Forestry BulletinForestry BulletinЛесной вестник2542-1468Bauman Moscow State Technical University70511110.18698/2542-1468-2026-1-138-146Math modelingМатематическое моделированиеResearch ArticleEducational process data formalization on basis of various fuzzy setsФормализация данных образовательного процесса на основе нечетких множеств разных типовPoleshchukOl’ga M.ПолещукОльга Митрофановна
Russian Federation

Dr. Sci. (Tech.), Professor, Head of Higher Mathematics and Physics Department

д-р. техн. наук, профессор, зав. кафедрой «Высшая математика и физика»

poleshhukom@bmstu.ruBMSTU (Mytishchi branch)ФГАОУ ВО «Московский государственный технический университет имени Н.Э. Баумана (национальный исследовательский университет)» (Мытищинский филиал)270320263011381462703202627032026Copyright ©; 2026, Poleshchuk O.M.Copyright ©; 2026, Полещук О.М.2026Poleshchuk O.M.Полещук О.М.https://journals.eco-vector.com/2542-1468/article/view/705111

The paper develops models for formalizing the data of the educational process in the conditions of various initial information. For modeling, the paper uses first type fuzzy sets, interval second type fuzzy sets, and Z-numbers. All the constructed models have been analyzed and recommended for use in solving various practical problems. First type fuzzy sets are recommended to be used to formalize statistical data of the educational process, as well as data obtained from a single expert (examiner). Interval second type fuzzy sets are recommended to be used to formalize statistical and expert data of the educational process with random errors, as well as data obtained from a group of experts. Z-numbers are recommended to be used to formalize the data of the educational process, taking into account their reliability. The numerical examples given in the article, together with theoretical justifications, provide an opportunity to choose a model for further data analysis in order to obtain sustainable final results and control decisions based on them.

Разработаны модели формализации данных образовательного процесса в условиях различной исходной информации, для создания которых были использованы нечеткие множества первого типа, интервальные нечеткие множества второго типа и Z-числа. По всем построенным моделям даны анализ и рекомендации для использования их в практических целях. Нечеткие множества первого типа рекомендуется использовать для формализации статистических данных образовательного процесса, а также данных, полученных от единственного эксперта (экзаменатора). Интервальные нечеткие множества второго типа рекомендуется использовать для формализации статистических и экспертных данных образовательного процесса со случайными ошибками, а также данных, полученных от группы экспертов. Z-числа рекомендуется использовать для формализации данных образовательного процесса с учетом их достоверности. Приведенные числовые примеры в совокупности с теоретическими обоснованиями предоставляют возможность выбора модели для дальнейшего анализа данных в целях получения устойчивых конечных результатов и управляющих решений на их основе.

educational processdata formalizationfuzzy setZ-numberобразовательный процессформализация данныхнечеткое множествоZ-числаZadeh L.A. Fuzzy logic and approximate reasoning. Synthese, 1975, v. 80, pp. 407–428.Zadeh L.A. Fuzzy logic and approximate reasoning // Synthese, 1975, v. 80, pp. 407–428.Poleshchuk O.M., Komarov E.G., Chernova T.V. Influence of research and development activities on professional performance of aerospace students. AIP Conference Proceedings, 2019, no. 2171(1), p. 140003. DOI: 10.1063/1.5133293Poleshchuk O.M., Komarov E.G., Chernova T.V. Influence of research and development activities on professional performance of aerospace students // AIP Conference Proceedings, 2019, no. 2171(1), p. 140003. DOI: 10.1063/1.5133293Poleshchuk O.M., Tumor S.V. The Analysis of Student Performance During Face-to-Face and Distance Learning under Z-Information. Lecture Notes in Electrical Engineering, 2022, v. 857, pp. 393–402. DOI: 10.1007/978-3-030-94202-1_37Poleshchuk O.M., Tumor S.V. The Analysis of Student Performance During Face-to-Face and Distance Learning under Z-Information // Lecture Notes in Electrical Engineering, 2022, v. 857, pp. 393–402. DOI: 10.1007/978-3-030-94202-1_37Poleshchuk O., Komarov E. Expert Fuzzy Information Processing. Studies in Fuzziness and Soft Computing, 2011, v. 268, pp. 1–239.Poleshchuk O., Komarov E. Expert Fuzzy Information Processing // Studies in Fuzziness and Soft Computing, 2011, v. 268, pp. 1–239.Ryjov A. Fuzzy Linguistic Scales: Definition, Properties and Applications. Soft Computing in Measurement and Information Acquisition. Studies in Fuzziness and Soft Computing // Reznik L., Kreinovich V. (eds), 2003, v. 127. DOI: 10.1007/978-3-540-36216-6_3Ryjov A. Fuzzy Linguistic Scales: Definition, Properties and Applications. Soft Computing in Measurement and Information Acquisition. Studies in Fuzziness and Soft Computing // Reznik L., Kreinovich V. (eds). 2003, v. 127. DOI: 10.1007/978-3-540-36216-6_3Poleshchuk O.M. Creation of linguistic scales for expert evaluation of parameters of complex objects based on semantic scopes. International Russian Automation Conf. (RusAutoCon–2018), 2018, pp. 1–6.Poleshchuk O.M. Creation of linguistic scales for expert evaluation of parameters of complex objects based on semantic scopes // International Russian Automation Conf. (RusAutoCon–2018), 2018, pp. 1–6.Runkler T.A., Katz C. Fuzzy clustering by particle swarm optimization // Proceedings of the IEEE Int. Conf. on Fuzzy Systems, 2006, pp. 34–41.Liu H.C., Yih J.M., Wu D.B., Liu S.W. Fuzzy C-mean clustering algorithms based on Picard iteration and particle swarm optimization. Proceedings of the Int. Workshop on Geoscience and Remote Sensing (ETT and GRS–2008), 2008, pp. 75–84.Liu H.C., Yih J.M., Wu D.B., Liu S.W. Fuzzy C-mean clustering algorithms based on Picard iteration and particle swarm optimization // Proceedings of the Int. Workshop on Geoscience and Remote Sensing (ETT and GRS–2008), 2008, pp. 75–84.Poleshchuk O, Komarov E. The determination of rating points of objects with qualitative characteristics and their usage in decision making problems. Int. J. of Computational and Mathematical Sciences, 2009, v. 3, no. 7, pp. 360–364.Poleshchuk O, Komarov E. The determination of rating points of objects with qualitative characteristics and their usage in decision making problems // Int. J. of Computational and Mathematical Sciences, 2009, v. 3, no. 7, pp. 360–364.Chen M., Ludwig A. Particle swarm optimization based fuzzy clustering approach to identify optimal number of clusters. J. of Artificial Intelligence and Soft Computing Research, 2014, v. 4, no. 1, pp. 43–56.Chen M., Ludwig A. Particle swarm optimization based fuzzy clustering approach to identify optimal number of clusters // J. of Artificial Intelligence and Soft Computing Research, 2014, v. 4, no. 1, pp. 43–56.Darwish A., Poleshchuk O. New models for monitoring and clustering of the state of plant species based on sematic spaces. J. of Intelligent and Fuzzy Systems, 2014, v. 3, no. 26, pp. 1089–1094.Darwish A., Poleshchuk O. New models for monitoring and clustering of the state of plant species based on sematic spaces // J. of Intelligent and Fuzzy Systems, 2014, v. 3, no. 26, pp. 1089–1094.Phyo O., Chaw E. Comparative Study of Fuzzy PSO (FPSO) Clustering Algorithm and Fuzzy C-Means (FCM) Clustering Algorithm. National J. of Parallel and Soft Computing, 2019, v. 1, no. 1, pp. 62–67.Phyo O., Chaw E. Comparative Study of Fuzzy PSO (FPSO) Clustering Algorithm and Fuzzy C-Means (FCM) Clustering Algorithm // National J. of Parallel and Soft Computing, 2019, v. 1, no. 1, pp. 62–67.Tanaka H, Ishibuchi H. Identification of possibilistic linear models. Fuzzy Sets and Systems, 1991, v. 41, pp. 145–160.Tanaka H, Ishibuchi H. Identification of possibilistic linear models // Fuzzy Sets and Systems, 1991, v. 41, pp. 145–160.Chang Y.-H. Hybrid fuzzy least- squares regression analysis and its reliability measures. Fuzzy Sets and Systems, 2001, v. 119, pp. 225–246. DOI:10.1016/S0165-0114(99)00092-5Chang Y.-H. Hybrid fuzzy least- squares regression analysis and its reliability measures // Fuzzy Sets and Systems, 2001, v. 119, pp. 225–246. DOI:10.1016/S0165-0114(99)00092-5Domrachev V.G., Poleshchuk O.M. On the construction of a regression model under fuzzy source data. Avtomatika I Telemehkanika, 2003, v. 11, pp. 74–83.Domrachev V.G., Poleshchuk O.M. On the construction of a regression model under fuzzy source data // Avtomatika I Telemehkanika, 2003, v. 11, pp. 74–83.Arefi M. Quantative fuzzy regression based on fuzzy outputs and fuzzy parameters. Soft Computing-A Fusion of Foundations, Methodologies and Applications, 2020, v. 24(1), pp. 311–320. DOI:10.1007/s00500-019-04424-2Liu F., Mendel J.M. Encoding words into interval Type-2 fuzzy sets using an interval approach. EEE Tranns. Fuzzy Systems, 2008, v. 16(6), pp. 1503–1521.Liu F., Mendel J.M. Encoding words into interval Type-2 fuzzy sets using an interval approach // EEE Tranns. Fuzzy Systems. 2008, v. 16(6), pp. 1503–1521.Poleshchuk O., Komarov E. A fuzzy linear regression model for interval type-2 fuzzy sets. Annual Conf. of the North American Fuzzy Information Processing Society – NAFIPS’2012, 2012, p. 6290970. DOI: 10.1109/NAFIPS.2012.6290970Poleshchuk O., Komarov E. A fuzzy linear regression model for interval type-2 fuzzy sets // Annual Conf. of the North American Fuzzy Information Processing Society – NAFIPS'2012, 2012, p. 6290970. DOI: 10.1109/NAFIPS.2012.6290970Poleshchuk O.M. Klasternyy analiz Z-informatsii na osnove etalonnoy sistemy nechetkikh opredeleniy prinadlezhnosti [Cluster analysis of Z-information based on a reference system of fuzzy identification]. Lesnoy vestnik / Forestry Bulletin, 2024, vol. 28, no. 2, pp. 150–155. DOI: 10.18698/2542-1468-2024-2-150-155Полещук О.М. Кластерный анализ Z-информации на основе эталонной системы нечетких определений принадлежности // Лесной вестник / Forestry Bulletin, 2024. Т. 28. № 2. С. 150–155. DOI: 10.18698/2542-1468-2024-2-150-155Zadeh L.A. A Note on Z-numbers. Information Sciences, 2011, v. 14, no. 181, pp. 2923–2932.Zadeh L.A. A Note on Z-numbers // Information Sciences, 2011, v. 14, no. 181, pp. 2923–2932.Sadikoglu F, Huseynov O, Memmedova K. Z-Regression analysis in psychological and educational researches. Procedia Comput. Sci., 2016, v. 102, pp. 385–389. DOI: 10.1016/J.PROCS.2016.09.416Sadikoglu F, Huseynov O, Memmedova K. Z-Regression analysis in psychological and educational researches // Procedia Comput. Sci., 2016, v. 102, pp. 385–389. DOI: 10.1016/J.PROCS.2016.09.416Poleshchuk O. Fuzzy regression model with input and output Z-numbers. IOP Conf. Series: Materials Science and Engineering, 2020, v. 919(5), p. 052041. DOI: 10.1088/1757-899X/919/5/052041Poleshchuk O. Fuzzy regression model with input and output Z-numbers // IOP Conf. Series: Materials Science and Engineering, 2020, v. 919(5), p. 052041. DOI: 10.1088/1757-899X/919/5/052041Poleshchuk O.M. Multiple Z-Regression with Fuzzy Coefficients. Advances in Intelligent Systems and Computing, 2021, v. 1306, pp. 63–70. DOI: 10.1007/978-3-030-64058-3_8Poleshchuk O.M. Multiple Z-Regression with Fuzzy Coefficients // Advances in Intelligent Systems and Computing, 2021, v. 1306, pp. 63–70. DOI: 10.1007/978-3-030-64058-3_8Poleshchuk O. Quintile multiple regression with fuzzy coefficients and initial Z-information. E3S Web of Conferences, 2023, v. 431, p. 05015. DOI: 10.1051/e3sconf/202343105015Poleshchuk O. Quintile multiple regression with fuzzy coefficients and initial Z-information // E3S Web of Conferences, 2023, v. 431, p. 05015. DOI: 10.1051/e3sconf/202343105015Jamal M., Khalif K., Mohamad S. The implementation of Z-numbers in fuzzy clustering algorithm for wellness of chronic kidney disease patients. J. of Physics: Conf. Series, 2018, v. 1366, p. 012058.Jamal M., Khalif K., Mohamad S. The implementation of Z-numbers in fuzzy clustering algorithm for wellness of chronic kidney disease patients // J. of Physics: Conf. Series, 2018, v. 1366, p. 012058.Aliev R., Guirimov B. Z-number clustering based on general Type-II fuzzy sets. Advances in Intelligent Systems and Computing, 2018, v. 896, pp. 270–278.Aliev R., Guirimov B. Z-number clustering based on general Type-II fuzzy sets // Advances in Intelligent Systems and Computing, 2018, v. 896, pp. 270–278.Aliev R.A., Pedrycz W., Guirimov B.G., Huseynov O.H. Clustering method for production of Z-numbers based if-then rules. Information Sciences, 2020, v. 520, pp. 155–176.Aliev R.A., Pedrycz W., Guirimov B.G., Huseynov O.H. Clustering method for production of Z-numbers based if-then rules // Information Sciences, 2020, v. 520, pp. 155–176.Poleshchuk O.M. Klasternyy analiz ekspertnoy informatsii na osnove Z-chisel [Cluster analysis of expert information based on Z-numbers]. Lesnoy vestnik / Forestry Bulletin, 2022, vol. 26, no. 1, pp. 143–148. DOI: 10.18698/2542-1468-2022-1-143-148Полещук О.М. Кластерный анализ экспертной информации на основе Z-чисел // Лесной вестник / Forestry Bulletin, 2022. Т. 26. № 1. С. 143–148. DOI: 10.18698/2542-1468-2022-1-143-148Poleshchuk O. Clustering Z-information based on a system of fuzzy reference requirements. E3S Web of Conferences, 2023, v. 420, p. 06022.Poleshchuk O. Clustering Z-information based on a system of fuzzy reference requirements // E3S Web of Conferences, 2023, v. 420, p. 06022.