## 3 MVO Algorithm and Its Application to Solve the RNP-WMN Problem
### MVO Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multirow{2}{*}{\(n\)} & \multicolumn{6}{c}{**Number of Connected Clients**} & \multicolumn{6}{c}{**Connected Client Ratio (\%)**} \\ \cline{3-10}  & & **MVO** & **WOA** & **GA** & **PSO** & **MVO** & **WOA** & **GA** & **PSO** \\ \hline \multirow{8}{*}{INS-3} & 50 & 46.2 & 40.3 & 41.9 & 42.6 & 92.3 & 80.6 & 83.9 & 85.2 \\  & 100 & 90.8 & 79.3 & 83.2 & 81.9 & 90.8 & 79.3 & 83.2 & 81.9 \\  & 150 & 132.5 & 113.1 & 122.0 & 125.9 & 88.3 & 75.4 & 81.3 & 83.9 \\  & 200 & 178.6 & 149.9 & 157.8 & 169.9 & 89.3 & 75.0 & 78.9 & 85.0 \\  & 250 & 220.7 & 181.0 & 190.5 & 206.7 & 88.3 & 72.4 & 76.2 & 82.7 \\  & 300 & 268.6 & 223.2 & 234.0 & 250.9 & 89.5 & 74.4 & 78.0 & 83.6 \\  & 350 & 314.9 & 253.4 & 263.2 & 292.8 & 90.0 & 72.4 & 75.2 & 83.6 \\  & 400 & 360.1 & 291.6 & 292.3 & 333.1 & 90.0 & 72.9 & 73.1 & 83.3 \\ \hline \multirow{8}{*}{INS-4} & 50 & 49.8 & 47.4 & 48.6 & 47.4 & 99.7 & 94.7 & 97.2 & 94.8 \\  & 100 & 98.7 & 93.0 & 95.3 & 95.1 & 98.7 & 93.0 & 95.3 & 95.1 \\ \cline{1-1}  & 150 & 146.5 & 134.9 & 136.0 & 138.6 & 97.7 & 89.9 & 90.7 & 92.4 \\ \cline{1-1}  & 200 & 196.4 & 179.9 & 189.9 & 188.5 & 98.2 & 89.9 & 94.9 & 94.3 \\ \cline{1-1}  & 250 & 246.0 & 221.3 & 233.8 & 230.4 & 98.4 & 88.5 & 93.5 & 92.1 \\ \cline{1-1}  & 300 & 294.6 & 268.0 & 279.5 & 270.0 & 98.2 & 89.3 & 93.2 & 90.0 \\ \cline{1-1}  & 350 & 343.9 & 311.9 & 325.3 & 315.1 & 98.2 & 89.1 & 92.9 & 90.0 \\ \cline{1-1}  & 400 & 391.6 & 353.0 & 361.1 & 351.7 & 97.9 & 88.3 & 90.3 & 87.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performance comparison of the number of connected clients and CCR when executing INS-3 and INS-4. Figure 8: Performance comparison of the CCR versus the number of clients. (**a**) INS-3. (**b**) INS-4. ### Impact of Coverage Radius of the Mesh Routers

Another metric also has a great influence on the performance of mesh routers placement algorithms in WMN; that is, the coverage radius of the mesh routers. This is more clearly visible in Figure 10, where we measure the CCR versus the coverage radius of the mesh routers for INS-6 and INS-8. We can observe that the wider the coverage area, the higher the CCR. This is obvious because the CCR is directly proportional to the width of the coverage radius of the mesh routers. Among the algorithms MVO, WOA, GA and PSO, MVO gives the highest efficiency for both INS-6 and INS-8. For INS-6 (Figure 10a), to be able to achieve 100% CCR, the required coverage radius for the MVO algorithm is 260 m. Meanwhile, this value for algorithms WOA, GA and PSO is greater than 300m. For INS-8 (Figure 10b), since the number of mesh routers in this instance is more than INS-6, the CCR is also better. The coverage radius of the mesh router only needs 220m, and the MVO algorithm can achieve a CCR of 100%. The details of the CCR values versus the coverage radius of the mesh routers for INS-6 and INS-8 are shown in Table 8.

Figure 10: Performance comparison of the CCR versus the coverage radius of the routers. (**a**) INS-6. (**b**) INS-8. Figure 9: Performance comparison of the path loss versus the number of clients. (**a**) INS-3. (**b**) INS-4. Figure 14: Performance comparison of the objective function value for INS-4 with 150 mesh clients. Figure 12: Performance comparison of the objective function value for INS-2 with 40 mesh routers. Figure 13: Performance comparison of the objective function value for INS-3 with 150 mesh clients. ## 5 Conclusions

The MRP-WMN has attracted many research groups recently. Because this is an NP-hard problem, approximate optimization algorithms are typically used to solve it. We used the MVO optimization algorithm to solve the MRP-WMN in this paper. A new objective function for the MRP-WMN is also proposed, which takes two important performance metrics into account: connected client ratio and connected router ratio. The simulation method on Matlab is used to evaluate the performance of the MVO algorithm in solving the MRP-WMN problem.