## 8 Results and Discussion
### Generational Distance (GD)  
The Generational Distance (GD) metric [15] calculates the distance related to the convergence between the Pareto front solutions of the current instance (\(PF_{known}\)) with respect to the actual Pareto \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline
**Ranking** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** & **17** & **18** & **19** & **20** \\ \hline
**Requirement** & 15 & 10 & 11 & 14 & 4 & 7 & 13 & 20 & 6 & 18 & 2 & 9 & 12 & 16 & 19 & 8 & 5 & 1 & 17 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ranking generated for the problem File 3. Figure 6: Graph of dominance for the File 3.  
front (\(PF_{true}\)) [45]. For each solution of \(PF_{known}\) is calculated the smallest distance with respect to the elements of \(PF_{true}\), defined by:  
\[d_{i}=\min_{j}(\left\|f_{true}^{i}-f_{known}^{i}\right\|)_{2}\]  
In this metric, \(d_{i}\) is multiplied by \(e_{i}^{GER}\), defining a weighting for the distance. The normalization is obtained by the application \(\left(d_{i}-\underline{f_{j}}\right)/\left(\overline{f_{j}}-\underline{f_{j }}\right)\). The metric is defined by:  
\[NDWD_{GER}=\sum_{i=1}^{N}e_{i}^{GER}\left(\frac{d_{i}-\underline{f_{j}}}{ \overline{f_{j}}-\underline{f_{j}}}\right)\]  
It is preferable for an algorithm to reach a low value for this metric. In Table 5, we compare the results obtained in [13; 14] with those obtained in this work. The lower the value presented, the closer the obtained solution to the actual Pareto front. A value of 0 means that the solution is already part of the real Pareto front. This occurs with the results from File 2. High values indicate that the solution is away from the Pareto front and, therefore, is not a good solution. Due to the high disparity and the difference of scales between the results obtained concerning the Pareto front, a normalization of the values of the solutions was applied to match the results. As shown in Table 5, the low values expressed by the GD metric demonstrate that the solutions obtained by ZAPROS III-\(i\) in this work surpassed the results found in [13; 14] in all the files, except File 2. In the previous results, the File 2 solution was already part of the real Pareto front and, hence, could not be further improved. In addition, we can see in Figures 7-10, the result of the four experiments concerning Files 1, 2, 3 and 4 (summarized in Table 1), respectively. The graphical difference between the results obtained in this and previous research is relatively small but, nonetheless, quite promising. The figures show the executions of NSGA-II, SPEA2, MOCell, ZAPROS III-\(i\), and the random search algorithm. Many methods based on metaheuristics are used to solve this type of problem, with quite quite similar solutions. The result obtained from ZAPROS III-\(i\) is represented by a single black dot in the graphs. This point represents the only solution available to the decision-maker from the ARNAAU tool. The other algorithms do not provide a single solution, but a set of solutions. The superiority of the results of metaheuristics relative to those pointed out by the decision-maker can be proven in experiments [46]. However, although metaheuristics may present better results, they do not represent the expressed will of the decision-maker as a whole. There are several methodologies in SBSE that incorporate the desire of the decision-maker. A more consistent explanation about this method of selection is given in [42]. The black dots shown in Figures 7-10 show that these are the best choices for the decision-maker, or the best solutions to the problem given the decision-maker's particularities and experience. In general, when we compare these results with those of [13; 14], we notice that there was an improvement in three of the four solutions tested by the ZAPROS III-\(i\) method.