# Research on an Evaluation Method of Snowdrift Hazard for Railway Subgrades
## 2 Methods
### Establishment of the FCE Model
The common ridge distribution (Figure 3) and trapezoidal distribution (Figure 4) fuzzy distributions and their graphs are shown below. Figure 4: Trapezoidal distribution or semi-trapezoidal distribution. (**a**) right-skewed; (**b**) left-skewed; (**c**) symmetric. Figure 3: Ridge distribution or semi-ridge distribution. (**a**) right-skewed; (**b**) left-skewed; (**c**) symmetric. (1) Ridge distribution. a, right-skewed  
\[A(x)= \begin{cases}1&x<a_{1}\\ \frac{1}{2}-\frac{1}{2}\sin\frac{\pi}{a_{2}-a_{1}}\Big{(}x-\frac{a_{1}+a_{2}}{2} \Big{)}&a_{1}\leq x\leq a_{2}\\ 0&x>a_{2}\end{cases}  \tag{5}\]  
b, left-skewed  
\[A(x)= \begin{cases}0&x<a_{1}\\ \frac{1}{2}+\frac{1}{2}\sin\frac{\pi}{a_{2}-a_{1}}\Big{(}x-\frac{a_{1}+a_{2}}{ 2}\Big{)}&a_{1}\leq x\leq a_{2}\\ 1&x>a_{2}\end{cases}  \tag{6}\]  
c, symmetric  
\[A(x)= \begin{cases}0&x<-a_{2}\\ \frac{1}{2}+\frac{1}{2}\sin\frac{\pi}{a_{2}-a_{1}}\Big{(}x+\frac{a_{1}+a_{2}}{ 2}\Big{)}&-a_{2}\leq x\leq-a_{1}\\ 1&-a_{1}<x<a_{1}\\ \frac{1}{2}-\frac{1}{2}\sin\frac{\pi}{a_{2}-a_{1}}\Big{(}x-\frac{a_{1}+a_{2}}{ 2}\Big{)}&a_{1}\leq x\leq a_{2}\\ 0&x>a_{2}\end{cases}  \tag{7}\]  
(2) Trapezoidal distribution. a, right-skewed  
\[A(x)= \begin{cases}1&x<a\\ \frac{b-x}{b-a}&a\leq x\leq b\\ 0&x>b\end{cases}  \tag{8}\]  
b, left-skewed  
\[A(x)= \begin{cases}0&x<a\\ \frac{x-a}{b-a}&a\leq x\leq b\\ 1&x>b\end{cases}  \tag{9}\]  
c, symmetric  
\[A(x)= \begin{cases}0&x<a\\ \frac{x-a}{b-a}&a\leq x<b\\ 1&b\leq x<c\\ \frac{d-x}{d-c}&c\leq x<d\\ 0&x\geq d\end{cases}  \tag{10}\]  
5. Step 5: Establish the fuzzy comprehensive evaluation matrix  
This process requires selecting appropriate membership functions for the evaluation indicators. Generally, both quantitative and qualitative indicators are involved in slope risk assessment. For quantitative indicators, membership vectors can be obtained by constructing triangular, trapezoidal, and other membership functions for the indicators. For qualitative indicators, we employ scoring methods to quantify them, then construct membership functions and obtain corresponding membership vectors. For the second-level evaluation indicators \(U_{ij}\), the membership vectors can be represented as:  
\[R_{ij}=(r_{ij1},r_{ij2},\cdots,r_{ijl}) \tag{11}\]For the first-level evaluation indicators \(U_{i}\), the membership matrix can be represented as:  
\[R_{i}= \begin{bmatrix}r_{i11}&r_{i11}&\cdots&r_{i1l}\\ r_{i21}&r_{i21}&\cdots&r_{i2l}\\ \vdots&\vdots&\ddots&\vdots\\ r_{in_{i}1}&r_{in_{i}1}&\cdots&r_{in_{i}l}\end{bmatrix}  \tag{12}\]  
For the overall risk assessment system, the membership matrix can be represented as:  
\[R= \begin{bmatrix}R_{1}\\ R_{2}\\ \vdots\\ R_{m}\end{bmatrix} = \begin{bmatrix}r_{111}&r_{111}&\cdots&r_{11l}\\ r_{121}&r_{121}&\cdots&r_{12l}\\ \vdots&\vdots&\ddots&\vdots\\ r_{mm_{1}}&r_{mm_{2}}&\cdots&r_{mm_{m}l}\end{bmatrix}  \tag{13}\]  
Performing fuzzy calculations on the membership matrix of the evaluation indicator system and the weight matrix of the indicators that have been clearly determined gives:  
\[B=A\times R= \begin{pmatrix}a_{i1},a_{i2},\cdots,a_{in_{l}}\end{pmatrix}\begin{bmatrix} r_{111}&r_{111}&\cdots&r_{11l}\\ r_{121}&r_{121}&\cdots&r_{12l}\\ \vdots&\vdots&\ddots&\vdots\\ r_{mm_{m}1}&r_{mm_{2}}&\cdots&r_{mm_{m}l}\end{bmatrix} =(b_{1},b_{2},\cdots,b_ {l}) \tag{14}\]  
Normalization processing of the obtained comprehensive evaluation set B yields the normalized evaluation set \(\overline{B}=(\overline{b}_{1},\overline{b}_{2},\cdots\overline{b}_{l})\). According to the principle of maximum membership, selecting the maximum value in the fuzzy comprehensive judgment set can determine the risk level.