Showing papers on "Fuzzy logic published in 1981"

Multiple Attribute Decision Making: Methods and Applications

Abstract: I. Introduction.- II. Multiple Attribute Decision Making - An Overview.- 2.1 Basics and Concepts.- 2.2 Classifications of MADM Methods.- 2.2.1 Classification by Information.- 2.2.2 Classification by Solution Aimed At.- 2.2.3 Classification by Data Type.- 2.3 Description of MADM Methods.- Method (1): DOMINANCE.- Method (2): MAXIMIN.- Method (3): MAXIMAX.- Method (4): CONJUNCTIVE METHOD.- Method (5): DISJUNCTIVE METHOD.- Method (6): LEXICOGRAPHIC METHOD.- Method (7): LEXICOGRAPHIC SEMIORDER METHOD.- Method (8): ELIMINATION BY ASPECTS (EBA).- Method (9): LINEAR ASSIGNMENT METHOD (LAM).- Method (10): SIMPLE ADDITIVE WEIGHTING METHOD (SAW).- Method (11): ELECTRE (Elimination et Choice Translating Reality).- Method (12): TOPSIS (Technique for Order Preference by Similarity to Ideal Solution).- Method (13): WEIGHTED PRODUCT METHOD.- Method (14): DISTANCE FROM TARGET METHOD.- III. Fuzzy Sets and their Operations.- 3.1 Introduction.- 3.2 Basics of Fuzzy Sets.- 3.2.1 Definition of a Fuzzy Set.- 3.2.2 Basic Concepts of Fuzzy Sets.- 3.2.2.1 Complement of a Fuzzy Set.- 3.2.2.2 Support of a Fuzzy Set.- 3.2.2.3 ?-cut of a Fuzzy Set.- 3.2.2.4 Convexity of a Fuzzy Set.- 3.2.2.5 Normality of a Fuzzy Set.- 3.2.2.6 Cardinality of a Fuzzy Set.- 3.2.2.7 The mth Power of a Fuzzy Set.- 3.3 Set-Theoretic Operations with Fuzzy Sets.- 3.3.1 No Compensation Operators.- 3.3.1.1 The Min Operator.- 3.3.2 Compensation-Min Operators.- 3.3.2.1 Algebraic Product.- 3.3.2.2 Bounded Product.- 3.3.2.3 Hamacher's Min Operator.- 3.3.2.4 Yager's Min Operator.- 3.3.2.5 Dubois and Prade's Min Operator.- 3.3.3 Full Compensation Operators.- 3.3.3.1 The Max Operator.- 3.3.4 Compensation-Max Operators.- 3.3.4.1 Algebraic Sum.- 3.3.4.2 Bounded Sum.- 3.3.4.3 Hamacher's Max Operator.- 3.3.4.4 Yager's Max Operator.- 3.3.4.5 Dubois and Prade's Max Operator.- 3.3.5 General Compensation Operators.- 3.3.5.1 Zimmermann and Zysno's ? Operator.- 3.3.6 Selecting Appropriate Operators.- 3.4 The Extension Principle and Fuzzy Arithmetics.- 3.4.1 The Extension Principle.- 3.4.2 Fuzzy Arithmetics.- 3.4.2.1 Fuzzy Number.- 3.4.2.2 Addition of Fuzzy Numbers.- 3.4.2.3 Subtraction of Fuzzy Numbers.- 3.4.2.4 Multiplication of Fuzzy Numbers.- 3.4.2.5 Division of Fuzzy Numbers.- 3.4.2.6 Fuzzy Max and Fuzzy Min.- 3.4.3 Special Fuzzy Numbers.- 3.4.3.1 L-R Fuzzy Number.- 3.4.3.2 Triangular (or Trapezoidal) Fuzzy Number.- 3.4.3.3 Proof of Formulas.- 3.4.3.3.1 The Image of Fuzzy Number N.- 3.4.3.3.2 The Inverse of Fuzzy Number N.- 3.4.3.3.3 Addition and Subtraction.- 3.4.3.3.4 Multiplication and Division.- 3.5 Conclusions.- IV. Fuzzy Ranking Methods.- 4.1 Introduction.- 4.2 Ranking Using Degree of Optimality.- 4.2.1 Baas and Kwakernaak's Approach.- 4.2.2 Watson et al.'s Approach.- 4.2.3 Baldwin and Guild's Approach.- 4.3 Ranking Using Hamming Distance.- 4.3.1 Yager's Approach.- 4.3.2 Kerre's Approach.- 4.3.3 Nakamura's Approach.- 4.3.4 Kolodziejczyk's Approach.- 4.4 Ranking Using ?-Cuts.- 4.4.1 Adamo's Approach.- 4.4.2 Buckley and Chanas' Approach.- 4.4.3 Mabuchi's Approach.- 4.5 Ranking Using Comparison Function.- 4.5.1 Dubois and Prade's Approach.- 4.5.2 Tsukamoto et al.'s Approach.- 4.5.3 Delgado et al.'s Approach.- 4.6 Ranking Using Fuzzy Mean and Spread.- 4.6.1 Lee and Li's Approach.- 4.7 Ranking Using Proportion to The Ideal.- 4.7.1 McCahone's Approach.- 4.8 Ranking Using Left and Right Scores.- 4.8.1 Jain's Approach.- 4.8.2 Chen's Approach.- 4.8.3 Chen and Hwang's Approach.- 4.9 Ranking with Centroid Index.- 4.9.1 Yager's Centroid Index.- 4.9.2 Murakami et al.'s Approach.- 4.10 Ranking Using Area Measurement.- 4.10.1 Yager's Approach.- 4.11 Linguistic Ranking Methods.- 4.11.1 Efstathiou and Tong's Approach.- 4.11.2 Tong and Bonissone's Approach.- V. Fuzzy Multiple Attribute Decision Making Methods.- 5.1 Introduction.- 5.2 Fuzzy Simple Additive Weighting Methods.- 5.2.1 Baas and Kwakernaak's Approach.- 5.2.2 Kwakernaak's Approach.- 5.2.3 Dubois and Prade's Approach.- 5.2.4 Cheng and McInnis's Approach.- 5.2.5 Bonissone's Approach.- 5.3 Analytic Hierarchical Process (AHP) Methods.- 5.3.1 Saaty's AHP Approach.- 5.3.2 Laarhoven and Pedrycz's Approach.- 5.3.3 Buckley's Approach.- 5.4 Fuzzy Conjunctive/Disjunctive Method.- 5.4.1 Dubois, Prade, and Testemale's Approach.- 5.5 Heuristic MAUF Approach.- 5.6 Negi's Approach.- 5.7 Fuzzy Outranking Methods.- 5.7.1 Roy's Approach.- 5.7.2 Siskos et al.'s Approach.- 5.7.3 Brans et al.'s Approach.- 5.7.4 Takeda's Approach.- 5.8 Maximin Methods.- 5.8.1 Gellman and Zadeh's Approach.- 5.8.2 Yager's Approach.- 5.9 A New Approach to Fuzzy MADM Problems.- 5.9.1 Converting Linguistic Terms to Fuzzy Numbers.- 5.9.2 Converting Fuzzy Numbers to Crisp Scores.- 5.9.3 The Algorithm.- VI. Concluding Remarks.- 6.1 MADM Problems and Fuzzy Sets.- 6.2 On Existing MADM Solution Methods.- 6.2.1 Classical Methods for MADM Problems.- 6.2.2 Fuzzy Methods for MADM Problems.- 6.2.2.1 Fuzzy Ranking Methods.- 6.2.2.2 Fuzzy MADM Methods.- 6.3 Critiques of the Existing Fuzzy Methods.- 6.3.1 Size of Problem.- 6.3.2 Fuzzy vs. Crisp Data.- 6.4 A New Approach to Fuzzy MADM Problem Solving.- 6.4.1 Semantic Modeling of Linguistic Terms.- 6.4.2 Fuzzy Scoring System.- 6.4.3 The Solution.- 6.4.4 The Advantages of the New Approach.- 6.5 Other Multiple Criteria Decision Making Methods.- 6.5.1 Multiple Objective Decision Making Methods.- 6.5.2 Methods of Group Decision Making under Multiple Criteria.- 6.5.2.1 Social Choice Theory.- 6.5.2.2 Experts Judgement/Group Participation.- 6.5.2.3 Game Theory.- 6.6 On Future Studies.- 6.6.1 Semantics of Linguistic Terms.- 6.6.2 Fuzzy Ranking Methods.- 6.6.3 Fuzzy MADM Methods.- 6.6.4 MADM Expert Decision Support Systems.- VII. Bibliography.

DETECTION AND CHARACTERIZATION OF CLUSTER SUBSTRUCTURE I. LINEAR STRUCTURE: FUZZY c-LINES*

Abstract: In Part I, a generalization of the Fuzzy c-Means (or Fuzzy ISODATA) clustering algorithms is developed. Necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a Picard iteration scheme which generates simultaneously (i) c fuzzy clusters in the data; (ii) a set of c prototypical straight lines in feature space which best fit the data in a well-defined sense; (iii) a set of c prototpyical centers of mass (on the c lines) which characterize the “core” of each linear fuzzy cluster. Theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices. A convergence theorem for each algorithm in the infinite family is given. The algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially “linear” substructure. In Part II, the Fuzzy c-Means and Fuzzy c-Lines algorithms are shown to be special cases of a more general class of fuzzy algorithms, the...

Image enhancement using smoothing with fuzzy sets

Abstract: A model for grey-tone image enhancement using the concept of fuzzy sets is suggested. It involves primary enhancement, smoothing, and then final enhancement. The algorithm for both the primary and final enhancements includes the extraction of fuzzy properties corresponding to pixels and then successive applications of the fuzzy operator "contrast intensifier" on the property plane. The three different smoothing techniques considered in the experiment are defocussing, averaging, and maxmin rule over the neighbors of a pixel. The reduction of the "index of fuzziness" and "entropy" for different enhanced outputs (corresponding to different values of fuzzifiers) is demonstrated for an English script input. Enhanced output as obtained by histogram modification technique is also presented for comparison.

Cooperative Fuzzy Games

Abstract: We derive versions of known results dealing with core, equilibria and Shapley-values of cooperative games in the case of cooperative fuzzy games, i.e., games defined on fuzzy subsets of the set of n players. A fuzzy coalition is an n-vector τ = τi associating with each player i his “rate of participation” τi ∈ [0, 1] in the fuzzy coalition and the real number v· is the coalition worth, assumed to be positively homogeneous. If also v is superadditive, or equivalently concave, the fuzzy game with side payments has a nonempty core. Associated with the coalition-worth function for any ordinary game with side payments is a fuzzy extension thereof, viz., the fuzzy game whose coalition-worth function is the least positively-homogeneous superadditive function majorizing the coalition-worth function of the original game. The fuzzy extension always has a nonempty core. Moreover, if the original game has a nonempty core, it coincides with that of its fuzzy extension. Analogous results are established for games without side payments. An axiomatization of “values” of fuzzy games with side payments is also given. The results are applied to show that the set of Walras equilibria coincides with the fuzzy core of an economy.

Additions of interactive fuzzy numbers

Abstract: This paper provides an account of an approach to modeling unknown data by means of fuzzy-set theory, and addresses the problem of deriving the uncertainty, on a sum of variables whose values lie within fuzzy intervals. The first part is an extensive presentation of the theoretical background of the approach: the extension principle is stated in terms of possibility of an event; the concept of variable interaction is investigated at length. Section II gives new results regarding the effective practical computation of additions of fuzzy numbers. Its originality lies in the introduction of interaction which enables to control the growth of uncertainty, in calculations. Moreover, the problem of computing mathematical expectations with fuzzy probabilities is solved. The results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed. Typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities. More generally, fuzzy arithmetic can be an important tool for sophisticated, computationally tractable sensitivity analysis in systems modeling, computer-aided design and operations research.

On fuzzy goal programming

Abstract: This paper illustrates how the goal programming problem with fuzzy goals having linear membership functions may be formulated as a single goal programming problem. Also, a previously defined method for dealing with fuzzy weights for each of the goals is re-examined.

Objective Function Clustering

Abstract: S8 illustrates some of the difficulties inherent with cluster analysis; its aim is to alert investigators to the fact that various algorithms can suggest radically different substructures in the same data set. The balance of Chapter 3 concerns objective functional methods based on fuzzy c-partitions of finite data. The nucleus for all these methods is optimization of nonlinear objectives involving the weights u ik ; functionals using these weights will be differentiable over M fc —but not over M c —a decided advantage for the fuzzy embedding of hard c-partition space. Classical first- and second-order conditions yield iterative algorithms for finding the optimal fuzzy c-partitions defined by various clustering criteria.

Fuzzy Sets: Theory of Applications to Policy Analysis and Information Systems

Abstract: In this volume, we present a spectrum of original research works ranging from the very basic properties and characteristics of fuzzy sets to specific areas of applications in the fields of policy analysis and information systems. The first part, theory, presents some fine added contributions toward a deeper basic understanding of fuzzy set theory and serves to enrich set theory in the direction of maturity and completeness of its theoretical development.

The IF THEN ELSE statement and interval-valued fuzzy sets of higher type

Abstract: A new fuzzy relation which represents an IF THEN ELSE (abbreviated to “ITE”) statement is constructed. It is shown that a relation which (a) always gives correct inference and (b) does not contain false information which is not present in the ITE statement, must be of a higher degree of fuzziness than the antecedents and consequents of the ITE statement. Three different ways of increasing the fuzziness of the relation are used here: (1) the fuzzy relation is of higher type; (2) it is interval-valued; (3) contains a BLANK or “don't know” component. These three types of fuzziness come about naturally because the relation is a restriction of an initial relation which represents the second approximation to the state of complete ignorance. There exist successive approximations to the state of complete ignorance, each of them being an intervalvalued fuzzy set BLANK of one type higher than the previous one. Similar representations of the zeroth and first approximation to the state of ignorance have been used in the theory of probability, though in a rather heuristic fashion. The assignment of a value to a variable is represented as a complete restriction of the BLANK state of type N; the “value” being any pure (non-interval-valued) fuzzy set of type N. With the new relation, the inferred set is a superposition of the consequents of the ITE statement and of the BLANK state, each of these components being multiplied by an interval-valued coefficient. In the case of modus ponens inference, the component with the highest coefficient (determined from a specially defined ordering relation for interval-values) is always the correct consequent, provided that the original ITE statement is logically consistent. A mathematical test for logical consistency of the (possibly fuzzy) ITE statement is given. Disjointness of fuzzy sets is defined and connected up with logical consistency. The paradoxes of the implication of mathematical logic disappear in the fuzzy set treatment of the ITE statement. Subnormal fuzzy singletons find their natural interpretation. When used as an antecedent in an ITE statement, such a singleton does not have enough strength to induce the consequent with complete certainty. Instead it induces a superposition of an interval-valued fuzzy set and the BLANK state. New definitions for union, intersection and complementation of fuzzy sets of higher type are suggested. A new interpretation of an interval-valued fuzzy set of type N as a collection of fuzzy sets of type N, not as a type N+1 set, is given. There exist two types of union, intersection and complementation for intervalvalued fuzzy sets. They are called the fuzzy and the crisp operations, respectively. It is suggested that the negation be represented by an interval-valued fuzzy set. We conclude that increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner.

On fuzzy goal programming—some comments

Abstract: This paper pertains to goal programming with fuzzy goals and fuzzy priorities. Hannan [1], in his paper on fuzzy goal programming, alludes to the difficulty of handling fuzzy priorities and further notes that a method that this author proposed [2] may lead to incorrect results. In this note, the general problem of goal programming with fuzzy priorities is reexamined, along with the solution to the specific example presented in my original paper [2]. It is shown that the method for handling fuzzy priorities originally proposed by this author does indeed capture the relative importance of goals.

An approach to the analysis of fuzzy systems

Abstract: This paper deals with a formal description of ill-defined processes (fuzzy systems) by the use of fuzzy relational equations. It is pointed out that fuzzy relational equations form a generalized version of the difference equations widely considered in control theory. Some equivalence between these two kinds of description is presented. Basic problems of fuzzy systems e.g. identification, prediction, sensitivity and stability are shown and numerical algorithms are given. Indices of each method are introduced (especially the degree of fuzziness, the sensitivity index) which makes it possible to express the quality of each of them.

Note on the arithmetic rule by zadeh for fuzzy conditional inference

Abstract: This paper shows that Zadeh's arithmetic rule for fuzzy conditional propositions “If x is A then y is B” and “If x is A then y is B else y is C” can infer quite reasonable consequences in a fuzzy conditional inference if new compositions of “max-[Odot] composition” and “max- composition” are used in the compositional rule of inference, though, as was pointed out before, this arithmetic rule cannot get suitable consequences in the compositional rule of inference which uses max-min composition. Moreover, it is shown that the arithmetic rule satisfies a syllogism under these two compositions.