## 3 MVO Algorithm and Its Application to Solve the RNP-WMN Problem
### MVO Algorithm
\begin{table}
\begin{tabular}{l l} \hline \hline
**MVO Algorithm** & **RNP-WMN Problem** \\ \hline Search space & WMN deployment area of dimensions \(W\times H\) \\ \hline Universe & Position of routers \\ \hline Solution (\(X^{best}\)) & Set of optimal mesh routers locations \\ \hline Inflation rate of universal & Objective function value \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Settings of the MVO algorithm for the RNP-WMN problem. \begin{table}
\begin{tabular}{c l c} \hline \hline
**Algorithm** & **Parameter** & **Setting** \\ \hline \multirow{3}{*}{MVO} & Universes number & 50 \\  & \(WEP\) & Increase from 0.2 to 1 \\  & \(TDR\) & Decrease from 0.6 to 0 \\ \hline \multirow{3}{*}{WOA} & Search-agent Number & 50 \\  & \(a\) & Decrease from 2 to 0 \\ \hline \multirow{3}{*}{GA} & Population size & 50 \\  & Crossover Rate & 0.7 \\ \cline{1-1}  & Mutation Rate & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of algorithms. Figure 4: The network topologies of WMN obtained using algorithms MVO, WOA, GA and PSO. (**a**) GA. (**b**) PSO. (**c**) WOA. (**d**) MVO. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameter** & **Setting** \\ \hline \multirow{3}{*}{PSO} & Population size & 50 \\  & \(c_{1}\) & 2 \\  & \(c_{2}\) & 2 \\ \cline{1-1}  & Inertia weight & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._

### Performance Metrics and Network Instances

In our simulation models, the metrics of the connected client ratio (CCR), path loss and objective function value are used to evaluate and analyze the performance of GA, WOA, PSO and MVO algorithms in solving the mesh router node placement problem. The CCR is determined according to (4). Path loss (PL) is the signal power loss over the transmission medium. In the context of this paper, the free space transmission medium is considered for WMN, and the \(PL\) is defined as follows [28]:

\[PL\ (dB)=10 log_{10}\Big{[}\Big{(}\frac{4\pi f_{c}d}{c}\Big{)}^{2}\Big{]} \tag{15}\]

where \(f_{c}\) is the carrier frequency, \(c\) is the speed of light (\(\simeq 3\times 10^{8}\) m/s) and \(d\) is the distance between the transmitter and the receiver. In this paper, we focus on analyzing the \(PL\) between the mesh client and the nearest mesh router. These performance metrics are evaluated through eight network instances (INS-1 to INS-8), as described in Table 5. The network instances of INS-1 to INS-2, INS-3 to INS-4 and INS-6 to INS-8 are used to evaluate the effect of the number of mesh routers, the number of mesh clients and the coverage radius of the mesh routers on network performance, respectively. ### Impact of the Number of Mesh Routers

The results obtained in Figure 5 show the effect of the number of mesh routers on CCR, where we plot the CCR as a function of the number of the mesh routers. These results are obtained when executing INS-1 and INS-2. We can observe that the higher the number of mesh routers, the higher the CCR for all algorithms, where the CDR is the best for the case of the MVO algorithm. For example, consider the INS-1 (Figure 5a) with 30 mesh routers; \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Setting** \\ \hline \(n\) & [100, 300] nodes \\ \hline \(m\) & [10, 50] nodes \\ \hline \(k\) & 1 node \\ \hline \(W\) & 2000 m \\ \hline \(H\) & 2000 m \\ \hline \(d_{CR}\) & [50, 200] m \\ \hline \(\lambda\) & [0, 1] \\ \hline Number of run & 30 \\ \hline Number of iteration & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instance** & \(m\) **(Routers)** & \(n\) **(Clients)** & _CR_ **(m)** \\ \hline INS-1 & [10, 45] & 150 & 200 \\ \hline INS-2 & [10, 45] & 350 & 200 \\ \hline INS-3 & 30 & [50, 400] & 200 \\ \hline INS-4 & 45 & [50, 400] & 200 \\ \hline INS-5 & 30 & 150 & [100, 300] \\ \hline INS-6 & 30 & 350 & [100, 300] \\ \hline INS-7 & 45 & 150 & [100, 300] \\ \hline INS-8 & 45 & 350 & [100, 300] \\ \hline \hline \end{tabular}
\end{table}