# A Fuzzy Path Selection Strategy for Aircraft Landing on a Carrier
## 3 Landing Path Selection Method Based on FMAGDM
### The PRCF and the Current Environmental Vector  
In the process of building a decision-making model, the PRCF or the current environmental vector is described by the fuzzy linguistic variables in this study. In this problem, each candidate landing path corresponds to an ideal environmental vector, and the triangular membership function is fit for this characteristic and is used to define the value of each fuzzy linguistic variable. For example, \"low\", \"medium\" or \"high\" is used to express the performance rating of \(l_{1}\) with respect to the weather, and the triangular membership function is used to define \"low\", \"medium\" or \"high\". Therefore, the triangular membership function is the foundation of establishing the decision-making model. The concept and attributes of the triangular membership function are given below to ensure a better understanding of its application in the path selection problem [27].  
**Definition 1**.: _Let \(a=[a^{L},a^{M},a^{U}]\) be a triangular membership function with \(0<a^{L}\leq a^{M}\leq a^{U}\), as shown in Figure 6._  
An example is presented to provide  
**Definition 2**.: _The mean value of the triangular membership function is defined as_  
\[s(a)=(a^{L}+2a^{M}+a^{U})/4 \tag{1}\]  
This definition is used to rank the triangular membership functions. Assume that \(a\) and \(b\) are two triangular membership functions; the following are defined: \(a\geq b\) if \(s(a)\geq s(b)\) and \(a<b\) if \(s(a)<s(b)\). For example, if \(a\) and \(b\) describe different performances of an aircraft, \(a\geq b\) indicates that the performance described by \(a\) is generally better than that described by \(b\).  
**Definition 3**.: _Assume that \(U\) is a universe of discourse, and the \(\alpha\)-cut set (denoted by \(a_{\alpha}\)) is defined as_  
\[a_{\alpha}=\{x\in U|\mu_{\lambda}(x)\geq\alpha\},\alpha\in[0,1] \tag{2}\]  
_where \(a_{\alpha}\) can be further expressed as \(a_{\alpha}=[a^{L}(\alpha),a^{R}(\alpha)]\), as illustrated in Figure 7. If it presents moderate weather, \(a_{\alpha}\) indicates that the degree of moderate weather is more than that of \(\alpha\)._  
**Definition 4**.: _If \(a\geq b\) (a and \(b\) are two triangular membership functions and can be denoted as \(a=[a^{L},a^{M},a^{U}]\) and \(b=[b^{L},b^{M},b^{U}]\)), and the fuzzy distance between them can be defined as_  
\[d(a,b)=[\max\{\int_{0}^{1}d^{L}(\alpha)da,0\},d^{L}(1),\int_{0}^{1}d^{R}(\alpha )da] \tag{3}\]  
Figure 6: The triangular membership function.  
Figure 7: Definition of \(\alpha\)-cut set in the triangular membership function.  
_where \(d^{L}(\alpha)\) and \(d^{R}(\alpha)\) are written as_  
\[\left\{\begin{array}{l}d^{L}(\alpha)=a^{L}(\alpha)-b^{R}(\alpha)\\ d^{R}(\alpha)=a^{R}(\alpha)-b^{L}(\alpha)\end{array}\right. \tag{4}\]  
Similar to the definition of distance in a geometric space, the fuzzy distance also represents the degree of approximation between these two triangular membership functions. For example, suppose that \(a\) and \(b\) describe good and bad air traffic situations, respectively; then, \(d(a,b)\) represents how large the difference is between these two air traffic conditions. It should be noted that \(d(a,b)\) is also a triangular membership function.  
Until now, knowledge of the triangular membership function to describe the fuzzy linguistic variables when establishing a decision-making model has been presented. Some main operations associated with triangular membership functions are listed below (\(a\) and \(b\) are two random triangular membership functions, \(k\) is a random real number):  
\[a+b=[a^{L},a^{M},a^{U}]+[b^{L},b^{M},b^{U}]=[a^{L}+b^{L},a^{M}+b^{M},a^{U}+b^{U }]; \tag{5}\]  
\[a\times b=[a^{L},a^{M},a^{U}]\times[b^{L},b^{M},b^{U}]=[a^{L}\times b^{L},a^{M }\times b^{M},a^{U}\times b^{U}]; \tag{6}\]  
\[k\times a=[k\times a^{L},k\times a^{M},k\times a^{U}]; \tag{7}\]  
\[a^{-1}=[1/a^{U},1/a^{M},1/a^{L}]. \tag{8}\]