## 3 The Proposed Sensor Deployment Algorithms
### Problem Analysis  
In an \(M*N\) rectangular area, different types of sensors need to be deployed, with each sensor using a probabilistic perception model. Heterogeneous sensors are selected for deployment, and each type of sensor has different costs and various parameters, including perception radius, perception error, and communication radius. There are \(W\) targets \(t_{j}(j=1,2,3\cdots W)\) to be monitored and \(T\) positions \(p_{i}(i=1,2,3\cdots T)\) available for deploying sensor nodes within the deployment area. The location of each target \(t_{j}\) to be monitored is known, and each candidate deployment position can accommodate only one sensor node. Assume that the performance metrics of sensor node \(S_{i,\mu}\) (\(i\) indicate which candidate location a sensor is placed at, \(i=1,2,3\cdots T\)), and \(u\) stands for which type of sensor is used for deployment, \(u=1,2,3\cdots U\)). For the node \(S_{i,\mu}\), the perception radius \(R_{s}\), perception error \(R(s,e)\), communication radius \(R_{e}\), and deployment cost are all known. Sensors are classified into \(U\) types based on these performance metrics. Within the rectangular area, there are certain special regions where deploying sensors is more challenging, requiring an additional deployment cost factor \(d_{i}(i=1,2,3\cdots T)\) to be applied to the original cost. The deployment cost \(d_{i}\) for each potential deployment location is known in advance. Therefore, the sensor deployment cost is calculated as:  
\[\sum\limits_{i=1}^{T}d_{i}p_{i,u}b_{u} \tag{2}\]  
where  
\[p_{i,u}=\left\{ \begin{array}{ll}1&\quad\text{$f$ $ $p_{i}$ is selected to type $u$}\\ 0&\quad\text{$f$ $ non node placed yet}\end{array} \right. \tag{3}\]  
The problem we need to study is how to select the appropriate candidate positions from the existing \(T\) candidate locations for deploying suitable sensor nodes. This involves setting minimum requirements for factors such as connectivity and perception probability, and then performing a comprehensive optimization of four metrics: deployment cost, connectivity, perception probability, and delay. Under the probabilistic perception model, since a target to be monitored may be perceived by multiple sensor nodes simultaneously, let \(pr(s_{i},t_{j})\) represent the probability that a target \(t_{j}\) can be perceived by a single sensor node \(s_{i}\). The joint perception probability is defined as the probability that the target \(t_{j}\) can be perceived by at least one sensor node. The calculation formula is as follows:  
\[PR(s_{i},t_{j})=1-\prod\limits_{i=1}^{T}\left[1-pr(s_{i},t_{j})\right] \tag{4}\]  
In this study, the energy consumption model follows the model proposed in references [49; 50]. The energy consumption of a single node in the network coverage is:  
\[E=\mu R_{S} \tag{5}\]  
where \(\mu\) is the energy parameter, and \(R_{S}\) is the sensor node's sensing radius. After the node is deployed, the energy consumption per unit area in the monitoring area is:  
\[E_{0}=\frac{\sum\limits_{i=1}^{T}\mu P_{i,u}R_{s_{i,u}}^{2}}{N} \tag{6}\]  
\(R_{s_{i,u}}\) represents sensing radius of the \(u\) type of sensor deployed at the \(i\) candidate location. Therefore, the four objective optimization functions are as follows:  
\[f_{1}=1-\frac{\sum\limits_{i=1}^{M}d_{i}P_{i,u}b_{i,u}}{\sum\limits_{i=1}^{M} \max(d_{i}P_{i,u}\max(b_{i,u}))} \tag{7}\]  
\[f_{2}=\frac{1}{W}\sum\limits_{i=1}^{W}PR(s_{i},t_{i}) \tag{8}\]  
\[f_{3}=\frac{1}{S}\sum\limits_{i=1}^{S}y(s_{i,u}) \tag{9}\]  
\[f_{4}=1-\frac{\sum\limits_{i=1}^{T}\mu P_{i,u}R_{s_{i,u}}^{2}}{\sum\limits_{i= 1}^{T}\mu P_{i,u}R_{s_{i,u}}^{2}}=1-\frac{\sum\limits_{i=1}^{T}P_{i,u}R_{s_{i, u}}^{2}}{\sum\limits_{i=1}^{T}P_{i,u}R_{s_{\max}}^{2}} \tag{10}\]where \(R_{s_{\max}}\) stands for the maximum sensing radius corresponding to each sensor type. \(S\) is the total number of deployed sensor nodes in the network. \(y(s_{i},\mu)\) indicates whether the sensor node \(s_{i}\) is connected to another node \(\mu\), and if not connected, \(y(s_{i},\mu)=0\).