Fuzzy Multisets Research Articles

Application of Intuitionistic Fuzzy Multisets in Appointment Process

In this paper, a precise note on intuitionistic fuzzy multisets is given and the concept is applied to appointment process. This process was carried out assuming three sets of 10-man committees screened five candidates vying for positions in an organization independently to obtain intuitionistic fuzzy multi-data. The obtained data are compared with the organization requirements of appointments via a new distance measure.

A study on m-polar fuzzy planar graphs

In many practical applications with a graph structure, there may exist crossing between edges which is not allowed in a crisp planar graph. Considering that graphic structures can be used to describe the crossing of edges in an m-polar fuzzy multi-graph with certain amount of m-polar fuzzy planarity value, first of all the notion of m-polar fuzzy multi-set is introduced. Then m-polar fuzzy multi-graphs, m-polar fuzzy planar graphs and m-polar fuzzy strong edges are defined. Several properties of m-polar fuzzy planar graphs are also studied.

Type 2 fuzzy multisets and its applications in decision making problems

The purpose of this paper is to introduce the concept of type 2 fuzzy multiset by integrating the type 2 fuzzy set and multiset models. Basic set operations of union, intersection and complement are extended to type 2 fuzzy multisets and made a comparative study on various set theoretic laws satisfied by these operations with set theoretic operations on type 1 and type 2 fuzzy sets. The notion of (α, λ) cut set is defined and some properties of cut sets are studied. Furthermore, we investigate the decision making based on type 2 fuzzy multisets. By means of (α, λ) cut sets, we propose an adjustable approach to type 2 fuzzy multiset based decision making. Moreover, we introduce the weighted type 2 fuzzy multiset and examine its application to decision making. Finally, by means of some illustrative examples we show the validity of type 2 fuzzy multiset, weighted type 2 fuzzy multiset in decision making problems.

On fuzzy multiset automata

The objective of this paper is to study fuzzy multiset automata, deterministic fuzzy multiset automata and fuzzy multiset languages. We show that a fuzzy multiset language is accepted by a fuzzy multiset finite automaton iff it is accepted by a deterministic fuzzy multiset finite automaton. We also introduce and study minimal realizations (minimal deterministic fuzzy multiset automata) for a given fuzzy multiset language.

Medical Diagnosis Using Distance-Based Similarity Measures of Single Valued Neutrosophic Multisets

This paper proposes a generalized distance measure and its similarity measures between single val- ued neutrosophic multisets (SVNMs). Then, the similari- ty measures are applied to a medical diagnosis problem with incomplete, indeterminate and inconsistent infor- mation. This diagnosis method can deal with the diagno- sis problem with indeterminate and inconsistent infor- mation which cannot be handled by the diagnosis method based on intuitionistic fuzzy multisets (IFMs).

Dice Similarity Measure between Single Valued Neutrosophic Multisets and Its Application in Medical Diagnosis

This paper introduces the concept of a single valued neutrosophic multiset (SVNM) as a generalization of an intuitionistic fuzzy multiset (IFM) and some basic operational relations of SVNMs, and then proposes the Dice similarity measure and the weighted Dice similarity measure for SVNMs and investigates their properties. Finally, the Dice similarity measure is applied to a medical diagnosis problem with SVNM information. This diagnosis method can deal with the medical diagnosis problem with indeterminate and inconsistent information which cannot be handled by the diagnosis method based on IFMs.

A multi-attribute decision-making method with prioritization relationship and dual hesitant fuzzy decision information

Dual hesitant fuzzy sets, encompassing fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases, are very useful in dealing with uncertainty considering the membership and nonmembership degrees by a set of possible values. This paper aims to develop a prioritized multi-attribute decision-making method to solve dual hesitant fuzzy decision problems. We first propose a correctional score function of dual hesitant fuzzy elements (DHFEs) for characterizing hesitant degrees in DHFEs more effectively. Then we apply the correctional score function and the dice similarity measure of dual hesitant fuzzy sets (DHFSs) to solve multi-attribute decision-making problems in which the attributes are in different priority levels and the attribute values take the form of DHFEs. Finally, an illustrative example is employed to show the feasibility of the proposed method in its practical applications.

Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets

Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented.

Fuzzy Multisets in Granular Hierarchical Structures Generated from Free Monoids

Fuzzy multisets defined by Yager take multisets on interval (0,1] as grades of membership. As Miyamoto later pointed out, the fuzzy multiset operations originally defined by Yager are not compatible with those of fuzzy sets as special cases. Miyamoto proposed different definitions for fuzzy multiset operations. This paper focuses on the two definitions of fuzzy multiset operations, one by Yager and the other by Miyamoto. It examines their differences in the framework of granular hierarchical structures generated from the free monoids as proposed in our previous papers. In order to define basic order between multisets on interval (0,1], Yager uses the natural order on the rangeN, the set of natural numbers, whereas Miyamoto newly introduces an order generated frombothdomain (0,1] and rangeNthrough the notion of cuts.

On Algebraic Properties of Fuzzy Membership Sequenced Multisets

Since the original paper by Yager, different definitions of the concept of fuzzy multisets (bags), and the corresponding operations are available in the literature, as well as some extensions of the union, intersection and difference operators of sets, and new algebraic operators. In this paper, a study of some algebraic properties of these operations including idempotency, identity, absorption, associativity, distributivity, demorgan’s laws and the principle of Inclusion/Exclusion is presented in the context of membership sequence of fuzzy multisets. Also, the compatibility of the union and intersection on these structures and sets via their root sets is established.

The Multi-Interval-Valued Fuzzy Soft Set with Application in Decision Making

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. By combining the multi-fuzzy set and soft set models, Y. Yang, X. Tan and C. Meng introduced the concept of multi-fuzzy soft sets and studied some of its operations, such as complement, “AND”, “OR”, Union and Intersection. They also gave an algorithm to analyze a decision problem using multi-fuzzy soft set. In this paper, we introduce the concept of multi-interval-valued fuzzy soft set (M-IVFSS). We also define its basic operations, namely complement, union, intersection, AND and OR. Finally, we give an application of this concept in decision-making problem.

Special Issue on Forum for Interdisciplinary Mathematics 2013

Special Issue on Forum for Interdisciplinary Mathematics 2013

Crisp and Fuzzy Granular Hierarchical Structures Generated from a Free Monoid

The granular hierarchical structure that we propose has the limitation that the set of truth values is not described explicitly. By regarding “sets of putting” as a special class of direct sets, we introduce crisp and fuzzy granular hierarchical structures. We found that there are two kinds of different preextensions between Yager’s fuzzy multisets and Zadeh’s fuzzy sets.

Fuzzy multiset hyperstructures

One studies connections between fuzzy multisets and hyperstructures. One determines conditions such that the hyperstructure associated with a fuzzy multiset may be a quasi-join space.

A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure

Zhu et al. (2012) proposed dual hesitant fuzzy set as an extension of hesitant fuzzy sets which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand these sets are more flexible, and provides much more information about the situation. In this paper, the axiom definition of a similarity measure between dual hesitant fuzzy sets is introduced. A new similarity measure considering membership and nonmembership degrees of dual hesitant fuzzy sets has been presented and also it is shown that the corresponding distance measures can be obtained from the proposed similarity measures. To check the effectiveness, the proposed similarity measure is applied in a bidirectional approximate reasoning systems. Mathematical formulation of dual hesitant fuzzy assignment problem with restrictions is presented. Two algorithms based on the proposed similarity measure, are developed to finds the optimal solution of dual hesitant fuzzy assignment problem with restrictions. Finally, the proposed method is illustrated by numerical examples.

Generalized Atanassov’s operators defined on lattice fuzzy multisets

In this paper the concept of an ordered weighted quasi-average operator (QOWA) on the one hand and that of an n-dimensional Atanassov’s operator on the other, are extended from fuzzy multisets on [0,1] to fuzzy multisets on any complete lattice endowed with a t-norm and a t-conorm. We show that, in the case of a distributive lattice, both operators provide particular cases of OWA operators defined on the lattice.In addition, a class of operators including Atanassov’s ones is defined on fuzzy lattice multisets. This new approach allows us to build a kind of n-ary aggregation functions for complete lattices which generalizes OWA operators.

CORRELATION MEASURE FOR INTUITIONISTIC FUZZY MULTI SETS

In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and only if the two IFMS are the same.

English

As Similarity measure for fuzzy sets is one of the important research topics of fuzzy set theory, there are several methods to measure similarity between two fuzzy sets (FS), Intuitionistic fuzzy sets (IFS) and Intuitionistic fuzzy multi sets (IFMS). In this paper, the Normalized Hamming Similarity measure of Intuitionistic Fuzzy Multi sets (IFMS) is introduced. This new measure for IFMS is based on the geometrical interpretation of IFS which involves both similarity and dissimilarity. Using this measure, the application of medical diagnosis and pattern recognition are shown.

OMFM: A Framework of Object Merging Based on Fuzzy Multisets

Information fusion is a process of merging information from multiple sources into a new set of information. Existing work on information fusion is applicable in various scenarios such as multiagent system, group decision making, and multidocument summarization. This paper intends to develop an effective framework to solve object merging problem based on fuzzy multisets. The objects defined in this paper are data segments in document fusion task, referring to the concepts with semantic-related terms of different semantic relations embedded. The fundamental operation is the merge function mapping data segments in multiple fuzzy multisets onto one object, which is a solution. Under this framework, we define quality measures of purity and entropy to quantify the quality of the solutions, balancing accurateness, and completeness of the results. Merge function that yields this kind of solutions is VI-optimal merge function and a series of theoretical properties concerning it are studied. Finally, we investigate the proposed framework in a special application scenario (i.e., document fusion) which is related to the task of multidocument summarization and show how the framework works with illustrative example.

Rough approximations of multi-fuzzy sets

Rough sets have been invented to cope with uncertainty and to deal with intelligent systems characterised by insufficient and incomplete information. This paper proposes a general framework for the study of relation-based multi-fuzzy rough approximation operators. We define rough multi-fuzzy sets and study their properties. An arbitrary multi-fuzzy relation is used to approximate multi-fuzzy sets. A pair of lower and upper rough approximation operators is hence obtained and basic properties of these approximation operators will be examined. By introducing the cut sets of multi-fuzzy sets, we study the connections between multi-fuzzy rough approximation operators and crisp approximation operators. Finally, the multi-fuzzy approximation operators will be defined by axioms.