## 5 Numerical Experiments
### Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{_UF1 (bi-objective)_} & \multicolumn{4}{c}{_UF2 (bi-objective)_} \\ \cline{2-10} _IGD_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ \\ \hline Average & **0.002318** & 0.1144 & 0.1370 & 0.1871 & **0.002213** & 0.0582 & 0.0604 & 0.1223 \\ Median & **0.002299** & 0.113 & 0.1317 & 0.1828 & **0.0021** & 0.0577 & 0.0483 & 0.1201 \\ STD & **0.0001827** & 0.0195 & 0.0441 & 0.0507 & **0.0004524** & 0.0073 & 0.0276 & 0.0107 \\ Worst & **0.002549** & 0.1577 & 0.2278 & 0.2464 & **0.003084** & 0.0732 & 0.1305 & 0.14369 \\ Best & **0.00212** & 0.0802 & 0.0899 & 0.1265 & **0.001783** & 0.0498 & 0.0369 & 0.1048 \\ \hline \multicolumn{10}{c}{_UF3 (bi-objective)_} & \multicolumn{4}{c}{_UF4 (bi-objective)_} \\ \cline{2-10} _IGD_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ \\ \hline Average & **0.009766** & 0.2556 & 0.3139 & 0.2886 & **0.001829** & 0.0586 & 0.1363 & 0.0681 \\ Median & **0.009917** & 0.2509 & 0.3080 & 0.2892 & **0.001828** & 0.0586 & 0.1343 & 0.0684 \\ STD & **0.0004257** & 0.0807 & 0.0447 & 0.0159 & **4.0192 \(\times\) 10\({}^{-5}\)** & 0.0004 & 0.0073 & 0.0021 \\ Worst & **0.01021** & 0.3678 & 0.3773 & 0.3129 & **0.001894** & 0.0593 & 0.1518 & 0.0703 \\ Best & **0.009194** & 0.1295 & 0.2564 & 0.2634 & **0.001781** & 0.0579 & 0.1273 & 0.0646 \\ \hline \multicolumn{10}{c}{_UF5 (bi-objective)_} & \multicolumn{4}{c}{_UF6 (bi-objective)_} \\ \cline{2-10} _IGD_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ \\ \hline Average & **0.07289** & 0.7970 & 2.2023 & 1.2914 & **0.01039** & 0.2793 & 0.6475 & 0.6881 \\ Median & **0.08149** & 0.6994 & 2.1257 & 1.3376 & **0.0106** & 0.2443 & 0.5507 & 0.6984 \\ STD & **0.03034** & 0.3785 & 0.5530 & 0.1348 & **0.001497** & 0.1044 & 0.2661 & 0.0553 \\ Worst & **0.09776** & 1.7385 & 3.0383 & 1.4674 & **0.01175** & 0.5504 & 1.2428 & 0.7401 \\ Best & **0.01682** & 0.4679 & 1.4647 & 1.1230 & **0.007643** & 0.1934 & 0.3793 & 0.5524 \\ \hline \multicolumn{10}{c}{_UF7 (bi-objective)_} & \multicolumn{4}{c}{_UF8 (tri-objective)_} \\ \cline{2-10} _IGD_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ \\ \hline Average & **0.002116** & 0.1603 & 0.3539 & 0.4552 & **0.003607** & 2.0577 & 0.5367 & _NA_ \\ Median & **0.002123** & 0.0734 & 0.3873 & 0.4377 & **0.003225** & 2.3359 & 0.5364 & _NA_ \\ STD & **0.0001523** & 0.1391 & 0.2044 & 0.1898 & **0.0008838** & 1.1455 & 0.1825 & _NA_ \\ Worst & **0.002331** & 0.4014 & 0.6151 & 0.677 & **0.005344** & 3.8789 & 0.7964 & _NA_ \\ Best & **0.001889** & 0.0628 & 0.054 & 0.029 & **0.003021** & 0.4613 & 0.2453 & _NA_ \\ \hline \multicolumn{10}{c}{_UF9 (tri-objective)_} & \multicolumn{4}{c}{_UF10 (tri-objective)_} \\ \cline{2-10} _IGD_ & _MOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ & _MOWOWOATS_ & _MOGWO_ & _MOPSO_ & _MOEA/D_ \\ \hline Average & **0.001473** & 0.1917 & 0.4885 & _NA_ & **0.005116** & 3.5945 & 1.6372 & _NA_ \\ Median & **0.001406** & 0.166 & 0.4145 & _NA_ & **0.005558** & 2.8255 & 1.5916 & _NA_ \\ STD & **0.0002609** & 0.0925 & 0.1445 & _NA_ & **0.0008621** & 3.4882 & 0.2988 & _NA_ \\ Worst & **0.001814** & 0.4479 & 0.7221 & _NA_ & **0.005628** & 12.9564 & 2.1622 & _NA_ \\ Best & **0.001133** & 0.1291 & 0.3336 & _NA_ & **0.003464** & 1.0431 \(\times\) 4 & 1.22008 & _NA_ \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _IGD_ statistical results for the _CEC2009_ test functions _UF1_ to _UF10_. Next, a comparison among _MOWOATS_, the cultural _MOQPSO_, and other modified _EAs_[24] is presented in Tables 3 and 4 for the following benchmark test functions: bi-objective _ZDT_, tri-objective _DTLZ_, and _CEC2009_ test problems. Tables 3 and 4 report results computed according to _IGD_ Equation (26) [24]. Again, the obtained results in Table 3 show that _MOWOATS_ outperforms the other algorithms for all test functions except _UF5_, where it reaches only the third best result, just behind _MTS_ and _GDE3_.