## 3 Improved VFS Method  
### Simplifying the RMD Calculation Process  
In the actual calculation process, combining Equations (5)-(8) and (11)-(13), it can be found that, when the measured value \(x\) of index \(i\) is located in different levels, the RMDs exhibit some general characteristics. Detailed analysis is as follows:  
(1) When \(x\) is in the minimum level interval, \(L_{i1}\leq x\leq L_{i2}\) (Figure 3a). When \(L_{i1}\leq x\leq L_{i2}\), only RMD functions of level 1 and level 2 are activated; for other levels, RMDs are equal to 0 because \(x\) is not in the \([c,d]\) range. According to Equations (11) and (12), the parameters of the VFS model with levels 1 and 2 can be obtained as follows: \(c_{i1}=a_{i1}=M_{i1}=L_{i1}\), \(b_{i1}=L_{i2}\), \(d_{i1}=L_{i3}\) and \(c_{i2}=L_{i1}\), \(a_{i2}=L_{i2}\), \(M_{2}=(L_{i2}+L_{i3})/2\), \(b_{i2}=L_{i3}\), \(d_{i2}=L_{i4}\). Thus, a linear relationship between \(x\) and parameters c, a, M, b, and d for levels 1 and 2 can be established as shown in Figure 3b. From Figure 3b, it can be seen that \(x\) is in the \([M_{i1},b_{i1}]\) interval for level 1, and for level 2, \(x\) is in the \([c_{i2},a_{i2}]\) interval. Therefore, the RMD of \(x\) for levels 1 and 2 can be calculated by Equations (6) and (7) as follows:  
\[\left\{ \begin{array}{l}u_{i1}(x)=0.5\Big{[}1+\frac{x-b_{i2}}{M_{i1}-b_{i1}} \Big{]}=0.5\Big{[}1+\frac{x-L_{i2}}{L_{i1}-L_{i2}}\Big{]}\,L_{i1}\leq x\leq L _{i2}\\ u_{i2}(x)=0.5\Big{[}1-\frac{x-L_{i2}}{c_{i2}-a_{i2}}\Big{]}=0.5\Big{[}1-\frac{ x-L_{i2}}{L_{i1}-L_{i2}}\Big{]}\,L_{i1}\leq x\leq L_{i2}\end{array} \right. \tag{19}\]  
In Equation (19), \(0\leq\frac{x-L_{i2}}{L_{i1}-L_{i2}}\leq 1\), hence, \(0.5\leq u_{i1}(x)\leq 1\) and \(0\leq u_{i2}(x)\leq 0.5\). In addition, it can be easily found that \(u_{i1}(x)+u_{i2}(x)=1\). The above features can also be presented intuitively in Figure 3c. Based on the above analysis, when the measured value \(x\) of index \(i\) is within level 1, the characteristics of RMD can be summarized as follows:  
\[\left\{ \begin{array}{l}u_{ij}(x)=[u_{i1}(x),u_{i2}(x),0,\cdots,0]\\ u_{i1}(x)+u_{i2}(x)=1\\ 0.5\leq u_{i1}(x)\leq 1\\ 0\leq u_{i2}(x)\leq 0.5\end{array} \right. \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Factors**} & \multicolumn{2}{c}{**Weight Determination Method**} & \multirow{2}{*}{**Reference**} \\ \cline{2-2} \cline{4-5}  & \(\sigma_{\text{d}}/\sigma_{\text{c}}\) & \(\sigma_{\text{d}}/\sigma_{\text{t}}\) & \(w_{\text{et}}\) & & \\ \hline
1 & 0.400 & 0.300 & 0.300 & Fuzzy mathematics & [8] \\
2 & 0.427 & 0.302 & 0.271 & Back analysis & [26] \\
3 & 0.365 & 0.313 & 0.322 & Entropy method & [27] \\
4 & 0.361 & 0.325 & 0.314 & Combination weighting & [28] \\
5 & 0.162 & 0.675 & 0.162 & Delphi method & [29] \\
6 & 0.266 & 0.413 & 0.321 & Entropy method & [30] \\
7 & 0.250 & 0.250 & 0.500 & Rough set theory & [31] \\
8 & 0.235 & 0.295 & 0.470 & Rough set theory & [32] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The weights of the rockburst prediction factor obtained by different methods. 2. When \(x\) is located in the interval of a mid-level j (\(\mathrm{j}=2,3,\cdots,m-1\)), \(L_{ij}\leq x\leq L_{i(j+1)}\) (Figure 4a). When \(L_{ij}\leq x\leq L_{i(j+1)}\), only RMD functions of level j and adjacent levels j - 1 and j + 1 are activated; for other levels, RMDs are equal to 0 because \(x\) is not in the \([c,d]\) range. According to Equation (12), the parameters of the VFS model with levels j, j - 1, and j + 1 can be obtained as follows: \[\mathrm{Level\;j}\left\{ \begin{array}{c}c_{ij}=L_{i(j-1)}\\ a_{ij}=L_{ij}\\ M_{ij}=(L_{ij+}L_{i(j+1)})/2\ \ \ i=1,2,\cdots,n\\ b_{ij}=L_{i(j+1)}\\ d_{ij}=L_{i(j+2)}\end{array} \right.\] (21) \[\mathrm{Level\;j-1}\left\{ \begin{array}{c}c_{i(j-1)}=L_{i(j-2)}\\ a_{i(j-1)}=L_{i(j-1)}\\ M_{i(j-1)}=\big{(}L_{i(j-1)}+L_{ij}\big{)}/2\ \ \ i=1,2,\cdots,n\\ b_{i(j-1)}=L_{ij}\\ d_{i(j-1)}=L_{i(j+1)}\end{array} \right.\] (22) \[\mathrm{Level\;j+1}\left\{