Contents I. Introduction
In The real world, information related to human cognition is characterized by uncertainty. To model and process this type of uncertain information, researchers have successively proposed various theories and tools, such as fuzzy set theory [1], evidence theory [2], [3], interval values [4], Z-numbers [5], and D number [6], [7]. These mathematical tools are widely used in many practical decision-making problems [8]–[15]. As one of the most popular mathematical tools in uncertain environments, fuzzy numbers can effectively reflect the fuzzy aspects of information, but the reliability of information is ignored. Currently, we are aware that there is a class of real-world problems whose related information is a combination of fuzzy and probability uncertainty. In order to create a formal basis for dealing with such uncertain information, Zadeh [5] proposed the concept of Z-number. A Z-number is an ordered pair of fuzzy numbers related to the real-valued uncertainty variable . is the fuzzy constraint on , and is a measure of the reliability of . Since the proposal of the Z-number, it has continued to attract the attention of scholars [12], [13], [16]–[26]. Aliev et al. [16], [18] borrowed the principle of extension and considered factors such as compatibility and consistency to define the arithmetic algorithm of Z-numbers and the measurement of the difference of Z-numbers. Liu et al. [23] proposed the concept of the negation of discrete Z-numbers, which opened a new door for processing Z-number information. Massanet et al. [26] introduced a novel concept of mixed-discrete Z-numbers to reduce the computational complexity of operations in the decision-making process. Recently, Aliev’s team [21] has been dedicated to studying the eigenvalues and eigenvectors of the Z-matrix to solve the partial reliability of the information contained in the Z-matrix in practical problems.
