## 5 Proposed Methodology
### Mathematical Analysis and Proof
It might restrict the MR from using its remaining energy to get back to the place of origin within the specified time. Therefore, this makes it essential to perform re-planning during the surveillance operation. When \(R\
eq 0\), then the surveillance process will start. As per the trajectory in \(Q\), the MR reaches the subsequent surveillance location. The route taken through the preceding surveillance spot to the present surveillance spot is deducted from \(Q\), which means \(\big{(}Q-l_{ij}\big{)}\), whenever the surveillance work has been obtained. The accessible surveillance site will be eliminated from R, which means \(R=(R-r_{i})\), soon after the information about a prospective position has been successfully gathered. The surveillance task is completed after \(R=0\). Subsequently, it is essential to check the remaining time \(\big{(}t_{initial}-t_{U}\big{)}\) and energy \(\big{(}E_{initial}-E_{U}\big{)}\), where \(t_{U}\) and \(E_{U}\) are the used time and energy consecutively before MR reaches a surveillance place. The loss of power is substantial whenever the MR's energy reserve is unable to facilitate a trip back toward its place of origin. The amount of time left over will not be adequate to gather information from all the surveillance zones because the MR takes up a large portion of time. For these circumstances, re-planning is necessary. This is required to determine the MR's latest acquisition position; we suppose it is \(r_{L}\). The surveillance place for re-planning is reserved in \(R^{\prime}\), and the starting \(R\) is reformed. The re-planned path is reserved in \(Q^{\prime}\), and the starting \(Q\) is reformed. The obtaining for possibilities facing obstacles is updated concurrently. Thus, a group that includes re-planning routes remains in \(P^{\prime}\), and initial \(P\) is updated as the \(P^{\prime}\). The MR surveillance work formulation has been described in Equations (12)-(14) [36]. \[max\ I_{M}\big{(}R^{n}_{,}Q^{n}_{,}P^{n}\big{)}=\underset{R^{n}\subseteq R,Q^{n }\subseteq Q,P^{n}\subseteq P}{\max}\sum_{i=1}^{n}F(x_{i}y_{i})A_{r_{i}} \tag{12}\]  
\[min\ dist_{M}\big{(}R^{n}_{,}Q^{n}_{,}P^{n}\big{)}=\underset{R^{n}\subseteq R,Q^ {n}\subseteq Q,P^{n}\subseteq P}{\min}\sum_{i,j}\in Q^{n}l_{i,j} \tag{13}\]  
\[min\ obs_{M}\big{(}R^{n}_{,}Q^{n}_{,}P^{n}\big{)}=\underset{R^{n}=R,Q^{n} \subseteq Q,P^{n}\subseteq P}{\min}\frac{\sum_{P^{\prime}_{i,j}\in P^{n}}R^{* }_{i,j}}{n-1} \tag{14}\]  
Equation (12) is used for the purpose of maximizing the gathered information from the surveillance zone, Equation (13) is used for minimizing the distance traveled by MR during surveillance, and Equation (14) is used to minimize the possibility of encountering obstacles. When obstacles are likely to be encountered frequently, both time and energy become wasted, which ultimately lowers the quantity of data acquired during surveillance. As a result, we must design a route that has plenty of data and little chance of hitting obstacles. Equations (15) and (16) show the energy and time constraints respectively [35; 36]. \[\sum_{l_{i,j}\in Q^{n}}^{n-1}l_{i,j}E_{l}+\sum_{i=1}^{n}E_{r}\leq(E_{initial}- E_{U}) \tag{15}\]  
\[\sum_{l_{i,j}\in Q^{n}}^{n-1}\frac{l_{i,j}}{v_{n}}+\sum_{i=1}^{n}t_{r}\leq(t_{ initial}-t_{U}) \tag{16}\]  
where the consumption of energy while the MR travels along the planned path for surveillance is \(\sum_{l_{i,j}\in Q^{n}}^{n-1}l_{i,j}E_{l}\) and energy required by the MR at the acquisition place during surveillance is \(\sum_{i=1}^{n}E_{r}\). The remaining energy and time for accomplishing the surveillance task are denoted by \((E_{initial}-E_{U})\) and \((t_{initial}-t_{U})\). It is clear from Equation (15) that the required energy is less or the same as the remaining energy. The time used by the MR while moving at the predefined path is \(\sum_{l_{i,j}\in Q^{n}}^{n-1}\frac{l_{i,j}}{v_{n}}\) and time utilized by the MR during the acquisition of data from the surveillance site is \(\sum_{i=1}^{n}t_{r}\).