# An Assessment Model for Air Passenger Risk Classification
## 2 Assessment Model for Air Passenger Risk Classification
### Improved Fuzzy Comprehensive Evaluation Method
#### 2.3.4 Improved Fuzzy Relation Matrices  
The fuzzy relation matrix \(R= \begin{bmatrix}r_{11}&\cdots&r_{1n}\\ \vdots&\ddots&\vdots\\ r_{m1}&\cdots&r_{mn}\end{bmatrix} \) is a mapping from the factor set to the evaluation set, and is composed of the single factor evaluation vector \(r_{i}=[r_{i1},r_{i2},\cdots,r_{in}]\) (\(i=1,2,\cdots,m\)), where \(r_{ij}\) represents the degree of membership of factor \(u_{i}\) belonging to grade \(v_{j}\), \(\sum\limits_{j=1}^{n}r_{ij}=1,i=1,2,\cdots,m\). The degree of membership function \(r_{ij}\) can be calculated by Equation (5), where \(t_{ij}\) is the number of experts who judge that factor \(u_{i}\) belongs to grade \(v_{j}\), and \(T\) is the number of experts participating in the evaluation. \[r_{ij}=\frac{t_{ij}}{T} \tag{5}\]  
A questionnaire is used to determine the evaluation results of 10 aviation security experts on the attribute values of some indexes, which is shown in Appendix B. The data obtained from the questionnaire are shown in Appendix C. According to the evaluation results, the standard of single factor evaluation is established for each index, and different methods are adopted to evaluate different kinds of indexes. (1) The single factor evaluation vector \(r_{i}=[r_{i1},r_{i2},\cdots,r_{in}](\sum\limits_{j=1}^{n}r_{ij}=1)\) of the index \(u_{i}\) whose nature is a discrete variable can be formulated according to the statistical results of expert evaluation. The degree of membership \(r_{ij}\) is determined by Equation (5). Then, the single factor evaluation vector of index \(u_{i}\) can be calculated by Equation (6):  
\[r_{i}=\left[\frac{t_{i1}}{T},\frac{t_{i2}}{T},\cdots,\frac{t_{in}}{T}\right] \tag{6}\]  
Taking the index of gender as an example, it can be seen from the statistical results that the single factor evaluation vector of men belonging to high-risk, medium-risk and low-risk is [0.4, 0.3, 0.3], and that of women belonging to high-risk, medium-risk and low-risk is [0.2, 0.3, 0.5]. (2) The single factor evaluation vector \(r_{k}=[r_{k1},r_{k2},\cdots,r_{kn}](\sum\limits_{l=1}^{n}r_{kl}=1)\) of the index \(u_{k}\) whose nature is a continuous variable can be determined after obtaining the fitting function of the degree of membership \(r_{kl}\) through statistical data. First, the functions \(f_{kl}(x)(l=1,2,\cdots,n-1)\) corresponding to the first \((n-1)\)\(r_{kl}\) are obtained by fitting. In addition, then the fitting function of \(r_{kn}\) can be calculated by Equation (7), where \(x\) is the attribute value of the index \(u_{k}\), and \(n\) is the number of comments. \[f_{kn}(x)=1-\sum\limits_{l=1}^{n-1}f_{kl}(x) \tag{7}\]  
When \(x=x_{i},f_{kl}(x_{i})(l=1,2,\cdots,n)\) is calculated, then \(r_{kl}=f_{kl}(x_{i})\), the single factor evaluation vector of the index \(u_{k}\) can be calculated by Equation (8):  
\[r_{k}=[f_{k1}(x_{i}),f_{k2}(x_{i}),\cdots,f_{kn}(x_{i})] \tag{8}\]  
Taking the index of age as an example, this continuous variable is divided into six intervals: under 14 years old, 14-18 years old, 19-30 years old, 31-50 years old, 51-60 years old and over 60 years old. Then, the data collected in these six intervals are fitted, respectively, to obtain the single factor evaluation of different age intervals. In the interval of under 14 years old, the single factor evaluation vector is \(r_{1}=[r_{11},r_{12},r_{13}]\), where \(r_{11}\), \(r_{12}\) and \(r_{13}\) are the degrees of belonging to high-risk, medium-risk and low-risk, respectively, and \(r_{11}\)\(+\)\(r_{12}\)\(+\)\(r_{13}\)\(=1\). Let \(y_{1}\), \(y_{2}\), and \(y_{3}\) be the fitting functions corresponding to \(r_{11}\), \(r_{12}\), and \(r_{13}\)According to the existing statistical data, the fitting results of \(y_{1}\) and \(y_{2}\) are shown in Figure 1, where \(y_{1}=0\), \(y_{2}=0.0022x^{2}-0.0112x+0.01\). Then, \(y_{3}=-0.0022x^{2}+0.0112x+0.99\) can be calculated by Equation (7).