## 3 Optimization Methodology
### Experiments and Statistical Analysis  
This study analyzed various combinations of DE algorithm parameters, which resulted in sixteen versions of DE. The different versions of DE were named according to the following code DE\(xy\), where \(x\) represents the variation of mutation parameter and \(y\) the combination of the \(F\) and \(CR\) parameters. Table 3 shows the parameters investigated in this work, and the versions of the DE algorithm with the DE\(xy\) encode. According to Table 3, the algorithm version with the name DE14 means that the parameters of this version were: _rand/1/bin_, \(CR=0.9\), and \(F=1.0\).  
The experiments consist of thirty runs of each algorithm to achieve the dimensionless maximum excess of temperature \(n\) times minimized (\(\theta_{max}\))\({}_{hm}\) for each optimization level, as well as the statistical analysis of results to identify the better versions of the DE. The statistical analysis starts with the use of the Shapiro-Wilk test to determine the population distribution, Shapiro and [PERSON] [28]. If the test validates the free distribution, then the non-parametric test of Kruskal-Wallis (KW) [29] is performed to identify if exists a significant difference between the algorithms' results. The KW test verifies the null hypotheses \(h_{0}\) represented by:  
\[\begin{array}{l}h_{0}=\mu_{1}=\mu_{2}=\ldots=\mu_{a}\\ h_{1}=\mu_{i}\ \
eq\ \mu_{a}\end{array} \tag{28}\]  
If the \(h_{0}\) is accepted, there is evidence that the algorithms results come from the same population. Otherwise, the alternative hypotheses \(h_{1}\) is accepted and there is at least one algorithm whose results are different from the other.  
The KW test helps to find a difference in one of the algorithms evaluated. However, it is not enough to know which one of the versions are different. For this purpose, the post hoc test of Dunn is applied to identify differences between the DE algorithm versions. Dunn test implements multiple comparisons based on rank sums [30]. The significance level used for all statistical test was \(\alpha_{\rm s}=0.05\). All tests were performed in the Software-R environment [27].  
The Dunn test uses the following equation to determine the difference between the analyzed factors (algorithm):  
\[\begin{array}{l}\big{|}\ R-R^{\prime}\big{|}\ \geq\ Z_{\alpha^{\prime}}\ \Big{[}\frac{N(N+1)}{12}\Big{]}^{1/2}\times\Big{(}\frac{1}{n}+\frac{1}{n} \Big{)}^{1/2}\end{array} \tag{29}\]  
where \(R\) and \(R^{\prime}\) are the mean of ranks of each factor (algorithm), \(N\) is the total number of observations, \(n\) and \(n^{\prime}\) are the numbers of observations for each of the algorithm version. The \(\alpha\)'represents \(\alpha_{\rm s}\) (\(a-1\)), where \(\alpha_{\rm s}\) is the significance level and \(a\) is the number of factors (algorithm versions). \(Z\) is the normal distribution. The right term of the equation is the minimal significance difference between the mean of ranks of the factors for a given \(\alpha_{\rm s}\). If the absolute difference between \(R\) and \(R^{\prime}\) is greater or equal than the minimal significance  
\begin{table}
\begin{tabular}{c c c} \hline
**DExy Encoded Name** & **Crossover** & **Amplification Factor** \\ \hline DE\(x1\) & 0.7 & 1.5 \\ DE\(x2\) & 0.7 & 1 \\ DE\(x3\) & 0.9 & 1.5 \\ DE\(x4\) & 0.9 & 1 \\ \hline \multicolumn{4}{c}{**Mutation Operator**} \\ \hline DE\(1y\) & & _rand/1/bin_ \\ DE\(2y\) & & _best/1/bin_ \\ DE\(3y\) & & _best/2/bin_ \\ DE\(4y\) & & _current-to-best/1/bin_ \\ \hline \end{tabular}
\end{table}
Table 3: Parameters combinations and versions of DE algorithm.  
difference, right-hand term of the Equation (29), then the \(h_{0}\) is rejected, and the compared factors (algorithms) have statistically significant difference.