## 5 Results and Discussion
### Algorithm Performance Evaluation
#### 5.1.2 Convergence Performance Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & **Evaluation** & \multirow{2}{*}{**GWO**} & \multirow{2}{*}{**SSA**} & \multirow{2}{*}{**WOA**} & \multirow{2}{*}{**DBO**} & \multirow{2}{*}{**HHO**} & \multirow{2}{*}{**IGWOHHO**} \\  & **Indicators** & & & & & & & \\ \hline \multirow{3}{*}{Schwefel 2.22} & Best & \(4.80\times 10^{-24}\) & \(1.82\times 10^{-245}\) & \(3.75\times 10^{-60}\) & \(1.77\times 10^{-96}\) & \(1.90\times 10^{-61}\) & 0 \\  & Mean & \(1.19\times 10^{-23}\) & \(3.14\times 10^{-234}\) & \(2.51\times 10^{-56}\) & \(1.74\times 10^{-77}\) & \(3.37\times 10^{-55}\) & 9.03 \(\times 10^{-238}\) \\  & St.dev & \(6.62\times 10^{-24}\) & 0 & \(5.02\times 10^{-56}\) & \(3.89\times 10^{-77}\) & \(7.32\times 10^{-55}\) & 0 \\ \hline \multirow{3}{*}{Rosenbrock} & Best & \(2.56\times 10^{1}\) & \(2.92\times 10^{-7}\) & \(2.63\times 10^{1}\) & \(2.41\times 10^{1}\) & \(8.21\times 10^{-5}\) & \(2.82\times 10^{-11}\) \\  & Mean & \(2.64\times 10^{1}\) & \(2.94\times 10^{-1}\) & \(2.66\times 10^{1}\) & \(2.47\times 10^{1}\) & \(4.17\times 10^{-4}\) & \(2.05\times 10^{-6}\) \\  & St.dev & \(6.52\times 10^{-1}\) & \(3.01\times 10^{-6}\) & \(2.84\times 10^{-1}\) & \(4.24\times 10^{-1}\) & \(3.43\times 10^{-4}\) & \(3.17\times 10^{-10}\) \\ \hline \multirow{3}{*}{Ackley} & Best & \(2.58\times 10^{-14}\) & \(8.88\times 10^{-16}\) & \(8.88\times 10^{-16}\) & \(8.88\times 10^{-16}\) & \(8.88\times 10^{-16}\) & \(8.88\times 10^{-16}\) \\  & Mean & \(2.86\times 10^{-14}\) & \(8.88\times 10^{-16}\) & \(3.02\times 10^{-15}\) & \(8.88\times 10^{-16}\) & \(8.88\times 10^{-16}\) & \(8.88\times 10^{-16}\) \\  & St.dev & \(1.58\times 10^{-15}\) & 0 & \(3.18\times 10^{-15}\) & 0 & 0 & 0 \\ \hline \multirow{3}{*}{Penalized} & Best & \(6.57\times 10^{-3}\) & \(2.45\times 10^{-13}\) & \(3.02\times 10^{-4}\) & \(7.26\times 10^{-16}\) & \(2.55\times 10^{-11}\) & \(7.25\times 10^{-16}\) \\  & Mean & \(1.92\times 10^{-2}\) & \(1.86\times 10^{-12}\) & \(1.56\times 10^{-3}\) & \(3.18\times 10^{-8}\) & \(8.70\times 10^{-7}\) & \(8.23\times 10^{-14}\) \\  & St.dev & \(1.61\times 10^{-2}\) & \(2.15\times 10^{-12}\) & \(2.54\times 10^{-3}\) & \(7.11\times 10^{-8}\) & \(1.67\times 10^{-6}\) & \(1.27\times 10^{-13}\) \\ \hline \multirow{3}{*}{Foxholes} & Best & \(9.98\times 10^{-1}\) & \(9.98\times 10^{-1}\) & \(9.98\times 10^{-1}\) & \(9.98\times 10^{-1}\) & \(9.98\times 10^{-1}\) & \(9.98\times 10^{-1}\) \\  & Mean & \(3.33\times 10^{0}\) & \(3.73\times 10^{0}\) & \(9.98\times 10^{-1}\) & \(9.98\times 10^{-1}\) & \(1.20\times 10^{0}\) & \(9.98\times 10^{-1}\) \\  & St.dev & \(5.22\times 10^{0}\) & \(5.07\times 10^{0}\) & \(3.79\times 10^{-10}\) & \(2.89\times 10^{-14}\) & \(4.45\times 10^{-1}\) & 0 \\ \hline \multirow{3}{*}{Kowalik} & Best & \(3.31\times 10^{-4}\) & \(3.31\times 10^{-4}\) & \(3.24\times 10^{-4}\) & \(3.08\times 10^{-4}\) & \(3.08\times 10^{-4}\) & \(3.07\times 10^{-4}\) \\  & Mean & \(3.07\times 10^{-4}\) & \(3.08\times 10^{-4}\) & \(8.20\times 10^{-4}\) & \(8.07\times 10^{-4}\) & \(4.96\times 10^{-4}\) & \(3.07\times 10^{-4}\) \\ \cline{1-1}  & St.dev & \(2.54\times 10^{-9}\) & \(4.58\times 10^{-8}\) & \(4.30\times 10^{-4}\) & \(4.32\times 10^{-4}\) & \(4.07\times 10^{-4}\) & \(2.05\times 10^{-12}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Algorithm optimality results. Figure 11: Convergence trends of algorithms on benchmark test functions. (**a**) Schwefel 2.22 function; (**b**) Rosenbrock function; (**c**) Ackley function; (**d**) Penalized function; (**e**) Foxholes function; (**f**) Kowalik function. The performance evaluation confirms that the IGWOHHO algorithm outperforms several swarm intelligence algorithms across various benchmark functions. It achieves high solution accuracy, strong robustness, and stable convergence in both unimodal and multimodal scenarios. The dynamic adjustment mechanism ensures a balanced exploration and exploitation process, enhancing adaptability to different problem landscapes.