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Article

A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory

by *,† and
Departament de Matemàtica Econòmica, Financera i Actuarial, Universitat de Barcelona, 08034 Barcelona, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2017, 9(10), 198; https://doi.org/10.3390/sym9100198
Received: 8 September 2017 / Revised: 15 September 2017 / Accepted: 16 September 2017 / Published: 21 September 2017
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name “modal interval trapezoidal fuzzy numbers”. 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. View Full-Text
Keywords: modal intervals; fuzzy numbers; fuzzy relations; semantic interpretation modal intervals; fuzzy numbers; fuzzy relations; semantic interpretation
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MDPI and ACS Style

Jorba, L.; Adillon, R. A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory. Symmetry 2017, 9, 198. https://doi.org/10.3390/sym9100198

AMA Style

Jorba L, Adillon R. A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory. Symmetry. 2017; 9(10):198. https://doi.org/10.3390/sym9100198

Chicago/Turabian Style

Jorba, Lambert, and Romà Adillon. 2017. "A Generalization of Trapezoidal Fuzzy Numbers Based on Modal Interval Theory" Symmetry 9, no. 10: 198. https://doi.org/10.3390/sym9100198

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