scholarly journals New Pythagorean Entropy Measure with Application in Multi-Criteria Decision Analysis

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1600
Author(s):  
Neeraj Gandotra ◽  
Bartłomiej Kizielewicz ◽  
Abhimanyu Anand ◽  
Aleksandra Bączkiewicz ◽  
Andrii Shekhovtsov ◽  
...  
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Entropy ◽  
Entropy Measure ◽  
Pythagorean Fuzzy Set ◽  
Additional Advantage ◽  
Pythagorean Fuzzy Sets ◽  
Intuitionistic Fuzzy

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods.

scholarly journals On Intuitionistic Fuzzy Entropy of Order-α

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Rajkumar Verma ◽  
BhuDev Sharma
Keyword(s):  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Entropy ◽  
Entropy Measure ◽  
Fuzzy Sets Theory ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Entropy ◽  
Renyi’S Entropy ◽  
Sets Theory

Using the idea of Rènyi’s entropy, intuitionistic fuzzy entropy of order-α is proposed in the setting of intuitionistic fuzzy sets theory. This measure is a generalized version of fuzzy entropy of order-α proposed by Bhandari and Pal and intuitionistic fuzzy entropy defined by Vlachos and Sergiadis. Our study of the four essential and some other properties of the proposed measure clearly establishes the validity of the measure as intuitionistic fuzzy entropy. Finally, a numerical example is given to show that the proposed entropy measure for intuitionistic fuzzy set is reasonable by comparing it with other existing entropies.

scholarly journals A p-Ideal in BCI-Algebras Based on Multipolar Intuitionistic Fuzzy Sets

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 993
Author(s):  
Jeong-Gon Lee ◽  
Mohammad Fozouni ◽  
Kul Hur ◽  
Young Bae Jun
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Finite Degree ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Ideal ◽  
The Relationship

In 2020, Kang, Song and Jun introduced the notion of multipolar intuitionistic fuzzy set with finite degree, which is a generalization of intuitionistic fuzzy set, and they applied it to BCK/BCI-algebras. In this paper, we used this notion to study p-ideals of BCI-algebras. The notion of k-polar intuitionistic fuzzy p-ideals in BCI-algebras is introduced, and several properties were investigated. An example to illustrate the k-polar intuitionistic fuzzy p-ideal is given. The relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal is displayed. A k-polar intuitionistic fuzzy p-ideal is found to be k-polar intuitionistic fuzzy ideal, and an example to show that the converse is not true is provided. The notions of p-ideals and k-polar ( ∈ , ∈ ) -fuzzy p-ideal in BCI-algebras are used to study the characterization of k-polar intuitionistic p-ideal. The concept of normal k-polar intuitionistic fuzzy p-ideal is introduced, and its characterization is discussed. The process of eliciting normal k-polar intuitionistic fuzzy p-ideal using k-polar intuitionistic fuzzy p-ideal is provided.

scholarly journals Intuitionistic Fuzzy Planar Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Noura Alshehri ◽  
Muhammad Akram
Keyword(s):  
Graph Theory ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Planar Graphs ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Dual Graphs ◽  
Intuitionistic Fuzzy

Graph theory has numerous applications in modern sciences and technology. Atanassov introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Intuitionistic fuzzy set has shown advantages in handling vagueness and uncertainty compared to fuzzy set. In this paper, we apply the concept of intuitionistic fuzzy sets to multigraphs, planar graphs, and dual graphs. We introduce the notions of intuitionistic fuzzy multigraphs, intuitionistic fuzzy planar graphs, and intuitionistic fuzzy dual graphs and investigate some of their interesting properties. We also study isomorphism between intuitionistic fuzzy planar graphs.

scholarly journals A New Similarity Measure between Intuitionistic Fuzzy Sets and Its Application to Pattern Recognition

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yafei Song ◽  
Xiaodan Wang ◽  
Lei Lei ◽  
Aijun Xue
Keyword(s):  
Pattern Recognition ◽  
Fuzzy Sets ◽  
Similarity Measure ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Similarity Measures ◽  
Intuitionistic Fuzzy Sets ◽  
Membership Degree ◽  
Intuitionistic Fuzzy ◽  
Axiomatic Definition

As a generation of ordinary fuzzy set, the concept of intuitionistic fuzzy set (IFS), characterized both by a membership degree and by a nonmembership degree, is a more flexible way to cope with the uncertainty. Similarity measures of intuitionistic fuzzy sets are used to indicate the similarity degree between intuitionistic fuzzy sets. Although many similarity measures for intuitionistic fuzzy sets have been proposed in previous studies, some of those cannot satisfy the axioms of similarity or provide counterintuitive cases. In this paper, a new similarity measure and weighted similarity measure between IFSs are proposed. It proves that the proposed similarity measures satisfy the properties of the axiomatic definition for similarity measures. Comparison between the previous similarity measures and the proposed similarity measure indicates that the proposed similarity measure does not provide any counterintuitive cases. Moreover, it is demonstrated that the proposed similarity measure is capable of discriminating difference between patterns.

scholarly journals Distance Based Entropy Measure of Interval-Valued Intuitionistic Fuzzy Sets and Its Application in Multicriteria Decision Making

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Tabasam Rashid ◽  
Shahzad Faizi ◽  
Sohail Zafar
Keyword(s):  
Decision Making ◽  
Fuzzy Set ◽  
Distance Measure ◽  
Intuitionistic Fuzzy Set ◽  
Multicriteria Decision Making ◽  
Fuzzy Entropy ◽  
Entropy Measure ◽  
Multicriteria Decision ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

Fuzzy entropy means the measurement of fuzziness in a fuzzy set and therefore plays a vital role in solving the fuzzy multicriteria decision making (MCDM) and multicriteria group decision making (MCGDM) problems. In this study, the notion of the measure of distance based entropy for uncertain information in the context of interval-valued intuitionistic fuzzy set (IVIFS) is introduced. The arithmetic and geometric average operators are firstly used to aggregate the interval-valued intuitionistic fuzzy information provided by the decision makers (DMs) or experts corresponding to each alternative, and then the fuzzy entropy of each alternative is calculated based on proposed distance measure. Several numerical examples are solved to demonstrate the application to MCDM and MCGDM problems to show the effectiveness of the proposed approach.

scholarly journals Note on one inequality and its application in intuitionistic fuzzy sets theory. Part 1

2021 ◽  
Vol 27 (1) ◽  
pp. 53-59
Author(s):  
Mladen V. Vassilev-Missana
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Sets Theory ◽  
Membership Functions ◽  
Real Numbers ◽  
Intuitionistic Fuzzy ◽  
Sets Theory

The inequality \mu^{\frac{1}{\nu}} + \nu^{\frac{1}{\mu}} \leq 1 is introduced and proved, where \mu and \nu are real numbers, for which \mu, \nu \in [0, 1] and \mu + \nu \leq 1. The same inequality is valid for \mu = \mu_A(x), \nu = \nu_A(x), where \mu_A and \nu_A are the membership and the non-membership functions of an arbitrary intuitionistic fuzzy set A over a fixed universe E and x \in E. Also, a generalization of the above inequality for arbitrary n \geq 2 is proposed and proved.

scholarly journals New Distance Measure for Atanassov’s Intuitionistic Fuzzy Sets and Its Application in Decision Making

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 429 ◽  
Author(s):  
Di Ke ◽  
Yafei Song ◽  
Wen Quan
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Distance Measure ◽  
Intuitionistic Fuzzy Set ◽  
Research Area ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy ◽  
The Difference ◽  
Atanassov’S Intuitionistic Fuzzy Sets

The intuitionistic fuzzy set introduced by Atanassov has greater ability in depicting and handling uncertainty. Intuitionistic fuzzy measure is an important research area of intuitionistic fuzzy set theory. Distance measure and similarity measure are two complementary concepts quantifying the difference and closeness of intuitionistic fuzzy sets. This paper addresses the definition of an effective distance measure with concise form and specific meaning for Atanassov’s intuitionistic fuzzy sets (AIFSs). A new distance measure for AIFSs is defined based on a distance measure of interval values and the transformation from AIFSs to interval valued fuzzy sets. The axiomatic properties of the new distance measure are mathematically investigated. Comparative analysis based in numerical examples indicates that the new distance measure is competent to quantify the difference between AIFSs. The application of the new distance measure is also discussed. A new method for multi-attribute decision making (MADM) is developed based on the technique for order preference by similarity to an ideal solution method and the new distance measure. Numerical applications indicate that the developed MADM method can obtain reasonable preference orders. This shows that the new distance measure is effective and rational from both mathematical and practical points of view.

scholarly journals A Novel Approach to Generalized Intuitionistic Fuzzy Sets Based on Interpolative Boolean Algebra

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2115
Author(s):  
Pavle Milošević ◽  
Bratislav Petrović ◽  
Ivana Dragović
Keyword(s):  
Boolean Algebra ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy ◽  
Novel Approach ◽  
Problematic Situations ◽  
Special Case

One of the main issues in IFS theory are generalizations of intuitionistic fuzzy set (IFS) definition as well as IFS operations. In this paper, we present the LBIFS-IBA approach by applying operations based on interpolative Boolean algebra (IBA) on generalized IFS. Namely, LBIFS are defined as a special case of Liu’s generalized IFS with the maximal interpretational surface. By extending the interpretational surface, the descriptive power of the approach is enhanced, and therefore the problematic situations when μA+νA>1 can be modeled. In addition, IBA-based algebra secures Boolean properties of the proposed approach. Considerable attention is given to comprehension of uncertainty within LBIFS-IBA, i.e., we propose a novel manner of uncertainty interpretation by treating values from [−1,1] interval. In order to prove its importance, we compare LBIFS-IBA with several well-known IFS generalizations, showing that only our approach offers meaningful uncertainty interpretation is all selected cases. Additionally, we illustrate the practical benefits of LBIFS-IBA by applying it to an example of modeling Japanese candlesticks for price charting and paying special attention to uncertainty interpretation.

scholarly journals Intuitionistic Fuzzy Sets Based Inpainting for Reconstruction of Heritage Images

2021 ◽  
Vol 12 (3) ◽  
pp. 5556-5565
Author(s):  
Shweta Dhondse Et. al.
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Image Inpainting ◽  
Assessment Method ◽  
Intuitionistic Fuzzy Sets ◽  
Characteristic Functions ◽  
Intuitionistic Fuzzy ◽  
Full Structure ◽  
Quality Assessment Method

Image inpainting is a process of reconstructing damaged or missing part of an image. Initially the work of restoration was performed by skilled artists and was limited to paintings and other artwork of eminence but this process was very tedious and strenuous and therefore digital image inpainting was introduced. The application of inpainting in reconstruction of heritage images is garnering a lot of attention from researchers. The reconstruction of heritage images using inpainting poses a challenging task because of its very high resolution and high meaning full structure content. In traditional exemplar based algorithms the patch with the minimum distance or high similarity may not always be the best match patch.   One of the characteristic functions of the Intuitionistic fuzzy set is its definition of degree of hesitancy. In the proposed paper this concept is used to find best exemplars. The mathematical model of intuitionistic fuzzy set is studied and a modified exemplar algorithm based on Intuitionistic fuzzy sets is proposed to solve image inpainting problem. The proposed intuitionistic based inpainting algorithm is applied to heritage images. The results are compared with existing and earlier proposed algorithms on the basis of subjective inpainting quality assessment method

scholarly journals An Effective Interval-Valued Intuitionistic Fuzzy Entropy to Evaluate Entrepreneurship Orientation of Online P2P Lending Platforms

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaohong Chen ◽  
Li Yang ◽  
Pei Wang ◽  
Wei Yue
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Entropy ◽  
Evaluation Procedure ◽  
Entropy Measure ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Entropy ◽  
Essential Properties ◽  
P2p Lending ◽  
Interval Valued

This paper describes an approach to measure the entrepreneurship orientation of online P2P lending platforms. The limitations of existing methods for calculating entropy of interval-valued intuitionistic fuzzy sets (IVIFSs) are significantly improved by a new entropy measure of IVIFS considered in this paper, and then the essential properties of the proposed entropy are introduced. Moreover, an evaluation procedure is proposed to measure entrepreneurship orientation of online P2P lending platforms. Finally, a case is used to demonstrate the effectiveness of this method.

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