0 (2.81e − 1)
155.784e + 0 (7.08e − 1) −9.537e + 0 (4.60e − 1) −3.226e + 0 (2.48e − 1) −5.694e + 0 (2.75e − 1) −2.829e + 0 (1.45e − 1) −1.229e + 1 (2.95e + 0) −2.640e + 0 (4.70e − 1)
WFG432.816e − 1 (8.35e − 3) −2.512e − 1 (3.64e − 3) −3.248e − 1 (1.22e − 2) −2.478e − 1 (2.46e − 3) =2.626e − 1 (3.78e − 3) −1.972e − 1 (5.39e − 3) +2.407e − 1 (3.55e − 3)
59.940e − 1 (5.27e − 3) −1.033e + 0 (9.27e − 3) −1.098e + 0 (2.28e − 2) −1.057e + 0 (5.30e − 3) −9.397e − 1 (7.75e − 3) =1.391e + 0 (9.82e − 2) −9.865e − 1 (5.77e − 3)
83.151e + 0 (8.52e − 2) −3.050e + 0 (3.27e − 2) −3.219e + 0 (6.05e − 2) −3.507e + 0 (1.71e − 1) −2.938e + 0 (9.92e − 3) =4.110e + 0 (9.22e − 2) −2.985e + 0 (1.58e − 2)
104.430e + 0 (1.82e − 1) −4.368e + 0 (5.06e − 2) −4.458e + 0 (1.05e − 1) −4.424e + 0 (7.57e − 2) =4.335e + 0 (2.19e − 2) =5.886e + 0 (2.28e − 1) −4.300e + 0 (5.79e − 2)
159.522e + 0 (7.92e − 1) −9.014e + 0 (3.68e − 1) −9.325e + 0 (2.04e − 1) −9.363e + 0 (8.00e − 1) =8.384e + 0 (1.43e − 2) +1.070e + 1 (1.17e − 1) −8.888e + 0 (1.17e − 1)
WFG532.840e − 1 (1.06e − 2) −2.837e − 1 (1.27e − 2) −3.226e − 1 (1.51e − 2) −2.523e − 1 (3.85e − 3) =1.767e − 1 (4.10e − 3) −2.857e − 1 (5.13e − 3) =2.463e − 1 (2.35e − 3)
51.015e + 0 (8.88e − 3) −1.093e + 0 (1.84e − 2) −1.041e + 0 (1.75e − 2) −1.051e + 0 (6.78e − 3) −9.864e − 1 (5.11e − 3) =1.311e + 0 (1.17e − 1) −9.842e − 1 (8.25e − 3)
83.257e + 0 (4.78e − 2) −3.102e + 0 (3.54e − 2) −3.257e + 0 (5.83e − 2) −3.443e + 0 (1.15e − 1) −2.912e + 0 (2.35e − 2) =3.902e + 0 (1.61e − 1) =2.999e + 0 (1.74e − 2)
104.169e + 0 (2.21e − 2) +4.459e + 0 (8.45e − 2) −4.540e + 0 (1.23e − 1) −4.472e + 0 (1.01e − 1) =4.324e + 0 (1.67e − 2) =6.169e + 0 (1.46e − 1) −4.266e + 0 (4.17e − 2)
158.616e + 0 (3.72e − 1) =9.206e + 0 (2.96e − 1) −9.056e + 0 (7.88e − 2) −9.132e + 0 (6.34e − 1) −8.047e + 0 (8.66e − 2) −1.104e + 1 (2.19e − 1) −5.491e + 0 (5.43e − 2)
WFG633.475e − 1 (1.15e − 2) −2.528e − 1 (6.39e − 3) −3.671e − 1 (3.40e − 2) −3.273e − 1 (4.27e − 2) =2.101e − 1 (1.10e − 2) −2.562e − 1 (6.03e − 3) =2.729e − 1 (8.43e − 3)
51.074e + 0 (1.65e − 2) −1.056e + 0 (1.08e − 2) −1.198e + 0 (3.44e − 2) −1.095e + 0 (1.09e − 2) =9.695e − 1 (1.48e − 2) =1.780e + 0 (1.40e − 1) =1.047e + 0 (1.15e − 2)
83.378e + 0 (7.79e − 2) −3.058e + 0 (4.75e − 2) −3.274e + 0 (6.16e − 2) −3.363e + 0 (6.95e − 2) =2.989e + 0 (2.55e − 2) =4.771e + 0 (2.08e − 1) =3.044e + 0 (1.51e − 2)
104.221e + 0 (4.75e − 2) +6.636e − 1 (2.32e − 2) −4.588e + 0 (1.98e − 1) −4.487e + 0 (7.94e − 2) =4.406e + 0 (2.91e − 2) −6.915e + 0 (5.56e − 2) −4.340e + 0 (4.32e − 2)
158.478e + 0 (4.55e − 1) −1.024e + 1 (4.25e − 1) −9.376e + 0 (9.01e − 2) −8.911e + 0 (1.81e − 1) −8.161e + 0 (9.22e − 2) −1.279e + 1 (2.08e − 1) −5.619e + 0 (4.12e − 1)
WFG733.382e − 1 (3.13e − 2) −2.984e − 1 (8.05e − 3) −3.286e − 1 (1.14e − 2) −2.677e − 1 (1.59e − 2) −1.524e − 1 (3.15e − 3) −2.018e − 1 (7.59e − 3) −1.235e − 1 (3.12e − 4)
51.037e + 0 (1.43e − 2) −1.089e + 0 (1.31e − 2) −1.133e + 0 (2.00e − 2) −1.083e + 0 (9.64e − 3) =9.253e − 1 (3.99e − 3) =1.771e + 0 (1.28e − 1) −9.132e − 1 (7.13e − 3)
83.216e + 0 (6.04e − 2) −3.404e + 0 (2.32e − 1) −3.245e + 0 (4.64e − 2) −3.279e + 0 (5.50e − 2) =2.897e + 0 (3.33e − 2) −4.807e + 0 (1.21e − 1) −2.411e + 0 (2.95e − 2)
106.658e − 1 (2.40e − 2) −4.936e + 0 (2.81e − 1) =6.574e − 1 (1.76e − 2) −4.403e + 0 (5.19e − 2) −4.299e + 0 (1.62e − 2) −6.881e + 0 (2.49e − 1) −5.117e − 1 (1.60e − 2)
159.214e + 0 (1.72e − 1) −1.090e + 1 (7.17e − 1) −9.536e + 0 (2.47e − 1) −9.318e + 0 (3.92e − 1) −8.304e + 0 (6.85e − 2) −1.062e + 1 (1.90e − 1) −5.814e + 0 (5.17e − 1)
WFG833.484e − 1 (1.09e − 2) −2.850e − 1 (1.70e − 2) −4.027e − 1 (1.65e − 2) −3.429e − 1 (2.42e − 2) −2.3762e − 1 (5.96e − 3) +2.510e − 1 (6.15e − 3) +2.913e − 1 (7.74e − 3)
51.082e + 0 (8.50e − 3) =1.091e + 0 (2.83e − 2) =1.175e + 0 (1.30e − 2) =1.154e + 0 (2.36e − 2) =1.120e + 0 (9.66e − 1) −1.622e + 0 (6.49e − 2) =1.049e + 0 (6.13e − 3)
83.264e + 0 (9.49e − 2) =3.205e + 0 (7.42e − 2) =3.642e + 0 (3.95e − 1) −3.397e + 0 (3.91e − 2) =3.205e + 0 (1.65e − 2) =4.547e + 0 (2.61e − 1) =3.181e + 0 (1.91e − 2)
104.756e + 0 (1.55e − 1) =4.611e + 0 (9.60e − 2) −6.357e + 0 (3.26e − 1) −4.581e + 0 (3.99e − 2) =4.781e + 0 (4.29e − 2) =6.335e + 0 (1.22e − 1) −4.762e + 0 (8.61e − 2)
159.213e + 0 (6.79e − 1) =9.433e + 0 (3.07e − 1) −9.600e + 0 (1.93e − 1) −9.308e + 0 (3.98e − 1) =9.047e + 0 (2.10e − 1) =9.756e + 0 (2.89e − 1) −9.367e + 0 (2.90e − 1)
WFG933.481e − 1 (3.38e − 2) −2.307e − 1 (2.88e − 2) =3.063e − 1 (1.02e − 2) −2.651e − 1 (1.86e − 2) −1.682e − 1 (2.26e − 2) −2.338e − 1 (4.28e − 3) −1.321e − 1 (5.13e − 3)
51.073e + 0 (2.43e − 2) −9.430e − 1 (2.99e − 1) =1.022e + 0 (1.74e − 2) −1.091e + 0 (1.99e − 2) −9.323e − 1 (8.43e − 3) =1.281e + 0 (3.44e − 2) =9.316e − 1 (5.66e − 3)
83.161e + 0 (5.55e − 2) =2.331e + 0 (1.27e + 0) +3.263e + 0 (5.85e − 2) =3.362e + 0 (5.21e − 2) −3.055e + 0 (8.28e − 2) =3.680e + 0 (1.81e − 1) =3.104e + 0 (4.92e − 2)
104.365e + 0 (1.40e − 1) =3.608e + 0 (1.06e + 0) +4.821e + 0 (3.50e − 1) −4.479e + 0 (9.37e − 2) =4.413e + 0 (6.04e − 2) =5.541e + 0 (2.92e − 2) =4.231e + 0 (7.16e − 2)
158.560e + 0 (2.38e − 1) =1.111e + 1 (2.84e + 0) −9.011e + 0 (6.74e − 2) −8.953e + 0 (2.49e − 1) =8.493e + 0 (1.18e − 1) −9.714e + 0 (2.07e − 1) −5.793e + 0 (6.84e − 1)
 2/31/124/32/61/40/40/23/226/19/203/31/11