Showing papers on "Membership function published in 1996"

Fuzzy sets

Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

Fuzzy Multiple Objective Decision Making: Methods And Applications

Abstract: 1 Introduction.- 1.1 Objectives of This Study.- 1.2 (Fuzzy) Multiple Objective Decision Making.- 1.3 Classification of (Fuzzy) Multiple Objective Decision Making.- 1.4 Applications of (Fuzzy) Multiple Objective Decision Making.- 1.5 Literature Survey.- 1.6 Fuzzy Sets.- 2 Multiple Objective Decision Making.- 2.1 Introduction.- 2.2 Goal Programming.- 2.2a A Portfolio Selection Problem.- 2.2b An Audit Sampling Problem.- 2.3 Fuzzy Programming.- 2.3.1 Max-Min Approach.- 2.3.1a A Trade Balance Problem.- 2.3.1b A Media Selection Problem.- 2.3.2 Augmented Max-Min Approach.- Example.- 2.3.2a A Trade Balance Problem.- 2.3.2b A Logistics Planning Model.- 2.3.3 Parametric Approach.- Example.- 2.4 Global Criterion Approach.- 2.4.1 Global Criterion Approach.- 2.4.1a A Nutrition Problem.- 2.4.2 TOPSIS for MODM.- 2. .2a A Water Quality Management Problem.- 2.5 Interactive Multiple Objective Decision Making.- 2.5.1 Optimal System Design.- 2.5.1a A Production Planning Problem.- 2.5.2 KSU-STEM.- 2.5.2a A Nutrition Problem.- 2.5.2b A Project Scheduling Problem.- 2.5.3 ISGP-II.- 2.5.3a A Nutrition Problem.- 2.5.3b A Bank Balance Sheet Management Problem.- 2.5.4 Augmented Min-Max Approach.- 2.5.4a A Water Pollution Control Problem.- 2.6 Multiple Objective Linear Fractional Programming.- 2.6.1 Luhandjula's Approach.- Example.- 2.6.2 Lee and Tcha's Approach.- 2.6.2a A Financial Structure Optimization Problem.- 2.7 Multiple Objective Geometric Programming.- Example.- 2.7a A Postal Regulation Problem.- 3 Fuzzy Multiple Objective Decision Making.- 3.1 Fuzzy Goal Programming.- 3.1.1 Fuzzy Goal Programming.- 3.1.1a A Production-Marketing Problem.- 3.1.1b An Optimal Control Problem.- 3.1.1c A Facility Location Problem.- 3.1.2 Preemptive Fuzzy Goal Programming.- Example: The Production-Marketing Problem.- 3.1.3 Interpolated Membership Function.- 3.1.3.1 Hannan's Method.- Example: The Production-Marketing Problem.- 3.1.3.2 Inuiguchi, Ichihashi and Kume's Method.- Example: The Trade Balance Problem.- 3.1.3.3 Yang, Ignizio and Kim's Method.- Example.- 3.1.4 Weighted Additive Model.- 3.1.4.1 Crisp Weights.- 3.1.4.1a Maximin Approach.- Example: The Production-Marketing Problem.- 3.1.4.1b Augmented Maximin Approach.- 3.1.4.1c Supertransitive Approximation.- Example: The Production-Marketing Problem.- 3.1.4.2 Fuzzy Weights.- Example: The Production-Marketing Problem.- 3.1.5 A Preference Structure on Aspiration Levels.- Example: The Production-Marketing Problem.- 3.1.6 Nested Priority.- 3.1.6a A Personnel Selection Problem.- 3.2 Fuzzy Global Criterion.- Example.- 3.3 Interactive Fuzzy Multiple Objective Decision Making.- 3.3.1 Werners's Method.- Example: The Trade Balance Problem.- 3.3.1a An Aggregate Production Planning Problem.- 3.3.2 Lai and Hwang's Method.- 3.3.3 Leung's Method.- Example.- 3.3.4 Fabian, Ciobanu and Stoica's Method.- Example.- 3.3.5 Sasaki, Nakahara, Gen and Ida's Method.- Example.- 3.3.6 Baptistella and Ollero's Method.- 3.3.6a An Optimal Scheduling Problem.- 4 Possibilistic Multiple Objective Decision Making.- 4.1 Introduction.- 4.1.1 Resolution of Imprecise Objective Functions.- 4.1.2 Resolution of Imprecise Constraints.- 4.2 Possibilistic Multiple Objective Decision Making.- 4.2.1 Tanaka and His Col1eragues' Methods.- Example.- 4.2.1.1 Possibilistic Regression.- Example 1.- Example 2.- 4.2.1.2 Possibilistic Group Method of Data Handling.- Example 28.- 4.2.2 Lai and Hwang's Method.- 4.2.3 Negi's Method.- Example.- 4.2.4 Luhandjula's Method.- Example.- 4.2.5 Li and Lee's Method.- Example.- 4.2.6 Wierzchon's Method.- 4.3 Interactive Methods for PMODM.- 4.3.1 Sakawa and Yano's Method.- Example.- 4.3.2 Slowinski's Method.- 4.3.2a A Long-Term Development Planning Problem of a Water Supply System.- 4.3.2b A Land-Use Planning Problem.- 4.3.2c A Farm Structure Optimization Problem.- 4.3.3 Rommelranger's Method.- Example.- 4.4 Hybrid Problems.- 4.4.1 Tanaka, Ichihashi and Asai's Method.- Example.- 4.4.2 Inuiguchi and Ichihashi's Method.- Example.- 4.5 Possibilistic Multiple Objective Linear Fractional Programming.- 4.6 Interactive Possibilistic Regression.- 4.6.1 Crisp Output and Crisp Input.- Example.- 4.6.2 Imprecise Output and Crisp Input.- Example.- 4.6.3 Imprecise Output and Imprecise Input.- Example.- 5 Concluding Remarks.- 5.1 Future Research.- 5.2 Fuzzy Mathematical Programming.- 5.3 Multiple Attribute Decision Making.- 5.4 Fuzzy Multiple Attribute Decision Making.- 5.5 Group Decision Making under Multiple Criteria.- Books, Monographs and Conference Proceedings.- Journal Articles, Technical Reports and Theses.- Appendix: Stochastic Programming.- A.1 Stochastic Programming with a Single Objective Function.- A.1.1 Distribution Problems.- A.1.2 Two-Stage Programming.- A.1.3 Chance-Constrained Programming.- A.2 Stochastic Programming with Multiple Objective Functions.- A.2.1 Distribution Problem.- A.2.2 Goal Programming Problem.- A.2.3 Utility Function Problem.- A.2.4 Interactive Problem.- References.

Fuzzy SETS AND FUZZY LOGIC

Abstract: Another approach to reasoning about uncertainty, with a different mathematical basis, is fuzzy logic. Brief history: Standard classical (Boolean) logic (Aristotle, c 50BC; Boole, 1854) uses two possible truth values: • A statement may be true (truth value 1) or false (truth value 0) Łukasiewicz logic (early 20th century): three truth values: • 2, 1 and 0 represent, respectively, “true”, “false” and “unknown” or “irrelevant” • This was further extended to an infinite-valued logic, where real numbers in the range [0,1] represent varying degrees of truth. Only of academic interest, until... Lotfi Zadeh (1965) introduced fuzzy set theory and fuzzy logic, and promoted these as a way of reasoning about uncertainty in computer systems.

Fuzzy sets and systems

Abstract: The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...

Local and fuzzy logics

Abstract: Fuzzy logic differs from conventional logical systems in that it aims at providing a model for approximate rather than precise reasoning.

Ranking fuzzy numbers through the comparison of its expected intervals

Abstract: This paper presents a method for ranking fuzzy numbers based on the comparison of expected intervals of these numbers. This relation is fuzzy and it verifies the distinguishability, rationality and robustness qualities. The term expected interval is extended to no normal fuzzy numbers, and this method then it allows to compare these type sets.

Fuzzy sets and their application to pattern classification and clustering analysis

Abstract: Publisher Summary This chapter provides an overview of a conceptual framework for pattern classification and cluster analysis based on the theory of fuzzy sets. There is a connection between the theory of fuzzy sets and pattern classification. The development of the theory of Fuzzy drew much of its initial inspiration from problems relating to pattern classification, especially the analysis of proximity relations and the separation of subsets of Rn by hyperplanes. However, in a more fundamental way, the intimate connection between the theory of fuzzy sets and pattern classification rests on the fact that most real-world classes are fuzzy in nature, in the sense that the transition from membership to nonmembership in such classes is gradual rather than abrupt. Most of the practical problems in pattern classification do not lend themselves to a precise mathematical formulation, with the consequence that the less precise methods based on the linguistic approach prove to be better matched to the imprecision that is intrinsic in such problems.

A fuzzy-Gaussian neural network and its application to mobile robot control

Abstract: A fuzzy-Gaussian neural network (FGNN) controller is described by applying a Gaussian function as an activation function. A specialized learning architecture is used so that membership function can be tuned without using expert's manipulated data. As an example of the application, a tracking control problem for the speed and azimuth of a mobile robot driven by two independent wheels is solved by using the FGNN controller. To simplify the implementation of the FGNN controller for the two-input/two-output controlled system, a learning controller is utilized consisting of two FGNN's based on independent reasoning and a connection net with fixed weights. The effectiveness of the proposed method is illustrated by performing the simulation of a circular or square trajectory tracking control.

Interval valued strict preference with Zadeh triples

Abstract: Preference modelling and choice theory are common to many different areas including operational research, economics, artificial intelligence and social choice theory. We consider “vague preferences” and introduce a new technique to model this vagueness with the aim of making a choice at the final stage. Our basic tools of modelling will be fuzzy relations and interval valued fuzzy sets. Specifically, we propose that the initial vagueness in the weak preferences of a decision maker is represented by a fuzzy relation and further constructs from this concept introduce a higher-order vagueness which is represented by interval valued fuzzy sets. We derive necessary and sufficient conditions on the representation of this initial vagueness such that a complete ranking of the alternatives is possible. It is shown that conditions weaker than min-transitivity on the representation of initial vagueness are necessary and sufficient for the alternatives to be partially ranked. Furthermore, two linearity conditions are shown to make the ordering of the alternatives a complete order. Conditions for the existence of unfuzzy non-dominated alternatives are also explored.

What is the best shape for a fuzzy set in function approximation

Abstract: The choice of fuzzy set functions affects how well fuzzy systems approximate functions. The most common fuzzy sets are triangles, trapezoids, and Gaussian bell curves. We compared these sets with many others on a wide range of approximand functions in one, two, and three dimensions. Supervised learning tuned the IF-part set functions and the centroids and volumes of the THEN-part sets. We compared the set functions based on how closely the adaptive fuzzy system converged to the approximand. The sinc function sin(x)/x performed best or nearly best in most cases.

Fuzzy Set Theory: Basic Concepts, Techniques and Bibliography

Abstract: List of Figures. Preface. 1: Elementary Set Theory. 1. Sets and subsets. 2. Functions and relations. 3. Partially ordered sets. 4. The lattice of subsets of a set. 5. Characteristic functions. 6. Notes. 2: Fuzzy Sets. 1. Definitions and examples. 2. Lattice theoretical operations on fuzzy sets. 3. Pseudocomplementation. 4. Fuzzy sets, functions and fuzzy relations. 5. alpha-levels. 6. Notes. 3: t-Norms, t-Conorms and Negations. 1. Pointwise extensions. 2. t-Norms and t-Conorms. 3. Negations. 4. Notes. 4: Special Types of Fuzzy Sets. 1. Normal fuzzy sets. 2. Convex fuzzy sets. 3. Piecewise linear fuzzy sets. 4. Compact fuzzy sets. 5. Notes. 5: Fuzzy Real Numbers. 1. The probabilistic view. 2. The non-probabilistic view. 3. Interpolation. 4. Notes. 6: Fuzzy Logic. 1. Connectives in classical logic. 2. Fundamental classical theorems. 3. Basic principles of fuzzy logic. 4. Lattice generated fuzzy connectives. 5. t-Norm generated fuzzy connectives. 6. Probabilistically generated fuzzy connectives. 7. Notes. 7: Bibliography. 1. Books. 2. Articles. Index.

Learning fuzzy inference systems using an adaptive membership function scheme

Abstract: An adaptive membership function scheme for general additive fuzzy systems is proposed in this paper. The proposed scheme can adapt a proper membership function for any nonlinear input-output mapping, based upon a minimum number of rules and an initial approximate membership function. This parameter adjustment procedure is performed by computing the error between the actual and the desired decision surface. Using the proposed adaptive scheme for fuzzy system, the number of rules can be minimized. Nonlinear function approximation and truck backer-upper control system are employed to demonstrate the viability of the proposed method.

A general interpolation technique in fuzzy rule bases with arbitrary membership functions

Abstract: Presents a general multidimensional fuzzy rule interpolation method. This method, compared to the existing interpolation methods, can be applied to arbitrary type of fuzzy sets, and does not require convex and normal sets in the rules. Another important difference is the new method gives an interpretable conclusion in every case, unlike the previously published methods. As a matter of course, to apply arbitrary type of sets, the general method makes calculation necessary for "every point" of the sets. A special method, based on the theory of the general method, is introduced for application in practice, which needs low computational capacity. The specialised method uses three of the most wide spread set types in practice: the crisp, the triangular, and the trapezoidal fuzzy sets. The difference between the new and the former methods is pointed out by examples and the results of different formal methods.

Designing neuro-fuzzy systems through backpropagation

Abstract: The goal of neuro-fuzzy combinations is to obtain adaptive systems that can use prior knowledge, and can be interpreted by means of linguistic rules. Neuro-fuzzy models can be divided into cooperative models, which use neural networks to determine fuzzy system parameters, and hybrid models which create a new architecture using concepts from both worlds. Besides this, there are concurrent neural/fuzzy models that use neural networks and fuzzy systems separately. Most approaches adapt the backpropagation learning rule [33] for neural networks. Some of these systems are discussed in the following pages.