## 3 Model
### The Optimization Mode  
#### 3.4.1 The Objective Function  
The purpose of route optimization in this paper is to achieve the reduction of air emission, noise level and cost on the basis of ensuring safety and improving efficiency. Therefore, this paper establishes the optimization objective from two perspectives of reducing air emissions and reducing noise.  
Figure 4: Selection of the flight candidate points.  
Figure 5: Horizontal route candidate extension method.  
1 Minimum aviation pollutant emission  
Reducing the air emissions of pollution gas is one of the important goals of the development of green aviation. Because the emission level of each pollution gas is too different, the air emission need to be normalized. Therefore, the objective of minimizing the normalized air emissions can be expressed as:  
\[Min\ \left(EN\right)\ =Min\left(\sum_{j}E_{j}\right), \tag{12}\]  
\[E_{j}=\sum_{i}\sum_{k}\left\{\sum_{m}\left[n_{i}^{j}\cdot Normal\left(\chi_{u,v }^{k}\cdot\gamma_{o}^{k}\cdot\left(\lambda_{j}^{D}\cdot E_{i,k,m}^{D}+\lambda_{ j}^{C}\cdot E_{i,k,m}^{C}+\lambda_{j}^{L}\cdot E_{i,k,m}^{L}\right)\right)\right] \right\}, \tag{13}\]  
\[\chi_{u,v}^{k}=\begin{cases}1&\text{the candidate point of segment $k$ is the start point of next segment}\\ 0&\text{otherwise}\end{cases}, \tag{14}\]  
\[\gamma_{o}^{k}=\mathfrak{s}_{o}^{k}\oplus\mathfrak{c}_{o}^{k}, \tag{15}\]  
\[E_{i,k,m}^{D}=t_{i,k}^{D}\cdot FF_{i,k}^{D}\cdot E_{i,m}^{D}, \tag{16}\]  
\[E_{i,k,m}^{C}=t_{i,k}^{C}\cdot FF_{i,k}^{C}\cdot E_{i,m^{\prime}}^{C} \tag{17}\]  
\[E_{i,k,m}^{L}=t_{i,k}^{L}\cdot FF_{i,k}^{L}\cdot E_{i,m^{\prime}}^{L} \tag{18}\]  
\[\lambda_{j}^{D}=\begin{cases}1&\text{$\varphi_{u,v}^{k}<0$}\\ 0&\text{or}\end{cases},\ \ \ \lambda_{j}^{C}=\begin{cases}1&\text{$\varphi_{u,v}^{k}>0$}\\ 0&\text{or}\end{cases},\ \ \ \lambda_{j}^{L}=\begin{cases}1&\text{$\varphi_{u,v}^{k}=0$}\\ 0&\text{or}\end{cases}, \tag{19}\]  
\[\mathfrak{s}_{o}^{k}=\begin{cases}1,&\text{$d_{o}^{k}<w$}\\ 0,&\text{otherwise}\end{cases},\ \ \ \varepsilon_{o}^{k}=\begin{cases}1&\text{$h_{o}^{k}<h_{MOC}$}\\ 0&\text{otherwise}\end{cases}. \tag{20}\]  
where, \(n_{i}^{j}\) is the number of \(i\)-type-aircraft in route \(j\), \(E_{j}\) represents the total air emissions in route \(j\), \(E_{i,k,m}^{D}\), \(E_{i,k,m}^{C}\), \(E_{i,k,m}^{L}\) respectively represent the air emissions of pollutant gas \(m\) of \(i\)-type-aircraft during descending, level flight, and climbing in segment \(k\), \(t_{i,k}^{D}\), \(t_{i,k}^{C}\), \(t_{i,k}^{L}\) respectively represent the flight time of \(i\)-type-aircraft during descending, level flight, and climbing in segment \(k\), \(F_{i,k}^{D}\), \(F_{i,k}^{C}\), \(F_{i,k}^{L}\) respectively represent the fuel flow rate of \(i\)-type-aircraft during descending, level flight, and climbing in segment \(k\), \(E_{i,m}^{D}\), \(E_{i,m}^{L}\), \(E_{i,m}^{C}\) respectively represent the emission index of pollutant gas \(m\) of \(i\)-type-aircraft during descending, level flight, and climbing in segment \(k\). \(\chi_{u,v}^{k}\), \(\gamma_{o}^{k}\), \(\mathfrak{s}_{o}^{k}\) and \(\varepsilon_{o}^{k}\) are decision variables, \(d_{o}^{k}\) represents the ground distance from the vertical projection of segment k to obstacle \(o\), \(h_{o}^{k}\) is the vertical distance from segment \(k\) to obstacle \(o\), \(w\) is the width of route protection zone, \(h_{MOC}\) is the minimum obstacle clearance.  
2 Minimum Noise Level  
ICAO specifies that 65 dB is the maximum threshold of noise level in the residential areas, and noise below 65 dB is considered as a comfortable living environment. Therefore, this paper aims to minimize the total exposure population affected by noise level above 65 dB. The objective function is expressed as:  
\[MinQ(L)=Min\left\{\chi_{u,v}^{k}\cdot\gamma_{o}^{k}\cdot Q(65)\right\}. \tag{21}\]