## 4 Experimental Results and Discussion of IHSSAO Performance Verification
### Experiment I: Classical Benchmark Function
#### 4.1.3 Comparison with IHSSAO and Other Algorithms
Moreover, the proposed IHSSAO ranked third when solving \(F_{14}\) and \(F_{19}\), and ranked fourth on \(F_{20}\) in Std performance. In summary, the IHSSAO proposed in this paper has a strong global search capability compared with the SSA and the AO, as well as significant advantages compared with the other five intelligent algorithms. Figure 6 shows the boxplot of seven different algorithms, namely, WOA, HHO, SSA, MVO, TSA, AO, and IHSSAO on 23 benchmark functions, and it can be noticed that the IHSSAO has a relatively narrower box plot compared to other algorithms in most cases, which indicates good consistency in terms of median, maximum, and minimum values. According to the convergence curves of the above several different algorithms on 23 benchmark functions represented in Figure 7, we can observe that the IHSSAO converges more rapidly than the other algorithms in most cases. IHSSAO displays tremendous superiority over the other advanced algorithms in the optimization process, with three main differences appearing in convergence behavior. The first behavior is that convergence of IHSSAO has an obvious advantage. For \(F_{1}\)-\(F_{4}\), the IHSSAO converges more rapidly than other algorithms and can obtain optimal values which cannot be obtained by other algorithms. The second behavior is the extremely quick convergence, as observed in \(F_{8}\), \(F_{9}\)-\(F_{11}\), and \(F_{14}\)-\(F_{23}\). For these functions, IHSSAO can find the optimal value extremely fast during the iteration and the exact approximation of the global optimal value is almost the best. The last behavior mainly shows the local best avoidance ability of IHSSAO. For \(F_{5}\), \(F_{6}\), \(F_{7}\), and \(F_{13}\), the algorithm jumps out of the local optimum after several stops, because it benefits from the influence of the PIOBL strategy. In addition, we introduce a nonparametric statistical test in this section, namely the Wilcoxon rank-sum test. This test is utilized to calculate the difference in statistical performance between algorithms for demonstrating the significance of the proposed IHSSAO. In this study, we set the significance level at 0.05. The obtained \(p\)-values and statistical results of IHSSAO using the Wilcoxon rank-sum test are listed in Table 6. In this table, the \"\(+\)\" sign denotes that the IHSSAO has a better performance than the comparison algorithms, the \"\(-\)\" sign denotes that the IHSSAO has a worse performance than the comparison algorithms, and \"\(=\)\" represents the IHSSAO is similar to the algorithms in the comparison. The last three rows of this table indicate the times of IHSSAO obtained \"\(+\)\", \"\(=\)\", and \"\(-\)\" compared with each algorithm in the Wilcoxon rank-sum test. Among the 23 benchmark functions, IHSSAO, respectively, outperformed STOA and TSA 23 times; outperformed WOA, SSA, and MVO 22 times; and outperformed HHO and AO 19 times. Based on the above statistics, it is evident that the IHSSAO proposed in this paper is significantly enhanced compared to the original SSA and AO and is the best optimizer among the seven advanced algorithms.