Decomposition-Based Multiobjective Evolutionary Optimization with Adaptive Multiple Gaussian Process Models

Table 4

Performance comparison of MOEA/D-FMG with various Gaussian distributions and MOEA/D-AMG in terms of the average IGD values and standard deviations on F, UF, and WFG problems.

Problems	MOEA/D-FMG (0, 0.6^2)	MOEA/D-FMG (0, 0.8^2)	MOEA/D-FMG (0, 1.0^2)	MOEA/D-FMG (0, 1.2^2)	MOEA/D-FMG (0, 1.4^2)	MOEA/D-AMG
F1	1.29E − 03 (8.00E − 04)+	1.44E − 03 (1.04E − 04)−	1.46E − 03 (2.36E − 05)−	1.60E − 03 (3.15E − 05)−	2.10E − 03 (2.04E − 05)−	1.35E − 03 (1.28E − 05)
F2	5.05E − 03 (5.13E − 02)−	4.03E − 03 (2.24E − 03)−	5.17E − 03 (1.24E − 04)−	6.30E − 03 (5.10E − 03)−	9.40E − 03 (6.51E − 03)−	2.85E − 03 (2.95E − 04)
F3	5.53E − 03 (3.03E − 02)−	3.45E − 03 (1.59E − 03)−	3.81E − 03 (2.11E − 04)−	5.20E − 03 (1.90E − 03)−	7.40E − 03 (5.86E − 03)−	2.55E − 03 (3.59E − 04)
F4	6.10E − 03 (4.18E − 02)−	4.72E − 03 (1.19E − 03)−	5.76E − 03 (5.22E − 04)−	6.60E − 03 (4.15E − 03)−	9.70E − 03 (4.03E − 03)−	2.93E − 03 (7.23E − 04)
F5	1.21E − 02 (2.45E − 02)−	8.80E − 03 (1.24E − 03)−	8.97E − 03 (1.46E − 04)−	1.03E − 02 (1.21E − 03)−	1.31E − 02 (4.72E − 03)−	7.94E − 03 (1.27E − 03)
F6	5.93E − 02 (1.75E − 02)−	5.93E − 02 (2.80E − 03)−	4.70E − 02 (7.65E − 03)+	6.96E − 02 (6.89E − 02)−	8.44E − 02 (3.00E − 02)−	6.54E − 02 (8.13E − 03)
F7	2.59E − 01 (2.08E − 02)−	1.27E − 01 (4.20E − 02)−	2.57E − 02 (3.11E − 02)+	3.72E − 02 (4.01E − 01)+	1.13E − 01 (7.48E − 02)−	4.21E − 02 (7.84E − 02)
F8	5.10E − 02 (5.24E − 02)−	1.25E − 02 (1.82E − 02)+	1.15E − 02 (2.94E − 03)+	1.38E − 02 (5.44E − 03)+	2.88E − 02 (8.89E − 03)−	1.55E − 02 (1.07E − 02)
F9	8.31E − 03 (3.84E − 02)−	8.93E − 03 (2.190E − 03)−	8.00E − 03 (1.17E − 03)−	1.04E − 02 (3.54E − 03)−	1.47E − 02 (2.61E − 02)−	5.14E − 03 (2.12E − 03)
UF1	2.45E − 03 (2.71E − 02)−	2.77E − 03 (8.93E − 04)−	2.88E − 03 (5.02E − 04)−	3.60E − 03 (3.21E − 03)−	5.00E − 03 (1.07E − 02)−	2.45E − 03 (1.42E − 03)
UF2	6.95E − 03 (2.14E − 02)−	6.40E − 03 (1.27E − 03)−	6.98E − 03 (1.35E − 03)−	7.50E − 03 (1.59E − 03)−	9.00E − 03 (2.96E − 03)−	6.68E − 03 (8.37E − 04)
UF3	2.06E − 03 (4.31E − 02)−	5.43E − 03 (1.94E − 02)+	5.60E − 03 (3.10E − 03)+	5.40E − 03 (6.02E − 03)+	6.51E − 03 (1.21E − 02)+	1.74E − 02 (1.99E − 02)
UF4	7.09E − 02 (3.61E − 03)−	6.26E − 02 (2.89E − 03)−	5.49E − 02 (2.81E − 03)+	5.35E − 02 (9.49E − 03)+	5.13E − 02 (6.77E − 03)+	6.98E − 02 (5.78E − 03)
UF5	3.41E − 01 (8.29E − 02)+	3.08E − 01 (7.50E − 02)+	3.17E − 01 (5.20E − 02)+	3.31E − 01 (5.26E − 01)+	3.85E − 01 (3.63E − 01)−	4.08E − 01 (9.50E − 02)
UF6	2.33E − 01 (1.32E − 01)−	2.26E − 01 (4.81E − 02)−	1.65E − 01 (2.66E − 01)∼	1.15E − 01 (3.70E − 02)+	1.41E − 01 (3.57E − 02)+	1.62E − 01 (2.06E − 01)
UF7	5.93E − 03 (2.27E − 01)−	5.82E − 03 (2.11E − 03)−	6.00E − 03 (3.71E − 04)−	6.90E − 03 (2.62E − 03)−	6.50E − 03 (3.27E − 02)−	5.13E − 03 (6.70E − 04)
UF8	6.09E − 02 (6.44E − 03)−	6.30E − 02 (3.23E − 03)−	6.59E − 02 (4.34E − 03)−	7.65E − 02 (3.53E − 02)−	8.73E − 02 (3.58E − 03)−	5.74E − 02 (8.29E − 03)
UF9	7.13E − 02 (5.02E − 02)−	4.34E − 02 (4.99E − 03)−	3.22E − 02 (3.12E − 03)−	4.50E − 02 (5.92E − 02)−	4.60E − 02 (4.87E − 02)−	2.76E − 02 (4.37E − 03)
UF10	6.19E − 01 (8.23E − 02)+	8.89E − 01 (7.95E − 02)+	1.70E + 00 (2.21E − 01)−	2.43E + 00 (1.19E − 01)−	2.58E + 00 (4.09E − 03)−	8.03E − 01 (1.71E − 01)
WFG1	1.12E + 00 (5.32E − 02)−	1.16E + 00 (8.02E − 03)−	1.16E + 00 (8.15E − 03)−	1.19E + 00 (9.32E − 03)−	1.22E + 00 (8.90E − 02)−	1.11E + 00 (1.17E − 02)
WFG2	1.55E − 02 (2.12E − 02)∼	1.55E − 02 (5.20E − 04)∼	1.57E − 02 (3.63E − 04)−	1.59E − 02 (1.31E − 03)−	1.62E − 02 (1.78E − 03)−	1.55E − 02 (3.24E − 04)
WFG3	4.45E − 01 (5.70E − 04)∼	4.45E − 01 (5.04E − 04)∼	4.59E − 01 (1.71E − 04)−	4.62E − 01 (2.21E − 04)−	4.64E − 01 (1.76E − 03)−	4.45E − 03 (1.42E − 04)
WFG4	5.00E − 02 (2.45E − 04)−	5.34E − 02 (8.10E − 04)−	6.03E − 02 (2.39E − 03)−	6.70E − 02 (6.18E − 04)−	7.50E − 02 (4.31E − 03)−	4.81E − 02 (1.64E − 03)
WFG5	6.55E − 02 (2.94E − 03)−	6.55E − 02 (2.10E − 03)−	6.56E − 02 (4.29E − 05)−	6.57E − 02 (1.94E − 04)−	6.56E − 02 (3.02E − 03)−	6.52E − 02 (4.14E − 04)
WFG6	5.13E − 03 (1.29E − 02)+	2.77E − 02 (6.98E − 02)−	6.21E − 02 (9.88E − 02)−	6.44E − 02 (2.15E − 04)−	1.31E − 01 (1.25E − 03)−	1.62E − 02 (5.02E − 02)
WFG7	5.50E − 03 (1.55E − 04)∼	5.60E − 03 (2.94E − 04)∼	5.81E − 03 (5.83E − 05)−	6.40E − 03 (3.49E − 04)−	8.30E − 03 (7.98E − 04)−	5.53E − 03 (2.13E − 05)
WFG8	8.28E − 02 (6.66E − 03)−	8.20E − 02 (1.96E − 03)−	8.34E − 02 (9.81E − 04)−	8.59E − 02 (1.38E − 03)−	8.96E − 02 (3.16E − 03)−	5.77E − 02 (1.60E − 03)
WFG9	1.59E − 02 (1.99E − 02)−	1.63E − 02 (8.29E − 02)−	1.73E − 02 (2.40E − 04)−	1.90E − 02 (1.01E − 04)−	2.11E − 02 (2.64E − 03)−	1.52E − 02 (2.92E − 04)
Total	4/3/21	4/3/21	6/1/21	6/0/22	3/0/25

The better average values of these algorithms for each instance are highlighted in bold italics.