New approaches for the comparison of [formula omitted] fuzzy numbers: a theoretical and operational analysis

Abstract

A key issue in operationalizing fuzzy set theory (particularly in decision analysis) is how to compare fuzzy numbers. In this paper, the case of L–R fuzzy numbers, i.e. the most general form of fuzzy numbers, is considered. In particular here, L–R fuzzy numbers represented by continuous, convex membership functions allowing also definite integration is taken into consideration, normality is not required. Traditional comparison methods are generally limited to the use of triangular fuzzy numbers, and often the shape of the membership function is not taken into account or only a part of it is used (leading to a loss of information). Most of the approaches one can find in the literature are characterised by the use of α-cuts and credibility levels, the use of areas for comparing fuzzy numbers has been proposed only recently. In particular, in the so-called NAIADE method a new semantic distance able to compare crisp numbers, fuzzy numbers and density functions has been developed. The basic idea underlying this paper is that, if only L–R fuzzy numbers are considered, other methodologies for comparing fuzzy numbers can be developed. Three indices based on the use of areas are studied, i.e. the expected value, the variance (with its decomposition into positive and negative semivariances) and the degree of coincidence of two fuzzy numbers. A justification of the use of these indices and a first tentative of axiomatisation is given. A short discussion on the issue of possible aggregation conventions of these indices is presented, and an empirical example is examined too.