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Table 9 Results of twenty benchmark problems (generations = 5000 and dimensions = 30)

From: Particle swarm optimization using multi-information characteristics of all personal-best information

Name

 

PSO-API

PSO

LPSO-API

LPSO

f 1(x)

AVE/rank

1.499e−082/2

1.500e+003/4

2.291e−159/1

1.184e−028/3

 

MED/SD

9.419e−084/3.947e−082

1.782e−003/3.663e+003

6.706e−163/9.471e−159

2.128e−029/3.017e−028

f 2(x)

AVE/rank

4.736e−042/2

5.000e+000/3

5.600e−082/1

1.450e+001/4

 

MED/SD

2.354e−042/6.150e−042

4.860e−004/6.069e+000

2.822e−082/6.334e−082

1.500e+001/9.986e+000

f 3(x)

AVE/rank

1.395e−043/2

1.510e+004/3

2.721e−108/1

1.660e+004/4

 

MED/SD

1.715e−044/2.946e−043

1.580e+004/6.208e+003

1.049e−109/5.623e−108

1.666e+004/5.629e+003

f 4(x)

AVE/rank

5.252e−032/2

1.769e+001/4

1.338e−066/1

1.968e+000/3

 

MED/SD

1.688e−032/8.290e−032

1.719e+001/3.485e+000

8.649e−068/3.874e−066

1.846e+000/7.950e−001

f 5(x)

AVE/rank

0.000e+000/1

4.500e−001/4

0.000e+000/1

5.000e−002/3

 

MED/SD

0.000e+000/0.000e+000

0.000e+000/8.255e−001

0.000e+000/0.000e+000

0.000e+000/2.236e−001

f 6(x)

AVE/rank

5.199e−001/2

2.045e+000/3

4.889e−001/1

2.307e+000/4

 

MED/SD

5.157e−001/3.248e−001

1.456e+000/1.958e+000

4.368e−001/3.293e−001

6.323e−001/4.144e+000

f 7(x)

AVE/rank

0.000e+000/1

6.980e+001/3

0.000e+000/1

7.686e+001/4

 

MED/SD

0.000e+000/0.000e+000

7.259e+001/2.296e+001

0.000e+000/0.000e+000

7.522e+001/2.848e+001

f 8(x)

AVE/rank

0.000e+000/1

1.018e+002/4

0.000e+000/1

8.270e+001/3

 

MED/SD

0.000e+000/0.000e+000

9.411e+001/3.081e+001

0.000e+000/0.000e+000

8.001e+001/3.588e+001

f 9(x)

AVE/rank

3.552e−015/1

7.264e−001/3

3.552e−015/1

3.807e+000/4

 

MED/SD

3.552e−015/0.000e+000

7.494e−003/3.189e+000

3.552e−015/0.000e+000

2.131e−014/6.874e+000

f 10(x)

AVE/rank

0.000e+000/1

5.139e−002/3

3.720e−004/2

1.354e+001/4

 

MED/SD

0.000e+000/0.000e+000

4.340e−002/5.443e−002

0.000e+000/1.664e−003

1.845e−002/3.301e+001

f 11(x)

AVE/rank

0.000e+000/1

6.248e−001/3

0.000e+000/1

1.805e+000/4

 

MED/SD

0.000e+000/0.000e+000

2.362e−002/1.952e+000

0.000e+000/0.000e+000

2.395e−003/3.033e+000

f 12(x)

AVE/rank

1.078e−001/3

2.256e−001/4

1.067e−001/2

5.183e−003/1

 

MED/SD

1.040e−001/1.480e−002

1.098e−002/5.944e−001

1.030e−001/1.504e−002

1.634e−028/2.318e−002

f 13(x)

AVE/rank

−3.000e+000/1

−2.999e+000/3

−3.000e+000/1

−2.940e+000/4

 

MED/SD

−3.000e+00/0.000e+000

−2.999e+000/7.685e−006

−3.000e+00/0.000e+000

−3.000e+00/2.683e−001

f 14(x)

AVE/rank

8.703e+001/1

2.200e+002/4

8.865e+001/2

1.319e+002/3

 

MED/SD

9.867e+001/4.002e+001

2.177e+002/3.608e+001

1.101e+002/4.941e+001

1.186e+002/6.466e+001

f 15(x)

AVE/rank

9.987e−002/1

2.446e+000/3

9.987e−002/1

1.025e+001/4

 

MED/SD

9.987e−002/5.355e−010

2.350e+000/6.174e−001

9.987e−002/7.443e−010

1.529e+001/9.398e+000

f 16(x)

AVE/rank

2.869e+001/1

1.573e+009/3

2.870e+001/2

2.542e+009/4

 

MED/SD

2.869e+001/1.449e−002

1.095e+006/3.957e+009

2.870e+001/1.788e−002

1.082e+009/4.418e+009

f 17(x)

AVE/rank

1.786e−036/2

4.287e+003/3

3.634e−104/1

9.561e+003/4

 

MED/SD

1.651e−037/3.264e−036

3.560e+003/2.774e+003

2.468e−110/1.625e−103

7.780e+003/6.709e+003

f 18(x)

AVE/rank

−3.972e+002/3

−4.016e+002/2

-4.127e+002/1

−3.880e+002/4

 

MED/SD

−3.965e+02/3.643e+000

−4.060e+02/1.933e+001

−4.127e+02/4.270e+000

−3.966e+02/2.133e+001

f 19(x)

AVE/rank

−1.190e+002/4

−1.190e+002/1

−1.190e+002/3

−1.190e+002/2

 

MED/SD

−1.190e+002/5.490e−002

−1.190e+02/6.539e−002

−1.190e+002/4.999e−002

−1.190e+002/1.005e−001

f 20(x)

AVE/rank

1.244e+002/3

1.153e+002/1

1.270e+002/4

1.172e+002/2

 

MED/SD

1.239e+002/2.857e+000

1.154e+002/3.703e+000

1.266e+002/2.562e+000

1.166e+002/2.931e+000

 

AR

1.75

3.05

1.45

3.4

 

FR

2

3

1

4

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