Table 4 Comparison of results of ARACO, RACO, ACO_R, and metaheuristics developed for combinatorial optimization and adapted to continuous domains

Branin RCOS \( \overrightarrow{x} \):[−5,15]ⁿ, n = 2

1.0 (17.8)

4.48

48.17

34.41

13.76

-

-

B₂ \( \overrightarrow{x} \):[−100,100]ⁿ, n = 2

1.0 (19.8)

4.09

28.23

21.71

-

-

-

Easom \( \overrightarrow{x} \):[−100,100]ⁿ, n = 2

1.0 [96%] (49.7)

1.11 [70%]

15.53 [98%]

29.51

-

-

-

Goldstein and Price \( \overrightarrow{x} \):[−2,2]ⁿ, n = 2

1.0 (17)

2.88

23.1

24.45

13.58

46.2

-

Rosenbrock (R₂) \( \overrightarrow{x} \):[−5,10]ⁿ, n = 2

1.0 (74.8)

1.05

10.9

12.83

6.41

10.9

8.34

Zakharov (Z₂) \( \overrightarrow{x} \):[−5,10]ⁿ, n = 2

1.0 (15.5)

1.29

18.87

40.25

12.58

1019

-

De Jong \( \overrightarrow{x} \):[−5.12,5.12]ⁿ, n = 3

1.0 (11)

3.81

35.63

67.70

-

-

35.63

Hartmann (H_3,4) \( \overrightarrow{x} \):[0,1]ⁿ, n = 3

1.0 (10.8)

2.48

31.66

53.83

50.66

63.33

-

Shekel (S_4,5) \( \overrightarrow{x} \):[0,10]ⁿ, n = 4

6.06 [87%]

1.0 [56%] (47.9)

16.55 [57%]

12.73 [76%]

17.82 [75%]

24.19 [54%]

-

Shekel (S_4,7) \( \overrightarrow{x} \):[0,10]ⁿ, n = 4

3.32

1.0 [92%] (48.9)

15.29 [79%]

13.9 [83%]

18.07 [80%]

25.03 [54%]

-

Shekel (S_4,10) \( \overrightarrow{x} \):[0,10]ⁿ, n = 4

2.97

1.0 [97%] (49.6)

14.41 [81%]

13.1 [83%]

18.34 [80%]

23.58 [50%]

-

Rosenbrock (R₅) \( \overrightarrow{x} \):[−5,10]ⁿ, n = 5

2.01

1.0 (184.3)

13.94 [97%]

22.08

11.62

29.05

-

Zakharov (Z₅) \( \overrightarrow{x} \):[−5,10]ⁿ, n = 5

1.0 (36.3)

1.7

20.02

38.05

62.08

1922.64

-

Hartmann (H_6,4) \( \overrightarrow{x} \):[0,1]ⁿ, n = 6

3.31 [97%]

1.0 [50%] (28.6)

25.24

32.81

53.01

93.40

-

Griewangk \( \overrightarrow{x} \):[−5.12,5,12]ⁿ, n = 10

16.78

1.0 [97%] (29.7)

46.8 [61%]

-

-

-

430.57