## 5 Experimental Results
### Performance
#### 5.2.2 Fraction of Dominated Solutions
As it was shown in Section 5.2.1, the solutions of _NSGA-III_ cover less hypervolume than the solutions of _NSGA-II_. However, as we can see now, approximately 40% of the solutions of _NSGA-II_ are dominated by the solutions of _NSGA-III_. Thereby, despite covering less hypervolume, _NSGA-III_ should be preferable in practice. A considerable fraction of solutions of _PO-prob_* is dominated by _NSGA-II_ (19.00% and 15.69%) and by _PO-prob_ (48.65% and 39.24%). At the same time, only a tiny fraction of solutions of _PO-prob_ is dominated by solutions produced by other algorithms. The largest fraction of solutions of _PO-prob_ are dominated by _NSGA-III_: 0.08% for random selection and 0.53% for tournament selection. **Results for \(n_{k}=25\). ** When the number of knapsacks increases even further, we can notice that all algorithms tend to produce more distinct sets of solutions, as the domination fractions reduce. However, the general pattern stays the same. The solutions of _NSGA-II_ and _PO-count_ are more often dominated by the solutions of other algorithms. The solutions of _PO-prob_ are almost never dominated. The next best performance is demonstrated by _NSGA-III_, with 9.59% and 3.60% of solutions dominated by _PO-prob_ for random and tournament selection, respectively. _PO-prob_* has approximately 20% of solutions dominated by _PO-prob_ and 10% of solutions dominated by _NSGA-III_. **Distribution of \(\theta\)**. To further analyze how the domination fraction changes for different numbers of knapsacks, we demonstrate the distribution of \(\theta\) for different values of \(n_{k}\) in Figure 10. Recall that \(\theta\) shows the fraction of dominated solutions averaged over different dominating algorithms, and its values are present in bold in Tables 4 and 5. We can notice similar tendencies for both random and tournament selection. _NSGA-II_ and _PO-count_ behave very similarly. For \(n_{k}=2\), the value of \(\theta\) for these algorithms is around 23%. After that, it starts increasing and reaches its peak of approximately 45% for \(n_{k}=7\). Finally, it gradually decreases to 20% for \(n_{k}=25\). _NSGA-III_ starts at a similar level. It reaches its peak of approximately 30% for \(n_{k}=5\) for random selection and \(n_{k}=3\) for tournament selection. After, it decreases below 10% for \(n_{k}=8\) and stays below this value for random selection, and relatively close to zero in the case of tournament selection. These results once again demonstrate the superiority of _NSGA-III_ over _NSGA-II_ for large numbers of objectives. _PO-prob_ starts at around 15% for random selection and 45% for tournament selection. However, the value of \(\theta\) drops to 0 very fast. This shows that the solutions produced by this algorithm are almost never dominated by any other algorithm for large numbers of knapsacks. _PO-prob\({}^{*}\)_ behaves similarly to _PO-prob_ until \(n_{k}=5\). Next, \(\theta\) starts going up and reaches its maximum at \(n_{k}=7\) for random selection and at \(n_{k}=8\) for tournament selection. For larger values of \(n_{k}\), \(\theta\) gradually decreases, but it is always larger than the corresponding value for _NSGA-III_.