## 3 Improved VFS Method
### Simplifying the RMD Calculation Process
\begin{array}{c}c_{i(j+1)}=L_{ij}\\ a_{i(j+1)}=L_{i(j+1)}\\ M_{i(j+1)}=\big{(}L_{i(j+1)}+L_{i(j+2)}\big{)}/2\ \ \ i=1,2,\cdots,n\\ b_{i(j+1)}=L_{i(j+2)}\\ d_{i(j+1)}=L_{i(j+3)}\end{array} \right.\] (23)  
Based on Equations (21)-(23), a linear relationship between \(x\) and c, a, M, b, and d for levels j, j - 1, and j + 1 can be established as shown in Figure 4b. From Figure 4b, it can be seen that there are two cases for level j when \(x\) is in the \(\big{[}a_{ij},b_{ij}\big{]}\) interval: (1) \(a_{ij}\leq x\leq M_{ij}\) and (2) \(M_{ij}\leq x\leq b_{ij}\). Combined with Equations (5) and (7), the RMD of \(x\) for level j can be calculated as follows:  
\[\left\{ \begin{array}{c}u_{ij}(x)=0.5\Big{[}1+\frac{x-a_{ij}}{M_{ij}-a_{ij}} \Big{]}=0.5\Big{[}1+\frac{x-L_{ij}}{L_{i(j+1)}-L_{ij}}\Big{]}\ L_{ij}\leq x\leq \frac{L_{ij}+L_{i(j+1)}}{2}\\ u_{ij}(x)=0.5\Big{[}1+\frac{x-b_{ij}}{M_{ij}-b_{ij}}\Big{]}=0.5\Big{[}1+\frac{x -L_{i(j+1)}}{L_{ij}-L_{i(j+1)}}\Big{]}\ \frac{L_{ij}+L_{i(j+1)}}{2}\leq x\leq L_{i(j+1)}\end{array} \right. \tag{24}\]  
In Equation (24), \(0\leq 2\frac{x-L_{ij}}{L_{i(j+1)}-L_{ij}}\leq 1\) and \(0\leq 2\frac{x-L_{i(j+1)}}{L_{i(j-1)}-L_{i(j+1)}}\leq 1\) such that \(0.5\leq u_{ij}(x)\leq 1\). Figure 3: **(a)** The linear relationship between measured value \(x\) and standard value \(L_{ij}\); **(b)** the linear relationship between \(x\) and the variable fuzzy set (VFS) model parameters c, a, M, b, and d on activation levels 1 and 2; **(c)** the position of the relative membership degree (RMD) value corresponding to \(x\) on the RMD function of activation levels 1 and 2. Figure 4b also shows that, whether \(x\) is in the \(\left[a_{ij},M_{ij}\right]\) or \(\left[M_{ij},b_{ij}\right]\) intervals of level j, in the VFS model of j - 1 and j + 1, \(x\) is located in the intervals of \(\left[b_{i(j-1)},d_{i(j-1)}\right]\) and \(\left[c_{i(j+1)},a_{i(j+1)}\right]\), respectively, and RMD can be calculated according to Equations (6) and (8):  
\[\left\{ \begin{array}{c}u_{i(j-1)}(x)=0.5\bigg{[}1-\frac{x-b_{i(j-1)}}{d_{i( j-1)}-b_{i(j-1)}}\bigg{]}=0.5\bigg{[}1-\frac{x-L_{ij}}{L_{i(j+1)}-L_{ij}} \bigg{]}L_{ij}\leq x\leq L_{i(j+1)}\\ u_{i(j+1)}(x)=0.5\bigg{[}1-\frac{x-a_{i(j+1)}}{c_{i(j+1)}-a_{i(j+1)}}\bigg{]} =0.5\bigg{[}1-\frac{x-L_{ij(j+1)}}{L_{ij}-L_{i(j+1)}}\bigg{]}L_{ij}\leq x\leq L _{i(j+1)}\end{array} \right. \tag{25}\]  
In Equation (25), \(0\leq\frac{x-L_{ij}}{L_{i(j+1)}-L_{ij}}\leq 1\) and \(0\leq\frac{x-L_{ij(j+1)}}{L_{ij}-L_{i(j+1)}}\leq 1\); hence, \(0\leq u_{i(j-1)}(x)\leq 0.5\) and \(0\leq u_{i(j+1)}(x)\leq 0.5\). In addition, \(1-\frac{x-L_{ij}}{L_{i(j+1)}-L_{ij}}+1-\frac{x-L_{i(j+1)}}{L_{ij}-L_{i(j+1)}} =2-1=1\) such that \(u_{i(j-1)}(x)+u_{i(j+1)}(x)=0.5\). The above features can also be clearly seen in Figure 4c, which can be summarized as follows:  
\[\left\{ \begin{array}{c}u_{ij}(x)=\Big{[}0,\cdots,0,u_{i(j-1)}(x),u_{ij}(x),u _{i(j+1)}(x),0,\cdots,0\Big{]}\\ 0.5\leq u_{ij}(x)\leq 1\\ 0\leq u_{i(j-1)}(x)\leq 0.5\\ 0\leq u_{i(j+1)}(x)\leq 0.5\\ u_{i(j-1)}(x)+u_{i(j+1)}(x)=0.5\end{array} \right. \tag{26}\]  
(3) When \(x\) is within the maximum level m interval, \(L_{im}\leq x\leq L_{i(m+1)}\) (Figure 5a). When \(L_{im}\leq x\leq L_{i(m+1)}\), only RMD functions of level m and level m - 1 are activated; for other levels, RMDs are equal to 0 because \(x\) is not in the \([c,d]\) range. According to Equations (12) and (13), the parameters of the VFS model for levels m and m - 1 can be obtained as follows: \(c_{im}=L_{i(m-1)}\), \(a_{im}=L_{im}\), \(M_{im}=b_{im}=d_{im}=L_{i(m+1)}\) and \(c_{i(m-1)}=L_{i(m-2)}\), \(a_{i(m-1)}=L_{i(m-1)}\), \(M_{i(m-1)}=\left(L_{i(m-1)}+L_{im}\right)/2\), \(b_{i(m-1)}=L_{im}\), \(d_{i(m-1)}=L_{i(m+1)}\). Thus, a linear relationship between \(x\) and c, a, M, b, and d for levels m and m - 1 can be established as shown in Figure 5b.