Functions defined on a set of Z-numbers

A wide class of problems in science and engineering requires mathematical formalization. One of the important aspects in this regard is that we should account for various types of uncertainties, which are simultaneously present in real-world problems and result from imprecise and partially reliable information. This mandates a need for a theoretical basis of uncertain functional dependencies. The theories of interval-valued functions, functions of random variables, and fuzzy functions are well-developed. The main disadvantage of these theories is that each of them can deal with only a single type of uncertainty. In order to ground the formal basis for dealing with real-world information, Zadeh suggested the concept of a Z-number as an ordered pair Z=(A,B) of fuzzy numbers used to describe a value of a random variable X, where A is a fuzzy constraint on values of X and B is a fuzzy value of probability measure of A. Unfortunately, there is no study on functions of Z-numbers in existence. In this paper, for the first time, we propose a general methodology for a construction of such functions on the basis of the extension principle applied to Z-numbers. The proposed methodology is helpful in restricting the increase of uncertainty in computation the values of Z-valued functions. Some properties of functions of Z-numbers are considered. We construct typical and commonly encountered functions such as addition of Z-numbers, minimum of Z-numbers, square root of Z-numbers and others. A spectrum of examples is provided to illustrate the proposed approach.