A new linear ordering of fuzzy numbers on subsets of ${{\mathcal F}({\pmb{\mathbb{R}}}})$

Abstract

We introduce a novel linear order on every family of fuzzy numbers which satisfies the assumption that their modal values must be all different and must form a compact subset of ${\mathbb{R}}$. A distinct new feature is that our linear determined procedure employs the corresponding order of a class interval associated with a confidence measure which seems intuitively anticipated. It is worthy noting that although we start from an entirely different rationale, we introduce a fuzzy ordering which initially coincides with the one established earlier by Ramik and Rimanek. However, this fuzzy ordering does not apply when the supports of the fuzzy numbers overlap. In order to cover such cases we extent this initial fuzzy ordering to the “extended fuzzy order” (XFO). This new XFO method includes a possibility and a necessity measure which are compared with the widely accepted PD and NSD indices of D. Dubois and H. Prade. The comparison shows that our possibility and necessity measures comply better with our intuition.