Search Results (507)

The rheological behaviour of complex fluids such as blood and polymer melts is governed by viscosities that are inherently subject to epistemic uncertainty arising from incomplete knowledge of evolving small scales rather than intrinsic randomness. Classical continuum models assume precisely known viscosity functions, an assumption that is physically unjustifiable in such systems, while existing fuzzy approaches have failed to integrate rigorously with the full conservation laws of continuum mechanics. To address this gap, we introduce Fuzzy–Viscous Fluid Dynamics (FVFD), a novel framework in which dynamic viscosity is governed by Dynamic Interval-Valued Intuitionistic Fuzzy Sets (DIVIFS), with membership functions grounded in Coleman–Gurtin internal-variable thermodynamics and evolution equations derived from a Lyapunov dissipation postulate. Employing the parabolic comparison principle together with Galerkin–Leray–Hopf theory, we establish that intuitionistic ordering constraints are preserved over time and prove the existence of global weak solutions to the coupled fuzzy Navier–Stokes equations (FNSEs). An exact analytical solution for fuzzy Couette flow is derived, recovering the classical Newtonian limit and shown, via a structural argument, to be non-linear precisely because and only because FVFD departs from the purely local generalised-Newtonian closure shared by the Power-law, Carreau–Yasuda, and Cross models. Three governing dimensionless parameters, the Reynolds number ($Re$), Damköhler number ($Da$), and fuzzy number ($Fz$), are identified and justified to characterise distinct flow regimes. This framework provides a rigorous, physically grounded alternative to stochastic and data-driven methods for explicitly tracking epistemic uncertainty through interval-valued hesitancy parameters, enabling more accurate modelling of complex fluids whose internal aggregation states remain inaccessible to direct observation. Full article

We introduce and study bipolar complex intuitionistic fuzzy Lie algebras (BCIFLAs), a new algebraic framework merging bipolar fuzzy theory, complex-valued membership functions in Cartesian form, and intuitionistic fuzzy Lie algebra theory. Adopting the Cartesian coordinate formulation of bipolar complex intuitionistic fuzzy sets (BCIFSs), we define bipolar complex intuitionistic fuzzy Lie subalgebras (BCIFLSAs) and ideals (BCIFLIs) and establish their fundamental properties, including closure under arbitrary meets. We prove that images and preimages of BCIFLSAs and BCIFLIs are preserved under Lie algebra homomorphisms, characterize them via level sets, and show that the sum of two BCIFLIs is, again, a BCIFLI. Non-degenerate three-level examples on $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{t}(2,\mathbb{R})$ illustrate the gap between the subalgebra and ideal conditions. This framework strictly generalizes fuzzy, intuitionistic fuzzy, complex intuitionistic fuzzy, and bipolar fuzzy Lie algebras. Full article

Complex interval-valued fuzzy sets are powerful tools for representing uncertainty and periodicity that occur in many real-life problems. They not only take both periodicity semantics and uncertainty into account but also describe the information using interval-valued membership grades, which gives experts more freedom to solve complex real-life problems effectively. Complex interval-valued fuzzy sets have been successfully employed many times in medical diagnosis and pattern recognition problems. In this article, two novel methods for computing the correlation coefficient and weighted correlation coefficient of complex interval-valued fuzzy sets are proposed. The methods employed fuzzy statistical parameters such as mean, variance, and covariance of complex interval-valued fuzzy sets. Several important mathematical properties are rigorously established. In order to establish the efficacy and implementation of the methods, a real-life application related to pattern recognition for mineral identification is discussed in detail. Furthermore, based on the proposed correlation and weighted correlation, a classification algorithm is developed. The time and space complexities of the proposed algorithm are analyzed. The proposed algorithm is validated through experiments conducted on two real benchmark datasets. The results demonstrate that the proposed approaches are more reliable and accurate than several existing approaches by achieving classification accuracies exceeding 98%. Full article

In existing research on non-contact damage by high-power microwave (HPM) systems to electronic systems such as seekers, challenges include high risk, high cost, and a scarcity of HPM-specific damage assessment models. To address this, this paper proposes a multi-port HPM damage level assessment model based on interval-valued intuitionistic fuzzy (IVIF) sets. Firstly, electromagnetic modeling and field-circuit coupling simulation of multi-target structures provide input data from field-circuit simulations. Secondly, a multi-indicator fuzzy matrix reflecting system damage uncertainty is formed using interval fuzzy membership functions. Thirdly, an information entropy-driven dynamic weighting mechanism weights each indicator’s fuzzy distribution characteristics, and hybrid weights are constructed by dynamic and fixed weights to optimize parameter range rationality. Finally, using hybrid weights and the exponential damage mapping function, we achieve probabilistic fusion of multiple parameters and determine damage levels from the probability results. Assessment shows that under the same incident field strength of 25 kV/m, electromagnetic simulation software-based simulation assigns a composite damage probability of 0.85 to L-band Port 5 and 0.096 to X-band Port 6, highlighting the contrast between the dominant L-band hotspot (Port 5) and the low X-band response at Port 6 under identical irradiation. Full article

Pullbacks are fundamental limit constructions in category theory. Their study within intuitionistic fuzzy module categories, which provide a framework for handling mathematical uncertainty, remains underdeveloped. This work establishes their theoretical foundation in such categories. We first investigate the explicit construction of pullbacks and then study the properties of double pullbacks, monomorphisms, and epimorphisms. A classification of different pullback categories is provided. Key results include characterising pullback construction conditions, describing features of double pullbacks and related morphisms, and rigorously establishing the relationship between pullbacks and exactness. This work deepens the understanding of limit theory in intuitionistic fuzzy settings and provides a foundation for future research. Full article

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

Bipolar fuzzy sets provide an effective framework for representing both positive and negative aspects of information. The necessity of a mathematically rigorous and valid distance measure in bipolar fuzzy environments motivates us to introduce a new real-valued function on the set of bipolar fuzzy sets defined over both discrete and continuous universes of discourse. The proposed function is shown to define a valid metric on the set of bipolar fuzzy sets, as it satisfies all the metric axioms. The metric induced by the real-valued function is inspired by the Canberra distance, and it can effectively quantify the dissimilarity between bipolar fuzzy sets in a normalized and interpretable manner. The practical utility of the proposed metric is demonstrated in a pattern recognition problem, where it successfully recognizes an unknown pattern using known bipolar fuzzy patterns. Using the proposed metric, a bipolar fuzzy C-means clustering algorithm is developed for sentiment analysis. The time complexity of the aforementioned algorithm is also analysed. Experiments conducted on the IMDb Movie Review Dataset demonstrate that the proposed algorithm outperforms k-means, fuzzy C-means, and intuitionistic fuzzy C-means algorithms. The proposed bipolar fuzzy C-means algorithm achieves an accuracy of 90.04%, a precision of 90.51%, a recall of 89.01%, an F1-score of 89.75%, a Root mean square error of 0.1191, and a Silhouette score of 0.75. The findings establish that the proposed metric and the associated bipolar fuzzy clustering approach provide a robust and effective framework of handling sentiment data associated with simultaneous positive and negative opinions. Full article

Acquisition costs account for a significant share of total construction project costs, underscoring the importance of purchasing and supply management for organizational success. Supplier selection and evaluation are particularly critical because they involve multiple criteria, qualitative and quantitative attributes, and several decision-makers. In the construction industry, these activities become even more complex due to sector-specific characteristics such as convergent material flows, temporary facilities, buyer–supplier conflicts, price-oriented decisions, and the volatility of project-based markets. This paper investigates the supplier evaluation process in a construction company and identifies the company’s requirements and decision-makers’ expectations. Based on the collected data, this research proposes a model aligned with the company’s characteristics and the decision-makers’ expectations. The model combines two methods: the Intuitionistic Fuzzy approach to aggregate decision-makers’ opinions and ELECTRE TRI to classify suppliers based on predefined criteria and thresholds. The proposed model handles different weights assigned to each decision-maker for each criterion without allowing compensation among criteria. This model also explores the role of symmetry in multicriteria decision-making by combining Intuitionistic Fuzzy Sets with the ELECTRE TRI method. Decision-makers validated the proposal and emphasized its simplicity and flexibility, which allow future adjustments to both the criteria weights and the decision-makers’ assigned weights. Full article

A central challenge in complex group decision-making is how to integrate heterogeneous types of information. Experts differ in background and experience, which leads to variation in their understanding of assessment attributes and in the forms of information they provide. Such information may include fuzzy semantic information, fuzzy semantic interval information, and uncertain information, increasing the complexity of the decision process. Traditional approaches commonly employ fuzzy set (FS) and intuitionistic fuzzy set (IFS) models to address group decision-making problems involving human cognitive judgments. These models constrain the sum of the membership degree (MD) and the non-membership degree (non-MD) to be equal to 1 and less than or equal to 1, respectively. However, when assessment information is insufficient, the MD and non-membership degree provided by experts may exceed this constraint. In addition, the score function (SF) and accuracy function (AF) used in FS and IFS do not account for indeterminacy, making them unsuitable for handling incomplete and hesitation information. To overcome these limitations, this study proposes a Pythagorean fuzzy stepwise weight assessment ratio analysis-based method and introduces a new score function (NSF) and a new accuracy function (NAF) within the Pythagorean fuzzy set framework for complex group decision-making. An illustrative case on raw material vendor selection for shipbuilding enterprises is used to validate the effectiveness of the proposed method. The results demonstrate that the method produces more reasonable and accurate vendor ranking outcomes. Full article

The recycling of power batteries is a key measure for improving the new energy industry chain and achieving green circular economy goals. However, the process of evaluating and selecting recycling schemes is influenced by multiple complex factors and often involves a significant amount of ambiguous and uncertain decision-making information. As an important extension of intuitionistic fuzzy sets, picture fuzzy sets characterize fuzzy information through three distinct dimensions: membership, neutrality, and non-membership. This three-dimensional structure offers unique advantages in addressing uncertain and ambiguous decision-making problems, where traditional fuzzy sets may lose valuable information. Drawing on the Bray–Curtis distance measure, this paper proposes a novel picture fuzzy distance measure that captures differences across all three dimensions more comprehensively. By combining the weighted form of the proposed picture fuzzy distance measure with the classical TOPSIS method, a new multi-attribute decision-making model is established under the picture fuzzy framework. The effectiveness and feasibility of the proposed method are demonstrated through a case study on the recycling of power batteries for electric vehicles. A sensitivity analysis of relevant parameters is conducted, confirming the stability of the model against variations in parameter settings. Comparative results indicate that the proposed novel picture fuzzy distance measure exhibits superior robustness compared to existing similar distance measures. Furthermore, the constructed decision-making model can provide reliable and practical support for uncertain multi-attribute decision-making problems in real-world applications. Full article

Objects in nature tend to occupy the minimum volume (or minimum area in two-dimensional space). In decision making processes, decisions involve the grouping of decision points within a given environment, and the existence of a closed-form representing this group decision is noteworthy. Current group decision-making models often rely on arithmetic and geometric operators, neglecting the inherent spatial information embedded within the decision space. When the structure formed by decision groups is associated with a convex structure representing it that occupies the least volume or area; the decision-making structure also needs to be re-evaluated geometrically in addition to operations in the literature. In this context, this article is a novel perspective on the introduction of the Shapeless Intuitionistic Fuzzy Set (S-IFS) which is an extension of intuitionistic fuzzy sets. The operations and structure of these sets, which are a new extension with the convex structure formed by the IF decision points, are examined. A numerical micromobility risk assessment case is presented to demonstrate the application of S-IFS in multi criteria decision making procedures. To compare S-IFS with the literature, a new score function has also been proposed to Circular Intuitionistic Fuzzy Set (C-IFS) with the perspective of the proposed fuzzy set. The effect of including the uncertainty in the geometric structure of group decisions into the result is clearly revealed by comparing the point group decisions of IFS, the circle covering group decisions of C-IFS, and the convex group decisions occupying the smallest area in two dimensions of the proposed S-IFS. This article aims to lead the examination of the geometric structure of fuzzy sets in group decisions, the inclusion of uncertainty in decision-making processes in a geometric sense, and the structure of group decisions in n-dimensions according to convex and concave formations for future studies. Full article

In this paper, we propose a new class of intuitionistic fuzzy b-metric spaces in the sense of Romaguera and investigate their fixed-point properties. Within this framework, we define and analyze Hicks-type contraction mappings. In addition, the concept of K-stationary intuitionistic fuzzy b-metrics is introduced and examined through illustrative examples. Our findings generalize classical results in fuzzy b-metric spaces and extend fixed-point theorems to the intuitionistic fuzzy setting. This study enriches fixed-point theory in intuitionistic fuzzy environments and provides a basis for further theoretical investigations and applications. Full article

Despite advances in using multi-criteria decision-making (MCDM) methods and their fuzzy set extensions for human evaluations of large language models (LLMs), several gaps remain in the literature, particularly in task-specific evaluations that offer a more tractable and interpretable approach. Thus, this work develops a generalized intuitionistic fuzzy MCDM approach that bridges methodological gaps by outlining two contributions. First, the integration of SWARA (Stepwise Weight Assessment Ratio Analysis) and WINGS (Weighted Influence Non-linear Gauge System) is demonstrated to compute the priority weights of the evaluation criteria, thereby augmenting the independence limitation in prior relevant studies. Second, we introduce a newly improved IF-EDAS (intuitionistic fuzzy Evaluation based on Distance from Average Solution) that preserves more uncertain information and provides a more natural extension of the canonical EDAS framework, starting with the adoption of the IFWAM (intuitionistic fuzzy weighted arithmetic mean) operator for a more intuitive approach in generating the intuitionistic fuzzy average solution vector. Also, the proposed IF-EDAS variant employs three decision rules and the Hamming distance metric in its novel computational approach. The proposed hybrid approach was deployed in two case studies evaluating five popular LLMs for text summarization across seven interdependent criteria. Results show that SWARA initially prioritizes accuracy, coherence, and consistency, but these were revised when accounting for criteria interdependence, with coherence and language quality emerging as the most preferred criteria. Both case studies suggest that Gemini may perform favorably, while Copilot may consistently rank last. The findings of the case studies share similar insights with those of three other similar IF-EDAS variants, although our claims may have limited external validity, which requires more case studies and experts in future task-specific human evaluations. The proposed approach, along with its deployment in two case studies, demonstrates human evaluations of LLMs with greater computational interpretability, which contribute to the general MCDM literature. Full article

Over the past few decades, extensive attention has been given by researchers and practitioners to the development and application of multi-criteria decision-making (MCDM) methods within intuitionistic fuzzy environments across a wide range of fields and disciplines. This challenging research area has emerged as one of the most prominent topics, and its importance and popularity are expected to continue growing in the future. The elliptic intuitionistic fuzzy set (EIFS) addresses complex, multidimensional, non-symmetrical vagueness and uncertainty more effectively than other traditional intuitionistic fuzzy sets (IFSs). Sustainable renewable energy source selection is a critical decision-making (DM) process aiming to identify the most suitable energy alternative. The process of selecting sustainable renewable energy sources necessitates a comprehensive assessment of numerous criteria, which encompass environmental ramifications, economic feasibility, and societal acceptance. Contemporary research suggests novel methodologies to enhance this selection process, highlighting the need for an MCDM framework that integrates a variety of factors. This study presents an innovative DM framework for sustainable renewable energy source selection based on EIFS and a newly developed aggregation operator, the Elliptic Intuitionistic Fuzzy Weighted Muirhead Mean Aggregation (EIFWMMA) operator. These mechanisms expand upon conventional intuitionistic fuzzy frameworks by employing an elliptical portrayal of membership and non-membership degrees, facilitating a more accurate and lifelike representation of uncertainty and hesitation in evaluations by experts. To enhance computational efficiency, the framework weaves together machine learning-driven dimensionality reduction and weight optimization strategies of principal component analysis (PCA) for DM. The suggested operators are employed in an MCDM scenario centered around the selection of sustainable renewable energy sources, where the hierarchy of alternatives is established through score values derived from EIFWMMA. A comparative exploration of Circular Intuitionistic Fuzzy Sets (C-IFSs) and Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) uncovers that the elliptical formulation yields consistently reliable, precise, and geometrically comprehensible results. The findings affirm that EIFS-based operators offer a resilient, adaptable, and broadly applicable strategy for tackling MCDM challenges amidst uncertainty. The Min–Max normalization method is employed to validate our proposed methodology for identifying alternatives within the MCDM paradigm. It also improves accuracy, stability, and scalability in comparison to conventional approaches. Full article

Transportation problems constitute a fundamental class of optimization models; however, real-world applications involve uncertainty, hesitation, and expert disagreement that cannot be adequately captured by deterministic or classical fuzzy approaches. This paper proposes a three-dimensional circular intuitionistic fuzzy potential method (3D–CIFMODI), which extends the classical MODI framework to Circular Intuitionistic Fuzzy Triples (C-IFTs) through radius-aware operations and indexed matrix representations. Unlike existing circular intuitionistic fuzzy transportation methods, which are primarily feasibility-driven, the proposed approach introduces a dual-based optimality framework based on circular reduced costs, preserving the full structure of uncertainty without reducing it to crisp equivalents. The method retains polynomial-time computational complexity $O\left(mn\right(m+n\left)\right)$, i.e., $O\left(n^3\right)$ for square problems, with only a constant computational overhead due to circular operations. A numerical case study demonstrates the effectiveness and robustness of the proposed framework. Furthermore, a comparative analysis between classical intuitionistic fuzzy (IFS) and circular intuitionistic fuzzy (C-IFS) representations shows that incorporating the radius parameter significantly improves discrimination capability, solution stability, and interpretability. The results confirm that the proposed method provides a unified, interpretable, and computationally efficient framework for solving multi-layer transportation problems under circular intuitionistic fuzzy uncertainty. Full article