## 4 Experimental Results and Discussion of IHSSAO Performance Verification
### Experiment I: Classical Benchmark Function
#### 4.1.3 Comparison with IHSSAO and Other Algorithms  
The Avg and Std obtained by the IHSSAO after running in 23 benchmark functions are shown in Table 5. Based on the results, we can learn that the IHHSAO outperforms the other algorithms in most cases. The unimodal benchmark function \(F_{1}\)-\(F_{7}\) with only one global optimum can be used to measure the exploitation capability of the algorithm. IHSSAO can obtain the theoretical optimal solutions on \(F_{1}\), \(F_{2}\), \(F_{3}\), and \(F_{4}\), while the other algorithms cannot find the optimal solutions. Although IHSSAO cannot find the theoretical optimal solutions on \(F_{5}\) and \(F_{7}\), its convergence accuracy and robustness are better than the other algorithms. For \(F_{6}\), IHSSAO performs slightly worse than SSA, ranking second among the seven different algorithms. This indicates that IHSSAO has good exploitation capability. Functions \(F_{8}\)-\(F_{13}\) are multimodal benchmark functions that contain a large number of locally optimal solutions, so that these functions can be utilized to analyze the capability of the algorithm to escape from local optima. From the results shown in Table 5, the Avg and Std of IHSSAO outperform other algorithms in all the above multimodal cases. Among them, the identical global optimal minima were also obtained by the HHO and AO on \(F_{9}\) and \(F_{11}\). In addition, for \(F_{11}\), WOA, HHO, and AO have a similar performance to the IHSSAO. Through the above tests, it can be tentatively verified that the IHSSAO has an excellent ability to avoid falling into local optima. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters** \\ \hline WOA [13] & \(b=1\); \(a_{1}=[2,\ 0]\); \(a_{2}=[-1,-2]\) \\ HHO [14] & \(\beta=1.5\); \(E_{0}\in[-1,\ 1]\) \\ SSA [16] & \(c_{1}=[1,\ 0]\); \(c_{2},c_{3}\in[0,\ 1]\) \\ MVO [34] & \(WEP\in[0.2,\ 1]\); \(TDR\in[0,\ 1]\); \(r_{1},r_{2},r_{3}\in[0,\ 1]\) \\ STOA [35] & \(C_{f}=2\); \(C_{b}\in[0,\ 0.5]\); \(u,v=1\) \\ TSA [36] & \(P_{max}=4,\ P_{min}=1\) \\ AO [23] & \(U=0.00565\); \(r_{3}=10\); \(\alpha=0.1\); \(\delta=0.1\); \(\omega=0.005\); \(G_{1}\in[-1,\ 1]\); \(G_{2}=[2,\ 0]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter settings for the optimization algorithms. Functions \(F_{14}\)-\(F_{23}\) belong to the fix-dimension multimodal benchmark functions, which consist of a few locally optimal solutions and are implemented to evaluate the algorithm performance in converting between exploration and exploitation phases. In terms of average fitness values, the IHHSAO runs better than the remaining seven algorithms on functions \(F_{15}\), \(F_{17}\), \(F_{21}\), \(F_{22}\), and \(F_{23}\), while achieving the same best results as some of the algorithms on \(F_{16}\), \(F_{18}\), and \(F_{19}\). In addition, for \(F_{14}\), the proposed IHSSAO algorithm performs worse than MVO, but it can be ranked in second place. For \(F_{20}\), the performance of IHSSAO is weaker than the MVO, TSA, and SSA ranking only fourth but with little difference in results. The above results show, to some extent, that the proposed algorithm has significantly improved performance compared with the original SSA and AO, and performs better in most cases in comparison with the other five advanced algorithms. On the other hand, IHSSAO attains the best standard deviation in all cases except for \(F_{14}\), \(F_{18}\), \(F_{19}\), and \(F_{20}\), which demonstrates that the IHSAAO can maintain a much better balance between exploration and exploitation.