## Abstract

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.

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## Notes

- 1.
In particular, \(\le _1 \ne \lnot (>_1)\).

- 2.
It assumes a continuous density, i.e., with no mass concentrated at a point.

- 3.
The joint distribution of the random set is obtained from a joint probability space and the corresponding multimappings. It is not equivalent to taking the joint of probabilities, one in each credal set.

- 4.
If \(\mu _{\tilde{A}}\) is not continuous, these results hold with strict \(\alpha \)-cuts \( \tilde{A}^\alpha = \{ r : \mu _{\tilde{A}}(r) > \alpha \}, \alpha < 1\).

- 5.
Defining a fuzzy set by the \(\alpha \)-cut multimapping is at the origin of fuzzy sets (Zadeh 1965) and has been studied especially by Negoita and Ralescu (1975), and more recently extended under the popular name “soft set”, dropping the nestedness assumption (Molodtsov 1999). However, soft sets are often only general multimappings or relations between two sets, without direct connection to fuzzy sets.

- 6.
Namely 6 possible interval comparisons, and 7 stochastic gradual number comparisons as the combination of sign dominance with probabilistic orderings leads to one method appearing in Table 3.

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## Acknowledgements

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

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Aiche, F., Dubois, D. Comparing the magnitude of fuzzy intervals and fuzzy random variables from the standpoint of gradual numbers.
*Soft Comput* **23, **5975–5990 (2019). https://doi.org/10.1007/s00500-018-3526-x

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### Keywords

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