## 3 Implementation
### Evaluating the Enhanced Algorithm Through Tests and Simulations
#### 3.1.2 Evaluating IGWO Against Competing Algorithms
In contrast, the basic GWO, WOA, and NGO algorithms exhibit more significant variations and fluctuations in their fitness value trajectories on certain unimodal functions, indicating that these algorithms are more susceptible to becoming trapped in local optima or encountering difficulties in overcoming saddle points. To further extend the comparative analysis to multimodal test functions, Figure 5 displays the trends in fitness values of the four algorithms on functions \(\mathrm{f_{6}}-\mathrm{f_{10}}\), illustrating their optimization performance in these scenarios. The analysis presented in Figure 5 clearly shows that the IGWO algorithm outperforms the GWO, WOA, and NGO algorithms in optimizing multimodal functions, achieving \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & & **GWO** & **WOA** & **NGO** & **IGWO** \\ \hline \multirow{2}{*}{\(\mathrm{f_{1}}\)} & Mean & 3.9762 \(\times\)\(10^{-6}\) & 47,898.1385 & 1.1477 \(\times\)\(10^{-22}\) & **0** \\  & Standard Deviation & 9.4465 \(\times\)\(10^{-6}\) & 10,779.6838 & 4.0563 \(\times\)\(10^{-22}\) & **0** \\  & Mean & 0.0019989 & 0.0035984 & 0.00070632 & **7.1205 \(\times\)\(10^{-5}\)** \\  & Standard Deviation & 0.0012469 & 0.0054488 & 0.00030836 & **6.8094 \(\times\)\(10^{-5}\)** \\  & Mean & 1.0125 \(\times\)\(10^{-27}\) & 2.3525 \(\times\)\(10^{-72}\) & 7.7076 \(\times\)\(10^{-88}\) & **0** \\  & Standard Deviation & 1.5709 \(\times\)\(10^{-27}\) & 1.2881 \(\times\)\(10^{-71}\) & 1.2394 \(\times\)\(10^{-87}\) & **0** \\  & Mean & 6.7115 \(\times\)\(10^{-7}\) & 38.1611 & 1.2477 \(\times\)\(10^{-37}\) & **3.1068 \(\times\)\(10^{-279}\)** \\  & Standard Deviation & 4.8223 \(\times\)\(10^{-7}\) & 29.044 & 1.023 \(\times\)\(10^{-37}\) & **0** \\  & Mean & 1.06 \(\times\)\(10^{-16}\) & 6.8394 \(\times\)\(10^{-52}\) & 1.2728 \(\times\)\(10^{-45}\) & **7.4703 \(\times\)\(10^{-305}\)** \\  & Standard Deviation & 8.6006 \(\times\)\(10^{-17}\) & 2.2572 \(\times\)\(10^{-51}\) & 1.3106 \(\times\)\(10^{-45}\) & **0** \\  & Mean & \(-\)5818.9213 & \(-\)**10**,**32**,**6**818 & \(-\)7637.5386 & \(-\)3466.9346 \\  & Standard Deviation & 795.4492 & **1763.6363 & 613.4378 & 118.4035 \\  & Mean & 1.176 & **0** & **0** & **0** \\  & Standard Deviation & 2.3394 & **0** & **0** & **0** \\  & Mean & 1.019 \(\times\)\(10^{-13}\) & 4.6777 \(\times\)\(10^{-15}\) & 6.6909 \(\times\)\(10^{-15}\) & **8.8818 \(\times\)\(10^{-16}\)** \\  & Standard Deviation & 1.694 \(\times\)\(10^{-14}\) & 2.9405 \(\times\)\(10^{-15}\) & 1.7413 \(\times\)\(10^{-15}\) & **0** \\  & Mean & 0.0022884 & 0.010723 & **0** & **0** \\  & Standard Deviation & 0.006078 & 0.040847 & **0** & **0** \\  & Mean & 5.7 & 3 & **3** & 599.8247 \\  & Standard Deviation & 14.7885 & 8.3808 \(\times\)\(10^{-5}\) & **1.2802 \(\times\)\(10^{-15}\)** & 0.019301 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results of different GWO algorithms for 10 test functions. exceptionally good results. This advantage is notably significant for functions f\({}_{7}\), f\({}_{8}\), and f\({}_{10}\), where it managed a standard deviation of 0. Except in the f\({}_{10}\) case, the IGWO algorithm has shown superior optimization capabilities. These findings underscore the IGWO algorithm's robust global search potential and its proficiency in overcoming local optima. It proves effective and exhibits high performance across a range of optimization challenges, regardless of their dimensionality or complexity, showcasing the efficacy of the enhancements made to the original algorithm. Figure 4: Cont. Figure 4: Comparison of function plot and convergence curve of single–peaked function. (**a**) f\({}_{1}\)–Schwefel 1.2 Function; (**b**) f\({}_{2}\)–Quartic Function; (**c**) f\({}_{3}\)–Sphere Function; (**d**) f\({}_{4}\)–Schwefel 2.21 Function; (**e**) f\({}_{5}\)–Schwefel 2.22 Function. Figure 5: _Cont._