## 3 Rationality of other protective facilities \(B_{33}\)
Since different quantitative indicators have different meanings and mathematical models, their dimensions are also different. There are three types of nondimensionalization processing methods: straight line, broken line and curved line. In this study, the linear standardization method is used to carry out the nondimensionalization processing of each quantitative indicator. There are three main types of traffic safety risk evaluation indicators on long-steep downgrade sections: the smaller it is the higher the risk, the larger it is the higher the risk, and the higher the risk of the attribute value in a certain range [17]. Assuming that \(u_{i}(i=1,2,3)\) are used to represent three types of evaluation indicator values, \(U\) satisfies [18]:  
\[U=\mathop{\cup}\limits_{i=1}^{3}u_{i}\quad u_{r}\cap u_{s}=\varnothing\quad r,s\in\{1,2,3\}\]  
For \(u_{i}\in U\), let its domain of discourse be \(d_{i}=[m_{i},M_{i}]\), among them: \(m_{i}\) and \(M_{i}\) represent the minimum and maximum values of \(u_{i}\), respectively. Definition: the nondimensionalization value of the attribute value \(x_{i}\) of the evaluation indicator \(u_{i}\) by the decision maker is \(r_{i}=u_{d_{i}}(x_{i}),i=1,2,\cdots,n\) and \(r_{i}\in[0,1]\). Among them: \(u_{d_{i}}(x_{i})\) is the nondimensionalization standard function defined by the indicator \(u_{i}\); on the domain of discourse \(d_{i}\). Therefore, three nondimensionalization standard functions are given:  
The \"smaller the higher the risk\" indicator's standard function of nondimensionalization (\(u_{i}\in U_{1}\))  
\[r_{i}=u_{d_{i}}(x_{i})=\left\{ \begin{array}{c}1,x_{i}\leq m_{i}\\ \frac{M_{i}-x_{i}}{M_{i}-m_{i}},x_{i}\in d_{i}\\ 0,x_{i}\geq M_{i}\end{array} \right. \tag{13}\]  
Applicable indicators: pavement skid-resistant performance, pavement roughness. Among them: the domain of discourse of the pavement skid-resistant performance and pavement roughness are both set as \(d_{i}=[60,100]\). The \"larger the higher the risk\" indicator's standard function of nondimensionalization (\(u_{i}\in U_{2}\))  
\[r_{i}=u_{d_{i}}(x_{i})=\left\{ \begin{array}{c}0,x_{i}\leq m_{i}\\ \frac{x_{i}-m_{i}}{M_{i}-m_{i}},x_{i}\in d_{i}\\ 1,x_{i}\geq M_{i}\end{array} \right. \tag{14}\]  
Applicable indicators: Accident rate per 100 million vehicle kilometers, proportion of sharp curves and steep slopes, rationality of setting up extra-large bridges and tunnels, proportion of large trucks, speed management and overload management. Except for the accident rate per 100 million vehicle kilometers, the domain of discourse for each of the above indicators is set as \(d_{i}=[0,60\%]\). The \"higher the risk of the attribute value in a certain range\" indicator's standard function of nondimensionalization (\(u_{i}\in U_{2}\))  
\[r_{i}=u_{d_{i}}(x_{i})=\left\{ \begin{array}{c}1-\frac{L_{1}-x_{i}}{\max\{L _{1i}-m_{i},M_{i}-L_{2}\}},x_{i}\leq L_{i1}\\ 1,x_{i}\leq[L_{i1},L_{i2}]\\ 1-\frac{x_{i}-L_{2}}{\max\{L_{1i}-m_{i},M_{i}-L_{2}\}},x_{i}\geq L_{i2}\end{array} \right. \tag{15}\]  
Applicable indicator: service level. Its domain of discourse is \(d_{i}=[0,1]\), \(L_{i1}=0.75\), \(L_{i2}=0.90\). Through the above transformation, each quantitative indicator is standardized to the interval of [0,1]. In order to be unified with the nondimensionalization result of the qualitative indicators so as to calculate the membership, the \(r_{i}\) is transformed as follows:  
\[f_{i}=r_{i}\times 100 \tag{16}\]  
The meaning of \(f_{i}\in[0,100]\) at this time is the degree of traffic safety risk for the long-steep downgrade sections represented by the evaluation indicator. However, sometimes the \(m\), \(M\) values are difficult to obtain, and a unified standard may not be obtained. Therefore, when it is difficult to obtain the \(m\), \(M\) values of the quantitative indicators, the same method as the qualitative indicators can be selected for processing.