On a total order on the set of Z-numbers based on discretefuzzy numbers

Abstract

[eng] Z -numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as ameasure of certainty or sureness of A. From the initial proposal, several other approaches have been introduced in order to reduce the computational cost of the involved operations. One of such approaches is called discrete Z -numbers where A and B are modelled as discrete fuzzy numbers. In this paper, the construction of total orders on the set of discrete Z-numbers isinvestigated for the first time. Specifically, the total order is designed for discrete Z -numbers where the second component has membership values belonging to a finite and prefixed setof values. The method relies on solid and coherent linguistic criteria and several linguistic properties are analyzed. The order involves the transformation of the first components of the discrete Z -numbers by using the credibility of the second components in the sense that a lower credibility enlarges in a greater extent the uncertainty of the first component. Then atotal order on the set of discrete fuzzy numbers is applied. Finally, a practical example on how to order discrete Z -numbers is presented and a comparison with other ranking methodsis performed from which the strengths of our method are stressed.