Soft Computing

, Volume 23, Issue 15, pp 5975–5990 | Cite as

Comparing the magnitude of fuzzy intervals and fuzzy random variables from the standpoint of gradual numbers

  • Farid Aiche
  • Didier DuboisEmail author


Gradual numbers have been introduced to separate fuzziness (understood as gradation) from uncertainty in so-called fuzzy numbers. Then a fuzzy number can be viewed as a standard interval of functions, each interpreted as a gradual number. Gradual numbers are naturally met when representing probabilities of fuzzy events, midpoints of fuzzy intervals, etc. They can be viewed as a non-monotonic generalization of cumulative probability distributions. This paper presents three methods for comparing gradual numbers that generalize stochastic orderings to such non-monotonic functions. Then it proposes joint extensions of stochastic dominance and statistical preference to random fuzzy intervals when the fuzzy intervals are understood as intervals of gradual numbers. This approach, which combines known probabilistic orderings with known forms of interval orderings, can be viewed as a systematic way of constructing methods for ranking fuzzy random variables.


Gradual numbers Stochastic dominance Statistical preference Probability Interval orderings Fuzzy intervals Fuzzy random variables 



The first author benefited from visit scholarships granted by University of Tizi-Ouzou, and E.N.P.E.I., Algiers.

Compliance with ethical standards

Conflict of interest

The authors declare that none of them has any conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ecole Nationale Préparatoire aux Etudes d’Ingéniorat (E.N.P.E.I.) Badji MokhtarRouiba, AlgiersAlgeria
  2. 2.Université Mouloud Mammeri de Tizi-OuzouTizi-OuzouAlgeria
  3. 3.IRITCNRS and Université de ToulouseToulouse Cedex 9France

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