Table 11 Results of twenty benchmark problems (generations = 5000 and dimensions = 50)

f ₁(x)

AVE/rank

8.111e−069/2

4.571e+003/4

6.638e−146/1

3.000e+003/3

MED/SD

2.816e−069/1.213e−068

1.378e+002/6.841e+003

4.585e−152/2.968e−145

9.040e−012/4.701e+003

f ₂(x)

AVE/rank

1.339e−035/2

2.635e+001/3

1.770e−076/1

3.800e+001/4

MED/SD

6.996e−036/1.559e−035

3.020e+001/1.696e+001

5.086e−078/6.311e−076

3.000e+001/1.735e+001

f ₃(x)

AVE/rank

7.320e−029/2

6.095e+004/3

6.921e−091/1

7.035e+004/4

MED/SD

4.714e−031/3.121e−028

6.084e+004/1.020e+004

2.327e−092/1.678e−090

6.774e+004/2.032e+004

f ₄(x)

AVE/rank

9.832e−024/2

4.042e+001/4

1.467e−055/1

2.482e+001/3

MED/SD

3.494e−024/1.388e−023

4.021e+001/4.275e+000

1.603e−057/5.564e−055

2.621e+001/3.161e+000

f ₅(x)

AVE/rank

0.000e+000/1

1.106e+003/3

0.000e+000/1

5.000e+003/4

MED/SD

0.000e+000/0.000e+000

1.130e+002/3.090e+003

0.000e+000/0.000e+000

1.000e+000/6.069e+003

f ₆(x)

AVE/rank

4.829e−001/1

1.498e+001/3

5.144e−001/2

1.795e+001/4

MED/SD

4.569e−001/2.745e−001

1.618e+001/9.801e+000

5.114e−001/2.600e−001

1.068e+001/1.779e+001

f ₇(x)

AVE/rank

0.000e+000/1

2.677e+002/4

0.000e+000/1

2.329e+002/3

MED/SD

0.000e+000/0.000e+000

2.671e+002/5.346e+001

0.000e+000/0.000e+000

2.276e+002/4.966e+001

f ₈(x)

AVE/rank

1.259e+001/2

2.729e+002/4

0.000e+000/1

2.725e+002/3

MED/SD

0.000e+000/5.634e+001

2.803e+002/5.489e+001

0.000e+000/0.000e+000

2.680e+002/5.470e+001

f ₉(x)

AVE/rank

4.973e−015/2

9.682e+000/3

4.618e−015/1

1.185e+001/4

MED/SD

3.552e−015/1.785e−015

1.316e+001/5.389e+000

3.552e−015/1.670e−015

1.370e+001/7.483e+000

f ₁₀(x)

AVE/rank

0.000e+000/1

2.885e+001/3

0.000e+000/1

3.159e+001/4

MED/SD

0.000e+000/0.000e+000

2.355e+000/4.211e+001

0.000e+000/0.000e+000

1.599e−002/5.296e+001

f ₁₁(x)

AVE/rank

0.000e+000/1

1.107e+001/4

0.000e+000/1

7.587e+000/3

MED/SD

0.000e+000/0.000e+000

1.186e+001/3.728e+000

0.000e+000/0.000e+000

6.002e+000/4.384e+000

f ₁₂(x)

AVE/rank

2.453e−001/3

5.749e+003/4

2.361e−001/2

9.655e−002/1

MED/SD

2.475e−001/3.344e−002

1.219e+002/1.672e+004

2.427e−001/2.283e−002

6.220e−002/1.673e−001

f ₁₃(x)

AVE/rank

−5.000e+000/1

−4.573e+000/4

−5.000e+000/1

−4.611e+000/3

MED/SD

−5.000e+00/0.000e+000

−4.771e+000/4.670e−001

−5.000e+00/0.000e+000

−4.926e+000/8.788e−001

f ₁₄(x)

AVE/rank

1.381e+002/2

6.056e+002/4

1.368e+002/1

4.808e+002/3

MED/SD

2.115e+002/1.298e+002

5.709e+002/1.049e+002

2.102e+002/1.290e+002

4.337e+002/2.018e+002

f ₁₅(x)

AVE/rank

9.987e−002/1

1.897e+001/3

9.987e−002/1

2.485e+001/4

MED/SD

9.987e−002/5.210e−010

1.899e+001/6.756e+000

9.987e−002/5.908e−010

2.559e+001/7.030e+000

f ₁₆(x)

AVE/rank

4.862e+001/1

1.186e+010/3

4.865e+001/2

1.631e+010/4

MED/SD

4.862e+001/1.551e−002

4.668e+009/1.816e+010

4.864e+001/2.043e−002

1.336e+010/1.745e+010

f ₁₇(x)

AVE/rank

6.371e−031/2

8.266e+004/3

7.198e−097/1

1.111e+005/4

MED/SD

1.426e−031/1.347e−030

7.732e+004/4.103e+004

4.644e−104/3.219e−096

8.578e+004/8.496e+004

f ₁₈(x)

AVE/rank

−3.782e+002/3

−3.824e+002/2

−3.911e+002/1

−3.586e+002/4

MED/SD

−3.788e+002/1.934e+000

−3.834e+002/1.983e+001

−3.909e+002/3.716e+000

−3.539e+002/1.941e+001

f ₁₉(x)

AVE/rank

−1.1883e+002/3

−1.1890e+002/1

−1.1882e+002/4

−1.1889e+002/2

MED/SD

−1.1883e+02/3.887e−002

−1.1889e+02/3.944e−002

−1.1881e+02/4.171e−002

−1.1887e+02/7.086e−002

f ₂₀(x)

AVE/rank

1.565e+002/3

1.435e+002/2

1.597e+002/4

1.426e+002/1

MED/SD

1.566e+002/3.440e+000

1.421e+002/5.957e+000

1.590e+002/3.475e+000

1.436e+002/3.741e+000

AR

1.8

3.2

1.4

3.25

FR

2

3

1

4