Interval-valued Fuzzy Sets Research Articles

Counterfactuals in fuzzy relational models

Given the pressing need for explainability in Machine Learning systems, the studies on counterfactual explanations have gained significant interest. This research delves into this timely problem cast in a unique context of relational systems described by fuzzy relational equations. We develop a comprehensive solution to the counterfactual problems encountered in this setting, which is a novel contribution to the field. An underlying optimization problem is formulated, and its gradient-based solution is constructed. We demonstrate that the non-uniqueness of the derived solution is conveniently formalized and quantified by admitting a result coming in the form of information granules of a higher type, namely type-2 or interval-valued fuzzy set. The construction of the solution in this format is realized by invoking the principle of justifiable granularity, another innovative aspect of our research. We also discuss ways of designing fuzzy relations and elaborate on methods of carrying out counterfactual explanations in rule-based models. Illustrative examples are included to present the performance of the method and interpret the obtained results.

Multi-Attribute Three-Way Decision Approach Based on Ideal Solutions under Interval-Valued Fuzzy Soft Environment

The interval-valued fuzzy soft set (IVFSS) model, which combines the benefits of the soft set model with the interval-valued fuzzy set (IVFS) model, is a growing and effective mathematical tool for processing hazy data. In detail, this model is characterized by symmetry, which has the lower and upper membership degree. The study of decision-making based on IVFSS has picked up more steam recently. However, existing multi-attribute decision-making (MADM) methods can only sort alternative schemes, but are not able to classify them, which is detrimental to decision-makers’ efficient decision-making. In this paper, we propose a multi-attribute three-way decision-making (MATWDM) algorithm based on ideal solutions for IVFSS. MATWDM is extended to the IVFSS environment by incorporating the concept of the ideal solution, offering a more adaptable and comprehensive approach for addressing uncertain MADM issues. The method not only obtains the ranking results of the alternatives, but also divides them into acceptance domain, rejection domain, and delayed-decision domain, which makes the decision results more reasonable and effective, facilitating decision-makers to make better decisions. We apply the proposed three-way decision algorithm to two practical cases as diverse as mine emergency decision and Homestay selection decision. Additionally, the effectiveness and viability of the suggested method are confirmed by experimental findings.

Fuzzy implications-based transformation approaches from semi-three-way decision spaces to three-way decision spaces and their applications

Three-way decision spaces, as an important component of three-way decisions, greatly enrich their theoretical development and application prospects. Meanwhile, fuzzy implications, as a vital class of fuzzy logic connectives, have made great contributions to the solution of practical problems, especially complex decision-making problems. This paper considers the collaborative effect of the two, which inject new vitality into the theoretical development and application prospects of fuzzy implications and three-way decision spaces. As a vital component of three-way decision spaces, (semi-)decision evaluation functions have been widely studied based on fuzzy logic connectives and become a research hotspot. Specifically, this paper focuses on fuzzy implications-based transformation approaches from semi-three-way decision spaces to three-way decision spaces and their applications. Firstly, we present some novel fuzzy implications-based transformation approaches from semi-decision evaluation functions to decision evaluation functions, and construction approaches of semi-decision evaluation functions involving the existing semi-decision evaluation functions, fuzzy sets, interval-valued fuzzy sets and fuzzy relations. Secondly, we discuss the relationship between our approaches and the known construction approaches of three-way decision spaces. Notably, our approaches cover all existing approaches except the uninorms-based approaches. Finally, by the experiment results, we obtain our approaches are feasible, effective, superior to the known three-way decision spaces approaches and have good anti-noise ability. And, the parameter ρ of our approaches is also effective and stable.

An Innovative Algorithm Based on Octahedron Sets via Multi-Criteria Decision Making

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.

Analyzing risk factors of tuberculosis using type-2 interval-valued trapezoidal fuzzy numbers with Einstein aggregation operators extended to MCDM

The principal motive of this work is to evolve and initiate an extension from interval-valued fuzzy sets to type-2 interval-valued fuzzy sets (T2IVFS) related to weighted aggregation functions containing the Einstein operator. The chief reason for this extension is that the constancy of the terms can also be taken into data during the aggregation operation. The main goal of this article is to compose the aggregation operators and their characteristics such as the Type-2 interval-valued fuzzy Einstein weighted arithmetic aggregating operator (T2IVFEWA), Type-2 interval-valued fuzzy Einstein weighted geometric aggregating operator (T2IVFEWG), and the characteristics are expressed. At last, to intimate the effectiveness of the suggested approach and explicate the purpose of these operators, a hybrid multi-criteria decision-making problem (MCDM) to select the best risk factor for Tuberculosis (TB) is considered and the result is compared with the outcome of the existing operators and methods. Additionally, a sensitivity analysis was conducted to verify the robustness of the proposed decision-making process.

Promoting inclusivity in education amid the post-COVID-19 challenges: An interval-valued fuzzy model for pedagogy method selection

The advent of the COVID-19 pandemic has brought about unprecedented disruptions in the field of education, necessitating a reevaluation of pedagogical approaches in the post-COVID era. This research paper introduces a novel interval-valued fuzzy simple additive weighting (SAW), weighted product model (WPM), and weighted aggregates sum product assessment (WASPAS) multi-criteria decision-making (MCDM) framework to address the challenge of promoting inclusivity in education amidst the post-COVID era. Leveraging the uncertainty inherent in the post-pandemic educational landscape, the proposed method offers a comprehensive pedagogy selection approach incorporating interval-valued fuzzy sets to account for imprecise and ambiguous data. By integrating the principles of inclusivity and diversity, the method evaluates various pedagogical approaches and their effectiveness in fostering an inclusive learning environment for diverse student populations. The study showcases the application of pedagogy selection in real-world educational scenarios, demonstrating its potential to inform policy decisions and enable educational institutions to adapt and cater to the evolving needs of learners in the aftermath of the COVID-19 pandemic.

Fusing multiple interval-valued fuzzy monotonic decision trees

As a powerful knowledge mining technique for ordinal classification tasks, dominance-based rough set theory has many advantages but also some issues. Sensitivity to noisy information means that even a single mislabeled sample can lead to substantial fluctuations in the approximation calculations. Another issue is that most monotonic classifiers can only handle real-valued data and cannot directly deal with interval-valued data, which is common in practical applications. To address these issues, a tree-based fusion learning method for monotonic classification of interval-valued attributes, named FM-IFMDT, has been proposed. Its functional structure comprises three components: (i) The proposed robust β-precision interval-valued fuzzy dominance neighborhood rough set model (β-IFDNRS) can adaptively identify and filter noise samples. (ii) The interval dominance discernibility matrix based on β-IFDNRS is developed for feature selection and can generate a set of complete and diverse feature subsets. (iii) The novel interval-valued fuzzy monotonic decision tree (IFMDT) based on the probability distribution is trained on each feature subset and is used as the base classifier of the weighted voting fusion model. Extensive experiments show the proposed fusion learning method has significant advantages.

Interval-valued bipolar complex fuzzy soft sets and their applications in decision making

In this manuscript we demonstrate interval-valued bipolar complex fuzzy set (IVBCFS) and then interval-valued bipolar complex fuzzy soft set (IVBCFSS), as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set and soft set. We also initiate operational laws and basic results and properties for IVBCFS and IVBCFSS. Further explanation is given for the basic algebraic operations like complement, extended union, extended intersection, restricted union, and restricted intersection, AND product and OR product for IVBCFSS. Moreover, we demonstrate some fundamental aggregation operators like IVBCFS average aggregation, IVBCFS geometric and as well as their properties. To emphasize the usefulness and application of the system, we also develop the decision-making method and joint instances of the IVBCFSS (set-ups 1 and 2). In order to describe the effectiveness and influence of the approaching novel work, this study uses a comparative analysis of the new creating concept with prevailing ideas.

N-Ary aggregation operators on function spaces: perspective of construction

For disposing numerous practical application problems involving expert systems, decision-making, image processing, classifications and etc, the investigations on the constructions and basic properties of n-ary aggregation operators (nAAOs) have always been a hot research topic with important research value and significance at theoretical investigations on aggregation operators (AOs). Herein, first, we propose a method for constructing nAAOs on function spaces via a family of known ones defined on a bounded poset, where those function spaces are composed by all fuzzy sets with that bounded poset as the truth values set. This method is different from the existing construction methods of nAAOs on bounded posets and provides a unified way of constructing usual nAAOs (like t-norms, uninorms, overlap functions, etc.) on function spaces via a family of known ones. Second, we present notion of representable nAAOs on function spaces and afford their equivalent characterization. Third, we discuss some vital properties of representable nAAOs on function spaces. Fourth, it is worth noticing that the obtained results cover the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when underlying bounded poset is taken as the corresponding truth values set, respectively. As a consequence, the theoretical results obtained herein have certain promotion and basic theoretical value for the mining of new potential applications of nAAOs in real problems, especially in expert systems, decision-making, image processing and etc.

Complex Pythagorean Normal Interval-Valued Fuzzy Aggregation Operators for Solving Medical Diagnosis Problem

This paper presents a new methodology for solving multiple-attribute decision-making problems (MADMs) using a complex Pythagorean normal interval-valued fuzzy set (CPNIVFS), which is an extended concept of a complex Pythagorean fuzzy set. Four types of different aggregating operations (AOs), including CPNIVF weighted averaging (CPNIVFWA), CPNIVF weighted geometric (CPNIVFWG), generalized CPNIVFWA (CGPNIVFWA), and generalized CPNIVFWG (CGPNIVFWG), are discussed. The scoring function, accuracy function, and operational laws of the CPNIVFS are defined. Algebraic structures, such as associative, distributive, idempotent, bounded, commutativity, and monotonicity properties, are also shown to be satisfied by complex Pythagorean normal interval-valued fuzzy numbers. Furthermore, an algorithm is proposed to solve the MADM problems based on the defined AOs. The proposed approach is then used for a medical diagnosis problem about brain tumors because computer science and machine tool technology are among the most important components of brain tumor research. The five types of brain tumors diagnosed in these patients are gliomas, meningiomas, metastases, embryonal tumors, and ependymomas. Several types of treatments are available, which are often combined as part of an overall treatment plan. Brain tumors can be treated in various ways, including surgery, radiation therapy, chemotherapy, immunotherapy, and clinical trials. Based on the comparisons and options gathered, the most suitable treatment can be chosen. In this regard, it is evident that the value of the integer ⅁\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Game $$\\end{document} plays a significant role in determining the model. The candidate models under consideration can be validated by comparing them with the previously proposed ones. The proposed technique is compared with the existing method to demonstrate its superiority and validity, and the results conclude that the former is more reliable and effective than the latter. Finally, the criteria are evaluated by expert assessments to determine the most appropriate options.

Novel knowledge and accuracy measures for interval-valued fuzzy sets with applications in cluster analysis and pattern detection

Novel knowledge and accuracy measures for interval-valued fuzzy sets with applications in cluster analysis and pattern detection

An attribute ranking method based on rough sets and interval-valued fuzzy sets

Feature importance is a complex issue in machine learning, as determining a superior attribute is vague, uncertain, and dependent on the model. This study introduces a rough-fuzzy hybrid (RAFAR) method that merges various techniques from rough set theory and fuzzy set theory to tackle uncertainty in attribute importance and ranking. RAFAR utilizes an interval-valued fuzzy matrix to depict preference between attribute pairs. This research focuses on constructing these matrices from datasets and identifying suitable rankings based on these matrices. The concept of interval-valued weight vectors is introduced to represent attribute importance, and their additive and multiplicative compatibility is examined. The properties of these consistency types and the efficient algorithms for solving related problems are discussed. These new theoretical findings are valuable for creating effective optimization models and algorithms within the RAFAR framework. Additionally, novel approaches for constructing pairwise comparison matrices and enhancing the scalability of RAFAR are suggested. The study also includes experimental results on benchmark datasets to demonstrate the accuracy of the proposed solutions.

A new decision analysis framework for multi-attribute decision-making under interval uncertainty

In decision-making analysis, the description on decision makers' risk attitude under uncertainty is a focus topic. This paper presents a novel decision analysis framework that considers uncertainty and risk using the cumulative prospect theory. The proposed approach describes uncertain preference information using interval-valued fuzzy sets, maintaining this representation without converting intervals into crisp numbers throughout the entire process. To quantify the distance between interval-valued fuzzy sets, we introduce a comprehensive interval-valued distance model that accounts for multiple situations determined based on the relative positions between intervals. Afterwards, some related theorems about the proposed interval-valued distance model are explored mathematically. Then, with the aid of different reference points and interval-valued distance model, the overall interval-valued prospect value of each alternative is investigated. Finally, an Enhanced Minimax Regret-based (EMR) approach is developed to compare and rank the obtained overall interval-valued prospect values. An illustrative example, accompanied by detailed discussions, is provided to demonstrate the flexibility and superiority of the proposed decision analysis framework.

Fuzzy and rough approach to the problem of missing data in fall detection system

Two new methods for mining incomplete data based on interval-valued fuzzy set theory and rough set theory, particularly maximal consistent blocks were proposed, and their application to the fall detection system, exactly to the posture detection system. The suggested methods are based on interval-valued aggregation operators supporting decision-making involving uncertainty. Additionally, the new measure of knowledge was used in two aspects: in the imputation of missing data and during the inference process. The usage of the knowledge measure allowed for a reduction of the uncertainty due to incompleteness and imprecision of data, while increasing the efficiency of the decision-making process, through the additional selection of objects for which the degree of the information measure is the highest. The developed hybrid approach to mining incomplete data and its application in a real decision-making problem yield stable performance despite the increase in missing data.

Interval Valued Neutrosophic INK-Ideal Via INK-Algebra

Based on Zadeh’s notion of interval-valued fuzzy set, we develop in this work the notion of interval-valued neutrosophic INK-ideals (briefly, IV N INK-ideals) in INK-algebra and evaluate some of its properties. We show that all N INK-ideals of INK-algebra A can be realized as IV level INK-ideals of INK-algebra A. We then deduce numerous related results, which are summarized in the abstract.

Generated admissible orders for intervals by matrices and continuous functions

In this paper, we systematically study the algebraic structures of the spaces L([0,1]), encompassing all closed subintervals of [0,1], under the generated admissible orders. We first prove that the admissible order on L([0,1]) generated by a non-degenerate matrix must be the form of two weighted averaging operators. As a corollary, we deduce that each admissible order on L([0,1]) generated by a non-degenerate matrix and the standard order ≤ on [0,1] are not isomorphic. Furthermore, we show that each admissible order on L([0,1]) derived from two continuous mappings and the standard order ≤ on [0,1] are not isomorphic, partially answering a conjecture proposed by Santana et al. (2020) [38]. Besides, we prove that L([0,1]) is a complete lattice under the admissible order generated by two continuous mappings. This is the first result regarding the completeness of L([0,1]). Finally, we apply the admissible orders to solve a minimal path problem within the context of interval-valued fuzzy weighted graph. The above results theoretically refine the study of the classification, non-isomorphism, and completeness of admissible orders, while expanding the scope of interval-valued fuzzy sets in practical applications.

A bipolar-valued fuzzy set is an intersected interval-valued fuzzy set

This paper shows that a bipolar-valued fuzzy set and a special interval-valued fuzzy set, whose cut-off point of left and right ends of the interval-valued membership is λ(0<λ<1), are identical from a mathematical point of view. All results on bipolar-valued fuzzy sets can be obtained by corresponding results of interval-valued fuzzy sets or fuzzy sets. This paper does not completely reject bipolar-valued fuzzy sets, but points the way for such research.

Type-2 Intuitionistic Interpolation Cubic Fuzzy Bézier Curve Modeling using Shoreline Data

The notion of fuzzy sets is fast becoming a key instrument in defining the uncertainty data and has increasingly been recognised by practitioners and researchers across different disciplines in recent decades. The uncertainty data cannot be modeled directly and this causes hindrance in obtaining accurate information for analysis or predictions. Hence, this paper contributes to another approach in which an application of type-2 intuitionistic fuzzy set (T-2IFS) in geometric modeling onto complex uncertainty data where the data are defined using the type-2 fuzzy concept. T-2IFS is the generalized forms of fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and interval-valued intuitionistic fuzzy sets. Based on the concept of T2IFS, type-2 intuitionistic fuzzy point (T-2IFP) is defined in order to generate a type-2 intuitionistic fuzzy control point (T-2IFCP). Following, the T-2IFCP will be blended with the Bernstein blending function through the interpolation method, resulting to a type-2 intuitionistic interpolation cubic fuzzy Bézier curve. Shoreline data is used as the data and further verifies that the model can be conceivably accepted. In conclusion, the proposed methods are reliable and can be expanded to many other areas.

Novel transformation methods from semi-three-way decision spaces to three-way decision spaces and their applications

As a vital theoretical research of three-way decision (3WD), three-way decision space (3WDS) further enriches the theory on 3WD and enhances its application. In 3WDS, decision evaluation function (DEF) is an indispensable component and plays a crucial role in practical decision problems. But, a large number of valuable functions are the semi-decision evaluation functions (S-DEFs). For this problem, Hu gave many S-DEFs and presented a transformation method from S-DEFs to DEFs. Since then, it has led to a research hotspot that is the introduction of the usual aggregation functions (AFs) into the transformation method of Hu's form. This paper focuses on the new form of transformation methods from semi-three-way decision spaces (S-3WDSs) to 3WDSs and their applications. Firstly, we present a new form of transformation method with parameter σ from S-DEFs to DEFs. And, we verify both theoretically and experimentally that this introduced parameter σ provides a valid window to dynamically adjust 3WDSs by changing the parameter σ, respectively. Notably, this new form of transformation method covers all existing 3WDSs transformation methods except the representable uninorms-based methods. Secondly, we give a more general function named with F-function, which can be introduced to these construction methods in the same way as usual AFs. This enables decision-makers to select more suitable functions for solving practical problems. Thirdly, based on the new form, usual AFs and F-functions, we present some new construction methods of S-DEFs (resp. DEFs) related to fuzzy sets, interval-valued fuzzy sets and random sets. Finally, to make up for the application of the theory of 3WDSs in complex datasets, we select 7 complex datasets for experimenting and conduct comparative and sensitivity analysis to test the viability and efficacy of our methods. Notably, we also give an algorithm to calculate the thresholds α and β, which can adjust the optimal thresholds α and β according to different 3WDSs.

ωA-IvE methodology: Admissible interleaving entropy methods applied to video streaming traffic classification

This work focuses on the width-based interval fuzzy entropy notion, considering the interval data diameter as a measure of the lack of knowledge and uncertainty related to the precise membership degrees of elements in an interval-valued fuzzy set. The w-preserving notion relates the uncertainty from input to output in system information. Such a concept generates a new entropy by applying width-based average functions and admissible order to compare interval data and define width-based fuzzy connectives in data fuzzy computations. A new admissible total order is introduced, requesting just one injective and increasing function, illustrated by a Decimal Digit Interleaving (DDI) function. The proposal methods are based on the admissible interleaving order and related expressions for width-based interval entropy considering different conditions for composition among width-based interval fuzzy operators, as negations with equilibrium and average functions, also including idempotent aggregation function and restricted equivalence functions. Finally, we illustrate the application of the proposed methods for solving a video streaming traffic classification problem. The admissible interleaving entropy analysis is performed over the attributes modeling the FuzzyNetClass approach, a computational model for traffic classification related to video streaming, exploring the integration of inference systems based on interval-valued fuzzy logic and machine learning algorithms. The results by comparison with other width-based interval fuzzy entropy methods were promising and pointed to continuing study and research efforts.