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

Table 1

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 IGD 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	3.43E − 03 (8.00E − 04)−	3.66E − 03 (1.94E − 04)−	1.36E − 03 (1.36E − 05)∼	1.57E − 03 (2.95E − 05)−	1.27E − 03 (2.16E − 05)+	2.41E − 02 (2.52E − 02)−	1.35E − 03 (1.28E − 05)
F2	1.05E − 01 (5.13E − 02)−	1.26E − 02 (1.34E − 03)−	3.47E − 03 (1.84E − 04)−	4.03E − 02 (4.90E − 03)−	6.86E − 02 (7.76E − 03)−	5.21E − 02 (1.47E − 02)−	2.85E − 03 (2.95E − 04)
F3	4.93E − 02 (3.03E − 02)−	1.32E − 02 (2.59E − 03)−	3.07E − 03 (2.21E − 04)−	1.37E − 02 (2.30E − 03)−	2.83E − 02 (5.86E − 03)−	1.38E − 02 (4.10E − 03)−	2.55E − 03 (3.59E − 04)
F4	6.07E − 02 (4.18E − 02)−	1.30E − 02 (1.16E − 03)−	3.16E − 03 (5.22E − 04)−	2.11E − 02 (4.25E − 03)−	2.09E − 02 (5.43E − 03)−	1.00E − 02 (3.71E − 03)−	2.93E − 03 (7.23E − 04)
F5	4.21E − 02 (2.45E − 02)−	1.42E − 02 (1.24E − 03)−	8.47E − 03 (8.86E − 04)−	1.43E − 02 (1.91E − 03)−	2.30E − 02 (5.78E − 03)−	1.57E − 02 (3.71E − 03)−	7.94E − 03 (1.27E − 03)
F6	7.30E − 02 (1.75E − 02)−	8.45E − 02 (2.90E − 03)−	6.26E − 02 (7.65E − 03)+	2.18E − 01 (6.89E − 02)−	2.20E − 01 (3.00E − 02)−	8.69E − 02 (5.60E − 03)−	6.54E − 02 (8.13E − 03)
F7	2.79E − 01 (2.08E − 02)−	2.02E − 01 (4.70E − 02)−	1.42E − 02 (3.11E − 02)+	1.38E + 00 (4.11E − 01)−	3.01E − 01 (4.48E − 02)−	3.11E − 01 (4.50E − 02)−	4.21E − 02 (7.84E − 02)
F8	2.10E − 01 (5.24E − 02)−	8.31E − 02 (1.82E − 02)−	8.32E − 03 (4.24E − 03)+	1.20E − 02 (5.14E − 03)+	7.74E − 02 (8.89E − 03)−	6.47E − 02 (2.17E − 02)−	1.55E − 02 (1.07E − 02)
F9	1.11E − 01 (3.84E − 02)−	1.53E − 02 (2.30E − 03)−	4.87E − 03 (1.17E − 03)−	3.95E − 02 (4.54E − 03)−	6.22E − 02 (1.23E − 02)−	5.49E − 02 (1.57E − 02)−	5.14E − 03 (2.12E − 03)
UF1	1.05E − 01 (2.71E − 02)−	1.27E − 02 (9.03E − 04)−	2.88E − 03 (5.02E − 04)−	3.89E − 02 (2.91E − 03)−	6.51E − 02 (1.16E − 02)−	4.68E − 02 (1.05E − 02)−	2.45E − 03 (1.42E − 03)
UF2	3.95E − 02 (2.14E − 02)−	1.40E − 02 (1.17E − 03)−	7.08E − 03 (1.05E − 03)−	1.30E − 02 (1.40E − 03)−	2.55E − 02 (7.06E − 03)−	1.67E − 02 (5.66E − 03)−	6.68E − 03 (8.37E − 04)
UF3	2.16E − 01 (4.31E − 02)−	8.60E − 02 (1.54E − 02)−	5.41E − 03 (3.30E − 03)+	1.53E − 02 (6.36E − 03)+	8.47E − 02 (1.21E − 02)−	6.62E − 02 (1.74E − 02)−	1.74E − 02 (1.99E − 02)
UF4	5.19E − 02 (3.61E − 03)+	4.86E − 02 (2.91E − 03)+	5.39E − 02 (2.94E − 03)+	9.46E − 02 (9.84E − 03)−	7.68E − 02 (8.00E − 03)−	9.30E − 02 (1.05E − 02)−	6.98E − 02 (5.78E − 03)
UF5	4.41E − 01 (8.29E − 02)−	2.78E − 01 (7.49E − 02)+	3.14E − 01 (5.19E − 02)+	1.04E + 00 (5.06E − 01)−	9.41E − 01 (1.63E − 01)−	7.26E − 01 (1.34E − 01)−	4.08E − 01 (9.50E − 02)
UF6	3.29E − 01 (1.32E − 01)−	1.56E − 01 (4.89E − 02)∼	2.91E − 01 (2.66E − 01)−	4.26E − 01 (3.72E − 02)−	2.28E − 01 (3.57E − 02)−	1.84E − 01 (1.22E − 01)−	1.62E − 01 (2.06E − 01)
UF7	2.53E − 01 (2.27E − 01)−	1.12E − 02 (2.21E − 03)−	4.12E − 03 (3.71E − 04)+	1.77E − 02 (2.62E − 03)−	3.47E − 02 (3.37E − 02)−	4.89E − 02 (1.05E − 01)−	5.13E − 03 (6.70E − 04)
UF8	7.02E − 02 (6.44E − 03)−	8.80E − 02 (3.73E − 03)−	6.50E − 02 (4.84E − 03)−	1.56E − 01 (3.56E − 02)−	1.65E − 01 (3.58E − 03)−	8.85E − 02 (4.38E − 03)−	5.74E − 02 (8.29E − 03)
UF9	1.13E − 01 (5.02E − 02)−	8.34E − 02 (4.49E − 03)−	3.32E − 02 (3.92E − 03)−	1.89E − 01 (5.92E − 02)−	1.56E − 01 (5.27E − 02)−	7.62E − 02 (1.06E − 02)−	2.76E − 02 (4.37E − 03)
UF10	2.79E − 01 (8.23E − 02)+	4.69E − 01 (7.35E − 02)+	1.79E + 00 (2.41E − 01)−	2.49E + 00 (1.16E − 01)−	1.77E − 01 (4.32E − 03)+	1.80E + 00 (3.11E − 01)−	8.03E − 01 (1.71E − 01)
WFG1	7.71E − 01 (5.32E − 02)+	1.06E + 00 (7.62E − 03)+	1.16E + 00 (9.55E − 03)−	1.17E + 00 (9.36E − 03)−	1.44E + 00 (9.80E − 02)−	1.83E + 00 (1.86E − 01)−	1.11E + 00 (1.17E − 02)
WFG2	5.79E − 02 (2.12E − 02)−	1.64E − 02 (5.10E − 04)−	1.57E − 02 (3.63E − 04)∼	6.27E − 03 (1.04E − 03)+	1.71E − 02 (1.78E − 03)−	1.26E − 02 (1.75E − 02)+	1.55E − 02 (3.24E − 04)
WFG3	6.68E − 03 (5.70E − 04)−	7.45E − 03 (5.04E − 04)−	5.05E − 03 (1.71E − 04)−	8.29E − 03 (2.31E − 04)−	1.22E − 02 (1.26E − 03)−	8.55E − 03 (2.38E − 04)−	4.45E − 03 (1.42E − 04)
WFG4	5.71E − 03 (2.45E − 04)+	7.84E − 03 (8.40E − 04)+	5.91E − 02 (2.39E − 03)−	4.54E − 02 (7.98E − 04)∼	3.89E − 02 (4.07E − 03)+	2.51E − 02 (5.94E − 03)+	4.81E − 02 (1.64E − 03)
WFG5	6.34E − 02 (2.94E − 03)+	6.46E − 02 (2.00E − 03)+	6.53E − 02 (4.09E − 05)∼	6.92E − 02 (1.44E − 04)−	6.94E − 02 (2.95E − 03)−	6.81E − 02 (3.23E − 04)−	6.52E − 02 (4.14E − 04)
WFG6	6.73E − 02 (1.29E − 02)−	2.07E − 01 (6.98E − 02)−	5.53E − 02 (9.88E − 02)−	7.71E − 03 (2.35E − 04)+	1.04E − 02 (1.35E − 03)+	6.73E − 03 (3.48E − 04)+	1.62E − 02 (5.02E − 02)
WFG7	6.10E − 03 (1.55E − 04)−	6.65E − 03 (1.94E − 04)−	5.79E − 03 (5.83E − 05)−	7.32E − 03 (3.43E − 04)−	1.11E − 02 (8.98E − 04)−	7.25E − 03 (2.37E − 04)−	5.53E − 03 (2.13E − 05)
WFG8	9.08E − 02 (6.66E − 03)−	6.60E − 02 (4.96E − 03)−	5.94E − 02 (9.81E − 04)−	6.65E − 02 (1.36E − 03)−	8.86E − 02 (3.06E − 03)−	6.49E − 02 (2.04E − 03)−	5.77E − 02 (1.60E − 03)
WFG9	1.72E − 02 (1.99E − 02)−	5.45E − 02 (9.49E − 02)−	1.73E − 02 (2.40E − 04)−	1.51E − 02 (1.83E − 04)∼	1.83E − 02 (2.60E − 03)−	1.61E − 02 (8.66E − 04)−	1.52E − 02 (2.92E − 04)
Total	5/0/23	6/1/21	7/3/18	4/2/22	4/0/24	3/0/25

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