## 4 Statistical Testing of ENMRA Algorithm
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Function** & **Metric** & **FPA** & **GWO** & **SSA** & **CS** & **DA** & **FDO** & **ASSA** & **CV L0** & **NMRA** & **NMRA** & **NMRA** & **NMRA** & **NMRA** \\ \hline  & Best & 2.02E + 01 & 2.02E + 01 & 2.00E + 01 & 2.02E + 01 & 2.00E + 01 & **2.34E-08** & 2.03E + 01 & 2.07E + 01 & 1.97E + 01 & 1.44E + 01 & 6.35E + 00 \\  & Median & 2.06E + 01 & 2.05E + 01 & 2.00E + 01 & 2.03E + 01 & 2.02E + 01 & **2.00E + 01** & 2.05E + 01 & 2.11E + 01 & 2.05E + 01 & 2.05E + 01 & 2.05E + 01 \\  & Worst & 2.06E + 01 & 2.06E + 01 & 2.06E + 01 & 2.04E + 01 & 2.05E + 01 & 2.02E + 01 & 2.07E + 01 & 2.15E + 01 & 2.02E + 01 & 2.06E + 01 \\ CEC10 & Mean & 2.04E + 01 & 2.05E + 01 & 2.00E + 01 & 2.03E + 01 & 2.02E + 01 & **1.93E + 01** & 2.05E + 01 & 2.11E + 01 & 2.04E + 01 & 2.02E + 01 & 2.00E + 01 \\  & Std & 9.37E 02 & 9.62E-02 & 1.05E-01 & 6.07E-02 & 1.51E+01 & 3.65E + 00 & **7.44E-02** & 2.07E-01 & 1.55E-01 & 1.23E + 00 & 2.57E + 00 \\  & p-rank & - & - & - & - & - & + & - & - & - & - & - \\  & \(f\)-rank & 7 & 9 & 3 & 6 & 5 & 1 & 10 & 11 & 8 & 4 & 2 \\
**Average \(f\)-rank** & 6.5 & 4.8 & 3.5 & 4.2 & 7.1 & 3.5 & 8.8 & 10.7 & 7.8 & 6.2 & **2.8** \\ \hline
**Overall \(f\)-rank** & 7 & 5 & 2 & 4 & 8 & 2 & 10 & 11 & 9 & 6 & **1** \\ \hline
**win(w)1 loss(0) (1 diet)** & 4 129412 & 28 127210 & 53 122512 & 21 29810 & 5 129213 & 56 124014 & 4 129313 & 8 129210 & 12 126812 & 8 1 28913 & 165 1119116 \\ \hline
**OE\%** & 2\% & 9.33\% & 18.33\% & 0.67\% & 2.67\% & 20\% & 2.33\% & 2.67\% & 4.67\% & 3.67\% & **60.33\%** \\ \hline \hline \end{tabular}   
Figure 2: _Cont_. When the outcome of the two algorithms was compared, the Wilcoxon rank-sum test was employed to get the p-rank. In this scenario, the statistical efficiency of the suggested ENMRA was demonstrated in contrast to other methods. For each CEC function, the rank of distinct algorithms is shown in the sixth row of Table 2 and is represented in terms of win(w)/loss(l)/tie(t). The win(w) condition occurred when the tested method outperformed the suggested one and was given the + sign. The performance of the algorithm was worse in the loss(l) situation than in the proposed approach, and the - sign was utilized for this scenario. Both methods under test were statistically significant in the last tie(t) condition, as indicated by the = symbol. As one can observe from the w/l/t row in Table 2, the proposed ENMRA outperformed competitive algorithms in most cases. The f-rank test additionally evaluated the suggested ENMRA's performance after the Wilcoxon rank-sum test. The f-rank test assigned a unique rank to each algorithm under comparison based on its performance, and the total f-rank was generated from all the functions as shown in Table 2. The table shows that ENMRA had the best performance and was ranked first majority of cases among all the other algorithms tested. Figure 2: Convergence graphs and boxplots for CEC 2019 benchmark functions. Table 2 (last row) displays the overall effectiveness (OE) of the ENMRA and competitor algorithms based on their complete performance to summarize the performance evaluation results. Again, Equation (14) is used to calculate the OE of each method [41], where \(L\) is the total number of losing tests and \(N\) represents the total number of tests for each algorithm. \[OE=\left(\frac{N-L}{N}\right)\times 100 \tag{14}\]  
Here, the total number of tests \(N\) is 300 as each algorithm runs 30 times for 10 CEC 2019 benchmark functions. Real-Time Application: Energy-Throughput-Efficient Cross-Layer Solution for Wireless Underground Sensor Networks (WUSNs)  
In the wide range of new applications, WUSNs [42; 43; 44] have attracted a great deal of attention that includes monitoring the soil and electricity grid, preventing/saving of mine disasters, extracting oil, predicting earthquakes and landslides, regulating border and security, and a lot more.