## 4 The Proposed ECSO-OBL Algorithm
### WSN Node Coverage Model  
Let \"\(N\)\" be number of sensor nodes deployed in a two-dimensional WSN with an area of \((W\times L)\) m\({}^{2}\). The set of sensor nodes is represented as \(S=\{s_{1},s_{2},s_{3},\ldots\ldots\ldots s_{N}\) \begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Ref.** & **Authors** & **Objective** & **Methodology** & **Limitations** \\ \hline
[19] & [PERSON] et al. (2021) & Regional coverage control in high-density WSNs is addressed. & Elite parallel CSO, which is randomized swarm optimization, is introduced. & It has a slow convergence rate and is not suitable for combinatorial optimization and discrete optimization problems. \\ \hline
[20] & [PERSON] et al. (2021) & Proposed to maximize network lifetime with variable ranges. & Hybrid combination of “Genetic Algorithm” and “Tabu Search” used to define cover sets with suitable sensing ranges. & This algorithm is not suitable for uniform WSNs. \\ \hline
[21] & [PERSON] et al. (2020) & Optimize the sensing range by minimizing redundancy in WSNs. & Based on the sensing range, the CSO algorithm classifies the sensor nodes into non-disjoint subsets. At a given time, only one subset is activated to cover the network area. The activation of each subset is in network lifetime. & \\ \hline
[22] & [PERSON] et al. (2025) & Enhance and optimize coverage in 2D and 3D WSNs. & Multi-strategy dung beetle optimization algorithm that uses a random number to change the inertia factor. & The maximum coverage is limited to 77.7\%. \\ \hline
[23] & [PERSON] and [PERSON] (2023) & Energy-efficient target coverage in WSNs. & Energy-efficient coverage is described as a set k-cover problem with independent sensor sets, and it is solved using graph theory. & The range of sensing is fixed, and it leads to lower energy efficiency. \\ \hline
[24] & [PERSON] et al. (2025) & Deploy minimum count of sensor nodes to fully cover the ROI and enhance connectivity. & The proposed method is described as a “connectivity problem” and a “covering problem”. It is solved using decomposition techniques based on the “Miller-Tucker-Zemlin” model. & It takes a greater number of iterations and execution time to achieve optimal results. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
**Step 2:**: In the given N-dimensional search space, generate the initial population (\"S\" number of nests) using Equation (4), and evaluate their individual fitness values. \[P=X+rand(N)\times(Y-X) \tag{4}\]  
In Equation (4), \(X\) and \(Y\) are the lower and upper bounds of search space and \(P\) is the set of nests, generated randomly. The function \(rand(N)\) gives an \(N\)-dimensional random number, where each dimension value falls between 0 and 1. **Step 3:**: Update the nest positions using Levy flight, and retain the nests with higher fitness values as per Equation (5). \[p_{i}^{a+1}=p_{i}^{a}+\Delta\otimes Levy(\gamma),\;\;\text{where}\;i=1,2,3, \ldots\ldots\,S \tag{5}\]  
where \(p_{i}^{a}\) is the position of the \(i^{th}\) nest for the \(t^{th}\) generation and \(p_{i}^{a+1}\) is the position of the \(i^{th}\) nest for the \((t+1)^{th}\) generation. \(\Delta\) is the step control volume; \(Levy(\gamma)\sim\mu=a^{-\gamma}\), where \(\gamma\) lies between 1 and 3, represents the random search; \(\mu\) is the random step size following a normal distribution; and \(\otimes\) is the pointwise multiplication operator. **Step 4:**: Integrate the CSO with adaptive PSO to update the nest positions using the sequential update mechanism, as shown in Equations (6) and (7), and retain the nests with higher fitness values.