Approximate Reasoning on a Basis of <italic>Z</italic>-Number-Valued If–Then Rules

Approximate reasoning is about reasoning with imperfect information. Nowadays, a large diversity of approaches to approximate reasoning with fuzzy information and fuzzy type-2 information exists. It should be stressed, however, that real-world imperfect information is characterized by combination of fuzzy and probabilistic uncertainties, which is referred to as bimodal information. In view of this, Zadeh introduced the concept of a Z-number regarded as an ordered pair Z = (A, B) of fuzzy numbers A and B, where A is a linguistic value of a variable of interest, and B is a linguistic value of probability measure of A, playing a role of its reliability. Unfortunately, up to day, there is no research on approximate reasoning realized on the basis of if-then rules with Z-number-valued antecedents and consequents, briefly, Z- rules. Zadeh addressed this problem as related to an uncharted territory. In this paper, a new approach is developed to study approximate reasoning with Z-rules on a basis of linear interpolation. We provide an application of the approach to job satisfaction evaluation and to students' educational achievement evaluation problems related to psychological and perceptual issues naturally characterized by imperfect information. The obtained results show applicability and validity of the proposed approach.