## 3 Computational Experiments Comparing MOMmCAA to Other Algorithms
### DLTZ Instances  
Tables 2-4 present the results of the metric values obtained by algorithms.  
As shown in Table 2, the MOMmCAA obtains significantly better \(HV\) values than MOLAPO, GS, MOPSO, NSGA-II, and MNMA for four, four, one, five, and seven out of the seven instances, respectively. Regarding the overall mean rankings, the MOMmCAA obtains the second optimal mean rank value, below MOPSO, followed by GS, NSGA-II, MOLAPO, and MNMA. The MOMmCAA has poor performance on the DLTZ1 and DLTZ3 test instances. In summary, the MOMmCAA is superior to the other four MOEAs on this metric. Table 3 shows that the MOMmCAA achieves seven, seven, six, six, and seven better \(C\) metric values than MOLAPO, GS, MOPSO, NSGA-II, and MNMA, respectively. This indicates the quality of the solutions obtained by the MOMmCAA. Table 4 summarizes the overall performance of six algorithms in terms of EI metric values. The MOMmCAA yields significantly better \(EI\) values than MOLAPO, GS, MOPSO, NSGA-II, and MNMA for six, four, one, seven, and five out of the seven instances, respectively. Overall, the EI statistics are similar to those for HV. Figure 4 plots the representative PFs obtained by the six comparison MOEAs. In summary, the MOMmCAA shows competitive performance on the DLTZ benchmark.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Pa** & **MOLAPO** & **CS** & **MOPSO** & **NSCGA-II** & **MNMA** & **MOMmCAA** \\ \hline DLZ1 & 0.9952 (0.0006) [3]- & 9.9966 (0.0033) [2]+ & 0.9996 (0.0019) [1]+ & 0.8791 (0.0095) [6] & -0.9955 (0.0140) [1] & -0.9974 (0.0095) [4] \\ DLTZ2 & 0.9986 (0.0006) [3]- & 9.971 (0.0089) [1]+ & 0.9988 (0.0009) [1]+ & 0.9991 (0.0076) [1] & -0.9869 (0.0039) [2] \\ DLTZ3 & 0.9988 (0.0006) [3]- & 0.9791 (0.0089) [1]+ & 0.9988 (0.0009) [1]+ & 0.9991 (0.0076) [1] & -0.9869 (0.0039) [2] \\ DLTZ4 & 0.8158 (0.0084) [3]- & 0.957 (0.0099) [1]+ & 0.9988 (0.0019) [1]+ & 0.9991 (0.0076) [1] & -0.9986 (0.0039) [2] \\ DLTZ5 & 0.9988 (0.0024) [4]- & 0.9702 (0.0229) [2]+ & 0.9998 (0.0020) [1]+ & 0.9993 (0.0076) [1] & -0.9991 (0.0076) [1] & -0.9994 (0.0039) [2] \\ DLTZ6 & 0.8839 (0.1128) [4]- & 0.9947 (0.0229) [2]+ & 0.9997 (0.0090) [1]+ & 0.9750 (0.1544) [5] & -0.9780 (0.1501) [6] & -0.9843 (0.1170) [3] \\ DLTZ8 & 0.8790 (0.0006) [3]- & 0.972 (0.9 \(\times\) 10) [6]- & 0.9911 (0.0033) [1]+ & 0.9997 (0.0076) [1] & -0.9994 (0.0146) [4] & -0.9994 (0.0039) [2] \\ DLTZ9 & 0.8797 (0.0006) [3]- & 0.8499 (0.0046) [3]+ & 0.9995 (0.0070) [2]+ & 0.9998 (0.0009) [1]+ & 0.9998 (0.0092) [4] & -0.9994 (0.0030) [3] \\ \hline Mean rank & 4.14 & 3.71 & 1.57 & 4.00 & 4.05 & 4.85 & 2.71 \\ \(*/-\gamma*\) & 2.4/1 & 2/4/1 & 4/1/2 & 0/5/2 & 0/7/7 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistics (mean (std. dev.)) of Hypervolume metric values of the final populations obtained by MOLAPO, GS, MOPSO, NSGA-II, MNMA, and MOMmCAA over 30 independent runs on the DLTZ benchmark. The ranking of each algorithm is in braces.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Pa** & **MOLAPO** & **CS** & **MOPSO** & **NSCGA-II** & **MNMA** & **MOMmCAA** \\ \hline DLZ1 & 0.9982 (0.0019) [3]- & 0.9996 (0.0033) [2]+ & 0.9996 (0.0157) [2]+ & 0.8881 (0.0027) [6] & 0.9597 (0.0121) [6] & -0.9887 (0.0205) [4] \\ DLTZ2 & 0.9986 (0.0006) [3]+ & 0.9709 (0.002 Figure 4: Representative \(PF\)s obtained by the seven MOEAs on the DLTZ benchmark.