## 4 Experimental Results
### Generality Analysis  
In order to perform a cross-domain evaluation of algorithms, we followed [77] and generated the average and aligned Friedman ranking considering both Hypervolume and IGD+ values. These rankings consider which position each algorithm takes on each problem. We also concatenated Hypervolume (Table 3) and IGD+ (Table 5) tables, both with 18 lines of data, in order to create a new table of mixed quality indicators with 36 lines of data. As best Hypervolume and IGD+ values are, respectively, the highest and the lowest ones, we have used \(\overline{IGD+}=1-IGD+\) values instead.  
Table 8 presents the statistical evaluation. In this table, smaller statistical values are considered as better.  
Considering just MOEA results, SPEA2 is the best single MOEA according to almost all statistical scores. The second best is GDE3, while IBEA and mIBEA are the worst-performing MOEAs according to this cross-domain analysis, with mIBEA performing'slightly' better than IBEA.  
Considering all of the nine studied algorithms, _MOABHH is the algorithm which performs better in a cross-domain perspective_, as highlighted (in grey) in Table 8. HHRL is the second-best except considering mixed average ranking values (5 th column of the table): in this specific case, HHLA is considered the second-best algorithm.  
Another interesting result is that it is not the case that an HH always gets better results than the LLHs that it is composed of; one can notice that, in general, SPEA2 and GDE3 got better statistical values when compared both to HHLA and HHCF.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **One Elitist** & **Two Elitist** & **Three Elitist** & **Not Elitist** \\ \hline MOABHH & P01, P03, P04, P05, P09, P12 & P02, P07, P08, P10, P11, P13, P14 & P15, P16 & P06, P17, P18 \\ HHLA & P01, P03, P07, P08, P09, P11, P12, P13, P17 & P04, P05, P10 & P14, P16, P18 & P02, P06, P15 \\ HHCF & P01, P06, P11, P12, P16, P17, P18 & P04, P07 & P05, P08, P10, P15 & P02, P03, P09, P13, P14 \\ HHRL & P01, P08, P11, P12, P13 & P04, P05, P06, P07, P09 & P02, P03, P10, P14 & P15, P16, P17, P18 \\ \hline \hline \end{tabular}
\end{table}
Table 7: How elitist HHs are on selection LLHs.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\
**Algorithm** & **Hypervolume** & **Hypervolume** & **IGD+** & **IGD+** & **Mixed** & **Mixed** \\  & **Rank** & **Aligned Rank** & **Rank** & **Aligned Rank** & **Rank** & **Aligned Rank** \\ \hline GDE3 & 4.22 & 69.72 & 4.88 & 77.05 & 4.55 & 145.08 \\ IBEA & 6.61 & 109.33 & 5.78 & 97.61 & 6.19 & 208.22 \\ NSGAII & 4.5 & 83.56 & 5.22 & 78.27 & 4.86 & 160.69 \\ SPEA2 & 4.17 & 69.94 & 4.56 & 72.27 & 4.36 & 142.11 \\ mIBEA & 6.72 & 108.72 & 5.44 & 86.22 & 6.08 & 196.77 \\ MOABHH & 3.77 & 60.72 & 4.05 & 73.34 & 3.91 & 131.00 \\ HHLA & 4.94 & 78.44 & 4.88 & 81.34 & 4.25 & 156.83 \\ HHRL & 4.11 & 66.44 & 4.38 & 78.34 & 5.86 & 141.75 \\ HHCF & 5.94 & 86.55 & 5.77 & 89.05 & 5.47 & 180.02 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Friedman Ranking and Aligned Friedman Rank of the algorithms for Hypervolume and IGD+. Highlighted values are the best values among all nine algorithms.