The arithmetic of discrete Z-numbers

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, L.A. Zadeh suggested the concept of a Z-number as a more adequate concept for description of real-world information. A Z-number is an ordered pair Z=(A,B) of fuzzy numbers A and B used to describe a value of a random variable X, where A is an imprecise estimation of a value of X and B is an imprecise estimation of reliability of A. The main critical problem that naturally arises in processing Z-numbers-based information is computation with Z-numbers. The general ideas underlying computation with continuous Z-numbers (Z-numbers with continuous components) is suggested by the author of the Z-number concept. However, as he mentions, “Problems involving computation with Z-numbers is easy to state but far from easy to solve”. Nowadays there is no arithmetic of Z-numbers suggested in the existing literature. Taking into account the fact that real problems are characterized by linguistic information which is, as a rule, described by a discrete set of meaningful linguistic terms, in our study we consider discrete Z-numbers. We suggest theoretical aspects of such arithmetic operations over discrete Z-numbers as addition, subtraction, multiplication, division, square root of a Z-number and other operations. The validity of the suggested approach is demonstrated by a series of numerical examples.