## 4 Results and Discussion
### Comparison of the Proposed Approaches with the Literature  
The results of the literature articles are compared with the proposed HHHOWOA1 and HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) results in this section. To make a precise comparison with the literature, articles that used the same criteria are found as much as possible. Therefore, the literature algorithms with population, maximum iteration, and independent run numbers of 30, 500, and 30, respectively, were chosen. These are the HHO, GWO, PSO, WOA, and WOALFVWPSO algorithms [5; 17; 25; 28]. Table 5 presents a comparison of the HHHOWOA1 and HHHOWOA2 PSO results with the literature algorithms for the F1-F23 functions. According to the literature, the HHHOWOA1 and HHHOWOA2 PSO approaches achieve the most optimal result, either better or with the same average fitness values, in 15 of the 23 functions and in 18 of the 23 functions, respectively. The proposed HHHOWOA1 algorithms achieved optimal results except for F1-F5, F12, F17, and F19. The proposedHHHOWOA2 PSO algorithms achieved optimal results except for F2-F4, F17, and F19. The HHO algorithm for F17 and F19 and the WOALFVWPSO algorithm for F2-F4 are better than the HHHOWOA2 PSO algorithm. But the results of the HHO algorithm for F17 and F19 have been rounded. For the F1-F23 functions, Friedman rank tests have been performed on the average optimization results of all algorithms, and the average values of Friedman rank tests have been calculated. According to Friedman mean rank tests, it is seen that the HHHOWOA2 PSO (\(w\) = 0.1, \(W_{c}\) = 0.99) algorithm has the best value in ranking for the F1-F23 functions. Then, the following algorithms come in order: HHHOWOA1, HHO, WOALFVWPSO, GWO, the WOA, and PSO. When the proposed HHHOWOA2 PSO and HHHOWOA1 algorithms are compared with the HHO algorithm over the F1-F7 benchmark functions, it is observed that the HHHOWOA2 PSO algorithm outperforms the HHO algorithm in all seven functions, while the HHHOWOA1 algorithm outperforms the HHO algorithm in six out of these seven functions. Therefore, the strength and proficiency of these two proposed algorithms in the exploitation phase are better than those of the HHO algorithm. When the proposed HHHOWOA2 PSO and HHHOWOA1 algorithms are compared with the HHO algorithm over the F8-F13 benchmark functions, they achieve the most optimal results, either better or with the same average fitness values, in six out of six functions and in five out of six functions, respectively. Therefore, the exploration stage and the ability of these two proposed algorithms to avoid getting stuck in local optima are better than those of the HHO algorithm. When the proposed HHHOWOA2 PSO and HHHOWOA1 algorithms are compared with the HHO algorithm over the F14-F23 benchmark functions, they achieve the most optimal results, either better or with the same average fitness values, in 8 out of 10 functions and in 8 out of 10 functions, respectively. Therefore, the equilibrium between the exploitation and exploration stages of the optimization of these two proposed algorithms is better than that of the HHO algorithm. In terms of the convergence curves, a comparison of the proposed HHHOWOA1 and HHHOWOA2 PSO algorithms with the literature algorithms for some benchmark problems is shown in Figure 5. Especially, from the convergence curve graphs of F1 and F4, it can be observed that the proposed HHHOWOA1 and HHHOWOA2 PSO algorithms converge better than the HHO algorithm and other algorithms at the point where it stagnates. When comparing the convergence curves of the two proposed methods, it can be seen that adding PSO allows the HHHOWOA2 PSO algorithm to converge more effectively without stagnation, thus outperforming HHHOWOA1.