## 4 Improved Fuzzy Analytic Hierarchy Process (IFAHP)
### Improved Method of Consistency Test  
Assuming the participation of \(K\) experts in the evaluation, the complementary judgment matrix for each expert can be established as \(\widetilde{\mathbf{X}}(\widetilde{\mathbf{x}}_{ij}^{k})_{n\times n}\) based on Table 7, where \(\widetilde{\mathbf{x}}_{ij}^{k}=\left[a_{ij}^{k},b_{ij}^{k},c_{ij}^{k},d_{ij}^{k}\right]\) represents a trapezoidal fuzzy number, \(k\in\{1,2,\cdots,K\},i,j\in N\). The kernel and kernel operator are introduced to compute the kernel matrix of the complementary judgment matrix for trapezoidal fuzzy numbers, aiming to test the consistency [67]. The kernel of trapezoidal fuzzy numbers \(\widetilde{\mathbf{x}}_{ij}^{k}\) is defined as follows. \[\widetilde{\mathbf{x}}_{\text{ker},ij}^{k}=\left\{ \begin{array}{ll}a_{ij}^{k},& \text{when }a_{ij}^{k}=b_{ij}^{k}=c_{ij}^{k}=d_{ij}^{k}\\ \frac{\left(c_{ij}^{k}\right)^{2}+\left(d_{ij}^{k}\right)^{2}-\left(d_{ij}^{k }\right)^{2}-\left(b_{ij}^{k}\right)^{2}+a_{ij}^{k}c_{ij}^{k}-d_{ij}^{k}d_{ij} ^{k}}{3\left(c_{ij}^{k}+d_{ij}^{k}-d_{ij}^{k}-b_{ij}^{k}\right)},&\text{ otherwise}\end{array} \right. \tag{1}\]  
The kernel matrix of \(\widetilde{\mathbf{X}}(\widetilde{\mathbf{x}}_{ij}^{k})_{n\times n}\) can be obtained as \(\mathbf{X}_{\text{ker}}^{k}=\left(x_{\text{ker},ij}^{k}\right)_{n\times n}\). The process of consistency testing is outlined as follows. 1. The kernel matrix of the \(k\)-th expert is obtained as \(\mathbf{X}_{\text{ker}}^{k}\), in accordance with Equation (1). 2. According to Equation (2), the consistency index \(\rho\) of \(\mathbf{X}_{\text{ker}}^{k}\) can be determined. \[\rho=\frac{2}{n(n-1)(n-2)}\sum_{i=1}^{n-1}\sum_{j=i+1}^{n} \begin{array}{ll} \sum_{\begin{array}{c}f=1\\ f\
eq i,j\end{array}}^{k}&\left|\widetilde{\mathbf{x}}_{\text{ker},ij}^{k}-\left( \widetilde{\mathbf{x}}_{\text{ker},if}^{k}+\widetilde{\mathbf{x}}_{\text{ker},fi}^{k} -\frac{1}{2}\right)\right|\\ f\
eq i,j\end{array} \] (2) If \(\rho<\varepsilon\), then \(\tilde{\mathbf{X}}^{k}\) satisfies the consistency criterion and proceed to step 7; otherwise, proceed to the next step. \(\varepsilon=0.2\) can be identified as the critical value of the consistency indexes. 3. The eigenvalue matrix \(\mathbf{X}_{\text{ker}^{*}}^{k}=\left(x_{\text{ker}^{*},ij}^{k}\right)_{n\times n}\) can be calculated for the kernel matrix \(\mathbf{X}_{\text{ker}}^{k}\), denoted as follows. \[x_{\text{ker}^{*},ij}^{k}=\chi\Big{(}\varphi_{i}^{k}-\varphi_{j}^{k}\Big{)}+0.5\] (3) \begin{table}
\begin{tabular}{c c} \hline \hline
**Qualitative Language** & **Trapezoidal Fuzzy Number** \\ \hline Highly significant & (0.8, 1, 1, 1) \\ Significant & (0.7, 0.9, 1, 1) \\ Relatively significant & (0.6, 0.8, 0.8, 1) \\ Average & (0.3, 0.5, 0.5, 0.7) \\ Less significant & (0, 0.2, 0.2, 0.4) \\ Insignificant & (0, 0.0, 0.1, 0.3) \\ Highly insignificant & (0, 0, 0, 0.2) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Corresponding table of qualitative analysis and quantitative calculation. where the vector \(\varphi_{i}^{k}=\chi n/(n-1)+\sum\limits_{j=1}^{n}x_{\text{ker},ij}^{k}/\chi n\) represents the sorted values, where \(\chi\geq(n-1)/2\). 4. The deviation matrix \(O\) can be calculated for the kernel matrix \(\mathbf{X}_{\text{ker}}^{k}\), where \(\mathbf{O}=\left(\mathbf{o}_{ij}\right)_{n\times n}=\mathbf{X}_{\text{ker}}^{k}-\mathbf{X}_{ \text{ker}^{*}}^{k}\), \(|\mathbf{o}_{st}|=\max\bigl{\{}\left|\mathbf{o}_{ij}\right|\,:\,i<j,i,j\in n\bigr{\}}\). 5. If \(o_{st}<0\), proceed to step 6; if \(o_{st}>0\), appropriately decrease \(\widetilde{x}_{st}^{k}\) by a value of \(\alpha\), that is, \(\widetilde{x}_{st,\text{ker}}^{k}=\widetilde{x}_{st}^{k}-\alpha\), and correspondingly increase \(\widetilde{x}_{st}^{k}\) by the same value, that is, \(\widetilde{x}_{st,\text{ker}}^{k}=\widetilde{x}_{st}^{k}+\alpha\). Where \(\alpha\) represents the adjustment quantity with \(0<\alpha<1\). 6.