## 3 Improved VFS Method
### Simplifying the RMD Calculation Process
Figure 5b shows that  
Figure 4: **(a)** The linear relationship between measured value \(x\) and standard value \(L_{ij}\); **(b)** the linear relationship between \(x\) and the VFS model parameters c, a, M, b, and d on activation levels j, j - 1, and j + 1; **(c)** the position of the RMD value corresponding to \(x\) on the RMD function of activation levels j, j - 1, and j + 1. \(x\) is in the \([a_{im},M_{im}]\) interval for level m and that, for level m \(-1\), x is in the \(\left[b_{i(m-1)},d_{i(m-1)}\right]\) interval. Therefore, the RMD of \(x\) for levels m and m \(-1\) can be calculated by Equations (5) and (8) as follows:  
\[\left\{ \begin{array}{c}u_{im}(x)=0.5\Big{[}1+\frac{x-a_{im}}{M_{m}-a_{im}} \Big{]}=0.5\Big{[}1+\frac{x-L_{im}}{L_{i(m+1)}-L_{im}}\Big{]}\,L_{im}\leq x\leq L _{i(m+1)}\\ u_{i(m-1)}(x)=0.5\Big{[}1-\frac{x-b_{i(m-1)}}{d_{i(m-1)}-b_{i(m-1)}}\Big{]}=0.5 \Big{[}1-\frac{x-L_{im}}{L_{i(m+1)}-L_{im}}\Big{]}\,L_{ij}\leq x\leq L_{i(j+1)} \end{array} \right. \tag{27}\]  
In Equation (27), \(0\leq\frac{x-L_{im}}{L_{i(m+1)}-L_{im}}\leq 1\); hence, \(0.5\leq u_{im}(x)\leq 1\) and \(0\leq u_{i(m-1)}(x)\leq 0.5\). In addition, \(1+\frac{x-L_{im}}{L_{i(m+1)}-L_{im}}+1-\frac{x-L_{im}}{L_{i(m+1)}-L_{im}}=2\) such that \(u_{im}(x)+u_{i(m-1)}(x)=1\). When x is within level m, these features can also be visually found in Figure 5c, summarized as follows:  
\[\left\{ \begin{array}{c}u_{ij}(x)=\Big{[}0,0,\cdots,u_{i(m-1)}(x),u_{im}(x) \Big{]}\\ u_{i(m-1)}(x)+u_{im}(x)=1\\ 0.5\leq u_{im}(x)\leq 1\\ 0\leq u_{i(m-1)}(x)\leq 0.5\end{array} \right. \tag{28}\]  
In the above analysis, only considering the change of measured value \(x\) between the interval \(\left[L_{i1}L_{i(m+1)}\right]\), the situation of \(x\leq L_{i1}\) or \(x\geq L_{i(m+1)}\) is not considered because the cognition is exact and not fuzzy in these two cases; its corresponding RMD value is equal to \(1\), which is expressed as follows:  
\[\left\{ \begin{array}{c}u_{ij}(x)=\left[1,0,\cdots,0\right]\,x\leq L_{i1}\\ u_{ij}(x)=\left[0,0,\cdots,1\right]\,x\geq L_{i(m+1)}\end{array} \right. \tag{29}\]  
According to the above characteristics (Equations (20), (26), (28), and (29)), combined with Equations (5)-(8), a simplified method of RMD calculation is proposed. The detailed steps are as follows:  
1. The VFS model parameters (c, a, M, b, d) corresponding to each level are established according to the criteria of indicator division, and the position of measured value \(x\) in the level interval needs to be identified. 2. According to the level interval where \(x\) is located, the corresponding activation RMD function is determined. If \(x\) is in the minimum or maximum levels, it only needs to run the RMD function  
Figure 5: **(a)** The linear relationship between measured value \(x\) and standard value \(L_{ij}\); **(b)** the linear relationship between \(x\) and the VFS model parameters c, a, M, b, and d on activation levels m and m \(-1\); **(c)** the position of the RMD value corresponding to \(x\) on the RMD function of activation levels m and m \(-1\). once. For level 1, RMD can be calculated by Equation (7), and for the maximum level, RMD can be obtained by Equation (5). If \(x\) is in the middle level, only the RMD corresponding to the level itself and one of the adjacent levels need to be calculated for RMD of the level itself, which can be calculated by selecting an appropriate function from Equations (5) and (7), while for the RMD of the adjacent level, one of Equations (6) and (8) can be selected for calculation. The RMD corresponding to other levels can be obtained by feature Equations (20), (26), and (28). In addition, when \(x\) is greater than \(L_{i(m+1)}\) or less than \(L_{i1}\), the RMD function does not need to be run and the RMD is determined directly by Equation (29). 3. By repeating the above steps, the RMD values for other indicators can be calculated and the final RMD matrix can be obtained.