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: rt.ryoji.tanabe@gmail.com (Ryoji Tanabe), hisao@sustc.edu.cn

(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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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(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

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(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

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(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

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(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

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(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

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(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

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(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

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(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

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(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

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(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

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(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