{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T06:50:41Z","timestamp":1787640641993,"version":"build-2736575974"},"reference-count":54,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,9,21]],"date-time":"2017-09-21T00:00:00Z","timestamp":1505952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name \u201cmodal interval trapezoidal fuzzy numbers\u201d. In this generalization, we accept that the alpha cuts associated with a trapezoidal fuzzy number can be modal intervals, also allowing that two interval modalities can be associated with a trapezoidal fuzzy number. In this context, it is difficult to maintain the traditional graphic representation of trapezoidal fuzzy numbers and we must use the interval plane in order to represent our modal interval trapezoidal fuzzy numbers graphically. Using this representation, we can correctly reflect the modality of the alpha cuts. We define some concepts from modal interval analysis and we study some of the related properties and structures, proving, among other things, that the inclusion relation provides a lattice structure on this set. We will also provide a semantic interpretation deduced from the modal interval extensions of real continuous functions and the semantic modal interval theorem. The application of modal intervals in the field of fuzzy numbers also provides a new perspective on and new applications of fuzzy numbers.<\/jats:p>","DOI":"10.3390\/sym9100198","type":"journal-article","created":{"date-parts":[[2017,9,21]],"date-time":"2017-09-21T12:17:40Z","timestamp":1505996260000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2729-1552","authenticated-orcid":false,"given":"Lambert","family":"Jorba","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tica Econ\u00f2mica, Financera i Actuarial, Universitat de Barcelona, 08034 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7045-4663","authenticated-orcid":false,"given":"Rom\u00e0","family":"Adillon","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tica Econ\u00f2mica, Financera i Actuarial, Universitat de Barcelona, 08034 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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Symmetry, 9.","DOI":"10.3390\/sym9040052"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hou, S., Wang, H., and Feng, S. (2016). Attribute control chart construction based on fuzzy score number. Symmetry, 8.","DOI":"10.3390\/sym8120139"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sa\u0142abun, W., and Piegat, A. (2016). Comparative analysis of MCDM methods for the assessment of mortality in patients with acute coronary syndrome. Artif. Intell. Rev., 1\u201315.","DOI":"10.1007\/s10462-016-9511-9"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1109\/TFUZZ.2004.825969","article-title":"Complex dynamics through fuzzy chains","volume":"12","author":"Bucolo","year":"2004","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_9","unstructured":"Garde\u00f1es, E., and Mielgo, H. (1986). Modal intervals: Functions. Polish Symposium on Interval and Fuzzy Mathematics, Zamenkhov\u2019s University of Poznan."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"601","DOI":"10.3233\/JIFS-16631","article-title":"Quantified trapezoidal fuzzy numbers","volume":"33","author":"Adillon","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.fss.2012.02.014","article-title":"Real vector space of LR-fuzzy intervals with respect to the shape-preserving t-norm-based addition","volume":"200","year":"2012","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","first-page":"253","article-title":"Calculus of Approximations","volume":"4","author":"Warmus","year":"1956","journal-title":"Bull. Acad. Pol. Sci."},{"key":"ref_13","unstructured":"Sunaga, T. (1958). Theory of interval algebra and its application to numerical analysis. Research Association of Applied Geometry (RAAG) Memoirs, Ggujutsu Bunken Fukuy-kai."},{"key":"ref_14","unstructured":"Moore, R.E. (1966). Interval Analysis, Prentice Hall."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Nickel, K. (1975). Verbandtheoretische grundlagen der intervallmathematik. Lecture Notes in Computer Science 29, Springer.","DOI":"10.1007\/3-540-07170-9_25"},{"key":"ref_16","unstructured":"Alefeld, G., and Herzberger, J. (1983). Introduction to Interval Computations, Academic."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/978-3-7091-8471-4_7","article-title":"Algebraische erweiterungen der intervallrechnung unter erhaltung der ordnungs und verbandsstrukturen","volume":"1","author":"Kaucher","year":"1977","journal-title":"Comput. Suppl."},{"key":"ref_18","unstructured":"Garde\u00f1es, E., Mielgo, H., and Trepat, A. (1985). Modal intervals: Reasons and ground semantics, interval mathematics. Lecture Notes in Computer Science 212, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1080\/00207160.2014.905681","article-title":"Stochastiics dynamics simulation with generalized interval probability","volume":"92","author":"Wang","year":"2015","journal-title":"Int. J. Comput. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1023\/A:1011465930178","article-title":"Modal intervals","volume":"7","author":"Sainz","year":"2001","journal-title":"Reliab. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF01001956","article-title":"Rough sets","volume":"11","author":"Pawlak","year":"1982","journal-title":"Int. J. Comput. Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/S0020-0255(98)10023-3","article-title":"A comparative study of fuzzy sets and rough sets","volume":"109","author":"Yao","year":"1998","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.3233\/IFS-141455","article-title":"A new rough set theory: rough soft hemirings","volume":"28","author":"Zhan","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sainz, M., Armengol, J., Calm, R., Herrero, P., Jorba, L., and Vehi, J. (2014). Modal interval analysis: New tools for numerical information. Lecture Notes in Mathematics 2091, Springer.","DOI":"10.1007\/978-3-319-01721-1"},{"key":"ref_25","first-page":"274","article-title":"Numerical clouds. A treatment for indiscernibility","volume":"15","author":"Adillon","year":"2013","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1080\/00207727808941724","article-title":"Operations on fuzzy numbers","volume":"9","author":"Dubois","year":"1987","journal-title":"Int. J. Syst. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/0165-0114(87)90028-5","article-title":"The mean value of a fuzzy number","volume":"24","author":"Dubois","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/S0165-0114(02)00098-2","article-title":"Nearest interval approximation of a fuzzy number","volume":"130","author":"Grzegorzewski","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0165-0114(92)90062-9","article-title":"The expected value of a fuzzy number","volume":"47","author":"Heilpern","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1142\/S0218488515500415","article-title":"A new point of view for fuzzy numbers and their defuzzification","volume":"23","author":"Adillon","year":"2015","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.jmaa.2007.04.003","article-title":"Continuous minimax optimization using modal intervals","volume":"339","author":"Sainz","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/j.amc.2003.07.025","article-title":"The nearest trapezoidal fuzzy number to a fuzzy quantity","volume":"156","author":"Abbasbandy","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1016\/j.fss.2006.11.015","article-title":"Trapezoidal approximation of fuzzy numbers-revisited","volume":"158","author":"Grzegorzewski","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1354","DOI":"10.1016\/j.fss.2007.12.001","article-title":"Trapezoidal approximations of fuzzy numbers preserving the expected interval\u2014Algorithms and properties","volume":"159","author":"Grzegorzewski","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1016\/j.fss.2006.11.017","article-title":"A note on trapezoidal approximations of fuzzy numbers","volume":"158","author":"Yeh","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.1016\/j.fss.2007.09.010","article-title":"Trapezoidal and triangular approximations preserving the expected interval","volume":"159","author":"Yeh","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1142\/S0218488513500372","article-title":"Nearest symmetric trapezoidal fuzzy number approximation preserving expected interval","volume":"21","author":"Veeramani","year":"2013","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.fss.2015.03.012","article-title":"Conditioned weighted L-R approximations of fuzzy numbers","volume":"283","author":"Ban","year":"2016","journal-title":"Fuzzy Sets Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1007\/s13370-014-0285-4","article-title":"An improved ranking method for generalized fuzzy numbers based on Euclidean distance concept","volume":"26","author":"Eslamipoor","year":"2015","journal-title":"Afrik. Mat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1007\/s00521-013-1541-5","article-title":"A developed distance method for ranking generalized fuzzy numbers","volume":"25","author":"Zare","year":"2014","journal-title":"Neural Comput. Appl."},{"key":"ref_41","unstructured":"Wagenknecht, M., and Schneider, V. (2005, January 7\u20139). Inclusion of fuzzy numbers. The variable Plateau Case. Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications, Barcelona, Spain."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3846\/16111699.2016.1278559","article-title":"Fuzzy extension of the CODAS method for multi-criteria market segment evaluation","volume":"18","author":"Ghorabaee","year":"2017","journal-title":"J. Bus. Econ. Manag."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Zhang, X., Xu, Z., and Liu, M. (2016). Hesitant trapezoidal fuzzy QUALIFLEX method and its application in the evaluation of green supply chain initiatives. Sustainability, 8.","DOI":"10.3390\/su8090952"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2016\/1956303","article-title":"An ELECTRE approach for multicriteria interval-valued intuitionistic trapezoidal fuzzy group decision making problems","volume":"2016","author":"Veeramachaneni","year":"2016","journal-title":"Adv. Fuzzy Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2013.12.012","article-title":"An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets","volume":"263","author":"Chen","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4179","DOI":"10.1007\/s12665-014-3314-6","article-title":"Site selection in municipal solid waste management with extended VIKOR method under fuzzy environment","volume":"72","author":"Liu","year":"2014","journal-title":"Environ. Earth Sci."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Mardani, A., Zavadskas, E.K., Govindan, K., Senin, A.A., and Jusoh, A. (2016). VIKOR technique: A systematic review of the state of the art literature on methodologies and applications. Sustainability, 8.","DOI":"10.3390\/su8010037"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.eswa.2012.07.066","article-title":"Group multi-criteria decision making based upon interval-valued fuzzy numbers: An extension of the MULTIMOORA method","volume":"40","author":"Zeng","year":"2013","journal-title":"Expert Syst. Appl."},{"key":"ref_49","first-page":"76","article-title":"Evaluation by an area-based method of ranking interval type-2 fuzzy sets (EAMRIT-2F) for multi-criteria group decision-making","volume":"15","author":"Ghorabaee","year":"2016","journal-title":"Transform. Bus. Econ."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1007\/s13762-016-0977-4","article-title":"An extended fuzzy TOPSIS\u2013GRA method based on different separation measures for green logistics service provider selection","volume":"13","author":"Celik","year":"2016","journal-title":"Int. J. Environ. Sci. Technol."},{"key":"ref_51","first-page":"48","article-title":"Multi-criteria group decision-making using an extended EDAS method with internal type-2 fuzzy sets","volume":"20","author":"Ghorabaee","year":"2017","journal-title":"Ekon. Manag."},{"key":"ref_52","first-page":"39","article-title":"A new method of assessment based on fuzzy ranking and aggregated weights (AFRAW) for MCDM problems under type-2 fuzzy environment","volume":"50","author":"Ghorabaee","year":"2016","journal-title":"Econ. Comput. Econ. Cybern. Stud. Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"453","DOI":"10.3846\/20294913.2016.1171262","article-title":"Multi-criteria decision-making method based on intuitionistic trapezoidal fuzzy prioritised owa operator","volume":"22","author":"Liu","year":"2016","journal-title":"Technol. Econ. Dev. Econ."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.jclepro.2016.07.031","article-title":"Multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy sets","volume":"137","author":"Ghorabaee","year":"2016","journal-title":"J. Clean. Prod."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:45:31Z","timestamp":1760208331000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,21]]},"references-count":54,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2017,10]]}},"alternative-id":["sym9100198"],"URL":"https:\/\/doi.org\/10.3390\/sym9100198","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,9,21]]}}}