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Intuitionistic Fuzzy Sets and Applications
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Intuitionistic Fuzzy Sets and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Sets, Systems and Decision Making".

Deadline for manuscript submissions: closed (31 July 2021) | Viewed by 34326

Special Issue Editor

Dr. Vassia Atanassova

E-Mail Website
Guest Editor
Associate Professor, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria
Interests: intuitionistic fuzzy sets; intercriteria analysis; decision making under uncertainty
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The present Special Issue aims to collect the most recent and notable advancements of the theory and application of intuitionistic fuzzy sets. Originally defined in 1983 in Bulgaria by Krassimir Atanassov (1954), intuitionistic fuzzy sets have grown from a “happenstance” and “a mathematical game” devised in а hospital bed to make time pass faster, to one of the most powerful, practically usable, and actively investigated extensions of Zadeh’s fuzzy sets, researched by thousands of people around the world.

Historically speaking, for the first twenty years of their existence, intuitionistic fuzzy sets were mostly developed by Atanassov and a tiny circle of researchers around him, and mainly from the perspective of the mathematical logic and mathematics behind the concept, namely, the aspects of algebra, analysis, geometry, and others. Since then, with the development of information technologies and decision science, things have significantly changed, and interest in intuitionistic fuzzy sets has steeply increased, with around a thousand papers being published today on an annual basis in top-ranking journals and conferences in the fields of mathematics, engineering, and various other sciences. The community of theoretical researchers is growing, as well as the pool of practitioners who are using the theoretically developed concept in different applications in medicine, industry, economics, artificial intelligence, and others.

Literally an idea that appeared well before its time, intuitionistic fuzzy sets are today giving us valuable tools to handle the inherent uncertainty that defines the time we live in.

Assoc. Prof. Vassia Atanassova
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Intuitionistic fuzzy sets
  • Intuitionistic fuzzy logic
  • Interval-valued intuitionistic fuzzy sets
  • Extensions of intuitionistic fuzzy sets
  • Applications of intuitionistic fuzziness in artificial intelligence, medicine, industry, economics, etc.

Published Papers (14 papers)

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Research

13 pages, 4570 KiB  
Article
Socially Responsible Portfolio Selection: An Interactive Intuitionistic Fuzzy Approach
by Yahya Hanine, Youssef Lamrani Alaoui, Mohamed Tkiouat and Younes Lahrichi
Mathematics 2021, 9(23), 3023; https://doi.org/10.3390/math9233023 - 25 Nov 2021
Cited by 6 | Viewed by 1941
Abstract
In this study, we address the topic of sustainable and responsible portfolio investments (SRI). The selection of such portfolios is based, in addition to traditional financial variables, on environmental, social, and governance (ESG) criteria. The interest of our approach resides in allowing socially [...] Read more.
In this study, we address the topic of sustainable and responsible portfolio investments (SRI). The selection of such portfolios is based, in addition to traditional financial variables, on environmental, social, and governance (ESG) criteria. The interest of our approach resides in allowing socially responsible (SR) portfolio investors to select their optimal portfolios by considering their individual preferences for each objective and simultaneous definition of the degrees of acceptance and rejection. In particular, we consider socially responsible portfolio selection as an optimization problem with multiple objectives before applying interactive intuitionistic fuzzy method to solve the portfolio optimization. The robustness of our approach is tested through an empirical study on the top 10 Stocks for ESG values worldwide. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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11 pages, 277 KiB  
Article
How to Assess Different Algorithms Using Intuitionistic Fuzzy Logic
by Tania Pencheva, Maria Angelova, Evdokia Sotirova and Krassimir Atanassov
Mathematics 2021, 9(18), 2189; https://doi.org/10.3390/math9182189 - 07 Sep 2021
Cited by 1 | Viewed by 1337
Abstract
Intuitionistic fuzzy logic is the main tool in the recently developed step-wise “cross-evaluation” procedure that aims at the assessment of different optimization algorithms. In this investigation, the procedure previously applied to compare the effectiveness of two or three algorithms has been significantly upgraded [...] Read more.
Intuitionistic fuzzy logic is the main tool in the recently developed step-wise “cross-evaluation” procedure that aims at the assessment of different optimization algorithms. In this investigation, the procedure previously applied to compare the effectiveness of two or three algorithms has been significantly upgraded to evaluate the performance of a set of four algorithms. For the first time, the procedure applied here has been tested in the evaluation of the effectiveness of genetic algorithms (GAs), which are proven as very promising and successful optimization techniques for solving hard non-linear optimization tasks. As a case study exemplified with the parameter identification of a S. cerevisiae fed-batch fermentation process model, the cross-evaluation procedure has been executed to compare four different types of GAs, and more specifically, multi-population genetic algorithms (MGAs), which differ in the order of application of the three genetic operators: Selection, crossover and mutation. The results obtained from the implementation of the upgraded intuitionistic fuzzy logic-based procedure for MGA performance assessment have been analyzed, and the standard MGA has been outlined as the fastest and most reliable one among the four investigated algorithms. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
21 pages, 3481 KiB  
Article
A Novel Approach to Generalized Intuitionistic Fuzzy Sets Based on Interpolative Boolean Algebra
by Pavle Milošević, Bratislav Petrović and Ivana Dragović
Mathematics 2021, 9(17), 2115; https://doi.org/10.3390/math9172115 - 01 Sep 2021
Cited by 6 | Viewed by 5572
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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11 pages, 346 KiB  
Article
On the Operation ∆ over Intuitionistic Fuzzy Sets
by Lilija Atanassova and Piotr Dworniczak
Mathematics 2021, 9(13), 1518; https://doi.org/10.3390/math9131518 - 29 Jun 2021
Cited by 4 | Viewed by 1689
Abstract
Recently, the new operation ∆ was introduced over intuitionistic fuzzy sets and some of its properties were studied. Here, new additional properties of this operations are formulated and checked, providing an analogue to the De Morgan’s Law (Theorem 1), an analogue of the [...] Read more.
Recently, the new operation ∆ was introduced over intuitionistic fuzzy sets and some of its properties were studied. Here, new additional properties of this operations are formulated and checked, providing an analogue to the De Morgan’s Law (Theorem 1), an analogue of the Fixed Point Theorem (Theorem 2), the connections between the operation ∆ on one hand and the classical modal operators over IFS Necessity and Possibility, on the other (Theorems 3 and 4). It is shown that it can be used for a de-i-fuzzification. A geometrical interpretation of the process of constructing the operator ∆ is given. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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7 pages, 239 KiB  
Article
ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem
by Stefka Fidanova and Krassimir Todorov Atanassov
Mathematics 2021, 9(13), 1456; https://doi.org/10.3390/math9131456 - 22 Jun 2021
Cited by 4 | Viewed by 1472
Abstract
Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between [...] Read more.
Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between the best methods, that solves combinatorial optimization problems. The method mimics behavior of the ants in the nature, when they look for a food. One of the algorithm parameters is called pheromone, and it is updated every iteration according quality of the achieved solutions. The intuitionistic fuzzy (propositional) logic was introduced as an extension of Zadeh’s fuzzy logic. In it, each proposition is estimated by two values: degree of validity and degree of non-validity. In this paper, we propose two variants of intuitionistic fuzzy pheromone updating. We apply our ideas on Multiple-Constraint Knapsack Problem (MKP) and compare achieved results with traditional ACO. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
8 pages, 287 KiB  
Article
Four Distances for Circular Intuitionistic Fuzzy Sets
by Krassimir Atanassov and Evgeniy Marinov
Mathematics 2021, 9(10), 1121; https://doi.org/10.3390/math9101121 - 15 May 2021
Cited by 32 | Viewed by 2400
Abstract
In the paper, for the first time, four distances for Circular Intuitionistic Fuzzy Sets (C-IFSs) are defined. These sets are extensions of the standard IFS that are extensions of Zadeh’s fuzzy sets. As it is shown, the distances for the C-IFS are different [...] Read more.
In the paper, for the first time, four distances for Circular Intuitionistic Fuzzy Sets (C-IFSs) are defined. These sets are extensions of the standard IFS that are extensions of Zadeh’s fuzzy sets. As it is shown, the distances for the C-IFS are different than those for the standard IFSs. At the moment, they do not have analogues in fuzzy sets theory. Examples, comparing the proposed distances, are given and some ideas for further research are formulated. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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16 pages, 3393 KiB  
Article
InterCriteria Analysis: Application for ECG Data Analysis
by Irena Jekova, Peter Vassilev, Todor Stoyanov and Tania Pencheva
Mathematics 2021, 9(8), 854; https://doi.org/10.3390/math9080854 - 14 Apr 2021
Cited by 9 | Viewed by 1932
Abstract
The InterCriteria Analysis (ICrA) is based on the mathematical formalisms of index matrices and intuitionistic fuzzy sets. It has been elaborated to discern possible similarities in the behavior of criteria pairs when multiple objects are considered, allowing also the accounting of information uncertainty. [...] Read more.
The InterCriteria Analysis (ICrA) is based on the mathematical formalisms of index matrices and intuitionistic fuzzy sets. It has been elaborated to discern possible similarities in the behavior of criteria pairs when multiple objects are considered, allowing also the accounting of information uncertainty. The focus of this study is to validate the applicability of ICrA over a large set of ECG criteria extracted for arrhythmia analysis and to evaluate its ability to support the pre-selection of criteria that could be further involved in decision making procedures. ICrA is applied over 88 ECG criteria (resulting in 3828 criteria pairs) calculated for 8528 ECGs from PhysioNet/CinC Challenge 2017 database. Three criteria pairs show strong positive consonance, another 26—positive consonance, while another 15 are in negative consonance. ICrA also reveals lack of dependencies in 98 criteria pairs. The correspondence between our observations (high degrees of agreement/disagreement and lack of dependencies) and our expectations based on knowledge of the principles involved in the computation of the ECG criteria validates the application of ICrA for reliable evaluation of the relation between different criteria. This potential of ICrA to highlight useful relations between ECG criteria makes it suitable in the ECG pre-processing stage for criteria pre-selection. Thus, optimization of the feature space could be achieved together with minimization of the computations’ complexity. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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11 pages, 231 KiB  
Article
On an Intuitionistic Fuzzy Form of the Goguen’s Implication
by Krassimir Atanassov, Nora Angelova and Vassia Atanassova
Mathematics 2021, 9(6), 676; https://doi.org/10.3390/math9060676 - 22 Mar 2021
Viewed by 1664
Abstract
In the present paper we construct a new intuitionistic fuzzy implication, giving intuitionistic fuzzy form to Goguen’s implication. Some of its basic properties are studied and illustrated with examples. Geometrical interpretations of the different forms of the new implication are given. Other forms [...] Read more.
In the present paper we construct a new intuitionistic fuzzy implication, giving intuitionistic fuzzy form to Goguen’s implication. Some of its basic properties are studied and illustrated with examples. Geometrical interpretations of the different forms of the new implication are given. Other forms of the Goguen’s implication are discussed. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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9 pages, 298 KiB  
Article
Third Zadeh’s Intuitionistic Fuzzy Implication
by Krassimir Atanassov
Mathematics 2021, 9(6), 619; https://doi.org/10.3390/math9060619 - 15 Mar 2021
Cited by 3 | Viewed by 1909
Abstract
George Klir and Bo Yuan named after Lotfi Zadeh the implication p→q=max(1−p,min(p,q)) (also Early Zadeh implication). In a series of papers, the author introduced two intuitionistic [...] Read more.
George Klir and Bo Yuan named after Lotfi Zadeh the implication p→q=max(1−p,min(p,q)) (also Early Zadeh implication). In a series of papers, the author introduced two intuitionistic fuzzy forms of Zadeh’s implication and studied their basic properties. In the present paper, a new (third) intuitionistic fuzzy form of Zadeh’s implication is proposed and some of its properties are studied. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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17 pages, 535 KiB  
Article
Towards Better Concordance among Contextualized Evaluations in FAST-GDM Problems
by Marcelo Loor, Ana Tapia-Rosero and Guy De Tré
Mathematics 2021, 9(1), 93; https://doi.org/10.3390/math9010093 - 04 Jan 2021
Cited by 3 | Viewed by 2208
Abstract
A flexible attribute-set group decision-making (FAST-GDM) problem consists in finding the most suitable option(s) out of the options under consideration, with a general agreement among a heterogeneous group of experts who can focus on different attributes to evaluate those options. An open challenge [...] Read more.
A flexible attribute-set group decision-making (FAST-GDM) problem consists in finding the most suitable option(s) out of the options under consideration, with a general agreement among a heterogeneous group of experts who can focus on different attributes to evaluate those options. An open challenge in FAST-GDM problems is to design consensus reaching processes (CRPs) by which the participants can perform evaluations with a high level of consensus. To address this challenge, a novel algorithm for reaching consensus is proposed in this paper. By means of the algorithm, called FAST-CR-XMIS, a participant can reconsider his/her evaluations after studying the most influential samples that have been shared by others through contextualized evaluations. Since exchanging those samples may make participants’ understandings more like each other, an increase of the level of consensus is expected. A simulation of a CRP where contextualized evaluations of newswire stories are characterized as augmented intuitionistic fuzzy sets (AIFS) shows how FAST-CR-XMIS can increase the level of consensus among the participants during the CRP. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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10 pages, 265 KiB  
Article
Martingale Convergence Theorem for the Conditional Intuitionistic Fuzzy Probability
by Katarína Čunderlíková
Mathematics 2020, 8(10), 1707; https://doi.org/10.3390/math8101707 - 03 Oct 2020
Cited by 1 | Viewed by 1671
Abstract
For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional [...] Read more.
For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state m. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
9 pages, 230 KiB  
Article
Intuitionistic Fuzzy Normed Subrings and Intuitionistic Fuzzy Normed Ideals
by Nour Abed Alhaleem and Abd Ghafur Ahmad
Mathematics 2020, 8(9), 1594; https://doi.org/10.3390/math8091594 - 16 Sep 2020
Cited by 11 | Viewed by 1771
Abstract
The main goal of this paper is to introduce the notion of intuitionistic fuzzy normed rings and to establish basic properties related to it. We extend normed rings by incorporating the idea of intuitionistic fuzzy to normed rings, we develop a new structure [...] Read more.
The main goal of this paper is to introduce the notion of intuitionistic fuzzy normed rings and to establish basic properties related to it. We extend normed rings by incorporating the idea of intuitionistic fuzzy to normed rings, we develop a new structure of fuzzy rings which will be called an intuitionistic fuzzy normed ring. As an extension of intuitionistic fuzzy normed rings, we define the concept of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals. Some essential operations specially subset, complement, union, intersection and several properties relating to the notion of generalized intuitionistic fuzzy normed rings are identified. Homomorphism and isomorphism of intuitionistic fuzzy normed subrings are characterized. We identify the image and the inverse image of intuitionistic fuzzy normed subrings under ring homomorphism and study their elementary properties. Some properties of intuitionistic fuzzy normed rings and relevant examples are presented. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
14 pages, 307 KiB  
Article
Bases of G-V Intuitionistic Fuzzy Matroids
by Yonghong Li, Li Li, Jiang Li, Dong Qiu and Huiming Duan
Mathematics 2020, 8(9), 1392; https://doi.org/10.3390/math8091392 - 20 Aug 2020
Cited by 1 | Viewed by 1510
Abstract
The purpose of this paper is to study intuitionistic fuzzy bases (IFBs) and the intuitive structure of a G−VIFM. Firstly, the intuitionistic fuzzy basis (IFB) of a [...] Read more.
The purpose of this paper is to study intuitionistic fuzzy bases (IFBs) and the intuitive structure of a G−VIFM. Firstly, the intuitionistic fuzzy basis (IFB) of a G−VIFM is defined; then the h-range and properties of an IFB are presented and a necessary and sufficient condition of a closed G−VIFM is studied. Especially, a necessary and sufficient condition of judging an IFB is presented and the intuitive tree structure of a closed G−VIFM is proposed and its properties are discussed. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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14 pages, 808 KiB  
Article
A p-Ideal in BCI-Algebras Based on Multipolar Intuitionistic Fuzzy Sets
by Jeong-Gon Lee, Mohammad Fozouni, Kul Hur and Young Bae Jun
Mathematics 2020, 8(6), 993; https://doi.org/10.3390/math8060993 - 17 Jun 2020
Cited by 2 | Viewed by 1473
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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