Algorithm 4 Pseudocode of AO.

1:                  Initialization part.

2: Set up parameters of AO algorithm.

3: Initialize population in a random way $\{{\mathbf{x}}^1,{\mathbf{x}}^2,\dots ,{\mathbf{x}}^N\}$.

4:                   Iterative part.

5: for $\mathrm{iteration}i=1,2,\dots ,I$ do

6:  Determine values of the fitness function F for each agent in the population.

7:  Establish the best solution ${\mathbf{x}}^{best}$ in the population.

8:  for $k=1,2,\dots ,N$ do

9:    Calculate mean solution ${\mathbf{x}}^{mean}$ in the population.

10:    Improve parameters $G_1,G_2,QF$ of the algorithm.

11:    if $\mathrm{iteration}i\le \frac{2}{3}I$ then

12:      if $rd<0.5$ then

13:        Perform step expanded exploration (17) by updating solution ${\mathbf{x}}^k$.

14:        In the result solution ${\mathbf{x}}^{new,k}$ is obtained.

15:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^k\right)$ then make substitution ${\mathbf{x}}^k={\mathbf{x}}^{new,k}$

16:        end if

17:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$

18:        end if

19:      else

20:        Perform step narrowed exploration (19) by updating solution ${\mathbf{x}}^k$.

21:        In the result solution ${\mathbf{x}}^{new,k}$ is obtained.

22:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^k\right)$ then make substitution ${\mathbf{x}}^k={\mathbf{x}}^{new,k}$.

23:        end if

24:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$.

25:        end if

26:      end if

27:    else

28:      if $rd<0.5$ then

29:        Perform step Expanded exploitation (23) by updating solution ${\mathbf{x}}^k$.

30:        In the result solution ${\mathbf{x}}^{new,k}$ is obtained.

31:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^k\right)$ then make substitution ${\mathbf{x}}^k={\mathbf{x}}^{new,k}$.

32:        end if

33:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$.

34:        end if

35:      else

36:        Perform step narrowed exploitation (24) by updating solution ${\mathbf{x}}^k$.

37:        In the result solution ${\mathbf{x}}^{new,k}$ is obtained.

38:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^k\right)$ then make substitution ${\mathbf{x}}^k={\mathbf{x}}^{new,k}$.

39:        end if

40:        if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$.

41:        end if

42:      end if

43:    end if

44:  end for

45: end for

46: return ${\mathbf{x}}^{best}$.