On Quantile Dominance and the Acceptability Index in Interval Comparison

S. Fernández Alonso, S. Díaz Vázquez, S. Montes Rodriguez

In many applications involving uncertainty, interval-valued quantities represent imprecise information. Comparing intervals is a fundamental problem in decision making and optimization.
We introduce a superiority degree between intervals based on their quantile functions. Given two intervals, we consider the set of α ∈ [0,1] for which the α-quantile of one does not exceed that of the other, and define the degree as its Lebesgue measure. This yields a measure of dominance.
The degree admits a closed-form expression and induces binary relations parameterized by k ∈ [0,1], ranging from trivial relations to preorders depending on k.
We establish a connection with the acceptability index of Sengupta and Pal (2000), showing that the proposed measure is an affine transformation of this index.
The proposed framework offers a flexible, theoretically grounded tool for interval comparison, with applications in decision making under uncertainty and optimization problems involving interval data.

Palabras clave: Interval-valued data, Interval comparison, Quantile functions, Superiority degree, Acceptability index

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