# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 2 Fuzzy Characterization Method for Geotechnical Parameters
### Fuzzy Sets and Fuzzy Cut set Theory  
Given a fuzzy set \(A\in U\), for any \(\lambda\in[0,1]\), let \(A_{\lambda}=\left\{x\Big{|}x\in U,\mu_{A}(x)\geq\lambda\right\}\) be the \(\lambda\)-cut set of fuzzy set \(A\), which is the set of all elements in the region \(U\) whose affiliation to fuzzy set \(A\) is more than or equal to \(\lambda\). By taking different values of \(\lambda\), different \(\lambda\)-cut sets can be constructed, and thus the fuzzy set \(A\) can be transformed into several cut sets \(A\) corresponding to different values of \(\lambda\). In each cut set \(A\), the minimum value \(\underline{x}_{\lambda}\) and the maximum value \(\overline{x}_{\lambda}\) of the elements can be determined, and thus the corresponding interval \([\underline{x}_{\lambda},\overline{x}_{\lambda}]\) of the elements is obtained. For the triangular fuzzy numbers (as shown in Figure 1a), by taking the \(\lambda\)-cut set of fuzzy set \(A\), we can obtain the corresponding interval \([\underline{x}_{\lambda},\overline{x}_{\lambda}]\). The values of the lower bound \(\underline{x}_{\lambda}\) and the upper bound \(\overline{x}_{\lambda}\) are as follows:  
\[\left. \begin{array}{l}\underline{x}_{\lambda}=\mu+(\lambda-1)k\sigma\\ \overline{x}_{\lambda}=\mu+(1-\lambda)k\sigma\end{array} \right\} \tag{1}\]  
Figure 1: Four common types of fuzzy affiliation functions (**a**) The triangular fuzzy affiliation function; (**b**) The trapezoidal fuzzy affiliation function; (**c**) The normal fuzzy affiliation function; (**d**) The log-normal fuzzy affiliation function. where \(\mu\) is the mean value of parameter \(x\), \(\sigma\) is the standard deviation of parameter \(x\), and \(k\) is the shape parameter (depending on the data available and accuracy of the results desired [6]) of the fuzzy numbers, \(k\in[0.5,3]\). For the trapezoidal fuzzy numbers (as shown in Figure 1b), by taking the \(\lambda\)-cut set of fuzzy set \(\Delta\), we can obtain the corresponding interval \([\underline{x}_{\lambda},\,\overline{x}_{\lambda}]\). The values of the lower bound \(\underline{x}_{\lambda}\) and the upper bound \(\overline{x}_{\lambda}\) are as follows:  
\[ \begin{array}{l}\underline{x}_{\lambda}=\mu+(\lambda-1)k_{2}\sigma-\lambda k _{1}\sigma\\ \overline{x}_{\lambda}=\mu+(1-\lambda)k_{2}\sigma+\lambda k_{1}\sigma\end{array}  \tag{2}\]  
where \(\mu\) and \(\sigma\) have the same meanings as above, \(k_{1}\) and \(k_{2}\) are the shape parameters (depending on the data available and the accuracy of the results desired [6]) of the trapezoidal fuzzy numbers, \(k_{1},\,k_{2}\in[0.5,3]\). For the normal fuzzy numbers (as shown in Figure 1c), by taking the \(\lambda\)-cut set of fuzzy set \(\Delta\), we can obtain the corresponding interval \([\underline{x}_{\lambda},\,\overline{x}_{\lambda}]\). The values of the lower bound \(\underline{x}_{\lambda}\) and the upper bound \(\overline{x}_{\lambda}\) are as follows:  
\[ \begin{array}{l}\underline{x}_{\lambda}=\mu-\sigma\sqrt{-2\text{ln}[\lambda+ (1-\lambda)\text{exp}(-k^{2}/2)]}\\ \overline{x}_{\lambda}=\mu+\sigma\sqrt{-2\text{ln}[\lambda+(1-\lambda)\text{ exp}(-k^{2}/2)]}\end{array}  \tag{3}\]  
where \(\mu\), \(\sigma\) and \(k\) have the same meanings as above. For the log-normal fuzzy numbers (as shown in Figure 1d), by taking the \(\lambda\)-cut set of fuzzy set \(\Delta\), we can obtain the corresponding interval \([\underline{x}_{\lambda},\,\overline{x}_{\lambda}]\). The values of the lower bound \(\underline{x}_{\lambda}\) and the upper bound \(\overline{x}_{\lambda}\) are as follows:  
\[ \begin{array}{l}\underline{x}_{\lambda}=\text{exp}\left(\text{ln}\mu- \frac{1}{2}\text{ln}\Big{(}1+(\sigma/\mu)^{2}\Big{)}-\sqrt{\text{ln}\Big{(}1+ (\sigma/\mu)^{2}\Big{)}}\sqrt{-2\text{ln}[\lambda+(1-\lambda)\text{exp}(-k^{2} /2)]}\right)\\ \overline{x}_{\lambda}=\text{exp}\left(\text{ln}\,\mu-\frac{1}{2}\,\text{ln} \Big{(}1+(\sigma/\mu)^{2}\Big{)}+\sqrt{\text{ln}\Big{(}1+(\sigma/\mu)^{2} \Big{)}}\sqrt{-2\,\text{ln}[\lambda+(1-\lambda)\,\text{exp}(-k^{2}/2)]}\right) \end{array}