## 4 Statistical Testing of ENMRA Algorithm  
The 100-Digit Challenge [38], a set of 10 CEC 2019 benchmark functions, was nominated for ENMRA evaluation. All the test functions are scalable. Comparison with FPA [4], DA [25], SSA [26], CS [27], FDO [28], GWO [10], ASSA [32], CV 1.0 [34], NMRA [5], and NMRV 3.0 [35] was performed to test the efficacy of ENMRA. The ENMRA and competitive algorithms were coded using MATLAB version R2020a. All compared methods were run under the same configuration, an Intel(r) Core(tm), 2.4 GHz (i5 CPU) with 4 GB memory to ensure a fair comparison. Each algorithm underwent 500 iterations with the help of 30 search agents. The parameter settings of competitive algorithms are given in Table 1. The findings are reported in terms of the performance metric such as best, worst, mean, and standard deviation values for 30 independent runs for each algorithm under consideration. Except in CEC02, CEC06-07, and CEC09-10 (see Table 2), ENMRA outperformed competitive algorithms. The box plots and convergence graphs of various methods for various test functions are shown in Figure 2.  
ENMRA's results for CEC01 function were better in terms of the best, median, worst, and mean values obtained. For CEC02 function, ENMRA, CS, and GWO identified an optimal solution, but FDO was found to be the best. Most algorithms produced a nearly globally optimal solution for the CEC03, but ENMRA was proven to be better than other methods in terms of least standard deviation. In the case of CEC04 and CEC05, ENMRA outperformed the competition in terms of fitness performance. When compared to competitive algorithms, ENMRA's results for CEC06 and CEC07 functions were the worst. In terms of the best-acquired fitness performance, ENMRA's results for function CEC08 were better. For CEC09 and CEC10 functions, ENMRA's findings were comparable to those of other methods. The box plots of competing algorithms for the benchmark test suite are shown in Figure 2. In this case, the box plots are utilized to assess algorithm efficiency concerning the fitness value attained. It was demonstrated that the proposed ENMRA was cost-effective in terms of fitness values in the majority of cases, as the median of fitness values for ENMRA was lower. As a result, ENMRA's overall performance was determined to be consistent to that of other optimization techniques. The performance of the competitive algorithms was statistically assessed using two nonparametric statistical tests, namely the Wilcoxon rank-sum test [39] and Friedman rank (f-rank) test [40], because of the stochastic nature of the above algorithms. \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Parameters** \\ \hline FPA [3] & \(p_{s}=0.5\) \\ GWO [8] & \(A=\) linearly decreasing from 2 \(to\) 0 \\ SSA [24] & \(c_{1}=c_{2}=c_{3}=rand[0, 1]\) \\ CS [25] & _probability_ (\(p_{a}\)) = 0.25_ \\ DA [23] & _w = [0.4–0.9], \(s=0.1\), \(a=0.1\), \(c=0.7\), \(f=1\), \(e=1\)_ \\ FDO [26] & _weight factor_ (\(W\!f\!p\!=0\); \(r\) is \(rand[-1,1]\) \\ ASSA [29] & \(c_{1}=c_{2}=c_{3}=rand[0, 1]\); \(\lambda=3\) \\ CV 1.0 [31] & \(A=\) linearly decreasing from 2 \(to\) 0, \(r_{1}=r_{2}=rand[0, 1]\) \\ NMRA [4] & \(\text{bp}=0.5\); \(\lambda=rand[0, 1]\); number of breeders B = population(n)/5 \\ NMRV 3.0 [32] & \(\text{bp}=0.5\); \(\lambda=rand[0, 1]\); B = n/5 \\ ENMRA & \(\text{bp}=0.5\); \(\lambda=rand[0, 1]\); B = n/5 ; \(a=linearly decreasing from 2 to 0; \(r_{1}=r_{2}=rand[0, 1]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter settings of competitive algorithms.