## 5 Experimental Results
### Characteristics of the Proposed Approach
#### 5.1.1 Evolution of Solutions for \(n_{k}=2\)  
For a random initialization, we plot the evolution of solutions and the first front of the last generation for \(n_{k}=2\) in Figure 4 (random selection) and Figure 5 (tournament selection). We choose to use the same initialization for all algorithms to show the difference in the resulting solutions. _NSGA-II_ and _NSGA-III_ are well-studied, so we concentrate more on the behavior of the algorithms based on \(PO\). Figure 4: _Cont._  
We start with the analysis of the results for **random selection**. Figure 4c,d show three generations of solutions for _PO-count_ and _PO-prob:__gen_\(=30\), _gen_\(=150\), and _gen_\(=300\). As _PO-prob\({}^{*}\)_ performs identically to _PO-prob_ until _gen_ reaches \(350\), its intermediate solutions are not plotted. Figure 4a,b show the first front of the final generation _gen_\(=500\) for all algorithms. We can notice interesting behavior of the _PO-prob_ algorithm: after a certain number of generations, the produced solutions tend to move to extreme values with very few solutions concentrated in the middle part of the Pareto-frontier, see Figure 4d for _gen_\(=300\). This pattern was observed in all \(30\) independent runs. It can be explained by the nature  
Figure 4: Visualization of solutions for \(n_{k}=2\) and random selection. \(y\)-axis is shared among plots. (**a**) First front for _gen_\(=500\), counting algorithms; (**b**) First front for _gen_\(=500\), probabilistic algorithms; (**c**) Evolution of solutions for _PO-count_; (**d**) Evolution of solutions for _PO-prob_. Figure 5: Visualization of solutions for \(n_{k}=2\) and tournament selection. \(y\)-axis is shared among plots. (**a**) First front for _gen_\(=500\), counting algorithms; (**b**) First front for _gen_\(=500\), probabilistic algorithms; (**c**) Evolution of solutions for _PO-count_; (**d**) Evolution of solutions for _PO-prob_. of the probabilistic sorting used in _PO-prob_ and by the fact that probabilities are multiplied. Indeed, a solution will be ranked high if its probability to be dominated by any other solution is low. In this case, the extreme solutions have little probability to be dominated with respect to one of the objectives. This contributes to their higher rankings in comparison to the solutions from the middle of the Pareto-frontier. Thereby, such solutions are advanced to the next generations and have higher probability to pass their characteristics to offspring. At the same time, in the case of _NSGA-II_, all solutions from the first non-dominated front with similar values of crowding distance are equally preferable. This observation is consistent with the computational example presented in Section 3. As we can see from Table 1, the extreme solutions \(A\) and \(B\) are more preferable than the solution from the center, \(C\), for _PO - prob_-based sorting. This tendency is, however, \"repaired\" by _PO-prob_*. The solutions of the latter first reach the extreme values using sorting by _PO-prob_ and then cover the whole length of the Pareto-frontier using _PD_ sorting from _NSGA-II_. We can also see that continuing evolution with _PO-prob_ allows it to reach even more extreme solutions during the last 150 generations. The solutions in the middle of the Pareto frontier produced by _PO-prob_ are clearly dominated by the solutions of other algorithms. At the same time, only _PO-prob_* can reach extreme solutions almost as well as _PO-prob_. Overall, _NSGA-II_ covers the middle part of the Pareto frontier better than all other algorithms. _NSGA-III_ and _PO-count_ demonstrate behavior similar to _NSGA-II_. In the case of **tournament selection**, _PO-prob_ behaves similarly to other algorithms and produces non-fragmented fronts, see Figure 5. At the same time, tournament selection results in worse coverage of the Pareto-frontier than random selection, compare Figures 4 and 5. The maximum achieved values of objective 2 with tournament selection are around \(0.92\times 10^{4}\).