Computational Intelligence and Neuroscience

Research Article

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

Table 1

The statistical results (mean and standard deviation) of the HV values obtained by each algorithm on DTLZ1 to DTLZ4 and DTLZ7. The best results are italicized.
ProblemMMOEADdMOPSOMOMBI2ϵ-MOEAOBEA
DTLZ133.994e − 1 (3.36e − 1) −7.015e − 3 (3.84e − 2) −5.819e − 1 (3.01e − 1) −2.931e − 1 (2.89e − 1) −8.471e − 1 (6.13e − 3)
55.010e − 1 (3.49e − 1) −2.856e − 4 (1.56e − 3) −1.406e − 1 (2.68e − 1) −8.164e − 2 (2.00e − 1) = 9.761e − 1 (4.15e − 4)
85.824e − 1 (2.20e − 1) −5.418e − 3 (2.51e − 2) −2.923e − 1 (3.37e − 1) −3.102e − 1 (5.34e − 3) −9.524e − 1 (6.84e − 3)
104.907e − 1 (2.76e − 1) −1.031e − 2 (2.31e − 2) −9.039e − 1 (9.79e − 2) = 8.356e − 2 (1.53e − 1) −9.571e − 1 (4.18e − 3)
153.566e − 1 (2.33e − 1) −5.092e − 3 (1.76e − 2) −2.195e − 1 (2.87e − 1) −4.015e − 1 (6.12e − 2) −9.456e − 1 (4.46e − 6)
DTLZ235.551e − 1 (7.76e − 4) −3.671e − 1 (1.60e − 2) −5.572e − 1 (5.51e − 4) +5.409e − 1 (2.93e − 3) −5.563e − 1 (5.70e − 4)
57.823e − 1 (5.77e − 3) = 4.986e − 1 (1.69e − 2) −7.881e − 1 (3.38e − 3) = 8.022e − 1 (5.35e − 3) +7.829e − 1 (3.48e − 3)
88.541e − 1 (1.27e − 2) −3.945e − 1 (2.67e − 2) −8.643e − 1 (1.33e − 2) −8.422e − 1 (2.30e − 2) −8.981e − 1 (4.96e − 3)
108.384e − 1 (2.34e − 2) −4.536e − 1 (2.17e − 2) −7.993e − 1 (2.28e − 2) −8.799e − 1 (1.83e − 2) −9.071e − 1 (1.22e − 2)
158.428e − 1 (6.76e − 2) −2.953e − 1 (2.68e − 2) −7.273e − 1 (6.12e − 2) −7.828e − 1 (4.00e − 2) −9.459e − 1 (7.02e − 3)
DTLZ333.806e − 1 (6.33e − 2) −1.775e − 3 (5.02e − 3) −1.001e − 1 (1.38e − 1) −4.173e − 1 (5.42e − 1) −5.212e − 1 (4.53e − 2)
52.780e − 1 (2.820e − 1) −2.563e − 1 (2.52e − 1) −6.709e − 1 (1.95e − 1) = 2.316e − 1 (3.55e − 1) −7.688e − 1 (5.15e − 2)
82.736e − 4 (1.50e − 3) −4.563e − 1 (2.63e − 2) = 2.670e − 1 (3.22e − 1) −4.621e − 1 (4.13e − 2) = 5.475e − 1 (3.15e − 1)
103.501e − 2 (2.42e − 1) −2.511e − 1 (3.44e − 1) −4.073e − 1 (2.61e − 1) = 4.012e − 1 (7.27e − 1) −4.912e − 1 (4.38e − 1)
155.370e − 1 (6.42e − 1) −2.036e − 1 (3.54e − 1) −1.996e − 1 (2.01e − 1) −5.722e − 1 (5.15e − 1) −9.879e − 1 (6.17e − 2)
DTLZ433.630e − 1 (1.69e − 1) −3.023e − 1 (4.80e − 2) −5.359e − 1 (6.60e − 2) +3.963e − 1 (1.53e − 1) −4.954e − 1 (1.03e − 1)
55.863e − 1 (1.70e − 1) −3.276e − 1 (5.26e − 2) −7.872e − 1 (2.44e − 2) +7.973e − 1 (4.00e − 2) +7.814e − 1 (8.95e − 3)
86.231e − 1 (1.33e − 1) −3.726e − 1 (5.68e − 2) −9.102e − 1 (3.20e − 2) −9.199e − 1 (1.62e − 2) +9.113e − 1 (6.95e − 3)
106.928e − 1 (1.79e − 1) −3.958e − 1 (7.36e − 2) −9.505e − 1 (8.43e − 3) +9.589e − 1 (5.99e − 3) +9.396e − 1 (6.00e − 3)
156.959e − 1 (1.80e − 1) −3.394e − 1 (1.02e − 1) −9.769e − 1 (9.59e − 3) −9.819e − 1 (5.12e − 3) −9.875e − 1 (8.11e − 4)
DTLZ732.288e − 1 (1.06e − 2) −2.398e − 1 (1.27e − 2) −2.620e − 1 (1.66e − 2) +2.357e − 1 (2.32e − 2) −2.526e − 1 (6.40e − 3)
51.993e − 2 (2.06e − 2) −1.805e − 1 (1.10e − 2) +2.069e − 1 (1.34e − 2) +1.694e − 1 (2.42e − 2) +1.320e − 1 (2.08e − 2)
82.283e − 3 (6.87e − 3) −1.044e − 1 (1.27e − 2) +9.639e − 2 (2.69e − 2) =5.824e − 2 (3.18e − 2) −9.463e − 2 (1.67e − 2)
108.257e − 5 (1.41e − 4) −1.194e − 1 (1.63e − 2) +1.503e − 1 (3.59e − 3) +6.109e − 4 (2.42e − 3) −8.100e − 2 (1.87e − 3)
158.583e − 6 (2.73e − 5) −8.524e − 2 (2.63e − 2) +1.159e − 1 (3.30e − 4) +4.012e − 4 (5.11e − 3) −5.400e − 2 (3.06e − 4)
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