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Intuitionistic fuzzy sets
Abstract:A definition of the concept 'intuitionistic fuzzy set' (IFS) is given, the latter being a generalization of the concept 'fuzzy set' and an example is described. Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
摘要:给出了直觉模糊集概念的定义,后者是模糊集概念的推广,并给出了一个实例。证明了与集合上的运算和关系以及定义在迭代函数系统集合上的模算子和拓扑算子有关的各种性质。
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Bi-fuzziness, Incompleteness, Inconsistency, Truth and Falsity Based on Saturation and Ignorance Functions. A New Approach of Penta-Valued Knowledge Representation
MAICS
2012
摘要
This paper presents a new five-valued knowledge representation of bipolar information. This representation is related to a five-valued logic that uses two logical values of truth (true, false) and three logical values of uncertainty (incomplete, inconsistent and fuzzy). The new approach is based on the concept of saturation function and ignorance function. In the framework of five-valued representation new formulae for union and intersection are constructed. Also, the paper presents a short application related to fuzzy preference modeling and decision making. Introduction Let X be a set of objects. We consider a property A , an object X x and the following sentence ) (x PA : x has the property A . We want to know if the sentence ) (x PA is true or false. After an evaluation, the information about logical value of sentence ) (x PA is described by a scalar ] 1 , 0 [ ) ( x TA . For the considered sentence, ) (x TA represents its truth degree. In the same time, the function ] 1 , 0 [ : X TA defines a Zadeh fuzzy set associated to the property A (Zadeh 1965). Then, we compute the degree of falsity: ) ( 1 ) ( x T x F A A (1) Using the scalar ) (x TA , we have obtained the following representation of information about sentence ) (x PA . )) ( ), ( ( ) ( x F x T x W A A A (2) This information is normalized because the components of vector ) (x WA verify the condition of partition of unity: 1 ) ( ) ( x F x T A A (3) The representation (3) is related to a bi-valued logic based on true and false. The next step was done by Atanassov (Atanassov 1986). He considered that after evaluation, the information about logical value of sentence ) (x PA is described by a vector with two components ) ( ), ( ) ( x F x T x V A A A (4) and supplementary these two components verify the inequality: 1 ) ( ) ( x F x T A A (5) The information represented by vector ) (x VA is not normalized but, Atanassov has introduced the intuitionistic index: ) ( ) ( 1 ) ( x F x T x U A A A (6) Using the vector ) (x VA , we have obtained an intuitionistic representation of information about sentence ) (x PA . )) ( ), ( ), ( ( ) ( x F x U x T x W A A A A (7) This information is normalized because the components of vector ) (x WA verify the condition of partition of unity: 1 ) ( ) ( ) ( x F x U x T A A A (8) The representation (8) is related to a three-valued logic based on true, neutral and false. In this paper we will consider the bipolar representation (Benferhat et al. 2006; Cornelis et al. 2003; Dubois et al. 2004) without having the condition (5). In this case, we cannot obtain immediately a normalized variant like (8). In the following, we present a method for obtaining a normalized representation of bipolar information. The paper has the following structure: section two presents the concepts of saturation, ignorance and bi-fuzziness. Section three presents the construction method of fivevalued representation. Section four presents a five-valued logic based on true, false, incomplete, inconsistent and fuzzy. Section five presents some operators for the fivevalued structure. Section six presents the using of fivevalued knowledge representation for fuzzy modeling of pairwise comparisons. Finally we present some conclusions. Saturation, Ignorance and Bi-fuzziness Functions In this section, firstly, we introduce the concepts of saturation function and ignorance function. These two functions are complementary. Both functions are essentially characterized by symmetry, boundary and monotonicity properties. Secondly, we introduce the concept of bi-fuzziness related to the index of indeterminacy (Patrascu 2008). Definition 1: A saturation function is a mapping ] 1 , 0 [ ] 1 , 0 [ : 2 S such that: i) ) , ( ) , ( x y S y x S ii) 0 ) , ( y x S if and only if ) 0 , 0 ( ) , ( y x iii) 1 ) , ( y x S if and only if ) 1 , 1 ( ) , ( y x iv) ) , ( y x S increases with respect to x and y The property a) describes the commutativity and the property d) describes the monotonicity. From property b) it results that the saturation value is low if and only if both arguments have low value and from property c) it results that the saturation value is high if and only if both arguments have high value. Example 1: 2 ) , ( y x y x S .
Neutrosophic Sets and Systems, Vol. 12, 2016
2016
摘要
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Solving Multi Objective Linear Programming Problems Using Intuitionistic Fuzzy Optimization Method: A Comparative Study
2014
摘要
—The paper aims to give computational algorithm to solve a multi objective linear programming problem using intuitionistic fuzzy optimization method. It also includes some basic properties of intuitionistic fuzzy set and operations on it. The development of algorithm is based on principle of optimal decision set obtained by intersection of various intuitionistic fuzzy decision sets which are obtained corresponding to each objective function. Further, as the intuitionistic fuzzy optimization technique utilizes degree of belonging and degree of non-belonging, we made a comparative study of linear and nonlinear membership function for belonging and non-belonging to see its impact on optimization and to get insight in such optimization process. The developed algorithm has been illustrated by a numerical example.
On Intuitionistic Fuzzy Soft Groups
2013
摘要
–In this paper, we give some basic properties ofintuitionistic fuzzy soft sets and (α, β)-level sets. Also we defineimage of an intuitionistic fuzzy soft set under a function, productof two intuitionistic fuzzy soft sets and obtain some results. Inaddition, we define intuitionistic fuzzy soft group and investigatetheir properties
Mapping on Intuitionistic Fuzzy Soft Expert Sets
2015
摘要
– We introduce the mapping on intuitionistic fuzzy soft expert set and its operations are studied. The basic operations of mapping on intuitionistic fuzzy soft expert set theory are defined
Relations on Interval Valued Neutrosophic Soft Sets
2014
摘要
– Mukherjee [34] introduced the concept of interval valued intuitionistic fuzzy soft relation. In this paper we will extend this concept to the case of interval valued neutrosophic soft relation (IVNSS relation for short) which can be discussed as a generalization of soft relations, fuzzy soft relation, intuitionistic fuzzy soft relation, interval valued intuitionistic fuzzy soft relations and neutrosophic soft relations. Basic operations are presented and the various properties like reflexivity, symmetry, transitivity of IVNSS relations are also studied
Interval Valued Neutrosophic Parameterized Soft Set Theory and its Decision Making
2014
摘要
– In this work, we present definition of interval valued neutrosophic parameterized (IVNP-)soft set and its operations. Then we define parameter reduction method for IVNP-soft set.We also give an example which shows that they can be successfully applied to problem that contains indeterminacy
Generalized Interval Neutrosophic Soft Set and its Decision Making Problem
2014
摘要
– In this work, we introduce the concept of generalized interval neutrosophic soft set and study their operations. Finally, we present an application of generalized interval neutrosophic soft set in decision making problem
Hesitant fuzzy soft sets
2013
摘要
– A new hybrid structure- hesitant fuzzy soft set, involving soft set is introduced. Soft set is relatively a new approach initiated by Molodtsov in 1999 to deal impreciseness and uncertainty. Hesitant fuzzy set is a generalization of fuzzy set whose membership is a subset of [0,1]. This paper is an endeavor to establish a link between soft sets and hesitant fuzzy sets. Basic operation such as intersection, union, compliment is defined and De Morgan’s law is also proved. It also discusses its use in decision making problem
Systematic Review of Decision Making Algorithms in Extended Neutrosophic Sets
Symmetry
2018
摘要
© 2018 by the authors. The Neutrosophic set (NS) has grasped concentration by its ability for handling indeterminate, uncertain, incomplete, and inconsistent information encountered in daily life. Recently, there have been various extensions of the NS, such as single valued neutrosophic sets (SVNSs), Interval neutrosophic sets (INSs), bipolar neutrosophic sets (BNSs), Refined Neutrosophic Sets (RNSs), and triangular fuzzy number neutrosophic set (TFNNs). This paper contains an extended overview of the concept of NS as well as several instances and extensions of this model that have been introduced in the last decade, and have had a significant impact in literature. Theoretical and mathematical properties of NS and their counterparts are discussed in this paper as well. Neutrosophic-set-driven decision making algorithms are also overviewed in detail.
Geometric-arithmetic energy and atom bond connectivity energy of dual hesitant q-rung orthopair fuzzy graphs
J. Intell. Fuzzy Syst.
2021
摘要
q-Rung orthopair fuzzy sets (q-ROFSs), originally proposed by Yager, can powerfully modify the range of indication of decision information by changing a parameter q based on the different hesitation degree, and the dual hesitant q-rung orthopair fuzzy set (DHq-ROFS), a new technique to consider human’s hesitance, can be more substantial of dealing with real multi-attribute decision making (MADM) problems. Inspired by DHq-ROFSs, in this article, we extend the concept of q-rung orthopair fuzzy graphs to dual hesitant q-rung orthopair fuzzy context and introduce the innovative concept of a dual hesitant q-rung orthopair fuzzy graphs based on Hamacher operator called dual hesitant q-rung orthopair fuzzy Hamacher graphs (DHq-ROFHGs). We propose the new concepts of geometric-arithmetic energy and atom bond connectivity energy of a DHq-ROFHG and determine its upper and lower bounds. Moreover, on the basis of the proposed concept of DHq-ROFHGs, we introduce a new approach to solve the MADM problems with dual hesitant q-rung orthopair fuzzy information. At the end, we give a numerical model related to the selection of most significant defensive factor to illustrate the applicability of the developed approach, and exhibit its viability. Comparative analysis is conducted and the superiorities are illustrated.
Multi-Criteria Decision Making Techniques Based on Some Extensions of Fuzzy Set
2019
摘要
xiv List of Publication From Thesis xvi Other Publications xvii Abbreviations and Nomenclature xviii
Decision Analysis, Uncertainty Theories and Aggregation Operators in Financial Selection Problems
2017
摘要
v RESUM ix ACKNOWLEDGEMENTS xiii LIST OF FIGURES xix LIST OF TABLES xix CHAPTER
Intuitionistic Fuzzy Equipotent Sublattices of Lattice Ordered Groups with Respect to S-Norms
2011
摘要
this paper, we introduce the notion of intuitionistic fuzzy equipotent lattice in a fuzzy lattice and then some basic properties are investigated. Characterization of intuitionistic fuzzy equipotent lattices are given. Using a collection of lattices, an intuitionistic fuzzy equipotent lattice is established. The notion of fuzzy equipotent lattice relation on the family of all intuitionistic fuzzy sub lattices of L are discussed upper and lower level sets of fuzzy equipotent lattices are studied.
Grey Relational Analysis based Intuitionistic Fuzzy Multi-Criteria Group Decision-Making Approach for Teacher Selection in Higher Education
2011
摘要
selection is a group decision-making process under multiple criteria involving subjectivity, imprecision, and vagueness, which are best represented by intuitionistic fuzzy sets. An intuitionistic fuzzy set, which is characterized by membership function (degree of acceptance), non-membership function (degree of rejection) and the degree of indeterminacy or the degree of hesitancy, is a more general and suitable way to deal with imprecise information, when compared to a fuzzy set. The purpose of this study is to develop an intuitionistic fuzzy multi criteria group making method with grey relational analysis for teacher selection in higher education. Intuitionistic fuzzy weighted averaging operator is used to aggregate individual opinions of decision makers into a group opinion. Eight criteria obtained from expert opinions are considered for selection process. The criteria are namely academic performances, teaching aptitude, research experience, leadership quality, personality, management capacity, and values. Weights of the criteria are obtained by using a questionnaire. The weights of decision makers are considered as equal i.e. their importance are equal. The rating of an alternative with respect to certain criteria offered by decision maker is represented by linguistic variable that can be expressed by intuitionistic fuzzy sets. Grey relational analysis is used for ranking and selection of alternatives to constitute a panel of selected candidates. An educational problem for teacher selection is provided to illustrate the effectiveness of the proposed model.
Intuitionistic Fuzzy Jensen-Rényi Divergence: Applications to Multiple-Attribute Decision Making
Informatica
2013
摘要
s need to associate measures which can measure vagueness and differences in the underlying charact erizing IFSs. In the present paper we introduce an information theoretic divergence measure, called in tuitionistic fuzzy Jensen-Renyi divergence. It is a difference measure in the setting of intuitionistic fuzzy set theory, involving parameters that provid e flexibility and choice. The strength of the new mea sure lies in its properties and applications. An approach to multiple-attribute decision making base d on intuitionistic fuzzy Jensen-Renyi divergence i s proposed. A numerical example illustrates the appli cation of the new measure and the role of various parameters therein to multipleattribute decision ma king problem formulated in terms of intuitionistic fuzzy sets. Povzetek: Razvita je nova verzija intuitivne mehke logike za uporabo v procesu odlocanja.
Logical Foundations of Fuzzy Mathematics
2009
摘要
s 253 English abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Czech abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Some q‐rung orthopair uncertain linguistic aggregation operators and their application to multiple attribute group decision making
Int. J. Intell. Syst.
2019
摘要
q‐Rung orthopair fuzzy sets (q‐ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the qth power of the membership and the qth power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q‐rung orthopair uncertain linguistic sets (q‐ROULSs) based on the q‐ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q‐ROULSs, several q‐rung orthopair uncertain linguistic aggregation operators are developed, including the q‐rung orthopair uncertain linguistic weighted arithmetic average operator, the q‐rung orthopair uncertain linguistic ordered weighted average operator, the q‐rung orthopair uncertain linguistic hybrid weighted average operator, the q‐rung orthopair uncertain linguistic weighted geometric average operator, the q‐rung orthopair uncertain linguistic ordered weighted geometric operator, and the q‐rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q‐rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision‐making steps and explain the validity and feasibility of two methods by comparing with other methods.
Exponential operation and aggregation operator for q‐rung orthopair fuzzy set and their decision‐making method with a new score function
Int. J. Intell. Syst.
2018
摘要
q‐Rung orthopair fuzzy set (q‐ROFS) is a powerful tool that attracts the attention of many scholars in dealing with uncertainty and vagueness. The aim of paper is to present a new score function of q‐rung orthopair fuzzy number (q‐ROFN) for solving the failure problems when comparing two q‐ROFNs. Then a new exponential operational law about q‐ROFNs is defined, in which the bases are positive real numbers and the exponents are q‐ROFNs. Meanwhile, some properties of the operational law are investigated. Later, we apply them to derive the q‐rung orthopair fuzzy weighted exponential aggregation operator. Additionally, an approach for multicriteria decision‐making problems under the q‐rung orthopair fuzzy data is explored by applying proposed aggregation operator. Finally, an example is investigated to illustrate the feasibility and validity of the proposed approach. The salient features of the proposed method, compared to the existing q‐rung orthopair fuzzy decision‐making methods, are (1) it can obtain the optimal alternative without counterintuitive phenomena; (2) it has a great power in distinguishing the optimal alternative.
Specific Types of q-Rung Picture Fuzzy Yager Aggregation Operators for Decision-Making
Int. J. Comput. Intell. Syst.
2020
摘要
q-rung picture fuzzy sets can handle complex fuzzy and impression information by changing a parameter q based on the different hesitation degree, and Yager operator is a useful aggregation technology that can control the uncertainty of valuating data from some experts and thus get intensive information in the process of decision-making. Thus, in this paper, we develop specific types of operators, namely, q-rung picture fuzzy Yager weighted average, q-rung picture fuzzy Yager ordered weighted average, q-rung picture fuzzy Yager hybrid weighted average, q-rung picture fuzzy Yager weighted geometric, q-rung picture fuzzy Yager ordered weighted geometric and q-rung picture fuzzy Yager hybrid weighted geometric operators. We propose q-rung picture fuzzy Yager aggregation operators to handle multiple attribute decision-making problems in a modernize way. Moreover, we discuss the effect of parameter on the decision-making results. To demonstrate the superiority and advantage of our proposed method, a comparison with existing methods is presented.