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Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "F: Engineering and Materials".

Deadline for manuscript submissions: 31 December 2026 | Viewed by 6081

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Editor

Prof. Dr. Cengız Kahraman

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Guest Editor

Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey
Interests: engineering economics; quality control and management; statistical decision making; multicriteria decision making; fuzzy decision making
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue covers symmetry and asymmetry phenomena occurring in the recent theory and practices of fuzzy set extensions. We invite authors to submit their theoretical, experimental, and practical research presenting engineering models under fuzziness dealing with the symmetry or asymmetry of different types of information.

This Special Issue is focused on the recent theoretical and practical developments of fuzzy set extensions for modeling under vague and imprecise conditions. Topics of interest include, but are not limited to, the following theoretical and/or practical developments in the modeling under fuzziness:

Type-2 fuzzy sets;
Hesitant fuzzy sets;
Intuitionistic fuzzy sets;
Spherical fuzzy sets;
Picture fuzzy sets;
Pythagorean fuzzy sets;
Q-rung orthopair fuzzy sets;
Neutrosophic sets;
Fermatean fuzzy sets;
Circular intuitionistic fuzzy sets;
Proportional fuzzy sets;
Decomposed fuzzy sets;
Other fuzzy set extensions.

Prof. Dr. Cengız Kahraman
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-anonymized 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

neutrosophic sets
symmetry/asymmetry
fuzzy sets

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Published Papers (5 papers)

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Research

28 pages, 1960 KB

Open AccessArticle

A Proportional Intuitionistic Fuzzy AHP–EDAS Framework for Symmetric Risk Assessment in Automotive Assembly Lines: A 12-Failure-Mode PFMEA Study

by Doğan Şengül and Fatma Kaymaz Karahan

Symmetry 2026, 18(7), 1115; https://doi.org/10.3390/sym18071115 - 30 Jun 2026

Viewed by 316

Abstract

Process Failure Mode and Effects Analysis (PFMEA) is the standard technique for proactive risk assessment in automotive assembly lines. To support differentiated criterion weighting, hesitancy-aware linguistic evaluation and rank stability validation, this paper proposes a symmetric extension of PFMEA that integrates Proportional Intuitionistic Fuzzy Sets (PIFSs) with the Analytic Hierarchy Process (AHP) and the Evaluation Based on Distance from Average Solution method (PIF-EDAS). PIFSs introduce a proportional balance between membership μ, non-membership ν and hesitancy π that is mathematically symmetric under linguistic-pair interchange and that preserves a constant hesitancy budget (π = 1/(1 +

k_{1}

k_{2}

)) across the nine-point linguistic scale. The framework is applied to an automotive Original Equipment Manufacturer (OEM) assembly line in Türkiye, on an inventory of twelve failure modes spanning torque, fastening, welding, panel alignment, harness, sealant, paint, ECU, trim, electrical connector, part variant and tightening sequence operations. Consistency of the PIF-AHP pairwise comparisons is confirmed (CR = 0.0017 ≪ 0.1), yielding criterion weights

w_{S}

= 0.598,

w_{O}

= 0.245 and

w_{D}

= 0.156. Comparative cross-method analysis against PIF-TOPSIS, PIF-VIKOR and a classical-style ordinal RPN benchmark indicates strong cross-method agreement: Spearman rank correlations range from 0.865 to 0.972, and Wrong Torque Application remains the unanimous top-priority failure across all four methods. Five sensitivity scenarios (proposed weights, equal weights, and severity-, occurrence- and detection-dominant) confirm that FM1 (Wrong Torque) remains the top-priority failure and FM9 (Damaged Interior Trim) remains the lowest-priority failure across all five scenarios; the composition of the upper-priority set is criterion-sensitive, with FM10 rising under equal-weight, occurrence- and detection-dominant scenarios and FM8 rising under the severity-dominant scenario. The proposed framework incorporates differentiated criterion weights, expert hesitancy and rank stability validation within a symmetric PIFS-based MCDM structure. The contribution of this study is therefore three-fold: (i) a symmetric PIFS formulation that enforces mirror symmetry under linguistic-pair interchange and a constant hesitation budget on the nine-point scale; (ii) a case-based assessment of twelve automotive PFMEA failure modes; and (iii) a transparent rank stability protocol for symmetric MCDM benchmarking. The framework integrates directly with existing FMEA workflows and scales linearly in computational complexity with the number of failure modes. Full article

(This article belongs to the Special Issue Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets)

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25 pages, 982 KB

Open AccessArticle

A Novel Multi-Criteria Decision-Making Methodology: The Presence–Absence Synthesis Method

by Mustafa Bal, Irem Ucal Sari and Özgür Kabak

Symmetry 2026, 18(2), 268; https://doi.org/10.3390/sym18020268 - 31 Jan 2026

Cited by 1 | Viewed by 637

Abstract

Traditional multi-criteria decision-making methods often operate on the assumption of symmetry, presupposing that the positive impact of a criterion’s presence is perfectly complementary to the negative impact of its absence. However, in real-world decision problems, this relationship is frequently asymmetric; some criteria act merely as “delighters,” while others represent “must-have” constraints. This study proposes a novel methodology, the Presence–Absence Synthesis (PAS) Method, which addresses this asymmetry by treating the “Presence Effect” and “Absence Effect” of criteria as two independent dimensions. The method is built upon intuitionistic fuzzy sets (IFSs) to effectively model the uncertainty and hesitation inherent in expert evaluations. The applicability of the proposed approach is demonstrated through a real-world workforce management problem aimed at assigning employees to the most suitable tasks based on their competencies in a retail store. In the study, the suitability scores derived from the PAS method are integrated into a mathematical optimization model for weekly employee scheduling, presenting a two-stage decision support framework. The results and comparisons with the Technique for Order Preference by Similarity to Ideal Solution method reveal that the PAS method more effectively distinguishes critical competency gaps (i.e., criteria with high absence effects), leading to more realistic task assignments and a measurable reduction in operational risks, such as skill mismatches and infeasible schedules. Furthermore, sensitivity analysis confirms that the proposed model yields consistent and robust results under varying conditions. Beyond the retail context, the proposed PAS framework is applicable to a wide range of decision-making problems, including healthcare staff allocation, project team formation, supplier selection, and other resource allocation settings where their presence cannot compensate for the absence of critical criteria. Full article

(This article belongs to the Special Issue Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets)

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38 pages, 441 KB

Open AccessArticle

Modeling Uncertainty with Interval-Valued Intuitionistic Fuzzy Filters in Hoop Algebras

by Amal S. Alali, Tahsin Oner, Ravikumar Bandaru, Neelamegarajan Rajesh and Ibrahim Senturk

Symmetry 2025, 17(9), 1411; https://doi.org/10.3390/sym17091411 - 30 Aug 2025

Viewed by 958

Abstract

This paper systematically investigates interval-valued intuitionistic fuzzy (IVIF) sets and filters within the framework of hoop algebras, unifying and extending classical fuzzy set theory and intuitionistic fuzzy sets (IFS) in algebraic logic. We clarify the foundational relationships among fuzzy sets, IFS, and hoop algebras, and introduce novel characterizations of IVIF filters, including necessary and sufficient conditions for their existence and structure. Theoretical advancements include the demonstration that IVIF filters can be described via their endpoint functions, the establishment of a bounded distributive lattice of IVIF filters, and the identification of congruence relations induced by these filters. Algorithmic and numerical aspects are addressed through explicit pseudocode and detailed examples, illustrating how the verification and construction of IVIF filters can be performed in finite hoop algebras. Practical implications are highlighted in decision-making scenarios where modeling uncertainty and vagueness with interval-valued membership and non-membership degrees offers enhanced flexibility and robustness. Our results lay a rigorous foundation for further applications of IVIF filters in fuzzy logic, artificial intelligence, and multi-criteria decision analysis. Full article

(This article belongs to the Special Issue Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets)

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15 pages, 295 KB

Open AccessArticle

Neutrosophic Quadruple Metric Spaces

by Memet Şahin and Arif Sarıoğlan

Symmetry 2025, 17(7), 1096; https://doi.org/10.3390/sym17071096 - 8 Jul 2025

Viewed by 980

Abstract

Instead of measuring the distance between two points with a positive real number, determining the degree to which the distance between these two points is close, not close, or uncertain allows for more detailed measurement. Recently, researchers have overcome this grading problem by using probability distribution functions, along with fuzzy, intuitionistic fuzzy, and neutrosophic sets. This study pioneers neutrosophic quadruple metric spaces as a powerful new tool for quantifying distances under complex, multi-dimensional uncertainty. It provides a comprehensive mathematical structure, including topology, convergence theory, and completeness, and handles both symmetric and asymmetric cases, generalising previous neutrosophic metric results. For this purpose, neutrosophic quadruple metric spaces were derived from neutrosophic metric spaces in order to better model situations involving uncertainty. Also, we generalised the findings obtained with the neutrosophic metric to the quadruple neutrosophic metric. Full article

(This article belongs to the Special Issue Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets)

22 pages, 5367 KB

Open AccessArticle

An Improved Bee Colony Optimization Algorithm Using a Sugeno–Takagi Interval Type-2 Fuzzy Logic System for the Optimal Design of Stable Autonomous Mobile Robot Controllers

by Leticia Amador-Angulo, Patricia Melin and Oscar Castillo

Symmetry 2025, 17(5), 789; https://doi.org/10.3390/sym17050789 - 20 May 2025

Cited by 1 | Viewed by 2068

Abstract

This study proposes an enhanced Sugeno–Takagi interval type-2 fuzzy logic system (SIT2FLS) to find the best values for two important parameters in Bee Colony Optimization (BCO). The aim of this study was to develop a stable controller for a mobile robot utilizing BCO in the fuzzy controller and to determine the best membership functions (MFs) in a type-1 fuzzy logic system (T1FLS) for control. Another objective was to use an SIT2FLS to find the best

α

and

β

parameters for BCO to enhance the robot trajectory, which was evaluated through an analysis of the mean squared errors. Three types of perturbations were analyzed and simulated. The performance of the SIT2FLS-FBCO was evaluated and compared to that of the T1FLS-FBCO. In addition, a comparative study was performed to demonstrate that the improved BCO works well when there are disturbances affecting the controller. Finally, it was compared with the Mamdani approach, and an FBCO with an interval type-3 FLS was also developed. Full article

(This article belongs to the Special Issue Theory and Applications on the Recent Extensions of Ordinary Fuzzy Sets)

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