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.

In this article, we introduce the interval-valued Intuitionistic Fuzzy set (\textbf{IVIFS}), which are generalized forms of Intuitionistic Fuzzy set (\textbf{IFS}) and Fuzzy set, this is because in Intuitionistic Fuzzy sets the non-membership function also applies to evaluations, and these sets are useful for modelling ambiguous concepts that abound in real problems. Here we try to look for new methods for more practical solutions in optimization problems for various sciences such as computer science, mathematics, engineering, medicine, psychology, climate and etc. First, with the introduction of t-norm Frank, an action we construct some Frank aggregation operators on interval-valued Intuitionistic Fuzzy Numbers (\textbf{IVIFN}s), including the Frank weighted averaging operator, Frank-ordered weighted averaging operator, Frank hybrid weighted averaging operator, Frank geometric weighted averaging operator, Frank geometric-ordered weighted averaging operator, and Frank geometric hybrid weighted averaging operator. Also, examine some of the characteristics of these operators. In the following, we introduce two multiple attribute group decision-making methods (\textbf{MAGDM}) based on such operators. Finally, we provide illustrative examples of these methods.

It is important to have an Intuitionistic Fuzzy set that allows each set element to have a membership value, a non-membership value, and a hesitancy value. This is because each element of the set can possess all three values. We will focus on one type of continuous Intuitionistic Fuzzy number, called trapezoidal Intuitionistic Fuzzy Numbers, because they are more flexible in representing information about membership and non-membership functions and are continuous. This research aims to introduce a parametric ranking and distance measure to compare and obtain the distinction value between Intuitionistic trapezoidal Fuzzy Numbers. Parametric measures offer more flexibility than deterministic measurement tools in modeling real-world problems by considering a suitable variety of responses based on different levels of parameters. After presenting the structure and effective indicators of the proposed tool, we have detailed its features and basic principles. Moreover, based on this measure, a hybrid process is designed for multi-criteria group decision-making (MCGDM) problems with trapezoidal Intuitionistic Fuzzy data. A numerical example is also examined to elucidate the implementation process of this integrated methodology. Additionally, comparative analysis with some related methods confirms the adequate performance of the new parametric measure in combined methods with similar subjects.

Triangular Intuitionistic Fuzzy Numbers (TIFNs) is a special case of Intuitionistic Fuzzy (IF) set and the ranking of TIFNs is an important problem. The aim of this paper is to develop a new methodology for ranking TIFNs by using multiattribute decision making methods (MADM). In this methodology, the value and ambiguity indices of TIFNs may be considered as the attributes and the TIFNs in comparison are seen as the alternatives. A compromise ratio method for Fuzzy MADM is developed based on the concept that larger TIFN should close to the maximum value index and is far away from the minimum ambiguity index simultaneously. The proposed ranking method is applied to solve multiattribute decision making problems in which the ratings of alter- natives on attributes are expressed by using TIFNs. Numerical examples are examined to demonstrate the implementation process and applicability of the proposed method in this paper. Furthermore, a comparison analysis of the proposed method is conducted to show its advantages over other methods.