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Table 4 Statistical comparisons of proposed Vs other algorithms for UBFs

From: An advanced hybrid meta-heuristic algorithm for solving small- and large-scale engineering design optimization problems

Vs

Criteria

Algorithm

Traditional algorithms

DE variants

PSO variants

Hybrid variants

Proposed algorithms

EO

HHO

JADE

SHADE

HEPSO

RPSOLF

PSOSCALF

FAPSO

aPSO

haDEPSO

aDE

Better

21

15

20

13

21

21

18

14

15

0

 

Equal

2

8

3

9

2

2

5

9

8

16

 

Worst

0

0

0

1

0

0

0

0

0

7

 

R+

313

416

345

355

329

465

323

377

342

315

 

R

152

49

120

130

136

79

142

88

123

150

 

p value

5.2e−10

5.6e−10

8.2e−10

5.8e−10

6.2e−10

6.9e−07

4.3e−09

6.2e−11

5.1e−10

6.9e−07

 

t test

a

a

a

a+

a

a

a

a+

a+

a+

 

Decision

+

+

+

+

+

+

+

Vs

 

EO

HHO

JADE

SHADE

HEPSO

RPSOLF

PSOSCALF

FAPSO

aDE

haDEPSO

aPSO

Better

21

11

20

13

20

19

19

15

0

0

 

Equal

2

4

2

9

2

2

3

4

7

8

 

Worst

0

8

1

1

1

2

1

4

16

15

 

R+

387

293

335

305

312

323

382

300

350

400

 

R

78

172

130

160

153

142

83

165

115

65

 

p value

5.3e−10

5.1e−09

6.2e−10

4.6e−08

5.7e−10

5.1e−09

5.6e−10

5.8e−10

6.2e−09

5.3e−10

 

t test

a

a

a

a+

a

a+

a+

a+

a

a+

 

Decision

+

+

+

+

+

+

+

Vs

 

EO

HHO

JADE

SHADE

HEPSO

RPSOLF

PSOSCALF

FAPSO

aDE

aPSO

haDEPSO

Better

0

14

20

13

20

20

15

14

8

15

 

Equal

6

7

3

10

3

3

8

9

15

8

 

Worst

17

2

0

0

0

0

0

0

0

0

 

R+

294

321

329

367

330

313

377

293

323

304

 

R

171

144

136

98

135

152

88

172

142

161

 

p value

5.1e−10

6.2e−10

4.6e−08

5.7e−10

5.1e−07

5.1e−10

5.3e−08

6.2e−09

4.6e−10

5.7e−07

 

t test

a

a

a

a+

a

a+

a+

a

a+

a

 

Decision

+

+

+

+

+

+

+