# Efficient Node Deployment of Large-Scale Heterogeneous Wireless Sensor Networks
## 4 Problem Formulation for Optimal Solution
The constraint in Equation (8) is used to ensure that the number of moves made by any node \(s\) is less than or equal to the maximum number of moves allowed for that node. The constraints in Equations (9) and (10) guarantee that the movement of a node \(s\) from zone \(i\) to zone \(j\) does not make zone \(i\) empty. Equations (11) and (12) ensure that the zone \(j\) which the node s will move to is not covered by any node, whether stationary or moving node. The constraints in Equations (13)-(15) handle a special case in which the number of nodes in zone \(j\) is 0 but \(\omega_{j}^{t}<\omega_{i}^{t}\). The constraint in Equation (16) ensures that the number of moves made by a mobile node \(s\) during time interval \(t\) should be less than or equal to 1. **(1)**: **State-Switching Constraints:**:  
The state-switching constraints involve the following:  
The constraints in Equations (17)-(20) determine the state switching of a sensor node from active to inactive state and vice versa. For each mobile node, the total number of state switching should be less than or equal to the maximum number of switches allowed for such node as given in Equation (21). **(2)**: **Distance Constraints:**:  
The distance constraints are designed to guarantee that moved node \(s\) from zone \(i\) to zone \(j\) is close as much as possible to zone \(j\). That's to say, the selected mobile node \(s\) has the minimum distance between its location on zone \(i\) and the new location at the centre of zone \(j\) as illustrated in Equations (22)-(24). **(3)**: **Energy Constraints:**:  
The energy constraints illustrated in Equations (25)-(29) are used to conserve energy and prolong the network lifetime. They involve the following:  
The mobile node \(s\) is utilized when its expected residual energy after the movement from zone \(i\) to zone \(j\) is greater than or equal to the threshold value, and it has the maximum expected residual energy compared to the others at the same time as given in Equations (25)-(28). 2. The state of the mobile node \(s\) is changed from _off_ to _on_ when its residual energy after the state-switching is greater than or equal to the threshold value, as illustrated in Equation (29). 3. **Decision Variable Constraint:** The decision variable constraint is illustrated in Equation (30), where all the decision variables are equal to 0 or 1. 4. **Walk-Through Example:** In this section, a simplified example shown in Figure 6 is explained walking through node deployment step by step to understand better how the constraints work and how the optimal deployment is performed. The example is used to illustrate the deployment of four sensors, \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) of four zones \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), and \(Z_{4}\), which is monitored for two-time intervals \(t_{1}\) and \(t_{2}\). Table 3 shows the residual energy of each sensor node before and after movement, providing the movement distance and residual energy. Figure 6 represents the initial deployment of the four sensors at the first-time interval, \(t_{1}\). Assume that relation between the priority of each zone and the others during the second time interval \(t_{2}\), is described as \(\omega_{1}^{t_{2}}<\omega_{3}^{t_{2}}<\omega_{2}^{t_{2}}<\omega_{4}^{t_{2}}\). Three combinations of the proposed optimal solution parameters are selected, listed in Table 4, to understand and evaluate their effects on network performance. We refer to these parameter combinations as Case 1-3, respectively. As the field coverage is used as the optimization function, all cases consider the weight function for each zone, \(\omega_{i}^{t}\) (zone priority). Regarding Case 1, the deployment point is only determined by the zone priority and the movement distance, making the zone with the highest priority and the shortest movement distance is selected as the candidate deployment point. Figure 7a shows the movement operation during the time interval, \(t_{2}\), under the parameter combination of Case 1.