# Connectivity Maintenance Based on Multiple Relay UAVs Selection Scheme in Cooperative Surveillance
## 3 Multiple Relay UAVs Selection Scheme Based on Fuzzy Optimum Selection
### Membership Degree of UAV Alternatives Assessment Based on Fuzzy Optimum Selection
The set of attributes can be expressed as \(\left\{ \begin{array}{c}RV(i)\quad MH(i)\quad RMH(i)\quad AD\ (i)\end{array} \right\}\) for UAV \(i\), and the values of four attributes in \(N_{U}\) UAV alternatives can form an objective value matrix as follows:  
\[\mathbf{X}=\left(x_{ij}\right)_{\mathbf{4}\times N_{U}}=\left[ \begin{array}{c}RV(i) \\ MH(i)\\ RMH(i)\\ AD(i)\end{array} \right]_{\mathbf{4}\times N_{U}}=\left[ \begin{array}{cccc}x_{11} &\ldots&x_{1i}&\ldots&x_{1N_{U}}\\ x_{21}&\ldots&x_{2i}&\ldots&x_{2N_{U}}\\ x_{31}&\ldots&x_{3i}&\ldots&x_{3N_{U}}\\ x_{41}&\ldots&x_{4i}&\ldots&x_{4N_{U}}\end{array} \right]_{\mathbf{4}\times N_{U}} \tag{14}\]  
where \(x_{ij}\) is the value of attribute \(i\) for the \(j\) UAV alternative. Due to the different units among four attributes in above matrix, and the optimum value is decided according to the positive and negative correlation for the attributes. Hence, the normalized matrix of \(\mathbf{X}\) can be obtained according to following criteria. If the attribute value is bigger, the membership degree to the optimum alternative is larger. The values of optimum and attribute are positive correlation, such as RV and RMH, and the normalizing formula is denoted as:  
\[r_{ij}=\frac{x_{ij}-x_{imin}}{x_{imax}-x_{imin}}\]  
In contrast, the optimum and attribute values are negative correlation: when the attribute value is smaller, the membership degree to the optimum alternative is larger, such as MH and AD. Correspondingly, the normalizing formula can be expressed as:  
\[r_{ij}=\frac{x_{imax}-x_{ij}}{x_{imax}-x_{imin}}\]  
where \(x_{imin},\ x_{imax}\) are, respectively, the corresponding minimum and maximum values for all alternatives of attribute \(i\), and the normalized matrix can be denoted as follow:  
\[\mathbf{R}=\left(r_{ij}\right)=\left[ \begin{array}{cccc}r_{11}&\ldots&r_{1i}& \ldots&r_{1N_{U}}\\ r_{21}&\ldots&r_{2i}&\ldots&r_{2N_{U}}\\ r_{31}&\ldots&r_{3i}&\ldots&r_{3N_{U}}\\ r_{41}&\ldots&r_{4i}&\ldots&r_{4N_{U}}\end{array} \right] \tag{15}\]  
In above matrix, the entry \(r_{ij}\) is the relative value to optimum alternative in the interval [0,1]. \(r_{ij}=1\) represents that the UAV alternative \(j\) is the optimum alternative just according to the attribute \(i\), otherwise \(r_{ij}=0\) is the worst alternative. Correspondingly, the ideal optimum alternative can be calculated as \(\mathbf{g}^{T}=\left(g_{1},\ldots,g_{4}\right)=\left(\underset{j=1,\ldots,N_{U} }{\max}\left(r_{ij}\right),\ldots,\underset{j=1,\ldots,N_{U}}{\max}\left(r_{4 j}\right)\right)\), and the ideal worst alternative can be expressed as \(\mathbf{b}^{T}=(b_{1},\ldots,b_{4})=\big{(}\min_{j=1,\ldots,N_{U}}(r_{1j}),\ldots, \min_{j=1,\ldots,N_{U}}(r_{4j})\big{)}\). Hence, the membership degree matrix to the optimum and worst alternatives are defined as follows:  
\[\mathbf{U}_{2\times N_{U}}=\left[ \begin{array}{cccc}u_{11}&u_{12}&\ldots&u_{1N_{ U}}\\ u_{21}&u_{22}&\ldots&u_{2N_{U}}\end{array} \right] \tag{16}\]  
Correspondingly, the following constraints should be satisfied. \[0\leq u_{kj}\leq 1(k=1,2;j=1,\ldots,N_{U}),\ \sum_{k=1}^{2}u_{kj}=1(j=1, \ldots,N_{U}),\ \sum_{j=1}^{N_{U}}u_{kj}>0(k=1,2)\]  
where \(u_{1j}\) is the membership degree to the ideal optimum alternative and \(u_{2j}\) is the one to the ideal worst alternative. Considering the four assessment attributes, role values is employed to prevent frequently role converting in cooperative surveillance, thus role values is considered as the most important attribute. Similar to shortest paths, the importance of the minimum hops to ground station is same as the ratio of minimum hops channels via current UAV between all UAVs and ground station, so they are more important criteria. In addition, actual distance to ground station is an auxiliary attribute for UAV alternatives assessment and reflects less information of communication network. In this work, the relevant assessment attributes are ordered as RV, MH, RMH, AD.