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As the rapidly progressing applications of uncertainty theories, the need for modifications to some of their existing mathematical tools or creating new tools to deal correctly with them in various environments is also exposed. Hesitant fuzzy numbers (HFNs), as a particular case of fuzzy numbers, are not an exception to this rule. Considering the necessity of determining the distance between given HFNs in many of their practical applications, this article shows that the existing methods either do not provide correct results or are not able to meet the needs of users. This paper aims to present new methods for distance measures of hesitant fuzzy numbers. To do them, three prevalent distance measures, i.e., the generalized distance measure, the Hamming distance measure, and the Euclidean distance measure, will be optimized into three distinct trinal categories. With the approach of reducing error propagation via reducing some unnecessary mathematical computations, new distance measures on HFNs will be introduced, first. The middle is the modification of the first category, which is more suitable when the given HFNs are equal-distance by the previous formula. Also, as the third category, the weighted form of these distance measures has been proposed, to be used where the real and membership parts of HFNs are not of equal importance. As an application of these, a TOPSIS-based technique for solving multi-attribute group decision-making problems with HFNs has been proposed. A numerical example will be implemented to describe the presented method. Finally, along with the validation of the proposed method, its numerical comparison with some other existing methods will be discussed in detail. [ABSTRACT FROM AUTHOR]

Uncertainty has long been explored as an objective and inalienable reality, and then modeled via different theories such as probability theory, fuzzy sets (FSs) theory, vague sets, etc. Hesitant fuzzy sets (HFSs) as a generalization of FSs, because of their flexibility and capability, extended and applied in many practical problems very soon. However, the above theories cannot meet all the scientific needs of researchers. For example, in some decision-making problems we encounter predetermined definite data, which have inductive uncertainties. In other words, the numbers themselves are crisp in nature, but are associated with varying degrees of satisfaction or fairness from the perspective of each decision-maker/judge. To this end, in this article, hesitant fuzzy numbers as a generalization of hesitant fuzzy sets will be introduced. Some concepts such as the operation laws, the arithmetic operations, the score function, the variance of hesitant fuzzy numbers, and a way to compare hesitant fuzzy numbers will be proposed. Mean-based aggregation operators of hesitant fuzzy numbers, i.e. hesitant fuzzy weighted arithmetic averaging (HWAA), hesitant fuzzy weighted geometric averaging (HWGA), hesitant fuzzy ordered weighted arithmetic averaging (HOWAA), and hesitant fuzzy ordered weighted geometric averaging (HOWGA) operators have been discussed in this paper, too. These new concepts will be used to model, and solve an uncertain multi-attribute group decision making (MAGDM) problem. The proposed method will be illustrated by a numerical example and the validity of the obtained solution will be checked by test criteria. [ABSTRACT FROM AUTHOR]

Hesitant fuzzy set, as an extension of fuzzy set, has absorbed many researchers from various scientific backgrounds to deal with uncertainty. Solving decision-making problems is one of the most important applications of hesitant fuzzy set. To do so, it is needed to rank hesitant fuzzy elements (HFEs) and to define some mathematical operations on them. In this paper, a new method based on the power average operator is firstly suggested to compare the HFEs. Given that different HFEs may have different numbers of elements, new procedures are developed to make them uniform. Then, four kinds of hesitant power average operators, including the hesitant power average operator, the weighted hesitant power average operator, the ordered weighted hesitant power average operator and the hybrid weighted hesitant power average operator, are introduced. Finally, a new approach to solve the multiple attribute decision-making problems is introduced via the proposed operators. Numerical examples are given to illustrate and compare the proposed methods. [ABSTRACT FROM AUTHOR]