Search Results (88)

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

This paper introduces the concept of interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras and explores their fundamental properties. We establish several new and equivalent characterizations, investigate their closure properties, and provide explicit algorithms for their construction and verification. Furthermore, we examine the relationships between interval-valued intuitionistic fuzzy n-fold implicative filters, interval-valued intuitionistic fuzzy filters, and classical n-fold implicative filters. The results presented here extend beyond straightforward generalizations, offering both practical tools and theoretical insights that are not previously available in the literature. The results presented here build upon earlier studies by systematically characterizing interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras. Several new equivalent conditions and computational methods are introduced, and relationships with existing filter concepts are clarified. Full article

In this article, a new similarity measure is discussed on interval-valued intuitionistic fuzzy values (IVIFVs). Here, the proposed similarity measure has been derived based on transformed intervals and its probability density functions, mean values, and standard deviations of IVIFVs. Based on the proposed similarity measure, several essential properties have been illustrated in this paper. Additionally, a new algorithm has been developed using the similarity measure of interval-valued intuitionistic fuzzy values (IVIFVs) to solve multi-attribute decision-making (MADM) problem. The proposed method is highly effective for solving various types of MADM problems. To demonstrate the effectiveness of the proposed similarity measure, a car selection problem has been considered, where the objective is to choose a suitable car for a decision maker from a set of alternatives evaluated under multiple criteria. In car selection, different features often involve conflicting criteria with imprecise data. Therefore, the proposed similarity measure of interval-valued intuitionistic fuzzy values assists in determining the best alternative among these conflicting criteria. Full article

To address the issues of imprecise user segmentation, inadequate handling of fuzzy evaluation information, and low recommendation accuracy in current electricity retail package recommendations, a novel recommendation method based on K-prototypes clustering and interval-valued intuitionistic fuzzy theory is proposed. First, a multi-dimensional user profile is constructed, incorporating five numerical tags—such as monthly average electricity consumption and monthly load factor—and two categorical tags: industry characteristics and value-added service demand. The K-prototypes algorithm is employed to cluster users, effectively resolving the profile distortion problem caused by the neglect of categorical features in traditional K-means clustering. Second, interval-valued intuitionistic fuzzy numbers are introduced to transform user linguistic evaluations into quantitative indicators. A projection measure-based model is established to objectively determine attribute weights, thereby eliminating subjective weighting bias. Finally, a comprehensive ranking of electricity retail packages is generated by integrating satisfaction levels of similar users and similar measures of new users. The recommendation performance is validated using Root Mean Square Error (RMSE), Kendall’s τ, Normalized Discounted Cumulative Gain (NDCG@5), and Discrimination Index (S). A case study involving users from a region in China demonstrates that the proposed method reduces the Root Mean Square Error (RMSE) to 0.32, which is 31.25% lower than the next best traditional method (K-prototypes + equal weight clustering with RMSE = 0.48), accurately addresses the core demands of diverse user groups, significantly improves recommendation precision and user satisfaction, and exhibits substantial practical application value. Full article

Investment decision-making is often characterized by uncertainty and the subjective weighting of criteria. This study aims to develop a more robust decision support framework by integrating the Generalized Interval-Valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS) with the Analytic Hierarchy Process (AHP) to objectively weight criteria and handle multi-evaluator hesitancy. In the proposed GIVHIFSS-AHP model, the AHP is employed to derive mathematically consistent criterion weights, which are subsequently embedded into the GIVHIFSS structure to accommodate interval-valued and hesitant evaluations from multiple decision-makers. The model is applied to a numerical case study evaluating five investment alternatives. Its performance is assessed through a comparative analysis with standard GIVHIFSS and GIFSS models, as well as a sensitivity analysis. The results indicate that the model produces financially rational rankings, identifying blue-chip technology stocks as the optimal choice (score: +2.4). The comparative analysis confirms its superiority over existing models, which yielded less-stable rankings. Moreover, the sensitivity analysis demonstrates the robustness of the results against minor perturbations in criterion weights. This research introduces a novel and synergistic integration of the AHP and GIVHIFSS. The key advantage of this approach lies in its ability to address the long-standing issue of arbitrary criterion weighting in Fuzzy Soft Set models by embedding the AHP as a foundational mechanism for ensuring validation and objectivity. This integration results in mathematically derived, consistent weights, thereby yielding empirically validated, more reliable, and defensible decision outcomes compared with existing models. Full article

The main focus of this paper is to introduce an algorithm that enhances the outcomes of multiple-attribute decision making by harnessing the adaptability of interval-valued intuitionistic fuzzy ($IVIF$) sets ($IVIFS$s). This algorithm utilizes $IVIF$ numbers ($IVIFN$s) to represent attribute values and attribute weights, enabling the decision maker to account for the intricate nuances and uncertainties that are inherent in the decision-making process. We introduce a novel generalized score function ($GSF$) designed to overcome the limitations of previous functions. This function incorporates two parameters, denoted as ${\gamma }_1\mathrm{and}{\gamma }_2({\gamma }_1+{\gamma }_2=1)$ with ${\gamma }_1\in (0,0.5)$. The core concept of this algorithm centers around the computation of the p-distance for each alternative relative to the positive ideal alternative. The p-distance is derived from the p-norm associated with each alternative’s score matrix, providing the decision maker ($DM$) with a tool to rank the available alternatives. Various examples are given to demonstrate the practicality and effectiveness of the proposed algorithm. Additionally, we apply the algorithm to a real event-based multiple-attribute decision-making ($MADM$) problem—the investment company problem—to identify the optimal alternatives through a comparative analysis. Full article

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

For interval-valued intuitionistic fuzzy sets featuring complementary symmetry in evaluation relations, this paper proposes a novel, complete fuzzy multi-attribute group decision-making (MAGDM) method that optimizes both expert weights and attribute weights. First, an optimization model is constructed to determine expert weights by minimizing the cumulative difference between individual evaluations and the overall consistent evaluations derived from all experts. Second, based on the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS), the improved closeness index for evaluating each alternative is obtained. Finally, leveraging entropy theory, a concise and interpretable optimization model is established to determine the attribute weight. This weight is then incorporated into the closeness index to enable the ranking of alternatives. Integrating these features, the complete fuzzy MAGDM algorithm is formulated, effectively combining the strengths of subjective and objective weighting approaches. To conclude, the feasibility and effectiveness of the proposed method are thoroughly verified and compared through detailed examination of two real-world cases. Full article

Intuitionistic fuzzy sets (IFSs), proposed in 1983, are one of the most viable and widely explored extensions of Zadeh’s fuzzy sets. In the decade following their introduction, they were extended to interval-valued IFSs (IVIFSs), temporal IFSs, IFSs of the second type (incorrectly called “Pythagorean fuzzy sets” by some authors) IFSs of n-th type, and IFSs over different universes. For each of these extensions, at least one geometrical interpretation has been defined, and for IVIFSs, at least seven different interpretations are known. In the present paper, revisiting some existing results on IVIFSs, some necessary modifications, additions, and corrections to the planar and spatial geometrical interpretations are introduced here for the first time. A new, eighth, geometrical interpretation of IVIFSs is proposed. A basic logic operation and two modal operators are illustrated and a comparison is made between the planar and the new “two-rods” geometrical interpretations of identical IVIFS elements. Finally, a new operator over IVIFSs is proposed for the first time, some of its properties are proven, and its geometrical interpretations are described. Full article

As a non-associative connective in fuzzy logic, the analysis and research of overlap functions have been extended to many generalized cases, such as interval-valued and intuitionistic fuzzy overlap functions (IFOFs). However, overlap functions face challenges in the Pythagorean fuzzy (PF) environment. This paper first extends overlap functions to the PF domain by proposing PF overlap functions (PFOFs), discussing their representable forms, and providing a general construction method. It then introduces a new PF similarity measure which addresses issues in existing measures (e.g., the inability to measure the similarity of certain PF numbers) and demonstrates its effectiveness through comparisons with other methods, using several examples in fractional form. Based on the proposed PFOFs and their induced residual implication, new generalized PF rough sets (PFRSs) are constructed, which extend the PFRS models. The relevant properties of their approximation operators are explored, and they are generalized to the dual-domain case. Due to the introduction of hesitation in IF and PF sets, the approximate accuracy of classical rough sets is no longer applicable. Therefore, a new PFRS approximate accuracy is developed which generalizes the approximate accuracy of classical rough sets and remains applicable to the classical case. Finally, three multi-criteria decision-making (MCDM) algorithms based on PF information are proposed, and their effectiveness and rationality are validated through examples, making them more flexible for solving MCDM problems in the PF environment. Full article

In a world rich with linguistic-valued data, traditional methods often lead to significant information loss when converting such data into other formats. This paper presents a novel approach for constructing an interval linguistic-valued intuitionistic fuzzy concept lattice, which adeptly manages qualitative linguistic information by leveraging the strengths of interval-valued intuitionistic fuzzy sets to represent both fuzziness and uncertainty. First, the interval linguistic-valued intuitionistic fuzzy concept lattice is constructed by integrating interval intuitionistic fuzzy sets, capturing the bidirectional fuzzy linguistic information between objects, which encompasses both positive and negative aspects. Second, by analyzing the expectations of concept extent relative to intent, and considering both the membership and non-membership perspectives of linguistic expressions, we focus on the extraction of linguistic association rules. Finally, comparative analyses and examples demonstrate the effectiveness of the proposed approach, showcasing its potential to advance the management of linguistic data in various domains. Full article

This paper presents an original and comprehensive investigation into the Taxonomic Measure of Development (TMD), introduced by Hellwig in 1968, enriching both its theoretical foundations and practical applications. It provides an overview of various variants of the Hellwig method, including their extensions and applications, while also exploring recent trends across multiple research domains. Primarily developed as a method for multidimensional analysis, TMD has evolved into a pivotal tool in multi-criteria decision-making. It is widely used for evaluating and ranking alternatives, particularly in the analysis of complex socio-economic phenomena and decision-making scenarios involving multiple criteria. This study systematically reviews the original algorithm and its subsequent extensions and modifications, including adaptations for fuzzy sets, intuitionistic fuzzy sets, and interval-valued fuzzy sets. Furthermore, it explores an integrated multi-criteria approach based on Hellwig’s method and its practical applications across various domains. This paper introduces an original approach by conducting a detailed, step-by-step analysis of the TMD framework. This process-oriented analysis is a novel contribution that sets this study apart from typical reviews based on statistical or bibliometric data. By examining key steps in the TMD framework—such as data collection, criterion weighting, data normalization, ideal value determination, distance calculation, and normalization factor—this paper highlights the method’s versatility in addressing complex, real-world decision-making problems. Although similar to the widely used Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method in its reliance on distance to evaluate alternatives, Hellwig’s approach is unique in focusing exclusively on proximity to an ideal solution, without considering distance from a negative ideal. This distinctive emphasis has led to numerous adaptations and extensions that address specific issues such as criterion dependencies, uncertainty, and rank reversal. The findings underscore the continued relevance of the Hellwig method, its recent extensions, and its growing international recognition. Full article

Effective franchise selection is crucial for global brands like Pizza Hut to maintain consistent quality and operational excellence amidst a competitive landscape. This paper introduces a novel confidence-interval circular intuitionistic fuzzy set (CIC-IFS) framework, designed to address the intricate challenges of master and sub-franchise selection in the European market. By integrating competence coefficients of decision-makers into the final evaluations, the model allows for a more accurate representation of expert judgments. Decision-makers can choose from various scenarios, ranging from super pessimistic to super optimistic, using ten forms of aggregation operations over index matrices. The proposed approach leverages confidence intervals within the circular intuitionistic fuzzy set paradigm to capture the uncertainty, vagueness, and hesitancy inherent in the decision-making process. A case study involving Pizza Hut’s European operations demonstrates the model’s efficacy in differentiating potential franchisees and identifying those best aligned with the brand’s values. The results indicate a significant improvement in selection accuracy compared to traditional methods and other fuzzy approaches, thereby enabling Pizza Hut to make more informed decisions and solidify its market position. Full article

Based on the approximation spaces, the interval-valued intuitionistic fuzzy rough set (IVIFRS) plays an essential role in coping with the uncertainty and ambiguity of the information obtained whenever human opinion is modeled. Moreover, a family of flexible t-norm (TNrM) and t-conorm (TCNrM) known as the Aczel–Alsina t-norm (AATNrM) and t-conorm (AATCNrM) plays a significant role in handling information, especially from the unit interval. This article introduces a novel clustering model based on IFRS using the AATNrM and AATCNrM. The developed clustering model is based on the aggregation operators (AOs) defined for the IFRS using AATNrM and AATCNrM. The developed model improves the level of accuracy by addressing the uncertain and ambiguous information. Furthermore, the developed model is applied to the segmentation problem, considering the information about the income and spending scores of the customers. Using the developed AOs, suitable customers are targeted for marketing based on the provided information. Consequently, the proposed model is the most appropriate technique for the segmentation problems. Furthermore, the results obtained at different values of the involved parameters are studied. Full article

Octahedron sets, which extend beyond the previously defined fuzzy set and soft set concepts to address uncertainty, represent a hybrid set theory that incorporates three distinct systems: interval-valued fuzzy sets, intuitionistic fuzzy sets, and traditional fuzzy set components. This comprehensive set theory is designed to express all information provided by decision makers as interval-valued intuitionistic fuzzy decision matrices, addressing a broader range of demands than conventional fuzzy decision-making methods. Multi-criteria decision-making (MCDM) methods are essential tools for analyzing and evaluating alternatives across multiple dimensions, enabling informed decision making aligned with strategic objectives. In this study, we applied MCDM methods to octahedron sets for the first time, optimizing decision results by considering various constraints and preferences. By employing an MCDM algorithm, this study demonstrated how the integration of MCDM into octahedron sets can significantly enhance decision-making processes. The algorithm allowed for the systematic evaluation of alternatives, showcasing the practical utility and effectiveness of octahedron sets in real-world scenarios. This approach was validated through influential examples, underscoring the value of algorithms in leveraging the full potential of octahedron sets. Furthermore, the application of MCDM to octahedron sets revealed that this hybrid structure could handle a wider range of decision-making problems more effectively than traditional fuzzy set approaches. This study not only highlights the theoretical advancements brought by octahedron sets but also provides practical evidence of their application, proving their importance and usefulness in complex decision-making environments. Overall, the integration of octahedron sets and MCDM methods marks a significant step forward in decision science, offering a robust framework for addressing uncertainty and optimizing decision outcomes. This research paves the way for future studies to explore the full capabilities of octahedron sets, potentially transforming decision-making practices across various fields. Full article