## 5 Development of the Execution of the Proposed Algorithm
### Analysis of the Hybrid Modified Sine Cosine Algorithm Cuckoo Search Algorithm (HMSCACSA) Results
In this case, five cycles of CPU time for 1500 iterations were used to avoid the immature convergence inaccuracy phase for several benchmark functions. Despite the challenging competition, the HMSCACSA had higher performance than the hybrid metaheuristics. Figure 6e shows that the proposed HMSCACSA achieved the fastest convergence by up to 63.89% and 60.83% after about 1200 iterations, with a minimization value up to 82.49% and 3.83% compared to other benchmarked hybrid metaheuristics and the MSCA for F5 (Table 3). These comparisons list two different percentages between HMSCACSA and another metaheuristics, that is, the iteration and minimization values, using Equation (6):  
\[\textit{Percentage\_Difference}=\frac{\left|\textit{highest}(\textit{another\_meta})\text{-}\textit{last}\ (\textit{another\_meta})\right|}{\textit{highest}\ (\textit{another\_meta})}\times 100 \tag{6}\]  
Based on the boxplots shown in Figure 7, several salient characteristics of the PSOGSA, PSOGWO, PSOABC, MSCA, and HMSCACSA search processes are presented. Taking the F5 function into consideration through this analysis, we attempted to highlight the overall performance of the HMSCACSA, which had a lower median compared to the other algorithms (Figure 7b). Although the MSCA exhibited a closed quartile bias range, it also comprised a larger interquartile range and a lower mean compared to the PSOGSA, PSOABC, and PSOGWO. However, the PSOGWO had a lower median than the other algorithms. It was also observed that all four algorithms exhibited similar results, except for PSOGWO, which had a bias towards the upper quartile. The statistical analysis indicated that the HMSCACSA exhibited the fastest convergence rate; this was primarily due to  
Figure 5: Convergence curves of the proposed Hybrid Modified Sine Cosine Algorithm Cuckoo Search Algorithm (HMSCACSA) using six mathematical functions. The MSCA, HMSCACSA, Particle Swarm Optimization–Artificial Bee Colony (PSOABC), Particle Swarm Optimization–Artificial Bee Colony (PSOGWO), and Particle Swarm Optimization–Gravitational Search Algorithm (PSOGSA) are presented as blue, green, cyan, black, and magenta smoothed lines, respectively. the advantages of the modification and hybridization processes in the proposed HMSCACSA that improved the investigation and utilization in global searching. To validate the processing time of the proposed variant methods, all algorithms were verified using the CPU time [29] deployed to reach the convergence. The mean and standard deviation were measured to analyze the performance of the proposed method for benchmarking of the mathematical function optimization. The results for both the mean and standard deviation are presented in Table 4, which also comprised 1500 iterations, a population of 30, and 20 dimensions. Furthermore, the mean values of the HMSCACSA optimization through the fourteen test functions indicated that the majority of the functions had low values (162.1748, 3,690,500,000, 666.0412, 2.6749, 14,274,000, and 224,550). Meanwhile, the standard deviations of the proposed HMSCACSA, which ranged from \(10^{3}\) to \(10^{10}\) for the different functions, were also low. These results indicated that the proposed HMSCACSA exhibited a higher performance than the other hybrid algorithms in the minimization for six test functions. Therefore, the proposed HMSCACSA enhanced the original SCA in terms of reaching optimal solutions and the ability to perform a local-global search. Figure 6: _Cont._  
Table 5 shows that the HMSCACSA had the best optimal solutions in most cases using the F1, F2, F3, F4, and F5 standard functions, reducing the minimum processing time by 23.5%, 31.32%, 29.1%, 29.0%, and 43.60%, respectively. Meanwhile, the PSOGSCA reached convergence faster than the MSCA by 27.69%, 29.47%, 26.44%, 29.49%, and 32.2% for functions F1, F2, F3, F4, and F5, respectively. Overall, it was indicated from the simulation outcomes that the proposed HMSCACSA had faster convergence than the MSCA.