## 5 Comprehensive Evaluation Example and Numerical Simulation
Therefore, the algorithm will not terminate until it reaches the convergence condition. At the same time, due to the a better initial value, the algorithm in this paper shows faster convergence speed than the classical eigenvector method. Figure 6: Comparison diagram of three consistency improvement methods. Use the aggregation method of Section 4.2 to obtain a comprehensive judgment matrix, and finally obtain the weight of each index to form the next generation relay satellite system development proposal. Parameter setting before aggregation: expert authority coefficient \(A1>A2>A3>A4\), and \(\alpha=\beta=0.5\); other parameters are shown in Table 3.  
The comprehensive judgment \(\overline{A}\) matrix would be  
\[\overline{A}=A_{1}^{0.2778}\circ A_{2}^{0.2952}\circ A_{3}^{0.1992 }\circ A_{4}^{0.2278}\] \[=\left[ \begin{array}{rrrrrr}1&0.5847&0.3040&1.0387&0.2311&0.6959&1. 6706\\ 1.7103&1&0.5983&2.8739&0.6384&2.5881&2.0944\\ 3.2899&1.6714&1&2.5328&1.6969&2.7672&4.6426\\ 0.9628&0.3480&0.3948&1&0.6701&0.8301&1.8242\\ 4.3280&1.5665&0.5893&1.4924&1&2.9778&4.0197\\ 1.4369&0.3864&0.3614&1.2047&0.3358&1&2.4724\\ 0.5986&0.4775&0.2154&0.5482&0.2488&0.4045&1\end{array} \right]\]  
After calculation, the maximum real eigenvalue \(\lambda_{max}=7.2075\). Then, \(\lambda_{max}\) is substituted into the CR calculation formula: \(CR=\frac{CI}{RI}=\frac{\lambda_{max}-n}{n-1}/RI(n)\); random consistency index (RI) [34] is shown in Table 4.  
The calculated consistency ratio \(CR_{1}=0.0258<0.1\). Therefore, the Comprehensive judgment matrix meets the consistency conditions. The eigenvector of \(\lambda_{max}\) is the final index weight, which is W = [0.1846 **0.4041 0.6359** 0.2166 **0.5333** 0.2260 0.1262 ]\({}^{T}\). It can be seen that the network capacity(C3), constellation coverage capability(C5), and channel support capability(C2) are the three indicators that the group of experts are most concerned in this evaluation example. In addition, for comparison, we calculated the case where only expert weights are considered as aggregation coefficients. That is, \(\overline{A}=A_{1}^{0.3077}\circ A_{2}^{0.2692}\circ A_{3}^{0.2308}\circ A_{4} ^{0.1923}\), \(\lambda_{max}=7.2105,CR_{2}<CR_{1}<0.1\). It can be seen from the comparison that the comprehensive judgment matrix generated according to the aggregation coefficient proposed in Section 4.2 has achieved a better consistency ratio in this example. However, it should be pointed out here that the comprehensive judgment matrix obtained according to the AIJ aggregation order is a typical convex combination. Therefore, after aggregation, the maximum real eigenvalues of the comprehensive judgment matrix \(\lambda_{\overline{A}}\) must satisfy \(\lambda_{\overline{A}}\leq\min(\lambda_{A1},\lambda_{A2},\lambda_{A3},\lambda _{AA})\). Therefore, the method proposed in this paper can make the comprehensive judgment matrix's consistency stronger when the initial input matrix consistency is poor. When the consistency of the initial input matrix is good, this improvement will become less obvious (as in this example, both aggregation coefficients are acceptable and the difference is very small). In order to explain this conclusion more intuitively, the aggregation coefficients of A1 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**n** & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
**RI(n)** & 0.52 & 0.88 & 1.11 & 1.25 & 1.34 & 1.40 & 1.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Random consistency index (RI). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) \\ \hline \(\lambda_{\max}\) & 7.6672 & 7.5153 & 7.9881 & 7.6284 \\ \(CI\) & 0.1112 & 0.0859 & 0.1647 & 0.1047 \\ \(\lambda_{1}^{2}\) & 0.2480 & 0.3211 & 0.1675 & 0.2633 \\ \(\lambda_{1}^{1}\) & 0.3077 & 0.2692 & 0.2308 & 0.1923 \\ \(\lambda_{i}\) & 0.2778 & 0.2952 & 0.1992 & 0.2278 \\ \hline \hline \end{tabular} \(\lambda_{i}\) is the hadamard aggregation coefficient.  
\end{table} 
Table 3: Calculation parameter.