## 6 Experimental Results
### Comparative study of the mean values:
#### Root-mean-square error or _RMSE_:  
Another general observation can be obtained if the root-mean-square error or \(RMSE\) is used as shown in Equation (36). The \(RMSE\) measures the average of the squared errors, that is, the difference between the estimator and what is estimated. In this case, the estimator considered is the average of the fitness of the reference function, determined by the optimization algorithm, and what is estimated is the value of the true optimum of the reference function. In Equation (36), \(MH\) is any of the metaheuristics indicated, i.e., VLE, PSO, GSA, DE or WOA, and \(F\) is the number of data considered. Table 12 shows the \(RMSE\) values calculated for the 15 classical functions and the five algorithms considered in this study. Table 12 also presents the \(RMSE\) values calculated for VLE and DE, excluding functions \(f_{14}\) and \(f_{15}\) due to lack of information for DE. With these results, it can be stated that, in general, the average fitness obtained by VLE is better adjusted to the true optimum of the reference functions than any of the published data sets for PSO, GSA, DE, and WOA. Considering the 15 classical functions, the \(RMSE\) value for VLE is 22.388, for PSO it is 1995.7, for GSA it is 2527.7, and for WOA 1933.6. Excluding functions f14 and f15, the \(RMSE\) values for VLE, PSO, GSA, DE and WOA are 24.048, 2143.7, 2715.2, 413.53 and 2077.0, respectively. According to this statistic metric, the \(RMSE\) obtained by VLE under these circumstances is lower than that obtained by the other algorithms.  
\[RMSE=\sqrt{\frac{(Avg_{MH}-Opt)^{2}}{F}} \tag{36}\]  
Similarly, Table 13 shows the statistical results obtained with the second set of optimization problems, i.e., the composite functions, which are the following: the benchmark function (BenFun), the average value (Avg), the standard deviation (StdDev), the median (Med), the minimum value achieved (Min), the optimal value published (Opt), the difference between the minimum value achieved (Min) and the optimal value published (\(DMO\)), the true percentage deviation (\(TPD\), %), and the maximum value reached (Max). Figure 19 shows the convergence graphics obtained by VLE for these benchmark functions. The number of iterations was fixed at 1000, 3000, 5000 and 10000 for these functions. Figure 20 shows the distribution graphics obtained by VLE for the composite functions chosen. All experiments were repeated at least 31 times to guarantee a meaningful statistical analysis.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**BenFun** & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **Rank** & **Subtotal** \\ \hline \(f_{1}\) & WOA & GSA & DE & VLE & PSO & 4 & \\ \(f_{2}\) & WOA & DE & VLE & PSO & GSA & 3 & \\ \(f_{3}\) & DE & WOA & VLE & PSO & GSA & 3 & \\ \(f_{4}\) & DE & WOA & GSA & VLE & PSO & 4 & 14 \\ \(f_{5}\) & VLE & DE & WOA & PSO & GSA & 1 & \\ \(f_{6}\) & WOA & GSA & VLE & PSO & DE & 3 & \\ \(f_{7}\) & DE & GSA & PSO & VLE & WOA & 4 & \\ \(f_{8}\) & DE & WOA & PSO & VLE & GSA & 4 & \\ \(f_{9}\) & DE & PSO & VLE & WOA & GSA & 3 & 15 \\ \(f_{10}\) & VLE & DE & WOA & PSO & GSA & 1 & \\ \(f_{11}\) & PSO,GSA,DE,WOA & VLE & & & & 2 & \\ \(f_{12}\) & PSO,GSA,DE & WOA & VLE & & & 3 & \\ \(f_{13}\) & PSO,GSA,DE,WOA & VLE & & & & 2 & \\ \(f_{14}\) & VLE,PSO,GSA,WOA & & & & & 1 & 10 \\ \(f_{15}\) & VLE & GSA & PSO & WOA & & & 39 \\  & & & & & \(f_{1}\)-\(f_{15}\): & \(39/15=2.6\) \\  & & & & & \(f_{1}\)-\(f_{4}\): & \(14/4=3.5\) \\  & & & & & \(f_{5}\)-\(f_{9}\): & \(15/5=3.0\) \\  & & & & & \(f_{10}\)-\(f_{15}\): & \(10/5=2.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 11: Ranking of the optimization results (average fitness) obtained applying VLE, PSO,GSA,DE and WOA to the classical benchmark functions considered; \(n=30\).  
Figure 19: Convergence graphics for composite functions tested, Table 6.  
Figure 20: Distribution graphics for composite functions tested, Table 6.