## 4 Results and Discussion
### Statistical Analysis  
The Friedman test is a non-parametric statistical technique used to compare ordinal data from multiple dependent groups. Its main purpose is to assess whether there are significant differences between the groups. A \(p\)-value below 0.05 suggests a significant difference between the groups.  
In this study, since the functions used to test the algorithms have different target values, the results of the functions have been subtracted from the target values, and the absolute value has been taken. The obtained absolute difference values have been used in the [PERSON]'s mean rank test. For each function, the algorithm with the lowest value receives the first rank, and the other algorithms are assigned the subsequent ranks accordingly. Next, the average ranks for each algorithm are computed, and the algorithm with the lowest average rank is deemed the best performer across the 23 benchmark functions.  
After calculating the absolute errors of the results of the proposed algorithms presented in Table 4 relative to the target value, the [PERSON] test was applied, and the \(p\)-value was found to be 9.9686 \(\times\) 10\({}^{-4}\). Therefore, it was understood that a significant difference existed between the groups. The [PERSON] mean results were found to be 3.6739 for HHHOWOA1, 3.7391 for HHHOWOA2, 4.1957 for HHHOWOA1 PSO (\(w\) = 0.1, \(W_{c}\) = 0.8), 4.0435 for HHHOWOA1 PSO (\(w\) = 0.3, \(W_{c}\) = 0.8), 2.8043 for HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.87), and  
Figure 5: Comparison of convergence curves of the proposed HHHOWOA1 and HHHOWOA2 PSO algorithms with the literature algorithms on some benchmark problems.  
2.5435 for HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99), respectively, as shown in Figure 6. According to the [PERSON] average ranking, the proposed HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) algorithm demonstrates the best performance, the proposed HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.87) algorithm demonstrates the second-best performance, and the HHHOWOA1 algorithm demonstrates the third-best performance across 23 benchmark functions.  
After calculating the absolute errors of the results of the best two proposed algorithms and the other literature algorithms presented in Table 5 relative to the target value, the Friedman test was applied, and the \(p\)-value was found to be 5.9890 \(\times\) 10\({}^{-16}\). Therefore, it was understood that a significant difference existed between the groups. The [PERSON] average results were found to be 5.1087 for GWO, 3.6087 for WOALFVWPSO, 6.0000 for PSO, 5.6304 for the WOA, 3.2391 for HHO, 2.5217 for HHHOWOA1, and 1.8913 for HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99), respectively, as shown in Figure 7. According to the [PERSON] average ranking, the proposed HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) algorithm demonstrates the best performance, the proposed HHHOWOA1 algorithm demonstrates the second-best performance, and the HHO algorithm demonstrates the third-best performance across 23 benchmark functions.  
Figure 6: Mean ranks of proposed algorithms among themselves.  
Figure 7: Mean ranks of the proposed algorithms and the literature algorithms.