A hesitant fuzzy number (HFN) is important as a generalization of the fuzzy number for hesitant fuzzy analysis and takes some applications that were discussed in recent literature. In this paper, we develop the hesitant fuzzy arithmetic, which is based on the extension principle for hesitant fuzzy sets. Employing this principle, standard arithmetic operations on fuzzy numbers are extended to HFNs and we show that the outcome of these operations on two HFNs are an HFN. Also we use the extension principle in HFSs for the ranking of HFNs, which may be an interesting topic. In this paper, we show that the HFNs can be ordered in a natural way. To introduce a meaningful ordering of HFNs, we use a new lattice operation on HFNs based upon extension principle and de ning the Hamming distance on them. Finally, the applications of them are explained on optimization and decision-making problems.

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the arithmetic and geometric aggregation operators with hesitant fuzzy linguistic information. Then, motivated by the idea of traditional arithmetic operation, we have developed some aggregation operators for aggregating hesitant fuzzy linguistic information: hesitant fuzzy linguistic weighted average (HFLWA) operator, hesitant fuzzy linguistic ordered weighted average (HFLOWA) operator and hesitant fuzzy linguistic hybrid average (HFLHA) operator. Furthermore, we propose the concept of the dual hesitant fuzzy linguistic set and develop some aggregation operators with dual hesitant fuzzy linguistic information. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy linguistic multiple attribute decision making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

In this paper, we have studied the basic arithmetic operations for developed parabolic fuzzy numbers by using the concept of the transmission average, which was already implied in [F. Abbasi et al., A new attitude coupled with fuzzy thinking to fuzzy rings and  elds, Journal of Intelligent and fuzzy Systems, 2015] in its rudimentary form and was  nally presented in its fully-edged form in [F. Abbasi et al., A new and e cient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers, Journal of fuzzy Set Valued Analysis, 2016]. The major advantage of these operations is that they  ndings are closer to reality than extension principle-based fuzzy arithmetic operations (in the domain of the membership function) or interval arithmetic (in the domain of -cuts). A technical example is given to illustrate applying the method. The proposed method can model and analyze the fuzzy system reliability in a more exible and intelligent manner in comparison with the other methods.

A bipolar fuzzy set is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to the fuzzy model. We define certain notions, including a bipolar fuzzy number and bipolar fuzzy arithmetic. In this paper we propose the new fuzzy arithmetic operations based on on bipolar fuzzy numbers from [1, 2]. We define 2-dipole fuzzy numbers and cutting them. The properties of these propose operations and their fundamental qualities are discussed in detail. Several illustrative examples were given to show the accomplishment and ability of the proposed method. At the end it is shown that the solution of the proposed method in comparison to other methods of solving fuzzy equations are more realistic, that is, they have smaller support. Further, we analyze our new approach to find the solutions of a bipolar fuzzy linear equations. In this paper, in addition to familiarity with bipolar fuzzy number operations based on transmission-average and presenting practical solutions for calculations in specific cases, this problem is identified.

Currently fuzzy set theory has a wide range to model real life problems with incomplete or vague information which perfectly suits the reality and its application is theatrically increasing. This work explored the basic fuzzy operations with the Gaussian Membership using the α-cut method. As it is known that, the Gaussian membership function has a great role in modelling the fuzzy problems this is what impelled to explore its operation which can further be used in analysis of fuzzy problems. Primarily the basic operations which has been discussed here are addition, subtraction, multiplication, division, reciprocal, exponential, logarithmic and nth power.

This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution Set (TSS)‎, Controllable ‎Solution Set (CSS)‎, ‎and Algebraic Solution Set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.

In this study fuzzy arithmetic is presented as a tool to tackle the prediction of the amount of barium, strontium and calcium sulfates scales in oilfield operations. Since the shape of fuzzy numbers’ membership functions is a spread representative of the whole possible values for a special model parameter, fuzzy numbers are able to consider the uncertainties in parameter determinations and thus give more real results than crisp values. Solubility product models and other required Equations for scale prediction contain uncertain parameters and therefore application of fuzzy numbers can be useful. LR fuzzy numbers and related primary arithmetical operations based on Zadeh's extension principle have been introduced and their use in predicting scale depositions has been investigated. Parameters such as solubility products, free sulfate concentration and scale mass have been determined as fuzzy numbers. As a case study, scale depositions of barium and strontium sulfate resulted from mixing two incompatible waters have been obtained and compared with none fuzzy approach. fuzzy computations are able to predict the maximum scale mass with respect to existing information.

In recent years, industrial clusters have received considerable attention from economists and industry analysts because they are seen as the main reason for certain economic regions' economic growth and success. For many Industrial States Organization, the selection of industrial clusters has become a critical strategic consideration due to the Budget allocation priority. In this paper, an extended qualitative flexible multiple (QUALIFLEX) methods is used to solve problems regarding the priority among this cluster using probability hesitant fuzzy information, which can lead to allocating the budget for industrial clusters more effectively. For more accuracy, we have applied a hesitant fuzzy Topsis for prioritizing. Both rankings have been aggregated by the Copeland method. From our research results, the Larestan Muscat is of great importance, and Abade Inlaid Wood, Citrus packaging, Shiraz Marquetry, and Niriz stone have ranked respectively.

In the real world, in most cases, such as industry, management, and even in daily life, we encounter optimization and decision-making problems that require the opinions of experts and masters on the problem to be able to make the best decision. In these cases, it is necessary to use an optimization problem with hesitant fuzzy parameters.There are few studies on hesitant fuzzy linear programming (HFLP) problems. Therefore,in this paper, we  consider such problems.Especially, we study HFLP problems with hesitant cost coefficients. For this purpose,we propose the simplex  method to solve the introduced optimization problems and draw a flowchart of the proposed  method.Finally, by solving two illustrative examples with hesitant fuzzy information, we examine the applicability of the proposed method.

Since the problems of everyday life are relative, so far various tools such as fuzzy sets, intuitive fuzzy sets, etc. have been expressed to express these ambiguities in mathematical modeling. In 2009, Torra introduced a new horizon for the discussion of hesitant fuzzy sets to discuss issues that are uncertain about decision making. In the course of his work, the quantitative and qualitative expansion of uncertain fuzzy sets is discussed. In this article, for the purpose of introducing more researchers to hesitant fuzzy sets, we review the types of hesitant fuzzy sets such as dual uncertain fuzzy sets, generalized hesitant fuzzy sets, and so on.