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Table 7 Results of twenty benchmark problems (generations = 5000 and dimensions = 20)

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

6.143e−101/2

3.714e−012/4

1.992e−180/1

5.879e−052/3

 

MED/SD

2.803e−102/1.498e−100

6.501e−013/8.397e−012

4.447e−183/2.124e−180

1.476e−053/1.638e−051

f 2(x)

AVE/rank

1.522e−050/2

1.000e+000/3

3.973e−091/1

3.000e+000/4

 

MED/SD

7.388e−051/2.773e−050

1.387e−010/3.077e+000

7.565e−092/1.174e−090

2.895e−033/4.701e+000

f 3(x)

AVE/rank

1.049e−060/2

3.145e+003/3

3.938e−132/1

8.000e+003/4

 

MED/SD

8.599e−062/3.394e−060

2.604e+003/3.448e+003

5.529e−134/1.488e−131

1.000e+004/5.342e+003

f 4(x)

AVE/rank

8.837e−042/2

2.666e+000/4

1.290e−078/1

5.100e−004/3

 

MED/SD

1.414e−042/1.569e−041

2.376e+000/1.136e+000

1.071e−079/4.198e−078

3.224e−004/6.771e−004

f 5(x)

AVE/rank

0.0e+000/1

0.0e+000/1

0.0e+000/1

0.0e+000/1

 

MED/SD

0.0e+000/0.0e+000

0.0e+000/0.0e+000

0.0e+000/0.0e+000

0.0e+000/0.0e+000

f 6(x)

AVE/rank

4.992e−001/3

7.748e−001/4

4.983e−001/2

4.625e−001/1

 

MED/SD

5.276e−001/3.032e−001

8.512e−001/3.869e−001

5.067e−001/2.680e−001

5.071e−001/2.670e−001

f 7(x)

AVE/rank

0.0e+000/1

1.411e+001/3

0.0e+000/1

2.490e+001/4

 

MED/SD

0.0e+000/0.0e+000

9.455e+00/1.125e+001

0.0e+00/0.0e+000

1.542e+01/1.996e+001

f 8(x)

AVE/rank

0.0e+000/1

2.034e+001/3

1.773e+000/2

2.530e+001/4

 

MED/SD

0.0e+000/0.0e+000

1.867e+01/1.158e+001

0.0e+00/6.846e+000

2.700e+01/1.999e+001

f 9(x)

AVE/rank

3.552e−015/1

2.440e−007/3

3.552e−015/1

7.676e−001/4

 

MED/SD

3.552e−015/0.0e+000

1.615e−007/2.245e−007

3.552e−015/0.0e+000

7.105e−015/3.432e+000

f 10(x)

AVE/rank

0.0e+000/1

4.861e−002/4

7.404e−004/2

3.588e−002/3

 

MED/SD

0.0e+000/0.0e+000

4.785e−002/3.737e−002

0.0e+000/3.311e−003

2.834e−002/3.710e−002

f 11(x)

AVE/rank

0.0e+000/1

2.615e−007/3

0.0e+000/1

2.000e−001/4

 

MED/SD

0.0e+000/0.0e+000

1.650e−008/9.516e−007

0.0e+000/0.0e+000

0.000e+000/8.944e−001

f 12(x)

AVE/rank

5.335e−002/4

3.893e−008/1

5.103e−002/3

7.775e−003/2

 

MED/SD

5.383e−002/9.641e−003

6.931e−014/1.741e−007

4.906e−002/1.051e−002

2.355e−032/3.477e−002

f 13(x)

AVE/rank

−2.0e+000/1

−2.0e+000/1

−2.0e+000/1

−2.0e+000/1

 

MED/SD

−2.0e+000/0.0e+000

−2.0e+000/3.520e−015

−2.0e+000/0.0e+000

−2.0e+000/0.0e+000

f 14(x)

AVE/rank

3.963e+001/1

8.864e+001/4

4.278e+001/2

5.657e+001/3

 

MED/SD

3.702e+01/1.884e+001

8.617e+01/3.462e+001

4.612e+01/1.646e+001

5.323e+01/2.357e+001

f 15(x)

AVE/rank

9.987e−002/1

1.370e+000/4

9.987e−002/1

1.199e+000/3

 

MED/SD

9.987e−002/2.883e−010

4.998e−001/3.682e+000

9.987e−002/4.206e−010

3.998e−001/3.579e+000

f 16(x)

AVE/rank

1.876e+001/1

3.382e+007/3

1.877e+001/2

4.691e+008/4

 

MED/SD

1.876e+001/2.135e−002

1.952e+03/1.374e+008

1.877e+001/1.374e−002

1.971e+08/5.627e+008

f 17(x)

AVE/rank

1.529e−012/2

1.252e+004/4

8.245e−035/1

8.891e+003/3

 

MED/SD

1.549e−032/6.838e−012

4.387e+03/1.629e+004

8.006e−096/3.687e−034

6.856e+03/8.742e+003

f 18(x)

AVE/rank

−4.224e+002/2

−4.161e+002/3

−4.288e+002/1

−4.051e+002/4

 

MED/SD

−4.229e+02/3.728e+000

−4.173e+02/1.819e+001

−4.290e+02/2.251e+000

−4.077e+02/1.290e+001

f 19(x)

AVE/rank

−1.1921e+2/4

−1.1931e+2/1

−1.1925e+2/3

−1.1928e+2/2

 

MED/SD

−1.1920e+2/6.117e−002

−1.1928e+2/1.049e−001

−1.1923e+2/5.970e−002

−1.1928e+2/7.102e−002

f 20(x)

AVE/rank

1.0953e+002/4

1.047e+002/2

1.0951e+002/3

1.042e+002/1

 

MED/SD

1.095e+02/2.107e+000

1.050e+02/2.457e+000

1.094e+002/2.029e+000

1.037e+02/2.714e+000

 

AR

1.85

2.9

1.55

2.9

 

FR

2

3

1

3