**5. Conclusions**

Logical analysis of data is a two-class learning method dealing with features that can be binarized. Logical analysis of data consists of several stages, and the formation of patterns is the most important. Finding pure patterns is a single-criteria optimization problem that consists of finding a pattern that covers as many positive observations as possible and does not cover negative observations. With a more general formulation, which allows making mistakes for a pattern, the problem turns into a search for a compromise between the completeness of the range of observations of some target class and the minimization of the coverage of observations that do not belong to the target class.

In this study, we used an approach to find patterns in the form of an approximation of the Pareto-optimal front and proposed evolutionary algorithms for solving such a problem due to their potential ability to cover the front. We modified the NSGA-II for pattern searching and tested it on several application problems from the repository.

Results of our work enabled us to find more informative patterns in the data, taking into account the coverage of objects of different classes. We compared our modified algorithm with commonly used machine learning algorithms on four classification problems. The results were comparable, and in some cases better than results of classical ML algorithms which do not meet the requirement of the interpretability of the result.

Our experiments discovered a significant influence of the probability of dropping one of the control variables of the initial population in the multi-criteria genetic algorithm. This parameter affects the selectivity of the selected patterns. So, the equal probability of zero and one entails a significant proportion of baseline observations, for which only patterns are found that cover only the baseline observation and no other observation of the training sample.

Since our two-criteria optimization model in a combination with the developed modification of the genetic algorithm does not require the number or ratio of observations of the opposite class to be pre-set. Thus, our approach is a more versatile tool of data analysis in this sense than known methods for the fuzzy patterns generation. The method considered in this paper could be useful for the classification tasks in, for instance, healthcare system, faults diagnosis and any problems in which the interpretability of results are of grea<sup>t</sup> importance.

**Author Contributions:** Conceptualization, I.S.M. and M.A.K.; methodology, I.S.M. and L.A.K.; software, M.A.K.; validation, I.S.M., M.K. and E.M.T.; formal analysis, P.S.S. and L.A.K.; investigation, I.S.M.; resources, I.S.M.; data curation, I.S.M. and M.A.K.; writing—original draft preparation, I.S.M., E.M.T. and L.A.K.; writing—review and editing, L.A.K., E.M.T., A.A.S. and P.S.S.; visualization, I.S.M., M.A.K. and A.A.S.; supervision, A.M.P., L.A.K. and P.S.S.; project administration, L.A.K. and A.M.P.; funding acquisition, L.A.K. and A.M.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the Ministry of Science and Higher Education of the Russian Federation, State Contract FEFE-2020-0013.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Initial data available at UCI Machine Learning Repository: https: //archive.ics.uci.edu/ml/datasets/breast+cancer+wisconsin+(original), access date: 13 February 2022; https://archive.ics.uci.edu/ml/datasets/Myocardial+infarction+complications, access date: 13 February 2022.

**Conflicts of Interest:** The authors declare no conflict of interest.
