Table 1 IGD metrics and HV metrics of each algorithm with 6 test functions.

From: Research and engineering application of layout optimization for lockbolt structures in railway wagons considering multidimensional failure modes based on MSNSGA-III

Test Function

\(M\)

\(D\)

Metrics

MSNSGA-III

NSGA-III

DLTZ1

4

13

IGD

3.2570e−01(1.916e−01) + 

3.9852e−01(2.928e−01)

HV

3.2084e−01(3.351e−01) + 

3.2673e−01(3.521e−01)

DLTZ2

4

13

IGD

1.2848e−01(1.456e−03) + 

1.4118e−01(3.080e−04)

HV

7.2513e−01(4.450e−03) + 

6.8325e−01(2.021e−03)

DLTZ4

4

13

IGD

2.2041e−01(1.361e−01) + 

2.4312e−01(1.478e−01)

HV

6.7899e−01(7.395e−02) + 

6.3334e−01(7.261e−02)

DLTZ5

4

13

IGD

5.8220e−02(1.004e−02) + 

6.6119e−02(1.215e−02)

HV

1.3447e−01(5.339e−03) + 

1.2735e−01(5.356e−03)

DLTZ6

4

13

IGD

4.5909e−01(3.937e−01) + 

4.8814e−01(4.410e−01)

HV

3.1342e−02(4.148e−02) + 

2.9201e−02(4.065e−02)

DTLZ7

4

13

IGD

1.9929e−01(2.046e−02) + 

2.2189e−01(1.365e−02)

HV

2.5397e−01(8.075e−03) + 

2.4567e−01(8.168e−03)