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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]

This paper intends to introduce mathematical tools for aggregation of the generalized hesitant fuzzy numbers in order to increase the use of them in the real world. The proposed operators, are based on general form of t-norm and t-conorm functions, enable us to do some mathematical computations and aggregate the given generalized hesitant fuzzy numbers. At first, some famous Archimedean t-norms and t-conorms, i.e., Algebraic, Einstein, Hamacher, and Frank t-norms and t-conorms, and their properties, have been developed to be employed with generalized hesitant fuzzy numbers. Then, several averaging and geometric-based aggregation operators for generalized hesitant fuzzy numbers have been proposed. Later on, a decision-making algorithm has been defined based on such operators to address the problems. The necessity and application of the proposed concepts have been explained by some numerical examples. [ABSTRACT FROM AUTHOR]

Generalized hesitant fuzzy numbers (GHFNs) are able to directly manage situations in which we may encounter a finite set of known values with a finite set of degrees of doubt as quantitative approximations of an uncertain situation/quantification of a linguistic expression. They are new extensions of hesitant fuzzy sets, which have been considered in this paper. In fact, in this paper, GHFNs will be utilized to model the uncertainty of the assessment values of options against criteria in multi-attribute decision making (MADM) problems. It means that all of the elements of decision matrix are GHFNs. Then, the technique for order of preference by similarity to ideal solution (TOPSIS) method, as a very successful method in solving MADM problems, will be updated to be used with GHFNs. To this end, the distance between GHFNs must be defined to obtain the distances between given alternatives from each of two subjective alternatives (positive/negative ideal solutions). Thus, three existing famous distance measures, i.e., general distance ( d g ), Hamming distance ( d h ), and Euclidean distance ( d e ) measures, have been updated for GHFNs firstly. Then, the new TOPSIS method will be proposed based on GHFNs. Finally, the numerical examples have been appointed to illustrate the proposed method, analyze comparatively and validate it. [ABSTRACT FROM AUTHOR]

In this paper, we considered the defined intuitionistic fuzzy numbers (IFNs) on real numbers and proposed a new method based on Hamacher operators to operate with membership and non-membership degrees of the given IFN in aggregation process. Also, a new method will be introduced to compute the distance between IFNs. These methods will be applied to solve multi-attribute group decision-making (MAGDM) problems, in which IFNs were used to model the arising uncertainty from human judgments or assessments. To do it, Choquet integral, as a powerful tool in both independent and interactive criteria, will be used individually or as a tool to provide appropriate conditions for application of TOPSIS method. The applicability of these methods is illustrated by numerical examples. [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]

This paper studies a linear programming (LP) and a multiple attribute decision-making (MADM) model with hesitant fuzzy numbers (HFNs) for solving group decision-making problems under uncertainty. We introduce a new type of HFN whose flexibility allows for its direct application to standard LP and MADM methods while facing credibility considerations or strategic reports from the experts evaluating the alternatives. These problems are initially formulated as models with uncertain, crisp data. Decision makers and experts are then invited to evaluate the data and propose HFNs representing its credibility. As a result, the crisp LP and MADM techniques are transformed into hesitant fuzzy LP (HFLP) and MADM (HF-MADM) models. The popular simplex method and the steps defining the corresponding MADM model, namely, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), are extended to implement the new HFNs. The strategic environment introduced allows for the definition of selection strategies and equilibria determined by the credibility profiles of experts providing evaluations. Developing a probabilistic framework applicable to these models would require defining convolution functions that become increasingly complex as further experts are added to the analysis. We present several numerical comparisons between the proposed hesitant fuzzy simplex and MADM results and those derived from the initial crisp models. The strategic importance of data credibility as a fundamental determinant of the optimality of results obtained will also be highlighted. [ABSTRACT FROM AUTHOR]

The paper aims are to present a method to solve the multiple-attribute decision-making (MADM) problems under the hesitant fuzzy set environment. In MADM problems, the information collection, aggregation, and the measure phases are crucial to direct the problem. However, to handle the uncertainties in the collection data, a hesitant fuzzy number is one of the most prominent ways to express uncertain and vague information in terms of different discrete numbers rather than a single crisp number. Additionally, to aggregate and to rank the collective numbers, a TOPSIS ("Technique for Order of Preference by Similarity to Ideal Solution") and the Choquet integral (CI) are the useful tools. Keeping all these features, in the present paper, we combine the TOPSIS and CI methods for hesitant fuzzy information and hence present a method named as TOPSIS-CI to address the MADM problems. The presented method has been described with a numerical example. Finally, the validity of the stated method as well as a comparative analysis with the existing methods is addressed in detail. [ABSTRACT FROM AUTHOR]

There are two very usful methods for solving MAGDM problems, called TOPSIS method and Choquet integral method. Because of the ability of Choquet integral method in managing of interaction criteria, we combined it with TOPSIS method to use from their benefits. First, we will introduce a new method for computing the distance between IT2FNs. Then, we proposed a hybrid method based on TOPSIS and Choquet integral in fuzzy type-2 environment. Numerical examples are considered to illustrate the proposed methods. [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]