Fuzzy Multisets Research Articles

Neutrosophic Multigroups and Applications

In recent years, fuzzy multisets and neutrosophic sets have become a subject of great interest for researchers and have been widely applied to algebraic structures include groups, rings, fields and lattices. Neutrosophic multiset is a generalization of multisets and neutrosophic sets. In this paper, we proposed a algebraic structure on neutrosophic multisets is called neutrosophic multigroups which allow the truth-membership, indeterminacy-membership and falsity-membership sequence have a set of real values between zero and one. This new notation of group as a bridge among neutrosophic multiset theory, set theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroups and give its applications to group theory.

On Multi-Fuzzy Rough Sets, Relations, and Topology

This article describes how rough set theory has an innate topological structure characterized by the partitions. The approximation operators in rough set theory can be viewed as the topological operators namely interior and closure operators. Thus, topology plays a role in the theory of rough sets. This article makes an effort towards considering closed sets a primitive concept in defining multi-fuzzy topological spaces. It discusses the characterization of multi-fuzzy topology using closed multi-fuzzy sets. A set of axioms is proposed that characterizes the closure and interior of multi-fuzzy sets. It is proved that the set of all lower approximation of multi-fuzzy sets under a reflexive and transitive multi-fuzzy relation forms a multi-fuzzy topology.

Synopsis of the Notions of Multisets and Fuzzy Multisets

This paper is an attempt to summarize the basic concepts of the theories of multiset and fuzzy multiset. We begin by describing multisets and the operations between them with some related results. In the same vein, the basic concepts of fuzzy multiset theory as well as the operations between fuzzy multisets are buttressed in relation to multiset theory. Finally, we present some properties of fuzzy multisets with some related results.

An Algorithmic Approach for Computing Unions and Intersections Between Fuzzy Multisets

Fuzzy multisets represent a particularly challenging generalization of the concept of fuzzy sets. The membership degrees of fuzzy multisets are given by multisets in [0,1] rather than single values. Mathematically, they can be also seen as a generalization of the hesitant fuzzy sets. But in this general setting, the information about repetition is not lost with fuzzy multisets; and so, the opinions given by the experts are more reliably accounted for. The definitions of the complement, union, and intersection operations for these sets and their relation with other extensions of fuzzy sets, however, is not straightforward. Aggregate unions and intersections have been shown to be equivalent to the standard definitions of union and intersection for the typical hesitant fuzzy sets. But computing them is not simple because the definitions of the aggregate operations as multiset unions of sequences based on permutations can potentially result in a huge number of operations. In this paper, we propose a new formulation for the aggregate union and intersection of fuzzy multisets that is computationally less intensive, thereby providing two algorithms amenable to computer-based calculations.

Reliability Analysis of Different Systems Using Triangular Multi-Fuzzy Sets

Reliability Analysis of Different Systems Using Triangular Multi-Fuzzy Sets

Complex Multi-Fuzzy Relation for Decision Making using Uncertain Periodic Data

We introduce a new type of relations called complex multi-fuzzy relation (CMFR). The novelty of CMFR lies in the ability of complex multi- membership functions to achieve more range of values while handling uncertainty of data that is periodic in nature. The application of complex multi-fuzzy sets is then discussed in determining: the influence of modern methods of education on student performance, and the time required for the former to affect the latter. A comparison between different existing relations and CMFR to show the ascendancy of our proposed CMFR is provided. Thereafter, a few related concepts such as complement, union, intersection and inverse along with several propositions are discussed, followed by the composition of CMFR along with some related theorems. Finally, the notions of symmetric, transitive, reflexive, and equivalence complex multi-fuzzy relations are established in our work.

Basic operations for fuzzy multisets

In their original and ordinary formulation, fuzzy sets associate each element in a reference set with one number, the membership value, in the real unit interval [0,1]. Among the various existing generalisations of the concept, we find fuzzy multisets. In this case, membership values are multisets in [0,1] rather than single values. Mathematically, they can also be seen as a generalisation of the hesitant fuzzy sets, but in this general environment, the information about repetition is not lost, so that, the opinions given by the experts are better managed. Thus, we focus our study on fuzzy multisets and their basic operations: complement, union and intersection. Moreover, we show how the hesitant operations can be worked out from an extension of the fuzzy multiset operations and investigate the important role that the concepts of order and sorting sequences play in the basic difference between these two related approaches.

Intuitionistic trapezoidal fuzzy multi-numbers and its application to multi-criteria decision-making problems

Intuitionistic trapezoidal fuzzy multi-numbers (ITFM-numbers) are a special intuitionistic fuzzy multiset on a real number set, which are very useful for decision makers to depict their intuitionistic fuzzy multi-preference information. In the ITFM-numbers, the occurrences are more than one with the possibility of the same or the different membership and non-membership functions. In this paper, we define ITFM-numbers based on multiple criteria decision-making problems in which the ratings of alternatives are expressed with ITFM-numbers. Firstly, some operational laws using t-norm and t-conorm are proposed. Then, some aggregation operators on ITFM-numbers are developed. Also, the ranking order of alternative is given according to the similarity of the alternative with respect to the positive ideal solution. Finally, a numerical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

FLM-TOPSIS: The fuzzy linguistic multiset TOPSIS method and its application in linguistic decision making

Linguistic decision making is an important subject in decision making, many interesting and important linguistic decision making methods have been proposed, in which, alternatives-criteria decision matrix are uniformly used to express linguistic assessments of alternatives provided by decision makers with respect to criteria. Alternatives-criteria decision matrixes have some limitations when we use them to distinguish distinct, partial unknown or hesitant linguistic decision making or carry out linguistic decision making in the huge amounts of decision information and alternatives. In this paper, we propose alternatives-linguistic terms decision matrix to represent linguistic assessments of alternatives, analyze advantages of the decision matrix in representing linguistic assessments and distinguishing distinct, partial unknown or hesitant linguistic decision making. To simple and fast fuse alternatives-linguistic terms decision matrixes, we further provide linguistic multiset or fuzzy linguistic multiset to represent linguistic assessments in alternatives-linguistic terms decision matrixes, analyze the function properties of the fuzzy linguistic multiset. Motivated by fuzzy multiset and the TOPSIS method, we develop the fuzzy linguistic multiset TOPSIS method for linguistic decision making, the method is mainly consisted of transformation, aggregation and exploitation phases. In transformation phase, linguistic assessments of alternatives are transformed into fuzzy linguistic multisets by using alternatives-linguistic terms decision matrixes. In aggregation phase, we use Union, Intersection and Sum operations of multisets to obtain the positive and negative ideal solutions of linguistic decision making, which are different with the positive and negative ideal solutions of the traditional TOPSIS method, in addition, we provide a pseudo-distance between two fuzzy linguistic multisets to fast fuse linguistic assessments of alternatives. In exploitation phase, we define a new closeness degree of alternative by using pseudo-distances between the alternative and the positive and negative ideal solutions, which can be used to obtain the set of most satisfying alternatives. We also design an algorithm to carry out linguistic decision making based on the proposed method. In cases study, we use two practical examples to illustrate the practicality of the proposed method and compare it with the symbolic aggregation-based method, the hesitant fuzzy linguistic TOPSIS method, the hesitant fuzzy linguistic VIKOR method and the probabilistic linguistic term sets TOPSIS method, results indicate that alternatives-linguistic terms decision matrix and fuzzy linguistic multiset are alternative, useful and flexible tools for linguistic decision method and the fuzzy linguistic multiset TOPSIS method is suitable to deal with partial unknown or hesitant linguistic decision making.

Intuitionistic Fuzzy Multisets in Medical Diagnosis

In this paper, sine inverse distance among intuitionistic fuzzy multi sets is proposed and some of its properties are discussed herein. The concept of the above method is an essential tool for dealing with uncertainties and shortcomings that affect the existing methods. Implementation of medical diagnosis is presented to find out the disease impacting the patient.

Complex Multi-Fuzzy Soft Set: Its Entropy and Similarity Measure

We develop a novel mathematical tool known as complex multi-fuzzy soft set (CMFSS) which has the ability to handle uncertainties, imprecision, and vagueness of information that are inherent in the data by considering the amplitude and phase terms of the complex numbers simultaneously. This CMFSS constitutes of a hybrid structure of multi-fuzzy set and soft set which are defined in a complex setting. The structure is flexible as it allows for a greater range of values for the membership function by extending them to the unit circle in a complex plane through modification of the multi-fuzzy soft set by the inclusion of an additional term called the phase term in order to take into account of the periodic nature of the data. Accordingly, the novelty of this paper lies in the complex multi-membership functions which consider more range of values while handling the uncertainty of the periodic data. In this paper, the concept of complex multi-fuzzy soft set is introduced. We then define its basic operations of complement, union and intersection and study some related properties, with supporting proofs. Subsequently, by means of level soft sets, we present an algorithm to solve a CMFSSs decision making problem, to illustrate the effectiveness and practicality of the proposed concept. Finally, we introduce axiomatic definitions of entropy and similarity measure for CMFSSs, and some formulas have also been put forward to calculate them. Numerical examples are given to demonstrate that the proposed entropy measure for CMFSSs is an important concept for measuring uncertainty in the information/data. Furthermore, some theorems are proposed showing how the entropy of CMFSS can be found from the similarity measure of CMFSS.

Multi-fuzzy Rough Sets based on Implicators and Continuous t-norms

This paper extends the study of multi-fuzzy rough sets using an implicator and a continuous t-norm and thus introduces multi-fuzzy rough sets based on fuzzy logical connectives. In this constructive approach, a pair of lower and upper approximation operators determined by an implicator and a triangular norm is defined. The fundamental properties of these approximation operators are examined. Connections between multi-fuzzy relations and the newly constructed multi-fuzzy rough approximation operators are also established. The theory of multi-fuzzy rough sets is analysed using an operator oriented view in the later sections. The lower and upper approximation operators are characterized by axioms. Various axiom sets of lower and upper multi-fuzzy set theoretic operators guarantee the existence of different types of multi-fuzzy relations which produce the same operators.

Operations on complex multi-fuzzy sets

In this paper, we introduce the concept of complex multi-fuzzy sets (CMkFSs) as a generalization of the concept of multi-fuzzy sets by adding the phase term to the definition of multi-fuzzy sets. In other words, we extend the range of multi-membership function from the interval [0,1] to unit circle in the complex plane. The novelty of CMkFSs lies in the ability of complex multi- membership functions to achieve more range of values while handling uncertainty of data that is periodic in nature. The basic operations on CMkFSs, namely complement, union, intersection, product and Cartesian product are studied along with accompanying examples. Properties of these operations are derived. Finally, we introduce the intuitive definition of the distance measure between two complex multi-fuzzy sets which are used to define δ-equalities of complex multi-fuzzy sets.

Topological Structures on Fuzzy Multisets

In this paper the concept of fuzzy topological space extended into fuzzy multiset topological space, closed fuzzy multiset, open fuzzy multisets, interior fuzzy multisets, closure of fuzzy multisets and their various properties discuss. Further, we extended the continuous functions to fuzzy multiset continuous function and also discussed subspace of fuzzy multiset topological space

EXTENDED HESITANT FUZZY SETS

Hesitant fuzzy sets (HFSs) are a useful tool to manage situations in which the decision makers (DMs) hesitate about several possible values for the membership to assess a variable, alternative, etc. However, HFSs have the information loss problem and cannot identify different DMs, which interferes with the application of HFSs in decision making. To overcome these limitations, we develop the extended hesitant fuzzy sets (EHFSs) in this paper. As an extension of HFSs, EHFSs have close relationships with existing fuzzy sets including intuitionistic fuzzy sets (IFSs), fuzzy multisets (FMSs), type-2 fuzzy sets (T2FSs), dual hesitant fuzzy sets (DHFSs), and especially HFSs. We propose a concept of extended hesitant fuzzy elements (EHFEs), then study the basic operations and the desirable properties of EHFEs in detail. Some extended hesitant distance measures are developed to illustrate their advantages comparing with the existing hesitant distance measures. To extend EHFSs to decision making, we combine the proposed distance measures with the Dempster-Shafer belief structure.

On Some Algebraic Structures of Type 2 Fuzzy Multisets

Many problems arise in practical applications when data is extracted from a group of experts, all of whom do not collectively agree. The theory of Type 2 Fuzzy Multisets is capable of modelling these types of uncertainties in decision making. The purpose of this paper is to introduce the study of various algebraic structures of Type 2 Fuzzy Multisets. As a beginning of this study, the concept of Type 2 Fuzzy Multigroup is introduced and its various properties are discussed. The authors also introduced the concept of normal Type 2 Fuzzy Multigroups which serves as a powerful tool in the study of group structures of Type 2 Fuzzy Multisets.

Correlation measure for fuzzy multisets

Abstract Fuzzy and multi set theories are recent approaches for information system and data mining. In this paper, we aim to construct correlation measure connected to these theories. We define fuzzy multi correlation measure and many of its properties are investigated, and some examples and applications are given. Also, we present new view of reduction for fuzzy multi information system.

A New Similarity Measure between Intuitionistic Fuzzy Multisets based on their Cardinality with Applications to Pattern Recognition and Medical Diagnosis

In this paper, a new similarity measure between Intuitionistic Fuzzy Multisets (IFMSs) based on their cardinality is proposed. Some properties of the proposed similarity measure between IFMSs are proved. The unique feature of the proposed measure is that the similarity measure can be applied for both intuitionistic fuzzy sets (IFS) and intuitionistic fuzzy multisets (IFMSs). Illustrative examples are taken to discuss that the proposed similarity measure can overcome the drawback of the existing similarity measure. Further the proposed similarity measure is also applied to pattern recognition and medical diagnosis problems.

Weighted fuzzy soft multiset and decision-making

Fuzzy soft multiset (FSMS) is a substantial and important fuzzy generalization of soft set and multiset (also called bag). In this paper, we have presented the idea of weighted fuzzy soft multiset (WFSMS) as a generalization of FSMS and its basic properties are to be studied. Also, we introduce a new adjustable approach to WFSMS based decision-making, for solving decision-making in an uncertain situation. The feasibility of our proposed WFSMS based decision-making procedure in practical application is read by a mathematical example.

Faces and dual of m-polar fuzzy planar graphs

Recently, Ghorai and Pal [12] introduced the notion of m-polar fuzzy planar graphs using the concept of m-polar fuzzy multisets. In this paper, m-polar fuzzy faces and strong m-polar fuzzy faces are defined. Then m-polar fuzzy dual graph of an m-polar fuzzy planar graph is introduced with example. Some isomorphism properties of m-polar fuzzy planar graphs are also studied.