## 3 \(k\)-Pareto Optimality-Based Sorting for Genetic Optimization
### Illustrative Example
Considering the second objective, only point \(A\) dominates \(B\). This gives us \(P\_dom_{f_{2}}(B)=1/6\approx 0.167\). Thus, the value of _PO-prob_ for point \(B\) is \(P\_dom_{f_{1}}(B)P\_dom_{f_{2}}(B)\approx 0.5\ast 0.167\approx 0.083\). The corresponding values for the rest of the points and the distribution of the points between fronts are presented in the sixth and the fifth columns of Table 1 respectively. As we can see, now the 6 points are distributed among 4 fronts, with \(B\) singled out into a separate front. A possible drawback of calculating _PO-prob_ as explained above can be multiplication by 0 when a point is non-dominated with respect to at least one objective. In this case, the value of _PO-prob_ is always equal to 0 no matter what the domination probability with respect to other objectives is. An easy solution to this problem is to replace the value 0 with  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Front**} & \multirow{2}{*}{**PD**} & \multicolumn{2}{c}{**PO-_count_**} & \multicolumn{2}{c}{_PO-prob_} & \multicolumn{2}{c}{_PO-prob_, \(\epsilon=0.1\)} \\ \cline{3-8}  & & **Points** & **Val. ** & **Points** & **Val. ** & **Points** & **Val. ** \\ \hline
1 & \(A\), \(B\), \(C\) & \(A\), \(B\), \(C\) & 0 & \(A\), \(C\) & 0.00 & \(C\) & 0.05 \\
2 & \(D\), \(E\) & \(E\) & 1 & \(B\) & 0.08 & \(A\) & 0.07 \\
3 & \(F\) & \(D\), \(F\) & 2 & \(E\) & 0.11 & \(B\) & 0a small number \(\epsilon\) in the calculations. The last two columns of Table 1 show the respective values of _PO-prob_ for \(\epsilon=0.1\). As we can see, now point \(C\) is more preferable to \(A\), as its domination probability with respect to \(f_{2}\) is less then the domination probability of \(A\) with respect to \(f_{1}\). Indeed, \(P\_dom_{f_{2}}(C)=3/6=0.5\) and \(P\_dom_{f_{1}}(A)=4/6\approx 0.667\). At the same time, \(P\_dom_{f_{2}}(C)<P\_dom_{f_{1}}(A)\) and \(P\_dom_{f_{1}}(C)=P\_dom_{f_{2}}(A)=0\). In the rest of the text and in our implementation, we use this version of calculating _PO-prob_ with \(\epsilon=\frac{1}{pop\_size}\), where _pop_size_ is the population size or the number of analyzed solutions. We refer to it simply as _PO-prob_. If we assume that there are no identical solutions, then _PO-prob_ is an estimation of the probability a randomly chosen point _non_-strictly dominates the analyzed one, see [14]. In the case of counting measure, sorting by _PO_ is equivalent to sorting by dominance rank relationship [33]. In addition, it was shown in previous works that sorting a random sample of independent continuously distributed points by _PD_ and _PO-count_ has the same limiting behaviour when the number of points becomes infinitely large [34]. Although _PO-count_ was already studied in the literature, _PO-prob_ was not. In addition, _PO-prob_ is not the only possible algorithm based on Pareto optimality, as similar approaches can be developed for different measures and relations expressing preference. That is why we present evaluation results for both _PO-prob_ and _PO-count_ for comparison and more detailed analysis. In the following sections, we evaluate how _PO_-based ranking affects the convergence properties of genetic optimization algorithms.