{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:20:15Z","timestamp":1760149215505,"version":"build-2065373602"},"reference-count":63,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T00:00:00Z","timestamp":1689292800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CONACyT"},{"name":"TecNM"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fuzzy systems are widely used in most fields of science and engineering, mainly because the models they produce are robust, accurate, easy to evaluate and capture real-world uncertainty better than do the classical alternatives. We propose a new methodology for structure and parameter tuning of Takagi\u2013Sugeno\u2013Kang fuzzy models using several optimization techniques. The output parameters are determined analytically, by finding the minimum of the root-mean-square error (RMSE) for a properly defined error function. The membership functions are simplified by considering symmetry and equispacing, to reduce the optimization problem of finding their parameters, and allow it to be carried out using the numerical method of gradient descent. Both algorithms are fast enough to finally implement a strategy based on the hill climbing approach to finding the optimal structure (number and type of membership functions) of the fuzzy system. The effectiveness of the proposed strategy is shown by comparing its performance, using four case studies found in current relevant works, to the popular adaptive network-based fuzzy inference system (ANFIS), and to other recently published strategies based on evolutionary fuzzy models. In terms of the RMSE, performance was at least 28% better in all case studies.<\/jats:p>","DOI":"10.3390\/sym15071417","type":"journal-article","created":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T08:35:47Z","timestamp":1689323747000},"page":"1417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Fuzzy Model Parameter and Structure Optimization Using Analytic, Numerical and Heuristic Approaches"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6887-8686","authenticated-orcid":false,"given":"Joel Artemio","family":"Morales-Viscaya","sequence":"first","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9151-0465","authenticated-orcid":false,"given":"Ad\u00e1n Antonio","family":"Alonso-Ram\u00edrez","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9777-4413","authenticated-orcid":false,"given":"Marco Antonio","family":"Castro-Liera","sequence":"additional","affiliation":[{"name":"Tecnologico Nacional de Mexico en La Paz (TecNM), Forjadores de Baja California Sur #4720, La Paz 23080, Mexico"}]},{"given":"Juan Carlos","family":"G\u00f3mez-Cort\u00e9s","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]},{"given":"David","family":"Lazaro-Mata","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]},{"given":"Jos\u00e9 Eleazar","family":"Peralta-L\u00f3pez","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8435-680X","authenticated-orcid":false,"given":"Carlos A.","family":"Coello Coello","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n y de Estudios Avanzados del Instituto Polit\u00e9cnico Nacional, Avenida Instituto Polit\u00e9cnico Nacional #2508, Ciudad de M\u00e9xico 07360, Mexico"}]},{"given":"Jos\u00e9 Enrique","family":"Botello-\u00c1lvarez","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5050-6208","authenticated-orcid":false,"given":"Alejandro Israel","family":"Barranco-Guti\u00e9rrez","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico en Celaya (TecNM), Antonio Garc\u00eda Cubas, Pte #600 Esquina. Av. Tecnol\u00f3gico, Celaya 38010, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1109\/5.949483","article-title":"Industrial applications of soft computing: A review","volume":"89","author":"Dote","year":"2001","journal-title":"Proc. IEEE"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kindo, A.A., Kaladzavi, G., Malo, S., Camara, G., Tapsoba, T.M.Y. (2020, January 1\u20133). Fuzzy logic approach for knowledge modeling in an Ontology: A review. 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