## 5 Experimental Results
### Performance
#### 5.2.2 Fraction of Dominated Solutions
This corresponds to +5% and +10% compared to the inverse domination relationship. **Results for \(n_{k}=2\) with tournament selection. ** For the same number of knapsacks, \(n_{k}=2\), and tournament selection, we can see that the probability-based algorithms are \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{4}{c}{Dominated algorithm, \(n_{k}=25\)} \\  & _NSGA-II_ & _NSGA-III_ & _PO-count_ & _PO-prob_ & _PO-prob\({}^{*}\) \\ _NSGA-II_ & & 0.00 & 8.28 & 0.00 & 0.15 \\ _NSGA-III_ & 18.97 & & 18.83 & 0.00 & 8.48 \\ _PO-count_ & 6.40 & 0.00 & & 0.00 & 0.20 \\ _PO-prob\({}^{*}\)_ & 36.55 & 9.59 & 36.52 & & 18.76 \\ _PO-prob\({}^{*}\)_ & 30.13 & 0.80 & 31.89 & 0.00 & \\ mean, \(\theta\) & 23.01 & 2.60 & 23.88 & 0.00 & 6.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{4}{c}{Dominated algorithm, \(n_{k}=2\)} \\  & _NSGA-II_ & _NSGA-III_ & _PO-count_ & _PO-prob_ & _PO-prob\({}^{*}\)_ \\ _NSGA-II_ & & 43.93 & 46.67 & 47.81 & 61.20 \\ _NSGA-III_ & & 35.99 & 43.17 & 45.71 & 59.33 \\ _PO-count_ & 35.83 & 36.70 & & & 42.32 & 57.43 \\ _PO-prob\({}^{*}\)_ & 6.61 & 7.75 & 7.53 & & 31.29 \\ _PO-prob\({}^{*}\)_ & 7.78 & 9.95 & 10.41 & 42.26 & \\ mean, \(\theta\) & 21.55 & 24.58 & 26.95 & 44.52 & 52.31 \\ \hline \hline  & \multicolumn{4}{c}{Dominated algorithm, \(n_{k}=7\)} \\  & _NSGA-II_ & _NSGA-III_ & _PO-count_ & _PO-prob_ & _PO-prob\({}^{*}\)_ \\ _NSGA-II_ & & 0.00 & 11.21 & 0.00 & 0.09 \\ _NSGA-III_ & 42.92 & & 44.00 & 0.53 & 15.69 \\ _PO-count_ & 12.16 & 0.00 & & 0.01 & 0.03 \\ _PO-prob\({}^{*}\)_ & 63.32 & 14.72 & 63.73 & & 39.24 \\ _PO-prob\({}^{*}\)_ & 62.91 & 1.71 & 61.52 & 0.11 & \\ mean, \(\theta\) & 45.33 & 4.11 & 45.12 & 0.16 & 13.76 \\ \hline \hline  & \multicolumn{4}{c}{Dominated algorithm, \(n_{k}=25\)} \\  & _NSGA-II_ & _NSGA-III_ & _PO-count_ & _PO-prob_ & _PO-prob\({}^{*}\)_ \\ _NSGA-II_ & & 0.00 & 6.69 & 0.03 & 0.32 \\ _NSGA-III_ & 17.35 & & 20.83 & 0.03 & 10.60 \\ _PO-count_ & 5.57 & 0.00 & & 0.01 & 0.12 \\ _PO-prob\({}^{*}\)_ & 29.77 & 3.60 & 30.92 & & 20.40 \\ _PO-prob\({}^{*}\)_ & 25.71 & 0.44 & 27.97 & 0.23 & \\ mean, \(\theta\) & 19.60 & 1.01 & 21.60 & 0.07 & 7.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Percentage of dominated solutions for tournament selection, %. dominated more often than the counting-based algorithms, see Table 5. The average number of dominated solutions through all algorithms, \(\theta\), is 44.52% for _PO-prob_ and 52.31% for _PO-prob_*. At the same time, the values of \(\theta\) for _NSGA-II_, _NSGA-III_, and _PO-count_ stay relatively close to the corresponding values for random selection. The results for this configuration, also confirm that _NSGA-II_ performs the best: it is dominated the least often and on average dominates more solutions that other algorithms. This difference becomes especially prominent when comparing it with probability based algorithms. _NSGA-II_ dominates 47.81% of the solutions of _PO-prob_ and 61.20% of solutions of _PO-prob_*. The latter algorithms, however, dominate only 6.61% and 7.78% of the solutions of _NSGA-II_, respectively. **Results for \(n_{k}=7\)**. However, this pattern changes when the number of objectives increases. Already for \(n_{k}=7\), _NSGA-II_ and _PO-count_ are substantially outperformed by other algorithms, both for random and tournament selection. More than 60% of solutions of these two algorithms are dominated by the probability-based algorithms _PO-prob_ and _PO-prob_*. The level of domination in the inverse direction is less than 1%. It is interesting to note that _NSGA-III_ also results in a set of solutions that are rarely dominated. The only algorithm capable of dominating a significant fraction of solutions of _NSGA-III_ is _PO-prob_. The corresponding values are 27.51% for random selection and 14.72% for tournament selection. This shows the superiority of _NSGA-III_ for many-objective optimization as compared to _NSGA-II_.