## 5 Experimental Results
### Performance
#### 5.2.2 Fraction of Dominated Solutions  
We calculate the percentage of dominated solutions as follows. For a given pair of algorithms, _algorithm1_ and _algorithm2_, we calculate how many solutions of _algorithm2_ (_dominated algorithm_) are dominated by solutions of _algorithm1_ (_dominating algorithm_). After that, we average the obtained results among all 30 independent runs. The detailed results for \(n_{k}=2\), \(n_{k}=7\), and \(n_{k}=25\) are shown in Tables 4 and 5 for random and tournament selection respectively. For example, for random selection and \(n_{k}=2\), on average, the solutions of _NSGA-II_ dominate 47.38% of the solutions produced by _PO-prob_*, see the third row and sixth column in Table 4. At the same time, on average, only 6.24% of solutions produced by _NSGA-II_ are dominated by the solutions of _PO-prob_*, see the seventh row and second column of the same table. Naturally, a better algorithm has a lower number of dominated solutions and a larger number of dominating solutions. This means that we want to achieve minimum per column and maximum per row. The rows in bold show the average number of dominated solutions for all dominating algorithms, denoted by \(\theta\). The value of \(\theta\) is calculated as a mean per column, and better algorithms have lower values of \(\theta\). For example, for the same setup, on average, 21.77% of solutions produced by _NSGA-II_ are dominated by other algorithms. The corresponding value for _PO-prob_* is 36.86%. This means that for this configuration, _NSGA-II_ performs better than _PO-prob_*. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{3}{c}{Dominated algorithm, \(n_{k}=2\)} \\  & _NSGA-II_ & _NSGA-III_ & _PO-count_ & _PO-prob_ & _PO-prob_* \\ _NSGA-II_ & & 44.24 & 39.21 & 13.52 & 47.38 \\ _NSGA-III_ & 37.50 & & 34.51 & 13.52 & 44.76 \\ _PO-count_ & 43.34 & 45.01 & & 13.52 & 46.94 \\ _PO-prob_ & 0.00 & 0.00 & 0.00 & & 8.37 \\ _PO-prob_* & 6.24 & 6.64 & 5.02 & 23.46 & \\ mean, \(\theta\) & 21.77 & 23.97 & 19.60 & 16.01 & 36.86 \\ \hline  & & \multicolumn{3}{c}{Dominated algorithm, \(n_{k}=7\)} \\  & _NSGA-II_ & _NSGA-III_ & _PO-count_ & _PO-prob_ & _PO-prob_* \\ _NSGA-II_ & & 0.00 & 8.57 & 0.00 & 0.04 \\ _NSGA-III_ & 39.72 & & 40.51 & 0.08 & 19.00 \\ _PO-count_ & 9.99 & 0.00 & & 0.00 & 0.05 \\ _PO-prob_ & 66.73 & 27.51 & 66.72 & & 48.65 \\ _PO-prob_* & 61.52 & 1.41 & 63.31 & 0.07 & \\ mean, \(\theta\) & 44.49 & 7.23 & 44.78 & 0.04 & 16.94 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Percentage of dominated solutions for random selection, %. **Results for \(n_{k}=2\) with random selection. ** Analyzing the results from Table 4, we can see that for two knapsacks and random selection, _PO-prob_ is dominated the least number of times. _NSGA-II_, _NSGA-III_, and _PO-count_, on average, dominate no more than 13.52% of the solutions of _PO-prob_. _PO-prob\({}^{*}\)_, however, dominates on average 23.46% of the solutions of _PO-prob_. At the same time, _PO-prob_ almost never dominates other algorithms. The only corresponding non-zero value is 8.37%, which represents the fraction of solutions of _PO-prob\({}^{*}\)_ dominated by _PO-prob_. This means that solution spaces of _PO-prob_ and other algorithms are distinct and do not intersect much. This supports our previous observation, see Figure 4.  
_PO-prob\({}^{*}\)_ is dominated the most often by others, with more than 40% of solutions being dominated by _NSGA-II_, _NSGA-III_, and _PO-count_. _NSGA-II_, _NSGA-III_, and _PO-count_ form a group of algorithms that are not usually dominated by the probability-based algorithms, _PO-prob_ and _PO-prob\({}^{*}\)_, but have high domination values among themselves. Among these three algorithms, _PO-count_ seems to provide the best configuration: it is dominated the least number of times by _NSGA-II_ and _NSGA-III_; the corresponding values are 39.21% and 34.51%, respectively. At the same time, _PO-count_ dominates 43.34% of the solutions of _NSGA-II_ and 45.01% of the solutions of _NSGA-III_.