INNER PRODUCT BASED ENTROPY IN THE INTUITIONISTIC FUZZY SETTING

2006 ◽  
Vol 14 (03) ◽  
pp. 351-366 ◽  
Author(s):  
IOANNIS K. VLACHOS ◽  
GEORGE D. SERGIADIS
Keyword(s):  
Fuzzy Sets ◽  
Correlation Coefficient ◽  
Intuitionistic Fuzzy Sets ◽  
Inner Product ◽  
Entropy Measure ◽  
Entropy Formula ◽  
Intuitionistic Fuzzy ◽  
New Correlation ◽  
Fuzzy Setting ◽  
Fuzzy Vector

In this paper, an entropy formula for intuitionistic fuzzy sets is presented. The notion of the intuitionistic fuzzy vector is introduced and an entropy measure is derived based on the normalized inner product between intuitionistic fuzzy vectors in the unit intuitionistic fuzzy cube. Some considerations regarding the geometrical representations of intuitionistic fuzzy sets are also stated and a connection between the different notions of entropy in the intuitionistic fuzzy setting is established. Finally, the relation of the proposed entropy measure to the concepts of correlation and informational energy for intuitionistic fuzzy sets is revealed and a new correlation coefficient is introduced.

Novel Similarity Measures, Entropy of Intuitionistic Fuzzy Sets and their Application in Software Quality Evaluation

2021 ◽  
Author(s):  
Xuan Thao Nguyen ◽  
Shuo Yan Chou
Keyword(s):  
Fuzzy Sets ◽  
Similarity Measures ◽  
Intuitionistic Fuzzy Sets ◽  
Entropy Measure ◽  
Uncertain Information ◽  
Membership Functions ◽  
Software Projects ◽  
Intuitionistic Fuzzy ◽  
Software Quality Evaluation

Abstract Intuitionistic fuzzy sets (IFSs), including member and nonmember functions, have many applications in managing uncertain information. The similarity measures of IFSs proposed to represent the similarity between different types of sensitive fuzzy information. However, some existing similarity measures do not meet the axioms of similarity. Moreover, in some cases, they could not be applied appropriately. In this study, we proposed some novel similarity measures of IFSs constructed by combining the exponential function of membership functions and the negative function of non-membership functions. In this paper, we also proposed a new entropy measure as a stepping stone to calculate the weights of the criteria in the proposed multi-criteria decision making (MCDM) model. The similarity measures used to rank alternatives in the model. Finally, we used this MCDM model to evaluate the quality of software projects.

Vague Correlation Coefficient of Interval Vague Sets and its Applications to Topsis in MADM Problems

2014 ◽  
pp. 140-173
Author(s):  
John Robinson P. ◽  
Henry Amirtharaj E. C.
Keyword(s):  
Fuzzy Sets ◽  
Correlation Coefficient ◽  
Intuitionistic Fuzzy Sets ◽  
New Method ◽  
Numerical Illustration ◽  
Vague Sets ◽  
Intuitionistic Fuzzy ◽  
Statistical Confidence ◽  
Order Preference ◽  
Novel Method

Various attempts are made by researchers on the study of vagueness of data through Intuitionistic Fuzzy sets and Vague sets, and also it is shown that Vague sets are Intuitionistic Fuzzy sets. However, there are algebraic and graphical differences between Vague sets and Intuitionistic Fuzzy sets. In this chapter, an attempt is made to define the correlation coefficient of Interval Vague sets lying in the interval [0,1], and a new method for computing the correlation coefficient of interval Vague sets lying in the interval [-1,1] using a-cuts over the vague degrees through statistical confidence intervals is also presented by an example. The new method proposed in this work produces a correlation coefficient in the form of an interval. The proposed method produces a correlation coefficient in the form of an interval from a trapezoidal shaped fuzzy number derived from the vague degrees. This chapter also aims to develop a new method based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to solve MADM problems for Interval Vague Sets (IVSs). A TOPSIS algorithm is constructed on the basis of the concepts of the relative-closeness coefficient computed from the correlation coefficient of IVSs. This novel method also identifies the positive and negative ideal solutions using the correlation coefficient of IVSs. A numerical illustration explains the proposed algorithms and comparisons are made with some existing methods.

Vague Correlation Coefficient of Interval Vague Sets and its Applications to Topsis in MADM Problems

Fuzzy Systems ◽  
2017 ◽  
pp. 1110-1149
Author(s):  
John Robinson P. ◽  
Henry Amirtharaj E. C.
Keyword(s):  
Fuzzy Sets ◽  
Correlation Coefficient ◽  
Intuitionistic Fuzzy Sets ◽  
New Method ◽  
Numerical Illustration ◽  
Vague Sets ◽  
Intuitionistic Fuzzy ◽  
Statistical Confidence ◽  
Order Preference ◽  
Novel Method

Various attempts are made by researchers on the study of vagueness of data through Intuitionistic Fuzzy sets and Vague sets, and also it is shown that Vague sets are Intuitionistic Fuzzy sets. However, there are algebraic and graphical differences between Vague sets and Intuitionistic Fuzzy sets. In this chapter, an attempt is made to define the correlation coefficient of Interval Vague sets lying in the interval [0,1], and a new method for computing the correlation coefficient of interval Vague sets lying in the interval [-1,1] using a-cuts over the vague degrees through statistical confidence intervals is also presented by an example. The new method proposed in this work produces a correlation coefficient in the form of an interval. The proposed method produces a correlation coefficient in the form of an interval from a trapezoidal shaped fuzzy number derived from the vague degrees. This chapter also aims to develop a new method based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to solve MADM problems for Interval Vague Sets (IVSs). A TOPSIS algorithm is constructed on the basis of the concepts of the relative-closeness coefficient computed from the correlation coefficient of IVSs. This novel method also identifies the positive and negative ideal solutions using the correlation coefficient of IVSs. A numerical illustration explains the proposed algorithms and comparisons are made with some existing methods.

Correlation in Interval-Valued Atanassov’s Intuitionistic Fuzzy Sets — Conjugate and Negation Operators

2017 ◽  
Vol 25 (05) ◽  
pp. 787-819 ◽  
Author(s):  
Renata Hax Sander Reiser ◽  
Benjamin Bedregal
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Correlation Coefficient ◽  
Intuitionistic Fuzzy Sets ◽  
Conjugate Functions ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Index ◽  
Intuitionistic Fuzzy Logic ◽  
Interval Valued ◽  
Atanassov’S Intuitionistic Fuzzy Sets

This paper studies the conjugate functions related to main connectives of the Intervalvalued Atanassov’s Intuitionistic Fuzzy Logic. The relationships among automorphism classes are formalized by the ϕ-representability theorem, passing from automorphisms to interval-valued intuitionistic automorphisms, also visiting other two ones, intuitionistic automorphisms and interval-valued automorphisms. Additionally, the ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation can be obtained either from an interval-valued fuzzy negation or from an Atanassov’s intuitionistic fuzzy negation, including a discussion presenting such reverse constructions. The ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation not only preserves the main properties of its corresponding fuzzy negation but also of two other ones, the intuitionistic fuzzy negation and interval-valued fuzzy negation. Moreover, an extension of the intuitionistic fuzzy index as well as the correlation coefficient is discussed in terms of fuzzy negations, by considering the Atanassov’s Intuitionistic Fuzzy Logic.

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