Showing papers on "Fuzzy number published in 1999"
01 Jan 1999
TL;DR: A new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by Saaty (1980) in the analytic hierarchy process is presented.
Abstract: We present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by Saaty (1980) in the analytic hierarchy process. We test our new procedure in two cases where there are formulas for the crisp weights. An example is presented where there are five criteria and three alternatives.
2,789 citations
Book•
19 Oct 1999
TL;DR: The basic definitions and properties of the Intuitionistic Fuzzy Sets (IFSs) are introduced in the book and readers will find discussions on some of the IFS extensions (for example, interval-values IFSs, temporal I FSs and others) and applications.
Abstract: The basic definitions and properties of the Intuitionistic Fuzzy Sets (IFSs) are introduced in the book. The IFSs are substantial extensions of the ordinary fuzzy sets. IFSs are objects having degrees of membership and of non-membership, such that their sum is exactly 1. The most important property of IFS not shared by the fuzzy sets is that modal-like operators can be defined over IFSs. The IFSs have essentially higher describing possibilities than fuzzy sets. In this book, readers will find discussions on some of the IFS extensions (for example, interval-values IFSs, temporal IFSs and others) and applications (e.g. intuitionistic fuzzy expert systems, intuitionistic fuzzy neural networks, intuitionistic fuzzy systems, intuitionistic fuzzy generalized nets, and other).
1,837 citations
TL;DR: The computational theory of perceptions (CTP) as mentioned in this paper is a methodology for reasoning and computing with perceptions rather than measurements, where words play the role of labels of perceptions and, more generally, perceptions are expressed as propositions in a natural language.
Abstract: Discusses a methodology for reasoning and computing with perceptions rather than measurements. An outline of such a methodology-referred to as a computational theory of perceptions is presented in this paper. The computational theory of perceptions, or CTP for short, is based on the methodology of CW. In CTP, words play the role of labels of perceptions and, more generally, perceptions are expressed as propositions in a natural language. CW-based techniques are employed to translate propositions expressed in a natural language into what is called the Generalized Constraint Language (GCL). In this language, the meaning of a proposition is expressed as a generalized constraint, N is R, where N is the constrained variable, R is the constraining relation and isr is a variable copula in which r is a variable whose value defines the way in which R constrains S. Among the basic types of constraints are: possibilistic, veristic, probabilistic, random set, Pawlak set, fuzzy graph and usuality. The wide variety of constraints in GCL makes GCL a much more expressive language than the language of predicate logic. In CW, the initial and terminal data sets, IDS and TDS, are assumed to consist of propositions expressed in a natural language. These propositions are translated, respectively, into antecedent and consequent constraints. Consequent constraints are derived from antecedent constraints through the use of rules of constraint propagation. The principal constraint propagation rule is the generalized extension principle. The derived constraints are retranslated into a natural language, yielding the terminal data set (TDS). The rules of constraint propagation in CW coincide with the rules of inference in fuzzy logic. A basic problem in CW is that of explicitation of N, R, and r in a generalized constraint, X is R, which represents the meaning of a proposition, p, in a natural language.
1,453 citations
Book•
31 Aug 1999
TL;DR: Fuzzy Logic: What, Why, for Which?
Abstract: Preface. 1. Fuzzy Logic: What, Why, for Which? 2. Algebraic Structures for Logical Calculi. 3. Logical Calculi and Model Theory. 4. Fuzzy Logic in Narrow Sense. 5. Functional Systems in Fuzzy Logic Theories. 6. Fuzzy Logic in Broader Sense. 7. Topoi and Categories of Fuzzy Sets. 8. Few Historical and Concluding Remarks. References. Index.
898 citations
TL;DR: This paper focuses on semantic approaches to approximate reasoning based on fuzzy sets, commonly exemplified by the generalized modus ponens, but also considers applications to current topics in Artificial Intelligence such as default reasoning and qualitative process modeling.
804 citations
TL;DR: The interval-valued possibilistic mean value is defined and the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory.
763 citations
TL;DR: The result shows that the approach developed is simple and comprehensible in concept, efficient in computation, and robust and flexible in modeling the human evaluation process, thus making it of general use for solving practical MA problems.
699 citations
TL;DR: The basic theory of the triangular fuzzy number is proved and the formulation of comparing the triangular fuzziness number's size is improved, and a practical example on petroleum prospecting is introduced.
472 citations
01 Oct 1999
TL;DR: In this article, a fuzzy genetics-based machine learning method for multidimensional pattern classification problems with continuous attributes is presented, where each fuzzy if-then rule is handled as an individual, and a fitness value is assigned to each rule.
Abstract: We examine the performance of a fuzzy genetics-based machine learning method for multidimensional pattern classification problems with continuous attributes In our method, each fuzzy if-then rule is handled as an individual, and a fitness value is assigned to each rule Thus, our method can be viewed as a classifier system In this paper, we first describe fuzzy if-then rules and fuzzy reasoning for pattern classification problems Then we explain a genetics-based machine learning method that automatically generates fuzzy if-then rules for pattern classification problems from numerical data Because our method uses linguistic values with fixed membership functions as antecedent fuzzy sets, a linguistic interpretation of each fuzzy if-then rule is easily obtained The fixed membership functions also lead to a simple implementation of our method as a computer program The simplicity of implementation and the linguistic interpretation of the generated fuzzy if-then rules are the main characteristic features of our method The performance of our method is evaluated by computer simulations on some well-known test problems While our method involves no tuning mechanism of membership functions, it works very well in comparison with other classification methods such as nonfuzzy machine learning techniques and neural networks
455 citations
TL;DR: Benefits of the methodology are illustrated in the process of classifying the iris data set and possible extensions of the methods are summarized.
Abstract: Evolutionary fuzzy systems are discussed in which the membership function shapes and types and the fuzzy rule set including the number of rules inside it are evolved using a genetic (evolutionary) algorithm. In addition, the genetic parameters (operators) of the evolutionary algorithm are adapted via a fuzzy system. Benefits of the methodology are illustrated in the process of classifying the iris data set. Possible extensions of the methods are summarized.
440 citations
Book•
18 Oct 1999
TL;DR: Fuzzy Relations: Solvability of Fuzzy Relation Equations On FuzzY Similiarity Relations and Approximate Reasoning MaximalSimiliarity and Fuzzed Reasoning Exercises.
Abstract: Residuated Lattices: Lattices and Equivalence Relations Lattice Filters Residuated Lattices BL-Algebras Exercises.- MV-Algebras: MV-Algebras and Wajsberg Algebras Complete MV-Algebras Pseudo-Boolean Algebras Exercises.- Fuzzy Propositional Logic: Semantics of Fuzzy Propositional Logic Exercises.- Fuzzy Relations: Solvability of Fuzzy Relation Equations On Fuzzy Similiarity Relations Fuzzy Similiarity and Approximate Reasoning Maximal Similiarity and Fuzzy Reasoning Exercises.- Solutions to Exercises.
TL;DR: The behaviour of a general reasoning method is described, six proposals for this general model are analyzed, and a method to learn the parameters of these FRMs by means of Genetic Algorithms is presented, adapting the inference mechanism to the set of rules.
TL;DR: A new method for evaluating weapon systems by analytical hierarchy process (AHP) based on linguistic variable weight, which possesses intuition, in accord with human rethinking-model, and is close to humanized uncertainty of language expression.
TL;DR: Stability theorems for a discrete-time system as well as for a continuous time system are given and a brief survey on the stability issues of fuzzy control systems is given.
Abstract: Addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents. It first presents two canonical forms of a fuzzy system: a parametric expression and a state-space expression. A fuzzy system with singleton consequents is found to be a piecewise-polytopic-affine system. Then the paper gives stability theorems for a discrete-time system as well as for a continuous time system. It also gives a brief survey on the stability issues of fuzzy control systems.
01 Jun 1999
TL;DR: The rule decoupled and one-input rule structures proposed in this paper provide greater flexibility and better functional properties than the conventional fuzzy PHD structures.
Abstract: The majority of the research work on fuzzy PID controllers focuses on the conventional two-input PI or PD type controller proposed by Mamdani (1974). However, fuzzy PID controller design is still a complex task due to the involvement of a large number of parameters in defining the fuzzy rule base. This paper investigates different fuzzy PID controller structures, including the Mamdani-type controller. By expressing the fuzzy rules in different forms, each PLD structure is distinctly identified. For purpose of analysis, a linear-like fuzzy controller is defined. A simple analytical procedure is developed to deduce the closed form solution for a three-input fuzzy inference. This solution is used to identify the fuzzy PID action of each structure type in the dissociated form. The solution for single-input-single-output nonlinear fuzzy inferences illustrates the effect of nonlinearity tuning. The design of a fuzzy PID controller is then treated as a two-level tuning problem. The first level tunes the nonlinear PID gains and the second level tunes the linear gains, including scale factors of fuzzy variables. By assigning a minimum number of rules to each type, the linear and nonlinear gains are deduced and explicitly presented. The tuning characteristics of different fuzzy PID structures are evaluated with respect to their functional behaviors. The rule decoupled and one-input rule structures proposed in this paper provide greater flexibility and better functional properties than the conventional fuzzy PHD structures.
TL;DR: This work and its conclusions may narrow the gap between the theoretical research on FDEs and FIEs and the practical applications already existing in the design of various fuzzy dynamical systems.
TL;DR: It is demonstrated how information about the noise in the training data can be incorporated into a type-2 FLS, which can be used to obtain bounds within which the true (noisefree) output is likely to lie.
TL;DR: The concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix and the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative.
TL;DR: It is proved that the hierarchical fuzzy systems are universal approximators and the sensitivity of the fuzzy system output with respect to small perturbations in its inputs is analyzed.
Abstract: In this letter, the hierarchical fuzzy systems are analyzed and designed. In the analysis part, we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs. In the design part, we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input-output pairs. The algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy.
TL;DR: A scheme based on the classical Euler method is discussed in detail, and this is followed by a complete error analysis, illustrated by solving several linear and nonlinear fuzzy Cauchy problems.
TL;DR: In this paper, a qualitative description and corresponding quantitative presentation of the first four phases that focus on the customer input are given, and Fuzzy and entropy methods are then proposed to obtain the final importance ratings of the customer needs in step 4.
Abstract: A typical quality function deployment (QFD) consists of four phases. Phase I (sometimes called house of quality), which translates customer needs for a product into technical measures (i.e. product design specifications), can be considered as a nine-step process. A concise and applicable qualitative description and the corresponding quantitative presentation of the first four steps that focus on the customer input are given. Fuzzy and entropy methods are then proposed to obtain the final importance ratings of the customer needs in Step 4. These ratings form the basis for the company to make the product more attractive to customers and thus more competitive. The proposed methods make fuller use of customer input to reveal the relative importance of their needs and are easy to apply. Step 1 is to collect and structure the customer needs. The fuzzy method is used in Step 2 to convert the customers' importance assessments of the needs to fuzzy numbers, and the relative importance ratings of the customer needs...
TL;DR: A new fuzzy arithmetic is defined and applied to fuzzy linear equations and fuzzy calculus based on a new parametric form presented in this paper.
01 Jan 1999
TL;DR: The result shows that the approach developed is simple and comprehensible in concept, efficient in computation, and robust and flexible in modeling the human evaluation process, thus making it of general use for solving practical MA problems.
Abstract: Presents an approach for solving qualitative multicriteria analysis (MA) problems using fuzzy pairwise comparison. Fuzzy numbers are used to approximate the decision-maker's (DM's) subjective assessments in assessing alternative performance and criteria importance. The concept of fuzzy extent analysis is applied for solving the reciprocal judgement matrices. To avoid the complex and unreliable process of comparing fuzzy utilities, the /spl alpha/-cut technique is applied to transform the fuzzy performance matrix into an interval matrix. Incorporated with the DM's attitude towards risk, an overall performance index is obtained for each alternative across all criteria in line with the ideal solution concept. An empirical study of a tender selection problem in Australia is conducted. The result shows that the approach developed is simple and comprehensible in concept, efficient in computation, and robust and flexible in modeling the human evaluation process, thus making it of general use for solving practical MA problems.
TL;DR: This paper discusses the voting by multiple fuzzy if-then rules, which is used as a fuzzy reasoning method for classifying input patterns in a single fuzzy rule-based classification system, and compares it with other classification methods such as neural networks and statistical techniques by computer simulations on some well-known test problems.
TL;DR: The present work characterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO).
Abstract: Introduces a singular value-based method for reducing a given fuzzy rule set. The method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membership functions to form new ones for the reduced set. The present work characterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO). Algorithms to preserve the SN and NN conditions in the new membership functions are presented. Preservation of the NO condition relates to a high-dimensional convex hull problem and is not always feasible in which case a closed-to-NO solution may be sought. The proposed method is applicable regardless of the adopted inference paradigms. With product-sum-gravity inference and singleton support fuzzy rule base, output errors between the full and reduced fuzzy set are bounded by the sum of the discarded singular values. The work discusses three specific applications of fuzzy reduction: fuzzy rule base with singleton support, fuzzy rule base with nonsingleton support (which includes the case of missing rules), and the Takagi-Sugeno-Kang (TSK) model. Numerical examples are presented to illustrate the reduction process.
TL;DR: A fuzzy model associated with the solution algorithm is proposed on the basis of an α-level weighted, fuzzy preference relation that addresses the problems of decision making with multiple judge, multiple criteria in a fuzzy environment.
Abstract: This paper investigates the problems of decision making with multiple judge, multiple criteria in a fuzzy environment, where the performance of alternatives and the importance of criteria are imprecisely defined and represented by fuzzy sets. A fuzzy model associated with the solution algorithm is proposed on the basis of an α-level weighted, fuzzy preference relation. A numerical example is solved for illustration.
TL;DR: A new definition of the relative position between two objects in a fuzzy set framework is proposed, based on a morphological and fuzzy pattern-matching approach, and consists of comparing an object to a fuzzy landscape representing the degree of satisfaction of a directional relationship to a reference object.
Abstract: In order to cope with the ambiguity of spatial relative position concepts, we propose a new definition of the relative position between two objects in a fuzzy set framework. This definition is based on a morphological and fuzzy pattern-matching approach, and consists of comparing an object to a fuzzy landscape representing the degree of satisfaction of a directional relationship to a reference object. It has good formal properties, it is flexible, it fits the intuition, and it can be used for structural pattern recognition under imprecision. Moreover, it also applies in 3D and for fuzzy objects issued from images.
TL;DR: The method utilizes fuzzy ratio scales 1 , 3 , 5 , 7 , 9 (the goal is normalizing heterogeneity into homogeneity) to indicate the relative strength of the factors in the corresponding criteria to determine the best weapon selection by ranking fuzzy numbers.
01 Dec 1999
TL;DR: It is verified that a FC(3) fuzzy system works very well both, for training and test driving situations, especially when the training data are insufficient, and is applied to the design of a distance controller for cars.
Abstract: Sophisticated fuzzy rule systems are supposed to be flexible, complete, consistent and compact (FC/sup 3/). Flexibility, and consistency are essential for fuzzy systems to exhibit an excellent performance and to have a clear physical meaning, while compactness is crucial when the number of the input variables increases. However, the completeness and consistency conditions are often violated if a fuzzy system is generated from data collected from real world applications. A systematic design paradigm is proposed using evolution strategies. The structure of the fuzzy rules, which determines the compactness of the fuzzy systems, is evolved along with the parameters of the fuzzy systems. Special attention has been paid to the completeness and consistency of the rule base. The completeness is guaranteed by checking the completeness of the fuzzy partitioning of input variables and the completeness of the rule structure. An index of inconsistency is suggested with the help of a fuzzy similarity which can prevent the algorithm from generating rules that seriously contradict with each other or with the heuristic knowledge. In addition, soft T-norm and BADD defuzzification are introduced and optimized to increase the flexibility of the fuzzy system. The proposed approach is applied to the design of a distance controller for cars. It is verified that a FC/sup 3/ fuzzy system works very well both, for training and test driving situations, especially when the training data are insufficient.