## 4 Service and Communication Technology Adaptation Method
### Determination of Index Weights Based on the FAHP-Improved CRITIC Combined Weighting Model
#### 4.2.1 Subjective Weighting  
The AHP method is a decision-making tool. It is widely used in multi-criteria decision-making problems. It relies on expert evaluation when conducting decision analysis, and the main characteristics of this subjective evaluation are fuzziness and uncertainty. FAHP takes into account the characteristics in the process of pairwise comparison of indexes and introduces fuzzy theory based on the traditional AHP method, which greatly reduces judgment errors. Figure 4: Subjective and objective combination weighting model. This paper uses the FAHP method based on triangular fuzzy numbers. The steps to weight the indexes of the adaptability evaluation system using the FAHP method are as follows. Step 1. Establishing a fuzzy judgment matrix. Experts and relevant practitioners score the selected communication indexes in pairs using a triangular fuzzy scale (as shown in Table 1), thereby establishing a fuzzy judgment matrix \(\tilde{A}^{k}=(\tilde{a}^{k}_{ij})_{n\times n}\), where \(\tilde{a}^{k}_{ji}=(\tilde{a}^{k}_{ij})^{-1}=(1/u_{ij},1/h_{ij},1/l_{ij})\):  
\[\tilde{A}^{k}=(\tilde{a}^{k}_{ij})_{n\times n}=\] \[ \begin{bmatrix}(1,1)&(u_{12},h_{12},l_{12})&\cdots&(u_{1n},h_{1n}, l_{1n})\\ (\frac{1}{u_{21}},\frac{1}{h_{21}},\frac{1}{l_{21}})&(1,1,1)&\cdots&(u_{2n},h_{2 n},l_{2n})\\ \vdots&\vdots&\ddots&\vdots\\ (\frac{1}{u_{1n}},\frac{1}{h_{n1}},\frac{1}{l_{n1}})&(\frac{1}{u_{n2}},\frac{ 1}{h_{n2}},\frac{1}{l_{n2}})&\cdots&(1,1,1)\end{bmatrix}  \tag{1}\]  
where \(k\) (\(k\) = 1, 2,..., \(K\)) represents the expert number and \(i\) (\(i\) = 1, 2,..., \(n\)) represents the index number. The judgment matrices from a total of \(K\) decision makers are averaged to update the decision matrix, obtaining \(\tilde{A}\):  
\[\tilde{a}_{ij}=\frac{\sum_{k=1}^{K}\tilde{a}^{k}_{ij}}{K} \tag{2}\]  
\[\tilde{A}=(\tilde{a}_{ij})_{n\times n}=\left(\frac{\sum_{k=1}^{K}\tilde{a}^{k}_ {ij}}{K}\right)_{n\times n} \tag{3}\]  
Step 2. Consistency check. For the updated decision matrix, the median of the elements represented by triangular fuzzy numbers is taken to form the median matrix \(m\ =\ (si)n\times n\). Calculate the largest eigenvalue \(\lambda\) max of matrix m, and then determine the consistency index CI using the formula. The random consistency index RI can be obtained from a lookup table. If the ratio of CI to RI is less than 0.1, the consistency check is passed. Otherwise, the fuzzy judgment matrix needs to be reconstructed. Step 3. Calculation of subjective weights. For triangular fuzzy numbers \(M1\ =\ (l1,h1,u1)\) and \(M2\ =\ (l2,h2,u2)\), the following computational rules are adhered to:  
\[ \begin{cases}M_{1}\oplus M_{2}=(l_{1}+l_{2},h_{1}+h_{2},u_{1}+u_{2})\\ M_{1}\otimes M_{2}=(l_{1}l_{2},h_{1}h_{2},u_{1}u_{2})\\ M_{1}^{-1}=(l_{1}^{-1},h_{1}^{-1},u_{1}^{-1})\end{cases}  \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scale** & **Definition** & **Triangular Fuzzy Numbers** \\ \hline
1 & equally important & (1,1,1) \\
3 & slightly important & (2,3,4) \\
5 & obviously important & (4,5,6) \\
7 & very important & (6,7,8) \\
9 & extremely important & (9,9,9) \\ \hline
2 & & (1,2,3) \\
4 & interval value between two adjacent levels & (3,4,5) \\
6 & & (5,6,7) \\
8 & & (7,8,9) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy triangular scale.