Computational Intelligence and Neuroscience / 2019 / Article / Tab 5

Research Article

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

Table 5

The statistical results (mean and standard deviation) of the IGD values obtained by MOEA/D, dMOPSO, MOBI2, ϵ-MOEA, and OBEA on WFG test suits. The best results are italicized.

ProblemMMOEADdMOPSOMOMBI2ɛ-MOEAOBEA

WFG131.299e + 0 (8.45e − 2) = 1.552e + 0 (7.51e − 3) −1.094e + 0 (7.15e − 2) −1.517e + 0 (6.32e − 2) −5.083e1 (7.94e2)
51.998e + 0 (9.68e − 2) −2.210e + 0 (3.04e − 2) −1.940e + 0 (9.49e − 2) −2.038e + 0 (1.74e − 2) −9.814e1 (1.56e1)
82.840e + 0 (1.72e − 1) −3.125e + 0 (5.07e − 2) −2.635e + 0 (2.10e − 1) −2.693e + 0 (3.05e − 2) −1.959e+0 (3.48e1)
103.344e + 0 (9.03e − 2) −3.566e + 0 (5.15e − 2) −3.070e + 0 (2.26e − 2) −3.051e + 0 (1.86e − 1) −2.088e+0 (4.77e2)
154.294e + 0 (8.37e − 2) −4.587e + 0 (5.58e − 2) −3.136e + 0 (3.54e − 1) = 3.972e + 0 (6.22e − 2) −3.115e + 0 (2.48e − 1)

WFG231.081e + 0 (8.61e − 2) −9.277e − 1 (1.79e − 1) −3.477e − 1 (1.76e − 2) −2.916e − 1 (5.33e − 2) = 2.747e − 1 (6.59e − 2)
55.835e + 0 (2.62e − 1) −3.811e + 0 (4.64e − 1) −1.348e + 0 (1.68e − 1) = 8.280e1 (2.05e1) +1.502e + 0 (5.72e − 1)
88.889e + 0 (6.72e − 1) −6.697e + 0 (4.23e − 1) −2.169e + 0 (4.20e − 1) +1.482e+0 (2.97e − 1) +3.321e + 0 (8.42e − 1)
101.696e + 1 (2.00e − 1) −1.309e + 1 (2.39e − 1) −3.028e+0 (4.88e1) +4.441e + 0 (1.41e + 0) = 4.661e + 0 (1.41e + 0)
152.766e + 1 (1.71e − 1) −2.274e + 1 (1.33e + 0) −1.142e + 1 (2.31e + 0) +3.401e+0 (6.14e1) +1.632e + 1 (3.19e + 0)

WFG334.820e − 1 (1.29e − 1) −4.060e − 1 (7.70e − 2) −1.453e1 (1.76e2) +3.051e − 1 (3.07e − 2) −1.730e − 1 (1.72e − 2)
51.468e + 0 (3.18e − 1) −7.413e − 1 (4.80e − 2) −9.781e − 1 (7.32e − 2) −7.482e − 1 (3.16e − 2) −5.355e1 (7.74e2)
84.279e + 0 (2.87e − 1) −2.304e + 0 (5.29e − 1) −3.362e + 0 (4.40e − 1) −1.107e+0 (3.13e2) +1.208e + 0 (2.60e − 1)
107.859e + 0 (1.16e + 0) −3.300e + 0 (8.09e − 1) −1.263e+0 (5.16e2) +3.607e + 0 (5.57e − 1) −1.464e + 0 (2.81e − 1)
151.337e + 1 (1.89e + 0) −8.238e + 0 (2.91e + 0) −1.035e + 1 (1.37e + 0) −1.493e+0 (8.05e2) +2.640e + 0 (4.70e − 1)

WFG433.073e − 1 (1.60e − 2) −4.085e − 1 (2.38e − 2) −2.616e − 1 (4.73e − 3) −1.967e1 (6.27e3) +2.407e − 1 (3.55e − 3)
52.095e + 0 (1.43e − 1) −1.648e + 0 (1.15e − 1) −1.851e + 0 (1.09e − 1) −8.292e1 (5.25e2) +9.865e − 1 (5.77e − 3)
87.380e + 0 (1.28e − 1) −7.561e + 0 (3.05e − 1) −4.017e + 0 (5.12e − 1) −3.549e + 0 (2.22e − 1) −2.985e+0 (1.58e2)
109.817e + 0 (2.22e − 1) −1.030e + 1 (2.35e − 1) −5.938e + 0 (3.00e − 1) −5.897e + 0 (3.90e − 1) −4.300e+0 (5.79e2)
151.713e + 1 (3.18e − 1) −1.672e + 1 (1.74e − 1) −2.061e + 1 (1.20e + 0) −1.316e + 1 (4.78e − 1) −8.888e+0 (1.17e1)

WFG533.099e − 1 (1.66e − 2) −4.132e − 1 (3.00e − 2) −2.748e − 1 (3.00e − 3) −2.138e1 (9.41e3) +2.463e − 1 (2.35e − 3)
52.166e + 0 (8.60e − 2) −1.314e + 0 (2.07e − 2) −2.062e + 0 (7.95e − 2) −9.772e1 (2.80e2) = 9.842e − 1 (8.25e − 3)
87.144e + 0 (7.49e − 2) −4.488e + 0 (2.12e − 1) −3.767e + 0 (3.57e − 2) −3.139e + 0 (1.64e − 1) −2.999e+0 (1.74e2)
109.567e + 0 (1.30e − 1) −5.826e + 0 (2.32e − 1) −4.820e + 0 (2.18e − 1) −5.511e + 0 (1.82e − 1) −4.266e+0 (4.17e2)
151.631e + 1 (1.30e − 1) −1.084e + 1 (3.02e − 1) −2.156e + 1 (1.93e + 0) −1.066e + 1 (6.76e − 1) −5.491e+0 (5.43e2)

WFG633.453e − 1 (2.07e − 2) −4.387e − 1 (5.19e − 2) −2.842e − 1 (7.16e − 3) −2.806e − 1 (1.48e − 2) −2.729e1 (8.43e3)
52.850e + 0 (3.33e − 1) −2.372e + 0 (1.64e − 1) −1.972e + 0 (1.01e − 1) −1.125e + 0 (2.99e − 2) −1.047e+0 (1.15e2)
87.849e + 0 (1.55e − 1) −8.386e + 0 (3.11e − 1) −3.760e + 0 (8.83e − 2) −3.268e + 0 (1.34e − 1) −3.044e+0 (1.51e2)
101.005e + 1 (2.43e − 1) −1.049e + 1 (2.93e − 1) −4.999e + 0 (2.82e − 1) −5.465e + 0 (9.81e − 2) −4.340e+0 (4.32e2)
151.672e + 1 (1.88e − 1) −1.686e + 1 (1.64e − 1) −1.929e + 1 (1.73e + 0) −1.108e + 1 (6.65e − 1) −5.619e+0 (4.12e1)

WFG734.836e − 1 (3.53e − 2) −4.494e − 1 (1.34e − 2) −2.607e − 1 (3.93e − 3) −2.035e − 1 (1.26e − 2) −1.235e1 (3.12e4)
52.716e + 0 (3.13e − 1) −1.523e + 0 (1.36e − 1) −2.024e + 0 (1.06e − 1) −9.271e − 1 (3.95e − 2) −9.132e1 (7.13e3)
87.776e + 0 (1.67e − 1) −6.017e + 0 (1.26e + 0) −3.846e + 0 (3.80e − 1) −3.218e + 0 (2.54e − 1) −2.411e+0 (2.95e2)
101.005e + 1 (1.88e − 1) −5.473e + 0 (4.05e − 1) −5.523e − 1 (1.38e − 2) = 7.827e − 1 (2.40e − 2) = 5.117e1 (1.60e2)
151.735e + 1 (1.36e − 1) −1.394e + 1 (1.98e + 0) −1.735e + 1 (2.38e + 0) −1.203e + 1 (7.03e − 1) −5.814e+0 (5.17e1)

WFG833.797e − 1 (2.78e − 2) −5.852e − 1 (3.90e − 2) −3.184e − 1 (7.85e − 3) −2.9223e − 1 (7.87e − 3) = 2.913e1 (7.74e3)
52.131e + 0 (2.08e − 1) −1.538e + 0 (4.79e − 2) −2.246e + 0 (6.38e − 2) −1.116e + 0 (3.62e − 2) −1.049e+0 (6.13e3)
86.814e + 0 (1.59e − 1) −6.756e + 0 (4.51e − 1) −3.973e + 0 (2.38e − 1) −3.523e + 0 (1.35e − 1) −3.181e+0 (1.91e2)
109.115e + 0 (2.44e − 1) −8.869e + 0 (4.10e − 1) −5.228e + 0 (2.71e − 1) −5.871e + 0 (5.23e − 1) −4.762e+0 (8.61e2)
151.324e + 1 (2.71e + 0) −1.545e + 1 (3.51e − 1) −1.970e + 1 (1.40e + 0) −1.138e + 1 (5.51e − 1) −9.367e+0 (2.90e1)

WFG934.680e − 1 (6.68e − 2) −3.495e − 1 (1.33e − 2) −2.710e − 1 (1.06e − 2) = 2.060e − 1 (3.04e − 2) +1.321e1 (5.13e3)
52.141e + 0 (1.10e − 1) −2.030e + 0 (8.89e − 2) −1.954e + 0 (9.93e − 2) −1.015e + 0 (4.68e − 2) −9.316e1 (5.66e3)
87.046e + 0 (2.36e − 1) −6.721e + 0 (4.80e − 1) −3.715e + 0 (4.70e − 2) −3.276e + 0 (1.53e − 1) −3.104e+0 (4.92e2)
109.240e + 0 (6.11e − 1) −8.124e + 0 (7.77e − 1) −5.123e + 0 (1.95e − 1) −5.399e + 0 (7.48e − 2) −4.231e+0 (7.16e2)
151.436e + 1 (2.01e + 0) −1.132e + 1 (5.41e − 1) −2.143e + 1 (1.63e + 0) −1.127e + 1 (6.58e − 1) −5.793e+0 (6.84e1)

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