# Efficient Node Deployment of Large-Scale Heterogeneous Wireless Sensor Networks
## 4 Problem Formulation for Optimal Solution
\\ \(y_{si}^{t}\) & 1 if node \(s\) exists in inactive state on zone \(i\) at time interval \(t\), and 0 otherwise, \(\forall s\in S\), \(\forall i\in A\), \(\  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]  
\[x_{si}^{t+1}y_{si}^{t}\leq m_{sij}^{t}\quad\forall s\in S,\forall i\in A,\forall t\in T \tag{6}\]  
\[1-\left(y_{si}^{t+1}\cdot x_{si}^{t}\right)\geq m_{sij}^{t}\quad\forall s\in S,\forall i\in A,\forall t\in T \tag{7}\]  
\[\sum_{i\in A}\sum_{j\in A}\sum_{i\in T}m_{sij}^{t}\leq M_{s}\quad\forall s\in S \tag{8}\]  
\[g_{i}\sum_{s\in\cup TN}\delta_{si}^{t}>1\quad\forall s\in S,i\in A,t\in T \tag{9}\]  
\[g_{i}\leq m_{sij}^{t}\quad\forall s\in S,\forall i,j\in A,\forall t\in T \tag{10}\]  
\[N_{j}\sum_{s\in\cup TN}\delta_{sj}^{t}>0\quad\forall s\in S,j\in A,t\in T \tag{11}\]  
\[2-N_{j}=m_{sij}^{t}+1\quad\forall s\in S,\forall i,j\in A,\forall t\in T \tag{12}\]  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]0  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]1  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]2  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]3  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]4  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]5  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]6  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]7  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]8  
\[Z_{IP}=\textbf{max}\underset{i\in T}{\sum}\sum_{i\in A}\sum_{s\in S}\omega_{i}^{t }x_{si}^{t}\ (\text{IP})\]9  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]0  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]1  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]2  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]3  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]4  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]5  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]6  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]7  
\[\sum_{i\in A}x_{si}^{t}\leq 1\quad\forall s\in S,\forall t\in T \tag{5}\]8For the reader to understand the constraints, they are divided into groups and described below:  
**(1)**: **Deployment Constraints:**:  
The deployment constraints involve the following:  
The constraint in Equation (3) is utilized to ensure that any mobile node \(s\) is either in an active or inactive state during any time interval. The constraint in Equation (4) is used to determine the value of the decision variable \(x_{si}^{t}\). The sensor node s is set to be active in zone \(i\) during time interval \(t\), \(x_{si}^{t}\) is enforced to 1, If that node is not used in any zone \(j\) during the next and previous time intervals. **(2)**: **Assignment Constraint:**:  
This constraint is utilized to ensure that any mobile node \(s\) is covering at most one zone at each time interval, as given in Equation (5). **(3)**: **Mobility Constraints:**:  
The mobility constraints are used to determine if the node \(s\) is moved from zone \(i\) to zone \(j\) at the end of interval \(t\). They involve the following:  
Equations (6) and (7) constraints compare zones where mobile node \(s\) is deployed during time intervals \(t\) and \(t+1\). The decision variable \(m_{sij}^{t}\) is enforced to 1 if they are different.