Showing papers on "Fuzzy number published in 2006"
TL;DR: In this article, a hierarchical multiple criteria decision-making (MCDM) model based on fuzzy-sets theory is proposed to deal with the supplier selection problems in the supply chain system.
1,559 citations
TL;DR: The aim of this paper is to extend the TOPSIS method to decision-making problems with fuzzy data, and the rating of each alternative and the weight of each criterion are expressed in triangular fuzzy numbers.
556 citations
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: A new method that transfers the house of quality (HOQ) approach typical of quality function deployment (QFD) problems to the supplier selection process is suggested and symmetrical triangular fuzzy numbers are suggested to capture the vagueness in people's verbal assessments.
352 citations
TL;DR: This paper presents the correct centroid formulae for fuzzy numbers and justify them from the viewpoint of analytical geometry and a numerical example demonstrates that Cheng's formULae can significantly alter the result of the ranking procedure.
336 citations
TL;DR: F fuzzy number logic is introduced in the pair wise comparison of AHP to make up for this deficiency in the conventional AHP, referred to as fuzzy AHP.
Abstract: Selecting process of a machine tool has been very important issue for companies for years, because the improper selection of a machine tool might cause of many problems affecting negatively on productivity, precision, flexibility and company’s responsive manufacturing capabilities. On the other hand, selecting the best machine tool from its increasing number of existing alternatives in market are multiple-criteria decision making (MCDM) problem in the presence of many quantitative and qualitative attributes. Therefore, in this paper, an analytic hierarchy process (AHP) is used for machine tool selection problem due to the fact that it has been widely used in evaluating various kinds of MCDM problems in both academic researches and practices. However, due to the vagueness and uncertainty on judgments of the decision-maker(s), the crisp pair wise comparison in the conventional AHP seems to insufficient and imprecise to capture the right judgments of decision-maker(s). That is why; fuzzy number logic is introduced in the pair wise comparison of AHP to make up for this deficiency in the conventional AHP. Shortly, in this study, an intelligent approach is proposed, where both techniques; fuzzy logic and AHP are come together, referred to as fuzzy AHP. First, the fuzzy AHP technique is used to weight the alternatives under multiple attributes; second Benefit/Cost (B/C) ratio analysis is carried out by using both the fuzzy AHP score and procurement cost, of each alternative. The alternative with highest B/C ratio is found out and called as the ultimate machine tool among others. In addition, a case study is also presented to make this approach more understandable for a decision-maker(s).
324 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 article, the authors introduce concepts of entropy of interval valued fuzzy set which is different from the entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, and discuss their relationship between similarity measure and entropy of intervals valued fuzzy sets in detail.
300 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.
01 Jan 2006
TL;DR: 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.
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.
TL;DR: The present paper proposes a new cluster validity measure, named Fuzzy Silhouette, a generalization to the fuzzy case of the Average Silhouettes Width Criterion, originally conceived to assess crisp (non-fuzzy) data partitions, which is designed to improve performance of the original silhouette criterion.
TL;DR: A theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model.
Abstract: Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work
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.
TL;DR: This paper proposes a new method to determine the membership function of the estimates of the parameters and the reliability function of multi-parameter lifetime distributions, and shows the effectiveness of this method with normal and Weibull distributions.
TL;DR: This paper presents, by means of Alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces, a systematic investigation of the interrelationship between fuzzy pre ordered sets, topological Spaces, and fuzzy topology spaces.
TL;DR: Three methods are presented to perform the center of gravity (COG) defuzzification method in the context of linguistic fuzzy models with t-norm-based inference: one well-known method, the discretisation method, and two new methods, the slope-based method and the modified transformation function method.
TL;DR: The purpose of this paper is to define the notion of intuitionistic fuzzy metric spaces due to Kramosil and Michalek, and the well-known fixed point theorems of Banach and Edelstein are extended to intuitionistic fuzzy metric spaces with the help of Grabiec.
Abstract: The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96], we define the notion of intuitionistic fuzzy metric spaces due to Kramosil and Michalek [Kramosil O, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetika 1975;11:326–34]. Further the well-known fixed point theorems of Banach and Edelstein are extended to intuitionistic fuzzy metric spaces with the help of Grabiec [Grabiec M. Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 1988;27:385–9].
TL;DR: The model is developed on the basis of a conceptual framework, which consists of seven major e‐service quality dimensions, including tangibles, reliability, responsiveness, confidence, empathy, quality of information, and integration of communication issues of Web sites.
Abstract: This article presents a quality evaluation model for measuring the performance of hospital Web sites. The model is developed on the basis of a conceptual framework, which consists of seven major e-service quality dimensions, including tangibles, reliability, responsiveness, confidence, empathy, quality of information, and integration of communication issues of Web sites. The dimensions and their associated attributes are first obtained from published articles in the health care and information technology literature and then adapted according to the suggestions of related domain experts. Two multicriteria decision-making methods are used in the evaluation procedure. Determined Web site evaluation dimensions and their relevant attributes are weighted using the Analytic Hierarchy Process (AHP) method. Vagueness in some stages of the evaluation required the incorporation of fuzzy numbers in the assessment process. Both fuzzy and crisp data are then synthesized using the fuzzy PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation) ranking method. The model is applied initially to measure the performance of the Web sites of Turkish hospitals. This study should be of interest to health care and technology practitioners and researchers, as the findings shed light on the further development of performance measurements for hospital Web sites. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1181–1197, 2006.
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.
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.
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.
TL;DR: This paper develops a systematic approach to the assessment of fuzzy association rules by partitioning the data stored in a database into examples of a given rule, counterexamples, and irrelevant data, and evaluation measures are derived from the cardinalities of the corresponding subsets.
Abstract: In order to allow for the analysis of data sets including numerical attributes, several generalizations of association rule mining based on fuzzy sets have been proposed in the literature. While the formal specification of fuzzy associations is more or less straightforward, the assessment of such rules by means of appropriate quality measures is less obvious. Particularly, it assumes an understanding of the semantic meaning of a fuzzy rule. This aspect has been ignored by most existing proposals, which must therefore be considered as ad-hoc to some extent. In this paper, we develop a systematic approach to the assessment of fuzzy association rules. To this end, we proceed from the idea of partitioning the data stored in a database into examples of a given rule, counterexamples, and irrelevant data. Evaluation measures are then derived from the cardinalities of the corresponding subsets. The problem of finding a proper partition has a rather obvious solution for standard association rules but becomes less trivial in the fuzzy case. Our results not only provide a sound justification for commonly used measures but also suggest a means for constructing meaningful alternatives.
TL;DR: Experimental results show that the proposed SVFNN for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions.
Abstract: Fuzzy neural networks (FNNs) for pattern classification usually use the backpropagation or C-cluster type learning algorithms to learn the parameters of the fuzzy rules and membership functions from the training data. However, such kinds of learning algorithms usually cannot minimize the empirical risk (training error) and expected risk (testing error) simultaneously, and thus cannot reach a good classification performance in the testing phase. To tackle this drawback, a support-vector-based fuzzy neural network (SVFNN) is proposed for pattern classification in this paper. The SVFNN combines the superior classification power of support vector machine (SVM) in high dimensional data spaces and the efficient human-like reasoning of FNN in handling uncertainty information. A learning algorithm consisting of three learning phases is developed to construct the SVFNN and train its parameters. In the first phase, the fuzzy rules and membership functions are automatically determined by the clustering principle. In the second phase, the parameters of FNN are calculated by the SVM with the proposed adaptive fuzzy kernel function. In the third phase, the relevant fuzzy rules are selected by the proposed reducing fuzzy rule method. To investigate the effectiveness of the proposed SVFNN classification, it is applied to the Iris, Vehicle, Dna, Satimage, Ijcnn1 datasets from the UCI Repository, Statlog collection and IJCNN challenge 2001, respectively. Experimental results show that the proposed SVFNN for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions.
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.
10 May 2006
TL;DR: The authors generalizes the intuitionistic fuzzy set (IFSFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS), and distinguishes between NS and IFS.
Abstract: In this paper one generalizes the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined.