The problem of ordering fuzzy numbers has been studied by many authors who have followed quite different methods. A straightforward idea is to transform fuzzy numbers into real numbers, with the purpose to induce on the set of the fuzzy numbers involved the order of the real line. The instruments of the transformation are called ranking functions and generally they arise from intuitions of different nature (geometrical, possibilistic et al). For instance, given an arbitrary triangular fuzzy number, we de ̄ne both its pessimistic and optimistic alternative as basic notions for generating an order. The aim of this paper is to formulate some axiomatic requirements in order to qualify an arbitrary ranking function through its behaviour on the remarkable subdomain of the triangular fuzzy numbers. A first attempt in this direction is present in [14], but here we generalize the class of functions considered and propose a different approach in determining some of the axioms. We will present four axioms divided into two groups: the first is conceived as really essential for any ranking function to be considered acceptable at least on the above cited subdomain; the second attempts to characterize, in a more specific way than the first, a reasonable behaviour of a ranking function through the idea to weight, in some way, the spread of a triangular fuzzy number.

### A Characterization of a general class of ranking functions on the triangular fuzzy numbers

#### Abstract

The problem of ordering fuzzy numbers has been studied by many authors who have followed quite different methods. A straightforward idea is to transform fuzzy numbers into real numbers, with the purpose to induce on the set of the fuzzy numbers involved the order of the real line. The instruments of the transformation are called ranking functions and generally they arise from intuitions of different nature (geometrical, possibilistic et al). For instance, given an arbitrary triangular fuzzy number, we de ̄ne both its pessimistic and optimistic alternative as basic notions for generating an order. The aim of this paper is to formulate some axiomatic requirements in order to qualify an arbitrary ranking function through its behaviour on the remarkable subdomain of the triangular fuzzy numbers. A first attempt in this direction is present in [14], but here we generalize the class of functions considered and propose a different approach in determining some of the axioms. We will present four axioms divided into two groups: the first is conceived as really essential for any ranking function to be considered acceptable at least on the above cited subdomain; the second attempts to characterize, in a more specific way than the first, a reasonable behaviour of a ranking function through the idea to weight, in some way, the spread of a triangular fuzzy number.
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2004
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11587/347237`
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