Table 3 Statistical results of 23 benchmark functions by the proposed LSDO technique and other recent algorithms.

F1

Min

8.3E−151

1.39E−55

3.79E−08

4.47E−12

1.03E−28

Average

4.6E−128

1.37E−51

3.92E−07

3.12E−11

1.37E−27

Median

4.4E−140

3.74E−54

1E−07

2.46E−11

4.77E−28

Max

9.1E−127

8.43E−51

4.09E−06

8.73E−11

1.04E−26

Std

2E−127

2.74E−51

9.2E−07

2.31E−11

2.38E−27

F2

Min

1.01E−69

1.83E−29

2.44E−06

1.42E−07

2.3E−15

Average

2E−60

3.76E−25

1.9E−05

2.77E−07

5.64E−15

Median

6.56E−63

1.13E−26

1.86E−05

2.66E−07

4.21E−15

Max

2.53E−59

3.98E−24

3.68E−05

4.78E−07

1.4E−14

Std

5.95E−60

9.1E−25

9.44E−06

9.9E−08

3.51E−15

F3

Min

2.2E−145

6.27E−46

0.027608

0.008462

9.18E−13

Average

8.7E−120

6.91E−34

1.122677

0.610441

4.18E−08

Median

9.3E−138

1.4E−39

0.772195

0.185412

4.13E−09

Max

1.7E−118

1.38E−32

3.914695

3.567009

3.9E−07

Std

3.9E−119

3.09E−33

1.096313

0.827115

9.17E−08

F4

Min

4.81E−73

1.11E−26

0.67531

0.002608

1.33E−13

Average

1.63E−67

4.52E−23

3.616654

0.008

5.45E−13

Median

5.11E−70

1.14E−23

3.022253

0.007092

4.09E−13

Max

1.67E−66

1.94E−22

9.361516

0.016667

1.87E−12

Std

4.02E−67

6.34E−23

2.343658

0.003845

4.55E−13

F5

Min

26.36703

27.90967

27.18973

25.92515

27.6644

Average

26.99756

28.65096

39.01094

27.18903

28.03399

Median

26.86409

28.74726

28.66203

27.09814

27.97984

Max

28.66131

28.98699

239.7785

28.79035

28.44604

Std

0.508238

0.295026

47.26339

0.72182

0.216873

F6

Min

0.020726

0.039957

2.886997

0.252254

0.080879

Average

0.141345

2.568541

3.800719

0.647554

0.292241

Median

0.082029

2.038779

3.736935

0.611378

0.255115

Max

0.549917

7.250251

4.850371

1.172757

0.75842

Std

0.154727

1.852701

0.527851

0.280888

0.181696

F7

Min

3.93E−05

8.66E−05

0.007604

0.001477

0.000168

Average

0.000179

0.002356

0.019206

0.004433

0.000708

Median

0.000139

0.001136

0.018479

0.003685

0.000688

Max

0.000513

0.013813

0.04436

0.01033

0.002187

Std

0.000146

0.003331

0.007628

0.002554

0.000488

F8

Min

−1733.12

−1655

−1394.45

−1495.31

−7613.49

Average

−1534.59

−1312.83

−1212.82

−1245.57

−6358.62

Median

−1536.16

−1385.86

−1232.52

−1224.18

−6324.46

Max

−1327.7

−598.802

−976.635

−1123.85

−5562.96

Std

94.79719

294.008

122.0762

104.0153

538.2484

F9

Min

0.00

4.33E−30

156.667

1.062467

0.00

Average

0.00

1.75E−22

228.0177

9.801018

0.00

Median

0.00

4.17E−25

228.634

9.824713

0.00

Max

0.00

3.02E−21

331.7581

24.96968

0.00

Std

0.00

6.75E−22

46.40919

5.565812

0.00

F10

Min

8.88E−16

8.88E−16

20.81133

20.76487

4.44E−15

Average

8.88E−16

8.88E−16

20.9608

20.92344

7.46E−15

Median

8.88E−16

8.88E−16

20.99356

20.94465

6.22E−15

Max

8.88E−16

8.88E−16

21.0961

21.06309

1.51E−14

Std

0.00

0.00

0.091505

0.083433

3.69E−15

F11

Min

0.00

0.00

1.3E−09

6.56E−13

0.00

Average

0.00

0.00

0.007018

0.009891

0.00

Median

0.00

0.00

1.44E−08

4.55E−12

0.00

Max

0.00

0.00

0.029126

0.055407

0.00

Std

0.00

0.00

0.010243

0.015766

0.00

F12

Min

0.000176

0.001152

0.374956

0.006066

0.000696

Average

0.003854

0.23467

2.805889

0.026151

0.00268

Median

0.002177

0.067805

2.009833

0.023474

0.00284

Max

0.01893

1.492821

7.656863

0.047176

0.004893

Std

0.004455

0.352063

2.128936

0.013414

0.001232

F13

Min

0.175696

0.046216

2.372295

0.09955

0.057519

Average

0.834851

1.867552

3.298085

0.613832

0.154385

Median

0.451512

1.934246

3.22876

0.609981

0.140323

Max

2.966605

2.999924

4.16073

1.044

0.378672

Std

0.957067

0.961284

0.565835

0.280029

0.077659

F14

Min

0.998004

0.998004

0.998004

0.998004

0.998004

Average

0.998004

3.494696

8.298683

3.892106

0.998004

Median

0.998004

1.495017

10.76318

2.982105

0.998004

Max

0.998004

12.67051

18.30431

12.67051

0.998004

Std

0.00

3.953203

5.533952

3.727681

1.02E−16

F15

Min

0.000307

0.000307

0.000308

0.00031

0.000308

Average

0.000308

0.00067

0.007136

0.003547

0.00035

Median

0.000307

0.000527

0.000505

0.000546

0.000313

Max

0.000309

0.002121

0.031699

0.020363

0.000582

Std

3.22E−07

0.000473

0.010606

0.007255

6.8E−05

F16

Min

−1.03163

−1.03163

−1.03163

−1.03163

−1.03163

Average

−1.03163

−1.03005

−1.0253

−1.03158

−1.03163

Median

−1.03163

−1.03163

−1.03163

−1.03163

−1.03163

Max

−1.03163

−1.00046

−0.99999

−1.03063

−1.03163

Std

2.28E−16

0.006966

0.012981

0.000223

1.53E−16

F17

Min

0.397887

0.397887

0.39789

0.397888

0.397887

Average

0.397887

0.397987

0.397927

0.397891

0.397887

Median

0.397887

0.397887

0.397907

0.397891

0.397887

Max

0.397887

0.399795

0.398082

0.397897

0.397887

Std

0.00

0.000426

4.53E−05

3.01E−06

0.00

F18

Min

0.00

3.00

3.000009

3.00

0.00

Average

0.00

3.00

8.400078

3.000068

0.00

Median

0.00

3.00

3.000084

3.000036

0.00

Max

0.00

3.00

84.00001

3.000238

0.00

Std

5.49E−16

5.21E−08

18.78799

6.53E−05

1.6E−15

F19

Min

−0.30048

−0.30048

−0.30048

−0.30048

−3.86278

Average

−0.30048

−0.2893

−0.30048

−0.30048

−3.86278

Median

−0.30048

−0.30038

−0.30048

−0.30048

−3.86278

Max

−0.30048

−0.19165

−0.30048

−0.30048

−3.86278

Std

1.14E−16

0.026531

1.14E−16

1.14E−16

2.22E−15

F20

Min

−3.322

−3.322

−3.32148

−3.32198

−3.322

Average

−3.23283

−3.09697

−3.07223

−3.22876

−3.29822

Median

−3.2031

−3.2031

−3.20118

−3.26239

−3.322

Max

−3.2031

−0.89904

−0.20816

−2.84039

−3.2031

Std

0.05282

0.550986

0.679321

0.125558

0.048793

F21

Min

−10.1532

−10.1532

−10.0895

−10.1502

−10.1532

Average

−10.1532

−8.703

−5.89545

−8.51218

−10.1532

Median

−10.1532

−10.1532

−4.90994

−10.1413

−10.1532

Max

−10.1532

−4.99677

−2.58642

−2.62918

−10.1532

Std

3.36E−15

2.23952

2.775111

2.963153

2.8E−12

F22

Min

−10.4029

−10.4029

−10.3637

−10.4024

−10.4029

Average

−10.4029

−8.45822

−7.02119

−10.0134

−10.4029

Median

−10.4029

−10.4029

−9.8942

−10.3959

−10.4029

Max

−10.4029

−1.0677

−1.82478

−2.76526

−10.4029

Std

3.13E−15

3.128689

3.57071

1.706042

5.02E−15

F23

Best

−10.5364

−10.5364

−10.4599

−10.5348

−10.5364

Mean

−10.266

−7.90449

−5.50502

−9.74305

−10.5364

Median

−10.5364

−10.5357

−2.83596

−10.5274

−10.5364

Worst

−5.12848

−3.79083

−1.66783

−2.42135

−10.5364

Std

1.20925

3.015319

3.728197

2.418464

2E−15