Computational Intelligence and Neuroscience / 2019 / Article / Tab 4

Research Article

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

Table 4

The statistical results (mean and standard deviation) of the IGD values obtained by MOEA/D, dMOPSO, MOBI2, ϵ-MOEA, and OBEA on DTLZ1 to DTLZ4 and DTLZ7. The best results are italicized.

ProblemMMOEADdMOPSOMOMBI2ϵ-MOEAOBEA

DTLZ132.569e − 1 (2.50e − 1) = 8.076e + 0 (5.99e + 0) −1.421e − 1 (1.63e − 1) −3.775e − 1 (4.66e − 1) −1.121e2 (5.13e4)
52.884e − 1 (2.88e − 1) −1.129e + 1 (4.53e + 0) −6.283e − 1 (3.21e − 1) −8.831e − 1 (5.86e − 1) −5.101e2 (4.27e4)
81.937e − 1 (9.66e − 2) −5.033e + 0 (3.95e + 0) −4.950e − 1 (2.04e − 1) −5.220e + 0 (2.22e + 0) −8.202e2 (2.51e3)
103.256e − 1 (3.79e − 1) −7.812e + 0 (3.13e + 0) −5.716e + 0 (2.09e + 0) −9.107e − 1 (5.31e − 1) −1.312e1 (3.09e3)
152.433e − 1 (1.20e − 1) = 7.121e + 0 (6.43e + 0) −7.148e − 1 (5.39e − 1) −6.084e + 0 (2.37e + 0) −1.214e1 (3.01e3)

DTLZ235.492e − 2 (2.34e − 4) −1.425e − 1 (8.05e − 3) −5.819e − 2 (1.15e − 3) −7.177e − 2 (2.78e − 3) −5.479e2 (9.14e5)
51.668e − 1 (1.91e − 3) = 2.619e − 1 (9.02e − 3) −2.094e − 1 (4.86e − 3) −1.289e1 (4.58e3) +1.696e − 1 (9.83e − 4)
83.260e − 1 (7.38e − 3) +5.501e − 1 (1.88e − 2) −4.079e − 1 (7.15e − 3) −3.102e − 1 (1.52e − 2) +3.335e − 1 (2.51e − 3)
104.185e1 (2.24e2)=6.212e − 1 (2.90e − 2) −4.373e − 1 (1.67e − 2) −5.073e − 1 (1.40e − 2) −4.215e − 1 (3.95e − 3)
157.053e − 1 (6.39e − 2) −9.488e − 1 (3.17e − 2) −8.965e − 1 (4.96e − 2) −6.734e − 1 (3.81e − 2) −5.885e1 (5.97e3)

DTLZ331.388e + 1 (9.64e + 0) −4.493e + 1 (3.14e + 1) −6.321e + 0 (5.45e + 0) −1.719e + 1 (7.00e + 0) −5.494e1 (3.21e2)
51.933e + 1 (1.38e + 1) −1.898e + 2 (3.36e + 1) −1.704e + 1 (5.60e + 0) −2.985e + 1 (1.07e + 1) −1.782e1 (2.51e3)
86.610e + 0 (4.88e + 0) −2.032e + 2 (9.75e + 0) −1.459e + 1 (5.64e + 0) −1.310e + 2 (2.52e + 1) −8.911e1 (8.01e1)
106.232e + 0 (2.12e + 0) = 2.031e + 2 (1.22e + 1) −1.530e + 2 (3.41e + 1) −3.215e + 1 (7.78e + 0) −1.197e + 0 (4.32e1)
156.469e + 0 (3.99e + 0) = 2.045e + 2 (7.22e + 0) −1.794e + 1 (9.92e + 0) −1.515e + 2 (3.44e + 1) −1.569e+0 (4.91e1)

DTLZ434.602e − 1 (3.30e − 1) −3.173e − 1 (2.84e − 2) −1.085e1 (1.47e1) +3.695e − 1 (3.02e − 1) −1.851e − 1 (2.20e − 1)
55.470e − 1 (2.31e − 1) −5.217e − 1 (2.20e − 2) −2.246e − 1 (5.14e − 2) −1.908e − 1 (9.86e − 2) −1.770e1 (1.44e2)
87.792e − 1 (1.35e − 1) −6.505e − 1 (2.01e − 2) −4.058e − 1 (6.38e − 2) −3.423e1 (5.58e2)=3.478e − 1 (1.95e − 3)
108.279e − 1 (1.38e − 1) −7.178e − 1 (1.26e − 2) −4.213e1 (2.13e2) +4.789e − 1 (1.80e − 2) −4.631e − 1 (3.80e − 3)
159.284e − 1 (1.14e − 1) −8.328e − 1 (2.13e − 2) −6.675e − 1 (2.36e − 2) −5.324e1 (1.82e2) +6.306e − 1 (1.38e − 3)

DTLZ731.907e − 1 (1.67e − 1) −1.705e − 1 (1.43e − 1) −1.840e − 1 (1.25e − 1) −2.591e − 1 (2.71e − 1) = 5.452e2 (3.12e3)
51.169e + 0 (2.71e − 1) −6.097e − 1 (2.00e − 1) −4.823e − 1 (1.64e − 1) = 6.667e − 1 (2.53e − 1) −4.141e1 (5.13e2)
81.659e + 0 (2.34e − 1) = 1.385e + 0 (2.73e − 1) = 4.324e + 0 (1.23e + 0) −2.151e + 0 (6.38e − 1) −9.146e1 (3.31e2)
102.704e + 0 (5.03e − 1) = 2.252e + 0 (4.09e − 1) = 1.877e + 1 (2.27e + 0) −1.007e + 1 (1.48e + 0) = 1.452e+0 (6.76e1)
153.181e + 0 (6.21e − 1) = 5.635e + 0 (1.69e + 0) = 2.844e + 1 (4.67e + 0) −3.766e + 1 (3.27e + 0) −1.953e+0 (3.87e1)

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