# A Fuzzy Path Selection Strategy for Aircraft Landing on a Carrier
## 3 Landing Path Selection Method Based on FMAGDM
### Landing Path Selection Process Based on the Fuzzy TOPSIS Approach
Higher performance ratings with respect to the air traffic situation are set higher by the LCO compared to the pilot. This is because the LCO grasps comprehensive traffic information, and he/she puts higher priority on the safety operation of the carrier. Step 2: Integration of the current environmental vector into \(X_{1}\) and \(X_{2}\)  
The environmental vector for the case under consideration can also be described by the linguistic variable, as listed in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Linguistic Variable** & **Too Poor** & **A Slightly Poor** & **Normal** & **Good** & **Very Good** \\ \hline Expression & TP & SP & NO & GO & VG \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Linguistic variable of environmental vector (for a case under consideration). Assume that the environmental evaluation of each contributing factor is \(\omega_{i}=[\omega^{L},\omega^{M},\omega^{U}]\) (\(i=1,2,...,5\)); the value of \(\omega_{i}\) is chosen from Table 3. After integrating the environmental vector, the normalized fuzzy decision matrix can be written as follows:  
\[Z=\left(z_{ij}\right)_{5\times 4}=\left(\omega_{i}x_{ij}\right)_{5\times 4}= \left[ \begin{array}{cccc}z_{11}&z_{12}&z_{13}&z_{14}\\ z_{21}&z_{22}&z_{23}&z_{24}\\ \vdots&\vdots&\vdots&\vdots\\ z_{51}&z_{52}&\ldots&z_{54}\end{array} \right] \tag{12}\]  
Step 3: Determination of the ideal solutions  
The ideal solutions refer to the fuzzy positive ideal reference point (FPIRP) and negative point (FNIRP) and are written as \(P\) and \(N\) for convenience. \(P\) is obtained when each PRCF chooses its highest value from the fuzzy decision matrix \(D\), and similarly \(N\) is obtained by choosing the lowest value for each performance rating. Therefore, \(P\) and \(N\) can be denoted as \(P=\left(p_{1},p_{2},...,p_{5}\right)\) and \(N=\left(n_{1},n_{2},...,n_{5}\right)\) with \(p_{i}=\underset{1\leq j\leq 4}{\max}\left\{z_{ij}\right\}\) and \(n_{i}=\underset{1\leq j\leq 4}{\min}\left\{z_{ij}\right\}\). These solutions are called the ideal solutions because the corresponding cases are almost impossible to happen in reality. Step 4: Calculation of the fuzzy distance between \(l_{j}(j=1,2,...,4)\) and the ideal solution  
The fuzzy distance is defined as follows using Equation (3):  
\[d(l_{j},P)=\sum_{i=1}^{5}d(z_{ij},p_{i}) \tag{13}\]  
\[d(l_{j},N)=\sum_{i=1}^{5}d(z_{ij},n_{i}) \tag{14}\]  
The optimal landing path is the one which is close to the FPIRP and far away from the FNIRP. According to different distances from the FPIRP, the ranking of all the alternative landing paths can be determined. Step 5: Calculation of the closeness coefficient  
The closeness coefficient denotes the degree of closeness between the alternative landing path and the fuzzy ideal solution, and a high value means a closer degree. The following formula is used to calculate the closeness coefficient. \[C(l_{j})=\mu\frac{d(l_{j},P)}{d(l_{j},P)+d(l_{j},N)}+(1-\mu)\frac{d(l_{j},N)}{ d(l_{j},P)+d(l_{j},N)} \tag{15}\]  
where \(\mu(0<\mu<1)\) is a variable parameter which denotes the weight of different distances from FPIRP and FNIRP, and it can be adjusted by the bias of the decision maker. For example, if the atmospheric environment is good in general, the decision maker usually puts more attention on FPIRP and sets \(\mu\) as a smaller value. A larger value will be set for \(\mu\) to put more attention on FNIRP if the air traffic situation is bad.