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40 Years of Intuitionistic Fuzzy Sets
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40 Years of Intuitionistic Fuzzy Sets

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: 30 June 2024 | Viewed by 3609

Special Issue Editors

Dr. Vassia Atanassova

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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
Prof. Dr. Tania Pencheva

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Guest Editor
Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria
Interests: drug design; molecular modelling; virtual ligand screening; modelling and optimization of biotechnological processes; genetic algorithms; generalized nets
Dr. Peter Vassilev

E-Mail Website
Guest Editor
Associate Professor, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria
Interests: generalized nets; index matrices; intuitionistic fuzzy sets; decision making and pattern recognition under uncertainty

Special Issue Information

Dear Colleagues,

The present Special Issue aims to commemorate the 40th anniversary of the invention of the concept and theory of intuitionistic fuzzy sets. Defined for the first time in 1983 in Bulgaria by Krassimir Atanassov (1954), intuitionistic fuzzy sets have grown from a “mathematical game”, devised in а hospital bed to make time pass faster, to one of the most powerful, theoretically backed up, practically usable, and actively researched extensions of Lotfi Zadeh’s fuzzy sets.

From a historical perspective, during the first twenty years of the existence of intuitionistic fuzzy sets, they were mostly developed by Atanassov and a small group of researchers around him, and mainly from the viewpoint of mathematical logic, algebra, analysis and geometry. With the advancement of information technologies and the development of the ideas of the applicability of IFS to decision science, the interest in the concept has steadily broadened, due to the efforts and creativity of a growing community of researchers and practitioners in the areas of decision making, artificial intelligence, engineering, medicine, economics and other areas.  

As an idea that appeared well before its time, intuitionistic fuzzy sets provide us with theoretically rigorous and practically useful instruments to handle the inherent uncertainty that defines the times we live in.

Dr. Vassia Atanassova
Prof. Dr. Tania Pencheva
Dr. Peter Vassilev
Guest Editors

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 (theory and applications)
  • intuitionistic fuzzy logic
  • uncertainty modeling
  • intuitionistic fuzzy set-based decision making

Published Papers (5 papers)

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Research

13 pages, 254 KiB  
Article
Intuitionistic Fuzzy Modal Multi-Topological Structures and Intuitionistic Fuzzy Multi-Modal Multi-Topological Structures
by Krassimir Atanassov
Mathematics 2024, 12(3), 361; https://doi.org/10.3390/math12030361 - 23 Jan 2024
Viewed by 464
Abstract
On the basis of K. Kuratowski’s definitions of a topological structure with a closure or interior operator, the concept of a modal topological structure (MTS) with one of these operators was introduced by the author. This new structure was illustrated with examples with [...] Read more.
On the basis of K. Kuratowski’s definitions of a topological structure with a closure or interior operator, the concept of a modal topological structure (MTS) with one of these operators was introduced by the author. This new structure was illustrated with examples with intuitionistic fuzzy topological operators from both examples, and for this reason, these structures were named intuitionistic fuzzy MTSs (IFMTSs). In a series of papers, the author introduced some modifications and extensions to the IFMTSs, e.g., intuitionistic fuzzy temporal topological structures, intuitionistic fuzzy level topological structures and others, and intuitionistic fuzzy multi modal topological structures and others. In the present paper, four new examples of intuitionistic fuzzy multi modal topological structures are given. On their base, the concepts of a modal multi-topological structure and of a multi-modal multi-topological structure are introduced and illustrated with examples from the area of the intuitionistic fuzzy sets—intuitionistic fuzzy modal multi-topological structure with a closure or an interior operator; and intuitionistic fuzzy multi-modal multi-topological structure with one of these operators. Two intuitionistic fuzzy topological operators are defined. Their basic properties are studied and they are used in the new structures. Full article
(This article belongs to the Special Issue 40 Years of Intuitionistic Fuzzy Sets)
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11 pages, 262 KiB  
Article
On Another Type of Convergence for Intuitionistic Fuzzy Observables
by Katarína Čunderlíková
Mathematics 2024, 12(1), 127; https://doi.org/10.3390/math12010127 - 30 Dec 2023
Viewed by 499
Abstract
The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In [...] Read more.
The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too. Full article
(This article belongs to the Special Issue 40 Years of Intuitionistic Fuzzy Sets)
13 pages, 1073 KiB  
Article
Intuitionistic Fuzzy Sets with Ordered Pairs and Their Usage in Multi-Attribute Decision Making: A Novel Intuitionistic Fuzzy TOPSIS Method with Ordered Pairs
by Cengiz Kahraman, Selcuk Cebi, Basar Oztaysi and Sezi Cevik Onar
Mathematics 2023, 11(18), 3867; https://doi.org/10.3390/math11183867 - 10 Sep 2023
Cited by 3 | Viewed by 837
Abstract
Intuitionistic Fuzzy Sets with Ordered Pairs (IFSOP) are the recent extension of intuitionistic fuzzy sets by incorporating functional and dysfunctional points of view into the definition of membership functions. This paper extends the Technique of Order Preference Similarity to the Ideal Solution (TOPSIS) [...] Read more.
Intuitionistic Fuzzy Sets with Ordered Pairs (IFSOP) are the recent extension of intuitionistic fuzzy sets by incorporating functional and dysfunctional points of view into the definition of membership functions. This paper extends the Technique of Order Preference Similarity to the Ideal Solution (TOPSIS) method to the Intuitionistic Fuzzy TOPSIS (IF TOPSIS) with ordered pairs method and applies it to a multi-criteria risk-based supplier selection problem under fuzziness. IF TOPSIS with ordered pairs involves finding a positive ideal solution and a negative ideal solution, and measuring the distance between each alternative and these solutions. The final ranking of the alternatives is obtained based on the proportion of distances between the positive and negative ideal solutions. By asking functional and dysfunctional questions in this ranking process, the developed IF TOPSIS with ordered pairs method incorporates the accuracy and consistency of expert judgments, enhancing the decision-making process. A sensitivity analysis is also presented in order to show the robustness of the rankings obtained by IF TOPSIS with ordered pairs. Full article
(This article belongs to the Special Issue 40 Years of Intuitionistic Fuzzy Sets)
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10 pages, 407 KiB  
Article
A Parametric Family of Fuzzy Similarity Measures for Intuitionistic Fuzzy Sets
by Madiha Qayyum, Etienne E. Kerre and Samina Ashraf
Mathematics 2023, 11(14), 3163; https://doi.org/10.3390/math11143163 - 19 Jul 2023
Viewed by 588
Abstract
Measuring the similarity between two objects and classifying them on the basis of their resemblance level has been a fundamental tool of the human mind. In an intuitionistic fuzzy environment, we find researchers that have attempted to generalize the fuzzy versions of similarity [...] Read more.
Measuring the similarity between two objects and classifying them on the basis of their resemblance level has been a fundamental tool of the human mind. In an intuitionistic fuzzy environment, we find researchers that have attempted to generalize the fuzzy versions of similarity measures between fuzzy sets to their intuitionistic forms for measuring the level of similarity between the intuitionistic fuzzy sets. Though many different forms of intuitionistic fuzzy similarity measures have been introduced so far, a comparative study reveals that among all these measures, it is difficult for one to claim the existence of a single measure that alone has the capability to recognize every single pattern assigned to it. This paper presents a four-parametric family of similarity measures for intuitionistic fuzzy sets employing weighted average cardinality and intuitionistic fuzzy t-norms along with their dual t-co-norms. A combinational variation of the parameters involved in this family resulted in some of the famous similarity measures having an intuitionistic version. These new measures are analyzed for their properties, and they have shown some remarkable results. Moreover, the proposed family has a practical advantage over the other measures in the existing literature because every member not only possesses the capability of successfully recognizing any pattern assigned to it up to a fine accuracy but also a choice of different t-norms and co-norms within a single measure equips it with the capacity to portray different mindsets of a decision-maker who, besides being unbiased, can possess a deep psychology of being an optimist, pessimist, or possessing neutral behavior in general. Lastly, the members of this family are tested for their feasibility in a sensitive medical decision process of detection of COVID-19. Full article
(This article belongs to the Special Issue 40 Years of Intuitionistic Fuzzy Sets)
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15 pages, 608 KiB  
Article
Orderings over Intuitionistic Fuzzy Pairs Generated by the Power Mean and the Weighted Power Mean
by Peter Vassilev, Todor Stoyanov, Lyudmila Todorova, Alexander Marazov, Velin Andonov and Nikolay Ikonomov
Mathematics 2023, 11(13), 2893; https://doi.org/10.3390/math11132893 - 27 Jun 2023
Viewed by 718
Abstract
In the present work, we prove a result concerning an ordering over intuitionistic fuzzy pairs generated by the power mean (Mp) for p>0. We also introduce a family of orderings over intuitionistic fuzzy pairs generated by the [...] Read more.
In the present work, we prove a result concerning an ordering over intuitionistic fuzzy pairs generated by the power mean (Mp) for p>0. We also introduce a family of orderings over intuitionistic fuzzy pairs generated by the weighted power mean (Mpα) and prove that a similar result holds for them. The considered orderings in a natural way extend the classical partial ordering and allow the comparison of previously incomparable alternatives. In the process of proving these properties, we establish some inequalities involving logarithms which may be of interest by themselves. We also show that there exists p>0 for which a finite set of alternatives, satisfying some reasonable requirements, some of which were not comparable under the classical ordering, has all its elements comparable under the new ordering. Finally, we provide some examples for the possible use of these orderings to a set of alternatives, which are in the form of intuitionistic fuzzy pairs as well as to results from InterCriteria Analysis. Full article
(This article belongs to the Special Issue 40 Years of Intuitionistic Fuzzy Sets)
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