ELSEVIER European Journal of Operational Research 110 (1998) 377-391 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology The fuzzy hyperbolic inequality index associated with fuzzy random variables Maria Angeles Gil *, Miguel Lfpez-Diaz, Hortensia L6pez-Garcia Departamento de Estadistica e Investigacifn Operativa y Dicklctica de la Matemftica, Universidad de Oviedo, Facultad de Ciencias. CICalvo Sotelo. s/n, 33007 Oviedo, Spain Received 2 April 1997;accepted 1 July 1997 Abstract The aim of this paper is focussed on the quantification of the extent of the inequality associated with fuzzy-valued random variables in general populations. For this purpose, the fuzzy hyperbolic inequality index associated with gen- eral fuzzy random variables is presented and a detailed discussion of some of the most valuable properties of this index (extending those for classical inequality indices) is given. Two examples illustrating the computation of the fuzzy in- equality index are also considered. Some comments and suggestions are finally included. © 1998 Elsevier Science B.V. Keywords: Fuzzy sets; Fuzzy random variable; Expected value of a fuzzy random variable; Hyperbolic inequality index 1. Introduction The measurement of inequality has become a topic with many valuable applications in fields like Economics and Industry. Inequality indices have been introduced in the literature to quantify the average relative variation of a population, with respect to a quantitative at- tribute which can be formalized as a random vari- able. Several indices have been suggested for this purpose (see, for instance, Eichhorn and Gehrig, 1982, for a review), and the most successful ones in the last two decades are those included in fam- ilies of measures that are stated on the basis of measures from Information Theory. These mea- * Corresponding author. Fax: +34-8510-3356; e-mail: angeles@pinon.ccu.uniovi.es. sures are those defined as the expected value of a certain function of the random variable involving its expected value (like the index of Theil, 1967, the families established by Atkinson, 1970; and Bourguignon, 1979, or the more general one stated by Gastwirth, 1975). The most common way to introduce some of the preceding families is that based on the statement of sets of desirable properties of inequality measures (like continuity, symmetry, minimality, mean inde- pendence, principles of transfers, among others). These sets are then used to give axiomatic character- izations of the measures (see, for instance, Bour- guignon, 1979; Cowell, 1980; Shorrocks, 1980; Cowell and Kuga, 1981; Eichhorn and Gehrig, 1982, for the so-called additively decomposable indices). In a previous paper (cf. Gil et al., 1989b) we have considered an alternative way to obtain some 0377-2217/98/$19.00 © 1998 ElsevierScienceB.V. All rights reserved. PHS0377-22 1 7(97)00252-X