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On the Search for a Measure to Compare Interval-Valued Fuzzy Sets

Author

Listed:
  • Susana Díaz-Vázquez

    (Department of Statistics and O. R. and Department of Computer Sciences, University of Oviedo, 33007 Oviedo, Spain
    These authors contributed equally to this work.)

  • Emilio Torres-Manzanera

    (Department of Statistics and O. R. and Department of Computer Sciences, University of Oviedo, 33007 Oviedo, Spain
    These authors contributed equally to this work.)

  • Irene Díaz

    (Department of Statistics and O. R. and Department of Computer Sciences, University of Oviedo, 33007 Oviedo, Spain
    These authors contributed equally to this work.)

  • Susana Montes

    (Department of Statistics and O. R. and Department of Computer Sciences, University of Oviedo, 33007 Oviedo, Spain
    These authors contributed equally to this work.)

Abstract

Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval.

Suggested Citation

  • Susana Díaz-Vázquez & Emilio Torres-Manzanera & Irene Díaz & Susana Montes, 2021. "On the Search for a Measure to Compare Interval-Valued Fuzzy Sets," Mathematics, MDPI, vol. 9(24), pages 1-30, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3157-:d:697181
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    References listed on IDEAS

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    1. Moshe Sniedovich, 2008. "Wald's maximin model: a treasure in disguise!," Journal of Risk Finance, Emerald Group Publishing, vol. 9(3), pages 287-291, May.
    2. Jay K. Satia & Roy E. Lave, 1973. "Markovian Decision Processes with Uncertain Transition Probabilities," Operations Research, INFORMS, vol. 21(3), pages 728-740, June.
    3. Chong Wu & Peng Luo & Yongli Li & Xuekun Ren, 2014. "A New Similarity Measure of Interval-Valued Intuitionistic Fuzzy Sets Considering Its Hesitancy Degree and Applications in Expert Systems," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-16, May.
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