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

Table 3

Performance comparison of MOEA/D-SBX, MOEA/D-DE, MOEA/D-GM, RM-MEDA, IM-MOEA, AMEDA, and MOEA/D-AMG in terms of the average R2 values and standard deviation on F, UF, and WFG problems.

Problems	MOEA/D−SBX	MOEA/D−DE	MOEA/D−GM	RM−MEDA	IM−MOEA	AMEDA	MOEA/D−AMG
F1	1.31E − 01 (1.91E − 04)−	1.32E − 01 (5.73E − 05)−	1.31E − 01 (8.83E − 06)∼	1.31E − 01 (3.00E − 05)−	1.31E − 01 (1.27E − 05)∼	1.31E − 01 (5.01E − 05)−	1.31E − 01 (8.34E − 06)
F2	1.82E − 01 (2.88E − 02)−	1.34E − 01 (5.40E − 04)−	1.31E − 01 (8.11E − 05)−	1.44E − 01 (1.76E − 03)−	1.57E − 01 (7.50E − 03)−	1.49E − 01 (4.50E − 03)−	1.31E − 01 (7.95E − 05)
F3	1.52E − 01 (1.58E − 02)−	1.34E − 01 (5.82E − 04)−	1.31E − 01 (7.06E − 05)−	1.34E − 01 (6.91E − 04)−	1.38E − 01 (2.25E − 03)−	1.35E − 01 (1.57E − 03)−	1.31E − 01 (7.05E − 05)
F4	1.55E − 01 (1.59E − 02)−	1.34E − 01 (4.52E − 04)−	1.31E − 01 (1.17E − 04)−	1.35E − 01 (4.76E − 04)−	1.37E − 01 (3.14E − 03)−	1.33E − 01 (1.45E − 03)−	1.31E − 01 (1.70E − 04)
F5	1.46E − 01 (1.03E − 02)−	1.34E − 01 (4.84E − 04)−	1.32E − 01 (6.57E − 04)−	1.34E − 01 (8.93E − 04)−	1.37E − 01 (3.04E − 03)−	1.35E − 01 (1.33E − 03)−	1.32E − 01 (3.81E − 04)
F6	8.17E − 02 (6.68E − 03)−	8.33E − 02 (6.11E − 04)−	7.91E − 02 (8.59E − 04)+	1.21E − 01 (2.49E − 02)−	1.18E − 01 (1.67E − 02)−	7.95E − 02 (8.90E − 04)−	7.89E − 02 (5.88E − 04)
F7	3.15E − 01 (1.99E − 02)−	2.33E − 01 (4.55E − 02)−	1.36E − 01 (1.47E − 02)+	7.13E − 01 (1.38E − 01)−	3.07E − 01 (2.12E − 02)−	3.51E − 01 (2.65E − 02)−	1.50E − 01 (3.87E − 02)
F8	2.95E − 01 (3.34E − 02)−	1.59E − 01 (5.86E − 03)−	1.33E − 01 (1.54E − 03)+	1.36E − 01 (2.27E − 03)−	1.58E − 01 (3.17E − 03)−	1.60E − 01 (1.33E − 02)−	1.35E − 01 (3.33E − 03)
F9	2.50E − 01 (2.34E − 02)−	2.00E − 01 (7.50E − 04)−	1.97E − 01 (2.04E − 04)∼	2.06E − 01 (6.14E − 04)−	2.18E − 01 (4.81E − 03)−	2.10E − 01 (3.38E − 03)−	1.97E − 01 (3.76E − 04)
UF1	1.81E − 01 (1.78E − 02)−	1.34E − 01 (3.13E − 04)−	1.31E − 01 (7.74E − 05)∼	1.43E − 01 (7.91E − 04)−	1.56E − 01 (9.45E − 03)−	1.48E − 01 (6.88E − 03)−	1.32E − 01 (6.57E − 04)
UF2	1.46E − 01 (9.19E − 03)−	1.34E − 01 (4.29E − 04)−	1.32E − 01 (4.82E − 04)∼	1.34E − 01 (4.08E − 04)−	1.37E − 01 (2.98E − 03)−	1.35E − 01 (2.24E − 03)−	1.32E − 01 (3.71E − 04)
UF3	2.88E − 01 (2.50E − 02)−	1.61E − 01 (4.57E − 03)−	1.32E − 01 (1.21E − 03)+	1.37E − 01 (3.92E − 03)+	1.63E − 01 (7.28E − 03)−	1.63E − 01 (1.15E − 020−	1.38E − 01 (1.02E − 02)
UF4	2.12E − 01 (1.59E − 03)+	2.09E − 01 (9.04E − 04)+	2.10E − 01 (9.10E − 04)+	2.23E − 01 (2.71E − 03)−	2.18E − 01 (1.93E − 03)−	2.23E − 01 (2.19E − 03)−	2.15E − 01 (1.50E − 03)
UF5	4.51E − 01 (6.76E − 02)−	2.63E − 01 (3.21E − 02)+	2.79E − 01 (1.76E − 02)+	5.75E − 01 (2.32E − 01)−	5.69E − 01 (5.63E − 02)−	4.83E − 01 (7.24E − 02)−	3.34E − 01 (6.39E − 02)
UF6	3.42E − 01 (6.58E − 02)−	2.42E − 01 (1.96E − 02)+	3.25E − 01 (1.34E − 01)−	3.16E − 01 (2.19E − 02)−	2.68E − 01 (1.41E − 02)−	2.53E − 01 (6.35E − 02)−	2.47E − 01 (8.10E − 02)
UF7	3.09E − 01 (1.28E − 01)−	1.68E − 01 (6.60E − 04)−	1.66E − 01 (1.98E − 04)∼	1.70E − 01 (9.38E − 04)−	1.77E − 01 (1.24E − 02)−	1.90E − 01 (6.55E − 02)−	1.66E − 01 (3.39E − 04)
UF8	8.01E − 02 (3.55E − 03)−	8.31E − 02 (7.51E − 04)−	7.93E − 02 (7.24E − 04)−	1.09E − 01 (2.20E − 02)−	1.24E − 01 (6.95E − 03)−	7.97E − 02 (4.53E − 04)−	7.83E − 02 (5.07E − 04)
UF9	8.20E − 02 (1.45E − 02)−	6.60E − 02 (1.80E − 03)−	5.85E − 02 (9.50E − 04)−	8.76E − 02 (1.08E − 02)−	9.33E − 02 (1.66E − 02)−	6.46E − 02 (2.95E − 03)−	5.94E − 02 (2.09E − 03)
UF10	1.91E − 01 (5.10E − 02)+	2.16E − 01 (1.29E − 02)+	4.55E − 01 (3.38E − 02)−	6.76E − 01 (2.13E − 02)−	1.28E − 01 (1.09E − 04)+	4.92E − 01 (5.74E − 02)−	3.05E − 01 (2.13E − 02)
WFG1	6.98E − 01 (2.59E − 02)−	8.27E − 01 (4.77E − 03)+	8.81E − 01 (1.30E − 02)−	9.33E − 01 (9.07E − 03)−	1.07E + 00 (4.40E − 02)−	1.20E + 00 (6.37E − 02)−	8.56E − 01 (4.93E − 03)
WFG2	5.24E − 01 (5.00E − 02)−	4.28E − 01 (1.17E − 04)∼	4.28E − 01 (3.25E − 05)∼	4.28E − 01 (9.78E − 05)∼	4.31E − 01 (1.04E − 03)−	4.49E − 01 (4.45E − 02)−	4.28E − 01 (2.66E − 05)
WFG3	4.52E − 01 (2.12E − 04)+	4.52E − 01 (1.57E − 04)−	4.51E − 01 (3.09E − 05)∼	4.53E − 01 (7.76E − 05)−	4.54E − 01 (3.97E − 04)−	4.53E − 01 (1.10E − 04)−	4.51E − 01 (2.29E − 05)
WFG4	5.81E − 01 (8.19E − 05)+	5.82E − 01 (1.75E − 04)+	5.96E − 01 (3.71E − 04)−	5.95E − 01 (4.35E − 04)−	5.91E − 01 (1.60E − 03)+	5.86E − 01 (1.60E − 03)+	5.93E − 01 (5.36E − 04)
WFG5	6.02E − 01 (1.66E − 03)∼	6.03E − 01 (5.86E − 04)+	6.03E − 01 (7.01E − 04)−	6.04E − 01 (7.93E − 04)−	6.02E − 01 (2.42E − 03)−	6.04E − 01 (1.68E − 03)−	6.02E − 01 (9.62E − 04)
WFG6	5.99E − 01 (3.55E − 03)−	6.41E − 01 (2.05E − 02)−	5.96E − 01 (2.91E − 02)−	5.82E − 01 (5.91E − 05)+	5.83E − 01 (5.16E − 04)+	5.82E − 01 (1.22E − 04)+	5.88E − 01 (2.10E − 02)
WFG7	5.82E − 01 (4.11E − 05)−	5.82E − 01 (3.76E − 05)−	5.81E − 01 (2.57E − 05)−	5.82E − 01 (1.06E − 04)−	5.83E − 01 (1.31E − 04)−	5.82E − 01 (8.75E − 05)−	5.81E − 01 (1.21E − 05)
WFG8	6.28E − 01 (8.08E − 04)−	6.26E − 01 (5.35E − 04)−	6.25E − 01 (1.71E − 04)−	6.30E − 01 (1.06E − 03)−	6.31E − 01 (1.03E − 03)−	6.27E − 01 (1.47E − 03)−	6.24E − 01 (6.73E − 05)
WFG9	5.91E − 01 (6.25E − 03)−	6.03E − 01 (3.17E − 02)−	5.91E − 01 (1.94E − 04)−	5.92E − 01 (6.75E − 04)−	5.91E − 01 (7.84E − 04)−	5.91E − 01 (9.08E − 04)−	5.90E − 01 (2.30E − 04)
Total	4/1/23	7/1/20	6/7/15	2/1/25	3/1/24	2/0/26

The best average values obtained by these algorithms for each instance are given in bold and italics.