## 5 Comparison of Results
### Results for the Benchmark Mathematical Functions  
Thirty experiments were performed with the proposed OFDE and OFHS methods applied to each of the eight benchmark mathematical functions shown in Section 4.1. The simulations were obtained as a result of software implemented in Matlab 2013 and 2015, from Mathworks, Los Angeles, CA, USA. The parameters used for the experiment for the OFDE method are shown in Table 10 and for the OFHS method are presented in Table 11.  
Table 10 shows the values of the population, dimension, generation, run, mutation rate (F), and crossover rate (CR) parameters used in the experiment for the proposed OFDE. Table 11 shows the values of the harmony, dimension, iteration, run, harmony memory accepting (HMR), and pitch adjustment rate (PArate) parameters used in the experiment for the proposed OFHS. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **OFDE** \\ \hline Population & 150 \\ Dimension & 50 \\ Generation & 2000 \\ Run & 30 \\ F & Dynamic \\ CR & Dynamic \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parameters used in the experiment for the OFDE method. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **OFDE** \\ \hline Harmony & 150 \\ Dimension & 50 \\ Iteration & 2000 \\ Run & 30 \\ HMR & Dynamic \\ PArate & Dynamic \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Parameters used in the experiment for the OFHS method. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Input 1** & **Input 2** \\ \hline \multirow{8}{*}{MFs Parameters} & Negative MF: & Negative MF: \\  & \(a_{0}=b_{0}=-50\) & \(a_{0}=b_{0}=-50\) \\  & \(-50<c_{0}<-15\) & \(-50<c_{0}<-5\) \\  & \(-15<d_{0}<-1\) & \(-5<d_{0}<-1\) \\ \cline{2-3}  & Zero MF: & Zero MF: \\  & \(-50<a_{1}<0\) & \(-50<a_{1}<0\) \\  & \(b_{1}=0\) & \(b_{1}=0\) \\  & \(0<c_{1}<50\) & \(0<c_{1}<50\) \\ \cline{2-3}  & Positive MF: & Positive MF: \\  & \(-1<a_{2}<5\) & \(-1<a_{2}<5\) \\  & \(0<b_{2}<50\) & \(0<b_{2}<50\) \\  & \(c_{2}=d_{2}=50\) & \(c_{2}=d_{2}=50\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Limits of the parameters of the membership functions. The results obtained by using the proposed OFDE method are shown in Table 12, and the results obtained by using the proposed OFHS method are shown in Table 13. These tables indicate the execution time of each mathematical function, where we can note lower times for OFHS. Tables 12 and 13 show the minimum and maximum obtained errors, the averages of the 30 experiments, and their standard deviations for all the mathematical functions. A statistical z-test was performed to validate the efficacy of the proposed methods. The claim is that the OFDE method has achieved lower values compared to the OFHS method. The parameters of the statistical test are as follows: Alpha 0.05, confidence level 95%, sample size is 30 and the critical value is \(-\)1,645. where  
\[\mu_{1}=\textit{mean of the OFDE algorithm}\mu_{2}=\textit{mean of the OFHS algorithm} \tag{20}\]  
The hypotheses would be as follows: \(H_{0}:\ \mu_{1}\geq\mu_{2}\) and \(H_{a}:\ \mu_{1}<\mu_{2}\ (\textit{Claim})\). We can say that there is enough statistical evidence for all values below the critical value Z or \(-\)1.645. Table 14 shows the results of the mean, standard deviation, and the Z-values obtained; in three functions, significant evidence is obtained, and in five, no significant evidence is obtained. We can indicate that the OFDE method is not better than OFHS in these mathematical benchmark functions.