# 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  
The weather condition, the height of clouds, the visibility, the performance of the aircraft and the air traffic condition are the five key contributing factors determining the result of the landing path selection and can be written as \(f_{1}\), \(f_{2}\), \(f_{3}\), \(f_{4}\) and \(f_{5}\), respectively. The PRCF are described by triangular membership functions and need to be unified into the same linguistic variables. Among the five contributing factors, only the performance ratings of cloud height and visibility can be described quantitatively, as shown in Tables 1 and 2. Note that the linguistic variables listed in Table 2 are described by the triangular membership functions. Assume that the performance rating of \(l_{j}(j=1,2,3,4)\) with respect to \(f_{i}(i=1,2,3,4,5)\) is \(e_{ij}=[e_{ij}^{L},e_{ij}^{M},e_{ij}^{U}]\) (the value of \(e_{ij}\) can be chosen from Table 2); the fuzzy decision matrix of the path selection problem can be written as \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**\(f_{1}\)** & **\(f_{2}\)** & **\(f_{3}\)** & **\(f_{4}\)** & **\(f_{5}\)** \\ \hline Good & \(>\)3000 feet & \(>\)5 km & Medium & Good \\ Bad & 1000–3000 feet & \(>\)5 km & Good & Good \\ Terrible & \(<\)1000 feet & \(<\)5 km & Good & Good \\ Indifferent & Indifferent & Bad & Bad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of performance ratings for the five contributing factors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Linguistic Variable** & **No** & **Low** & **Medium** & **High** & **Very High** \\ \hline Expression & NO & LO & ME & HI & VH \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Unified linguistic variables of performance ratings. \[D=\left[ \begin{array}{cccc}e_{11}&e_{12}&e_{13}&e_{14}\\ e_{21}&e_{22}&e_{23}&e_{24}\\ \vdots&\vdots&\vdots&\vdots\\ e_{51}&e_{52}&\ldots&e_{54}\end{array} \right] \tag{9}\]  
where matrix \(D\) is the base of priori knowledge of the decision makers. Having introduced the basics of the developed fuzzy TOPSIS approach for solving the path selection related decision making problem, the detailed procedure will be described below. Step 1: Normalization of the fuzzy decision matrix  
When the raw data are collected, different measurement units are used by contributing factors, e.g., the descriptions of \(f_{2}\) and \(f_{3}\), so normalization is needed to transform each element of \(D\) into the interval of [0, 1] to eliminate anomalies. Here, a linear transformation is used to calculate the normalized fuzzy decision matrix \(X\):  
\[X=\left[ \begin{array}{cccc}x_{11}&x_{12}&x_{13}&x_{14}\\ x_{21}&x_{22}&x_{23}&x_{24}\\ \vdots&\vdots&\vdots&\vdots\\ x_{51}&x_{52}&\ldots&x_{54}\end{array} \right] \tag{10}\]  
As the PRCF denote the efficiency in the path selection problem, \(x_{ij}\) is normalized using its largest value:  
\[x_{ij}=[e_{ij}^{L}/e_{i}^{\text{Imax}}\,e_{ij}^{M}/e_{i}^{\text{Imax}}\,e_{ij }^{\text{II}}/e_{i}^{\text{IImax}}] \tag{11}\]  
where \(e_{i}^{\text{Imax}}=\max\limits_{1\leq j\leq 4}\{e_{ij}^{\text{II}}\}(i=1,2,...,5,j=1,2,3,4)\). All linguistic variables can be transformed to the interval of [0, 1] by the same method used in Equation (11). Assume that the pilot and the LCO are decision makers No. 1 and No. 2, respectively, and the normalized fuzzy decision matrix \(X_{1}\) and \(X_{2}\) can be obtained according to prior knowledge of them. \[X_{1}=\left[ \begin{array}{cccc}H1&ME&LO&NO\\ VH&HI&LO&NO\\ VH&VH&LO&LO\\ ME&HI&HI&LO\\ ME&HI&HI&LO\end{array} \right],\,X_{2}=\left[ \begin{array}{cccc}H1&ME&NO&NO\\ HI&ME&LO&NO\\ HI&HI&LO&NO\\ \end{array} \right]\]  
Comparing \(X_{1}\) with \(X_{2}\), it can be found that the performance ratings of a pilot are higher than those of the LCO with respect to the atmospheric environment in general. This is because the pilot has an accurate sense of the atmospheric environment, and he/she is usually conservative for safety reasons when determining the landing path.