Showing papers on "Fuzzy number published in 2010"
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TL;DR: It is proved that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set, and it is proved also that the operations proposed are consistent with the ones of intuitionist fuzzy sets when applied to the envelope.
Abstract: Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.
2,232 citations
TL;DR: An evaluation model based on the fuzzy analytic hierarchy process and the technique for order performance by similarity to ideal solution, fuzzy TOPSIS is developed to help the industrial practitioners for the performance evaluation in a fuzzy environment.
Abstract: Multiple criteria decision-making (MCDM) research has developed rapidly and has become a main area of research for dealing with complex decision problems. The purpose of the paper is to explore the performance evaluation model. This paper develops an evaluation model based on the fuzzy analytic hierarchy process and the technique for order performance by similarity to ideal solution, fuzzy TOPSIS, to help the industrial practitioners for the performance evaluation in a fuzzy environment where the vagueness and subjectivity are handled with linguistic values parameterized by triangular fuzzy numbers. The proposed method enables decision analysts to better understand the complete evaluation process and provide a more accurate, effective, and systematic decision support tool.
619 citations
01 Mar 2010
TL;DR: The developed models and procedures based on I-IIFOWG and IIFWG (interval-valued intuitionistic fuzzy weighted geometric) operators are extended to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take theform of interval-valued intuistic fuzzy numbers.
Abstract: With respect to multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, some new group decision making analysis methods are developed. Firstly, some operational laws, score function and accuracy function of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers are introduced. Then two new aggregation operators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator are proposed, and some desirable properties of the I-IFOWG and I-IIFOWG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFOWG and IFWG (intuitionistic fuzzy weighted geometric) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers. Further, we extend the developed models and procedures based on I-IIFOWG and IIFWG (interval-valued intuitionistic fuzzy weighted geometric) operators to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of interval-valued intuitionistic fuzzy numbers. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.
591 citations
TL;DR: A hierarchy MCDM model based on fuzzy sets theory and VIKOR method is proposed to deal with the supplier selection problems in the supply chain system.
Abstract: During recent years, how to determine suitable suppliers in the supply chain has become a key strategic consideration. However, the nature of supplier selection is a complex multi-criteria problem including both quantitative and qualitative factors which may be in conflict and may also be uncertain. The VIKOR method was developed to solve multiple criteria decision making (MCDM) problems with conflicting and non-commensurable (different units) criteria, assuming that compromising is acceptable for conflict resolution, the decision maker wants a solution that is the closest to the ideal, and the alternatives are evaluated according to all established criteria. In this paper, linguistic values are used to assess the ratings and weights for these factors. These linguistic ratings can be expressed in trapezoidal or triangular fuzzy numbers. Then, a hierarchy MCDM model based on fuzzy sets theory and VIKOR method is proposed to deal with the supplier selection problems in the supply chain system. A numerical example is proposed to illustrate an application of the proposed model.
531 citations
TL;DR: The validity of the Roy-Maji method is discussed, the weighted fuzzy soft set is introduced and its application to decision making is also investigated.
469 citations
TL;DR: The proposed method provides a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of the attributes.
Abstract: Type-2 fuzzy sets involve more uncertainties than type-1 fuzzy sets. They provide us with additional degrees of freedom to represent the uncertainty and the fuzziness of the real world. In this paper, we present an interval type-2 fuzzy TOPSIS method to handle fuzzy multiple attributes group decision-making problems based on 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 provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of the attributes.
456 citations
TL;DR: The Choquet integral is used to propose some intuitionistic fuzzy aggregation operators that not only consider the importance of the elements or their ordered positions, but also can reflect the correlations among the elements and theirordered positions.
425 citations
TL;DR: This paper develops some new generalized aggregation operators, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval-valued intuitionism fuzzy sets.
Abstract: The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by Yager (Fuzzy Optim Decision Making 2004;3:93–107). However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval-valued intuitionistic fuzzy weighted averaging operator, generalized interval-valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval-valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval-valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. © 2009 Wiley Periodicals, Inc.
391 citations
TL;DR: In this paper, an intuitionistic fuzzy Choquet integral operator is proposed for multiple criteria decision making, where interactions phenomena among the decision making criteria are considered.
Abstract: For the real decision making problems, most criteria have inter-dependent or interactive characteristics so that it is not suitable for us to aggregate them by traditional aggregation operators based on additive measures. Thus, to approximate the human subjective decision making process, it would be more suitable to apply fuzzy measures, where it is not necessary to assume additivity and independence among decision making criteria. In this paper, an intuitionistic fuzzy Choquet integral is proposed for multiple criteria decision making, where interactions phenomena among the decision making criteria are considered. First, we introduced two operational laws on intuitionistic fuzzy values. Then, based on these operational laws, intuitionistic fuzzy Choquet integral operator is proposed. Moreover, some of its properties are investigated. It is shown that the intuitionistic fuzzy Choquet integral operator can be represented by some special t-norms and t-conorms, and it is also a generalization of the intuitionistic fuzzy OWA operator and intuitionistic fuzzy weighted averaging operator. Further, the procedure and algorithm of multi-criteria decision making based on intuitionistic fuzzy Choquet integral operator is given under uncertain environment. Finally, a practical example is provided to illustrate the developed approaches.
390 citations
TL;DR: A generalization of the Hukuhara difference of fuzzy numbers is proposed, using their compact and convex level-cuts to solve interval and fuzzy linear equations and fuzzy differential equations.
389 citations
TL;DR: Application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown and some of their properties are studied.
Abstract: In this paper, we define generalised fuzzy soft sets and study some of their properties. Application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown.
TL;DR: The concept of a triangular IFN (TIFN) is introduced as a special case of the IFN and a new methodology for ranking TIFNs is developed on the basis of a ratio of the value index to the ambiguity index and applied to multiattribute decision making problems in which the ratings of alternatives on attributes are expressed with TIFN.
Abstract: The concept of an intuitionistic fuzzy number (IFN) is of importance for quantifying an ill-known quantity, and the ranking of IFNs is a very difficult problem. The aim of this paper is to introduce the concept of a triangular IFN (TIFN) as a special case of the IFN and develop a new methodology for ranking TIFNs. Firstly the concepts of TIFNs and cut sets as well as arithmetical operations are introduced. Then the values and ambiguities of the membership function and the non-membership function for a TIFN are defined. A new ranking method is developed on the basis of the concept of a ratio of the value index to the ambiguity index and applied to multiattribute decision making problems in which the ratings of alternatives on attributes are expressed with TIFNs. The validity and applicability of the proposed method, as well as analysis of the comparison with other methods, are illustrated with a real example.
TL;DR: A fuzzy multicriteria decision-making methodology is suggested for the selection among energy policies, based on the analytic hierarchy process (AHP) under fuzziness, which determines the best energy policy for Turkey.
Abstract: Since the correct energy policy affects economic development and environment, the most appropriate energy policy selection is excessively important. Recently some studies have concentrated on selecting the best energy policy and determining the best energy alternatives. In most of these studies, multicriteria and fuzzy approaches to energy policy making are frequently used. The fuzzy set theory is a powerful tool to treat the uncertainty in case of incomplete or vague information. In this paper, a fuzzy multicriteria decision-making methodology is suggested for the selection among energy policies. The methodology is based on the analytic hierarchy process (AHP) under fuzziness. It allows the evaluation scores from experts to be linguistic expressions, crisp or fuzzy numbers. In the application of the proposed methodology, the best energy policy is determined for Turkey.
TL;DR: The proposed method provides a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.
Abstract: In this paper, we present a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.
TL;DR: In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed and the complement, ''and'', ''or'', union, intersection, necessity and possibility operations are defined on the interval -valued intuitionism fuzzy soft sets.
Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed. Our interval-valued intuitionistic fuzzy soft set theory is a combination of an interval-valued intuitionistic fuzzy set theory and a soft set theory. In other words, our interval-valued intuitionistic fuzzy soft set theory is an interval-valued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. The complement, ''and'', ''or'', union, intersection, necessity and possibility operations are defined on the interval-valued intuitionistic fuzzy soft sets. The basic properties of the interval-valued intuitionistic fuzzy soft sets are also presented and discussed.
TL;DR: Some relations and operations of interval-valued intuitionistic fuzzy numbers are introduced, some types of matrices are defined and a method based on distance measure for group decision making with interval-value intuitionistic fuzziness matrices is developed.
TL;DR: The aim of this paper is to investigate the intuitionistic fuzzy multiple attribute decision-making problems where the attribute values are expressed in intuitionists fuzzy numbers or interval-valued intuitionism fuzzy numbers.
Abstract: The aim of this paper is to investigate the intuitionistic fuzzy multiple attribute decision-making problems where the attribute values are expressed in intuitionistic fuzzy numbers or interval-val...
TL;DR: An evolution model that integrates triangular fuzzy numbers and analytic hierarchy process to develop a fuzzy evaluation model which prioritized the relative weights of course website quality factors is developed.
Abstract: Although previous studies have identified various influences on course website effectiveness, the evaluation of the relative importance of these factors across different online learning experience groups has not been empirically determined. This study develops an evolution model that integrates triangular fuzzy numbers and analytic hierarchy process to develop a fuzzy evaluation model which prioritized the relative weights of course website quality factors. Firstly, this study conducts a review of the literature on course website quality to generate 16 sub-criteria along with four criteria used to measure course website quality. Secondly, a fuzzy AHP approach is adopted to determine the relative weights linking the above criteria between high and low online learning experience groups. The results indicated that there are some similarities and differences between high- and low-experience groups with regard to the evaluation of course website quality. The evaluation model and results can provide a valuable reference for system designers seeking to enhance course website effectiveness.
TL;DR: This paper introduces the concept of fuzzy decision reducts, dependent on an increasing attribute subset measure, and presents a generalization of the classical rough set framework for data-based attribute selection and reduction using fuzzy tolerance relations.
TL;DR: This paper proposes a new method based on Fuzzy K-Means clustering for reducing the size of the concept lattices and demonstrates the implementation of proposed method on two application areas: information retrieval and information visualization.
Abstract: During the design of concept lattices, complexity plays a major role in computing all the concepts from the huge incidence matrix. Hence for reducing the size of the lattice, methods based on matrix decompositions like SVD are available in the literature. However, SVD computation is known to have large time and memory requirements. In this paper, we propose a new method based on Fuzzy K-Means clustering for reducing the size of the concept lattices. We demonstrate the implementation of proposed method on two application areas: information retrieval and information visualization.
TL;DR: It is seen that the proposed structure of a type 2 Takagi–Sugeno–Kang fuzzy neural system is a potential candidate for identification and control purposes of uncertain plants, with the uncertainties being handled adequately by type 2 fuzzy sets.
Abstract: In industry, most dynamical plants are characterized by unpredictable and hard-to-formulate factors, uncertainty, and fuzziness of information, and as a result, deterministic models usually prove to be insufficient to adequately describe the process In such situations, the use of fuzzy approaches becomes a viable alternative However, the systems constructed on the base of type 1 fuzzy systems cannot directly handle the uncertainties associated with information or data in the knowledge base of the process One possible way to alleviate the problem is to resort to the use of type 2 fuzzy systems In this paper, the structure of a type 2 Takagi–Sugeno–Kang fuzzy neural system is presented, and its parameter update rule is derived based on fuzzy clustering and gradient learning algorithm Its performance for identification and control of time-varying as well as some time-invariant plants is evaluated and compared with other approaches seen in the literature It is seen that the proposed structure is a potential candidate for identification and control purposes of uncertain plants, with the uncertainties being handled adequately by type 2 fuzzy sets
TL;DR: This work combines Gaussian kernel with fuzzy rough sets and proposes a Gaussian kernels approximation based fuzzy rough set model, which is proven that fuzzy relations with Gaussiankernel are reflexive, symmetric and transitive.
TL;DR: A new decision-making model with probabilistic information was developed and used the concept of the immediate probability to aggregate the information, which modifies the objective probability by introducing the attitudinal character of the decision maker using the ordered weighting average (OWA) operator.
TL;DR: A newly‐developed ARAS‐F method is presented to solve different problems in transport, construction, economics, technology and sustainable development, to help the stakeholders with the performance evaluation in an uncertain environment.
Abstract: The main approaches which are applied to select the logistic center are the methods of gravity center, analytic hierarchy process, similarity to ideal solution, fuzzy ranking, assessment, etc. Multiple Criteria Decision‐Making (MCDM) combines analytical and inductive knowledge, describing a domain problem, which can be fuzzy and/or incomplete. The fuzzy MCDM (FMCDM) approach can explain the problem more appropriately. The purpose of the paper is to select the most suitable site for logistic centre among a set of alternatives, to help the stakeholders with the performance evaluation in an uncertain environment, where the subjectivity and vagueness of criteria are described by triangular fuzzy numbers. The paper presents a newly‐developed ARAS‐F method to solve different problems in transport, construction, economics, technology and sustainable development.
Book•
04 Feb 2010
TL;DR: This monograph is suitable for researchers, graduate students and seminars of theoretical and applied mathematics, computer science, statistics and engineering, and it is the first one in Fuzzy Approximation Theory.
Abstract: This monograph belongs to the broader area of Fuzzy Mathematics and it is the first one in Fuzzy Approximation Theory. The chapters are self-contained with lots of applications to teach several advanced courses and the topics covered are very diverse. An extensive background of Fuzziness and Fuzzy Real Analysis is given. The author covers Fuzzy Differentiation and Integration Theory followed by Fuzzy Ostrowski inequalities. Then results on classical algebraic and trigonometric polynomial Fuzzy Approximation are presented. The author develops a complete theory of convergence with rates of Fuzzy Positive linear operators to Fuzzy unit operator, the so-called Fuzzy Korovkin Theory. The related Fuzzy Global Smoothness is included. Then follows the study of Fuzzy Wavelet type operators and their convergence with rates to Fuzzy unit operator. Similarly the Fuzzy Neural Network Operators are discussed followed by Fuzzy Random Korovkin approximation theory and Fuzzy Random Neural Network approximations. The author continues with Fuzzy Korovkin approximations in the sense of Summability. Finally fuzzy sense differences of Fuzzy Wavelet type operators are estimated. The monograph's approach is quantitative and the main results are given via Fuzzy inequalities, involving Fuzzy moduli of continuity, that is Fuzzy Jackson type inequalities. The exposed theory is destined and expected to find applications to all aspects of Fuzziness from theoretical to practical in almost all sciences, technology, finance and industry. Also it has its interest within Pure Mathematics. So this monograph is suitable for researchers, graduate students and seminars of theoretical and applied mathematics, computer science, statistics and engineering.
TL;DR: Out-of-sample forecasting of the stock index in Taiwan is performed and the results are compared with those of previous studies to demonstrate the performance of the proposed model.
Abstract: Neural networks have been popular due to their capabilities in handling nonlinear relationships Hence, this study intends to apply neural networks to implement a new fuzzy time series model to improve forecasting Differing from previous studies, this study includes the various degrees of membership in establishing fuzzy relationships, which assist in capturing the relationships more properly These fuzzy relationships are then used to forecast the stock index in Taiwan With more information, the forecasting is expected to improve, too In addition, due to the greater amount of information covered, the proposed model can be used to forecast directly regardless of whether out-of-sample observations appear in the in-sample observations This study performs out-of-sample forecasting and the results are compared with those of previous studies to demonstrate the performance of the proposed model
TL;DR: The proposed stability-analysis approach offers a nice property that includes the membership functions of both fuzzy model and fuzzy controller in the LMI-based stability conditions for a dedicated FMB control system.
Abstract: This paper presents the stability analysis of fuzzy-model-based (FMB) control systems. Staircase membership functions are introduced to facilitate the stability analysis. Through the staircase membership functions approximating those of the fuzzy model and fuzzy controller, the information of the membership functions can be brought into the stability analysis. Based on the Lyapunov-stability theory, stability conditions in terms of linear-matrix inequalities (LMIs) are derived in a simple and easy-to-understand manner to guarantee the system stability. The proposed stability-analysis approach offers a nice property that includes the membership functions of both fuzzy model and fuzzy controller in the LMI-based stability conditions for a dedicated FMB control system. Furthermore, the proposed stability-analysis approach can be applied to the FMB control systems of which the membership functions of both fuzzy model and fuzzy controller are not necessarily the same. Greater design flexibility is allowed to choose the membership functions during the design of fuzzy controllers. By employing membership functions with simple structure, it is possible to lower the structural complexity and the implementation cost. Simulation examples are given to illustrate the merits of the proposed approach.
TL;DR: An optimization model is developed by which a straightforward formula for deriving attribute weights can be obtained and an approach to ranking the given alternatives and then selecting the most desirable one(s) is given.
Abstract: The aim of this article is to investigate the approach to multiple attribute group decision making (MAGDM) with intuitionistic fuzzy information. We first introduce a deviation measure between two intuitionistic fuzzy numbers, and then utilize the intuitionistic fuzzy hybrid aggregation operator to aggregate all individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix. Based on the deviation measure, we develop an optimization model by which a straightforward formula for deriving attribute weights can be obtained. Furthermore, based on the intuitionistic fuzzy weighted averaging operator and information theory, we utilize the score function and accuracy function to give an approach to ranking the given alternatives and then selecting the most desirable one(s). In addition, we extend the above results to MAGDM with interval-valued intuitionistic fuzzy information.
01 Jan 2010
TL;DR: F fuzzy parameterized fuzzy soft (fpfs) sets are given and fpfs-aggregation operator is defined to form fpFS-decision making method that allows constructing more efficient decision processes.
Abstract: In this work, we give definition of fuzzy parameterized fuzzy soft (fpfs) sets and their operations. We then define fpfs-aggregation operator to form fpfs-decision making method that allows constructing more efficient decision processes.
TL;DR: A novel adaptive fuzzy controller is designed based on the Razumikhin function approach, which guarantees that the system output converges to a small neighborhood of the reference signal and all the signals in the closed-loop system remain bounded.