## 2 Establishment of Evaluation System
### Fuzzy Analytic Hierarchy Evaluation Algorithm  
The evaluation indexes are compared in pairs, and a fuzzy judgment matrix is constructed. When the fuzzy analytic hierarchy process is used to construct the matrix, the matrix is often expressed in the form of interval number or fuzzy number. The triangular fuzzy number (\(l\), \(m\), \(u\)) is used to represent the elements in the matrix because it contains the interval number and the way of comparing the indicators through the median value, where l and u are the upper and lower bounds, respectively, and m is the median value. The expression of the fuzzy set membership function is shown in Equation (1):  
\[\mu_{m}(x) \begin{cases}\left(x-l\right)/\left(m-l\right),&l\leq x\leq m\\ \left(x-u\right)/\left(m-u\right),&m\leq x\leq u\\ 0,&others\end{cases}  \tag{1}\]  
The value of the fuzzy comprehensive degree of the \(i\)th object is defined as shown in Equation (2):  
\[s_{i}=t_{i}\otimes\left(t_{1}\oplus t_{2}\oplus\cdots t_{n}\right)^{-1}=\left( \begin{matrix}\sum\limits_{j=1}^{n}l_{ij}\\ \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}u_{ij}\\ \end{matrix} ,\frac{\sum\limits_{j=1}^{n}m_{ij}}{\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{n}m_{ij}},\frac{\sum\limits_{j=1}^{n}u_{ij}}{\sum\limits_{i=1} ^{n}\sum\limits_{j=1}^{n}l_{ij}}\right)=\left(l_{i},m_{i},u_{i}\right) \tag{2}\]  
The possibility of \(M_{1}\) (\(l_{1}\), \(m_{1}\), \(u_{1}\)) \(\geq\)\(M_{2}\) (\(l_{2}\), \(m_{2}\), \(u_{2}\)) is shown as Equation (3):  
\[\mu_{M_{2}}(d) \begin{cases}1&m_{1}\geq m_{2}\\ 0&others\\ \frac{l_{2}-u_{1}}{\left(m_{1}-u_{1}\right)-\left(m_{2}-l_{2}\right)},&m_{1} \leq m_{2},\ u_{1}\geq l_{2}\end{cases}  \tag{3}\]  
The possibility degree of triangular fuzzy number \(M\) greater than \(k\) triangular fuzzy numbers \(M_{i}\) (\(i=1\), \(2\), \(\ldots\), \(k\)) is defined by Equation (4):  
\[V(M\geq M_{1},M_{2},\cdots,M_{k})=\min V(M\geq M_{i}) \tag{4}\]  
The pure good measure that \(A_{i}\) is superior to other schemes is shown in Equation (5):  
\[d^{\prime}(A_{i})=\min V(S_{i}\geq S_{k}) \tag{5}\]  
The standardized weight of the calculated index is Equation (6):  
\[W^{\prime}=\left[d^{\prime}(A_{1}),d^{\prime}(A_{2}),\cdots,d^{\prime}(A_{n}) \right]^{T} \tag{6}\]  
After linearization, the weight value of each index is shown in Equation (7):  
\[W_{i}=\frac{W^{\prime}}{\sum W^{\prime}} \tag{7}\]  
In order to reasonably establish the evaluation system for semi-trailer train braking timing, three experts were invited to score the evaluation indexes proposed in this paper. Information about each expert is shown in Table 1. Each expert applies [PERSON]'s 1-9 scale table [14] to construct a judgment matrix that compares two indicators. The judgment matrices given by the three experts are represented by \(D_{1}\), \(D_{2}\) and \(D_{3}\), respectively. The judgment matrix is shown in Equation (8). The judgment matrix scale, linguistic variables and their corresponding triangular fuzzy numbers are shown in Table 2.  
\[D_{1}= \begin{bmatrix}1&1&1/3&1&1\\ 1&1&1/3&1/3&1\\ 3&3&1&3&3\\ 1&3&1/3&1&1/3\\ 1&1&1/3&3&1\end{bmatrix} D_{2}= \begin{bmatrix}1&1/3&1/5&1&1\\ 3&1&1/3&1/3&3\\ 5&3&1&5&3\\ 1&1&1/5&1&1/3\\ 1&1/3&1/3&3&1\end{bmatrix} D_{3}= \begin{bmatrix}1&1/3&1/5&1&1\\ 3&1&1&1&3\\ 5&1&1&3&1\\ 1&1&1/3&1&1\\ 1&1/3&1&1\end{bmatrix}  \tag{8}\]  
After the matrix in Equation (8) passes the consistency test, the expert scoring matrix is converted into a triangular fuzzy number judgment matrix according to the triangular fuzzy number given in Table 2. The converted results are shown in Tables 3-5. The matrix utilization Equation (9) of Tables 3-5 is transformed into a fuzzy comprehensive evaluation matrix. In Equation (9), \(\widetilde{a}_{ij}\) is the fuzzy comparison value of evaluation index \(i\) and evaluation index \(j\), and \(\widetilde{d}_{st}\) is the geometric mean of the fuzzy comparison value of index \(i\) to all indexes \(j\).