Research Article

A Cross-Reference Line Method Based Multiobjective Evolutionary Algorithm to Enhance Population Diversity

Table 4

Statistical results (mean values and standard deviations) of HV value obtained by MOEA/D, NSGA-III, RVEA, KnEA, and MOEA-CRL on DTLZ1-DTLZ7, WFG1-WFG9, MaF3, MaF11, and MaF15 with 5 objectives and 10 objectives.

ProblemMMOEA/DNSGA-IIIRVEAKnEAMOEA-CRL

DTLZ159.0873e − 1 (8.52e − 2) −9.7979e − 1 (1.56e − 4) ≈9.7984e − 1 (1.55e − 4) ≈6.2616e − 1 (1.64e − 1) −9.7988e − 1 (1.63e − 4)
109.7273e − 1 (5.32e − 3) −9.8648e − 1 (3.26e − 2) −9.9967e − 1 (1.88e − 5) ≈0.0000e + 0 (0.00e + 0) −9.9973e − 1 (1.10e − 4)

DTLZ257.1112e − 1 (5.08e − 4) −8.1269e − 1 (4.52e − 4) −8.1252e − 1 (4.48e − 4) −7.9064e − 1 (3.54e − 3) −8.1626e − 1 (9.62e − 4)
106.2665e − 1 (2.70e − 2) −9.4539e − 1 (3.51e − 2) −9.6963e − 1 (1.72e − 4) −9.5650e − 1 (4.53e − 3) −9.7102e − 1 (1.70e − 3)

DTLZ354.3566e − 1 (1.27e − 1) −7.7543e − 1 (2.88e − 3) +7.7757e − 1 (1.36e − 3) +3.9573e − 1 (1.31e − 1) −7.3975e − 1 (3.36e − 3)
106.4046e − 1 (1.97e − 2) −3.3590e − 1 (4.26e − 1) −9.6443e − 1 (9.40e − 3) +0.0000e + 0 (0.00e + 0) −9.6234e − 1 (2.68e − 1)

DTLZ452.7690e − 1 (1.65e − 1) −7.3262e − 1 (6.50e − 2) +7.7683e − 1 (1.68e − 2) +7.6308e − 1 (4.31e − 3) +7.2000e − 1 (5.17e − 2)
106.2169e − 1 (2.11e − 2) −9.6721e − 1 (1.24e − 2) −9.6984e − 1 (1.99e − 4) −9.5637e − 1 (4.37e − 3) −9.7165e − 1 (1.51e − 3)

WFG456.2968e − 1 (8.71e − 4) −8.0469e − 1 (8.35e − 4) −8.0565e − 1 (1.10e − 3) ≈7.8730e − 1 (2.17e − 3) −8.0575e − 1 (1.74e − 3)
105.8107e − 1 (5.61e − 2) −9.4578e − 1 (3.92e − 3) +9.4326e − 1 (3.57e − 3) +9.5750e − 1 (1.56e − 3) +9.1518e − 1 (5.08e − 3)

WFG555.9294e − 1 (1.98e − 2) −7.6126e − 1 (3.68e − 4) +7.6092e − 1 (4.04e − 4) +7.4495e − 1 (2.82e − 3) +7.3465e − 1 (2.49e − 3)
105.3478e − 1 (1.75e − 2) −8.9907e − 1 (6.43e − 4) +8.9790e − 1 (1.17e − 3) +8.9664e − 1 (8.14e − 4) +8.4566e − 1 (3.56e − 3)

WFG655.5065e − 1 (3.13e − 2) −7.4242e − 1 (1.58e − 2) +7.4662e − 1 (1.44e − 2) +7.2041e − 1 (1.03e − 2) ≈7.2008e − 1 (1.20e − 2)
104.7917e − 1 (9.01e − 2) −8.6935e − 1 (1.66e − 2) ≈8.6291e − 1 (2.01e − 2) ≈8.6854e − 1 (1.59e − 2) ≈8.7014e − 1 (1.72e − 2)

WFG756.2952e − 1 (1.76e − 3) −8.0771e − 1 (6.15e − 4) +8.0685e − 1 (6.00e − 4) +7.9537e − 1 (2.41e − 3) +7.9284e − 1 (1.74e − 3)
106.1632e − 1 (3.69e − 2) −9.4638e − 1 (2.02e − 2) +9.4552e − 1 (2.93e − 3) −9.5737e − 1 (6.09e − 3) +9.4268e − 1 (2.70e − 3)

WFG853.2668e − 1 (1.10e − 2) −6.9440e − 1 (3.44e − 3) −6.9749e − 1 (1.55e − 3) ≈6.6088e − 1 (4.04e − 3) −6.9786e − 1 (2.26e − 3)
105.2900e − 1 (2.26e − 2) −8.3115e − 1 (2.65e − 2) −7.4198e − 1 (7.45e − 2) −8.1031e − 1 (5.88e − 2) −8.5967e − 1 (2.09e − 2)

WFG954.1801e − 1 (7.01e − 2) −7.6559e − 1 (4.98e − 3) +7.6952e − 1 (3.01e − 3) +7.6796e − 1 (3.00e − 3) +7.4665e − 1 (4.89e − 3)
105.3897e − 1 (5.22e − 2) -8.6420e − 1 (4.42e − 2) +8.7342e − 1 (1.19e − 2) +9.0581e − 1 (3.21e − 2) +8.3182e − 1 (1.92e − 2)
+/−/≈0/20/010/8/210/5/58/10/2

DTLZ551.0968e − 1 (8.30e − 3) ≈1.0483e − 1 (1.62e − 2) ≈9.1890e − 2 (1.22e − 3) −7.1580e − 2 (2.93e − 2) −1.0808e − 1 (5.72e − 3)
109.7922e − 2 (3.20e − 4) +2.8809e − 2 (3.32e − 2) −9.0906e − 2 (1.34e − 4) +3.5304e − 2 (3.14e − 2) −8.2365e − 2 (2.16e − 2)

DTLZ659.4634e − 2 (6.13e − 3) ≈5.7651e − 2 (4.09e − 2) −9.9670e − 2 (4.73e − 3) ≈9.1029e − 2 (2.57e − 3) −9.2650e − 2 (2.20e − 2)
109.8124e − 2 (2.62e − 4) +3.0230e − 3 (1.66e − 2) −9.2019e − 2 (9.23e − 4) +0.0000e + 0 (0.00e + 0) −9.1491e − 2 (3.74e − 2)

DTLZ751.7168e − 1 (5.33e − 2) −2.3157e − 1 (9.31e − 3) +2.1343e − 1 (8.52e − 3) −2.4843e − 1 (1.07e − 2) +2.2732e − 1 (3.03e − 3)
104.5806e − 3 (7.31e − 3) −1.7116e − 1 (7.07e − 3) +1.3371e − 1 (2.42e − 2) +9.1243e − 2 (2.89e − 2) +7.4082e − 2 (2.95e − 2)

WFG159.4334e − 1 (7.14e − 2) −9.7370e − 1 (2.11e − 2) −9.8276e − 1 (2.89e − 2) ≈9.9246e − 1 (1.41e − 3) −9.9835e − 1 (1.60e − 2)
105.0234e − 1 (1.69e − 1) −9.4008e − 1 (4.39e − 2) +9.9015e − 1 (2.42e − 2) +9.9756e − 1 (8.35e − 4) +7.4040e − 1 (5.17e − 2)

WFG259.6566e − 1 (3.42e − 2) −9.9555e − 1 (8.56e − 4) −9.9404e − 1 (1.15e − 3) −9.9320e − 1 (7.41e − 4) −9.9683e − 1 (7.33e − 4)
109.9648e − 1 (2.10e − 3) +9.9700e − 1 (1.46e − 3) +9.8471e − 1 (4.33e − 3) −9.9306e − 1 (1.47e − 3) +9.9191e − 1 (2.27e − 3)

WFG359.1998e − 2 (3.47e − 4) −1.6765e − 1 (1.32e − 2) ≈1.5967e − 1 (1.56e − 2) ≈7.7484e − 2 (1.82e − 2) −1.6083e − 1 (1.27e − 2)
107.5376e − 2 (7.87e − 3) +2.4516e − 4 (1.34e − 3) −0.0000e + 0 (0.00e + 0) −0.0000e + 0 (0.00e + 0) −4.0606e − 2 (7.87e − 3)

MaF359.9646e − 1 (1.07e − 4) −9.9870e − 1 (2.01e − 3) −9.9895e − 1 (4.31e − 4) −8.8686e − 1 (1.27e − 1) −9.9975e − 1 (7.14e − 4)
109.9993e − 1 (4.54e − 5) ≈2.9125e − 1 (4.53e − 1) −9.8343e − 1 (5.59e − 2) −0.0000e + 0 (0.00e + 0) −9.9997e − 1 (1.30e − 1)

MaF1159.7508e − 1 (2.07e − 2) −9.9583e − 1 (5.38e − 4) −9.9397e − 1 (1.11e − 3) −9.9299e − 1 (8.84e − 4) −9.9751e − 1 (5.92e − 4)
109.3320e − 1 (6.36e − 2) −9.9641e − 1 (2.18e − 3) −9.8256e − 1 (4.48e − 3) −9.9284e − 1 (8.14e − 4) −9.9818e − 1 (2.03e − 3)

MaF1552.9995e − 2 (1.11e − 2) +0.0000e + 0 (0.00e + 0) −2.4759e − 2 (1.06e − 2) +0.0000e + 0 (0.00e + 0) −1.0571e − 2 (5.05e − 3)
103.0314e − 11 (1.16e − 10) −0.0000e + 0 (0.00e + 0) −1.9696e − 7 (2.48e − 7) −0.0000e + 0 (0.00e + 0) −2.6803e − 7 (2.42e − 10)
+/−/≈5/10/34/12/25/10/34/14/0