A Supplementary File for”Review and Analysis of Three Components of the
Differential Evolution Mutation Operator in
MOEA/D-DE”
Ryoji Tanabe and Hisao Ishibuchi

1Table S.1: Comparison of MOEA/D-DE with the three index selection methods on the 16 problems(the four DTLZ and nine WFG problems) with M ∈ {2, 3, 4, 5}. The current/1 mutation strategy
and the replacement method were used in MOEA/D-DE. The means and standard deviations of
the hypervolume indicator values of 51 runs are shown (higher is better).
(a) M = 2

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

4.75e-01 (8.62e-02)
4.22e-01 (7.10e-06)
2.05e-01 (1.08e-01)
4.22e-01 (1.49e-05)
4.22e-01 (7.10e-06)
4.16e-01 (5.85e-03)
3.53e-01 (1.20e-01)
3.69e-01 (7.11e-02)

4.88e-01 (8.69e-02)
4.22e-01 (6.15e-06)
2.10e-01 (1.27e-01)

4.94e-01 (8.70e-02)

7.64e-01 (5.69e-04)

7.05e-01 (3.88e-04)
3.94e-01 (2.86e-03)
3.74e-01 (4.25e-04)

4.22e-01 (1.03e-05)

4.22e-01 (1.19e-05)
4.22e-01 (6.91e-06)

4.17e-01 (6.69e-03)

4.16e-01 (9.67e-03)

3.48e-01 (1.21e-01)
3.89e-01 (7.31e-02)
7.64e-01 (8.12e-04)
7.05e-01 (4.58e-04)

3.67e-01 (1.18e-01)

3.99e-01 (2.68e-03)
3.75e-01 (6.73e-04)

3.69e-01 (2.93e-03)

3.46e-01 (1.87e-02)

3.63e-01 (3.10e-02)

4.21e-01 (1.47e-04)

2.25e-01 (9.89e-02)

4.22e-01 (6.15e-06)

3.51e-01 (1.84e-02)
4.21e-01 (1.30e-04)
3.68e-01 (3.17e-03)

3.52e-01 (2.00e-02)

4.22e-01 (6.91e-06)

4.29e-01 (7.01e-02)

7.64e-01 (7.89e-04)
7.05e-01 (4.54e-04)

3.95e-01 (2.68e-03)
3.75e-01 (3.58e-04)
3.49e-01 (1.57e-02)
4.21e-01 (1.52e-04)

3.68e-01 (3.65e-03)
3.53e-01 (2.64e-02)(b) M = 3

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

9.03e-01 (1.32e-01)
7.38e-01 (1.66e-03)

9.06e-01 (1.19e-01)
7.39e-01 (1.48e-03)

9.05e-01 (1.64e-01)
7.39e-01 (1.55e-03)
2.71e-01 (2.41e-01)
7.32e-01 (3.32e-02)
2.65e-01 (3.32e-05)
2.54e-01 (7.59e-03)
3.11e-01 (3.57e-02)
4.34e-01 (7.82e-03)
1.21e+00 (7.24e-03)

3.13e-01 (2.37e-01)
7.37e-01 (2.58e-02)

2.65e-01 (3.89e-05)
2.55e-01 (5.93e-03)
3.08e-01 (3.90e-02)
4.31e-01 (5.02e-03)
1.21e+00 (5.69e-03)

8.41e-01 (3.98e-03)
6.48e-01 (7.41e-03)
6.64e-01 (2.49e-03)
6.44e-01 (1.75e-02)
7.04e-01 (4.96e-03)
5.79e-01 (1.23e-02)
6.19e-01 (7.39e-03)

2.67e-01 (2.37e-01)
7.27e-01 (3.84e-02)
2.65e-01 (2.10e-05)
2.57e-01 (6.60e-03)
3.18e-01 (4.26e-02)
4.38e-01 (1.05e-02)

1.20e+00 (6.19e-03)
8.41e-01 (6.36e-03)
6.61e-01 (7.31e-03)
6.65e-01 (2.38e-03)

8.42e-01 (3.64e-03)

6.50e-01 (6.75e-03)
6.65e-01 (2.96e-03)

6.45e-01 (1.58e-02)

6.47e-01 (1.74e-02)

7.13e-01 (4.83e-03)
5.84e-01 (1.20e-02)

7.06e-01 (5.99e-03)
5.75e-01 (1.03e-02)

6.17e-01 (7.11e-03)

6.19e-01 (9.59e-03)2(c) M = 4

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

1.09e+00 (3.90e-01)

1.15e+00 (2.69e-01)
9.32e-01 (3.01e-03)
3.19e-01 (3.43e-01)
9.27e-01 (2.72e-02)

1.15e+00 (3.17e-01)

2.10e-01 (1.03e-04)

2.10e-01 (1.15e-04)
2.03e-01 (1.29e-02)
3.49e-01 (1.18e-02)

9.33e-01 (2.33e-03)

3.58e-01 (3.38e-01)
9.27e-01 (2.55e-02)
2.10e-01 (1.30e-04)
2.05e-01 (8.50e-03)

2.03e-01 (7.53e-03)

3.49e-01 (1.06e-02)
4.80e-01 (2.71e-02)

3.50e-01 (1.32e-02)

1.34e+00 (1.55e-02)
9.24e-01 (1.30e-02)

7.55e-01 (2.15e-02)
7.82e-01 (1.47e-02)
8.09e-01 (6.59e-03)
8.16e-01 (2.15e-02)
6.23e-01 (2.17e-02)
7.34e-01 (1.18e-02)

4.86e-01 (2.38e-02)
1.33e+00 (1.75e-02)
9.18e-01 (1.58e-02)

9.32e-01 (3.04e-03)
4.61e-01 (3.75e-01)
9.28e-01 (2.09e-02)

4.88e-01 (2.59e-02)

7.65e-01 (1.69e-02)

1.34e+00 (1.80e-02)
9.19e-01 (1.39e-02)
7.56e-01 (1.74e-02)

7.83e-01 (1.35e-02)

7.85e-01 (1.79e-02)

8.13e-01 (1.09e-02)

8.11e-01 (5.96e-03)
8.11e-01 (2.18e-02)
6.22e-01 (2.42e-02)
7.31e-01 (1.04e-02)

8.22e-01 (2.22e-02)
6.25e-01 (2.19e-02)

7.30e-01 (9.96e-03)(d) M = 5

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

4.70e-01 (4.85e-01)
1.14e+00 (2.25e-03)
5.84e-02 (1.60e-01)
1.08e+00 (2.85e-02)
1.56e-01 (1.66e-04)
1.48e-01 (3.81e-03)
3.07e-01 (3.28e-02)
4.81e-01 (5.33e-02)
1.47e+00 (3.60e-02)

7.37e-01 (5.57e-01)

7.38e-01 (5.86e-01)

1.14e+00 (1.61e-03)

1.14e+00 (1.89e-03)

9.62e-01 (1.42e-02)

6.56e-01 (5.45e-02)
7.07e-01 (4.78e-02)
9.81e-01 (3.37e-02)
7.12e-01 (6.00e-02)

6.58e-02 (1.82e-01)

8.93e-02 (2.14e-01)

1.09e+00 (3.12e-02)

1.08e+00 (3.28e-02)
1.56e-01 (2.41e-04)
1.48e-01 (3.96e-03)
3.10e-01 (3.37e-02)
5.23e-01 (7.45e-02)

1.56e-01 (4.15e-03)
1.48e-01 (4.79e-03)
3.23e-01 (2.78e-02)
5.52e-01 (8.39e-02)

1.46e+00 (3.55e-02)
9.59e-01 (1.69e-02)
6.69e-01 (5.16e-02)

7.01e-01 (4.43e-02)
9.89e-01 (7.93e-03)
6.95e-01 (5.40e-02)

4.74e-01 (2.82e-02)

4.80e-01 (2.97e-02)

6.01e-01 (5.35e-02)

5.99e-01 (5.07e-02)

31.47e+00 (2.57e-02)

9.57e-01 (1.63e-02)
6.53e-01 (4.38e-02)
7.12e-01 (4.41e-02)
9.89e-01 (1.97e-02)

6.99e-01 (5.10e-02)
4.78e-01 (3.00e-02)
5.89e-01 (5.05e-02)

Table S.2: Comparison of MOEA/D-DE with the three index selection methods on the 16 problems(the four DTLZ and nine WFG problems) with M ∈ {2, 3, 4, 5}. The rand/1 mutation strategy
and the replacement method were used in MOEA/D-DE. The means and standard deviation of thehypervolume indicator values of 51 runs are shown (higher is better).
(a) M = 2

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

4.45e-01 (1.13e-01)
4.22e-01 (8.97e-06)

4.66e-01 (1.36e-01)
4.22e-01 (7.71e-06)

2.04e-01 (1.01e-01)

1.89e-01 (1.11e-01)

4.22e-01 (1.70e-05)
4.22e-01 (8.97e-06)

4.22e-01 (7.71e-06)

4.08e-01 (1.27e-01)
4.22e-01 (8.44e-06)
1.76e-01 (1.08e-01)
4.22e-01 (1.74e-05)
4.22e-01 (8.44e-06)
4.14e-01 (9.87e-03)

4.15e-01 (7.98e-03)

3.62e-01 (1.19e-01)
3.38e-01 (6.43e-02)

4.22e-01 (1.16e-05)

4.12e-01 (7.84e-03)
3.58e-01 (1.20e-01)

3.90e-01 (1.10e-01)

4.21e-01 (1.83e-04)

3.55e-01 (6.09e-02)
7.64e-01 (8.21e-04)
7.04e-01 (4.67e-04)
3.92e-01 (3.25e-03)
3.74e-01 (4.72e-04)
3.48e-01 (1.41e-02)
4.21e-01 (1.62e-04)

3.64e-01 (3.31e-03)

3.68e-01 (2.72e-03)

3.55e-01 (2.72e-02)

3.52e-01 (2.43e-02)

3.89e-01 (7.48e-02)

7.64e-01 (6.32e-04)

7.63e-01 (3.34e-03)

7.04e-01 (4.13e-04)
3.92e-01 (3.19e-03)

7.05e-01 (5.02e-04)

3.75e-01 (6.78e-04)

3.74e-01 (5.44e-04)

3.47e-01 (1.21e-02)
4.21e-01 (1.68e-04)
3.67e-01 (2.66e-03)
3.44e-01 (1.67e-02)

3.58e-01 (2.51e-02)

3.98e-01 (2.91e-03)(b) M = 3

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

8.10e-01 (2.73e-01)
7.39e-01 (1.68e-03)

8.96e-01 (1.26e-01)

2.73e-01 (2.19e-01)

2.69e-01 (2.57e-01)
7.32e-01 (3.33e-02)

7.41e-01 (1.87e-02)

4.34e-01 (6.03e-03)

8.57e-01 (1.79e-01)
7.39e-01 (1.39e-03)
2.27e-01 (2.40e-01)
7.37e-01 (2.57e-02)
2.65e-01 (4.88e-05)
2.56e-01 (7.13e-03)
2.94e-01 (3.40e-02)
4.28e-01 (3.71e-03)

1.20e+00 (6.57e-03)

1.20e+00 (5.95e-03)

8.40e-01 (4.66e-03)

7.03e-01 (6.95e-03)

8.40e-01 (3.48e-03)
6.46e-01 (6.33e-03)
6.64e-01 (2.80e-03)
6.45e-01 (1.82e-02)
6.95e-01 (5.79e-03)

5.74e-01 (1.32e-02)

5.76e-01 (1.01e-02)

6.19e-01 (7.96e-03)

6.18e-01 (6.45e-03)

7.39e-01 (1.34e-03)

2.65e-01 (5.56e-05)
2.57e-01 (6.18e-03)

2.65e-01 (2.62e-05)

2.95e-01 (3.16e-02)

2.90e-01 (3.11e-02)

4.28e-01 (3.13e-03)
1.20e+00 (6.32e-03)
8.39e-01 (3.96e-03)
6.45e-01 (6.70e-03)
6.64e-01 (2.46e-03)
6.46e-01 (1.76e-02)

6.98e-01 (6.47e-03)
5.74e-01 (9.94e-03)
6.18e-01 (8.41e-03)

2.57e-01 (5.92e-03)

6.55e-01 (7.76e-03)

6.63e-01 (2.32e-03)
6.44e-01 (1.49e-02)4

(c) M = 4

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

1.08e+00 (3.11e-01)

1.21e+00 (2.02e-01)

9.31e-01 (2.47e-03)

9.31e-01 (2.60e-03)
3.10e-01 (3.40e-01)
9.27e-01 (2.71e-02)

1.14e+00 (2.38e-01)
9.31e-01 (2.59e-03)

3.43e-01 (3.56e-01)
9.29e-01 (2.52e-02)

2.10e-01 (1.00e-04)
2.05e-01 (5.70e-03)
3.45e-01 (1.24e-02)
4.60e-01 (1.27e-02)
1.34e+00 (1.43e-02)
9.15e-01 (1.29e-02)
7.50e-01 (2.06e-02)
7.79e-01 (1.29e-02)

8.09e-01 (6.85e-03)
7.94e-01 (2.08e-02)
6.12e-01 (2.08e-02)
7.30e-01 (1.15e-02)

2.10e-01 (6.93e-05)

2.05e-01 (6.26e-03)
3.44e-01 (1.69e-02)

4.32e-01 (3.68e-01)

9.29e-01 (2.46e-02)
2.10e-01 (9.78e-05)
2.05e-01 (5.26e-03)
3.45e-01 (1.48e-02)

4.64e-01 (1.81e-02)

4.58e-01 (1.34e-02)

1.32e+00 (1.60e-02)

1.34e+00 (1.41e-02)

9.18e-01 (1.39e-02)

9.14e-01 (1.67e-02)
7.51e-01 (1.87e-02)
7.79e-01 (1.57e-02)

7.54e-01 (1.75e-02)

7.73e-01 (1.34e-02)
8.09e-01 (6.13e-03)

8.10e-01 (5.27e-03)

6.16e-01 (2.06e-02)

7.94e-01 (2.29e-02)
6.10e-01 (2.37e-02)

7.27e-01 (9.65e-03)

7.33e-01 (1.10e-02)

7.99e-01 (2.21e-02)(d) M = 5

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

WOR
Mean (Std.)WR
Mean (Std.)

WPR
Mean (Std.)

6.89e-01 (4.79e-01)

6.68e-01 (5.26e-01)

1.14e+00 (1.99e-03)
9.08e-02 (1.93e-01)

1.14e+00 (2.00e-03)

6.21e-01 (5.59e-01)
1.14e+00 (2.26e-03)

3.24e-01 (3.26e-02)

8.16e-02 (1.64e-01)
1.08e+00 (2.92e-02)
1.57e-01 (5.92e-03)
1.48e-01 (6.38e-03)
3.22e-01 (3.64e-02)

4.39e-01 (5.28e-02)

4.99e-01 (8.14e-02)

1.47e+00 (3.09e-02)

1.45e+00 (3.27e-02)
9.62e-01 (1.40e-02)

1.09e+00 (2.99e-02)

1.56e-01 (1.30e-04)
1.51e-01 (9.68e-03)

9.63e-01 (1.47e-02)
6.40e-01 (4.50e-02)
6.90e-01 (4.22e-02)
9.87e-01 (2.00e-02)
6.83e-01 (5.52e-02)
4.95e-01 (2.33e-02)

5.92e-01 (4.65e-02)

6.45e-01 (4.11e-02)
7.06e-01 (6.15e-02)

9.84e-01 (2.88e-02)
6.76e-01 (5.07e-02)
4.87e-01 (2.45e-02)
5.89e-01 (4.83e-02)

59.77e-02 (2.10e-01)

1.08e+00 (3.65e-02)
1.57e-01 (5.94e-03)

1.49e-01 (8.62e-03)
3.08e-01 (3.20e-02)
4.33e-01 (4.89e-02)
1.47e+00 (2.37e-02)
9.64e-01 (1.64e-02)

6.44e-01 (4.42e-02)
7.03e-01 (4.61e-02)
9.82e-01 (2.83e-02)
6.82e-01 (4.83e-02)
4.84e-01 (1.67e-02)
5.96e-01 (5.25e-02)

Table S.3: Comparison of MOEA/D-DE with the two mutation strategies on the 16 problems (thefour DTLZ and nine WFG problems) with M ∈ {2, 3, 4, 5}. The means and standard deviations of
the hypervolume indicator values of 51 runs are shown (higher is better).
(a) M = 2

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

current/1
Mean (Std.)

rand/1
Mean (Std.)

4.88e-01 (8.69e-02)

4.17e-01 (6.69e-03)

4.45e-01 (1.13e-01)
4.22e-01 (8.97e-06)
2.04e-01 (1.01e-01)
4.22e-01 (1.70e-05)
4.22e-01 (8.97e-06)
4.15e-01 (7.98e-03)

3.48e-01 (1.21e-01)

3.62e-01 (1.19e-01)

4.22e-01 (6.15e-06)
2.10e-01 (1.27e-01)
4.22e-01 (1.03e-05)
4.22e-01 (6.15e-06)

3.89e-01 (7.31e-02)

3.38e-01 (6.43e-02)

7.64e-01 (8.12e-04)

7.64e-01 (6.32e-04)

7.05e-01 (4.58e-04)

7.04e-01 (4.13e-04)
3.92e-01 (3.19e-03)
3.75e-01 (6.78e-04)
3.47e-01 (1.21e-02)
4.21e-01 (1.68e-04)
3.67e-01 (2.66e-03)
3.44e-01 (1.67e-02)

3.99e-01 (2.68e-03)
3.75e-01 (6.73e-04)
3.51e-01 (1.84e-02)
4.21e-01 (1.30e-04)
3.68e-01 (3.17e-03)
3.63e-01 (3.10e-02)(b) M = 3

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

current/1
Mean (Std.)

rand/1
Mean (Std.)

9.06e-01 (1.19e-01)

8.10e-01 (2.73e-01)
7.39e-01 (1.68e-03)

7.39e-01 (1.48e-03)

2.67e-01 (2.37e-01)
7.27e-01 (3.84e-02)

2.73e-01 (2.19e-01)

2.65e-01 (2.10e-05)

6.65e-01 (2.38e-03)

2.65e-01 (5.56e-05)
2.57e-01 (6.18e-03)
2.95e-01 (3.16e-02)
4.28e-01 (3.13e-03)
1.20e+00 (6.32e-03)
8.39e-01 (3.96e-03)
6.45e-01 (6.70e-03)
6.64e-01 (2.46e-03)

6.45e-01 (1.58e-02)

6.46e-01 (1.76e-02)

7.13e-01 (4.83e-03)
5.84e-01 (1.20e-02)

6.98e-01 (6.47e-03)
5.74e-01 (9.94e-03)

6.17e-01 (7.11e-03)

6.18e-01 (8.41e-03)

2.57e-01 (6.60e-03)
3.18e-01 (4.26e-02)
4.38e-01 (1.05e-02)
1.20e+00 (6.19e-03)
8.41e-01 (6.36e-03)
6.61e-01 (7.31e-03)

67.41e-01 (1.87e-02)

(c) M = 4

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

current/1
Mean (Std.)

rand/1
Mean (Std.)

1.15e+00 (2.69e-01)

1.08e+00 (3.11e-01)
9.31e-01 (2.47e-03)

9.32e-01 (3.01e-03)

3.19e-01 (3.43e-01)
9.27e-01 (2.72e-02)

3.43e-01 (3.56e-01)

2.10e-01 (1.03e-04)

2.10e-01 (1.00e-04)

9.29e-01 (2.52e-02)

2.03e-01 (7.53e-03)

2.05e-01 (5.70e-03)

3.50e-01 (1.32e-02)
4.86e-01 (2.38e-02)

3.45e-01 (1.24e-02)
4.60e-01 (1.27e-02)

1.33e+00 (1.75e-02)

1.34e+00 (1.43e-02)

9.18e-01 (1.58e-02)

6.25e-01 (2.19e-02)

9.15e-01 (1.29e-02)
7.50e-01 (2.06e-02)
7.79e-01 (1.29e-02)
8.09e-01 (6.85e-03)
7.94e-01 (2.08e-02)
6.12e-01 (2.08e-02)

7.30e-01 (9.96e-03)

7.30e-01 (1.15e-02)

7.65e-01 (1.69e-02)
7.83e-01 (1.35e-02)
8.13e-01 (1.09e-02)
8.22e-01 (2.22e-02)(d) M = 5

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

current/1
Mean (Std.)

rand/1
Mean (Std.)

7.37e-01 (5.57e-01)

6.89e-01 (4.79e-01)
1.14e+00 (1.99e-03)

1.14e+00 (1.61e-03)

6.58e-02 (1.82e-01)
1.09e+00 (3.12e-02)

1.09e+00 (2.99e-02)

1.56e-01 (4.15e-03)

1.56e-01 (1.30e-04)

1.48e-01 (4.79e-03)
3.23e-01 (2.78e-02)

1.51e-01 (9.68e-03)

5.52e-01 (8.39e-02)

4.39e-01 (5.28e-02)

1.46e+00 (3.55e-02)
9.59e-01 (1.69e-02)

1.47e+00 (3.09e-02)

6.69e-01 (5.16e-02)

9.08e-02 (1.93e-01)

3.24e-01 (3.26e-02)

9.63e-01 (1.47e-02)

6.95e-01 (5.40e-02)

6.40e-01 (4.50e-02)
6.90e-01 (4.22e-02)
9.87e-01 (2.00e-02)
6.83e-01 (5.52e-02)

4.80e-01 (2.97e-02)

4.95e-01 (2.33e-02)

5.99e-01 (5.07e-02)

5.92e-01 (4.65e-02)

7.01e-01 (4.43e-02)
9.89e-01 (7.93e-03)

7

Table S.4: Comparison of MOEA/D-DE with the five bound handling methods on the 16 problems(the four DTLZ and nine WFG problems) with M ∈ {2, 3, 4, 5}. The means and standard deviationsof the hypervolume indicator values of 51 runs are shown (higher is better).
(a) M = 2

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

replacement
Mean (Std.)reinitialization
Mean (Std.)reflection
Mean (Std.)r-reflection
Mean (Std.)

resampling
Mean (Std.)

4.88e-01 (8.69e-02)

5.90e-01 (1.14e-01)

4.22e-01 (6.15e-06)

4.22e-01 (5.34e-04)
7.29e-02 (1.40e-01)
4.22e-01 (1.27e-04)
4.22e-01 (5.34e-04)
3.01e-04 (1.87e-03)

5.79e-01 (1.16e-01)
4.22e-01 (1.26e-04)
1.07e-01 (1.57e-01)
4.22e-01 (2.02e-04)
4.22e-01 (1.26e-04)
2.64e-01 (4.84e-02)
3.06e-01 (1.15e-01)
3.87e-01 (6.60e-02)
7.60e-01 (8.14e-03)
7.04e-01 (4.16e-04)
3.99e-01 (2.56e-03)
3.74e-01 (1.60e-04)
4.04e-01 (3.17e-03)
4.20e-01 (2.33e-04)
3.66e-01 (3.49e-03)
4.03e-01 (9.51e-04)

4.93e-01 (1.47e-01)
4.22e-01 (3.89e-05)
3.63e-02 (7.31e-02)
4.22e-01 (4.30e-05)
4.22e-01 (3.89e-05)
2.81e-01 (4.91e-02)
3.06e-01 (1.16e-01)
3.71e-01 (6.87e-02)
7.63e-01 (3.26e-03)
7.04e-01 (4.39e-04)
3.99e-01 (2.46e-03)
3.74e-01 (5.24e-04)
4.04e-01 (1.34e-02)
4.20e-01 (2.57e-04)
3.66e-01 (2.97e-03)
4.03e-01 (1.06e-03)

5.60e-01 (1.38e-01)
4.22e-01 (2.05e-04)
1.53e-01 (1.54e-01)
3.91e-01 (9.37e-02)
4.22e-01 (2.05e-04)
3.71e-01 (2.93e-02)
3.01e-01 (1.14e-01)

2.10e-01 (1.27e-01)
4.22e-01 (1.03e-05)
4.22e-01 (6.15e-06)
4.17e-01 (6.69e-03)

3.48e-01 (1.21e-01)
3.89e-01 (7.31e-02)

4.58e-01 (1.12e-03)

3.75e-01 (6.73e-04)

3.51e-01 (5.10e-02)
7.60e-01 (8.77e-03)
7.04e-01 (4.13e-04)
3.99e-01 (2.67e-03)
3.68e-01 (1.23e-03)

3.51e-01 (1.84e-02)

4.08e-01 (3.48e-03)

4.21e-01 (1.30e-04)

4.20e-01 (3.66e-04)
3.65e-01 (3.22e-03)
4.03e-01 (1.41e-03)

7.64e-01 (8.12e-04)
7.05e-01 (4.58e-04)

3.99e-01 (2.68e-03)

3.68e-01 (3.17e-03)

3.63e-01 (3.10e-02)

4.04e-01 (5.16e-02)

7.58e-01 (1.01e-02)
7.04e-01 (4.68e-04)
4.00e-01 (2.14e-03)

3.74e-01 (3.42e-04)
4.06e-01 (3.75e-03)
4.20e-01 (1.67e-04)
3.64e-01 (2.59e-03)
4.04e-01 (8.95e-04)(b) M = 3

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

replacement
Mean (Std.)reinitialization
Mean (Std.)reflection
Mean (Std.)r-reflection
Mean (Std.)

resampling
Mean (Std.)

9.06e-01 (1.19e-01)

9.08e-01 (1.22e-01)
7.32e-01 (3.17e-03)
3.73e-02 (1.09e-01)

8.26e-01 (2.14e-01)
7.30e-01 (3.33e-03)
2.78e-02 (9.73e-02)
6.94e-01 (9.76e-02)
2.65e-01 (1.16e-03)
0.00e+00 (0.00e+00)
2.86e-01 (2.93e-02)
3.92e-01 (3.13e-02)
1.19e+00 (7.55e-03)
8.38e-01 (4.04e-03)
6.40e-01 (7.33e-03)
6.37e-01 (6.29e-03)
6.27e-01 (9.73e-03)
6.84e-01 (7.23e-03)
5.72e-01 (1.06e-02)
6.37e-01 (1.46e-02)

7.92e-01 (2.09e-01)
7.33e-01 (3.35e-03)
5.53e-03 (3.95e-02)
7.33e-01 (2.79e-02)
2.65e-01 (5.85e-04)
7.18e-05 (5.13e-04)
2.80e-01 (2.45e-02)
3.94e-01 (1.38e-02)
1.20e+00 (6.29e-03)
8.39e-01 (4.09e-03)
6.46e-01 (7.40e-03)
6.46e-01 (5.14e-03)
6.19e-01 (1.21e-02)
6.89e-01 (6.81e-03)
5.72e-01 (1.03e-02)
6.41e-01 (1.74e-02)

9.46e-01 (1.08e-01)

7.39e-01 (1.48e-03)
2.67e-01 (2.37e-01)

7.27e-01 (3.84e-02)
2.65e-01 (2.10e-05)
2.57e-01 (6.60e-03)

7.41e-01 (1.44e-03)

2.65e-01 (5.66e-04)
0.00e+00 (0.00e+00)

3.18e-01 (4.26e-02)

3.39e-01 (4.42e-03)

4.38e-01 (1.05e-02)

3.46e-01 (6.23e-02)
1.20e+00 (6.33e-03)
8.40e-01 (3.54e-03)
6.46e-01 (5.75e-03)
5.10e-01 (1.81e-02)
6.59e-01 (8.19e-03)
6.86e-01 (3.97e-03)
5.65e-01 (7.77e-03)
6.44e-01 (1.13e-02)

1.20e+00 (6.19e-03)

8.41e-01 (6.36e-03)
6.61e-01 (7.31e-03)
6.65e-01 (2.38e-03)

6.45e-01 (1.58e-02)
7.13e-01 (4.83e-03)
5.84e-01 (1.20e-02)

6.17e-01 (7.11e-03)

87.37e-01 (1.71e-03)
8.54e-02 (1.81e-01)
7.00e-01 (9.21e-02)
2.65e-01 (5.15e-04)

9.19e-06 (4.88e-05)
2.72e-01 (1.68e-02)
4.15e-01 (2.20e-02)
1.17e+00 (5.90e-02)
8.42e-01 (3.60e-03)

6.61e-01 (6.16e-03)
6.31e-01 (7.11e-03)
6.62e-01 (6.27e-03)

7.03e-01 (5.36e-03)
5.82e-01 (7.91e-03)
6.45e-01 (1.64e-02)

(c) M = 4

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

replacement
Mean (Std.)reinitialization
Mean (Std.)reflection
Mean (Std.)r-reflection
Mean (Std.)

resampling
Mean (Std.)

1.15e+00 (2.69e-01)

9.27e-01 (2.72e-02)

3.91e-01 (3.99e-01)
9.18e-01 (1.06e-02)
0.00e+00 (0.00e+00)
9.07e-01 (1.81e-02)

2.10e-01 (1.03e-04)

2.11e-01 (1.44e-03)

2.03e-01 (7.53e-03)

0.00e+00 (0.00e+00)
2.41e-01 (2.14e-02)
1.82e-01 (2.08e-02)

2.58e-01 (3.80e-01)
9.16e-01 (1.22e-02)
0.00e+00 (0.00e+00)
9.05e-01 (3.53e-02)
2.10e-01 (1.29e-03)
0.00e+00 (0.00e+00)
3.14e-01 (1.39e-02)
3.25e-01 (8.31e-02)
1.33e+00 (2.43e-02)
9.28e-01 (1.19e-02)
7.01e-01 (1.62e-02)
6.82e-01 (2.53e-02)
6.64e-01 (2.23e-02)
7.88e-01 (1.80e-02)
6.14e-01 (1.33e-02)
7.06e-01 (1.30e-02)

3.85e-01 (3.79e-01)
9.21e-01 (7.12e-03)
0.00e+00 (0.00e+00)
9.07e-01 (2.98e-02)
2.10e-01 (9.64e-04)
0.00e+00 (0.00e+00)
3.32e-01 (1.41e-02)
3.20e-01 (5.30e-02)
1.33e+00 (1.75e-02)
9.25e-01 (1.18e-02)
7.10e-01 (1.43e-02)
7.04e-01 (2.32e-02)
6.67e-01 (1.94e-02)
7.97e-01 (1.81e-02)
6.11e-01 (1.71e-02)
7.06e-01 (1.34e-02)

6.32e-01 (3.88e-01)
9.29e-01 (2.17e-03)
0.00e+00 (0.00e+00)
8.64e-01 (8.26e-02)
2.10e-01 (1.46e-04)
0.00e+00 (0.00e+00)
3.12e-01 (1.64e-02)
3.52e-01 (6.00e-02)
1.32e+00 (5.99e-02)

9.32e-01 (3.01e-03)
3.19e-01 (3.43e-01)

3.50e-01 (1.32e-02)
4.86e-01 (2.38e-02)

1.33e+00 (1.75e-02)
9.18e-01 (1.58e-02)
7.65e-01 (1.69e-02)
7.83e-01 (1.35e-02)
8.13e-01 (1.09e-02)
8.22e-01 (2.22e-02)

6.25e-01 (2.19e-02)
7.30e-01 (9.96e-03)

1.36e+00 (1.33e-02)

9.34e-01 (1.09e-02)
7.24e-01 (1.45e-02)
5.86e-01 (2.71e-02)
7.48e-01 (2.16e-02)
7.97e-01 (1.64e-02)
6.17e-01 (1.04e-02)
7.12e-01 (1.20e-02)

9.38e-01 (9.98e-03)

7.52e-01 (1.70e-02)
7.18e-01 (3.20e-02)
7.25e-01 (1.26e-02)
8.17e-01 (1.74e-02)
6.39e-01 (1.73e-02)

7.10e-01 (1.34e-02)(d) M = 5

DTLZ1
DTLZ2
DTLZ3
DTLZ4
DTLZ5
DTLZ6
DTLZ7
WFG1
WFG2
WFG3
WFG4
WFG5
WFG6
WFG7
WFG8
WFG9

replacement
Mean (Std.)reinitialization
Mean (Std.)reflection
Mean (Std.)r-reflection
Mean (Std.)

resampling
Mean (Std.)

7.37e-01 (5.57e-01)

2.80e-01 (3.47e-01)
8.97e-01 (5.07e-02)
0.00e+00 (0.00e+00)
1.01e+00 (4.24e-02)
1.70e-01 (1.46e-02)
0.00e+00 (0.00e+00)
2.81e-02 (2.03e-02)
1.83e-01 (3.51e-02)

4.11e-02 (8.77e-02)
8.75e-01 (5.26e-02)
0.00e+00 (0.00e+00)
9.80e-01 (5.10e-02)
1.85e-01 (5.52e-03)
0.00e+00 (0.00e+00)
9.87e-02 (4.95e-02)
3.41e-01 (1.07e-01)
1.46e+00 (3.77e-02)
1.02e+00 (1.61e-02)
5.90e-01 (2.77e-02)
5.91e-01 (4.44e-02)
5.30e-01 (4.05e-02)
6.39e-01 (3.21e-02)
4.86e-01 (2.33e-02)
5.23e-01 (2.88e-02)

4.19e-02 (1.06e-01)
9.62e-01 (6.65e-02)
0.00e+00 (0.00e+00)
1.01e+00 (3.90e-02)
1.77e-01 (1.29e-02)
0.00e+00 (0.00e+00)
1.54e-01 (5.11e-02)
2.89e-01 (4.54e-02)
1.46e+00 (3.06e-02)
1.01e+00 (2.00e-02)
5.92e-01 (3.34e-02)
6.10e-01 (4.13e-02)
5.28e-01 (4.27e-02)
6.53e-01 (3.94e-02)
4.81e-01 (1.87e-02)
5.39e-01 (3.66e-02)

3.93e-01 (3.43e-01)
1.13e+00 (7.81e-03)
1.84e-03 (1.32e-02)
9.49e-01 (8.54e-02)

1.14e+00 (1.61e-03)
6.58e-02 (1.82e-01)
1.09e+00 (3.12e-02)

1.56e-01 (4.15e-03)
1.48e-01 (4.79e-03)
3.23e-01 (2.78e-02)
5.52e-01 (8.39e-02)

1.46e+00 (3.55e-02)
9.59e-01 (1.69e-02)
6.69e-01 (5.16e-02)
7.01e-01 (4.43e-02)
9.89e-01 (7.93e-03)

6.95e-01 (5.40e-02)
4.80e-01 (2.97e-02)
5.99e-01 (5.07e-02)

1.52e+00 (1.51e-02)
1.04e+00 (8.92e-03)

6.58e-01 (2.34e-02)
5.11e-01 (3.38e-02)
6.83e-01 (3.94e-02)
7.52e-01 (5.16e-02)

5.08e-01 (1.28e-02)
5.50e-01 (4.19e-02)

91.88e-01 (4.31e-03)

0.00e+00 (0.00e+00)
2.92e-01 (4.01e-02)
3.00e-01 (5.03e-02)
1.42e+00 (9.84e-02)
1.03e+00 (6.67e-03)
6.86e-01 (3.55e-02)

5.65e-01 (4.25e-02)
6.20e-01 (4.00e-02)
6.69e-01 (2.98e-02)
5.67e-01 (3.95e-02)

5.40e-01 (4.06e-02)