## 6 Conclusions  
In this paper, we study the application of a novel sorting technique based on Pareto optimatily to genetic optimization. The proposed approach is very flexible and can be defined for various sets of elements, relations, and measures. In the case of genetic optimization, the set of elements is represented by a discrete set of points on a plane with the traditional numerical relationship \(\geq\). However, we consider two different measures: counting and probability, which result in two versions of the algorithm: _PO-count_ and _PO-prob_. The proposed ranking method was implemented in the framework of the _NSGA-II_ algorithm and was tested on the 0/1 knapsack problem with the number of objectives \(n_{k}\) ranging from 2 to 25.  
Our experimental results demonstrated that counting _PO_ performs very similar to the traditional non-dominated sorting. At the same time, probabilistic _PO_ ranking has multiple advantages. It allows the genetic algorithm to distinguish better between the solutions, which is particularly useful in the case of many-objective optimization. It also results in higher values of hypervolume both for large and small number of knapsacks as compared to _NSGA-II_ and _NSGA-III_. In the case of random selection, hypervolume is increased by 4% for 2 knapsacks and by 61% for 25 knapsacks. The respective values for tournament selection are 1% and 30%. For medium numbers of knapsacks, we observe a decrease in hypervolume, but it stays within \(-2\%\). Apart from increasing the hypervolume when the number of objectives becomes larger, _PO-prob_ also results in a set of solutions that are very rarely dominated by other algorithms. For example, for 10 knapsacks, none of the solutions of _PO-prob_ are dominated by _NSGA-II_. At the same time, _PO-prob_ dominates approximately 60% of the solutions produced by _NSGA-II_. Finally, we demonstrated both theoretically and practically the efficiency of _PO-prob_ in terms of execution time. Sorting based on probabilistic _PO_ allows decreasing the computational time from \(O(N^{2}M)\) to \(O(NMlog(N))\), where \(N\) and \(M\) stand for populations size and number of objectives, respectively. The fact that _PO-prob_ performs better than _PO-count_ indicates that the practical problem formalization might be crucial. In this case, the choice of the probability measure over the counting one improves the performance drastically.  
Although _PO-prob_-based sorting demonstrated good performance, this approach has some limitations. It was demonstrated that it introduces front fragmentation for \(n_{k}=2\) and slightly decreases hypervolume when the number of objectives is between five and eight. These problems can be partially solved by the _PO-prob_* algorithm, which is a combination of _PO-prob_ and the traditional _PD_ ranking used in the original version of _NSGA-II_. However, when the number of objectives is large, _PO-prob_ provides the best performance.  
As it was experimentally demonstrated in Section 5.1.3, sorting by _PO-prob_ results in almost no ties. In this sense, it can be considered as a linear extension of non-dominated sorting. However, compared to other popular linear extensions, such as sum, average, minimum, or maximum of objectives, _PO-prob_ has a clear **interpretation**. It represents the probability that a randomly chosen point will dominate the point under investigation, see Section 3. Thereby, the resulting algorithms are easier to manage and understand.