A large number of clustering methods exist including deterministic, probabilistic, and fuzzy clustering. All these methods are devoted to handling different types of uncertainty. No studies have been encountered on clustering taking into account a confluence of probabilistic and fuzzy information. In the existing studies, the reliability of extracted knowledge is one of the important issues to be investigated. The concept ofZ-number arises as a formal construct that expresses reliability of information under bimodal distribution. In this article, we propose an approach to construction ofZ-number-valued clusters of a dataset for evaluation of reliability of extracted data-driven knowledge. Real-world applications are given that confirm the usefulness of the proposed method.

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.

Linear programming (LP) is the operations research technique frequently used in the fields of science, economics, business, management science, and engineering. Although it is investigated and applied for more than six decades, and LP models with different level of generalization of information about parameters including models with interval, fuzzy, generalized fuzzy, and random numbers are considered, until now there is no approach to account for reliability of information within the framework of LP. Professor L. Zadeh introduced the concept of a Z-number to describe uncertain information, which is a more generalized notion closely related to reliability. The use of Z-information is more adequate and intuitively meaningful for formalizing information structure of a decision problem. In this paper, we suggest a study of fully Z-number based LP (Z-LP) model to better fit real-world problems within the framework of LP. We propose the method to solve Z-LP problems, which utilize differential evolution optimization and Z-number arithmetic developed by the authors. The suggested model and solution method for Z-LP are illustrated on the basis of a benchmark LP problem, where we conduct comparative analysis, which shows validity of the approach.

In decision analysis, uncertainty is usually described in the framework of probability. However, a large number of experimental and theoretical studies showed that a single nature of probability does not accurately capture human preferences. To avoid this drawback, they use imprecise probabilities. But, as decision maker is usually uncertain about first-order imprecise probabilities, imprecise hierarchical probability models are used. For most of such models, the second levels are precise. There also exist studies on two-level imprecise hierarchical models, which use imprecise probabilities or possibilities at the second level. Most of these works are based on lower prevision theory leading to a large number of optimization problems. In the present paper, we propose an imprecise hierarchical decision-making model where the first and the second level are described by interval probabilities. The method associates with the construction of a nonadditive measure as a lower prevision and uses this capacity in Choquet integral for constructing a utility function.