Table 10 Average and standard deviation of the IGD values obtained by the five algorithms on the WFG test suite with different numbers of objectives.
From: An improved farmland fertility algorithm for many-objective optimization problems
Text suite | M | NSGA-III | MOEA/D | RVEA | ARMOEA | MOIFF |
|---|---|---|---|---|---|---|
WFG1 | 3 | 1.48E−01 (3.61E−03) +  | 2.42E−01 (7.46E−03)− | 1.74E−01 (1.24E−02) +  | 1.48E−01 (2.84E−03) +  | 2.29E−01 (2.92E−02) |
5 | 3.63E−01 (3.59E−03) +  | 7.03E−01 (1.50E−02) +  | 3.67E−01 (4.56E−03) +  | 3.70E−01 (3.14E−03) +  | 7.78E−01 (1.52E−01) | |
8 | 8.78E−01 (2.33E−02) +  | 1.48E + 00 (5.59E−02) +  | 9.79E−−01 (3.64E−02) +  | 8.72E−01 (1.89E−02) +  | 2.76E + 00 (5.26E−01) | |
10 | 9.54E−01 (1.62E−02) +  | 1.69E + 00 (6.97E−02) +  | 1.06E + 00 (2.41E−02) +  | 1.02E + 00 (2.85E−02) +  | 2.61E + 00 (4.60E−01) | |
15 | 2.14E + 00 (4.13E−01) +  | 2.96E + 00 (2.43E−01) +  | 1.77E + 00 (8.15E−02) +  | 1.80E + 00 (4.60E−02) +  | 6.86E + 00 (8.41E−01) | |
WFG2 | 3 | 1.64E−01 (9.43E−04) +  | 2.36E−01 (1.52E−02) =  | 1.81E−01 (4.47E−03) +  | 1.64E−01 (1.13E−03) +  | 2.45E−01 (1.37E−02) |
5 | 3.91E−01 (1.57E−03) +  | 7.82E−01 (1.35E−01)− | 3.81E−01 (6.84E−03) +  | 3.92E−01 (1.98E−03) +  | 5.90E−01 (2.75E−02) | |
8 | 1.03E + 00 (2.03E−01) +  | 1.70E + 00 (1.78E−02)− | 9.89E−01 (3.46E−02) +  | 9.44E−01 (9.27E−03) +  | 1.27E + 00 (8.74E−02) | |
10 | 1.33E + 00 (1.86E−01) +  | 1.83E + 00 (1.24E−02)− | 1.06E + 00 (3.26E−02) +  | 1.01E + 00 (1.59E−02) +  | 1.50E + 00 (8.08E−02) | |
15 | 2.13E + 00 (1.39E−01) +  | 2.55E + 00 (1.84E−02) +  | 1.87E + 00 (6.72E−02) +  | 1.72E + 00 (3.47E−02) +  | 3.66E + 00 (2.36E−01) | |
WFG3 | 3 | 1.02E−01 (1.02E−02) +  | 1.57E−01 (1.12E−03) =  | 2.22E−01 (1.23E−02)− | 1.15E−01 (7.74E−03) +  | 1.53E−01 (1.45E−02) |
5 | 4.37E−01 (4.08E−02)− | 8.03E−01 (3.75E−01)− | 4.38E−01 (9.30E−03)− | 4.49E−01 (3.56E−02)− | 2.75E−01 (5.93E−02) | |
8 | 1.28E + 00 (5.07E−01)− | 3.62E + 00 (1.69E−01)− | 2.11E + 00 (2.57E−01)− | 1.81E + 00 (1.12E−01)− | 7.93E−01 (1.74E−01) | |
10 | 1.94E + 00 (4.25E−01)− | 1.83E + 00 (1.24E−02)− | 3.08E + 00 (3.85E−01)− | 2.45E + 00 (8.92E−02)− | 7.86E−01 (1.12E−01) | |
15 | 3.73E + 00 (1.54E + 00)− | 9.15E + 00 (2.29E−01)− | 6.82E + 00 (1.33E + 00)− | 5.59E + 00 (9.67E−02)− | 1.82E + 00 (2.84E−01) | |
WFG4 | 3 | 2.21E−01 (7.18E−05)− | 2.54E−01 (4.08E−03)− | 2.28E−01 (3.34E−03)− | 2.21E−01 (9.66E−05)− | 5.63E−02 (1.39E−02) |
5 | 9.68E−01 (4.56E−04)− | 1.57E + 00 (3.65E−01)− | 9.67E−01 (5.65E−04)− | 9.67E−01 (3.79E−04)− | 1.41E−01 (1.58E−02) | |
8 | 2.99E + 00 (1.04E−01)− | 6.47E + 00 (3.36E−01)− | 2.97E + 00 (9.52E−03)− | 2.96E + 00 (3.20E−03)− | 6.56E−01 (3.35E−01) | |
10 | 4.55E + 00 (1.77E−02)− | 8.44E + 00 (4.12E−01)− | 4.45E + 00 (2.65E−02)− | 4.55E + 00 (9.26E−03)− | 8.11E−01 (2.18E−02) | |
15 | 9.65E + 00 (3.97E−01)− | 1.56E + 01 (2.72E−01)− | 9.21E + 00 (9.97E−02)− | 9.38E + 00 (3.19E−02)− | 1.78E + 00 (1.38E + 00) | |
WFG5 | 3 | 2.30E−01 (3.88E−05)− | 2.48E−01 (2.37E−03)− | 2.31E−01 (3.77E−04)− | 2.30E−01 (4.74E−05)− | 1.37E−01 (3.75E−02) |
5 | 9.59E−01 (1.98E−04)− | 1.48E + 00 (3.59E−01)− | 9.59E−01 (1.92E−04)− | 9.59E−01 (1.68E−04)− | 1.88E−01 (3.14E−02) | |
8 | 2.94E + 00 (1.39E−03)− | 6.02E + 00 (2.11E−01)− | 2.95E + 00 (8.43E−03)− | 2.94E + 00 (1.99E−03)− | 5.93E−01 (6.54E−02) | |
10 | 4.53E + 00 (3.33E−03)− | 7.92E + 00 (2.76E−01)− | 4.42E + 00 (2.52E−02)− | 4.53E + 00 (7.72E−03)− | 8.30E−01 (4.33E−02) | |
15 | 9.31E + 00 (1.39E−01)− | 1.51E + 01 (1.21E−01)− | 9.16E + 00 (5.63E−02)− | 9.34E + 00 (5.68E−02)− | 3.02E + 00 (1.51E + 00) | |
WFG6 | 3 | 2.34E−01 (9.47E−03)− | 2.78E−01 (1.52E−02)− | 2.46E−01 (1.17E−02)− | 2.40E−01 (8.94E−03)− | 1.51E−01 (4.70E−02) |
5 | 9.60E−01 (7.65E−04)− | 1.57E + 00 (3.13E−01)− | 9.60E−01 (1.12E−03)− | 9.59E−01 (6.72E−04)− | 2.32E−01 (3.50E−02) | |
8 | 2.95E + 00 (4.20E−03)− | 6.93E + 00 (7.53E−02)− | 2.97E + 00 (1.12E−02)− | 2.95E + 00 (3.35E−03)− | 9.24E−01 (3.34E−01) | |
10 | 4.64E + 00 (4.87E−01)− | 9.16E + 00 (1.43E−01)− | 4.39E + 00 (2.96E−02)− | 4.53E + 00 (9.26E−03)− | 8.95E−01 (2.92E−02) | |
15 | 1.18E + 01 (8.86E−01)− | 1.61E + 01 (2.05E−01)− | 9.50E + 00 (3.69E−01)− | 9.41E + 00 (3.57E−02)− | 3.10E + 00 (1.30E + 00) | |
WFG7 | 3 | 2.21E−01 (4.27E−04)− | 2.91E−01 (1.76E−02)− | 2.24E−01 (1.58E−03)− | 2.21E−01 (1.48E−04)− | 2.67E−02 (9.61E−03) |
5 | 9.68E−01 (3.81E−04)− | 1.69E + 00 (1.49E−01)− | 9.69E−01 (4.68E−04)− | 9.69E−01 (6.97E−04)− | 8.72E−02 (1.52E−02) | |
8 | 2.97E + 00 (5.59E−03)− | 7.00E + 00 (1.03E−01)− | 2.99E + 00 (1.68E−02)− | 2.96E + 00 (4.89E−03)− | 7.23E−01 (2.39E−02) | |
10 | 4.61E + 00 (2.55E−01)− | 9.08E + 00 (2.58E−01)− | 4.45E + 00 (2.84E−02)− | 4.52E + 00 (1.94E−02)− | 8.64E−01 (1.56E−02) | |
15 | 9.44E + 00 (3.25E−01)− | 1.63E + 01 (8.25E−02)− | 9.33E + 00 (6.49E−02)− | 9.39E + 00 (4.26E−02)− | 3.16E + 00 (7.08E−01) | |
WFG8 | 3 | 2.82E−01 (2.57E−03)− | 3.06E−01 (5.35E−03)− | 2.92E−01 (3.19E−03)− | 2.73E−01 (1.79E−03) =  | 2.69E−01 (1.20E−02) |
5 | 9.95E−01 (6.40E−03)− | 1.33E + 00 (2.75E−01)− | 9.86E−01 (8.62E−04)− | 9.81E−01 (1.35E−03)− | 6.05E−01 (4.21E−02) | |
8 | 3.52E + 00 (4.10E−01)− | 6.32E + 00 (2.43E−01)− | 3.05E + 00 (2.54E−02)− | 3.02E + 00 (3.03E−02)− | 2.16E + 00 (4.80E−01) | |
10 | 4.94E + 00 (3.78E−01)− | 8.62E + 00 (2.94E−01)− | 4.40E + 00 (6.03E−02)− | 4.63E + 00 (4.38E−02)− | 3.08E + 00 (8.23E−01) | |
15 | 1.11E + 01 (7.78E−01)− | 1.46E + 01 (1.35E + 00)− | 9.37E + 00 (4.82E−01)− | 9.39E + 00 (4.23E−02)− | 6.16E + 00 (7.03E−01) | |
WFG9 | 3 | 2.23E−01 (1.12E−03)− | 2.68E−01 (2.91E−02)− | 2.24E−01 (2.05E−03)− | 2.25E−01 (2.17E−02)− | 9.18E−02 (5.04E−02) |
5 | 9.39E−01 (3.29E−03)− | 1.39E + 00 (1.41E−01)− | 9.48E−01 (1.26E−03)− | 9.47E−01 (2.11E−03)− | 3.23E−01 (3.64E−02) | |
8 | 2.93E + 00 (1.01E−02)− | 6.41E + 00 (1.62E−01)− | 2.95E + 00 (1.89E−02)− | 2.93E + 00 (5.57E−03)− | 1.03E + 00 (1.27E−01) | |
10 | 4.47E + 00 (1.23E−01)− | 8.61E + 00 (6.55E−01)− | 4.32E + 00 (3.44E−02)− | 4.50E + 00 (1.38E−02)− | 1.58E + 00 (4.21E−01) | |
15 | 9.11E + 00 (3.59E−01)− | 1.46E + 01 (2.03E + 00)− | 9.17E + 00 (8.39E−02)− | 9.12E + 00 (3.88E−02)− | 4.36E + 00 (3.85E−01) | |
+/=/− |  | 11/0/34 | 5/2/38 | 10/0/35 | 11/1/33 |  |
