## 3 Proposed Methodology
### A Hybrid Boundary Tour  
By assuming that the sensor nodes _SNs_ are able to transmit their information in all directions within an \(r\) meter radius, we propose a novel approach to determine the critical points that the AUV can visit within the transmission and reception boundaries with minimal energy consumption, time, and cost, taking into account the priority of capturing the most important information. In addition to MACO, we propose a hybrid algorithm that tries to determine the best location for the AUV to start aggregating sensor information from sensor nodes based on the best constructed \(\textit{NT}\) tour. The proposed evolutionary algorithm first detects the locations that the AUV can visit by detecting the intersections of the \(\textit{NT}\). Detection is performed at the edges of the transmission range of the sensor nodes based on the angle of the AUV path between the connected sensors. Each sensor node in the tour is represented by a 3D point; therefore, the _kth_ point on the boundary of node transmission can be given by  
\[P_{k}=\left(\left.\left[r+(mb-2r+1)\cdot U\right]\ \right|\ k\in R^{3}\right) \tag{9}\]where \(mb\) and \(r\) represent the maximum bound of the search space and the radius transmission distance of the node. \(U\) denotes a uniform distribution within [0, 1]. These points represent positions along the lines determined by the center points and are randomly distributed within the bounds defined by the problem space. In this regard, a vector space direction (\(DV\)) between the \(jth\) node and the \(kth\) point is given by the vector direction rule as follows:  
\[\overrightarrow{DV_{p}^{l}}=(x_{p}-x_{j},\ \ y_{p}-y_{j},\ \ z_{p}-z_{j}) \tag{10}\]  
where \(x_{j}\), \(y_{j}\), and \(z_{j}\) represent the coordinates of the \(jth\) node. Hence, the best location for the AUV to start aggregate information from sensor nodes can be calculated depending on Line Parametric rules \(L(\delta)\) as follows [19]:  
\[L(\delta)=\left(x_{j}+\delta\big{(}x_{p}-x_{j}\big{)},y_{j}+\delta\big{(}y_{p }-y_{j}\big{)},z_{j}+\delta\big{(}z_{p}-z_{j}\big{)}\right) \tag{11}\]  
where \(\delta\) is a control parameter that determines how far the sensor node is from its transmission range. When \(\delta=0\), the AUV can aggregate the node information exactly from the sensor node location. When \(\delta=1\), the AUV is at the boundary of node transmission. Therefore, when \(\delta\) varies within [0, 1], all coordinates between the sensor node and its transmission range along the vector direction can be traced, and this ensures that the AUV starts sensing information within the transmission range of the node. It should be noted that \(\delta\) is dynamically changed depending on the maximum transmission range and sensor position. This dynamic change guarantees that the parameter is customized to the particular surroundings and environmental circumstances. In such a setting, we assume that the \(x\), \(y\), and \(z\) coordinates represent the most likely locations for the \(jth\) sensor node to relay information. The 3D distance from the sensor node to its transmission range equals the radius r in meters and is given by  
\[\big{(}x-x_{j}\big{)}^{2}+\big{(}y-y_{j}\big{)}^{2}+\big{(}z-z_{j}\big{)}^{2}= r^{2}\]  
Therefore,  
\[ \begin{array}{c}\big{(}\big{(}x_{j}+\delta\big{(}x_{p}-x_{j}\big{)}\big{)}- x_{j}\big{)}^{2}+\big{(}\big{(}y_{j}+\delta\big{(}y_{p}-y_{j}\big{)}\big{)}-y_{j} \big{)}^{2}+\big{(}\big{(}z_{j}+\delta\big{(}z_{p}-z_{j}\big{)}\big{)}-z_{j} \big{)}^{2}=r^{2}\\ \delta^{2}\Big{[}\big{(}x_{p}-x_{c}\big{)}^{2}+\big{(}y_{p}-y_{c}\big{)}^{2}+ \big{(}z_{p}-z_{c}\big{)}^{2}\Big{]}=r^{2}\end{array}  \tag{12}\]  
where \(0<\delta\leq 1\), and when \(\delta>1\), no data transfer exists, which guarantees that AUV starts receiving information inside the allowed transmission range. The \(L\) coordinates represent the potential position in the space that the AUV can occupy, constrained by a transmission range from the center.