# Efficient Node Deployment of Large-Scale Heterogeneous Wireless Sensor Networks
## 4 Problem Formulation for Optimal Solution  
Our previous modelling indicates that the deployment problem can be solved optimally. Optimal solution will help in producing an efficient heuristic. This problem is mathematically formulated using ILP. Although this solution is used to confirm the optimal solution and fully understand the problem with its major constraints, it is intractable to solve large-scale problems in terms of the elapsed time and/or memory required for problem-solving. Hence, the ILP solution is utilized to solve small-scale problems and guarantee the proposed solution's efficiency. To fully understand the used notations used in our solutions, the reader is referred to Table 2 for the meaning of each symbol used in the formulation. As can be seen in Equation (2), the objective function maximizes the field coverage described as the sum over all time intervals of the products of the observation weight \(\omega_{i}^{t}\) and the decision variable \(x_{si}^{t}\). \[\text{\emph{The overall field coverage}}=\sum_{i\in T}\sum_{i\in A\leq S}\omega_ {i}^{t}x_{si}^{t} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Parameters**} \\
**Paper ID** & **Residual** & \begin{tabular}{c} **Movement** \\ **Consumed** \\ **Energy** \\ \end{tabular} & \begin{tabular}{c} **Movement** \\ **Distance** \\ \end{tabular} & \
\begin{tabular}{c} **Areas (Zones)** \\ **Priority** \\ \end{tabular} \\ \hline
\begin{tabular}{c} [6] \\ [13] \\ \end{tabular} & \(\surd\) & \(\times\) & \(\times\) & \(\surd\) \\ Proposed Algorithm & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A comparative summary of the algorithms mentioned above and the proposed ones. Based on this computation, the objective function is defined by the following equation:  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Given Parameters** & **Description** \\ \hline
**Notation** & **Description** \\ \hline \(S\) & The set of all mobile nodes. \\ \(A\) & The set of all zones in the monitored field. \\ \(T\) & The time horizon. \\ \(TN\) & The set of all stationary and mobile nodes in the monitored field. \\ \(\omega_{i}^{t}\) & The time-varying weight function for each zone \(i\in A\), where \(t\in T\). \\ \(M_{s}\) & The maximum number of moves allowed for such node \(s\), \(s\in S\). \\ \(P_{s}\) & The maximum number of switches allowed for such node \(s\), \(s\in S\). \\ \(d_{sij}^{t}\) & Is the Euclidean distance between the location of node \(s\) on zone \(i\) and its new location on zone \(j\)(center of zone \(j\)), where \(i,j\in A\), \(s\in S\) and \(t\in T\). \\ \(E_{sij}^{t}\) & The energy required for the movement of node \(s\) from zone \(i\) to zone \(j\) at time interval \(t\), where \(i,j\in A\), \(s\in S\) and \(t\in T\). \\ \(ES_{s}\) & The energy required for changing the state of node \(s\) from _off_ to _on_\(s\in S\). \\ \(RE_{s}^{t}\) & Is the residual energy of node \(s\) at time interval \(t\), where \(s\in S\) and \(t\in T\). \\ \hline \multicolumn{3}{p{113.8 pt}}{Indicator Parameter} \\ \hline \(\delta_{si}^{t}\) & 1 if node \(s\) is on zone \(i\) at time \(t\) and 0 otherwise. \\ \hline \multicolumn{3}{p{113.8 pt}}{Decision Variables} \\ \hline \(x_{si}^{t}\) & 1 if node \(s\) exists in active state on zone \(i\) at time interval \(t\), and 0 otherwise, \(\forall s\in S\), \(\forall i\in A\), \(\forall t\in T\).