Showing papers on "Fuzzy mathematics published in 2012"
TL;DR: This paper proposes dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy Sets, hesitant fuzzy set, and fuzzy multisets as special cases, and investigates the basic operations and properties of DHFSs.
Abstract: In recent decades, several types of sets, such as fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, type 𝑛 fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, use a notion of nested interval to reflect their common ground, then propose an extension principle of DHFSs. Additionally, we give an example to illustrate the application of DHFSs in group forecasting.
540 citations
Book•
14 Dec 2012
TL;DR: This book presents a mathematically-based introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory, and comprises a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types.
Abstract: This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight into Fuzzy Logic. Fuzzy Sets have been introduced by Lotfi Zadeh in 1965 and since then, they have been used in many applications. As a consequence, there is a vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. The main purpose of the present book is to reduce this gap by providing a theoretical introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory. Well-known applications, as for example fuzzy control, are also discussed in this book and placed on new ground, a theoretical foundation. Moreover, a few advanced chapters and several new results are included. These comprise, among others, a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types, that investigates their approximation capability by providing new error estimates.
370 citations
TL;DR: This paper discusses basic notions underlying fuzzy sets, especially gradualness, uncertainty, vagueness and bipolarity, in order to clarify the significance of using fuzzy sets in practice.
278 citations
17 Aug 2012
TL;DR: This paper compared grey systems with other kinds of uncertainty models such as stochastic probability, rough set theory, and fuzzy mathematics.
Abstract: Purpose – The purpose of this paper is to introduce the elementary concepts and fundamental principles of grey systems and the main components of grey systems theory. Also to discuss the astonishing progress that grey systems theory has made in the world of learning and its wide‐ranging applications in the entire spectrum of science.Design/methodology/approach – The characteristics of unascertained systems including incomplete information and inaccuracies in data are analysed and four uncertain theories: probability statistics, fuzzy mathematics, grey system and rough set theory are compared. The scientific principle of simplicity and how precise models suffer from inaccuracies are also shown.Findings – The four uncertain theories, probability statistics, fuzzy mathematics, grey system and rough set theory are examined with different research objects, different basic sets, different methods and procedures, different data requirements, different emphasis, different objectives and different characteristics....
240 citations
TL;DR: The proposed method is simpler than the methods presented in Chen and Lee (2010a, 2010b) and provides a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.
Abstract: In this paper, we present a new method to deal with fuzzy multiple attributes group decision-making problems based on ranking interval type-2 fuzzy sets. First, we propose a new method for ranking interval type-2 fuzzy sets. Then, we propose a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of interval type-2 fuzzy sets. We also use some examples to illustrate the fuzzy multiple attributes group decision-making process of the proposed method. The proposed method is simpler than the methods presented in Chen and Lee (2010a, 2010b) for fuzzy multiple attributes group decision-making based on interval type-2 fuzzy sets. It provides us with a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.
176 citations
TL;DR: A secondary fuzzy comprehensive evaluation system is constructed to evaluate the risk of floor water invasion in coal mines using fuzzy statistical method and expert evaluation method, and the membership degree of every index is constructed.
165 citations
TL;DR: This article investigates the group decision making problems in which all the information provided by the decision makers is expressed as IT2 fuzzy decision matrices, and the information about attribute weights is partially known, which may be constructed by various forms.
Abstract: Interval type-2 fuzzy sets (IT2 FSs) are a very useful means to depict the decision information in the process of decision making. In this article, we investigate the group decision making problems in which all the information provided by the decision makers (DMs) is expressed as IT2 fuzzy decision matrices, and the information about attribute weights is partially known, which may be constructed by various forms. We first use the IT2 fuzzy weighted arithmetic averaging operator to aggregate all individual IT2 fuzzy decision matrices provided by the DMs into the collective IT2 fuzzy decision matrix, then we utilize the ranking-value measure to calculate the ranking value of each attribute value and construct the ranking-value matrix of the collective IT2 fuzzy decision matrix. Based on the ranking-value matrix and the given attribute weight information, we establish some optimization models to determine the weights of attributes. Furthermore, we utilize the obtained attribute weights and the IT2 fuzzy weighted arithmetic average operator to fuse the IT2 fuzzy information in the collective IT2 fuzzy decision matrix to get the overall IT2 fuzzy values of alternatives by which the ranking of all the given alternatives can be found. Finally, we give an illustrative example.
158 citations
Book•
21 Nov 2012
TL;DR: Fuzzy Sets and Possibility Theory in Approximate and Plausible Reasoning and Fuzzy Set Techniques in Information Retrieval, Part I and Part II.
Abstract: Series Foreword. Contributing Authors. Introduction. Part I: Reasoning. 1. Fuzzy Sets and Possibility Theory in Approximate and Plausible Reasoning B. Bouchon-Meunier, et al. 2. Weighted Inference Systems V. Novak. 3. Closure Operators in Fuzzy set Theory L. Biacino, G. Gerla. Part II: Learning and Fusion. 4. Learning Fuzzy Decision Rules B. Bouchon-Meunier, C. Marsala. 5. Neuro-Fuzzy Methods in Fuzzy Rule Generation D. Nauck, R. Kruse. 6. Merging Fuzzy Information D. Dubois, et al. Part III: Fuzzy Information Systems. 7. Fuzzy Databases P. Bosc, et al. 8. Fuzzy Set Techniques in Information Retrieval D.H. Kraft, et al. Summary. References
150 citations
01 Mar 2012
TL;DR: A new algorithm is proposed for solving a special type of fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product.
Abstract: In the literature, several algorithms are proposed for solving the transportation problems in fuzzy environment but in all the proposed algorithms the parameters are represented by normal fuzzy numbers. Chen [Operations on fuzzy numbers with function principal, Tamkang Journal of Management Science 6 (1985) 13-25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new algorithm is proposed for solving a special type of fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product. In the proposed algorithm transportation costs are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed algorithm a numerical example is solved and the obtained results are compared with the results of existing approaches. Since the proposed approach is a direct extension of classical approach so the proposed approach is very easy to understand and to apply on real life transportation problems for the decision makers.
143 citations
TL;DR: A new definition of intuitionistic fuzzy rough sets is given with the analysis of its basic properties based on the notion of two universes, general binary relations, and a pair (T, I) of intuitionists fuzzy t-norm T and intuitionism fuzzy implicator I.
132 citations
TL;DR: This paper studies reduction of a fuzzy covering and fusion of multi-fuzzy covering systems based on the evidence theory and rough set theory, by using the method of maximum flow.
01 Oct 2012
TL;DR: Type-2 fuzzy logic systems T2FLSs offer opportunity to model levels of uncertainty which traditional fuzzy logic type1 struggles and provide the capability of handling a higher level of uncertainty and a number of missing components that have held back successful deployment of fuzzy systems in decision making.
Abstract: Fuzzy set theory has been proposed as a means for modeling the vagueness in complex systems. Fuzzy systems usually employ type-1 fuzzy sets, representing uncertainty by numbers in the range [0, 1]. Despite commercial success of fuzzy logic, a type-1 fuzzy set T1FS does not capture uncertainty in its manifestations when it arises from vagueness in the shape of the membership function. Such uncertainties need to be depicted by fuzzy sets that have blur boundaries. The imprecise boundaries of a type-2 fuzzy set T2FS give rise to truth/membership values that are fuzzy sets in [0], [1], instead of a crisp number. Type-2 fuzzy logic systems T2FLSs offer opportunity to model levels of uncertainty which traditional fuzzy logic type1 struggles. This extra dimension gives more degrees of freedom for better representation of uncertainty compared to type-1 fuzzy sets. A type-1 fuzzy logic system T1FLSs inference produces a T1FS and the result of defuzzification of the T1FS, a crisp number, whereas a T2FLS inference produces a type-2 fuzzy set, its type-reduced fuzzy set which is a T1FS and the defuzzification of the type-1 fuzzy set. The type-reduced fuzzy set output gives decision-making flexibilities. Thus, FLSs using T2FS provide the capability of handling a higher level of uncertainty and provide a number of missing components that have held back successful deployment of fuzzy systems in decision making.
01 Jan 2012
TL;DR: A new operation on Triangular Fuzzy Numbers is defined, where the method of subtraction and division has been modified, and these modified operators yield the exact inverse of the addition and multiplication operators.
Abstract: The fuzzy set theory has been applied in many fields such as operation research, control theory and management sciences etc. The fuzzy numbers and fuzzy values are widely used in engineering applications because of their suitability for representing uncertain information. In standard fuzzy arithmetic operations we have some problem in subtraction and division operations. In this paper, a new operation on Triangular Fuzzy Numbers is defined, where the method of subtraction and division has been modified. These modified operators yield the exact inverse of the addition and multiplication operators.
TL;DR: The proposed multicriteria fuzzy decision making method outperforms Ye's method (2009) due to the fact that the proposed method can overcome the drawback of Ye'smethod (2009).
Abstract: In this paper, we present a new method for multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, where interval-valued intuitionistic fuzzy values are used to represent evaluating values of the decision-maker with respect to alternatives. First, we propose a new method for ranking interval-valued intuitionistic fuzzy values. Based on the proposed fuzzy ranking method of interval-valued intuitionistic fuzzy values, we propose a new method for multicriteria fuzzy decision making. The proposed multicriteria fuzzy decision making method outperforms Ye's method (2009) due to the fact that the proposed method can overcome the drawback of Ye's method (2009), where the drawback of Ye's method is that it can not distinguish the ranking order between alternatives in some situations. The proposed method provides us with a useful way for dealing with multicriteria fuzzy decision making problems based on interval-valued intuitionistic fuzzy sets.
TL;DR: An interval-valued intuitionistic fuzzy multi-criteria decision-making approach is proposed based on the prospect score function, which gives a matrix of score function values and a comprehensive evaluation value of each alternative.
Abstract: This paper analyzes the limitations of existing score functions of intuitionistic fuzzy set. A new score function is defined based on the prospect value function, and examples are given to demonstrate the validity of this score function. As for the interval-valued intuitionistic fuzzy multi-criteria decision-making problems in which criteria weights are fixed, and the criteria values of alternatives are in the form of interval intuitionistic fuzzy numbers, an interval-valued intuitionistic fuzzy multi-criteria decision-making approach is proposed based on the prospect score function. This approach gives a matrix of score function values and a comprehensive evaluation value of each alternative. And the order of alternatives is listed by comparing the projection values of each alternative to the positive ideal solution. Finally, an example shows the feasibility and validity of this approach.
TL;DR: A new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionists' fuzzy values is presented.
Abstract: In this paper, we present a new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values. First, we briefly review the concepts of interval-valued intuitionistic fuzzy sets and the Karnik-Mendel algorithms. Then, we propose the intuitionistic fuzzy weighted average operator and interval-valued intuitionistic fuzzy weighted average operator, based on the traditional weighted average method and the Karnik-Mendel algorithms. Then, we propose a fuzzy ranking method for intuitionistic fuzzy values based on likelihood-based comparison relations between intervals. Finally, we present a new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values. The proposed method provides us with a useful way for multiattribute decision making based on interval-valued intuitionistic fuzzy values.
TL;DR: The solution methodology of Yang et al.
TL;DR: A new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment based on a new ranking method, in which the evaluated values of the same alternative with different attributes are considered as one unified entity.
Abstract: Highlights? A new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment. ? The same alternative with different attributes are considered as one unified entity. ? A revised score function and a revised accuracy function of intuitionistic fuzzy sets based on human intuition. ? The degree of membership, the degree of nonmembership and the degree of hesitation are with various importance. ? An optimization model is established to estimate the relative degree of importance. In this paper we present a new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment. This approach is based on a new ranking method of intuitionistic fuzzy sets, in which the evaluated values (in the form of intervals) of the same alternative with different attributes are considered as one unified entity. According to people's intuition, the ranking method proposed in this paper is mainly grounded on a revised score function and a revised accuracy function of intuitionistic fuzzy sets. Different from the traditional methods, in this new approach, the degree of membership, the degree of nonmembership and the degree of hesitation are considered with various importance in reflecting the true image of the respective alternative. Furthermore, an optimization model is established to estimate the relative degree of importance of each quantity. Finally, two practical examples are provided to illustrate our approach.
TL;DR: In order to rank all fuzzy numbers, the method of ''a new approach for ranking of trapezoidal fuzzy numbers'' by Abbasbandy and Hajjari (2009) is modified.
Abstract: In order to rank all fuzzy numbers, we modify the method of ''a new approach for ranking of trapezoidal fuzzy numbers'' by Abbasbandy and Hajjari (2009). Our proposed method is used for ranking symmetric fuzzy numbers. The advantage of this method is illustrated by some comparative examples.
Journal Article•
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TL;DR: It is shown that a fuzzy soft topological space gives a parametrized family of fuzzy topological spaces and that the constant mapping is not continuous in general.
Abstract: In the present paper we introduce the topological structure of fuzzy soft sets and fuzzy soft continuity of fuzzy soft mappings. We show that a fuzzy soft topological space gives a parametrized family of fuzzy topological spaces. Furthermore, with the help of an example it is shown that the constant mapping is not continuous in general. Then the notions of fuzzy soft closure and interior are introduced and their basic properties are investigated. Finally, the initial fuzzy soft topology and some properties of projection mappings are studied.
TL;DR: A concept of a granular representation of numeric membership functions of fuzzy sets is introduced, which offers a synthetic and qualitative view at fuzzy sets and their ensuing processing and helps regard the problem as a certain optimization task.
TL;DR: It is proved that the generalized Atanassov's operators also generalize OWA operators of any dimension by allowing negative weights and this work applies the results to a decision making problem.
TL;DR: A hybrid method of forecasting based on fuzzy time series and intuitionistic fuzzy sets is proposed that uses the degree of nondeterminacy to establish fuzzy logical relations on time series data.
Abstract: Fuzzy time series models are of great interest in forecasting when the information is imprecise and vague. However, the major problem in fuzzy time series forecasting is the accuracy of the forecasted values. In the present study we propose a hybrid method of forecasting based on fuzzy time series and intuitionistic fuzzy sets. The proposed model is a simplified computational approach that uses the degree of nondeterminacy to establish fuzzy logical relations on time series data. The developed model was implemented on the historical enrollment data for the University of Alabama and the forecasted values were compared with the results of existing methods to show its superiority. The suitability of the proposed method was also examined in forecasting market share prices of the State Bank of India on the Bombay Stock Exchange, India.
TL;DR: The aim of this paper is to develop an effective method for solving matrix games with payoffs of triangular fuzzy numbers (TFNs) which always assures that players’ gain-floor and loss-ceiling have a common TFN-type fuzzy value and hereby any matrix game with payoff of TFNs has aTFN- type fuzzy value.
29 May 2012
TL;DR: In this article, some interval-valued intuitionistic fuzzy geometric operators based on Einstein operations are developed, such as the interval- VALUE Einstein weighted geometric (IVIFWGε) operator, interval-VALUE Einstein ordered weighted geometric(IVIFOWG ε) operator and interval-VALUE Einstein hybrid geometric(ivIFHWGα) operator.
Abstract: The notion of interval-valued intuitionistic fuzzy set (IVIFS) is a generalization of that of Atanassov's intuitionistic fuzzy set (AIFS). The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. In this article, we develop some interval-valued intuitionistic fuzzy geometric operators based on Einstein operations, such as the interval-valued intuitionistic fuzzy Einstein weighted geometric (IVIFWGe) operator, interval-valued intuitionistic fuzzy Einstein ordered weighted geometric (IVIFOWGe) operator and interval-valued intuitionistic fuzzy Einstein hybrid geometric (IVIFHWGe) operator, which are the generalizations of the geometric aggregation operators based on AIFSs. Moreover, we establish various properties of these operators.
TL;DR: This work presents a construction method starting from fuzzy preference relations and taking into account the ignorance of the expert in the construction of the latter, and proposes two generalizations of the weighted voting strategy to work with Atanassov's intuitionistic fuzzy preference Relations.
Abstract: In this work we present a construction method for Atanassov's intuitionistic fuzzy preference relations starting from fuzzy preference relations and taking into account the ignorance of the expert in the construction of the latter. Moreover, we propose two generalizations of the weighted voting strategy to work with Atanassov's intuitionistic fuzzy preference relations. An advantage of these algorithms is that they start from fuzzy preference relations and their results can be compared with those of any other decision-making algorithm based on fuzzy sets theory. We verify that our proposal is able to provide a unique solution in some cases in which the voting strategy is not able to order the alternatives.
TL;DR: A non-nested level-based representation of fuzziness is described, closely related to some existing models and concepts in the literature, and it is claimed that fuzzy mathematical objects and operations are uniquely and easily defined as extensions of their crisp counterparts.
TL;DR: This paper generalizes the fuzzy rough set model on two different universes proposed by Sun and Ma on the basis of the bipolar fuzzy compatible relation R"("@a","@b") (@a,@[email protected]?(0,1]), and some related results are obtained.
Abstract: Pawlak initiated the concept of the rough set as a formal tool for modeling and processing incomplete information in information systems. Various fuzzy generalizations of the rough set have been proposed in the literature. In this paper we generalize the fuzzy rough set model on two different universes proposed by Sun and Ma. Concretely, based on the bipolar fuzzy compatible relation R"("@a","@b") (@a,@[email protected]?(0,1]), the bipolar fuzzy rough set model on two different universes is presented. Some properties of the bipolar fuzzy rough set model are discussed. Two extended models of the bipolar fuzzy rough set model are given, and some related results are obtained. Finally, an example is applied to illustrate the application of the bipolar fuzzy rough set model presented in this paper.
TL;DR: The goal is to think of the state of the art in mathematical fuzzy logic (MFL) and to outline some of the tasks on which, in the author's opinion, MFL should focus in the future.
TL;DR: A new ranking index to rank various fuzzy numbers effectively is given and several numerical examples following the procedure indicate the ranking results to be valid.
Abstract: Ranking fuzzy numbers plays a very important role in decision making and some other fuzzy application systems. Many different methods have been proposed to deal with ranking fuzzy numbers. Constructing ranking indexes based on the centroid of fuzzy numbers is an important case. But some weaknesses are found in these indexes. The purpose of this paper is to give a new ranking index to rank various fuzzy numbers effectively. Finally, several numerical examples following the procedure indicate the ranking results to be valid.