## 5 Numerical Experiments
### Results and Discussion
However, the results are far better for the rest test functions, thus giving strong support to the claim that our algorithm works better for most classes of _MOPs_. Likewise, results in Table 4 compare _MOWOATS_ to the cultural _MOQPSO_, and to some _EAs_ according to the _IGD_ metric over bi-objective _ZDT_[20], and tri-objective _DTLZ_[20] test functions. For these benchmarks, our algorithm has always produced better _IGD_ indices than the others for every test function. Figure 2 presents the distribution of the obtained solutions from _MOWOATS_ with respect to the true _PF_ points for some _DTLZ_ test functions. The figure shows the symmetry of the obtained solutions with respect to the true _PF_ solutions. It also presents the diversity of the obtained solutions over the whole solution space. This corroborates the low _IGD_ values obtained by _MOWOATS_. Figure 3 presents a depiction of the solutions obtained from _MOWOATS_ with respect to the true _PF_ solutions for some _ZDT_ test functions. The figure shows the symmetry of the obtained solutions with respect to the true _PF_ solutions, too, and it also clarifies the coverage of obtained solutions across the whole solution space. Figure 2: A depiction of the generated Pareto Optimal solutions for some _DTLZ_ test functions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & _FI(2 DTI)_ & _FI(2 DTI2)_ & _FI3(2 DTI3)_ & _FI(2 DTI4)_ & _FI(5 DTI6)_ & _F(DTL21)_ & _F(DTL22)_ & _F(DTL24)_ \\ \hline _Sigma MOPSO_ & \(6.07\times 10^{-1}\) & \(5.00\times 10^{-1}\) & \(2.50\times 10^{-1}\) & \(2.64\times 10^{1}\) & \(2.88\times 10^{-3}\) & \(2.23\times 10^{1}\) & \(2.84\times 10^{-1}\) & \(7.49\times 10^{-1}\) \\ _Cluster MOPSO_ & \(9.03\times 10^{-3}\) & \(9.19\times 10^{-2}\) & \(2.92\times 10^{-1}\) & \(9.89\times 10^{-3}\) & \(6.24\times 10^{-3}\) & \(1.39\times 10^{-1}\) & \(1.43\times 10^{-1}\) & \(8.93\times 10^{-1}\) \\ _MOPSO-CD_ & \(1.30\times 10^{-1}\) & \(6.60\times 10^{-3}\) & \(3.00\times 10^{-1}\) & \(1.53\times 10^{-1}\) & \(2.93\times 10^{-3}\) & \(2.80\) & \(9.86\times 10^{-2}\) & \(6.18\times 10^{-1}\) \\ _MOQPSO_ & \(2.17\times 10^{-2}\) & \(4.25\times 10^{-2}\) & \(1.87\times 10^{-1}\) & \(8.81\) & \(3.92\times 10^{-1}\) & \(1.38\times 10^{1}\) & \(6.94\times 10^{-2}\) & \(1.02\times 10^{-1}\) \\ _Perference-ender_ & \(6.94\times 10^{-3}\) & \(4.75\times 10^{-1}\) & \(1.90\times 10^{-1}\) & \(2.08\) & \(5.86\times 10^{-3}\) & \(1.39\times 10^{4}\) & \(2.29\times 10^{-1}\) & \(2.91\times 10^{-1}\) \\ _Dual GPSO_ & & & & & & & & \\ _Cultural_ & \(6.13\times 10^{-3}\) & \(4.87\times 10^{-3}\) & \(1.89\times 10^{-1}\) & \(5.26\times 10^{-3}\) & \(5.38\times 10^{-3}\) & \(4.70\times 10^{-2}\) & \(6.51\times 10^{-2}\) & \(6.70\times 10^{-2}\) \\ _MOQPSO_ & & & & & & & & \\ _MOWOATS_ & \(\mathbf{1.30\times 10^{-3}}\) & \(\mathbf{7.32\times 10^{-4}}\) & \(\mathbf{2.87\times 10^{-3}}\) & \(\mathbf{2.85\times 10^{-4}}\) & \(\mathbf{8.92\times 10^{-4}}\) & \(\mathbf{3.76\times 10^{-4}}\) & \(\mathbf{9.73\times 10^{-4}}\) & \(\mathbf{1.28\times 10^{-3}}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Mean of _IGD_ values for MOWOATS_, Cultural MOQPSO, and some population-based algorithms over _(2 DT, DTLZ)_ test problems.