Showing papers on "Fuzzy mathematics published in 2006"
TL;DR: A method of image compression and reconstruction on the basis of the F-transform, which is a fuzzy partition of a universe into fuzzy subsets (factors, clusters, granules etc.), is presented.
548 citations
TL;DR: The proposed fuzzy Lyapunov function is formulated as a line-integral of a fuzzy vector which is a function of the state, and it can be regarded as the work done from the origin to the current state in the fuzzy vector field.
440 citations
TL;DR: A new TOPSIS approach for selecting plant location under linguistic environments is presented, where the ratings of various alternative locations under various criteria, and the weights of various criteria are assessed in linguistic terms represented by fuzzy numbers.
Abstract: The selection of plant location plays a very important role in minimizing cost and maximizing the use of resources for many companies. In this paper, a new TOPSIS approach for selecting plant location under linguistic environments is presented, where the ratings of various alternative locations under various criteria, and the weights of various criteria are assessed in linguistic terms represented by fuzzy numbers. To avoid complicated fuzzy arithmetic operations, the linguistic variables, which are represented by triangular fuzzy numbers, are transformed into crisp numbers based on graded mean representation. The canonical representation of multiplication operations on triangular fuzzy numbers is used to obtain the “positive ideal solution” and the “negative ideal solution”. The closeness efficient is defined to determine the ranking order of all alternatives by calculating the distance to both the “positive-ideal solution” and the “negative-ideal solution” simultaneously. Compared with existing fuzzy TOPSIS methods, the proposed method can deal with group decision-making problems in a more efficient manner. A numerical example of plant location selection is used to illustrate the efficiency of the proposed method.
414 citations
TL;DR: This paper defines precompact set in intuitionistic fuzzy metric spaces and proves that any subset of an intuitionism fuzzy metric space is compact if and only if it is precompacts and complete.
Abstract: In this paper, we define precompact set in intuitionistic fuzzy metric spaces and prove that any subset of an intuitionistic fuzzy metric space is compact if and only if it is precompact and complete. Also we define topologically complete intuitionistic fuzzy metrizable spaces and prove that any G δ set in a complete intuitionistic fuzzy metric spaces is a topologically complete intuitionistic fuzzy metrizable space and vice versa. Finally, we define intuitionistic fuzzy normed spaces and fuzzy boundedness for linear operators and so we prove that every finite dimensional intuitionistic fuzzy normed space is complete.
322 citations
TL;DR: In this paper, a modification of the distance based approach called the sign distance is proposed, which is both efficient to evaluate and able to overcome the shortcomings of the previous techniques, such as coefficient of variation (CV index), distance between fuzzy sets, centroid point and original point, and weighted mean value.
292 citations
01 Jan 2006
TL;DR: A review of the fundamentals of fuzzy sets, fuzzy rules and fuzzy inference systems is provided in this chapter and the different defuzzification techniques and their processes are discussed with the same example step by step.
Abstract: A review of the fundamentals of fuzzy sets, fuzzy rules and fuzzy inference systems is provided in this chapter. Beginning with crisp or classical sets and their operations, we derived fuzzy sets and their operations. Classical set membership functions and fuzzy membership functions are discussed in detail following set theory. Fuzzy rules are described using an air conditioner control example. The different defuzzification techniques and their processes are discussed with the same example step by step. Finally, some other fuzzy techniques are discussed such as off-line and on-line fuzzy control systems as well as a fuzzy closed-loop control system including multiple lookup tables.
260 citations
TL;DR: Experimental results show that better control can be achieved using a type-2 FLC with fewer fuzzy sets/rules so one benefit of type-1 FLC is a lower trade-off between modeling accuracy and interpretability.
246 citations
TL;DR: A modified fuzzy LLSM, which is formulated as a constrained nonlinear optimization model, is suggested to tackle all problems of the fuzzy logarithmic least squares method and its advantages.
209 citations
TL;DR: The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers and some important and interesting results are obtained which in turn lead to a solution of fuzzylinear programming problems without converting them to crisp linear programming problems.
Abstract: The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting results are obtained which in turn lead to a solution of fuzzy linear programming problems without converting them to crisp linear programming problems.
207 citations
TL;DR: It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard, and some heuristics based methods on Dubois and Prade’s approach are employed, finding some positive fuzzy vector x which satisfies A ˜ x ˜ = b ˜, where A and b are a fuzzy matrix and a fuzzy vector respectively.
207 citations
Book•
01 Jan 2006
TL;DR: This book discusses the construction of Fuzzy Set Mathematics, the theory and practice of quantitative membership analysis, and some of the techniques used to derive membership functions.
Abstract: Series Editor's Introduction Acknowledgments 1. Introduction 2. An Overview of Fuzzy Set Mathematics 2.1 Set Theory 2.2 Why Fuzzy Sets? 2.3 The Membership Function 2.4 Operations of Fuzzy Set Theory 2.5 Fuzzy Numbers and Fuzzy Variables 2.6 Graphical Representations of Fuzzy Sets 3. Measuring Membership 3.1 Introduction 3.2 Methods for Constructing Membership Functions 3.3 Measurement Properties Required for Fuzzy Sets 3.4 Measurement Properties of Membership Functions 3.5 Uncertainty Estimates in Membership Assignment 4. Internal Structure and Properties of a Fuzzy Set 4.1 Cardinality: The Size of a Fuzzy Set 4.2 Probability Distributions for Fuzzy Sets 4.3 Defining and Measuring Fuzziness 5. Simple Relations Between Fuzzy Sets 5.1 Intersection, Union, and Inclusion 5.2 Detecting and Evaluating Fuzzy Inclusion 5.3 Quantifying and Modeling Inclusion: Ordinal Membership Scales 5.4 Quantified and Comparable Membership Scales 6. Multivariate Fuzzy Set Relations 6.1 Compound Set Indexes 6.2 Multiset Relations: Comorbidity, Covariation, and Co-Occurrence 6.3 Multiple and Partial Intersection and Inclusion 7. Concluding Remarks References Index About the Authors
TL;DR: The generality and the applicability of the proposed representation to a large class of problems are stressed, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics.
Book•
21 Feb 2006
TL;DR: This book combines material from the previous books FP and FS with three chapters on fuzzy maximum entropy (imprecise side conditions) estimators producing fuzzy distributions and crisp discrete/continuous distributions.
Abstract: This book combines material from our previous books FP (Fuzzy Probabilities: New Approach and Applications,Physica-Verlag, 2003) and FS (Fuzzy Statistics, Springer, 2004), plus has about one third new results. From FP we have material on basic fuzzy probability, discrete (fuzzy Poisson,binomial) and continuous (uniform, normal, exponential) fuzzy random variables. From FS we included chapters on fuzzy estimation and fuzzy hypothesis testing related to means, variances, proportions, correlation and regression. New material includes fuzzy estimators for arrival and service rates, and the uniform distribution, with applications in fuzzy queuing theory. Also, new to this book, is three chapters on fuzzy maximum entropy (imprecise side conditions) estimators producing fuzzy distributions and crisp discrete/continuous distributions. Other new results are: (1) two chapters on fuzzy ANOVA (one-way and two-way); (2) random fuzzy numbers with applications to fuzzy Monte Carlo studies; and (3) a fuzzy nonparametric estimator for the median.
TL;DR: A new fuzzy membership function is proposed to the nonlinear fuzzy support vector machine that gives good performance on reducing the effects of outliers and significantly improves the classification accuracy and generalization.
Abstract: It is known that with a proper fuzzy membership function, a fuzzy support vector machine can effectively reduce the effects of outliers when solving the classification problem. In this paper, a new fuzzy membership function is proposed to the nonlinear fuzzy support vector machine. The fuzzy membership is calculated in the feature space and is represented by kernels. This method gives good performance on reducing the effects of outliers and significantly improves the classification accuracy and generalization.
TL;DR: Three convergence theorems about the use of fuzzy simulation in computing the credibility of a fuzzy event, finding the optimistic value of a return function, and calculating the expected value of an fuzzy variable are proved.
Abstract: We discuss the convergence of fuzzy simulation as it is employed in fuzzy optimization problems. Several convergence concepts for sequences of fuzzy variables are defined such as convergence in optimistic value. A new approach to approximating essentially bounded fuzzy variables with continuous possibility distributions is introduced. Applying the proposed approximation method to our previous work, we prove three convergence theorems about the use of fuzzy simulation in computing the credibility of a fuzzy event, finding the optimistic value of a return function, and calculating the expected value of a fuzzy variable
TL;DR: The originality and value of IFT are here demonstrated, which show that the results obtained by various authors on intuitionistic fuzzy topological spaces (IFTSs), are not redundant with others for the ordinary fuzzy sense.
Abstract: Purpose – In 2000, Wang and He published an important result on the theory of intuitionistic fuzzy sets (IFSs) Indeed, they showed that every IFS may be regarded as an L‐fuzzy set for some appropriate lattice L This paper aims to show that, nevertheless, the results obtained by various authors on intuitionistic fuzzy topological spaces (IFTSs), are not redundant with others for the ordinary fuzzy senseDesign/methodology/approach – The most important definitions and results on intuitionistic fuzzy topology (IFT) one compared with the result obtained by Wang and HeFindings – That these results are not redundantResearch limitations/implications – Clearly, this paper is devoted to IFTSsPractical implications – The main applications are in the mathematical fieldOriginality/value – The originality and value of IFT are here demonstrated
TL;DR: Using the idea of quasi-coincidence of a fuzzy point with a fuzzy set, the concept of an (@a,@b)-fuzzy interior ideal is introduced, which is a generalization of a warm interior ideal, in a semigroup, and related properties are investigated.
Book•
20 Mar 2006
TL;DR: This paper presents a meta-modelling framework for fuzzy statistical analysis and estimation of random fuzzy sets and its applications to time series analysis and forecasting.
Abstract: Introduction.- Set-valued Data.- Modeling of fuzzy data.- Random fuzzy sets.- Aspect of statistical Inference.- Convergence of random fuzzy sets.- Fuzzy Statistical Analysis and Estimation.- Testing Hypothesis with Fuzzy Data.- Fuzzy Time Series Analysis and Forecasting.
Journal Article•
TL;DR: A new fuzzy rough set approach which, differently from most known fuzzy set extensions of rough set theory, does not use any fuzzy logical connectives (t-norm, t-conorm, fuzzy implication) and creates a base for induction of fuzzy decision rules having syntax and semantics of gradual rules.
Abstract: We propose a new fuzzy rough set approach which, differently from most known fuzzy set extensions of rough set theory, does not use any fuzzy logical connectives (t-norm, t-conorm, fuzzy implication). As there is no rationale for a particular choice of these connectives, avoiding this choice permits to reduce the part of arbitrary in the fuzzy rough approximation. Another advantage of the new approach is that it is based on the ordinal properties of fuzzy membership degrees only. The concepts of fuzzy lower and upper approximations are thus proposed, creating a base for induction of fuzzy decision rules having syntax and semantics of gradual rules. The proposed approach to rule induction is also interesting from the viewpoint of philosophy supporting data mining and knowledge discovery, because it is concordant with the method of concomitant variations by John Stuart Mill. The decision rules are induced from lower and upper approximations defined for positive and negative relationships between credibility degrees of multiple premises, on one hand, and conclusion, on the other hand.
TL;DR: A ''direct fuzzy approach (DFA)'' to fuzzy control charts for attributes under vague data is proposed without using any transformation method, and the unnatural patterns for the proposed fuzzycontrol charts are defined using the probabilities of fuzzy events.
TL;DR: This paper gives the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.
01 Jan 2006
TL;DR: Improving the fuzzy Analytic Hierarchy Process (AHP) method is proposed by using the approximate fuzzy eigenvector of such fuzzy symmetry matrix, which reflects the dispersed projection of decision information in general.
Abstract: For fuzzy multi-attribute decision-making, a fuzzy symmetry matrix, by referring to covariance definition of random variables, is constructed as attribute evaluation space based on fuzzy decision-making matrix. Improving the fuzzy Analytic Hierarchy Process (AHP) method is proposed by using the approximate fuzzy eigenvector of such fuzzy symmetry matrix. This algorithm reflects the dispersed projection of decision information in general. It has better objectivity and resolving power for the decision-making. This algorithm is used for illustration and comparison with other methods. The results are applied in an example to illustrate that this algorithm is more efficient and objective for multi-attribute decision-making application.
TL;DR: It is shown that the observability of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor and it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable.
Abstract: Fuzzy discrete-event systems as a generalization of (crisp) discrete-event systems have been introduced in order that it is possible to effectively represent uncertainty, imprecision, and vagueness arising from the dynamic of systems. A fuzzy discrete-event system has been modeled by a fuzzy automaton; its behavior is described in terms of the fuzzy language generated by the automaton. In this paper, we are concerned with the supervisory control problem for fuzzy discrete-event systems with partial observation. Observability, normality, and co-observability of crisp languages are extended to fuzzy languages. It is shown that the observability, together with controllability, of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor. When a decentralized solution is desired, it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable. Moreover, the infimal controllable and observable fuzzy superlanguage, and the supremal controllable and normal fuzzy sublanguage are also discussed. Simple examples are provided to illustrate the theoretical development
TL;DR: The one-sample method of testing about the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space).
TL;DR: Within the effectively formalized representation developed here, based on a complete logical system, it is possible to reconstruct numerous well-known properties of CRI-related fuzzy inference methods, albeit not from the analytic point of view as usually presented, but as formal derivations of the logical system employed.
TL;DR: The objective of this paper is to introduce a fuzzy distance measure for generalized fuzzy numbers (GFN), which computes the fuzzy distance between two generalized fuzzyNumbers and also LR-type fuzzy numbers.
TL;DR: The definition of weakly commuting and R-weakly commuting mappings in intuitionistic fuzzy metric spaces are formulated and Pant’s theorem is proved.
Abstract: The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [2], we define the notion of intuitionistic fuzzy metric spaces (see, [1]) due to Kramosil and Michalek [17] and Jungck’s common fixed point theorem ([11]) is generalized to intuitionistic fuzzy metric spaces. Further, we first formulate the definition of weakly commuting and R-weakly commuting mappings in intuitionistic fuzzy metric spaces and prove the intuitionistic fuzzy version of Pant’s theorem ([21]).
TL;DR: Several guidelines based on Hajek's methodology in fuzzy logic are formulated, which enable us to follow closely the constructions and methods of classical mathematics recast in a fuzzy setting.
TL;DR: The compatibility index of the AHP is used to illustrate how the answers obtained by fuzzifying AHP judgments do not produce better results than direct derivation of the principal eigenvector.
Abstract: Fuzzy logic has difficulty producing valid answers in decision-making. Absent are theorems to prove that it works to produce results already known that are being estimated with judgments by transforming such judgments numerically. The numerical representation of judgments in the AHP is already fuzzy. Making fuzzy judgments more fuzzy does not lead to a better more valid outcome and it often leads to a worse one. The compatibility index of the AHP is used to illustrate how the answers obtained by fuzzifying AHP judgments do not produce better results than direct derivation of the principal eigenvector. Other authors who did experiments with given data in decision making quoted in the conclusions section of the paper, have observed that fuzzy sets gives the poorest answers among all methods used to derive best decisions.
TL;DR: In this paper, the concept of an intuitionistic fuzzy set was applied to H"v-modules, and some related properties were investigated, including properties of the notion of fuzzy subsets.