On Theory of Multisets and Applications

2016 ◽  
pp. 1-22
Author(s):  
B. K. Tripathy
Keyword(s):  
Fuzzy Sets ◽  
Model Uncertainty ◽  
Rough Sets ◽  
Real Life ◽  
Intuitionistic Fuzzy Sets ◽  
Current Status ◽  
Soft Sets ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Mathematics ◽  
Basic Sets

Although multiple occurrences of elements are immaterial in sets, in real life situations repetition of elements is useful. So, the notion of multisets (also called as bags) was introduced, where repetition of elements is taken into account. Fuzzy set, intuitionistic (a misnomer here as intuitionistic mathematics has nothing to do with its fuzzy counterpart) fuzzy sets, rough sets and soft sets are extensions of the basic notion of sets as they model uncertainty in data. Following this multisets have been extended to fuzzy multisets, intuitionistic fuzzy sets, rough multisets and soft multisets. Many properties of basic sets have been extended to the context of multisets, fuzzy multisets, intuitionistic fuzzy sets, rough multisets and soft multisets. Several applications of different multisets mentioned above are found in literature. In this chapter, it is our aim to introduce the different concepts of multisets, their properties, current status and highlight their applications.

Intuitionistic Fuzzy Soft Ideals

2020 ◽  
pp. 91-104
Author(s):  
Shuker Khalil
Keyword(s):  
Social Science ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Sets ◽  
Real Life ◽  
Soft Sets ◽  
Vague Sets ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Sets Theory

The basic notions of soft sets theory are introduced by Molodtsov to deal with uncertainties when solving problems in practice as in engineering, social science, environment, and economics. This notion is convenient and easy to apply as it is free from the difficulties that appear when using other mathematical tools as theory of theory of fuzzy sets, rough sets, and theory of vague sets. The soft set theory has recently gaining significance for finding rational and logical solutions to various real-life problems, which involve uncertainty, impreciseness, and vagueness. The concepts of intuitionistic fuzzy soft left almost semigroups and the intuitionistic fuzzy soft ideal are introduced in this chapter, and some of their basic properties are studied.

scholarly journals On Complex Intuitionistic Fuzzy Soft Sets with Distance Measures and Entropies

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Tanuj Kumar ◽  
Rakesh Kumar Bajaj
Keyword(s):  
Fuzzy Sets ◽  
Real Life ◽  
Multicriteria Decision Making ◽  
Intuitionistic Fuzzy Sets ◽  
Distance Measures ◽  
Soft Sets ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Fuzzy Soft Sets ◽  
Intuitionistic Fuzzy Soft Sets

We introduce the concept of complex intuitionistic fuzzy soft sets which is parametric in nature. However, the theory of complex fuzzy sets and complex intuitionistic fuzzy sets are independent of the parametrization tools. Some real life problems, for example, multicriteria decision making problems, involve the parametrization tools. In order to get their new entropies, some important properties and operations on the complex intuitionistic fuzzy soft sets have also been discussed. On the basis of some well-known distance measures, some new distance measures for the complex intuitionistic fuzzy soft sets have also been obtained. Further, we have established correspondence between the proposed entropies and the distance measures of complex intuitionistic fuzzy soft sets.

Refined Neutrosophic Sets and Refined Neutrosophic Soft Sets

2016 ◽  
pp. 321-343 ◽  
Author(s):  
Irfan Deli
Keyword(s):  
Fuzzy Sets ◽  
Medical Diagnosis ◽  
Real Life ◽  
Intuitionistic Fuzzy Sets ◽  
Soft Sets ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Formal Framework ◽  
Fuzzy Soft Sets ◽  
Intuitionistic Fuzzy Soft Sets

Refined neutrosophic sets (RNS) are a generalization of a neutrosophic sets, intuitionistic fuzzy sets, fuzzy sets, intuitionistic fuzzy multi-sets and fuzzy multi-sets. Similarly, refined neutrosophic soft sets (RNSS) are a generalization of a neutrosophic soft sets, intuitionistic fuzzy soft sets, fuzzy soft sets, intuitionistic fuzzy soft multi-sets and fuzzy soft multi-sets. These sets are a powerful general formal framework that has been proposed to present uncertainty, imprecise, incomplete, inaccurate and inconsistent information which exist in real life. This chapter will survey concept of RNS and concept of RNSS with basic definitions and will present an efficient approach for both RNS and RNSS. Also, the chapter will introduce an application of RNS in medical diagnosis problem, pattern recognition and an application of RNSS in decision making to illustrate the advantage of the proposed approach.

A New Ranking Approach for Interval Valued Intuitionistic Fuzzy Sets and its Application in Decision Making

2019 ◽  
Vol 8 (2) ◽  
pp. 110-125 ◽  
Author(s):  
Pranjal Talukdar ◽  
Palash Dutta
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Model Uncertainty ◽  
Real Life ◽  
Intuitionistic Fuzzy Sets ◽  
Multi Criteria Decision Making ◽  
Numerical Examples ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Interval Valued

Ranking of interval valued intuitionistic fuzzy sets (IVIFSs) plays an important role because of its attraction and applicability to model uncertainty in real life problems. In this article, an attempt has been made to devise a new method for ranking of IVIFSs based on exponential function. The significance of the method is illustrated with the help of some numerical examples and the results are compared with other existing methods. Furthermore, a multi criteria decision making method is presented here to evaluate the final ranking of the alternatives using the proposed ranking method and discussed the consistency of so obtained results.

Rough Sets and Approximate Reasoning

2014 ◽  
pp. 180-228 ◽  
Author(s):  
B. K. Tripathy
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Point Of View ◽  
Approximate Reasoning ◽  
Human Reasoning ◽  
Intuitionistic Fuzzy ◽  
Application Point ◽  
Future Work

Several models have been introduced to capture impreciseness in data. Fuzzy sets introduced by Zadeh and Rough sets introduced by Pawlak are two of the most popular such models. In addition, the notion of intuitionistic fuzzy sets introduced by Atanassov and the hybrid models obtained thereof have been very fruitful from the application point of view. The introduction of fuzzy logic and the approximate reasoning obtained through it are more realistic as they are closer to human reasoning. Equality of sets in crisp mathematics is too restricted from the application point of view. Therefore, extending these concepts, three types of approximate equalities were introduced by Novotny and Pawlak using rough sets. These notions were found to be restrictive in the sense that they again boil down to equality of sets and also the lower approximate equality is artificial. Keeping these points in view, three other types of approximate equalities were introduced by Tripathy in several papers. These approximate equalities were further generalised to cover the approximate equalities of fuzzy sets and intuitionistic fuzzy sets by him. In addition, considering the generalisations of basic rough sets like the covering-based rough sets and multigranular rough sets, the study has been carried out further. In this chapter, the authors provide a comprehensive study of all these forms of approximate equalities and illustrate their applicability through several examples. In addition, they provide some problems for future work.

On lattice ordered intuitionistic fuzzy soft sets

2018 ◽  
Vol 7 (1-2) ◽  
pp. 46-61 ◽  
Author(s):  
Tahir Mahmood ◽  
Muhammad Irfan Ali ◽  
Muhammad Aamir Malik ◽  
Waseem Ahmed
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Soft Sets ◽  
Intuitionistic Fuzzy ◽  
Decision Making Problem ◽  
Fuzzy Soft Sets ◽  
Intuitionistic Fuzzy Soft Sets

Lattices, soft sets, fuzzy sets and their generalizations have always been important for Mathematicians and the researchers working on uncertaities. In this paper our aim is to introduce the concept of lattice ordered intuitionistic fuzzy soft sets. After introducing extended union, extended intersection,  AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy soft sets, in this paper the affects of lattice ordered intuitionistic fuzzy soft sets and anti-lattice ordered intuitionistic fuzzy soft sets on restricted union, restricted intersection, extended union, extended intersection,AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy sets are discussed. Further a decision making problem is solved by using these concepts.

scholarly journals Intuitionistic Fuzzy Soft Rough Set and Its Application in Decision Making

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Haidong Zhang ◽  
Lan Shu ◽  
Shilong Liao
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Mathematical Tool ◽  
Intuitionistic Fuzzy ◽  
Rough Approximation ◽  
Soft Rough Set

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. In this paper, we present concepts of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, and investigate some properties of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets in detail. Furthermore, classical representations of intuitionistic fuzzy soft rough approximation operators are presented. Finally, we develop an approach to intuitionistic fuzzy soft rough sets based on decision making and a numerical example is provided to illustrate the developed approach.

Extension of TOPSIS for Multiple Attribute Decision Making Using Intuitionistic Fuzzy Sets

2012 ◽  
Vol 433-440 ◽  
pp. 4053-4058 ◽  
Author(s):  
Yuan Yuan ◽  
Li Yang He
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Quantitative Estimation ◽  
Ideal Point ◽  
Real Life ◽  
Multiple Attribute Decision Making ◽  
Intuitionistic Fuzzy Sets ◽  
Distance Measures ◽  
Intuitionistic Fuzzy ◽  
Multiple Attribute

This electronic document is a “live” template. The various components of your paper [title, text, heads, etc.] are already defined on the style sheet, as illustrated by the portions given in this document. Due to the nature of vagueness inherent to real-life situations, some fuzzy data are deemed to suitable enough to describe the qualitative and/or quantitative estimation for decision making problems. Therefore, a new method for multiple attribute decision making under fuzzy environment is discussed, in which the attribute values take the form of intuitionistic fuzzy numbers. To overcome some disadvantages of existing distance measures like indiscrimination, counterintuitive results and difficulty of interpretation, we introduce a new class of distance for describing the deviation degrees between intuitionistic fuzzy sets. Furthermore, the measure of similarity degree for each alternative to ideal point is calculated through using the new proposed fuzzy distance. A model of TOPSIS is designed with the introduction of the particular closeness coefficient composed of similarity degrees. Then, we extend the TOPSIS method to aggregate the fuzzy information corresponding to each alternative, and rank the alternatives according to their closeness coefficients. Finally, an illustrative example is given to demonstrate the proposed approach practicality and effectiveness.

scholarly journals Three-Way Decisions with Interval-Valued Intuitionistic Fuzzy Decision-Theoretic Rough Sets in Group Decision-Making

Symmetry ◽  
2018 ◽  
Vol 10 (7) ◽  
pp. 281 ◽  
Author(s):  
Dajun Ye ◽  
Decui Liang ◽  
Pei Hu
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Group Decision Making ◽  
Rough Sets ◽  
Group Decision ◽  
Intuitionistic Fuzzy Sets ◽  
Fuzzy Decision ◽  
Grey Correlation ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

In this article, we demonstrate how interval-valued intuitionistic fuzzy sets (IVIFSs) can function as extended intuitionistic fuzzy sets (IFSs) using the interval-valued intuitionistic fuzzy numbers (IVIFNs) instead of precision numbers to describe the degree of membership and non-membership, which are more flexible and practical in dealing with ambiguity and uncertainty. By introducing IVIFSs into three-way decisions, we provide a new description of the loss function. Thus, we firstly propose a model of interval-valued intuitionistic fuzzy decision-theoretic rough sets (IVIFDTRSs). According to the basic framework of IVIFDTRSs, we design a strategy to address the IVIFNs and deduce three-way decisions. Then, we successfully extend the results of IVIFDTRSs from single-person decision-making to group decision-making. In this situation, we adopt a grey correlation accurate weighted determining method (GCAWD) to compute the weights of decision-makers, which integrates the advantages of the accurate weighted determining method and grey correlation analysis method. Moreover, we utilize the interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operation to count the aggregated scores and the accuracies of the expected losses. By comparing these scores and accuracies, we design a simple and straightforward algorithm to deduce three-way decisions for group decision-making. Finally, we use an illustrative example to verify our results.

Soft Sets and Its Applications

2016 ◽  
pp. 65-85 ◽  
Author(s):  
B. K. Tripathy ◽  
K. R. Arun
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Soft Set ◽  
Soft Sets ◽  
Practical Applications ◽  
Set Operations ◽  
New Models ◽  
Pros And Cons

Uncertainty is an inherent characteristic of modern day databases. In order to handle such databases with uncertainty, several new models have been introduced in the literature. Some new models like fuzzy sets introduced by Zadeh (1965), rough sets invented by Z. Pawlak (1982) and intuitionistic fuzzy sets extended by K.T. Atanassov (1986). All these models have their own pros and cons. However, one of the major problems with these models is the lack of sufficient number of parameters to deal with uncertainty. In order to add adequate number of parameters, soft set theory was introduced by Molodtsov in 1999. Since then the theoretical developments on soft set theory has attracted the attention of researchers. However, the practical applications of any theory are of enough importance to make use of it. In this chapter, the basic definitions of soft set, operations and properties are discussed. Also, the aim in this chapter is to discuss on the different applications of soft sets; like decision making, parameter reduction, data clustering and data dealing with incompleteness.

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