Showing papers on "Fuzzy number published in 1995"
Book•
01 Jan 1995TL;DR: Fuzzy Sets and Fuzzy Logic is a true magnum opus; it addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic.
Abstract: Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty,
and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic
addresses practically every significant topic in the broad expanse of the union of fuzzy set theory
and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast
territory with impeccable authority, deep insight and a meticulous attention to detail.
To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point
of semantics which relates to the meanings of fuzzy sets and fuzzy logic.
A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The
problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow
sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization
of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous
with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much
broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different
from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is
usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW
In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However,
to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic.
Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely,
that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y
by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability
theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory,
fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network
theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to
fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical
programming. What is gained through fuzzification is greater generality, higher expressive
power, an enhanced ability to model real-world problems and, most importantly, a methodology for
exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,
7,131 citations
01 Mar 1995
TL;DR: After synthesizing a FLS, it is demonstrated that it can be expressed mathematically as a linear combination of fuzzy basis functions, and is a nonlinear universal function approximator, a property that it shares with feedforward neural networks.
Abstract: A fuzzy logic system (FLS) is unique in that it is able to simultaneously handle numerical data and linguistic knowledge. It is a nonlinear mapping of an input data (feature) vector into a scalar output, i.e., it maps numbers into numbers. Fuzzy set theory and fuzzy logic establish the specifics of the nonlinear mapping. This tutorial paper provides a guided tour through those aspects of fuzzy sets and fuzzy logic that are necessary to synthesize an FLS. It does this by starting with crisp set theory and dual logic and demonstrating how both can be extended to their fuzzy counterparts. Because engineering systems are, for the most part, causal, we impose causality as a constraint on the development of the FLS. After synthesizing a FLS, we demonstrate that it can be expressed mathematically as a linear combination of fuzzy basis functions, and is a nonlinear universal function approximator, a property that it shares with feedforward neural networks. The fuzzy basis function expansion is very powerful because its basis functions can be derived from either numerical data or linguistic knowledge, both of which can be cast into the forms of IF-THEN rules. >
2,024 citations
TL;DR: A genetic-algorithm-based method for selecting a small number of significant fuzzy if-then rules to construct a compact fuzzy classification system with high classification power is proposed.
Abstract: This paper proposes a genetic-algorithm-based method for selecting a small number of significant fuzzy if-then rules to construct a compact fuzzy classification system with high classification power. The rule selection problem is formulated as a combinatorial optimization problem with two objectives: to maximize the number of correctly classified patterns and to minimize the number of fuzzy if-then rules. Genetic algorithms are applied to this problem. A set of fuzzy if-then rules is coded into a string and treated as an individual in genetic algorithms. The fitness of each individual is specified by the two objectives in the combinatorial optimization problem. The performance of the proposed method for training data and test data is examined by computer simulations on the iris data of Fisher. >
765 citations
01 Apr 1995
TL;DR: A fundamental theoretical question on why fuzzy control has such a good performance for a wide variety of practical problems is considered, and it is proved that for each fixed fuzzy logic belonging to a wide class of fuzzy logics, the fuzzy logic control systems using these two and any method of defuzzification are capable of approximating any real continuous function on a compact set to arbitrary accuracy.
Abstract: In this paper, we consider a fundamental theoretical question on why does fuzzy control have such a good performance for a wide variety of practical problems. We try to answer this fundamental question by proving that for each fixed fuzzy logic belonging to a wide class of fuzzy logics, and for each fixed type of membership function belonging to a wide class of membership functions, the fuzzy logic control systems using these two and any method of defuzzification are capable of approximating any real continuous function on a compact set to arbitrary accuracy. On the other hand, this result can be viewed as an existence theorem of an optimal fuzzy logic control system for a wide variety of problems. >
625 citations
TL;DR: A method is presented for tuning fuzzy control rules by genetic algorithms to make the fuzzy logic control systems behave as closely as possible to the operator or expert behavior in a control process.
381 citations
TL;DR: The concepts of correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets are introduced and their first properties are studied and two decomposition theorems of the correlation are introduced.
371 citations
TL;DR: The aim of this paper is to show how to automatically build this fuzzy controller, and the proposed design methodology is detailed for the synthesis of a Sugeno or Mamdani type fuzzy controller precisely equivalent to a given PI controller.
Abstract: It has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy. In particular, any given linear control can be achieved with a fuzzy controller for a given accuracy. The aim of this paper is to show how to automatically build this fuzzy controller. The proposed design methodology is detailed for the synthesis of a Sugeno or Mamdani type fuzzy controller precisely equivalent to a given PI controller. The main idea is to equate the output of the fuzzy controller with the output of the PI controller at some particular input values, called modal values. The rule base and the distribution of the membership functions can thus be deduced. The analytic expression of the output of the generated fuzzy controller is then established. For Sugeno-type fuzzy controllers, precise equivalence is directly obtained. For Mamdani-type fuzzy controllers, the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values. The usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed. >
239 citations
Book•
01 Jan 1995
TL;DR: Fuzzy sets: Basic Notions Examples of Fuzzy Sets Operations with FuzzY Sets t-norm-Based Operations Fuzzi Numbers and Their Arithmetic Fuzzifified Relationships: Fuzzed Relations Properties of FBuzzy Relations Fuzzing Relationships between Variables
Abstract: Fuzzy Sets: Basic Notions Examples of Fuzzy Sets Operations with Fuzzy Sets t-norm-Based Operations Fuzzy Numbers and Their Arithmetic Fuzzifified Relationships: Fuzzy Relations Properties of Fuzzy Relations Fuzzy Relationships between Variables Fuzzy Programming Linguistic Variables: The Notion of a Linguistic Variable Fuzzy Control Relational Equations and Fuzzy Control Approximate Reasoning Examples for Applications of Fuzzy Controllers Measure Theory and Fuzzy Sets: Fuzzy Measures for Crisp Sets Fuzzy Measures for Fuzzy Sets Fuzziness and Probability Some Applications Fuzziness Measures Fuzzy Data Analysis: Data and Their Analysis Qualitative Data Analysis Quantitative Data Analysis Evaluation of Methods.
237 citations
Proceedings Article•
01 Jan 1995
TL;DR: The connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets is elucidated and it is shown that the indistinguishedability inherent to fuzzy sets can be computed and that this indistinguishesability cannot be overcome in approximate reasoning.
Abstract: Fuzzy set theory is based on a `fuzzification' of the predicate in (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL-monoids that can be understood as a generalization of MV-algebras. Residuation is a basic concept in GL-monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval.
222 citations
TL;DR: The author provides a theoretical justification for the fuzzy identifiers by proving that they are capable of following the output of a general nonlinear dynamic system to arbitrary accuracy in any finite time interval.
Abstract: Uses fuzzy systems as identifiers for nonlinear dynamic systems. The author provides a theoretical justification for the fuzzy identifiers by proving that they are capable of following the output of a general nonlinear dynamic system to arbitrary accuracy in any finite time interval. The fuzzy identifiers are constructed from a set of adaptable fuzzy IF-THEN rules and can combine both numerical information (in the form of input-output pairs obtained by exciting the system with an input signal and measuring the corresponding outputs) and linguistic information (in the form of IF-THEN rules about the behavior of the system in terms of vague and fuzzy words) into their designs in a uniform fashion. The author develops two fuzzy identifiers. The first one is designed through the following four steps: 1) define some fuzzy sets in the state space U/spl sub/R/sup n/ of the system; these fuzzy sets do not change; 2) construct fuzzy rule bases of the fuzzy identifier which comprise rules whose IF parts constitute all the possible combinations of the fuzzy sets defined in 1); 3) design the fuzzy systems in the fuzzy identifier based on the fuzzy rule bases of 2); and 4) develop an adaptive law for the free parameters in the fuzzy identifier. The second fuzzy identifier is designed in a similar way as the first one except that: a) the parameters characterizing the fuzzy sets in the state space change during the adaptation procedure; and b) the fuzzy systems and the adaptive law are different. The author proves that: 1) both fuzzy identifiers are globally stable in the sense that all variables in the fuzzy identifiers are uniformly bounded, and 2) under some conditions the identification errors of both fuzzy identifiers converge to zero asymptotically. Finally, the author simulates the fuzzy identifiers for identifying the chaotic glycolytic oscillator, and the results show that: 1) the fuzzy identifiers can approximate the chaotic system at a reasonable speed and accuracy without using any linguistic information, and 2) by incorporating some fuzzy linguistic IF-THEN rules about the behavior of the system into the fuzzy identifiers, the speed and accuracy of the fuzzy identifiers are greatly improved. >
Book•
30 Sep 1995
TL;DR: The Basics of FBuzzy Set Theory Fuzzy Phenomena and FuzzY Concepts Naive Thoughts of FZZy Sets Definition of Fuzzi Sets Basic Operations of F Buzzy Sets The Resolution Theorem A Representation Theorem Extension Principles References Factor Spaces.
Abstract: The Basics of Fuzzy Set Theory Fuzzy Phenomena and Fuzzy Concepts Naive Thoughts of Fuzzy Sets Definition of Fuzzy Sets Basic Operations of Fuzzy Sets The Resolution Theorem A Representation Theorem Extension Principles References Factor Spaces What are "Factors"? The State Space of Factors Relations and Operations Between Factors Axiomatic Definition of Factor Spaces Describing Concepts in a Factor Space References The Basics of Fuzzy Decision-Making Feedback Extension and Its Applications Feedback Ranks and Degrees of Coincidence Equivalence Between Sufficient Factors and Coincident Factors How to Improve the Precision of a Feedback Extension Representation of the Intention of a Concept Basic Forms of Fuzzy Decision-Making Limitations of the Weighted Average Formula References Determination of Membership Functions A General Method for Determining Membership Functions The Three-Phase Method The Incremental Method The Multiphase Fuzzy Statistical Method The Method of Comparisons The Absolute Comparison Method The Set-Valued Statistical Iteration Method Ordering by Precedence Relations The Relative Comparison Method and the Mean Pair-Wise Comparison Method References Multifactorial Analysis Background of the Problem Multifactorial Functions Axiomatic Definition of Additive Standard Multifactorial Functions Properties of ASMm-funcs Generations of ASMm-funcs Applications of ASMm-funcs in Fuzzy Decision-Making A General Model of Multifactorial Decision-Making References Variable Weights Analysis Defining the Problem An Empirical Variable Weight Formula Principles of Variable Weights References Multifactorial Decision-Making with Multiple Objectives Background and Models Multifactorial Evaluation The Multifactorial Evaluation Approach to the Classification of Quality Incomplete Multifactorial Evaluation Multi-Level Multifactorial Evaluation An Application of Multifactorial Evaluation in Textile Engineering References Set-Valued Statistics and Degree Analysis Fuzzy Statistics and Random Sets The Falling Shadow of Random Sets Set-Valued Statistics Degree Analysis Random and Set-Valued Experiments A Mathematical Model for Employee Evaluation References Refinements of Fuzzy Operators The Axiomatic Structure of Zadeh's Operators Common Fuzzy Operators Generalized Fuzzy Operators The Strength of Fuzzy Operators "AND" and "OR" Fuzzy Operators Based on the Falling Shadow Theory References Multifactorial Decision Based on Theory of Evidence A Brief Introduction to Theory of Evidence Composition of Belief Measures Multifactorial Evaluation Based on the Theory of Evidence Two Special Types of Composition Functions The Maximum Principle for Multiple Object Evaluations References
TL;DR: This paper proposes to design a fuzzy controller with the fuzzy sliding surface, and the output of the fuzzy controller is inferred by the proper compositional rule of inference, and shows the stability of this fuzzy control system and the boundedness of the tracking error by using the Lyapunov theory.
TL;DR: The present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures.
Abstract: Many engineering systems are too complex to be defined in precise mathematical terms. They often contain information and features that are vague, imprecise, qualitative, linguistic, or incomplete. The traditional deterministic and probabilistic techniques are not adequate to analyze such systems. This paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems. The development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation, manipulation, and solution of the finite element equations. Simple stress analysis problems involving vaguely defined geometry, material properties, external loads, and boundary conditions are solved to establish and to illustrate the new procedure. The approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information. If complete data are known, the method handles the information similar to that of a probabilistic approach. The present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures.
01 Jan 1995
TL;DR: The position articulated in this article is that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real-world settings, and it needs an infusion of concepts and techniques drawn from fuzzy logic to enhance its effectiveness.
Journal Article•
TL;DR: In this article, the reliability of each system component is represented by a triangular fuzzy number, and the proposed method uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated interval arithmetic operations.
TL;DR: This paper argued that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real-world settings and proposed an infusion of concepts and techniques drawn from fuzzy logic, such as a linguistic variable and the calculus of fuzzy if-then rules.
Abstract: The relationship between probability theory and fuzzy logic has long been an object of discussion and some controversy. The position articulated in this article is that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real-world settings. To enhance its effectiveness, probability theory needs an infusion of concepts and techniques drawn from fuzzy logic—especially the concept of a linguistic variable and the calculus of fuzzy if–then rules. In the final analysis, probability theory and fuzzy logic are complementary rather than competitive.
TL;DR: This paper presents a method for combining multiple networks based on fuzzy logic, especially the fuzzy integral, which non-linearly combines objective evidence, in the form of a network output, with subjective evaluation of the importance of the individual neural networks.
Abstract: Multiplayer feedforward networks trained by minimizing the mean squared error and by using a one of c teaching function yield network outputs that estimate posterior class probabilities. This provides a sound basis for combining the results from multiple networks to get more accurate classification. This paper presents a method for combining multiple networks based on fuzzy logic, especially the fuzzy integral. This method non-linearly combines objective evidence, in the form of a network output, with subjective evaluation of the importance of the individual neural networks. The experimental results with the recognition problem of on-line handwriting characters show that the performance of individual networks could be improved significantly. >
TL;DR: This paper proposes an architecture of fuzzy neural networks with triangular fuzzy weights that can handle fuzzy input vectors as well as real input vectors and derives a learning algorithm from the cost function for adjusting three parameters of each triangular fuzzy weight.
Journal Article•
TL;DR: In this paper, the concepts of statistically convergent and statis tically Cauchy sequences of fuzzy numbers have been introduced and discussed and also /(p)-spaces of sequences of fuzzy numbers have be introduced.
Abstract: In this paper, the concepts of statistically convergent and statis tically Cauchy sequences of fuzzy numbers have been introduced and discussed. Also /(p)-spaces of sequences of fuzzy numbers have been introduced.
TL;DR: The proposed method constructs an optimal structure of the simplified fuzzy inference that minimizes model errors and the number of the membership functions to grasp nonlinear behavior of power system short-term loads.
Abstract: This paper proposes an optimal fuzzy inference method for short-term load forecasting. The proposed method constructs an optimal structure of the simplified fuzzy inference that minimizes model errors and the number of the membership functions to grasp nonlinear behavior of power system short-term loads. The model is identified by simulated annealing and the steepest descent method. The proposed method is demonstrated in examples.
01 Jan 1995
TL;DR: This paper extends the method for extracting fuzzy rules directly from numerical input-output data for pattern classification to function approximation, and compares the approximation performance of the fuzzy system with the function approximation approach based on neural networks.
Abstract: In our previous work (1993) we developed a method for extracting fuzzy rules directly from numerical input-output data for pattern classification. In this paper we extend the method to function approximation. For function approximation, first, the universe of discourse of an output variable is divided into multiple intervals, and each interval is treated as a class. Then the same as for pattern classification, using the input data for each interval, fuzzy rules are recursively defined by activation hyperboxes which show the existence region of the data for the interval and inhibition hyperboxes which inhibit the existence region of data for that interval. The approximation accuracy of the fuzzy system derived by this method is empirically studied using an operation learning application of a water purification plant. Additionally, we compare the approximation performance of the fuzzy system with the function approximation approach based on neural networks. >
TL;DR: This paper proposes a new supervised self-tuning fuzzy modeling, which consist of some membership function expressed by the radial basis function with insensitive region with descent method, which is carried out by the genetic algorithms.
TL;DR: PID controllers can be realized by fuzzy control methods of product-sum-gravity method and simplified fuzzy reasoning method by extending membership functions of antecedent parts of fuzzy rules and extrapolative reasoning can be executed by the product- Summers gravity method.
TL;DR: An extension of the constraint-based approach to job-shop scheduling that accounts for the flexibility of temporal constraints and the uncertainty of operation durations is proposed, including fuzzy extensions of well-known look-ahead schemes.
Abstract: This paper proposes an extension of the constraint-based approach to job-shop scheduling, that accounts for the flexibility of temporal constraints and the uncertainty of operation durations. The set of solutions to a problem is viewed as a fuzzy set whose membership function reflects preference. This membership function is obtained by an egalitarist aggregation of local constraint-satisfaction levels. Uncertainty is qualitatively described in terms of possibility distributions. The paper formulates a simple mathematical model of job-shop scheduling under preference and uncertainty, relating it to the formal framework of constraint-satisfaction problems in artificial intelligence. A combinatorial search method that solves the problem is outlined, including fuzzy extensions of well-known look-ahead schemes.
Proceedings Article•
01 Jan 1995
TL;DR: This paper proposes to compute a function as a suitable weighted mean of the distances between the \ac s of the fuzzy numbers, which replaces the classical (x-y)² on the real line.
Abstract: In the course of the studies on fuzzy regression analysis, we encountered the problem of introducing a distance between fuzzy numbers, which replaces the classical (x - y)2 on the real line. Our proposal is to compute such a function as a suitable weighted mean of the distances between the a-cuts of the fuzzy numbers. The main difficulty is concerned with the definition of the distance between intervals, since the current definitions present some disadvantages which are undesirable in our context. In this paper we describe an approach which removes such drawbacks.
01 Mar 1995
TL;DR: F fuzzy logic, fuzzy arithmetic and linguistic variables are applied to the analysis of system structures, fault trees, event trees, the reliability of degradable systems, and the assessment of system criticality based on the severity of a failure and its probability of occurrence.
Abstract: The analysis of system reliability often requires the use of subjective-judgments, uncertain data, and approximate system models. By allowing imprecision and approximate analysis fuzzy logic provides an effective tool for characterizing system reliability in these circumstances; it does not force precision where it is not possible. Here we apply the main concepts of fuzzy logic, fuzzy arithmetic and linguistic variables to the analysis of system structures, fault trees, event trees, the reliability of degradable systems, and the assessment of system criticality based on the severity of a failure and its probability of occurrence. >
TL;DR: An auto-tuning method for the fuzzy neural network by genetic algorithm and delta rule is proposed to construct minimal and optimal structure of the fuzzy model.
Book•
01 Sep 1995
TL;DR: Fuzzy logic and genetic algorithms learning fuzzy and hybrid systems decision and aggregation techniques fuzzy logic in databases foundations of fuzzy logic applications of fuzzy sets.
Abstract: Fuzzy logic and genetic algorithms learning fuzzy and hybrid systems decision and aggregation techniques fuzzy logic in databases foundations of fuzzy logic applications of fuzzy sets.
20 Mar 1995
TL;DR: The author proposes a new and general decision making method for evaluating weapon systems using fuzzy AHP based on entropy weight based on symmetric triangular fuzzy numbers.
Abstract: The performance evaluation of weapon systems is a multiple criteria decision making problem. The descriptions and judgements on weapon systems are usually linguistic and fuzzy. The traditional methods using the analytic hierarchy process (AHP) are mainly used in crisp (non-fuzzy) decision applications with a very unbalanced scale of judgements. To overcome these problems the author proposes a new and general decision making method for evaluating weapon systems using fuzzy AHP based on entropy weight. The authors uses symmetric triangular fuzzy numbers . >