This paper first improves Chen and Hsieh’ s definition of Generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the Generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the -cutting sets or Generalized -cutting sets, so that the Generalized fuzzy numbers are integrated into a partial ordering set, a semi-ring and a complete non-separable metric space. Again, through two isomorphism theorem, the relationship between Generalized fuzzy number space and fuzzy number space is established. Finally, as an application of Generalized fuzzy numbers, the concept of continuous Generalized fuzzy number-valued functions is introduced. And it is pointed out that many results of trapezoidal Generalized fuzzy numbers can also be extended to Generalized fuzzy numbers.

Ranking fuzzy numbers(FNs) was a critical issue in fuzzy computing field. Generally, triangular FNs, trapezoidal FNs, and even interval-valued FNs(IVFNs) were often expressed in ranking. However, ranking intuitionistic FNs(IFNs) were less mentioned due to the complicated components in membership functions. Herein, we will develop fuzzy binary relation that is an extended fuzzy preference relation(EFPR) to express the preference degree of two IFNs, and then the relation is improved to be a relative preference relation(RPR) used to rank a set of IFNs. Since EFPR on IFNs is a total ordering relation, RPR will be also a total ordering relation. Based on belonging and non-belonging components of membership functions in IFNs, using EFPR being also fuzzy preference relation(FPR) is suitable to compare FNs on pairwise, but time complexity on fuzzy operation of comparison computing is complicated. Hence, RPR is developed to avoid comparing on pairwise. Through yielding RPR values for a set of IFNs, IFNs are effectively and efficiently ranked to utilize related decision-making problems.

In this paper I will introduce the trapezoidal intuitionistic fuzzy numbers (IF numbers) and will prove some operations for them. Also I am going to propose a new ordering method for IF numbers in which I will consider two characteristic values of membership and non-membership functions for an IF number. These values are defined by the integral of the inverse fuzzy membership and non-membership functions multiplied by the grade with powered parameter.

The concept of an intuitionistic fuzzy number (I F N) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an intuitionistic fuzzy number (IFN) is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.

Since the inception of intuitionistic fuzzy sets in 1986, many authors have proposed different methods for ranking intuitionistic fuzzy numbers (IFNs). However, due to the complexity of the problem, a method which gives a satisfactory result to all situations is a challenging task. Most of them contained some shortcomings, such as requirement of complicated calculations, inconsistency with human intuition and indiscrimination and some produce different rankings for the same situation and some methods cannot rank crisp numbers. For overcoming the above problems, in this paper, a new parametric ranking method for IFNs is proposed. It is developed based on the concept -cuts and -cuts and area on left side of IFNs. The proposed ranking method is applied to solve partner selection problem in which the rating of partner on attributes are expressed by using triangular IFNs. The proposed method is much simpler and more efficient than other methods in the literature. Some comparative examples are also given to illustrate the advantages of the proposed method.

Sustainable tourism is one of today's rapidly growing economic areas. Promotional activities should be carried out to increase the income from sustainable tourism. To carry out these studies effectively, it is necessary to choose the best centre. Multi-Criteria Decision-Making (MCDM) approaches can be used to select the best tourism centre. In this paper, a new Generalized similarity measure on intuitionistic trapezoidal fuzzy multi-numbers is developed for decision information. The desirable properties of this proposed measure are presented in detail. Further, we develop an approach to the MCDM problem based on the proposed measure. Finally, the effectiveness and applicability of our proposed MCDM model, as well as comparison analysis with other methods, are illustrated with a practical example.

Normal intuitionistic fuzzy Number (NIFN), which is introduced based on intuitionistic fuzzy sets and normal fuzzy numbers, is a useful tool for presenting uncertain information under complicated situations. This study focuses on the development of an e ective method by combining NIFNs with the power average and harmonic mean operators to address Multi-Criteria Group Decision-Making (MCGDM) problems, wherein weight information is completely unknown. First, an e ective ranking method for NIFNs is provided in view of defects of the existing comparison method of NIFNs. Subsequently, three normal intuitionistic Generalized power harmonic aggregation operators are proposed based on the operations of NIFNs. Next, a new MCGDM method is developed. Finally, a numerical example concerning coal mine safety evaluation is provided for demonstration. The feasibility and validity of the proposed method are further veri ed by sensitivity analysis and comparison with other existing methods.

The objective of this paper is to develop arithmetic operations between Generalized trapezoidal (triangular) fuzzy numbers so that the drawbacks of the existing works are overcome. In this respect, the extension principle has been used to calculate different arithmetic operations.

In this paper, newly de ned level operators and modal-like operators over extensional Generalized intuitionistic fuzzy sets (GIFSB) are proposed. Some of the basic properties of the new operators are discussed.