Evolutionary Search with Multiple Utopian Reference Points in Decomposition-Based Multiobjective Optimization
Table 3
Mean and standard deviation of IH values obtained by four classical algorithms and MOEA/D-MUP.
Problem
NSGA-II
SMS-EMOA
MOEA/D-DRA
MOEA/D-DRA-UT
MOEA/D-MUP
F1
4.70E-01 (1.59E-03)-
4.65E-01 (3.24E-03)-
4.80E-01 (3.07E-03)-
4.93E-01 (1.02E-03)-
4.95E-01 (2.93E-04)
F2
4.66E-01 (1.86E-03)-
4.58E-01 (1.63E-03)-
4.71E-01 (2.75E-03)-
4.93E-01 (7.69E-04)-
4.95E-01 (2.81E-04)
F3
4.50E-01 (3.39E-03)-
4.39E-01 (4.60E-03)-
4.64E-01 (7.66E-03)-
4.91E-01 (2.66E-03)-
4.93E-01 (4.57E-04)
F4
4.45E-01 (2.94E-03)-
4.31E-01 (2.38E-03)-
4.57E-01 (6.22E-03)-
4.91E-01 (2.38E-03)-
4.93E-01 (9.87E-04)
F5
4.31E-01 (1.22E-02)-
4.14E-01 (1.21E-02)-
4.42E-01 (2.13E-02)-
4.91E-01 (1.28E-03)-
4.92E-01 (7.98E-04)
F6
4.19E-01 (8.72E-03)-
3.98E-01 (6.95E-03)-
4.36E-01 (1.99E-02)-
4.89E-01 (2.22E-03)=
4.90E-01 (1.42E-03)
F7
3.27E-01 (4.10E-03)-
3.29E-01 (2.78E-03)-
3.35E-01 (1.07E-02)-
3.64E-01 (1.10E-02)+
3.56E-01 (4.01E-03)
F8
7.93E-01 (2.26E-03)-
7.81E-01 (2.23E-03)-
7.83E-01 (1.60E-02)-
8.25E-01 (2.79E-03)=
8.24E-01 (3.87E-03)
F9
5.72E-01 (3.92E-02)-
6.16E-01 (2.45E-02)-
6.76E-01 (2.27E-02)-
7.40E-01 (2.53E-02)=
7.38E-01 (7.61E-03)
F10
5.60E-01 (3.54E-02)-
5.97E-01 (1.08E-02)-
6.70E-01 (2.31E-02)-
6.99E-01 (1.08E-02)-
7.26E-01 (6.19E-03)
F11
5.21E-01 (5.42E-02)-
5.71E-01 (3.13E-02)-
6.34E-01 (3.49E-02)-
6.92E-01 (2.56E-02)-
7.00E-01 (8.77E-03)
F12
5.04E-01 (4.19E-02)-
5.57E-01 (1.89E-02)-
6.22E-01 (2.51E-02)-
6.73E-01 (1.42E-02)-
6.89E-01 (8.42E-03)
F13
4.54E-01 (7.91E-02)-
5.37E-01 (6.65E-02)-
5.80E-01 (5.73E-02)-
6.48E-01 (1.63E-02)=
6.50E-01 (7.62E-03)
F14
4.58E-01 (3.27E-02)-
5.02E-01 (3.22E-02)-
5.69E-01 (2.89E-02)-
6.36E-01 (1.54E-02)=
6.39E-01 (9.76E-03)
F15
4.13E-01 (1.40E-02)-
4.17E-01 (3.52E-03)-
4.75E-01 (1.57E-02)-
4.91E-01 (5.18E-03)=
4.90E-01 (4.29E-03)
F16
9.18E-01 (7.53E-03)-
4.65E-01 (3.24E-03)-
9.27E-01 (9.17E-03)-
9.51E-01 (3.97E-03)-
9.59E-01 (1.51E-03)
+/-/=
0/16/0
0/16/0
0/16/0
1/9/6
∖
According to Wilcoxon’s rank sum test with a significant level , +, − and = indicate that the corresponding MOEA is significantly better than, worse than or similar to MOEA/D-MUP.
