## 3 Proposed Method
### Fog Node Location Model  
Fog nodes, which are often located inside the transportation infrastructure, provide a distributed computing architecture for optimizing clustering operations by being strategically positioned close to the edge of the network. This approach reduces the need to transfer massive volumes of raw data to a central data center or cloud for processing. Rather, fog nodes enable the local and real-time execution of the vehicle-network-suited MDPC clustering technique. Fast decision-making is made possible by automobiles moving across the network, which is crucial for communication and security. Additionally, fog nodes reduce network traffic by only transmitting pertinent cluster information, conserving bandwidth. The scalability, adaptability to changing network conditions, and independence in the case of network failures all contribute to the reliability of clustering operations. In in-vehicle networks with limited energy supply, these nodes may also be modified for energy efficiency. By providing automotive networks with the processing power required for quick, dependable, and efficient clustering via MDPC, fog nodes provide enhanced network scalability, communication, and performance in dynamic vehicle situations. Each item designated as i in a fog computing environment with 'n' objects needs certain computing resources, \(C_{Ri}\), and storage resources, \(S_{Ri}\), in order to meet the service requirements. It is presumed that all entities may be serviced by a given number of virtual fog nodes, V, and have a shared maximum response latency, represented as _MRes_. Every virtual fog node, represented as _Fnod_, has W distinct device types attached to it, represented as _Kdev_. \(Q_{V}w\) denotes the unique number of each device type, \(Kdev_{V}w\), within these devices. Within this collection, a specific device is denoted as \(dev_{V}wj\), where j denotes the device type index and m denotes the device type. \(Res_{vwj}\) denotes the service delay for each item \(i\) where the jth device of type w serves that object. Furthermore, the unit resource costs are represented by the notation \(Cost_{v}wj=\{Cost_{S_{v}}wj,Cost_{C_{v}}wj\}\) and \(Re_{v}wj=\{Re_{S_{v}}wj,Res_{C_{v}}wj\}\), where \(Cost_{S}\) denotes the cost of storage resources, \(Res_{S}\), and \(Cost_{C}\) denotes the cost of computation resources, \(Re_{C}\). These expenses are related to the type w gadget j. The following formula may be used to construct the fog node placement and allocation problem:  
\[Min:\ K_{1}*\sum_{v=1}^{V}\sum_{v=n=1}^{vv}\sum_{j=1}^{Q_{V}w}\sum _{i=1}^{n}Res_{vwj}x_{i}+K_{2}*\sum_{v=1}^{V}\sum_{w=1}^{w}\sum_{j=1}^{Q_{V}w} Cost_{vwj}Re_{vwj}. \tag{1}\]  
\[s.t.\ \ \ \left\{ \begin{array}{l}x_{i}=1,\ if\ Thing\ i\ is\ served,\ otherwise,\ x_{i}=0\\ Res_{vwj}\leq MRes\\ \left[\sum_{v=1}^{V}\sum_{w=1}^{w}\sum_{j=1}^{Q_{V}w}Cost_{vwj}Re_{vwj}\right] \geq\sum_{i=1}^{n}(S_{Ri}+C_{Ri})\end{array} \right.. \tag{2}\]  
If there are 'n' points in an Euclidean space, and each point represents a \"Thing,\" then the resources needed to service each Thing and the related resource prices are regarded as characteristics of these points. Subsequently, the points are arranged into clusters by minimizing the weighted distance between the points and the centers of each cluster. To guarantee that the distance between each point and its cluster centroid stays below a certain threshold, a constraint is also applied to these clusters. The following may be used to summarize this clustering process:  
\[Min:\ K_{1}\sum_{i=1}^{W}\sum_{j=1}^{W}dj{y_{ij}}Cost_{i}{x_{i}}+K_{2}\sum_{i=1}^{W }\sum_{j=1}^{W}Q_{ij}y_{ij}\]  
\[s.t.\ \left\{ \begin{array}{l}x_{i},y_{ij}\ \in\{0,1\}\\ Q_{ij}=d_{i}O_{ij}\\ \left[Q_{ij}\right]\geq OThr\end{array} \right. \tag{3}\]  
The weights of the items are represented by \(K_{1}\) and \(K_{2}\). The selection point as a virtual fog node is denoted by \(x_{i}\), and the quality of service provided by point j or \(\left[Q_{ij}\right]\geq OThr\) is shown by \(y_{ij}\).