## 3 Improved VFS Method
### Optimizing SRMD  
In Equation (10), a weight is determined according to the actual situation. If the research target already has a reasonable weight assignment, it is taken as the initialization weight. If it does not exist, equal weight is preferable. Based on this, the obtained RMD is brought into Equation (10) to calculate an initial SRMD. Then, a more accurate result is obtained by optimizing the initial SRMD using the BP network model. In this way, the second issue raised in Section 2.3 can be effectively solved. Many studies have shown that the fusion of BP neural network and fuzzy set theory can give better play to their respective advantages [30; 31]. In this paper, we use the sigmoid function characteristics of the SRMD function in Equation (10) to establish a BP neural network to optimize the SRMD. The establishment process mainly includes three parts: (1) creating a network structure; (2) setting the activation function and calculating input and output of each layer; and (3) creating a weight adjustment model. The specific establishment process is as follows:  
(1) Creating a network structure  
For convenient discussion, a three-layer structure of BP neural network is constructed, as shown in Figure 6. Let the input layer have m nodes, i.e., the initial SRMD corresponds to m levels; the hidden layer have p nodes, i.e., p unit systems; and the output layer have also m nodes corresponding to the optimized SRMD. The error back propagation algorithm of BP network is used to update the connection weights (\(w_{ik}\) and \(w_{kh}\)) between network layers. (2) Setting the activation function  
Sigmoid function is a commonly used activation function of BP networks, which has the characteristics of being s-shaped, asymptotically bounded, completely monotone [32; 33]. When \(\alpha=2\)  
Figure 6: BP neural network structure. \(p=1\), let \(d_{jg}=\sum_{i=1}^{n}\left[w_{i}\left(1-u_{ij}(x)\right)\right]\) and \(d_{jb}=\sum_{i=1}^{n}w_{i}u_{ij}(x)\) such that \(d_{jg}=1-d_{jb}\). Submit \(d_{jb}\) into Equation (10) to obtain the following:  
\[v(x)_{j}=\left[1+\left(\frac{1-d_{jb}}{d_{jb}}\right)^{2}\right]^{-1} \tag{30}\]  
According to Equation (30), \(v(x)_{j}\) is a nonlinear function of \(d_{jb}\); its first and second derivatives can be expressed as follows:  
\[\frac{dv(x)_{j}}{dd_{jb}}=\frac{2d_{jb}\left(1-d_{jb}\right)}{\left(\left(1-d_ {jb}\right)^{2}+d_{jb}^{2}\right)^{2}} \tag{31}\]  
\[\frac{d^{2}v(x)_{j}}{dd_{jb}^{2}}=\frac{2\left(1-2d_{jb}\right)\left[\left(1-d _{jb}\right)^{2}+d_{jb}^{2}+4d_{jb}\left(1-d_{jb}\right)\right]}{\left[\left(1 -d_{jb}\right)^{2}+d_{jb}^{2}\right]^{2}} \tag{32}\]  
In Equation (31), \(0\leq d_{jb}\leq 1\); therefore \(\frac{dv(x)_{j}}{dd_{jb}}>0\) and then \(v(x)_{j}\) is a monotonically increasing function of \(d_{jb}\). According to Equation (32), when \(d_{jb}=0.5\), \(\frac{d^{2}v(x)_{j}}{dd_{jb}^{2}}=0\); when \(d_{jb}<0.5\), \(\frac{d^{2}v(x)_{j}}{dd_{jb}^{2}}>0\); and when \(d_{jb}>0.5\), \(\frac{d^{2}v(x)_{j}}{dd_{jb}^{2}}<0\). Therefore, the function curve of Equation (30) in the \([0,0.5]\) interval is concave and in the \([0.5,1]\) interval is convex, as shown in Figure 7. Based on the above analysis, \(d_{jb}=0.5\) is the only inflection point of Equation (30) in the interval \([0,1]\), so when \(\alpha=2\) and \(p=1\), the SRMD function (Equation (10)) can be transformed into a sigmoid function. 3. Creating a weight adjustment model  
Before creating a weight updating formula, the activation function (Equation (30)) is used to calculate the input and output of hidden and output layers. In the input layer, the input information (SRMD for each level) of node m is transmitted directly to the hidden layer node k, that is, the input and output of the node are equal (\(x_{j}=v(x)_{j}\)). For the hidden layer node k, the input and output are as follows:  
\[\left\{ \begin{array}{c}I_{k}=\sum\limits_{j=1}^{m}w_{jk}v(x)_{j}\\ u_{k}=\left[1+\left(\frac{1}{\sum_{j=1}^{n}w_{jk}v(x)_{j}}-1\right)^{2}\right] ^{-1}\end{array} \right.