A Supplementary File for“An Analysis of Control Parameters of MOEA/D
Under Two Different Optimization Scenarios”
Ryoji Tanabe, Hisao Ishibuchi∗
Shenzhen Key Laboratory of Computational Intelligence, Department of Computer Scienceand Engineering, Southern University of Science and Technology, Shenzhen, 518055, China
Abstract
This is a supplementary file for “An Analysis of Control Parameters of MOEA/DUnder Two Different Optimization Scenarios”.
∗Corresponding authorEmail addresses: [email protected] (Ryoji Tanabe), [email protected]
(Hisao Ishibuchi)
Preprint submitted to Applied Soft Computing March 5, 2018
10000 20000 30000 40000 500000.00
0.25
0.50µ = 200µ = 300µ = 100µ = 400µ = 50µ = 25DTLZ1 (M = 2)
10000 20000 30000 40000 500000.00
0.25
0.50µ = 50µ = 25µ = 100µ = 200µ = 300µ = 400DTLZ1 (M = 2)
10000 20000 30000 40000 500000.0
0.5
µ = 105µ = 55µ = 28µ = 300µ = 406µ = 210DTLZ1 (M = 3)
10000 20000 30000 40000 500000.0
0.5
µ = 105µ = 55µ = 28µ = 210µ = 300µ = 406DTLZ1 (M = 3)
10000 20000 30000 40000 500000.0
0.5
µ = 120µ = 56µ = 35µ = 455µ = 220µ = 286DTLZ1 (M = 4)
10000 20000 30000 40000 500000.0
0.5
1.0 µ = 120µ = 56µ = 35µ = 220µ = 455µ = 286DTLZ1 (M = 4)
10000 20000 30000 40000 500000.0
0.5
µ = 126
µ = 210
µ = 495
µ = 330DTLZ1 (M = 5)
10000 20000 30000 40000 500000.0
0.5
1.0 µ = 126
µ = 210
µ = 330
µ = 495DTLZ1 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.1: Performance of MOEA/D with various µ settings on the DTLZ1 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
2
10000 20000 30000 40000 50000
0.34
0.35 µ = 400µ = 300µ = 200µ = 100µ = 50µ = 25DTLZ2 (M = 2)
10000 20000 30000 40000 500000.340
0.345
0.350 µ = 25µ = 50µ = 100µ = 200µ = 300µ = 400DTLZ2 (M = 2)
10000 20000 30000 40000 50000
0.50
0.55
µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28DTLZ2 (M = 3)
10000 20000 30000 40000 50000
0.55
0.56
0.57
µ = 105µ = 55µ = 28µ = 210µ = 300µ = 406DTLZ2 (M = 3)
10000 20000 30000 40000 50000
0.5
0.6
0.7 µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35DTLZ2 (M = 4)
10000 20000 30000 40000 50000
0.65
0.70
µ = 120µ = 220µ = 286µ = 35µ = 56µ = 455DTLZ2 (M = 4)
10000 20000 30000 40000 50000
0.5
0.6
0.7µ = 495
µ = 330
µ = 210
µ = 126DTLZ2 (M = 5)
10000 20000 30000 40000 500000.6
0.7
0.8 µ = 126
µ = 210
µ = 330
µ = 495DTLZ2 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.2: Performance of MOEA/D with various µ settings on the DTLZ2 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
3
10000 20000 30000 40000 500000.0
0.2
µ = 100µ = 50µ = 200µ = 25µ = 300µ = 400DTLZ3 (M = 2)
10000 20000 30000 40000 500000.0
0.2
µ = 50µ = 25µ = 100µ = 200µ = 300µ = 400DTLZ3 (M = 2)
10000 20000 30000 40000 500000.00
0.25
0.50µ = 105µ = 55µ = 28µ = 406µ = 300µ = 210DTLZ3 (M = 3)
10000 20000 30000 40000 500000.00
0.25
0.50µ = 105µ = 28µ = 55µ = 406µ = 210µ = 300DTLZ3 (M = 3)
10000 20000 30000 40000 500000.00
0.25
0.50µ = 56µ = 35µ = 120µ = 286µ = 455µ = 220DTLZ3 (M = 4)
10000 20000 30000 40000 500000.0
0.5
µ = 56µ = 35µ = 120µ = 286µ = 455µ = 220DTLZ3 (M = 4)
10000 20000 30000 40000 500000.00
0.25
0.50µ = 210
µ = 330
µ = 495
µ = 126DTLZ3 (M = 5)
10000 20000 30000 40000 500000.0
0.5
µ = 210
µ = 330
µ = 126
µ = 495DTLZ3 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.3: Performance of MOEA/D with various µ settings on the DTLZ3 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
4
10000 20000 30000 40000 500000.1
0.2
0.3
µ = 400µ = 300µ = 200µ = 50µ = 100µ = 25DTLZ4 (M = 2)
10000 20000 30000 40000 500000.1
0.2
0.3
µ = 400µ = 200µ = 300µ = 100µ = 50µ = 25DTLZ4 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
0.6 µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28DTLZ4 (M = 3)
10000 20000 30000 40000 50000
0.2
0.4
0.6 µ = 406µ = 300µ = 105µ = 210µ = 55µ = 28DTLZ4 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35DTLZ4 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75 µ = 455µ = 286µ = 120µ = 220µ = 56µ = 35DTLZ4 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6µ = 495
µ = 330
µ = 210
µ = 126DTLZ4 (M = 5)
10000 20000 30000 40000 50000
0.4
0.6
µ = 495
µ = 330
µ = 126
µ = 210DTLZ4 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.4: Performance of MOEA/D with various µ settings on the DTLZ4 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
5
10000 20000 30000 40000 500000.00
0.25
0.50µ = 100µ = 50µ = 200µ = 300µ = 25µ = 400WFG1 (M = 2)
10000 20000 30000 40000 500000.00
0.25
0.50
µ = 100µ = 200µ = 50µ = 300µ = 25µ = 400WFG1 (M = 2)
10000 20000 30000 40000 500000.0
0.5
µ = 55µ = 28µ = 105µ = 210µ = 300µ = 406WFG1 (M = 3)
10000 20000 30000 40000 500000.0
0.5
µ = 28µ = 55µ = 105µ = 210µ = 300µ = 406WFG1 (M = 3)
10000 20000 30000 40000 500000.0
0.5
µ = 56µ = 35µ = 120µ = 220µ = 286µ = 455WFG1 (M = 4)
10000 20000 30000 40000 500000.0
0.5
µ = 35µ = 56µ = 120µ = 220µ = 286µ = 455WFG1 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
µ = 126
µ = 210
µ = 330
µ = 495WFG1 (M = 5)
10000 20000 30000 40000 50000
0.25
0.50
µ = 126
µ = 210
µ = 330
µ = 495WFG1 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.5: Performance of MOEA/D with various µ settings on the WFG1 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
6
10000 20000 30000 40000 50000
0.5
0.6µ = 200µ = 300µ = 400µ = 100µ = 50µ = 25WFG2 (M = 2)
10000 20000 30000 40000 50000
0.5
0.6µ = 200µ = 300µ = 400µ = 100µ = 50µ = 25WFG2 (M = 2)
10000 20000 30000 40000 50000
0.6
0.8
µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG2 (M = 3)
10000 20000 30000 40000 50000
0.6
0.8
µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG2 (M = 3)
10000 20000 30000 40000 50000
0.6
0.8µ = 286µ = 455µ = 220µ = 120µ = 56µ = 35WFG2 (M = 4)
10000 20000 30000 40000 50000
0.6
0.8
µ = 286µ = 455µ = 220µ = 120µ = 56µ = 35WFG2 (M = 4)
10000 20000 30000 40000 50000
0.7
0.8µ = 495
µ = 330
µ = 210
µ = 126WFG2 (M = 5)
10000 20000 30000 40000 50000
0.7
0.8
µ = 495
µ = 330
µ = 210
µ = 126WFG2 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.6: Performance of MOEA/D with various µ settings on the WFG2 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
7
10000 20000 30000 40000 50000
0.45
0.50
0.55
µ = 200µ = 100µ = 300µ = 400µ = 50µ = 25WFG3 (M = 2)
10000 20000 30000 40000 50000
0.45
0.50
0.55
µ = 100µ = 200µ = 50µ = 300µ = 400µ = 25WFG3 (M = 2)
10000 20000 30000 40000 500000.5
0.6
µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG3 (M = 3)
10000 20000 30000 40000 50000
0.55
0.60
µ = 210µ = 105µ = 300µ = 406µ = 55µ = 28WFG3 (M = 3)
10000 20000 30000 40000 50000
0.5
0.6
µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG3 (M = 4)
10000 20000 30000 40000 50000
0.55
0.60
0.65 µ = 286µ = 455µ = 220µ = 120µ = 56µ = 35WFG3 (M = 4)
10000 20000 30000 40000 50000
0.55
0.60
0.65µ = 495
µ = 330
µ = 210
µ = 126WFG3 (M = 5)
10000 20000 30000 40000 500000.55
0.60
0.65µ = 330
µ = 495
µ = 126
µ = 210WFG3 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.7: Performance of MOEA/D with various µ settings on the WFG3 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
8
10000 20000 30000 40000 500000.25
0.30
0.35 µ = 200µ = 300µ = 100µ = 400µ = 50µ = 25WFG4 (M = 2)
10000 20000 30000 40000 500000.25
0.30
0.35 µ = 25µ = 50µ = 100µ = 200µ = 300µ = 400WFG4 (M = 2)
10000 20000 30000 40000 500000.4
0.5
0.6 µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG4 (M = 3)
10000 20000 30000 40000 500000.4
0.5
0.6 µ = 28µ = 55µ = 105µ = 210µ = 300µ = 406WFG4 (M = 3)
10000 20000 30000 40000 500000.3
0.4
0.5
0.6
0.7
0.8 µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG4 (M = 4)
10000 20000 30000 40000 500000.3
0.4
0.5
0.6
0.7
0.8 µ = 35µ = 56µ = 120µ = 220µ = 286µ = 455WFG4 (M = 4)
10000 20000 30000 40000 50000
0.4
0.5
0.6µ = 495
µ = 330
µ = 210
µ = 126WFG4 (M = 5)
10000 20000 30000 40000 50000
0.6
0.8 µ = 210
µ = 330
µ = 126
µ = 495WFG4 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.8: Performance of MOEA/D with various µ settings on the WFG4 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
9
10000 20000 30000 40000 50000
0.25
0.30
µ = 400µ = 300µ = 200µ = 100µ = 50µ = 25WFG5 (M = 2)
10000 20000 30000 40000 500000.25
0.30
µ = 25µ = 50µ = 100µ = 200µ = 300µ = 400WFG5 (M = 2)
10000 20000 30000 40000 50000
0.4
0.5µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG5 (M = 3)
10000 20000 30000 40000 50000
0.4
0.5
µ = 28µ = 55µ = 105µ = 210µ = 300µ = 406WFG5 (M = 3)
10000 20000 30000 40000 50000
0.4
0.6µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG5 (M = 4)
10000 20000 30000 40000 50000
0.5
0.6
µ = 56µ = 35µ = 120µ = 220µ = 286µ = 455WFG5 (M = 4)
10000 20000 30000 40000 50000
0.4
0.5
0.6µ = 495
µ = 126
µ = 330
µ = 210WFG5 (M = 5)
10000 20000 30000 40000 50000
0.5
0.6
0.7µ = 126
µ = 210
µ = 330
µ = 495WFG5 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.9: Performance of MOEA/D with various µ settings on the WFG5 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
10
10000 20000 30000 40000 50000
0.25
0.30
µ = 200µ = 400µ = 300µ = 100µ = 50µ = 25WFG6 (M = 2)
10000 20000 30000 40000 50000
0.25
0.30
µ = 50µ = 200µ = 100µ = 300µ = 400µ = 25WFG6 (M = 2)
10000 20000 30000 40000 50000
0.4
0.5
µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG6 (M = 3)
10000 20000 30000 40000 50000
0.4
0.5
µ = 28µ = 55µ = 105µ = 210µ = 300µ = 406WFG6 (M = 3)
10000 20000 30000 40000 50000
0.4
0.6µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG6 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6
µ = 56µ = 35µ = 220µ = 120µ = 286µ = 455WFG6 (M = 4)
10000 20000 30000 40000 50000
0.4
0.5
0.6 µ = 495
µ = 126
µ = 210
µ = 330WFG6 (M = 5)
10000 20000 30000 40000 500000.4
0.6
µ = 210
µ = 126
µ = 330
µ = 495WFG6 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.10: Performance of MOEA/D with various µ settings on the WFG6 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
11
10000 20000 30000 40000 500000.25
0.30
0.35 µ = 300µ = 200µ = 400µ = 100µ = 50µ = 25WFG7 (M = 2)
10000 20000 30000 40000 500000.25
0.30
0.35 µ = 25µ = 50µ = 100µ = 200µ = 300µ = 400WFG7 (M = 2)
10000 20000 30000 40000 500000.3
0.4
0.5
µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG7 (M = 3)
10000 20000 30000 40000 500000.4
0.5
µ = 28µ = 55µ = 105µ = 210µ = 300µ = 406WFG7 (M = 3)
10000 20000 30000 40000 50000
0.4
0.6
µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG7 (M = 4)
10000 20000 30000 40000 50000
0.5
0.6
0.7µ = 120µ = 220µ = 286µ = 56µ = 455µ = 35WFG7 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6
µ = 495
µ = 330
µ = 210
µ = 126WFG7 (M = 5)
10000 20000 30000 40000 50000
0.6
0.8 µ = 210
µ = 126
µ = 330
µ = 495WFG7 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.11: Performance of MOEA/D with various µ settings on the WFG7 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
12
10000 20000 30000 40000 500000.20
0.25
0.30µ = 400µ = 200µ = 100µ = 300µ = 50µ = 25WFG8 (M = 2)
10000 20000 30000 40000 50000
0.25
0.30µ = 100µ = 400µ = 50µ = 200µ = 25µ = 300WFG8 (M = 2)
10000 20000 30000 40000 50000
0.3
0.4
0.5 µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG8 (M = 3)
10000 20000 30000 40000 50000
0.4
0.5 µ = 105µ = 55µ = 210µ = 300µ = 28µ = 406WFG8 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
0.5
µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG8 (M = 4)
10000 20000 30000 40000 50000
0.4
0.5
0.6 µ = 220µ = 286µ = 120µ = 455µ = 56µ = 35WFG8 (M = 4)
10000 20000 30000 40000 50000
0.3
0.4
0.5 µ = 495
µ = 330
µ = 210
µ = 126WFG8 (M = 5)
10000 20000 30000 40000 500000.4
0.5
0.6µ = 495
µ = 330
µ = 210
µ = 126WFG8 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.12: Performance of MOEA/D with various µ settings on the WFG8 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
13
10000 20000 30000 40000 50000
0.25
0.30
µ = 300µ = 100µ = 400µ = 200µ = 25µ = 50WFG9 (M = 2)
10000 20000 30000 40000 50000
0.25
0.30
µ = 300µ = 400µ = 100µ = 200µ = 25µ = 50WFG9 (M = 2)
10000 20000 30000 40000 500000.3
0.4
0.5µ = 406µ = 300µ = 210µ = 105µ = 55µ = 28WFG9 (M = 3)
10000 20000 30000 40000 50000
0.4
0.5µ = 406µ = 300µ = 28µ = 55µ = 105µ = 210WFG9 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
0.5
µ = 455µ = 286µ = 220µ = 120µ = 56µ = 35WFG9 (M = 4)
10000 20000 30000 40000 50000
0.5
0.6µ = 56µ = 120µ = 35µ = 220µ = 286µ = 455WFG9 (M = 4)
10000 20000 30000 40000 50000
0.3
0.4
0.5µ = 495
µ = 330
µ = 210
µ = 126WFG9 (M = 5)
10000 20000 30000 40000 50000
0.5
0.6
µ = 126
µ = 210
µ = 330
µ = 495WFG9 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
ervo
lum
eva
lues
acro
ssal
l31
runs
Figure S.13: Performance of MOEA/D with various µ settings on the WFG9 problem withM ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent the number of function evaluationsand the HV values, respectively. The shaded area indicates 25-75 percentiles.
14
0 10000 20000 30000 40000 500000.00
0.25
0.50gchm
gchd
gpbiDTLZ1 (M = 2)
0 10000 20000 30000 40000 500000.00
0.25
0.50gchd
gchm
gpbiDTLZ1 (M = 2)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gpbi
gchmDTLZ1 (M = 3)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gpbi
gchmDTLZ1 (M = 3)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gchm
gpbiDTLZ1 (M = 4)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gchm
gpbiDTLZ1 (M = 4)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gchm
gpbiDTLZ1 (M = 5)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gchm
gpbiDTLZ1 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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lum
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l31
runs
Figure S.14: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the DTLZ1 problem with M ∈ {2, 3, 4, 5}. The horizontal and verticalaxes represent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
15
0 10000 20000 30000 40000 50000
0.32
0.34
gchm
gchd
gpbiDTLZ2 (M = 2)
0 10000 20000 30000 40000 50000
0.32
0.34
gchdgchmgpbiDTLZ2 (M = 2)
0 10000 20000 30000 40000 50000
0.50
0.55gchd
gpbi
gchmDTLZ2 (M = 3)
0 10000 20000 30000 40000 50000
0.50
0.55
gchm
gchd
gpbiDTLZ2 (M = 3)
0 10000 20000 30000 40000 50000
0.6
0.7gpbi
gchd
gchmDTLZ2 (M = 4)
0 10000 20000 30000 40000 50000
0.6
0.7
gchm
gchd
gpbiDTLZ2 (M = 4)
0 10000 20000 30000 40000 50000
0.6
0.8gpbi
gchd
gchmDTLZ2 (M = 5)
0 10000 20000 30000 40000 50000
0.6
0.7
0.8gchm
gpbi
gchdDTLZ2 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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lum
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l31
runs
Figure S.15: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the DTLZ2 problem with M ∈ {2, 3, 4, 5}. The horizontal and verticalaxes represent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
16
0 10000 20000 30000 40000 500000.0
0.2
gchd
gchm
gpbiDTLZ3 (M = 2)
0 10000 20000 30000 40000 500000.0
0.2
gchd
gchm
gpbiDTLZ3 (M = 2)
0 10000 20000 30000 40000 500000.0
0.2
0.4gchd
gchm
gpbiDTLZ3 (M = 3)
0 10000 20000 30000 40000 500000.00
0.25
0.50 gchd
gchm
gpbiDTLZ3 (M = 3)
0 10000 20000 30000 40000 500000.0
0.2
gchd
gchm
gpbiDTLZ3 (M = 4)
0 10000 20000 30000 40000 500000.0
0.2
0.4gchd
gchm
gpbiDTLZ3 (M = 4)
0 10000 20000 30000 40000 500000.00
0.25
0.50 gchd
gchm
gpbiDTLZ3 (M = 5)
0 10000 20000 30000 40000 500000.00
0.25
0.50gchm
gchd
gpbiDTLZ3 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.16: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the DTLZ3 problem with M ∈ {2, 3, 4, 5}. The horizontal and verticalaxes represent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
17
0 10000 20000 30000 40000 500000.1
0.2
0.3gchm
gpbi
gchdDTLZ4 (M = 2)
0 10000 20000 30000 40000 500000.1
0.2
0.3gchm
gpbi
gchdDTLZ4 (M = 2)
0 10000 20000 30000 40000 50000
0.2
0.4
0.6 gpbi
gchd
gchmDTLZ4 (M = 3)
0 10000 20000 30000 40000 50000
0.2
0.4
gpbi
gchm
gchdDTLZ4 (M = 3)
0 10000 20000 30000 40000 50000
0.25
0.50
0.75 gpbi
gchm
gchdDTLZ4 (M = 4)
0 10000 20000 30000 40000 50000
0.25
0.50
gpbi
gchd
gchmDTLZ4 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6
0.8 gpbi
gchd
gchmDTLZ4 (M = 5)
0 10000 20000 30000 40000 50000
0.4
0.6
0.8gpbi
gchd
gchmDTLZ4 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.17: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the DTLZ4 problem with M ∈ {2, 3, 4, 5}. The horizontal and verticalaxes represent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
18
0 10000 20000 30000 40000 500000.0
0.2
0.4
gchm
gchd
gpbiWFG1 (M = 2)
0 10000 20000 30000 40000 500000.0
0.2
0.4
gchm
gchd
gpbiWFG1 (M = 2)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gchm
gpbiWFG1 (M = 3)
0 10000 20000 30000 40000 500000.0
0.5
gchd
gchm
gpbiWFG1 (M = 3)
0 10000 20000 30000 40000 500000.00
0.25
0.50
gchd
gchm
gpbiWFG1 (M = 4)
0 10000 20000 30000 40000 500000.00
0.25
0.50
gchd
gchm
gpbiWFG1 (M = 4)
0 10000 20000 30000 40000 500000.00
0.25
0.50gchm
gchd
gpbiWFG1 (M = 5)
0 10000 20000 30000 40000 500000.00
0.25
0.50gchm
gchd
gpbiWFG1 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.18: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG1 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
19
0 10000 20000 30000 40000 50000
0.5
0.6gchd
gchm
gpbiWFG2 (M = 2)
0 10000 20000 30000 40000 50000
0.5
0.6gchd
gchm
gpbiWFG2 (M = 2)
0 10000 20000 30000 40000 50000
0.6
0.7
0.8
gchd
gchm
gpbiWFG2 (M = 3)
0 10000 20000 30000 40000 50000
0.6
0.7
0.8
gchd
gchm
gpbiWFG2 (M = 3)
0 10000 20000 30000 40000 50000
0.6
0.8gchm
gchd
gpbiWFG2 (M = 4)
0 10000 20000 30000 40000 50000
0.6
0.7
0.8gchm
gchd
gpbiWFG2 (M = 4)
0 10000 20000 30000 40000 500000.4
0.6
0.8 gchm
gchd
gpbiWFG2 (M = 5)
0 10000 20000 30000 40000 50000
0.6
0.7
0.8gchm
gchd
gpbiWFG2 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.19: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG2 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
20
0 10000 20000 30000 40000 500000.4
0.5gchd
gchm
gpbiWFG3 (M = 2)
0 10000 20000 30000 40000 500000.4
0.5gchd
gchm
gpbiWFG3 (M = 2)
0 10000 20000 30000 40000 50000
0.5
0.6 gchm
gchd
gpbiWFG3 (M = 3)
0 10000 20000 30000 40000 50000
0.5
0.6 gchd
gchm
gpbiWFG3 (M = 3)
0 10000 20000 30000 40000 50000
0.5
0.6 gchm
gchd
gpbiWFG3 (M = 4)
0 10000 20000 30000 40000 50000
0.5
0.6 gchd
gchm
gpbiWFG3 (M = 4)
0 10000 20000 30000 40000 50000
0.25
0.50gchm
gchd
gpbiWFG3 (M = 5)
0 10000 20000 30000 40000 50000
0.5
0.6 gchm
gchd
gpbiWFG3 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.20: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG3 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
21
0 10000 20000 30000 40000 500000.20
0.25
0.30
0.35
gchd
gchm
gpbiWFG4 (M = 2)
0 10000 20000 30000 40000 500000.20
0.25
0.30
0.35
gchd
gchm
gpbiWFG4 (M = 2)
0 10000 20000 30000 40000 500000.3
0.4
0.5
0.6gchd
gchm
gpbiWFG4 (M = 3)
0 10000 20000 30000 40000 500000.3
0.4
0.5
0.6gchd
gchm
gpbiWFG4 (M = 3)
0 10000 20000 30000 40000 500000.4
0.5
0.6
0.7gpbi
gchm
gchdWFG4 (M = 4)
0 10000 20000 30000 40000 500000.4
0.5
0.6
0.7gchm
gchd
gpbiWFG4 (M = 4)
0 10000 20000 30000 40000 500000.3
0.4
0.5
0.6
0.7
0.8gpbi
gchd
gchmWFG4 (M = 5)
0 10000 20000 30000 40000 500000.3
0.4
0.5
0.6
0.7
0.8gchm
gchd
gpbiWFG4 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.21: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG4 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
22
0 10000 20000 30000 40000 50000
0.20
0.25
0.30gchm
gchd
gpbiWFG5 (M = 2)
0 10000 20000 30000 40000 50000
0.20
0.25
0.30gchd
gchm
gpbiWFG5 (M = 2)
0 10000 20000 30000 40000 50000
0.3
0.4
0.5 gchd
gchm
gpbiWFG5 (M = 3)
0 10000 20000 30000 40000 500000.3
0.4
0.5 gchd
gchm
gpbiWFG5 (M = 3)
0 10000 20000 30000 40000 50000
0.4
0.5
0.6 gchd
gpbi
gchmWFG5 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6 gchm
gchd
gpbiWFG5 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6gpbi
gchd
gchmWFG5 (M = 5)
0 10000 20000 30000 40000 500000.4
0.6
gchm
gchd
gpbiWFG5 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.22: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG5 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
23
0 10000 20000 30000 40000 50000
0.2
0.3 gchd
gchm
gpbiWFG6 (M = 2)
0 10000 20000 30000 40000 50000
0.2
0.3 gchd
gchm
gpbiWFG6 (M = 2)
0 10000 20000 30000 40000 50000
0.3
0.4
0.5 gchd
gchm
gpbiWFG6 (M = 3)
0 10000 20000 30000 40000 50000
0.3
0.4
0.5 gchd
gchm
gpbiWFG6 (M = 3)
0 10000 20000 30000 40000 50000
0.4
0.6gpbi
gchd
gchmWFG6 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6gchm
gchd
gpbiWFG6 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6gpbi
gchd
gchmWFG6 (M = 5)
0 10000 20000 30000 40000 50000
0.4
0.6
gchm
gchd
gpbiWFG6 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.23: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG6 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
24
0 10000 20000 30000 40000 500000.2
0.3gchd
gchm
gpbiWFG7 (M = 2)
0 10000 20000 30000 40000 500000.2
0.3gchd
gchm
gpbiWFG7 (M = 2)
0 10000 20000 30000 40000 500000.3
0.4
0.5gchd
gchm
gpbiWFG7 (M = 3)
0 10000 20000 30000 40000 500000.3
0.4
0.5gchd
gchm
gpbiWFG7 (M = 3)
0 10000 20000 30000 40000 50000
0.4
0.6 gchd
gpbi
gchmWFG7 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6gchm
gchd
gpbiWFG7 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6
gpbi
gchd
gchmWFG7 (M = 5)
0 10000 20000 30000 40000 50000
0.6
0.8gchm
gchd
gpbiWFG7 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.24: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG7 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
25
0 10000 20000 30000 40000 50000
0.20
0.25
0.30gchd
gchm
gpbiWFG8 (M = 2)
0 10000 20000 30000 40000 50000
0.20
0.25
0.30gchd
gchm
gpbiWFG8 (M = 2)
0 10000 20000 30000 40000 50000
0.3
0.4
gchd
gchm
gpbiWFG8 (M = 3)
0 10000 20000 30000 40000 50000
0.3
0.4
0.5gchd
gchm
gpbiWFG8 (M = 3)
0 10000 20000 30000 40000 50000
0.4
0.5gpbi
gchm
gchdWFG8 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6gchm
gchd
gpbiWFG8 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6 gpbi
gchm
gchdWFG8 (M = 5)
0 10000 20000 30000 40000 50000
0.4
0.6 gchm
gpbi
gchdWFG8 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.25: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG8 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
26
0 10000 20000 30000 40000 50000
0.2
0.3 gpbi
gchm
gchdWFG9 (M = 2)
0 10000 20000 30000 40000 50000
0.2
0.3 gpbi
gchd
gchmWFG9 (M = 2)
0 10000 20000 30000 40000 50000
0.3
0.4
gchd
gchm
gpbiWFG9 (M = 3)
0 10000 20000 30000 40000 50000
0.3
0.4
0.5gchd
gchm
gpbiWFG9 (M = 3)
0 10000 20000 30000 40000 500000.3
0.4
0.5 gpbi
gchd
gchmWFG9 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6gchm
gchd
gpbiWFG9 (M = 4)
0 10000 20000 30000 40000 50000
0.4
0.6 gpbi
gchd
gchmWFG9 (M = 5)
0 10000 20000 30000 40000 50000
0.4
0.6 gchm
gchd
gpbiWFG9 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.26: Performance of MOEA/D with the three scalarizing functions (gchm, gchd, andgpbi with θ = 5) on the WFG9 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axesrepresent the number of function evaluations and the HV values, respectively. The shadedarea indicates 25-75 percentiles.
27
10000 20000 30000 40000 500000.00
0.25
0.50θ = 3θ = 2θ = 5θ = 1θ = 8θ = 10θ = 0.1θ = 0.5
DTLZ1 (M = 2)
10000 20000 30000 40000 500000.00
0.25
0.50θ = 3θ = 2θ = 8θ = 10θ = 1θ = 5θ = 0.1θ = 0.5
DTLZ1 (M = 2)
10000 20000 30000 40000 500000.0
0.5
θ = 3θ = 2θ = 5θ = 8θ = 10θ = 1θ = 0.1θ = 0.5
DTLZ1 (M = 3)
10000 20000 30000 40000 500000.0
0.5
θ = 5θ = 3θ = 8θ = 10θ = 2θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 3)
10000 20000 30000 40000 500000.0
0.5
1.0 θ = 8θ = 10θ = 3θ = 5θ = 2θ = 1θ = 0.5θ = 0.1
DTLZ1 (M = 4)
10000 20000 30000 40000 500000.0
0.5
1.0 θ = 10θ = 8θ = 3θ = 5θ = 2θ = 1θ = 0.5θ = 0.1
DTLZ1 (M = 4)
10000 20000 30000 40000 500000.00
0.05
0.10 θ = 10θ = 8θ = 5θ = 3θ = 2θ = 1θ = 0.5θ = 0.1
DTLZ1 (M = 5)
10000 20000 30000 40000 500000.0
0.2
θ = 10θ = 8θ = 5θ = 3θ = 2θ = 1θ = 0.5θ = 0.1
DTLZ1 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.27: Performance of MOEA/D using the PBI function gpbi with various θ values onthe DTLZ1 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
28
10000 20000 30000 40000 50000
0.2
0.3
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
DTLZ2 (M = 2)
10000 20000 30000 40000 50000
0.25
0.30
0.35 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
DTLZ2 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 3)
10000 20000 30000 40000 50000
0.2
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ2 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 1θ = 0.5
DTLZ2 (M = 5)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 1θ = 0.5
DTLZ2 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.28: Performance of MOEA/D using the PBI function gpbi with various θ values onthe DTLZ2 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
29
10000 20000 30000 40000 500000.0
0.2
θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
DTLZ3 (M = 2)
10000 20000 30000 40000 500000.0
0.2
θ = 5θ = 2θ = 3θ = 1θ = 8θ = 10θ = 0.1θ = 0.5
DTLZ3 (M = 2)
10000 20000 30000 40000 500000.00
0.05
0.10 θ = 8θ = 3θ = 10θ = 1θ = 2θ = 5θ = 0.5θ = 0.1
DTLZ3 (M = 3)
10000 20000 30000 40000 500000.0
0.1
0.2θ = 8θ = 10θ = 5θ = 2θ = 0.5θ = 3θ = 1θ = 0.1
DTLZ3 (M = 3)
10000 20000 30000 40000 500000.00
0.05
θ = 8θ = 10θ = 3θ = 1θ = 2θ = 5θ = 0.5θ = 0.1
DTLZ3 (M = 4)
10000 20000 30000 40000 500000.0
0.1
θ = 2θ = 0.5θ = 3θ = 1θ = 5θ = 0.1θ = 10θ = 8
DTLZ3 (M = 4)
10000 20000 30000 40000 500000.00
0.05
θ = 10θ = 2θ = 3θ = 0.1θ = 1θ = 8θ = 5θ = 0.5
DTLZ3 (M = 5)
10000 20000 30000 40000 500000.0
0.1
θ = 0.1θ = 3θ = 2θ = 0.5θ = 8θ = 5θ = 1θ = 10
DTLZ3 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.29: Performance of MOEA/D using the PBI function gpbi with various θ values onthe DTLZ3 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
30
10000 20000 30000 40000 500000.1
0.2
0.3
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 2)
10000 20000 30000 40000 500000.1
0.2
0.3
θ = 5θ = 3θ = 8θ = 2θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 3)
10000 20000 30000 40000 50000
0.2
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ4 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 2θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 0.5θ = 1
DTLZ4 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 3θ = 5θ = 8θ = 10θ = 2θ = 0.1θ = 0.5θ = 1
DTLZ4 (M = 5)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 3θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ4 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.30: Performance of MOEA/D using the PBI function gpbi with various θ values onthe DTLZ4 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
31
10000 20000 30000 40000 500000.0
0.2
0.4 θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
WFG1 (M = 2)
10000 20000 30000 40000 500000.0
0.2
0.4 θ = 3θ = 5θ = 2θ = 8θ = 0.1θ = 10θ = 1θ = 0.5
WFG1 (M = 2)
10000 20000 30000 40000 500000.00
0.25
0.50θ = 3θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
WFG1 (M = 3)
10000 20000 30000 40000 500000.00
0.25
0.50θ = 3θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
WFG1 (M = 3)
10000 20000 30000 40000 500000.0
0.2
0.4
θ = 5θ = 8θ = 3θ = 10θ = 0.1θ = 2θ = 1θ = 0.5
WFG1 (M = 4)
10000 20000 30000 40000 500000.0
0.2
0.4
θ = 5θ = 8θ = 3θ = 10θ = 0.1θ = 2θ = 1θ = 0.5
WFG1 (M = 4)
10000 20000 30000 40000 500000.00
0.25
0.50θ = 8θ = 10θ = 5θ = 0.1θ = 3θ = 2θ = 1θ = 0.5
WFG1 (M = 5)
10000 20000 30000 40000 500000.00
0.25
0.50θ = 8θ = 10θ = 5θ = 0.1θ = 3θ = 2θ = 1θ = 0.5
WFG1 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.31: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG1 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
32
10000 20000 30000 40000 50000
0.25
0.50θ = 8θ = 3θ = 10θ = 5θ = 0.1θ = 2θ = 0.5θ = 1
WFG2 (M = 2)
10000 20000 30000 40000 50000
0.5
0.6θ = 8θ = 3θ = 0.1θ = 10θ = 5θ = 2θ = 0.5θ = 1
WFG2 (M = 2)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 0.1θ = 5θ = 8θ = 3θ = 10θ = 2θ = 0.5θ = 1
WFG2 (M = 3)
10000 20000 30000 40000 50000
0.6
0.8 θ = 0.1θ = 5θ = 10θ = 3θ = 8θ = 2θ = 0.5θ = 1
WFG2 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 0.1θ = 8θ = 10θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG2 (M = 4)
10000 20000 30000 40000 50000
0.6
0.8θ = 0.1θ = 8θ = 10θ = 5θ = 3θ = 2θ = 1θ = 0.5
WFG2 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 1θ = 0.5
WFG2 (M = 5)
10000 20000 30000 40000 50000
0.6
0.8 θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG2 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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Figure S.32: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG2 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
33
10000 20000 30000 40000 50000
0.4
0.6 θ = 2θ = 3θ = 1θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG3 (M = 2)
10000 20000 30000 40000 50000
0.4
0.5
θ = 0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1
WFG3 (M = 2)
10000 20000 30000 40000 50000
0.25
0.50
θ = 2θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 1θ = 0.5
WFG3 (M = 3)
10000 20000 30000 40000 50000
0.5
0.6
θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG3 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
θ = 5θ = 8θ = 10θ = 0.1θ = 3θ = 1θ = 2θ = 0.5
WFG3 (M = 4)
10000 20000 30000 40000 50000
0.5
0.6
θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 2θ = 1θ = 0.5
WFG3 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
θ = 10θ = 8θ = 0.1θ = 5θ = 2θ = 3θ = 0.5θ = 1
WFG3 (M = 5)
10000 20000 30000 40000 50000
0.5
0.6
θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG3 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.33: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG3 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
34
10000 20000 30000 40000 500000.15
0.20
0.25
0.30
0.35 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG4 (M = 2)
10000 20000 30000 40000 500000.15
0.20
0.25
0.30
0.35 θ = 0.5θ = 0.1θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG4 (M = 2)
10000 20000 30000 40000 500000.0
0.2
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG4 (M = 3)
10000 20000 30000 40000 500000.0
0.2
0.4
0.6 θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG4 (M = 3)
10000 20000 30000 40000 500000.00.1
0.3
0.5
0.7 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG4 (M = 4)
10000 20000 30000 40000 500000.00.1
0.3
0.5
0.7 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG4 (M = 4)
10000 20000 30000 40000 500000.0
0.2
0.4
0.6
0.8 θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5θ = 2
WFG4 (M = 5)
10000 20000 30000 40000 500000.0
0.2
0.4
0.6
0.8 θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 2θ = 1
WFG4 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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Figure S.34: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG4 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
35
10000 20000 30000 40000 50000
0.2
0.3θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG5 (M = 2)
10000 20000 30000 40000 50000
0.2
0.3θ = 0.1θ = 0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG5 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG5 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
0.5θ = 0.1θ = 0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG5 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
WFG5 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG5 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG5 (M = 5)
10000 20000 30000 40000 50000
0.4
0.6
0.8 θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG5 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
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hyp
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l31
runs
Figure S.35: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG5 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
36
10000 20000 30000 40000 50000
0.2
0.3θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG6 (M = 2)
10000 20000 30000 40000 50000
0.2
0.3θ = 1θ = 2θ = 3θ = 0.5θ = 5θ = 8θ = 10θ = 0.1
WFG6 (M = 2)
10000 20000 30000 40000 500000.2
0.4
θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG6 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG6 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG6 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG6 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 3θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG6 (M = 5)
10000 20000 30000 40000 50000
0.4
0.6
θ = 3θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG6 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
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lum
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l31
runs
Figure S.36: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG6 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
37
10000 20000 30000 40000 50000
0.2
0.3
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG7 (M = 2)
10000 20000 30000 40000 50000
0.2
0.3
θ = 1θ = 2θ = 0.5θ = 3θ = 5θ = 8θ = 10θ = 0.1
WFG7 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG7 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
0.5θ = 2θ = 3θ = 0.1θ = 0.5θ = 5θ = 8θ = 10θ = 1
WFG7 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG7 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG7 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
0.75 θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5θ = 2
WFG7 (M = 5)
10000 20000 30000 40000 50000
0.4
0.6
θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
WFG7 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
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l31
runs
Figure S.37: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG7 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
38
10000 20000 30000 40000 50000
0.1
0.2
0.3 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 2)
10000 20000 30000 40000 50000
0.2
0.3 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG8 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1θ = 1
WFG8 (M = 3)
10000 20000 30000 40000 50000
0.2
0.4
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG8 (M = 4)
10000 20000 30000 40000 50000
0.3
0.4
0.5
θ = 2θ = 3θ = 5θ = 0.1θ = 8θ = 10θ = 0.5θ = 1
WFG8 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 0.5θ = 2θ = 1
WFG8 (M = 5)
10000 20000 30000 40000 50000
0.4
0.6 θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
WFG8 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
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l31
runs
Figure S.38: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG8 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
39
10000 20000 30000 40000 50000
0.1
0.2
0.3θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 2)
10000 20000 30000 40000 50000
0.2
0.3θ = 1θ = 3θ = 2θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 2)
10000 20000 30000 40000 50000
0.2
0.4
θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 3)
10000 20000 30000 40000 50000
0.3
0.4
0.5 θ = 2θ = 0.5θ = 0.1θ = 1θ = 3θ = 5θ = 8θ = 10
WFG9 (M = 3)
10000 20000 30000 40000 50000
0.25
0.50θ = 2θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 0.5θ = 1
WFG9 (M = 4)
10000 20000 30000 40000 50000
0.4
0.6 θ = 0.1θ = 2θ = 3θ = 5θ = 10θ = 8θ = 0.5θ = 1
WFG9 (M = 4)
10000 20000 30000 40000 50000
0.25
0.50
θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
WFG9 (M = 5)
10000 20000 30000 40000 50000
0.4
0.6θ = 3θ = 5θ = 0.1θ = 8θ = 10θ = 2θ = 0.5θ = 1
WFG9 (M = 5)
(a) Final Population scenario (b) Reduced UEA scenario
Number of function evaluations Number of function evaluations
Med
ian
hyp
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lues
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l31
runs
Figure S.39: Performance of MOEA/D using the PBI function gpbi with various θ valueson the WFG9 problem with M ∈ {2, 3, 4, 5}. The horizontal and vertical axes represent thenumber of function evaluations and the HV values, respectively. The shaded area indicates25-75 percentiles.
40
25 100 200 300 400µ
0.2
0.3
0.4
0.5
0.6 gchd
gpbi
gchm
DTLZ1 (M = 2)
28 105 210 300 406µ
0.0
0.2
0.4
0.6
0.8gpbi
gchd
gchm
DTLZ1 (M = 3)
126 210 330 495µ
0.0
0.2
0.4
0.6
0.8gchd
gchm
gpbi
DTLZ1 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.50
0.52
0.54
0.56
0.58 gchd
gchm
gpbi
DTLZ1 (M = 2)
28 105 210 300 406µ
0.0
0.2
0.4
0.6
0.8gchd
gpbi
gchm
DTLZ1 (M = 3)
126 210 330 495µ
0.0
0.2
0.4
0.6
0.8
1.0 gchd
gchm
gpbi
DTLZ1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.40: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the DTLZ1 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
25 100 200 300 400µ
0.3360.3380.3400.3420.3440.3460.3480.350
gchm
gchd
gpbi
DTLZ2 (M = 2)
28 105 210 300 406µ
0.46
0.48
0.50
0.52
0.54
0.56
0.58 gchd
gpbi
gchm
DTLZ2 (M = 3)
126 210 330 495µ
0.72
0.74
0.76
0.78
0.80
0.82 gpbi
gchd
gchm
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.34550.34600.34650.34700.34750.34800.34850.3490
gchmgchdgpbi
DTLZ2 (M = 2)
28 105 210 300 406µ
0.566
0.568
0.570
0.572
0.574
gchd
gchm
gpbi
DTLZ2 (M = 3)
126 210 330 495µ
0.780
0.785
0.790
0.795
0.800
0.805gchm
gchd
gpbi
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.41: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the DTLZ2 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
41
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35 gchd
gchm
gpbi
DTLZ3 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5gchd
gchm
gpbi
DTLZ3 (M = 3)
126 210 330 495µ
0.10
0.15
0.20
0.25
0.30
0.35 gchd
gchm
gpbi
DTLZ3 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.15
0.20
0.25
0.30
0.35 gchd
gchm
gpbi
DTLZ3 (M = 2)
28 105 210 300 406µ
0.2
0.3
0.4
0.5
0.6 gchd
gpbi
gchm
DTLZ3 (M = 3)
126 210 330 495µ
0.150.200.250.300.350.400.450.50
gchd
gchm
gpbi
DTLZ3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.42: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the DTLZ3 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35 gchm
gchd
gpbi
DTLZ4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 gchd
gpbi
gchm
DTLZ4 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7
0.8gpbi
gchd
gchm
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35 gchm
gpbi
gchd
DTLZ4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 gchd
gpbi
gchm
DTLZ4 (M = 3)
126 210 330 495µ
0.4
0.5
0.6
0.7
0.8 gpbi
gchd
gchm
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.43: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the DTLZ4 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
42
25 100 200 300 400µ
0.250.300.350.400.450.500.55 gchm
gchd
gpbi
WFG1 (M = 2)
28 105 210 300 406µ
0.450.500.550.600.650.700.75 gchd
gchm
gpbi
WFG1 (M = 3)
126 210 330 495µ
0.300.350.400.450.500.550.600.65 gchm
gpbi
gchd
WFG1 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.250.300.350.400.450.500.55
gchm
gchd
gpbi
WFG1 (M = 2)
28 105 210 300 406µ
0.450.500.550.600.650.700.750.80 gchd
gchm
gpbi
WFG1 (M = 3)
126 210 330 495µ
0.350.400.450.500.550.600.65 gchm
gchd
gpbi
WFG1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.44: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG1 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
25 100 200 300 400µ
0.530.540.550.560.570.580.590.600.61 gchm
gchd
gpbi
WFG2 (M = 2)
28 105 210 300 406µ
0.6250.6500.6750.7000.7250.7500.775
gchd
gchm
gpbi
WFG2 (M = 3)
126 210 330 495µ
0.450.500.550.600.650.700.750.80
gchm
gchd
gpbi
WFG2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.540.550.560.570.580.590.600.61 gchm
gchd
gpbi
WFG2 (M = 2)
28 105 210 300 406µ
0.660.680.700.720.740.760.780.80 gchm
gchd
gpbi
WFG2 (M = 3)
126 210 330 495µ
0.6250.6500.6750.7000.7250.7500.7750.8000.825 gchm
gchd
gpbi
WFG2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.45: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG2 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
43
25 100 200 300 400µ
0.550
0.555
0.560
0.565
0.570
0.575 gchm
gchd
gpbi
WFG3 (M = 2)
28 105 210 300 406µ
0.570.580.590.600.610.620.630.64 gchd
gchm
gpbi
WFG3 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6
0.7 gchm
gchd
gpbi
WFG3 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.56000.56250.56500.56750.57000.57250.57500.57750.5800
gchm
gchd
gpbi
WFG3 (M = 2)
28 105 210 300 406µ
0.6000.6050.6100.6150.6200.6250.6300.635
gchd
gchm
gpbi
WFG3 (M = 3)
126 210 330 495µ
0.500.520.540.560.580.600.620.640.66 gchd
gchm
gpbi
WFG3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.46: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG3 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
25 100 200 300 400µ
0.32
0.33
0.34
0.35
gchm
gchd
gpbi
WFG4 (M = 2)
28 105 210 300 406µ
0.46
0.48
0.50
0.52
0.54
0.56 gchd
gchm
gpbi
WFG4 (M = 3)
126 210 330 495µ
0.50
0.55
0.60
0.65
0.70gpbi
gchm
gchd
WFG4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.32
0.33
0.34
0.35gchm
gchd
gpbi
WFG4 (M = 2)
28 105 210 300 406µ
0.48
0.50
0.52
0.54
0.56
0.58gchd
gchm
gpbi
WFG4 (M = 3)
126 210 330 495µ
0.6250.6500.6750.7000.7250.7500.775 gchm
gchd
gpbi
WFG4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.47: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG4 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
44
25 100 200 300 400µ
0.298
0.300
0.302
0.304
0.306
0.308
0.310
gchd
gchm
gpbi
WFG5 (M = 2)
28 105 210 300 406µ
0.42
0.44
0.46
0.48
0.50
0.52 gchd
gchm
gpbi
WFG5 (M = 3)
126 210 330 495µ
0.5250.5500.5750.6000.6250.6500.6750.700 gchd
gpbi
gchm
WFG5 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.304
0.306
0.308
0.310
0.312
gchd
gchm
gpbi
WFG5 (M = 2)
28 105 210 300 406µ
0.470.480.490.500.510.520.53 gchd
gchm
gpbi
WFG5 (M = 3)
126 210 330 495µ
0.600.620.640.660.680.700.720.74 gchm
gchd
gpbi
WFG5 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.48: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG5 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
25 100 200 300 400µ
0.30000.30250.30500.30750.31000.31250.31500.31750.3200
gchm
gchd
gpbi
WFG6 (M = 2)
28 105 210 300 406µ
0.42
0.44
0.46
0.48
0.50
0.52
0.54 gchd
gchm
gpbi
WFG6 (M = 3)
126 210 330 495µ
0.5250.5500.5750.6000.6250.6500.6750.700 gpbi
gchd
gchm
WFG6 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.30250.30500.30750.31000.31250.31500.31750.3200
gchm
gchd
gpbi
WFG6 (M = 2)
28 105 210 300 406µ
0.460.470.480.490.500.510.520.530.54 gchd
gchm
gpbi
WFG6 (M = 3)
126 210 330 495µ
0.600.620.640.660.680.700.720.74 gchm
gchd
gpbi
WFG6 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.49: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG6 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
45
25 100 200 300 400µ
0.3320.3340.3360.3380.3400.3420.3440.3460.348
gchd
gchm
gpbi
WFG7 (M = 2)
28 105 210 300 406µ
0.46
0.48
0.50
0.52
0.54
0.56gchd
gchm
gpbi
WFG7 (M = 3)
126 210 330 495µ
0.580.600.620.640.660.680.700.720.74 gchd
gpbi
gchm
WFG7 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.338
0.340
0.342
0.344
0.346
0.348
gchd
gchm
gpbi
WFG7 (M = 2)
28 105 210 300 406µ
0.50
0.52
0.54
0.56gchd
gchm
gpbi
WFG7 (M = 3)
126 210 330 495µ
0.640.660.680.700.720.740.760.78 gchm
gchd
gpbi
WFG7 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.50: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG7 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
25 100 200 300 400µ
0.2700.2750.2800.2850.2900.2950.300 gchm
gchd
gpbi
WFG8 (M = 2)
28 105 210 300 406µ
0.360.380.400.420.440.460.480.50 gchd
gchm
gpbi
WFG8 (M = 3)
126 210 330 495µ
0.300.350.400.450.500.550.600.65 gpbi
gchm
gchd
WFG8 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.2860.2880.2900.2920.2940.2960.2980.300
gchm
gchd
gpbi
WFG8 (M = 2)
28 105 210 300 406µ
0.44
0.45
0.46
0.47
0.48
0.49gchd
gchm
gpbi
WFG8 (M = 3)
126 210 330 495µ
0.54
0.56
0.58
0.60
0.62gchm
gchd
gpbi
WFG8 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.51: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG8 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
46
25 100 200 300 400µ
0.28
0.29
0.30
0.31
0.32 gchd
gchm
gpbi
WFG9 (M = 2)
28 105 210 300 406µ
0.340.360.380.400.420.440.460.480.50 gchm
gchd
gpbi
WFG9 (M = 3)
126 210 330 495µ
0.35
0.40
0.45
0.50
0.55
0.60gpbi
gchd
gchm
WFG9 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.28
0.29
0.30
0.31
0.32gchd
gchm
gpbi
WFG9 (M = 2)
28 105 210 300 406µ
0.46
0.47
0.48
0.49gchm
gchd
gpbi
WFG9 (M = 3)
126 210 330 495µ
0.56
0.58
0.60
0.62
0.64gchm
gchd
gpbi
WFG9 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.52: Influence of µ on the performance of MOEA/D with the three scalarizing func-tions (gchm, gchd, and gpbi) on the WFG9 problem with M ∈ {2, 3, 5}. The median HV valueat 50 000 evaluations among 31 runs is shown.
47
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 2)
28 105 210 300 406µ
0.0
0.2
0.4
0.6
0.8θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 3)
126 210 330 495µ
0.00
0.02
0.04
0.06
0.08
θ = 8θ = 10θ = 5θ = 3θ = 2θ = 1θ = 0.5θ = 0.1
DTLZ1 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 5θ = 3θ = 10θ = 0.1θ = 8θ = 1θ = 0.5
DTLZ1 (M = 2)
28 105 210 300 406µ
0.0
0.2
0.4
0.6
0.8θ = 5θ = 3θ = 8θ = 10θ = 2θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 3)
126 210 330 495µ
0.000.050.100.150.200.250.30
θ = 10θ = 8θ = 5θ = 0.1θ = 3θ = 2θ = 1θ = 0.5
DTLZ1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.53: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ1 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
25 100 200 300 400µ
0.1750.2000.2250.2500.2750.3000.3250.350
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
DTLZ2 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.80.9 θ = 3
θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.280.290.300.310.320.330.340.35
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
DTLZ2 (M = 2)
28 105 210 300 406µ
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 3)
126 210 330 495µ
0.20.30.40.50.60.70.8
θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.54: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ2 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
48
25 100 200 300 400µ
0.000.050.100.150.200.250.300.35 θ = 2
θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ3 (M = 2)
28 105 210 300 406µ
0.10
0.15
0.20
0.25
θ = 5θ = 8θ = 10θ = 3θ = 1θ = 2θ = 0.5θ = 0.1
DTLZ3 (M = 3)
126 210 330 495µ
0.084
0.085
0.086
0.087
0.088θ = 5θ = 2θ = 3θ = 0.5θ = 10θ = 0.1θ = 8θ = 1
DTLZ3 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.050.100.150.200.250.300.35 θ = 2
θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ3 (M = 2)
28 105 210 300 406µ
0.150.200.250.300.350.400.450.50
θ = 5θ = 10θ = 8θ = 3θ = 0.1θ = 2θ = 1θ = 0.5
DTLZ3 (M = 3)
126 210 330 495µ
0.1320.1340.1360.1380.1400.1420.1440.146
θ = 5θ = 0.5θ = 2θ = 0.1θ = 8θ = 3θ = 10θ = 1
DTLZ3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.55: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ3 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.80.9 θ = 3
θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35θ = 3θ = 5θ = 2θ = 8θ = 0.1θ = 10θ = 1θ = 0.5
DTLZ4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.56: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ4 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
49
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4θ = 0.1θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.5θ = 1
WFG1 (M = 2)
28 105 210 300 406µ
0.00.10.20.30.40.50.6
θ = 3θ = 2θ = 5θ = 0.1θ = 8θ = 10θ = 1θ = 0.5
WFG1 (M = 3)
126 210 330 495µ
0.0
0.1
0.2
0.3
0.4
0.5θ = 5θ = 8θ = 10θ = 3θ = 0.1θ = 2θ = 1θ = 0.5
WFG1 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4θ = 0.1θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.5θ = 1
WFG1 (M = 2)
28 105 210 300 406µ
0.00.10.20.30.40.50.60.7 θ = 3
θ = 2θ = 5θ = 0.1θ = 8θ = 10θ = 1θ = 0.5
WFG1 (M = 3)
126 210 330 495µ
0.0
0.1
0.2
0.3
0.4
0.5θ = 5θ = 8θ = 10θ = 3θ = 0.1θ = 2θ = 1θ = 0.5
WFG1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.57: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG1 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 0.1θ = 5θ = 3θ = 8θ = 10θ = 2θ = 0.5θ = 1
WFG2 (M = 2)
28 105 210 300 406µ
0.10.20.30.40.50.60.70.8
θ = 0.1θ = 5θ = 3θ = 8θ = 10θ = 2θ = 1θ = 0.5
WFG2 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8
θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.40
0.45
0.50
0.55
0.60θ = 0.1θ = 3θ = 2θ = 5θ = 0.5θ = 10θ = 8θ = 1
WFG2 (M = 2)
28 105 210 300 406µ
0.450.500.550.600.650.700.750.80 θ = 0.1
θ = 3θ = 8θ = 5θ = 10θ = 2θ = 0.5θ = 1
WFG2 (M = 3)
126 210 330 495µ
0.55
0.60
0.65
0.70
0.75
0.80θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.58: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG2 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
50
25 100 200 300 400µ
0.4000.4250.4500.4750.5000.5250.5500.575
θ = 2θ = 3θ = 1θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG3 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 2θ = 3θ = 5θ = 8θ = 0.1θ = 10θ = 0.5θ = 1
WFG3 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 8θ = 10θ = 5θ = 0.1θ = 3θ = 2θ = 0.5θ = 1
WFG3 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.53
0.54
0.55
0.56
0.57
0.58 θ = 0.5θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.1
WFG3 (M = 2)
28 105 210 300 406µ
0.45
0.50
0.55
0.60
0.65 θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG3 (M = 3)
126 210 330 495µ
0.4750.5000.5250.5500.5750.6000.6250.650
θ = 0.1θ = 5θ = 8θ = 10θ = 3θ = 2θ = 1θ = 0.5
WFG3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.59: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG3 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.1750.2000.2250.2500.2750.3000.3250.350 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG4 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8 θ = 3
θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 1θ = 0.5
WFG4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.32
0.33
0.34
0.35 θ = 0.5θ = 0.1θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG4 (M = 2)
28 105 210 300 406µ
0.2
0.3
0.4
0.5
0.6 θ = 0.1θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5
WFG4 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
WFG4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.60: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG4 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
51
25 100 200 300 400µ
0.1500.1750.2000.2250.2500.2750.3000.325 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG5 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG5 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.7
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG5 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.30000.30250.30500.30750.31000.31250.3150
θ = 0.5θ = 1θ = 2θ = 0.1θ = 3θ = 5θ = 8θ = 10
WFG5 (M = 2)
28 105 210 300 406µ
0.30
0.35
0.40
0.45
0.50
0.55 θ = 0.1θ = 0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG5 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7
θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG5 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.61: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG5 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.1500.1750.2000.2250.2500.2750.3000.325 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG6 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG6 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.7
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG6 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.240.250.260.270.280.290.300.310.32 θ = 1
θ = 0.5θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1
WFG6 (M = 2)
28 105 210 300 406µ
0.200.250.300.350.400.450.500.55 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG6 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7θ = 2θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG6 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.62: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG6 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
52
25 100 200 300 400µ
0.1750.2000.2250.2500.2750.3000.3250.350 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG7 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG7 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8 θ = 3
θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
WFG7 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.31
0.32
0.33
0.34
0.35 θ = 1θ = 2θ = 3θ = 0.5θ = 5θ = 8θ = 10θ = 0.1
WFG7 (M = 2)
28 105 210 300 406µ
0.200.250.300.350.400.450.500.550.60 θ = 1
θ = 2θ = 0.1θ = 0.5θ = 3θ = 5θ = 8θ = 10
WFG7 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7
0.8 θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 2θ = 0.5θ = 1
WFG7 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.63: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG7 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 2)
28 105 210 300 406µ
0.100.150.200.250.300.350.400.450.50 θ = 2
θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
WFG8 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.18
0.20
0.22
0.24
0.26
0.28
0.30θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 2)
28 105 210 300 406µ
0.200.250.300.350.400.450.50 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 3)
126 210 330 495µ
0.250.300.350.400.450.500.550.600.65 θ = 3
θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
WFG8 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.64: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG8 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
53
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6
0.7 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG9 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.28
0.29
0.30
0.31
0.32
0.33θ = 1θ = 2θ = 3θ = 5θ = 0.5θ = 8θ = 10θ = 0.1
WFG9 (M = 2)
28 105 210 300 406µ
0.150.200.250.300.350.400.450.50
θ = 1θ = 2θ = 0.5θ = 0.1θ = 3θ = 5θ = 8θ = 10
WFG9 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
θ = 2θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG9 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.65: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG9 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 2)
28 105 210 300 406µ
0.0
0.2
0.4
0.6
0.8θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 3)
126 210 330 495µ
0.00
0.02
0.04
0.06
0.08
θ = 8θ = 10θ = 5θ = 3θ = 2θ = 1θ = 0.5θ = 0.1
DTLZ1 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 5θ = 3θ = 10θ = 0.1θ = 8θ = 1θ = 0.5
DTLZ1 (M = 2)
28 105 210 300 406µ
0.0
0.2
0.4
0.6
0.8θ = 5θ = 3θ = 8θ = 10θ = 2θ = 0.1θ = 1θ = 0.5
DTLZ1 (M = 3)
126 210 330 495µ
0.000.050.100.150.200.250.30
θ = 10θ = 8θ = 5θ = 0.1θ = 3θ = 2θ = 1θ = 0.5
DTLZ1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.66: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ1 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
54
25 100 200 300 400µ
0.1750.2000.2250.2500.2750.3000.3250.350
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
DTLZ2 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.80.9 θ = 3
θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.280.290.300.310.320.330.340.35
θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
DTLZ2 (M = 2)
28 105 210 300 406µ
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ2 (M = 3)
126 210 330 495µ
0.20.30.40.50.60.70.8
θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.67: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ2 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
55
25 100 200 300 400µ
0.000.050.100.150.200.250.300.35 θ = 2
θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ3 (M = 2)
28 105 210 300 406µ
0.10
0.15
0.20
0.25
θ = 5θ = 8θ = 10θ = 3θ = 1θ = 2θ = 0.5θ = 0.1
DTLZ3 (M = 3)
126 210 330 495µ
0.084
0.085
0.086
0.087
0.088θ = 5θ = 2θ = 3θ = 0.5θ = 10θ = 0.1θ = 8θ = 1
DTLZ3 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.050.100.150.200.250.300.35 θ = 2
θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ3 (M = 2)
28 105 210 300 406µ
0.150.200.250.300.350.400.450.50
θ = 5θ = 10θ = 8θ = 3θ = 0.1θ = 2θ = 1θ = 0.5
DTLZ3 (M = 3)
126 210 330 495µ
0.1320.1340.1360.1380.1400.1420.1440.146
θ = 5θ = 0.5θ = 2θ = 0.1θ = 8θ = 3θ = 10θ = 1
DTLZ3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.68: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ3 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.80.9 θ = 3
θ = 2θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35θ = 3θ = 5θ = 2θ = 8θ = 0.1θ = 10θ = 1θ = 0.5
DTLZ4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 1θ = 0.5
DTLZ4 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.69: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the DTLZ4 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluationsamong 31 runs is shown.
56
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4θ = 0.1θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.5θ = 1
WFG1 (M = 2)
28 105 210 300 406µ
0.00.10.20.30.40.50.6
θ = 3θ = 2θ = 5θ = 0.1θ = 8θ = 10θ = 1θ = 0.5
WFG1 (M = 3)
126 210 330 495µ
0.0
0.1
0.2
0.3
0.4
0.5θ = 5θ = 8θ = 10θ = 3θ = 0.1θ = 2θ = 1θ = 0.5
WFG1 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.0
0.1
0.2
0.3
0.4θ = 0.1θ = 3θ = 5θ = 2θ = 8θ = 10θ = 0.5θ = 1
WFG1 (M = 2)
28 105 210 300 406µ
0.00.10.20.30.40.50.60.7 θ = 3
θ = 2θ = 5θ = 0.1θ = 8θ = 10θ = 1θ = 0.5
WFG1 (M = 3)
126 210 330 495µ
0.0
0.1
0.2
0.3
0.4
0.5θ = 5θ = 8θ = 10θ = 3θ = 0.1θ = 2θ = 1θ = 0.5
WFG1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.70: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG1 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 0.1θ = 5θ = 3θ = 8θ = 10θ = 2θ = 0.5θ = 1
WFG2 (M = 2)
28 105 210 300 406µ
0.10.20.30.40.50.60.70.8
θ = 0.1θ = 5θ = 3θ = 8θ = 10θ = 2θ = 1θ = 0.5
WFG2 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8
θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.40
0.45
0.50
0.55
0.60θ = 0.1θ = 3θ = 2θ = 5θ = 0.5θ = 10θ = 8θ = 1
WFG2 (M = 2)
28 105 210 300 406µ
0.450.500.550.600.650.700.750.80 θ = 0.1
θ = 3θ = 8θ = 5θ = 10θ = 2θ = 0.5θ = 1
WFG2 (M = 3)
126 210 330 495µ
0.55
0.60
0.65
0.70
0.75
0.80θ = 0.1θ = 10θ = 8θ = 5θ = 3θ = 2θ = 0.5θ = 1
WFG2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.71: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG2 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
57
25 100 200 300 400µ
0.4000.4250.4500.4750.5000.5250.5500.575
θ = 2θ = 3θ = 1θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG3 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 2θ = 3θ = 5θ = 8θ = 0.1θ = 10θ = 0.5θ = 1
WFG3 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 8θ = 10θ = 5θ = 0.1θ = 3θ = 2θ = 0.5θ = 1
WFG3 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.53
0.54
0.55
0.56
0.57
0.58 θ = 0.5θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.1
WFG3 (M = 2)
28 105 210 300 406µ
0.45
0.50
0.55
0.60
0.65 θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG3 (M = 3)
126 210 330 495µ
0.4750.5000.5250.5500.5750.6000.6250.650
θ = 0.1θ = 5θ = 8θ = 10θ = 3θ = 2θ = 1θ = 0.5
WFG3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.72: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG3 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.1750.2000.2250.2500.2750.3000.3250.350 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG4 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
0.6 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG4 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8 θ = 3
θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 1θ = 0.5
WFG4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.32
0.33
0.34
0.35 θ = 0.5θ = 0.1θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG4 (M = 2)
28 105 210 300 406µ
0.2
0.3
0.4
0.5
0.6 θ = 0.1θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5
WFG4 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
WFG4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.73: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG4 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
58
25 100 200 300 400µ
0.1500.1750.2000.2250.2500.2750.3000.325 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG5 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG5 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.7
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG5 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.30000.30250.30500.30750.31000.31250.3150
θ = 0.5θ = 1θ = 2θ = 0.1θ = 3θ = 5θ = 8θ = 10
WFG5 (M = 2)
28 105 210 300 406µ
0.30
0.35
0.40
0.45
0.50
0.55 θ = 0.1θ = 0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10
WFG5 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7
θ = 0.1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG5 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.74: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG5 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.1500.1750.2000.2250.2500.2750.3000.325 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG6 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG6 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.7
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG6 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.240.250.260.270.280.290.300.310.32 θ = 1
θ = 0.5θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1
WFG6 (M = 2)
28 105 210 300 406µ
0.200.250.300.350.400.450.500.55 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5
WFG6 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7θ = 2θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG6 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.75: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG6 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
59
25 100 200 300 400µ
0.1750.2000.2250.2500.2750.3000.3250.350 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG7 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5
θ = 2θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG7 (M = 3)
126 210 330 495µ
0.10.20.30.40.50.60.70.8 θ = 3
θ = 5θ = 8θ = 10θ = 0.1θ = 2θ = 0.5θ = 1
WFG7 (M = 5)H
yper
volu
me
(a) Final Population scenario
25 100 200 300 400µ
0.31
0.32
0.33
0.34
0.35 θ = 1θ = 2θ = 3θ = 0.5θ = 5θ = 8θ = 10θ = 0.1
WFG7 (M = 2)
28 105 210 300 406µ
0.200.250.300.350.400.450.500.550.60 θ = 1
θ = 2θ = 0.1θ = 0.5θ = 3θ = 5θ = 8θ = 10
WFG7 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
0.7
0.8 θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 2θ = 0.5θ = 1
WFG7 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.76: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG7 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 2)
28 105 210 300 406µ
0.100.150.200.250.300.350.400.450.50 θ = 2
θ = 1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6θ = 3θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
WFG8 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.18
0.20
0.22
0.24
0.26
0.28
0.30θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 2)
28 105 210 300 406µ
0.200.250.300.350.400.450.50 θ = 1
θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG8 (M = 3)
126 210 330 495µ
0.250.300.350.400.450.500.550.600.65 θ = 3
θ = 5θ = 10θ = 8θ = 0.1θ = 2θ = 0.5θ = 1
WFG8 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.77: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG8 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
60
25 100 200 300 400µ
0.10
0.15
0.20
0.25
0.30
0.35 θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 2)
28 105 210 300 406µ
0.1
0.2
0.3
0.4
0.5θ = 1θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 0.1
WFG9 (M = 3)
126 210 330 495µ
0.1
0.2
0.3
0.4
0.5
0.6
0.7 θ = 2θ = 3θ = 5θ = 8θ = 10θ = 0.1θ = 0.5θ = 1
WFG9 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
25 100 200 300 400µ
0.28
0.29
0.30
0.31
0.32
0.33θ = 1θ = 2θ = 3θ = 5θ = 0.5θ = 8θ = 10θ = 0.1
WFG9 (M = 2)
28 105 210 300 406µ
0.150.200.250.300.350.400.450.50
θ = 1θ = 2θ = 0.5θ = 0.1θ = 3θ = 5θ = 8θ = 10
WFG9 (M = 3)
126 210 330 495µ
0.3
0.4
0.5
0.6
θ = 2θ = 0.1θ = 3θ = 5θ = 8θ = 10θ = 0.5θ = 1
WFG9 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.78: Influence of µ on the performance of MOEA/D using gpbi with various θ valueson the WFG9 problem with M ∈ {2, 3, 5}. The median HV value at 50, 000 evaluations among31 runs is shown.
61
0.1 2.0 5.0 8.0 10.0θ
0.0
0.1
0.2
0.3
0.4
0.5
0.6 gchm
gchd
gpbi
DTLZ1 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.0
0.2
0.4
0.6
0.8gchd
gpbi
gchm
DTLZ1 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.00.10.20.30.40.50.6
gchd
gchm
gpbi
DTLZ1 (M = 5)H
yper
volu
me
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6 gchd
gchm
gpbi
DTLZ1 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.0
0.2
0.4
0.6
0.8gchd
gchm
gpbi
DTLZ1 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.0
0.2
0.4
0.6
0.8
gchd
gchm
gpbi
DTLZ1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.79: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the DTLZ1 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
0.1 2.0 5.0 8.0 10.0θ
0.1750.2000.2250.2500.2750.3000.3250.350 gchm
gchd
gpbi
DTLZ2 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6 gchd
gpbi
gchm
DTLZ2 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8
gpbi
gchd
gchm
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.290.300.310.320.330.340.35 gchd
gchm
gpbi
DTLZ2 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.200.250.300.350.400.450.500.550.60 gchm
gchd
gpbi
DTLZ2 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.20.30.40.50.60.70.8
gchm
gpbi
gchd
DTLZ2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.80: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the DTLZ2 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
62
0.1 2.0 5.0 8.0 10.0θ
0.1750.2000.2250.2500.2750.3000.3250.350 gchd
gchm
gpbi
DTLZ3 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.0750.1000.1250.1500.1750.2000.2250.2500.2750.300 gchd
gchm
gpbi
DTLZ3 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10
0.15
0.20
0.25
0.30 gchd
gchm
gpbi
DTLZ3 (M = 5)H
yper
volu
me
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.24
0.26
0.28
0.30
0.32
0.34gchd
gchm
gpbi
DTLZ3 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.2
0.3
0.4
0.5gchd
gchm
gpbi
DTLZ3 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.150.200.250.300.350.400.45
gchm
gchd
gpbi
DTLZ3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.81: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the DTLZ3 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
0.1 2.0 5.0 8.0 10.0θ
0.10
0.15
0.20
0.25
0.30
0.35 gchm
gpbi
gchd
DTLZ4 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6 gpbi
gchd
gchm
DTLZ4 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8
gpbi
gchd
gchm
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.10
0.15
0.20
0.25
0.30
0.35 gchm
gpbi
gchd
DTLZ4 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6 gpbi
gchm
gchd
DTLZ4 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8 gpbi
gchd
gchm
DTLZ4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.82: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the DTLZ4 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
63
0.1 2.0 5.0 8.0 10.0θ
0.0
0.1
0.2
0.3
0.4
0.5gchm
gchd
gpbi
WFG1 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.00.10.20.30.40.50.60.7 gchd
gchm
gpbi
WFG1 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.00.10.20.30.40.50.6
gchm
gchd
gpbi
WFG1 (M = 5)H
yper
volu
me
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.0
0.1
0.2
0.3
0.4
0.5gchm
gchd
gpbi
WFG1 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.00.10.20.30.40.50.60.7 gchd
gchm
gpbi
WFG1 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.00.10.20.30.40.50.6
gchm
gchd
gpbi
WFG1 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.83: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG1 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
0.1 2.0 5.0 8.0 10.0θ
0.460.480.500.520.540.560.580.600.62 gchd
gchm
gpbi
WFG2 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8 gchd
gchm
gpbi
WFG2 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8 gchm
gchd
gpbi
WFG2 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.52
0.54
0.56
0.58
0.60
0.62 gchd
gchm
gpbi
WFG2 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.55
0.60
0.65
0.70
0.75
0.80 gchd
gchm
gpbi
WFG2 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.60
0.65
0.70
0.75
0.80gchm
gchd
gpbi
WFG2 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.84: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG2 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
64
0.1 2.0 5.0 8.0 10.0θ
0.4000.4250.4500.4750.5000.5250.5500.575
gchd
gchm
gpbi
WFG3 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6gchm
gchd
gpbi
WFG3 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6gchm
gpbi
gchd
WFG3 (M = 5)H
yper
volu
me
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.53
0.54
0.55
0.56
0.57
0.58 gchd
gchm
gpbi
WFG3 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.4750.5000.5250.5500.5750.6000.6250.650 gchd
gchm
gpbi
WFG3 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.500.520.540.560.580.600.620.640.66 gchm
gchd
gpbi
WFG3 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.85: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG3 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
0.1 2.0 5.0 8.0 10.0θ
0.1750.2000.2250.2500.2750.3000.3250.350 gchd
gchm
gpbi
WFG4 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6 gchd
gchm
gpbi
WFG4 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8 gpbi
gchd
gchm
WFG4 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.32
0.33
0.34
0.35 gchd
gchm
gpbi
WFG4 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.3
0.4
0.5
0.6 gchd
gchm
gpbi
WFG4 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.3
0.4
0.5
0.6
0.7
0.8 gchm
gchd
gpbi
WFG4 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.86: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG4 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
65
0.1 2.0 5.0 8.0 10.0θ
0.1500.1750.2000.2250.2500.2750.3000.325 gchm
gchd
gpbi
WFG5 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.200.250.300.350.400.450.50
gchd
gchm
gpbi
WFG5 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.7 gpbi
gchd
gchm
WFG5 (M = 5)H
yper
volu
me
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.304
0.306
0.308
0.310
0.312 gchd
gchm
gpbi
WFG5 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.48
0.49
0.50
0.51
0.52gchd
gchm
gpbi
WFG5 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.3
0.4
0.5
0.6
0.7gchm
gchd
gpbi
WFG5 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.87: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG5 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
0.1 2.0 5.0 8.0 10.0θ
0.1500.1750.2000.2250.2500.2750.3000.325 gchd
gchm
gpbi
WFG6 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.200.250.300.350.400.450.500.55 gchd
gchm
gpbi
WFG6 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.7 gpbi
gchd
gchm
WFG6 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.250.260.270.280.290.300.310.32 gchd
gchm
gpbi
WFG6 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.400.420.440.460.480.500.520.54 gchd
gchm
gpbi
WFG6 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.3
0.4
0.5
0.6
0.7gchm
gchd
gpbi
WFG6 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.88: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG6 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
66
0.1 2.0 5.0 8.0 10.0θ
0.1750.2000.2250.2500.2750.3000.3250.350 gchd
gchm
gpbi
WFG7 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6 gchd
gchm
gpbi
WFG7 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.10.20.30.40.50.60.70.8 gpbi
gchd
gchm
WFG7 (M = 5)H
yper
volu
me
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.3200.3250.3300.3350.3400.3450.350 gchd
gchm
gpbi
WFG7 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.30
0.35
0.40
0.45
0.50
0.55gchd
gchm
gpbi
WFG7 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.3
0.4
0.5
0.6
0.7
0.8 gchm
gchd
gpbi
WFG7 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.89: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG7 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
0.1 2.0 5.0 8.0 10.0θ
0.10
0.15
0.20
0.25
0.30 gchd
gchm
gpbi
WFG8 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5 gchd
gchm
gpbi
WFG8 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6gpbi
gchm
gchd
WFG8 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.20
0.22
0.24
0.26
0.28
0.30 gchd
gchm
gpbi
WFG8 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.30
0.35
0.40
0.45
0.50 gchd
gchm
gpbi
WFG8 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.300.350.400.450.500.550.600.65 gchm
gpbi
gchd
WFG8 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.90: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG8 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
67
0.1 2.0 5.0 8.0 10.0θ
0.10
0.15
0.20
0.25
0.30
0.35 gchm
gchd
gpbi
WFG9 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.150.200.250.300.350.400.450.50 gchd
gchm
gpbi
WFG9 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.1
0.2
0.3
0.4
0.5
0.6gpbi
gchd
gchm
WFG9 (M = 5)
Hyp
ervo
lum
e
(a) Final Population scenario
0.1 2.0 5.0 8.0 10.0θ
0.28
0.29
0.30
0.31
0.32
0.33gchd
gchm
gpbi
WFG9 (M = 2)
0.1 2.0 5.0 8.0 10.0θ
0.4500.4550.4600.4650.4700.4750.4800.485 gchd
gchm
gpbi
WFG9 (M = 3)
0.1 2.0 5.0 8.0 10.0θ
0.3
0.4
0.5
0.6
gchm
gchd
gpbi
WFG9 (M = 5)
Hyp
ervo
lum
e
(b) Reduced UEA scenario
Figure S.91: Comparison of the two Chebyshev functions (gchm and gchd) and gpbi withvarious θ values on the WFG9 problem with M ∈ {2, 3, 5}. The median HV value at 50 000evaluations among 31 runs is shown.
68