## 5 Model Formulation and Description
### Model Formulation
\(\mathrm{f_{S}(X)}\) is the sub function that collects the interest detected by every node in the area given by Equation (8). \(\mathrm{f_{G}(X)}\), given by Equation (5), is the sub function that allows sensor nodes on the RoI boundary to move with \(\mu\) in the right direction to maximize the collected interest. By using the derivate of the reference function in the direction of \(\overset{\rightarrow}{i}\) and \(\overset{\rightarrow}{j}\). \(\mathrm{f_{TX}(X)}\), given by Equation (6), one finds the minimum number of nodes needed to ensure connectivity. \(\mathrm{f_{N}(X)}\) is given by Equation (7), to define the number of the nodes to deploy. In Algorithm 1, the number of iterations \(T\) depends on the application parameters, such as the size of the map, the sensing range, the initial minimal distance of deployment \(d_{0}\), and the final desired distance \(d_{N}\) between a sensor node and its first-degree neighbors. It can be calculated as follows: \(n=(d_{N}\ -\ d_{0})/\ \
u\), with \(\
u\) the added value to the actual minimal distance between nodes each time. We initialize the distance between close sensor nodes with \(R_{S}\), then we calculate the number of nodes needed to monitor the whole area. Since the nodes are too close to each other and overlapping between sensor nodes is penalized, the maximum value of the joint interest is achieved in the first iteration, and full coverage and connectivity are ensured. This method has allowed us to approach a multi-objective optimization problem by solving a mono-objective optimization problem, which maximizes the joint interest. The other objectives can be automatically reached by an upgrade of their values. Thus, the proposed algorithm allows finding the maximum joint interest that can be collected by all sensor nodes in the network while minimizing their number, and maximizing the distance between close nodes. ```
1.T = T0, is the initial temperature and d = d0 the initial distance of deployment. v and u denote respectively the added value to the actual distance of deployment d and the transition value for nodes at the boundary. 2. Let compute the initial number of nodes, \(\mathrm{N}_{0}=\frac{\mathrm{A}}{\mathrm{d}_{0}}\times\frac{\mathrm{B}}{\mathrm{d }_{0}}\)
3.\(\mathrm{X}_{0}\) = {x1, x2,..., xN0 } (i) corresponds to the index of the current node. Initial interest function value is \(\mathrm{f}_{\mathrm{S}}(\mathrm{X}_{0})\) = 0 and \(\mathrm{f}_{\mathrm{S}}(\mathrm{X})\) = 0. 4. While T > 0 \(\backslash\)Let distribute nodes line by line and save those with a collected interest value Si \(\geq\) 0, then compute \(\mathrm{f}_{\mathrm{S}}(\mathrm{X})\). The first position is chosen randomly. 5. While k < sensor nodes on x-axis,
6. While j < sensor nodes on y-axis
7. Define the cluster
8. Calculate the collected interest Si per node then the joint interest. 9. If Si > 0 then xi \(\epsilon\) (X); \(\mathrm{f}_{\mathrm{S}}(\mathrm{X})\) = \(\mathrm{f}_{\mathrm{S}}(\mathrm{X})+\mathrm{Si}\) End
10. Shift the sensor position horizontally by v;
11.1.5 Update j
12. end
13.2 Shift the sensor position vertically by v.\(\sqrt{3}/2\);
14. Update k
15. End \(\backslash\)Let search for sensors on the RoI boundary, shift their positions and calculate the sum of collected interest of sensor nodes. 16.\(\mathrm{3.2t=\left(d-\frac{\mathrm{Rs}}{\mathrm{d}}\right)/\mu}\), is the number of iterations needed for a node to move from its position on the Roi boundary until approximating its neighbour by \(\mathrm{Rs}/4\). 17. While t > 0
18. While I < N sensor
19. Detect sensors on the RoI boundary \(\mathrm{S}_{\mathrm{E}}\)
20. Shift \(\mathrm{S}_{\mathrm{E}}\) positions by u using \(\mathrm{f}_{\mathrm{G}}(\mathrm{X})\)
21. Define the cluster then calculate the collected interest
22. End
23. Calculate the joint interest \(\mathrm{f}_{\mathrm{S}}(\mathrm{X})\);
24. \(\mathrm{t=t-\gamma}\)
25. End
26.