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Symmetry
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Symmetry and Fuzzy Set
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Symmetry and Fuzzy Set

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 June 2026 | Viewed by 1811

Special Issue Editor

Dr. Panagiotis Chountas

E-Mail Website
Guest Editor
School of Computer Science and Engineering, University of Westminster, London, UK
Interests: machine learning; data mining; big data; data science; intuitionistic fuzzy sets

Special Issue Information

Dear Colleagues,

This Special Issue aims to provide a comprehensive platform for presenting cutting-edge research that bridges the gap between fuzzy logic and its extensions in the practical application of symmetry across various domains, including fuzzy measures, fuzzy relations, fuzzy control, fuzzy logic and data mining/machine learning, fuzzy decision making, fuzzy modeling, and fuzzy information retrieval. In recent years, the integration of fuzzy logic with multi-criteria methods has gained traction, improving the robustness of results obtained by MCDM/MCDA models. However, traditional MCDM methods encounter bottlenecks in high-dimensional problems. Limited research has evaluated the effectiveness of LLMs in tackling specific MCDM problems where symmetry may be present. We wish to further explore the capability of LLMs by proposing approaches for automatically addressing general, complex MCDM problems. Submissions involving applications of fuzzy theory and/or its extensions—where symmetry, or a deliberate lack thereof, is present—are also welcome. 

Dr. Panagiotis Chountas
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 250 words) can be sent to the Editorial Office for assessment.

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. Symmetry is an international peer-reviewed open access monthly 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 2400 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

  • fuzzy theory and symmetry
  • fuzzy decision making
  • intuitionistic fuzzy sets
  • multiple-criteria decision-making methods (MCDMs)
  • fuzzy fractional Brownian motion
  • large language models (LLMs)
  • data and knowledge engineering

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Published Papers (4 papers)

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Research

26 pages, 668 KB  
Article
A Novel Methodology for Distance and Similarity Measures in Hesitant Fuzzy Sets: Enhancing Pattern Recognition and Decision-Making
by Zahid Hussain, Sania Zahra, Rashid Hussain, Mehboob Ali and Panagiotis Chountas
Symmetry 2026, 18(6), 947; https://doi.org/10.3390/sym18060947 (registering DOI) - 31 May 2026
Abstract
Various distance and similarity measures have been proposed for hesitant fuzzy sets (HFSs) in the literature. However, some of these approaches are either inadequate or fail to produce reliable results for different scenarios. In this paper, we introduce a novel methodology for computing [...] Read more.
Various distance and similarity measures have been proposed for hesitant fuzzy sets (HFSs) in the literature. However, some of these approaches are either inadequate or fail to produce reliable results for different scenarios. In this paper, we introduce a novel methodology for computing distance and similarity measures between two HFSs based on an axiomatic framework. A key challenge arises when the lengths of two hesitant fuzzy elements (HFEs) differ. Traditionally, it is assumed that a pessimist would repeatedly add the minimum value, while an optimist would add the maximum value until the HFEs are of equal length. However, this approach may introduce bias and is not intuitively acceptable. To overcome this limitation, we propose an innovative and intuitive technique that ensures fairness by repeatedly adding zero to equalize HFE lengths. This method aligns with intuition and satisfies all axioms of distance and similarity measures. Several numerical examples demonstrate its effectiveness compared to existing methods. Additionally, we apply our approach to develop a hesitant fuzzy TODIM (HF-TODIM) model for interactive and multi-criteria decision-making. To validate its applicability, we use it to evaluate different livestock species and identify the most profitable option. The results confirm that our method is well-suited for handling complex and uncertain hesitant fuzzy information in a balanced and intuitive manner. Full article
(This article belongs to the Special Issue Symmetry and Fuzzy Set)
13 pages, 290 KB  
Article
Fuzzy Study Regarding the Fractional Integral Applied to the q-Multiplier Transformation
by Alina Alb Lupaş and Daria Lupaş
Symmetry 2026, 18(4), 549; https://doi.org/10.3390/sym18040549 - 24 Mar 2026
Viewed by 265
Abstract
q-calculus and fractional calculus combined with geometric function theory lead to remarkable results. The fractional integral introduced by Riemann–Liouville applied to the q-multiplier transformation is used in this research to study the two dual theories of fuzzy differential subordination and fuzzy [...] Read more.
q-calculus and fractional calculus combined with geometric function theory lead to remarkable results. The fractional integral introduced by Riemann–Liouville applied to the q-multiplier transformation is used in this research to study the two dual theories of fuzzy differential subordination and fuzzy differential superordination and to develop specific fuzzy results. In the theorems examining fuzzy differential subordinations and fuzzy differential superordinations, fuzzy best dominants and fuzzy best subordinants are also provided. In addition, the demonstrated outcomes reveal corollaries by taking specific functions with established geometric features into consideration as the fuzzy best subordinant and fuzzy best dominant. The work concludes with a fuzzy differential sandwich theorem and related corollaries that combine the findings of this research on fuzzy differential subordinations and superordinations. Full article
(This article belongs to the Special Issue Symmetry and Fuzzy Set)
19 pages, 844 KB  
Article
Parallels and Meridians in the Intuitionistic Fuzzy Triangle: A Confidence-Aware Framework for Decision Making
by Vassia Atanassova and Peter Vassilev
Symmetry 2026, 18(3), 468; https://doi.org/10.3390/sym18030468 - 9 Mar 2026
Viewed by 345
Abstract
The paper proposes a completely new geometric interpretation of intuitionistic fuzzy sets for confidence-aware decision making. Instead of directly using ordered pairs of membership and non-membership degrees, we reinterpret the intuitionistic fuzzy triangle as a structured coordinate framework that simultaneously represents the evaluation [...] Read more.
The paper proposes a completely new geometric interpretation of intuitionistic fuzzy sets for confidence-aware decision making. Instead of directly using ordered pairs of membership and non-membership degrees, we reinterpret the intuitionistic fuzzy triangle as a structured coordinate framework that simultaneously represents the evaluation outcomes and the evaluators’ expertise in the following manner: experts’ confidence levels are modelled with line segments parallel to the hypotenuse, while evaluation scores are represented by line segments radiating from the origin of the coordinate system toward the hypotenuse. Their intersections form a finite lattice of points whose total number depends on the chosen confidence and assessment scales. The proposed construction preserves the semantic foundations of intuitionistic fuzziness: points closer to the origin reflect higher uncertainty in the evaluator’s confidence, while points onto the hypotenuse represent determinate judgments (with varying degrees of positivity or negativity) based on the complete evaluator’s confidence. The geometric distances between intersections provide a formal explanation of varying discriminative power: assessments from highly confident reviewers are more distinguishable than those from less confident ones. In addition, a colour-based visualization further supports the intuitive interpretation of confidence-weighted evaluations. The proposed framework offers an alternative yet fully consistent way to model expertise-dependent decision processes within the intuitionistic fuzzy setting, bridging geometric insight and practical evaluation scenarios via a structured system of parallels and meridians. Full article
(This article belongs to the Special Issue Symmetry and Fuzzy Set)
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39 pages, 995 KB  
Article
Multi-Granulation Variable Precision Fuzzy Rough Set Based on Generalized Fuzzy Remote Neighborhood Systems and the MADM Application Design of a Novel VIKOR Method
by Xinyu Mei and Yaoliang Xu
Symmetry 2026, 18(1), 84; https://doi.org/10.3390/sym18010084 - 3 Jan 2026
Cited by 1 | Viewed by 540
Abstract
Variable precision fuzzy rough sets (VPFRSs) and multi-granulation fuzzy rough sets (MGFRSs) are both significant extensions of rough sets. However, existing variable precision models generally lack the inclusion property, which poses potential risks in applications. Meanwhile, multi-granulation models tend to emphasize either optimistic [...] Read more.
Variable precision fuzzy rough sets (VPFRSs) and multi-granulation fuzzy rough sets (MGFRSs) are both significant extensions of rough sets. However, existing variable precision models generally lack the inclusion property, which poses potential risks in applications. Meanwhile, multi-granulation models tend to emphasize either optimistic or pessimistic scenarios but overlook compromise situations. A generalized fuzzy remote neighborhood system is a symmetric union-fuzzified form of the neighborhood system, which can extend the fuzzy rough set model to a more general framework. Moreover, semi-grouping functions eliminate the left-continuity required for grouping functions and the associativity in t-conorms, making them more suitable for information aggregation. Therefore, to overcome the limitations of existing models, we propose an optimistic (OP), pessimistic (PE), and compromise (CO) variable precision fuzzy rough set (OPCAPFRS) based on generalized fuzzy remote neighborhood systems. The semi-grouping function and its residual minus are employed in the OPCAPFRS. We discuss the basic properties of the OPCAPFRS and prove that it satisfies the generalized inclusion property (GIP). This partially addresses the issue that a VPFRS cannot fulfill the inclusion property. A novel methodology for addressing multi-attribute decision-making (MADM) problems is developed through the fusion of the proposed OPCAPFRS framework and the VIKOR technique. The proposed method is applied to the problem of selecting an optimal CPU. Subsequently, comparative experiments and a parameter analysis are conducted to validate the effectiveness and stability of the proposed method. Finally, three sets of experiments are performed to verify the reliability and robustness of the new approach. It should be noted that the new method performed ranking on a dataset containing nearly ten thousand samples, obtaining both the optimal solution and a complete ranking, thereby validating its scalability. Full article
(This article belongs to the Special Issue Symmetry and Fuzzy Set)
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