## 3 Materials and Methods
In our assumption, the coverage area of one sensor is calculated as the area of the square described around a circle with a coverage radius \(r\) as follows:  
\[S_{1}=4r^{2}. \tag{3}\]  
This assumption was introduced specifically to take into account the uncovered area around the sensor when calculating the number of sensors. For example, we considered the case of coverage of the territory with four sensors with the intersection of coverage areas. This situation can be schematically represented as shown below in Figure 1. \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description** \\ \hline \(T_{s}\) & Square of the territory \\ \(S_{1}\) & Square covered by one sensor \\ \(P_{1}\) & Cost of the one sensor \\ \(S_{covered}\) & Square of the covered territory \\ \(S_{uncovered}\) & Square of the uncovered territory \\ \(S(c)\) & Square of the intersection of the coverage areas with intersection level \(c\) \\ \(c_{surv}\) & Total network’s survivability \\ \(c_{s}\) & Sensor network survivability relative to the sensors’ loss \\ \(c_{h}\) & Sensor network survivability relative to the loss of communication between the sensors \\ \(c_{p}^{h}\) & Sensor network survivability relative to the loss of maximum bandwidth sensors \\ \(c_{p}^{s}\) & Sensor network survivability relative to the loss of maximum bandwidth connections \\ \hline \end{tabular}
\end{table} 
Table 2: Notation used in this article. According to Figure 1a, the value of the uncovered territory can be calculated as the difference between the area of a square with a side of \(2r\) and the area of four segments with an angle of 90 degrees as follows:  
\[S_{uncovered}=\left(2r\right)^{2}-4\pi r^{2}\frac{90}{360}\approx 0.86r^{2}. \tag{4}\]  
In Figure 1b, the value \(c\) is an intersection level of the sensors' coverage areas. Taking into account value \(c\), values covered by one sensor area Equation (3) and uncovered territory Equation (4) can be calculated as follows:  
\[S_{1}=\left(2r-c\right)^{2}, \tag{5}\]  
\[S_{uncovered}=\left(2r-c\right)^{2}-\Big{(}\pi r^{2}-2S(c)\Big{)}. \tag{6}\]  
In Equation (6), \(S(c)\) is the value of the area of intersection of the coverage areas with intersection level \(c\). This value is equal to twice the area of the circle segment with radius \(r=OA\) and chord \(AB\) (Figure 2). Figure 1: Territory coverage with four sensors: without intersection level (**a**); with intersection level (**b**). Figure 2: Finding the value of the area of intersection of coverage areas. Thus, the value of \(S(c)\) can be calculated as follows:  
\[S(c)=2S\Big{(}\frac{c}{2}\Big{)}=2\bigg{(}\frac{1}{2}OA^{2}(\alpha-sin(\alpha)) \bigg{)}=OA^{2}(\alpha-sin(\alpha)), \tag{7}\]  
where \(\alpha=2 arcsin\Big{(}\frac{KB}{OA}\Big{)}\), \(KB=\sqrt{OA^{2}-\big{(}OA-\frac{\varsigma}{2}\big{)}^{2}}\), and \(OA=r\). As the coverage of the territory is maximized, the area of the uncovered territory decreases. In this regard, the condition of maximizing the coverage in problem formulation (1) can be replaced by the condition of minimizing the uncovered area. According to this replacement and taking into account Equations (2)-(7), problem (1) can be formulated as:  
\[\left\{ \begin{array}{c}P(r,c)\to min\\ S_{uncovered}(r,c)\to min\end{array} \right., \tag{8}\]  
\[r_{min}\leq r\leq r_{max}, \tag{9}\]  
\[0\leq c\leq c_{max}. \tag{10}\]  
The solution to the described problem is the set of Pareto optimal parameters of the coverage radius and the value of the intersection of the coverage areas of the sensors. Using the obtained parameters, the maximum coverage area is achieved at the minimum cost of the sensor network. The second step after the network structure has been built with a designed approach, is the network survivability check. In [16], an approach for estimating the survivability of a sensor network based on [PERSON] and [PERSON] algorithms is presented.