A new approach to Zadeh’s Z-numbers: Mixed-discrete Z-numbers

Abstract

One of the main goals of computing with words is the accurate modeling of natural language. In this direction, 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 a measure of certainty or sureness of A. This structure allows to model many imprecise sentences of the natural language, but has the drawback of the complexity and hight computational cost of their operations, because the second component is usually considered from a probabilistic point of view. Since the computational problems are caused by the second component, we present in this paper a new approach called mixed-discrete Z-numbers. In this new approach the first component will be managed as a usual fuzzy number, and the second one as a discrete fuzzy number with support in a finite chain. That is, the second component B of a Z-number is modeled as a linguistic valuation based on a discrete fuzzy number and the operations on these second components are managed through aggregation functions on discrete fuzzy numbers. Understanding B as a measure of certainty and not as a measure of probability, greatly improves experts’ flexibility, allows to model situations where no probability distribution is known, and reduces greatly the computational complexity of Z-numbers operations. After studying these new Z-numbers and their operations, an application to reach a decision from a group of experts is presented in order to show the potential of this approach.