## 2 Materials and Methods
### Mathematical Modelling of p-WOA  
In this study, we provide numerical examples to explain how to search for vehicles, how to construct clusters, and how to choose a cluster leader for optimal performance of clustering. The numerical analysis of enclosing a target, plotting an attack using a bubble circle, and searching a vehicle are all depicted in this section (Figure 3). (1) Encircling Prey  
After being assigned a cluster head, vehicles can locate it and establish communication with it by incorporating Equations (1) and (2), where D represents the distance between vehicle X(t) and cluster head X*(t), and X(t + 1) displays the next iteration toward the optimal solution (cluster head). Because p-WOA does not know where exactly exploration will take place, it will pick whichever vehicle has the most optimal cluster head arrangement as the target. In Equations (1) and (2), we see how, once the optimized investigation is set up, other vehicles looking for cluster heads inform the optimized vehicle specialist of the locations in which they agree:  
\[\overset{\rightarrow}{\mathrm{D}}=\left|\overset{\rightarrow}{\mathrm{C}}. \overset{\rightarrow}{\mathrm{X}}(\mathrm{t})-\overset{\rightarrow}{\mathrm{X }}(\mathrm{t})\right| \tag{1}\]  
\[\overset{\rightarrow}{\mathrm{X}}(\mathrm{t}+1)=\overset{\rightarrow}{\mathrm{ X}}^{\ast}(\mathrm{t})-\overset{\rightarrow}{\mathrm{A}}.\overset{\rightarrow}{ \mathrm{D}} \tag{2}\]  
Here, t represents the current iteration, A and C are coefficient vectors, X* is the vector indicating the position of the best possible arrangement found so far, X represents the location direction, \(|\ |\) is the highest possible esteem, and \".\" represents the growth in dot-product size. Each iteration toward the best solution requires X* to be refreshed. Equations (3) and (4) can be used to determine the trajectories A and C:  
\[\overset{\rightarrow}{\mathrm{A}}=2\overset{\rightarrow}{\mathrm{a}}. \overset{\rightarrow}{\mathrm{r}}-\overset{\rightarrow}{\mathrm{a}} \tag{3}\]  
\[\overset{\rightarrow}{\mathrm{C}}=2.\overset{\rightarrow}{\mathrm{r}} \tag{4}\]  
While optimization techniques have two basic stages: the exploration stage (where vehicles are searched) and the exploitation stage (formation of the cluster head). Finding all of the cars in a network based on their distance from one another requires first performing a search or exploration, and then grouping. The variable (a) in Equation (3) is used to toggle between the exploration and exploitation stages; it is reduced from 2 to 0 at each iteration, and (r) is a random vector in the interval [0, 1]. (2) Bubble-net Attacking Method (Exploitation Phase)  
Two strategies, based on vehicles' Bubble-net behavior, are offered:(3)  
(2.1) Shrinking Encircling Mechanism  
This strategy reduces the value of (a) in Equation (3). Additionally, (a) minimizes the path to the validation of A. The interval [\(-\)a, a] where a is minimized from 2 to 0 over a specific collection of emphases can be thought of as including arbitrary esteem in (a). The unused location of a search agent can be set anywhere between the true position of the operator and the position of the current best operator by altering the value of A between \(-\)1 and 1. (4) (2.2) Spiral Updating Position  
This technique measures the distance between the area of the vehicle (X, Y) and the desired area (X*, Y*).