Complexity

Research Article

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

Table 2

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 HV values and standard deviation on F, UF, and WFG problems.
ProblemsMOEA/D-SBXMOEA/D-DEMOEA/D-GMRM-MEDAIM-MOEAAMEDAMOEA/D-AMG
F18.70E − 01 (1.54E − 03)−8.71E − 01 (2.70E − 04)−8.74E − 01 (3.36E − 05)−8.74E − 01 (7.14E − 05)−8.75E − 01 (4.73E − 05)∼8.71E − 01 (2.26E − 04)−8.75E − 01 (3.88E − 05)
F27.20E − 01 (5.49E − 02)−8.54E − 01 (2.49E − 03)−8.66E − 01 (5.73E − 04)−7.98E − 01 (8.48E − 03)−7.60E − 01 (1.87E − 02)−7.88E − 01 (2.12E − 02)−8.70E − 01 (7.14E − 04)
F38.15E − 01 (3.06E − 02)−8.55E − 01 (3.68E − 03)−8.70E − 01 (4.58E − 04)∼8.55E − 01 (2.22E − 03)−8.39E − 01 (5.94E − 03)−8.53E − 01 (5.18E − 03)−8.72E − 01 (6.95E − 04)
F48.11E − 01 (3.32E − 02)−8.55E − 01 (2.17E − 03)−8.65E − 01 (1.28E − 03)−8.43E − 01 (5.58E − 03)−8.48E − 01 (7.24E − 03)−8.60E − 01 (3.73E − 03)−8.71E − 01 (1.91E − 03)
F58.29E − 01 (2.00E − 02)−8.53E − 01 (2.22E − 03)−8.60E − 01 (2.36E − 03)−8.54E − 01 (2.85E − 03)−8.44E − 01 (7.85E − 03)−8.51E − 01 (3.32E − 03)−8.63E − 01 (1.93E − 03)
F66.39E − 01 (2.89E − 02)−5.93E − 01 (1.23E − 02)−6.60E − 01 (1.19E − 02)+4.37E − 01 (6.99E − 02)−4.80E − 01 (3.16E − 02)−6.07E − 01 (1.37E − 02)−6.56E − 01 (4.48E − 03)
F74.66E − 01 (3.62E − 02)−5.91E − 01 (6.00E − 02)−8.56E − 01 (4.41E − 02)+1.99E − 02 (4.58E − 02)−3.73E − 01 (5.14E − 02)−3.60E − 01 (5.04E − 02)−8.19E − 01 (1.04E − 01)
F85.00E − 01 (5.69E − 02)−7.31E − 01 (3.17E − 02)−8.63E − 01 (6.60E − 03)+8.39E − 01 (1.30E − 02)−7.48E − 01 (1.28E − 02)−7.47E − 01 (3.07E − 02)−8.51E − 01 (1.69E − 02)
F93.78E − 01 (5.67E − 02)−5.14E − 01 (4.36E − 03)−5.31E − 01 (1.62E − 03)−4.69E − 01 (4.37E − 03)−4.31E − 01 (1.93E − 02)−4.55E − 01 (2.16E − 02)−5.35E − 01 (2.86E − 03)
UF17.09E − 01 (3.82E − 02)−8.54E − 01 (1.80E − 03)−8.61E − 01 (7.80E − 04)−8.01E − 01 (4.86E − 03)−7.64E − 01 (2.13E − 02)−7.94E − 01 (2.29E − 02)−8.69E − 01 (2.66E − 03)
UF28.29E − 01 (1.91E − 02)−8.54E − 01 (1.61E − 03)−8.65E − 01 (2.05E − 03)−8.56E − 01 (1.65E − 03)−8.42E − 01 (6.25E − 03)−8.52E − 01 (4.95E − 03)−8.70E − 01 (1.50E − 03)
UF35.12E − 01 (4.39E − 02)−7.25E − 01 (2.26E − 02)−8.67E − 01 (5.32E − 03)+8.29E − 01 (1.75E − 02)−7.34E − 01 (1.71E − 02)−7.42E − 01 (2.49E − 02)−8.50E − 01 (2.77E − 02)
UF44.45E − 01 (4.52E − 03)+4.60E − 01 (5.13E − 03)+4.49E − 01 (4.95E − 03)+3.75E − 01 (1.58E − 02)−4.06E − 01 (1.12E − 02)−3.79E − 01 (1.25E − 02)−4.24E − 01 (8.22E − 03)
UF51.71E − 01 (6.28E − 02)+2.07E − 01 (1.02E − 01)+1.30E − 01 (5.09E − 02)+9.14E − 03 (2.89E − 02)−4.72E − 03 (8.86E − 03)−1.97E − 02 (4.83E − 02)−9.30E − 02 (4.48E − 02)
UF63.28E − 01 (4.60E − 02)−3.18E − 01 (9.71E − 02)−2.73E − 01 (1.22E − 01)−1.15E − 01 (2.08E − 02)−3.87E − 01 (3.80E − 02)+4.28E − 01 (6.99E − 02)+3.63E − 01 (9.42E − 02)
UF74.43E − 01 (2.10E − 01)−6.88E − 01 (4.63E − 03)−7.01E − 01 (1.31E − 03)+6.77E − 01 (4.88E − 03)−6.51E − 01 (4.81E − 02)−6.48E − 01 (1.07E − 01)−6.99E − 01 (1.93E − 03)
UF86.42E − 01 (1.91E − 02)−5.89E − 01 (1.69E − 02)−6.61E − 01 (9.41E − 03)−4.49E − 01 (5.22E − 02)−4.68E − 01 (5.44E − 03)−6.03E − 01 (7.93E − 03)−6.66E − 01 (1.36E − 02)
UF99.00E − 01 (7.35E − 02)−9.51E − 01 (1.05E − 02)−1.04E + 00 (8.35E − 03)−6.97E − 01 (1.27E − 01)−7.74E − 01 (8.95E − 02)−9.60E − 01 (2.11E − 02)−1.04E + 00 (1.29E − 02)
UF102.49E − 01 (9.28E − 02)+5.23E − 02 (2.76E − 02)+0.00E + 00 (0.00E + 00)−0.00E + 00 (0.00E + 00)−4.62E − 01 (8.57E − 04)+0.00E + 00 (0.00E + 00)−1.13E − 02 (2.17E − 02)
WFG12.08E + 00 (1.66E − 01)+1.29E + 00 (1.90E − 02)+1.04E + 00 (2.63E − 02)−1.01E + 00 (2.45E − 02)−4.57E − 01 (1.89E − 01)−7.22E − 02 (1.53E − 01)−1.12E + 00 (2.50E − 02)
WFG24.41E + 00 (1.47E − 02)−4.45E + 00 (2.17E − 03)−4.46E + 00 (6.76E − 04)∼4.45E + 00 (1.33E − 03)∼4.41E + 00 (1.01E − 02)−4.45E + 00 (1.65E − 02)∼4.46E + 00 (8.70E − 04)
WFG33.96E + 00 (2.94E − 03)−3.96E + 00 (3.05E − 03)−3.97E + 00 (6.33E − 04)−3.96E + 00 (1.01E − 03)−3.93E + 00 (5.72E − 03)−3.95E + 00 (1.09E − 03)−3.98E + 00 (5.80E − 04)
WFG41.70E + 00 (1.32E − 03)+1.69E + 00 (3.06E − 03)+1.46E + 00 (7.99E − 03)−1.50E + 00 (4.23E − 03)∼1.54E + 00 (2.10E − 02)+1.62E + 00 (2.61E − 02)+1.50E + 00 (5.60E − 03)
WFG51.45E + 00 (1.07E − 02)∼1.45E + 00 (7.25E − 03)∼1.45E + 00 (3.20E − 04)∼1.43E + 00 (7.76E − 04)−1.43E + 00 (1.24E − 02)−1.44E + 00 (9.12E − 04)∼1.45E + 00 (2.75E − 04)
WFG61.43E + 00 (4.82E − 02)−9.99E − 01 (2.36E − 01)−1.52E + 00 (3.40E − 01)−1.68E + 00 (9.49E − 04)+1.67E + 00 (7.07E − 03)+1.69E + 00 (1.52E − 03)+1.62E + 00 (2.45E − 01)
WFG71.70E + 00 (6.59E − 04)∼1.69E + 00 (1.35E − 03)∼1.70E + 00 (4.60E − 04)−1.68E + 00 (1.44E − 03)−1.67E + 00 (2.47E − 03)−1.69E + 00 (8.86E − 04)−1.70E + 00 (2.40E − 04)
WFG81.34E + 00 (1.38E − 02)−1.39E + 00 (1.25E − 02)−1.41E + 00 (2.07E − 03)−1.39E + 00 (3.16E − 03)−1.34E + 00 (6.77E − 03)−1.39E + 00 (4.88E − 03)−1.41E + 00 (2.98E − 03)
WFG91.54E + 00 (7.89E − 02)−1.52E + 00 (3.15E − 01)−1.63E + 00 (1.30E − 03)−1.64E + 00 (8.96E − 04)∼1.63E + 00 (1.12E − 02)−1.64E + 00 (3.86E − 03)∼1.64E + 00 (1.46E − 03)
Total5/2/215/2/217/3/181/3/244/1/233/3/22 
The best average values obtained by these algorithms for each instance are given in bold and italics.