## 3 The Air Quality Fuzzy Comprehensive Evaluation (AQFCE) Paradigm
### Evaluation Module of the AQFCE
#### (3) Fuzzy matrix  
Membership degree is the basic idea of fuzzy math and is objectively existent. It can be denoted by \(r_{ij}\), which means the membership degree of factor \(u_{i}\) aiming at the comment \(v_{j}\). Membership degree is an effective solution to the problem of data near boundary (for the first limitation of the AQI). For air quality evaluation, a pollutant concentration value can belong to both possible levels to various extents rather than be defined as only one certain level, and further membership degree will be combined with factor weight to make a comprehensive evaluation. The membership function can calculate the membership degree of pollutant concentration to the evaluation level. Commonly employed membership functions include the rectangular distribution function, Cauchy distribution function, Gauss distribution function, ridge distribution function, K-order parabolic distribution function, etc. In view of this evaluation object, factor and criterion, the trapezoid-triangle distribution function is adopted in this study. Details are presented as Equations (4)-(6) and Figure 1.  
\[r_{ij}=\left\{ \begin{array}{cc}1&u_{i}\leq v_{ij}\\ \left(v_{i(j+1)}-u_{i}\right)/\left(v_{i(j+1)}-v_{ij}\right)&v_{ij}<u_{i}<v_{i (j+1)}\\ 0&u_{i}\geq v_{i(j+1)}\end{array} \right.\qquad j=1 \tag{4}\]  
\[r_{ij}=\left\{ \begin{array}{cc}\left(u_{i}-v_{i(j-1)}\right)/\left(v_{ij}-v _{i(j-1)}\right)&v_{i(j-1)}<u_{i}<v_{ij}\\ \left(v_{i(j+1)}-u_{i}\right)/\left(v_{i(j+1)}-v_{ij}\right)&v_{ij}<u_{i}<v_{i (j+1)}\\ 0&u_{i}\geq v_{i(j+1)}\end{array} \right.\qquad j=2,3,4 \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & \\ \cline{2-13} \multirow{2}{*}{**Level**} & \multicolumn{3}{c}{**SO\({}_{2}\) (ug/m\({}^{3}\))**} & \multicolumn{3}{c}{**NO\({}_{2}\) (ug/m\({}^{3}\))**} & \multicolumn{3}{c}{**CO (mg/m\({}^{3}\))**} & \multicolumn{3}{c}{**O\({}_{3}\) (ug/m\({}^{3}\))**} & \multicolumn{3}{c}{**PM\({}_{2.5}\) (ug/m\({}^{3}\))**} & \multicolumn{3}{c}{**PM\({}_{10}\) (ug/m\({}^{3}\))**} \\ \cline{2-13}  & **24 h** & **1 h** & **24 h** & **1 h** & **24 h** & **1 h** & **8 h** & **1 h** & **24 h** & **1 h** & **24 h** & **1 h** \\ \hline I & 50 & 100 & 40 & 100 & 2 & 5 & 100 & 150 & 25 & 37.5 & 50 & 75 \\ II & 125 & 350 & 80 & 200 & 4 & 10 & 160 & 200 & 75 & 100 & 150 & 200 \\ III & 200 & 500 & 134 & 376 & 8 & 20 & 200 & 245 & 110 & 150 & 220 & 300 \\ IV & 300 & 600 & 190 & 494 & 16 & 40 & 240 & 320 & 150 & 210 & 250 & 350 \\ V & 400 & 800 & 380 & 940 & 24 & 60 & 400 & 520 & 210 & 280 & 300 & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A new calculation grid for China. (5)  
The fuzzy matrix can be expressed by the matrix \(R\):  
\[R=\left(r_{ij}\right)_{m\times n}=\left[ \begin{array}{cccc}r_{11}&r_{12}&\cdots &r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ r_{m1}&r_{\ell 2}&\cdots&r_{mm}\end{array} \right] \tag{6}\]  
The fuzzy matrix can be expressed by the matrix \(R\):  
\[R=\left(r_{ij}\right)_{m\times n}=\left[ \begin{array}{cccc}r_{11}&r_{12}& \cdots&r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ r_{m1}&r_{\ell 2}&\cdots&r_{mm}\end{array} \right] \tag{7}\]  
(4) Factor weight  
Hazard of each evaluation factor needs to be quantified and compared, and a proper weighting method can calculate the weight of factor, namely its relative hazard on air quality. The multiple super-scale weighting method is a frequently used method in environment quality evaluations [33], but it can be further improved. Generally, there are some concentration limits (mainly higher value) without definite scientific bases in the calculation grid, which will significantly influence the evaluation result. Thus, the consideration of all limits in the weighting algorithm cannot accurately reflect the fatalness of each pollutant, while limits of the second level are more reliable.