Interval-valued Fuzzy Sets Research Articles

Granular Symmetric Implicational Method

As a means of describing realistic problems, fuzzy sets can be included into the category of information granules from a broader perspective. Then the interval-valued fuzzy set itself is an expression of information granule which has more elaborate and stronger characterization abilities than generic fuzzy sets. In this study, facing up with modeling situations involving the use of interval-valued fuzzy sets, we come up with the granular symmetric implicational (GSI) method of fuzzy inference in view of the symmetric implicational idea and granular computing, which includes the basic GSI method and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\zeta (w, z)$</tex-math></inline-formula> -GSI method. First, complete residuated lattices are employed as the structures of truth-values for interval-valued fuzzy sets. Second, unified expressions of optimal solutions to two GSI methods are gained for R-implications and (S, N)-implications. Lastly, it is shown through examples that the GSI method is superior over corresponding interval-valued fully implicational method. The originality of this work is three-fold. To begin with, the interval-valued fuzzy operators are introduced to the symmetric implicational mechanism, and novel symmetric implicational principles are presented which ameliorate the previous ones. Moreover, we offer a new construction method for interval-valued implications and corresponding adjoint couples, and on the strength of it we validate the reversibility and continuous properties of the GSI method. Finally, the hierarchical granular inference strategy is established for the GSI method in allusion to the circumstance of multiple rules.

Amplitude interval-valued complex Pythagorean fuzzy sets with applications in signals processing

In this paper, we introduce the notion of amplitude interval-valued complex Pythagorean fuzzy sets (AIVCPFSs). The motivation for this extension is the utility of interval-valued complex fuzzy sets in membership and non-membership degree which can express the two dimensional ambiguous information as well as the interaction among any set of parameters when they are in the form of interval-valued. The principle of AIVCPFS is a mixture of the two separated theories such as interval-valued complex fuzzy set and complex Pythagorean fuzzy set which covers the truth grade (TG) and falsity grade (FG) in the form of the complex number whose real part is the sub-interval of the unit interval. We discuss some set-theoretic operations and laws of the AIVCPFSs. We study some particular examples and basic results of these operations and laws. We use AIVCPFSs in signals and systems because its behavior is similar to a Fourier transform in certain cases. Moreover, we develop a new algorithm using AIVCPFSs for applications in signals and systems by which we identify a reference signal out of the large number of signals detected by a digital receiver. We use the inverse discrete Fourier transform for the membership and non-membership functions of AIVCPFSs for incoming signals and a reference signal. Thus a method for measuring the resembling values of two signals is provided by which we can identify the reference signal.

New concepts of domination in cubic graphs with application

The cubic set, introduced as a combination of a fuzzy set and an interval-valued fuzzy set, provided researchers with more flexibility than the previous two sets in dealing with complex and uncertain problems. Fuzzy graphs, based on this type of set, are among the emerging fuzzy graphs that have a great potential to model the surrounding phenomena. Consistent with the special role that cubic graphs play in decision-making and selecting superior options, dominating these graphs is of great importance and value. In this paper, we introduce the domination of the cubic graphs in terms of strong edges and examine their properties. In addition, we examine domination in terms of independent sets and since many of the phenomena surrounding us are hybrid, we also discuss the domination concept on its fuzzy operations. Finally, we present an application of this graph on the subject of domination.

Interval-Valued Intuitionistic Fuzzy Subalgebras/Ideals of Hilbert Algebras

In this paper, the concept of interval-valued intuitionistic fuzzy sets to subalgebras and ideals of Hilbert algebras is introduced. The inverse image of interval-valued intuitionistic fuzzy subalgebras and interval-valued intuitionistic fuzzy ideals of Hilbert algebras is studied and some related properties are investigated. Equivalence relations on interval-valued intuitionistic fuzzy ideals are discussed.

An Algebraic Approach to Modular Inequalities Based on Interval-Valued Fuzzy Hypersoft Sets via Hypersoft Set-Inclusions

Interval-valued fuzzy hypersoft set is an emerging field of study which is projected to address the limitations of interval-valued fuzzy soft set for the entitlement of multiargument approximate function. This kind of function maps the subparametric tuples to power set of universe. It emphasizes on the partitioning of attributes into their respective subattribute values in the form of disjoint sets. These features make it a completely new mathematical tool for solving problems dealing with uncertainties. In this study, after characterization of essential properties, operations, and set-inclusions ( L -inclusion and J -inclusion) of interval-valued fuzzy hypersoft set, some of its modular inequalities are discussed via set-inclusions. It is proved that all set-inclusion-based properties and inequalities are preserved when ordinary approximate function of interval-valued fuzzy soft set is replaced with multiargument approximate function of interval-valued fuzzy hypersoft set.

Interval-valued fuzzy sets aggregation and evaluation approaches

We review interval-valued fuzzy sets (IVFS) and develop their application for several areas. We show how to aggregate IVFSs and provide information measures to guide the aggregation. An example of the use of IVFS to represent the uncertainties in Covid-19 contact tracing is illustrated. Then we consider the application of IVFS in spatial data. By determining distance and area of IVFS data we can use aggregation to apply this to minimum bounding rectangles for GIS. Finally, IVFS can be applied in computing diversity in the process of selection of groups exhibiting diversity.

Topological Data Analysis with Cubic Hesitant Fuzzy TOPSIS Approach

A hesitant fuzzy set (HFS) and a cubic set (CS) are two independent approaches to deal with hesitancy and vagueness simultaneously. An HFS assigns an essential hesitant grade to each object in the universe, whereas a CS deals with uncertain information in terms of fuzzy sets as well as interval-valued fuzzy sets. A cubic hesitant fuzzy set (CHFS) is a new computational intelligence approach that combines CS and HFS. The primary objective of this paper is to define topological structure of CHFSs under P(R)-order as well as to develop a new topological data analysis technique. For these objectives, we propose the concept of “cubic hesitant fuzzy topology (CHF topology)”, which is based on CHFSs with both P(R)-order. The idea of CHF points gives rise to the study of several properties of CHF topology, such as CHF closure, CHF exterior, CHF interior, CHF frontier, etc. We also define the notion of CHF subspace and CHF base in CHF topology and related results. We proposed two algorithms for extended cubic hesitant fuzzy TOPSIS and CHF topology method, respectively. The symmetry of optimal decision is analyzed by computations with both algorithms. A numerical analysis is illustrated to discuss similar medical diagnoses. We also discuss a case study of heart failure diagnosis based on CHF information and the modified TOPSIS approach.

Decision-making strategy for slope stability using similarity measures between interval-valued fuzzy credibility sets

Decision-making strategy for slope stability using similarity measures between interval-valued fuzzy credibility sets

Time-sequential hesitant fuzzy set and its application to multi-attribute decision making

The hesitant fuzzy set has been an important tool to address problems of decision making. There are several various improved hesitant fuzzy sets, such as dual hesitant fuzzy set, hesitant interval-valued fuzzy set, and intuitionistic hesitant fuzzy set, however, no one kind of improved fuzzy sets could reflect attitude characteristics of decision makers on time-sequences. In reality, time-sequence is one important sector to reflect hesitant situations as decision makers might have different knowledges of the same alternative at different moments. To perfect the description of such hesitant situations and obtain more reasonable results of decision making, we define a new kind of hesitant fuzzy set, namely, time-sequential hesitant fuzzy set. Meanwhile, its corresponding basic operators, score function and distance measures are proposed. We also propose the concept of fluctuated hesitant information to describe hesitant degrees of decision makers on time-sequences. By comprehensively utilizing the score function, fluctuated hesitant information and distance measures under time-sequential hesitant fuzzy set, a synthetic decision model is proposed. Two illustrated examples and one real-application are utilized to illustrate the effectiveness and advantage of the synthetic decision model under time-sequential hesitant fuzzy set.

Group Decision-Making Problems Based on Mixed Aggregation Operations of Interval-Valued Fuzzy and Entropy Elements in Single- and Interval-Valued Fuzzy Environments

Fuzzy sets and interval-valued fuzzy sets are two kinds of fuzzy information expression forms in real uncertain and vague environments. Their mixed multivalued information expression and operational problems are very challenging and indispensable issues in group decision-making (GDM) problems. To solve single- and interval-valued fuzzy multivalued hybrid information expression, operations, and GDM issues, this study first presents the notion of a single- and interval-valued fuzzy multivalued set/element (SIVFMS/SIVFME) with identical and/or different fuzzy values. To effectively solve operational problems for various SIVFME lengths, SIVFMS/SIVFME is converted into the interval-valued fuzzy and entropy set/element (IVFES/IVFEE) based on the mean and information entropy of SIVFME. Then, the operational relationships of IVFEEs and the expected value function and sorting rules of IVFEEs are defined. Next, the IVFEE weighted averaging and geometric operators and their mixed-weighted-averaging operation are proposed. In terms of the mixed-weighted-averaging operation and expected value function of IVFEEs, a GDM method is developed to solve multicriteria GDM problems in the environment of SIVFMSs. Finally, the proposed GDM method was utilized for a supplier selection problem in a supply chain as an actual sample to show the rationality and efficiency of SIVFMSs. Through the comparative analysis of relative decision-making methods, we found the superiority of this study in that the developed GDM method not only compensates for the defects of existing GDM methods, but also makes the GDM process more reasonable and flexible.

Certain Structure of Lagrange’s Theorem with the Application of Interval-Valued Intuitionistic Fuzzy Subgroups

This paper presents the concept of an interval-valued intuitionistic fuzzy subgroup defined on interval-valued intuitionistic fuzzy sets. We study some of the fundamental algebraic properties of interval-valued intuitionistic fuzzy cosets and interval-valued intuitionistic fuzzy normal subgroup of a given group. This idea is used to describe the interval-valued intuitionistic fuzzy order and index of interval-valued intuitionistic fuzzy subgroup. We have created numerous algebraic properties of interval-valued intuitionistic fuzzy order of an element. We also prove the interval-valued intuitionistic fuzzification of Lagrange’s theorem.

Novel entropy and distance measures for interval-valued intuitionistic fuzzy sets with application in multi-criteria group decision-making

Entropy and distance are the most important information-theoretic measures. These measures have found useful applications in different areas. In the present communication, we study the entropy and distance measures under an interval-valued intuitionistic fuzzy (IVIF) environment using an exponential function. First, it presents the novel exponential entropy and distance measures for interval-valued intuitionistic fuzzy sets (IVIFSs) with proof of their authenticity. A method is offered to solve multi-criteria group decision-making (MCGDM) problems in the IVIF environment based on the weighted exponential entropy measure. The performance of the proposed IVIF MCGDM method is shown by taking two case studies. Second, the success and strength of the proposed IVIF distance measures are demonstrated by comparing them with the existing ones. Further, the paper advances an approach to solve multi-attribute decision-making problems under the IVIF environment. Finally, it considers a real-world example to illustrate the applicability and authenticity of the proposed approach. In doing so, the proposed approach is compared with existing methods to exhibit its advantages. Thus, the proposed IVIF information measures and multi-criteria group decision-making method are more suitable to solve real-life decision-making problems.

On inf -Hesitant Fuzzy Γ -Ideals of Γ -Semigroups

The notions of an inf -hesitant fuzzy Γ -ideal and a sup , inf -hesitant fuzzy Γ -ideal, which are a generalization of an interval-valued fuzzy Γ -ideal, of a Γ -semigroup are introduced and some properties are investigated. Characterizations of the notions are provided in terms of sets, fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. Furthermore, characterizations of a Γ -ideal of a Γ -semigroup are given in terms of inf -hesitant and sup , inf -hesitant fuzzy Γ -ideals.

Convexity and level sets for interval-valued fuzzy sets

Convexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-valued fuzzy sets in order to be able to cover some wider area of imprecision. We show some of its interesting properties, and study the preservation under the intersection and the cutworthy property. Finally, we applied convexity to decision-making problems.

Detection of outlier information using linguistically quantified statements – the state of the art

Over the past decade, outlier detection has become a prominent area of research. Much of the current literature on Knowledge Discovery in Databases (KDD) pays particular attention to the identification of rare, unusual data items or surprising observations that differ substantially from the majority of the data. In Data Mining (DM), outlier detection refers to the identification of unusual data records that might be interesting in the context of data pattern discovery or may represent potential data errors that require further investigation. Recent research has emphasized the need for KDD / DM systems capable of processing information given in a linguistic form and communicating the result to the user in a natural language. Such systems represent a valuable tool for intelligent data analysis tasks, including outlier detection. The paper provides an overview of the current state of research on the detection of outlier information using linguistically quantified statements based on diverse variants of fuzzy linguistic summarization. It pays special attention to the use of monotonic and non-monotonic quantifiers defined on interval-valued fuzzy sets. The outlier detection procedure using linguistic summaries is described in detail.

Applications of Cubic Schweizer–Sklar Power Heronian Mean to Multiple Attribute Decision‐Making

The cubic set (CS) is a basic simplification of several fuzzy notions, including fuzzy set (FS), interval‐valued FS (IVFS), and intuitionistic FS (IFS). By the degrees of IVFS and FS, CS exposes fuzzy judgement, and this is a much more potent mathematical approach for dealing with information that is unclear, ambiguous, or indistinguishable. The article provides many innovative operational laws for cubic numbers (CNs) drawn on the Schweizer–Sklar (SS) t‐norm (SSTN), and the SS t‐conorm (SSTCN), as well as several desired properties of these operational laws. We also plan to emphasise on the cubic Schweizer–Sklar power Heronian mean (CSSPHM) operator, as well as the cubic Schweizer–Sklar power geometric Heronian mean (CSSPGHM) operator, in order to maintain the supremacy of the power aggregation (PA) operators that seize the complications of the unsuitable information and Heronian mean (HM) operators that contemplate the interrelationship between the input data being aggregated. A novel multiple attribute decision‐making (MADM) model is anticipated for these freshly launched aggregation operators (AOs). Finally, a numerical example of enterprise resource planning is used to validate the approach’s relevance and usefulness. There is also a comparison with existing decision‐making models.

Selection of an alternative based on interval-valued hesitant picture fuzzy sets

Motivated by interval-valued hesitant fuzzy sets (IVHFSs) and picture fuzzy sets (PcFSs), a notion of interval-valued hesitant picture fuzzy sets (IVHPcFSs) is presented in this article. The concept of IVHPcFSs is put forward and some operational rules are developed to deal with it. The cosine similarity measures (SMs) are modified for IVHPcFSs to deal with interval-valued hesitant picture fuzzy (IVHPcF) data and the linear programming (LP) methodology is used to find out the criteria’s weights. A multiple criteria decision making (MCDM) approach is then developed to tackle the vague and ambiguous information involved in MCDM problems under the framework of IVHPcFSs. For the validation and strengthen of the proposed MCDM approach a practical example is put forward to select the educational expert at the end.

A new divergence measure of interval-valued pythagorean fuzzy sets and its application

As the extension of the Fuzzy sets (FSs) theory, the Interval-valued Pythagorean Fuzzy Sets (IVPFS) was introduced which play an important role in handling the uncertainty. The Pythagorean fuzzy sets (PFSs) proposed by Yager in 2013 can deal with more uncertain situations than intuitionistic fuzzy sets because of its larger range of describing the membership grades. How to measure the distance of Interval-valued Pythagorean fuzzy sets is still an open issue. Jensen–Shannon divergence is a useful distance measure in the probability distribution space. In order to efficiently deal with uncertainty in practical applications, this paper proposes a new divergence measure of Interval-valued Pythagorean fuzzy sets,which is based on the belief function in Dempster–Shafer evidence theory, and is called IVPFSDM distance. It describes the Interval-Valued Pythagorean fuzzy sets in the form of basic probability assignments (BPAs) and calculates the divergence of BPAs to get the divergence of IVPFSs, which is the step in establishing a link between the IVPFSs and BPAs. Since the proposed method combines the characters of belief function and divergence, it has a more powerful resolution than other existing methods.

On the Search for a Measure to Compare Interval-Valued Fuzzy Sets

Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval.

On Interval-Valued Fuzzy Soft Preordered Sets and Associated Applications in Decision-Making

Recently, using interval-valued fuzzy soft sets to rank alternatives has become an important research area in decision-making because it provides decision-makers with the best option in a vague and uncertain environment. The present study aims to give an extensive insight into decision-making processes relying on a preference relationship of interval-valued fuzzy soft sets. Firstly, interval-valued fuzzy soft preorderings and an interval-valued fuzzy soft equivalence are established based on the interval-valued fuzzy soft topology. Then, two crisp preordering sets, namely lower crisp and upper crisp preordering sets, are proposed. Next, a score function depending on comparison matrices is expressed in solving multi-group decision-making problems. Finally, a numerical example is given to illustrate the validity and efficacy of the proposed method.