## 2 The VFS Method
### Calculating Steps of VFS Method  
Step 1: Establishing parameters (c, a, M, b, d) of VFS model  
Assuming an indicator \(i\) is divided into m levels and the boundary eigenvalues of each level are \(\left(\big{[}L_{i1},L_{i2}\big{]},\big{[}L_{i2},L_{i3}\big{]},\cdots,\big{[}L_ {in},L_{i(m+1)}\big{]}\right)\) (Figure 2). According to previous literatures [13; 22; 25], the attracting interval \(\big{[}a_{ij},b_{ij}\big{]}\), extended interval \(\big{[}c_{ij},d_{ij}\big{]}\), and point \(M_{ij}\) for each level are calculated as follows:  
\[\text{Level 1}\left\{ \begin{array}{c}c_{i1}=L_{i1}\\ a_{i1}=L_{i1}\\ M_{i1}=L_{i1}\\ b_{i1}=L_{i2}\\ d_{i1}=L_{i3}\end{array} \right.\;i=1,2,\cdots,n \tag{11}\]  
(ii) For level j \((j=2,3,\cdots,\text{m}-1)\), the intervals \(\big{[}a_{ij},b_{ij}\big{]}\) and \(\big{[}c_{ij},d_{ij}\big{]}\) and point \(M_{ij}\) were expressed as follows:  
\[\text{Level j}\left\{ \begin{array}{c}c_{ij}=L_{i(j-1)}\\ a_{ij}=L_{ij}\\ M_{ij}=\Big{(}L_{ij}+L_{i(j+1)}\Big{)}/2\quad i=1,2,\cdots,n\\ b_{ij}=L_{i(j+1)}\\ d_{ij}=L_{i(j+2)}\end{array} \right. \tag{12}\](iii) For maximum level m, the intervals \([a_{im},b_{im}]\) and \([c_{im},d_{im}]\) and point \(M_{im}\) were denoted as follows:  
\[\text{Level m}\left\{ \begin{array}{l}c_{im}=L_{i(m-1)}\\ a_{im}=L_{im}\\ M_{im}=L_{i(m+1)}\\ b_{im}=L_{i(m+1)}\\ d_{im}=L_{i(m+1)}\end{array} \right.\quad i=1,2,\cdots,n \tag{13}\]  
Step 2: Calculating RMD  
Substituting the above parameters (c, a, M, b, d) into Equations (5)-(8) repeatedly can obtain RMD functions corresponding to each level. Then, the measured value \(x\) of index \(i\) is brought into the RMD function to calculate the RMD value of each level, which is expressed as follows:  
\[u_{ij}(x)=[u_{i1}(x),u_{i2}(x),\cdots,u_{im}(x)] \tag{14}\]  
Now, assuming that the prediction and assessment object has n indicators, the RMD matrix of n indicators to m levels can be obtained by repeating the above process n times, which can be expressed as follows:  
\[u_{ij}(x)=\left[ \begin{array}{cccc}u_{11}(x)&u_{12}(x)&\cdots&u_{1m}(x)\\ u_{21}(x)&u_{22}(x)&\cdots&u_{2m}(x)\\ \vdots&\vdots&\ddots&\vdots\\ u_{n1}(x)&u_{n2}(x)&\cdots&u_{mm}(x)\end{array} \right] \tag{15}\]  
Step 3: Calculating SRMD  
Based on the obtained RMD matrix, the SRMD of the prediction and assessment object to each level can be obtained by introducing \(u_{ij}(x)\) into Equation (10):  
\[v(x)_{j}=[v(x)_{1},v(x)_{2},\cdots,v(x)_{m}] \tag{16}\]  
After calculating the SRMD, the final prediction and assessment level eigenvalue H can be obtained:  
\[\left\{ \begin{array}{l}H=(1,2,\cdots,m)\cdot(v(x)_{j}^{\prime})^{T}\\ v(x)_{j}^{\prime}=\frac{v(x)_{j}}{\sum_{j=1}^{m}v(x)_{j}}\end{array} \right. \tag{17}\]  
The relationship between classification level and eigenvalue H is expressed as follows:  
\[\left\{ \begin{array}{l}1\leq\text{H}<1.5\;\in\;\text{Level 1}\\ \text{j}-0.5\leq\text{H}<\text{j}+0.5\;\in\;\text{Level j}\;(\text{j}=1,2, \cdots,\text{m}-1)\\ \text{m}-0.5\leq\text{H}<\text{m}+0.5\;\in\;\text{Level m}\end{array} \right. \tag{18}\]  
In the above calculation process, it can be clearly found that two issues need to be improved when employing the VFS method for multi-index prediction and evaluation. Firstly, how to simplify RMD calculation process? The traditional RMD calculation process is very complicated and time-consuming; a prediction object needs to run n \(\times\) m times Equations (5)-(8) repeatedly to get the RMD matrix. Although Equation (9) can reduce some, it still has a large computational burden. Secondly, how to  
Figure 2: The boundary eigenvalues of index \(i\) corresponding to m levels. consider the indicator weights of SRMD function so as to obtain high-precision prediction results? In Equation (10), weight assignment is the core issue, which is related to the prediction accuracy of SRMD. However, the weighting is generally carried out in a relatively specific situation in the actual forecasting process. As shown in Table 1, weights are assigned differently based on different occasions, objects, and methods.