# A Hybrid Traffic Sensor Deployment Model with Communication Consideration for Highways
## 2 The Optimization Model for the Hybrid Sensor Deployment
The basic idea is to find those \((i,j)\) pairs making \(\sum_{\{k\in S\}}\forall x_{ij}^{k}\) equal to \(1\). Feasible pairs of Figure 2 are labeled with red dashed lines. These dash lines joined together are similar to a smooth trajectory, where the destination cell of a dashed line is the origin cell of its downstream line. Sensors are selected to be deployed at the junction of two adjacent lines. So for each dashed line, it implies that the origin and destination cell of this line should be placed with sensors, i.e., cells \(3\), \(7\), and m equipped with sensors. According to the characteristic of variable \(x_{ij}^{k}\), some constraints related to trajectory optimization should be added. With these considerations, the objective function and constraints of the multi-sensor deployment will be stated. The main objective of this paper is to maximize the accuracy rate of monitoring traffic scenes, which can be briefly stated as Equation (1). \[max\sum_{i,j\in N\cup\{0,n+1\},k\in S}\pi_{ij}^{k}x_{ij}^{k} \tag{1}\]  
where \(0\) and \(n+1\) represent two dummy cells. To make solutions feasible, constraints for sensor and sever location are detailedly described as follows. (1) Dummy cell constraint  
Information flow in the network flows from the start cell to the target cell, so the direction of information transmission and traffic flow are fixed. Constraints need to be added to ensure that the trajectory starts from \(0\) and ends with \(n+1\). Equation (2) guarantees that feasible trajectories need start at dummy cell \(0\) and end at dummy cell \(n+1\), as shown in Figure 2.  
\[ \begin{split}\sum_{k\in S,j\in N\cup\{n+1\}}x_{0j}^{k}& =1\\ \sum_{k\in S,j\in N\cup\{0\}}x_{i,n+1}^{k}&=1\end{split}  \tag{2}\]  
(2) Road cell constraint  
Since sensors will not be placed on all road cells, a feasible trajectory could only contain partial road cells. For simplicity, it is assumed that each cell is equipped with one sensor at most. In fact, a monitor point along the road might use several sensors. Although Equation (3) does not satisfy all scenarios, it can be modified if necessary. This paper first formulates the model under a simple scenario, and a cell with multiple sensors will be considered as extended research. \[\sum_{k\in S,j\in N-\{i\}}x_{ij}^{k}\leq 1,\forall i\in N \tag{3}\]  
Figure 2: An example of finding sensors’ location. In addition, Equation (4) is added to avoid generating a detour trajectory. It implies that a trajectory should move along the direction of traffic flow. \[\sum_{k\in\mathcal{S},j\in\{1,2,...,j-1\}}x_{ij}^{k}=0,\forall i\in\mathbf{N} \tag{4}\]  
Except for eliminating detour, the consistency of a trajectory should also be considered, which is stated as Equation (5). It indicates that if a road cell is a destination cell of a trajectory, it must be an origin cell of another trajectory. To meet this requirement, an auxiliary parameter \(\Omega^{k}\) represents the set of connective combination types of the \(k\)th combination type. For example, if the combination type {camera, camera} is selected, the following combination type should be {camera, camera} or {camera, RSU}. So, for \(k=1\), the connective set \(\Omega^{1}\) is equal to \(\{1,2\}\). \[\sum_{i\in\mathbf{N}}x_{ij}^{k}=\sum_{i\in\mathbf{N},m\in\Omega^{k}}x_{ji}^{m},\forall j \in\mathbf{N},k\in\mathbf{S} \tag{5}\]  
(3) Investment cost constraint  
With economic considerations, the total cost of sensors and MEC servers should not be larger than the budget W, as shown in Equation (6). It should be noted that \(c^{k}\) denotes the first sensor cost of combination type \(k\). For example, \(c^{1}\) is equal to \(w_{c}\). Since the first sensor of type \(1\) is the camera, the value of \(c^{1}\) should be equal to the cost of the camera. \[\sum_{i,j\in\mathbf{N}\cup\{n+1\},k\in\mathbf{S}}c^{k}x_{ij}^{k}+\sum_{u\in\mathbf{N}} \varpi z_{u}\leq W \tag{6}\]  
where \(\varpi\) is the unit cost of an MEC server.