# An Innovative Hybrid Approach Producing Trial Solutions for Global Optimization
## 3 Experiments
### Experimental Results
The PROPOSED algorithm, with 361,454 evaluations, performs better than DE but does not reach the efficiency of IPSO. It shows moderate results in some problems, such as DIFFPOWER10, with 29,549 calls, limiting its overall performance. The improved PROPOSED (KMEANS) algorithm achieves better results, with 257,033 total evaluations. In many functions, such as SCHWEFEL222, with 22,058 evaluations, and DIFFPOWER10, with 26,615 evaluations, the addition of KMEANS significantly improves the performance, making it competitive with IPSO. In conclusion, IPSO is the most efficient algorithm, with the lowest total number of evaluations, achieving better results with fewer calls. PROPOSED (KMEANS) also performs well, approaching the performance of IPSO. On the other hand, GWO and WOA have the worst performances, requiring the highest number of evaluations. MEWOA shows some improvements over WOA but remains less efficient than the top-performing algorithms. In Figure 3, pairwise comparisons between the proposed methods and the other methods are presented. In all these comparisons, the critical parameter \(p<0.05\) confirms the statistical significance of the results. Overall, the proposed methods achieve a significant reduction in the required number of objective function calls and, in many cases, outperform other optimization techniques. Compared to the differential evolution (DE) algorithm, the proposed approaches perform better across all mathematical models, with the reduction in the number of calls exceeding 50%. This trend is confirmed by Table 3. Additionally, Figures 2,b and 3 present statistical comparisons between the global optimization methods used in the experiments, highlighting the superiority of our proposed approaches. Table 4 presents the number of objective function calls required for different values of the differential weight F. When F is randomly selected within the range [0.0, 1.0], the total number of calls is 361,454. When F = 0.5, the calls increase to 366,966, while when F = 0.9, they decrease to 342,529, indicating the best performance. This suggests that higher values of F improve convergence by reducing the total number of function calls. Although  
Figure 2: Comparison of the number of function calls. The test was performed using different optimization methods. random F values provide a good balance between exploration and exploitation, they do not outperform the fixed value F = 0.9. On the other hand, F = 0.5 appears to burden the algorithm with more function calls. Therefore, selecting a higher or adaptive F can enhance the algorithm's efficiency by lowering computational costs. From the data in Table 5, we observe that as the population size decreases, the number of function calls generally decreases as well, indicating increased efficiency with smaller populations. For example, the ACKLAY function requires 5818 calls with a population of 200, 4647 with a population of 150, and 2829 with a population of 100. A similar trend is seen with DIFFPOWER10, which reduces from 29,549 to 22,112 and then to 15,325 as the population size decreases. However, there is variation between functions. SCHWE-FEL222 shows very high call numbers, particularly with a population of 200 (78,734 calls), suggesting increased complexity. In contrast, functions like GKLS350 require fewer calls, even with larger populations. Although calls often decrease with a smaller population, this does not necessarily mean that the solution's quality is maintained. Choosing the right population size is crucial, as excessive reduction may limit the search space and degrade the optimization performance. Table 5 illustrates how population size impacts performance. Balancing population size and the number of calls is essential for achieving efficient and reliable results.