Table 5 The IGD + values of each algorithm under MAF test function set.

From: A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem

Problem

N

M

D

FEs

VMEF

BiGE-BEW

MOEA/DG

MOEA/D

LSMaODE

MaOEA/IT

MaOEA/IGD

MOEA/TS

MAF1

210

5

14

50,000

1.8543e-1 -

1.0517e-1 + 

2.3552e-1 -

1.6592e-1 -

1.2620e-1 -

3.3993e-1 -

2.8743e-1 -

1.2272e-1

120

8

17

80,000

2.7252e-1 -

2.6528e-1 -

5.2451e-1 -

4.7440e-1 -

2.4571e-1 -

5.4817e-1 -

3.7133e-1 -

2.2028e-1

220

10

19

100,000

2.9785e-1 -

2.3621e-1 + 

5.8867e-1 -

5.0942e-1 -

2.7471e-1 = 

4.9106e-1 -

3.5212e-1 -

2.7719e-1

120

15

24

150,000

3.3081e-1 -

3.4338e-1 -

6.1949e-1 -

5.7599e-1 -

3.6527e-1 -

7.5495e-1 -

4.1971e-1 -

3.1795e-1

210

20

29

200,000

4.4472e-1 -

3.8674e-1 + 

6.8756e-1 -

6.6912e-1 -

4.2792e-1 -

7.4730e-1 -

5.3303e-1 -

4.1199e-1

MAF2

210

5

14

50,000

1.1248e-1 -

1.0519e-1 -

9.2970e-2 = 

1.1076e-1 -

1.2270e-1 -

1.4516e-1 = 

3.3206e-1 -

9.3374e-2

120

8

17

80,000

2.3560e-1 -

1.6377e-1 + 

1.9319e-1 -

2.5650e-1 -

2.2422e-1 -

3.7710e-1 -

3.0168e-1 -

1.6924e-1

220

10

19

100,000

2.2526e-1 -

1.7358e-1 + 

1.9575e-1 = 

3.3234e-1 -

2.2195e-1 -

5.0052e-1 -

3.8235e-1 -

1.9938e-1

120

15

24

150,000

2.4890e-1 + 

2.2095e-1 + 

2.2024e-1 + 

5.2995e-1 -

2.6286e-1 + 

5.9051e-1 -

4.5057e-1 -

3.0070e-1

210

20

29

200,000

2.6609e-1 + 

1.8796e-1 + 

2.8531e-1 + 

5.6237e-1 -

2.1122e-1 + 

4.6866e-1 -

4.8668e-1 -

3.5580e-1

MAF3

210

5

14

50,000

9.7092e + 0 -

1.5617e + 0 -

7.4469e + 3 -

1.0426e-1 + 

8.7921e + 0 -

4.7530e + 3 -

6.2179e + 0 -

2.2111e-1

120

8

17

80,000

1.5009e + 2 -

4.4947e + 6 -

2.2763e + 3 -

1.7557e-1 + 

2.9299e + 5 -

7.5709e + 7 -

2.1420e + 1 -

6.8655e-1

220

10

19

100,000

1.3738e + 3 -

2.7237e + 6 -

4.1384e + 3 -

1.5972e-1 + 

3.4255e + 5 -

9.0692e + 8 -

2.0140e + 0 -

4.1766e-1

120

15

24

150,000

8.4474e + 1 -

5.1788e + 9 -

1.2061e + 3 -

1.5227e-1 + 

1.6296e + 5 -

5.3592e + 8 -

1.4639e + 1 -

9.1059e-1

210

20

29

200,000

4.3838e-1 + 

8.7536e + 9 -

1.9862e + 3 -

2.0936e-1 + 

2.9259e + 5 -

8.2083e + 8 -

5.3289e + 0 -

6.7356e-1

MAF4

210

5

14

50,000

5.8691e + 0 -

4.8536e + 0 + 

2.0693e + 3 -

1.2695e + 1 -

3.5458e + 0 + 

1.6962e + 2 -

1.9252e + 1 -

5.6357e + 0

120

8

17

80,000

3.2751e + 1 + 

3.7750e + 1 + 

1.2224e + 4 -

1.4499e + 2 -

3.2533e + 1 + 

3.4369e + 3 -

3.8028e + 2 -

4.4658e + 1

220

10

19

100,000

1.0387e + 2 + 

8.0779e + 1 + 

5.7546e + 4 -

5.2522e + 2 -

1.0131e + 2 + 

6.1958e + 3 -

4.7495e + 2 -

1.6806e + 2

120

15

24

150,000

4.6539e + 3 -

4.5320e + 3 -

1.4801e + 6 -

1.8904e + 4 -

4.2615e + 3 -

3.9958e + 5 -

3.4064e + 4 -

2.1009e + 3

210

20

29

200,000

1.3074e + 5 -

1.0864e + 5 -

5.3297e + 7 -

6.8845e + 5 -

1.4464e + 5 -

7.6476e + 6 -

7.5044e + 5 -

4.3143e + 4

MAF5

210

5

14

50,000

1.9708e + 0 -

2.0197e + 0 -

4.4149e + 0 -

7.5960e + 0 -

2.2678e + 0 -

9.5824e + 0 -

3.8332e + 0 -

1.9084e + 0

120

8

17

80,000

2.8444e + 1 -

2.4929e + 1 -

7.6746e + 1 -

8.3445e + 1 -

2.2141e + 1 -

1.8190e + 2 -

4.3980e + 1 -

2.0672e + 1

220

10

19

100,000

9.6961e + 1 -

8.1394e + 1 -

2.4374e + 2 -

3.0249e + 2 -

9.6314e + 1 -

7.9852e + 2 -

2.7764e + 2 -

6.1096e + 1

120

15

24

150,000

3.7343e + 3 -

2.2756e + 3 -

7.0997e + 3 -

7.3249e + 3 -

6.3433e + 3 -

2.6498e + 4 -

7.3247e + 3 -

2.2073e + 3

210

20

29

200,000

7.8956e + 4 -

2.9751e + 4 -

1.5089e + 5 -

1.7095e + 5 -

5.1353e + 4 -

1.2748e + 5 -

1.7095e + 5 -

2.4609e + 4

MAF6

210

5

14

50,000

1.8278e-2 + 

3.6632e-3 + 

7.2942e-1 -

1.9956e-1 = 

1.2605e-2 + 

4.7792e-2 + 

5.9641e-1 -

6.8462e-2

120

8

17

80,000

1.0519e-1 = 

1.4622e + 0 = 

5.9759e-1 -

3.6038e-1 -

7.3844e-2 + 

2.9806e + 0 -

5.5747e-1 -

8.1643e-2

220

10

19

100,000

4.3838e-1 -

8.1761e + 0 -

7.2456e-1 -

2.5294e-1 = 

3.6225e-1 -

2.6444e-1 = 

4.6633e-1 -

2.4844e-1

120

15

24

150,000

3.4727e-1 -

2.9589e + 1 -

3.9984e-1 -

4.7251e-1 = 

3.0975e-1 + 

1.4970e + 1 -

5.1156e-1 -

3.4322e-1

210

20

29

200,000

4.1648e-1 = 

4.8647e + 1 -

4.0177e-1 = 

4.0488e-1 = 

3.2122e-1 + 

1.6706e + 1 -

4.4754e-1 -

3.4743e-1

 + /-/ = 

6/22/2

11/18/1

2/25/3

5/21/4

9/20/1

1/27/2

0/30/0

 
  1. Significant values are in bold.
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