## 4 Results and Discussion
### Comparison of Proposed Algorithms Among Themselves  
In this section, the results attained with 30 runs of the F1-F23 functions using the proposed HHHOWOA1, HHHOWOA2, HHHOWOA1 PSO (\(w\) = 0.1, \(W_{c}\) = 0.8), HHHOWOA1 PSO (\(w\) = 0.3, \(W_{c}\) = 0.8), HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.87), and HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) algorithms are given in Table 4. In addition, internal comparisons among themselves and interpretation of the results are performed for the benchmark functions.  
In the comparison among themselves, method HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) achieved either better or equivalent average optimization values in 15 out of 23 functions, method HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.87) in 14 out of 23 functions, method HHHOWOA1 in 10 out of 23 functions, method HHHOWOA2 in 8 out of 23 functions, method HHHOWOA1 PSO (\(w\) = 0.3, \(W_{c}\) = 0.8) in 8 out of 23 functions, and method HHHOWOA1 PSO (\(w\) = 0.1, \(W_{c}\) = 0.8) in 6 out of 23 functions. In general, it seems that all algorithms for F1-F4 are very close to 0, and there is only an exponential difference between them. This shows that, according to these functions, all the proposed algorithms successfully find the global optimum. However, HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.87) provided the best results for functions F1 to F4 compared to the others.  
For all algorithms, both F8 and F10 achieved consistently good and equivalent optimization values. Additionally, all the suggested algorithms for F9 and F11 successfully obtain the desired result of zero. Compared to the other proposed algorithms, method HHHOWOA1 PSO (\(w\) = 0.1, \(W_{c}\) = 0.8) achieved the best optimization value for F12, and method HHHOWOA1 PSO (\(w\) = 0.3, \(W_{c}\) = 0.8) reached the best optimization value for F13. F8-F13 functions can measure the ability of algorithms to escape the local optimum. Especially, the HHHOWOA1 PSO algorithm can successfully escape from the local optimum according to the results. All the proposed algorithms obtain the best results for F14 and F18. For F19, method HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) provided the result closest to the goal value. For F15-F17 and F20-F23, methods HHHOWOA1 and HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) provided results closer to the goal value. As F14-F23 multimodal functions can measure the combination of the exploration and exploitation capabilities of an algorithm, this shows that the combination of these two phases in HHHOWOA1 and HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) is balanced.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Num. of** & **Function** & **Var.** & **Boundary** \\
**Func.** & **Num** & **(Lower–Upper)** & **\(f_{min}\)** \\ \hline \(F_{19}(z)\) & \(-\sum_{i=1}^{4}c_{i}\exp(-\sum_{j=1}^{3}a_{ij}(z_{j}-p_{ij})^{2})\) & 3 & \(LB=1\), \(UB=3\) & \(-3.86\) \\ \hline \(F_{20}(z)\) & \(-\sum_{i=1}^{4}c_{i}\exp(-\sum_{j=1}^{6}a_{ij}(z_{j}-p_{ij})^{2})\) & 6 & \(LB=0\), \(UB=1\) & \(-3.32\) \\ \hline \(F_{21}(z)\) & \(-\sum_{i=1}^{5}\left[(Z-a_{i})(Z-a_{i})^{T}+c_{i}\right]^{-1}\) & 4 & \(LB=0\), \(UB=10\) & \(-10.1532\) \\ \hline \(F_{22}(z)\) & \(-\sum_{i=1}^{7}\left[(Z-a_{i})(Z-a_{i})^{T}+c_{i}\right]^{-1}\) & 4 & \(LB=0\), \(UB=10\) & \(-10.4028\) \\ \hline \(F_{23}(z)\) & \(-\sum_{i=1}^{10}\left[(Z-a_{i})(Z-a_{i})^{T}+c_{i}\right]^{-1}\) & 4 & \(LB=0\), \(UB=10\) & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} & \multicolumn{3}{c}{**Foint**} \\ \hline
**Foint** & **Foint** &