## 3 The Air Quality Fuzzy Comprehensive Evaluation (AQFCE) Paradigm
### Evaluation Module of the AQFCE
#### (3) Fuzzy matrix
In general, the second level is not only the demarcation between pollution and nonpollution but also the target to protect human health, and it is usually employed by the government to evaluate whether air quality is up to standard. Most of all, they have more recommended limits from world authorities such as the WHO. Therefore, an improved method named the Normalized Secondary-Level Limit Weighting (NSLLW) algorithm is proposed as Equation (8). Additionally, due to the fact that factor weight will be combined with a fuzzy matrix to obtain an overall evaluation result of air quality, it can serve as an independent tool to obtain a great deal of useful information including chief pollutant, warning pollutant (\(u_{i}/v_{i2}>1\)), relative hazard and rank (for the second limitation of the AQI). Table 3 shows an example of evaluation results of NSLLW. In contrary, all evaluation results of the AQI are all decided by pollutant with the highest subindex, which is biased and insufficient. \[w_{i}=\left[u_{i}/(v_{i2})\right]/\sum_{i=1}^{m}\left[u_{i}/(v_{i2})\right] \tag{8}\]  
Figure 1: Membership function of air pollutant concentration. (5) Fuzzy operator  
The final evaluation result will be obtained by synthesizing the factor weight and fuzzy matrix with an appropriate composite operator which can be defined by the researcher according to the actual evaluation task. The fuzzy operator plays a significant role and determines the nature of a model. Common fuzzy math models can be divided into three categories:  
Principal factor decision model: \(M(\wedge,\vee)\)  
\[ \begin{array}{c}m\\ b_{j}=\begin{array}{c}\vee\\ i=1\end{array}\left(w_{i}\;\wedge\;r_{ij}\right)\left(j=1,2,\ldots,n\right)\\ \end{array}  \tag{9}\]  
Principal factor prominent model: \(M(\wedge,\oplus)\)  
\[ \begin{array}{c}b_{j}=\begin{array}{c}m\\ \oplus\\ i=1\end{array}\left(w_{i}\;\wedge\;r_{ij}\right)\;=\begin{array}{c}m\\ \sum\\ i=1\end{array}\left(w_{i}\;\wedge\;r_{ij}\right)\left(j=1,2,\ldots,n\right)\\ \end{array}  \tag{10}\]  
Weighted average model: \(M(\bullet,+)\)  
\[ \begin{array}{c}b_{j}=\begin{array}{c}m\\ \sum\\ i=1\end{array}\left(w_{i}\bullet r_{ij}\right)\;(j=1,2,\ldots,n)\\ \end{array}  \tag{11}\]  
\(M(\bullet,+)\) is the first choice for environment quality evaluations because this model is favorable for taking all factors into account and obtaining an integrated effect. This weighted average model seems to be reasonable, but diluting the influence of the chief pollutant makes its evaluation level far lower than that of the AQI (conservative strategy) even under the same conditions, which may affect the people who are sensitive to the chief pollutant. Therefore, we choose \(M(\bullet,\vee)\) which considers the integrated effect of all pollutants and meanwhile highlights the influence of the chief pollutant to ensure a smooth transition from the AQI to AQFCE (for the third limitation of the AQI). Its equation is as follows:  
\[ \begin{array}{c}m\\ b_{j}=\begin{array}{c}\vee\\ i=1\end{array}\left(w_{i}\bullet r_{ij}\right)\;(j=1,2,\ldots,n)\\ \end{array}  \tag{12}\]  
(6) Evaluation result  
According to the principle of maximum membership degree, the maximum value of \(B\) reflects the final evaluation result (level) of air quality. \[B=\left(w_{1},w_{2},\ldots w_{m}\right)\;(\bullet,\vee)\left[ \begin{array}{ cccc}r_{11}&r_{12}&\cdots&r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ r_{m1}&r_{m2}&\cdots&r_{mn}\end{array} \right] \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\mathbf{SO_{2}}\) & \(\mathbf{NO_{2}}\) & \(\mathbf{CO}\) & \(\mathbf{O_{3}}\) & \(\mathbf{PM_{2.5}}\) & \(\mathbf{PM_{10}}\) \\ \hline Concentration & 9 ug/m\({}^{3}\) & 54 ug/m\({}^{3}\) & 1 mg/m\({}^{3}\) & 189 ug/m\({}^{3}\) & 105 ug/m\({}^{3}\) & 137 ug/m\({}^{3}\) \\ Relative hazard (weight) & 0.02 & 0.15 & 0.06 & 0.26 & 0.31 & 0.20 \\ Rank & 6 & 4 & 5 & 2 & 1 & 3 \\ Warning pollutant & & & & ✓ & ✓ & \\ Chief pollutant & & & & & ✓ & \\ \hline \hline \end{tabular}
\end{table}