## 3 Validation
### Discussion  
Table 2 presents the OSP results obtained for the Canton Tower when using different approaches. The results from referenced papers are referred to with square brackets, followed by denotation to separate variants. References [27] and Reference [6] (0.1) used the same optimization objective to minimize the maximum off-diagonal element. The difference between the aforementioned references is related to the algorithm used for optimization. References [6] (0.1) and [6] (0.2) used the same algorithm for optimization, but Reference [6] (0.2) minimized the sum of the least squares of the off-diagonal elements; thus, Reference [6] (0.2) used the first objective function in this study.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Direction** & **MOP-No. 28** & **Reference [27]** & **Reference [6]** & **Reference [6]** \\  & & & **(o.1)** & **(o.2)** \\ \hline \multirow{2}{*}{x} & 5, 6, 10, 14, 16, 17, 22, 25, 26, 27 & 2, 6, 14, 20, 21, 22, 23, 24, 29 & 2, 3, 17, 18, 19, 20, 21, 26 \\ \hline \multirow{2}{*}{y} & 3, 4, 9, 11, 16, 20, 22, 25, 26, 27 & 1, 2, 4, 5, 6, 7, 11, 16, 20, 22, 23 & 1, 2, 3, 10, 17, 19, 20, 23, 27 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Optimal sensor placement (OSP) node ID of the Canton Tower when using 20 sensors with different methods.  
Figure 3: Visualization of Pareto Set No. 28: (**a**) Three objectives; (**b**) visualization of \(f_{1}\) and \(f_{2}\); (**c**) visualization of \(f_{1}\) and \(f_{3}\); and (**d**) visualization of \(f_{2}\) and \(f_{3}\).  
We added the different OSP solutions into the proposed objective function to compare values. Table 3 indicates that the method of minimizing the maximum off-diagonal elements has no obvious advantages over the method used in [6] (0.2) and the MOP. The results of the aforementioned solutions exhibit poor performance for \(f_{2}\); thus, the results have no comparative significance. The results of the single-objective optimization method used in [6] (0.2) are superior to those of the MOP method for \(f_{1}\). By contrast, for \(f_{3}\), the solution of the MOP method is superior to that of the single-objective method.  
We subsequently calculated the two solution groups for Pareto Set No. 28 simultaneously with the proposed MODM method. We used MODM analysis to compare the performance levels of the MOP method and the method used in [6] (0.2). The membership functions and closeness distance of [6] (0.2) indicated that its solution was inferior to that of the MOP method. For the first objective, the single-objective algorithm exhibited the best performance, which is in line with the expectation. Thus, the multiobjective optimization algorithm still has room for progress in single-objective convergence. The membership values in Table 3 indicate that the multiobjective optimization solution in the first and third objectives were notably close to the results of the ideal solution in the solution set. Moreover, the method used in [6] (0.2) provided the best results for the first objective; however, this method only provided average results for the third objective. Thus, the MOP method outperformed the method in [6] (0.2) for the third objective. The results also indicated that the aforementioned two optimization methods were not effective for the second objective.