## 3 Case Study
### OSP in Seamless Condition
#### 3.3.4 Evaluation of OSP Results  
To further analyze the differences between various intelligent algorithms, quantitative analysis is conducted using MAC, fitness values, and S criteria to determine the most suitable optimization algorithm for this arch dam [20; 31]. To describe the modal independence of solutions obtained by different algorithms, the maximum value of the off-diagonal elements of the MAC matrix (MAC-MAX), the average value of the off-diagonal elements of the MAC matrix (MAC-AVE), the root mean square of the off-diagonal elements of the MAC matrix (MAC-RMS), and the maximum singular value ratio (S) are chosen as evaluation criteria, and the smaller the value of these four indexes, the better the results of sensor placement. The evaluation metrics and MAC matrix diagrams of the layout results of different optimization methods are shown in Figures 12 and 13. From Figures 12 and 13, it is apparent that the results of sensor placement obtained by all improved intelligent algorithms are superior to the results before improvement. The four indicators of IAHA are all smaller than the other five intelligent algorithms, among which MAC-MAX has decreased by 50%, 45%, 53%, 47%, and 8% compared to the other five intelligent algorithms, respectively; MAC-AVE has reduced by 50%, 50%, 47%, 33%, and 11% compared to the other five intelligent algorithms, respectively; MAC-RMS has reduced by 45%, 42%, 44%, 34%, and 5% compared to the other five intelligent algorithms, respectively; S has reduced by 38%, 33%, 43%, 34%, and 7% compared to the other five intelligent algorithms, respectively. Among the six intelligent algorithms, IAHA shows the best performance, with its four indicators being smaller than the other five intelligent algorithms. Therefore, IAHA demonstrates the most effective optimization for the fitness function used in this paper. Figure 11: OSP results based on intelligent algorithms: (**a**) PSO, (**b**) IPSO, (**c**) SSA, (**d**) ISAMS, (**e**) AHA, and (**f**) IAHA. Figure 12: Evaluation indicators for the layout results of different optimization methods: (**a**) MAC-MAX, (**b**) MAC-AVE, (**c**) MAC-RMS, and (**d**) S.  
Figure 13: MAC matrix diagrams corresponding to different optimization methods: (**a**) PSO, (**b**) IPSO, (**c**) SSA, (**d**) ISSAMS, (**e**) AHA, and (**f**) IAHA. To validate the superiority of IAHA, a further analysis of six intelligent algorithms is conducted in terms of algorithm performance. Since the performance and solution outcomes of intelligent algorithms are significantly influenced by the initialization parameters of the program, efforts are made to align these six algorithms on a similar starting line as much as possible before testing and set the population size of each algorithm to 100, the number of iterations to 500, and the variable dimension to 20. The remaining parameter settings for each algorithm are shown in Table 4. Due to the inherent stochastic nature of intelligent algorithms, this paper runs 10 times on each of the six algorithms. The maximum, average, and minimum fitness values of each algorithm are compared, and the corresponding convergence curves are plotted in Figure 14.  
According to Figure 14, in terms of algorithm convergence accuracy, the convergence curves of the maximum, average, and minimum fitness values for the improved algorithms in the 10 runs all show better performance, and IAHA has the best convergence performance. According to Table 4, in terms of algorithm parameter settings, AHA and IAHA only need to set three parameters, while PSO, IPSO and SSA, ISSAMS need to set six and five parameters, respectively. From the convergence time of the algorithms, according to Table 5, it can be seen that the respective algorithms have increased in time compared to the time before the