## 4 WSN Deployment Approaches Based on the SA Algorithm  
In reality, temperature is the controlling mechanism used to convert material from a high energy state into a low energy, in solid conditions. This process is imitated in SA, where the temperature controls the number and spread of accessible solutions from a given solution in the search space. SA begins with a high initial temperature to allow a random walk in the search space. As the temperature gradually decreases, the system becomes more focused, only allowing moves in the search space which improve the performance of the solution. The process terminates when a temperature close to zero is reached [8]. If the temperature is not high enough, the SA approximates a local optimum, instead of the global optimum of the objective function. In our approach, we use the simulated annealing algorithm as an optimization method in order to find the optimal deployment. Two approaches were presented in [13] and in [14] using the same algorithm. In those approaches, the authors assumed that each point of the ROI is different from another since each point has its own importance, thus they used the term \"interest\" to refer to the importance given to supervise a point. In the first one, the objective was to find the optimal topology guaranteeing coverage, connectivity, and a maximum interest with a fixed number of nodes. In the second one, the objective was how to deploy a fixed number of nodes with the purpose of providing coverage, a maximum interest, and ensuring connectivity with the minimum number of hops between all nodes and the sink. Our goal is to find the minimum number of sensor nodes with the best topology providing the maximum interest with at least 1-coverage and 1-connectivity. In the approach presented in [13], the objective function computes the value of the interest collected by every set of nodes, guarantees connectivity, then provides the optimal topology, ensuring the maximum interest in the area. \(\left\{\text{ If }\text{f}^{\text{K}}(\text{X}^{*})\geq\text{f}^{\text{K}}( \text{X}),\text{ with }\text{f}^{\text{K}}(\text{X}^{*})\text{ the new output of }\text{f}^{\text{K}}(\text{X})\right\}\) the function's output is saved to C, otherwise it's equal to the optimum solution stored in C. As shown in Equation (2) this function has two sub-functions. The first one computes the minimal distance between nodes and the second calculates the joint interest collected in all the regions of nodes. \[\text{f}^{\text{K}}(\text{X})=\ \left\{ \begin{array}{cc}\gamma.\text{f}_{ \text{min}}\left(\text{X}\right)\ +\ \text{f}_{\text{s}}\left(\text{X}\right),\text{ if }\text{X is K }-\text{survivable}\\ \\ \text{C},\text{ otherwise}\end{array} \right. \tag{2}\]  
where: X is a set of N nodes. \(\text{X}\)\(=(\text{x}_{1},\text{ x}_{2},\ldots\text{x}_{\text{i}}\ldots\text{x}_{\text{N}})\in\text{R}^{2}\). C is a constant ensuring that only survivable solutions will be accepted. \(\gamma\) controls the dispersion of nodes, a small value makes sensor nodes to stick together, a larger one disperses nodes regardless of the importance of the RoI. \(\text{f}_{\text{s}}(\text{X})\) is the sub-function representing the joint interest of the sensor nodes and penalizes coverage overlapping: \(\text{f}_{\text{s}}(\text{X})=\ \sum_{i=1}^{\text{N}}\alpha_{\text{i}}.\text{area}( \text{S}_{\text{i}})-\beta.\text{ area}\bigcap_{i=1}^{\text{N}}\text{S}_{\text{i}}. \alpha_{\text{i}}\) is the given importance of location \(\text{x}_{\text{i}},\beta\) a penalty on sensor coverage overlapping and \(\text{r}_{\text{s}}\) is the sensing range. \(\text{S}_{\text{i}}\)denotes the open ball with center \(\text{x}_{\text{i}}\) and radius \(\text{r}_{\text{s}}\), and \(\delta(\text{x}^{*},\text{x}_{\text{i}})=\|\text{x}^{*}-\text{x}_{\text{i}}\|\) is the Euclidean distance between points \(\text{x}^{*}\) and \(\text{x}_{\text{i}}\). \(\text{S}_{\text{i}}=\{\text{x}^{*}\in\text{Z}:\ \delta(\text{x}^{*},\text{x}_{\text{i}})\leq\text{r}_{\text{s}}\}\).