# A Novel Hybrid Harris Hawk-Arithmetic Optimization Algorithm for Industrial Wireless Mesh Networks
## 4 Results and Discussion
### Simulation Analysis of Industrial WMNs  
Understanding the convergence behavior of optimization algorithms is crucial for obtaining optimal solutions. In the performance evaluation of the proposed HHAOA algorithm against other algorithms, the convergence behavior was analyzed using 500 iterations with 100 runs for 100 search agent sets, shown in Figure 8. Table 8 shows the statistical analysis of the convergence. By observing the convergence ability of these algorithms, their performance can be determined clearly. The objective function was utilized to find the minimum fitness value, which demonstrates improved connectivity and coverage.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **AOA** & **MFO** & **SCA** & **GWO** & **WOA** & **HHO** & **HHAOA** \\ \hline  
\hline
**Average** & **5.067** & **4.2** & **5.534** & **3.234** & **3.534** & **2.667** & **1.967** \\ \hline
**Final Rank** & **6** & **5** & **7** & **3** & **4** & **2** & **1** \\ \hline \hline \end{tabular}
\end{table}
Table 8: Convergence analysis for the WMN for all the algorithms.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimization** & **Mean** & **Best** & **Worst** & **Std. Dev** \\ \hline AOA & 0.531 & 0.485 & 0.673 & 0.0858 \\ MFO & 0.488 & 0.475 & 0.735 & 0.0841 \\ SCA & 0.472 & 0.46 & 0.524 & 0.0675 \\ GWO & 0.463 & 0.450 & 0.720 & 0.0587 \\ WOA & 0.456 & 0.435 & 0.634 & 0.0046 \\ HHO & 0.469 & 0.455 & 0.755 & 0.0674 \\ HHAOA & 0.449 & 0.435 & 0.613 & 0.0025 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Different benchmark functions of Friedman ranking test for the comparing algorithms.  
In the numerical analysis, the HHAOA method yielded the best outcomes with the least mean value of 0.499. Comparatively, the AOA came in second place with a mean value of 0.531. While the HHAOA method is 6.412% faster than AOA, it is worth noting that the WOA method is lagging with a minimal difference of 0.007. This resulted in WOA reaching a third place with a mean value of 0.456, which is 8.617% higher than the HHAOA method. The GWO, HHO, SCA, and MFO methods were followed in the remaining ranking order, which makes it clear that the proposed HHAOA method produced better results in the numerical analysis.  
Figure 9 shows the network connectivity and coverage region for the WMN for all the algorithms and the initial WMN topology. Figure 9a illustrates that the initial iteration resulted in insufficient mesh client coverage and high network congestion due to multiple overlapping mesh routers, which increased deployment costs. Even with redundant mesh routers, their connectivity did not reach the maximum capacity of 100%. Figure 9b-h shows the optimized WMN connectivity and coverage of the AOA, MFO, SCA, GWO, WOA, HHO, and the proposed HHAOA, respectively. In these, SCA has the maximum number of disconnected clients and more number of overlapped router placements. It has a 150% less efficient connectivity than the HHO, which secured second place, with four unconnected clients in its optimized WMN. Additionally, the AOA has eight disconnected clients, followed by MFO, WOA, and GWO, with seven, five, and four disconnected clients in the WMN network. However, Figure 9h shows that the proposed HHAOA has dramatically improved the client coverage and optimally deployed the mesh routers to achieve full network connectivity with the least number of routers and had only one disconnected client. Likewise, the proposed HHAOA significantly reduced network congestion by reducing the number of mesh routers by 31.7% while maintaining high coverage and connectivity. Furthermore, other comparing algorithms produced network topologies where mesh routers overlapped, resulting in more significant interference. In contrast, the topology formed by the HHAOA approach was more widely dispersed, resulting in improved client coverage.  
Figure 8: WMN convergence for different algorithms.