# 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
Considering the feature of UAV alternatives in communication network, the weighed vector reflecting different influence of the four attributes can be calculated by analytic hierarchy process. Correspondingly, the judgment matrix can be denoted as follow:  
\[\mathrm{A}=\left[ \begin{array}{ccccc}&\mathrm{RV}&\mathrm{MH}&\mathrm{RMH}& \mathrm{AD}\\ \mathrm{RV}&1&3&3&5\\ \mathrm{MH}&\frac{1}{3}&1&1&3\\ \mathrm{RMH}&\frac{1}{3}&1&1&3\\ \mathrm{AD}&\frac{1}{5}&\frac{1}{3}&1\end{array} \right] \tag{17}\]  
The judgment matrix is satisfied with the complete consistency. Correspondingly, the weighed vector is calculated as \(\mathbf{\omega}=(0.5205,0.2010,0.2010,0.0775)\), and the evaluation vector of UAV alternative \(j\) can be expressed as  
\[\mathbf{r}_{j}=\big{(}r_{1,j},r_{2,j},r_{3,j},r_{4,j}\big{)}^{T}\]  
In space \(\mathbf{R}^{4}\), the generalized superior distance of UAV alternative \(j\) to the ideal optimum alternative can be defined as:  
\[\|\mathbf{\omega}\cdot(\mathbf{r}_{j}-\mathbf{g})\|=\left\{\sum_{k=1}^{4}\left[\omega_{k }\cdot(r_{kj}-g_{k})\right]^{p}\right\}^{\frac{1}{p}} \tag{18}\]  
In contrast, the generalized inferior distance can be denoted as:  
\[\|\ \mathbf{\omega}\cdot(\mathbf{r}_{j}-\mathbf{b})\ \|=\left\{\sum_{k=1}^{4}\left[\omega_{k} \cdot(r_{kj}-b_{k})\right]^{p}\right\}^{\frac{1}{p}} \tag{19}\]  
When \(p=2\), the distance between the ideal optimum alternative and UAV alternative \(j\) is represented by Euclidean distance. Hence, the weighted generalized superior and inferior distances can be expressed as:  
\[ \begin{array}{l}D(\mathbf{r}_{j},\mathbf{g})=u_{1j}\cdot\|\mathbf{\omega}\cdot(\mathbf{r}_{j }-\mathbf{g})\|\\ D(\mathbf{r}_{j},\mathbf{b})=u_{2j}\cdot\|\mathbf{\omega}\cdot(\mathbf{r}_{j}-\mathbf{b})\|\end{array}  \tag{20}\]In order to solve optimal membership degree \(u_{1j}\), an objective function can be designed as:  
\[Min\bigg{\{}F(u_{1j})=\sum\limits_{i=1}^{N_{U}}[D^{2}(\mathbf{r}_{i},\mathbf{g})+D^{2}( \mathbf{r}_{i},\mathbf{b})]=\sum\limits_{i=1}^{N_{U}}\Big{[}u_{1i}{}^{2}\cdot\|\mathbf{ \omega}\cdot(\mathbf{r}_{1}-\mathbf{g})\|^{2}+(1-u_{1i})^{2}\cdot\|\mathbf{\omega}\cdot( \mathbf{r}_{1}-\mathbf{b})\|^{2}\Big{]}\bigg{\}},j=1,2,\ldots,N_{U} \tag{21}\]  
Define that \(\frac{dF(u_{1j})}{du_{1j}}=0\), the following result can be obtained. \[u_{1j}\cdot\|\mathbf{\omega}\cdot(\mathbf{r}_{j}-\mathbf{g})\|^{2}-(1-u_{1j})\cdot\|\mathbf{ \omega}\cdot(\mathbf{r}_{j}-\mathbf{b})\|^{2}=0 \tag{22}\]  
Then for all UAV alternatives, namely \(j=1,\ldots,N_{U}\), the optimal matched membership degree of alternative \(j\) can be calculated as follow:  
\[u_{1j}^{*}=\frac{1}{1+\frac{\|\mathbf{\omega}\cdot(\mathbf{r}_{j}-\mathbf{g})\|^{2}}{\| \mathbf{\omega}\cdot(\mathbf{r}_{j}-\mathbf{b})\|^{2}}}=\frac{1}{1+\frac{\sum\limits_{k= 1}^{4}\left[\omega_{k}\cdot(\mathbf{r}_{kj}-\mathbf{g}_{k})\right]^{2}}{\sum\limits_{k= 1}^{4}\left[\omega_{k}\cdot(\mathbf{r}_{kj}-\mathbf{b}_{k})\right]^{2}}} \tag{23}\]  
As final reference indicator, the membership degree of UAV alternative \(j\), reflects distances to the ideal optimum and worst alternatives. Obviously, greater \(u_{1j}\) corresponds to the more important UAV alternative \(j\), and vice versa. Thus, if \(u_{1j}\) is bigger, UAV alternative \(j\) is easily chosen as a relay UAV to construct communication links for connectivity maintenance. In multiple UAVs cooperative surveillance, each UAV can be chosen as a relay to establish communication network. According to the proposed four assessment attributes, multiple relay UAVs selection sequence is obtained by sorting the membership degrees in descending. According to the alternatives sequence, UAVs are appropriately allocated three different roles to balance surveillance payoff and network connectivity based on multiple relay UAVs selection scheme.