Showing papers on "Membership function published in 1980"

A Linguistic Approach to Decisionmaking with Fuzzy Sets

Abstract: A technique for making linguistic decsions is presented. Fuzzy sets are assued to be an appropriate way of dealng with uncertainty, and it b therefore cncluded that decisions taken on the basis of such infomation must themselves be fuzzy. It b inappriate then to present the decision in numencal form; a statement in natural angug is much better. For brevity only a single-stage multlabute decsion problem is considered. Solutions to such problems are shown using ideas in linguisc approximation and truth qualiction. An extensive example illuminates the basic ideas and techniques.

Fuzzy sets and systems

Abstract: R. Babuska, Fuzzy Modeling for Control, Kluwer Academic Publ., Dordrecht, 1998, 288 p. B. Bouchon-Meunier, L. Foulloy, M. Ramdani, Logique Floue Exercices Corriges et Exemples d' Applications, Cepadues Editions. Toulouse, 199B, 200 p. G.Q. Chen, Fuzzy Logic in Data Modeling Semantics, Constraints and Database Design, Kluwer Academic Publ., Boston. 1998, 240 p. K.J. Cios. W. Pedrycz, R.S. Swiniarski. Data Mining Methods for Knowledge Discovery, Kluwer Academic Publ., Dordrecht. 199B. J. Godjevac, Idees Nettes sur la Logique Floue, Presses Polytechniques et Universitaires Romandes. Lausanne, 1997,200 p. E. Hisdal, Logical structures for representation of knowledge and uncertainty, Physica-Verlag, Heidelberg, Germany, 199B. M. Jamshidi, A. Titli, L.A. Zadeh, S. Boverie, Applications of Fuzzy LogiC Towards High Machine Intelligence Quotient Systems, Prentice-Hall, Englewood Cliffs, NJ, 1997,423 p. F. Lootsma, Fuzzy Logic for Planning and Decision Making Applied Optimization 8. Kluwer Academic Publ., Dordrecht, 1998. W. Mielczarski, Fuzzy Logic Techniques in Power Systems, Physica-Verlag, Berlin, 1998, 456 p. J.N. Mordeson, P.S. Nair, Fuzzy Mathematics An Introduction for Engineers and Scientists. Physica-Verlag, Berlin, 1998,270 p. W. Pedrycz, Computational Intelligence An Introduction, CRC Press, Boca Raton, FL, 1997,304 p. M. Reghis, E. Roventa, Classical and Fuzzy Concepts in Mathematical Logic and Applications, CRC Press, Boca Raton, FL, 1998. L. Reznik, Fuzzy Controllers, Newness Press, UK, 1997. Schindler, Fuzzy-Datenanalyse durch kontextbasierte Datenbankanfragen, Deutscher Universitiits Verlag, Leverkusen, Germany, 1998. O. Wolkenhauer, Possibility Theory with Applications to Data Analysis. Research Studies Press, Hertfordshire, UK, 1998, 290 p. Edited volumes

A fuzzy sets based linguistic approach: Theory and applications

Abstract: Fuzzy sets theory and fuzzy logic constitute the basis for the linguistic approach. Under this approach, variables can assume linguistic values. Each linguistic value is characterized by a label and a meaning. The label is a sentence of a language. The meaning is a fuzzy subset of a universe of discourse. Models, based on this approach, can be constructed to simulate approximate reasoning. The implementation of these models presents two major problems, namely how to associate a label to an unlabelled fuzzy set on the basis of semantic similarity (linguistic approximation) and how to perform arithmetic operations with fuzzy numbers. For each problem a solution is proposed. Two illustrative applications are discussed.

Fuzzy subsets of type ii in decisions

Abstract: We are interested in the construction of an ideal decision function from its approximation in terms of other decision functions. We investigate fuzzy subsets of type II, those with linguistic grades of membership. We also discuss fuzzy subsets whose elements are fuzzy subsets of another set.

New Results about Properties and Semantics of Fuzzy Set-Theoretic Operators

Abstract: This communication is a first step towards a general approach to fuzzy set-theoretic operators, i.e. algebraic operators which coincide with set-operators when membership values are crisp. Some properties of subclasses of such fuzzy set-theoretic operators are investigated. Specific examples are given. An attempt to discuss a possible interpretation of these operators is proposed. The choice of a good operator in a given practical situation can be crucial in decision analysis, information retrieval, pattern recognition for the purpose of aggregating several pieces of information.

On choosing between fuzzy subsets

Abstract: The problem of selecting the highest grade of membership of fuzzy subsets of type 2 and of choosing the most truthful of a group of fuzzy propositions involve making a choice among fuzzy subsets on the unit interval. A procedure is proposed for the selection of fuzzy subsets on the unit interval. This procedure involves selecting the subset closest to a linear membership function on the unit interval.

Fuzzy Sets and Decision Analysis

Abstract: The applicability of fuzzy set theory to decision analysis (DA) is examined. It extends the ideas of an earlier paper "Fuzzy decision analysis," by Watson et al. [33]. Particular emphasis is placed on justifying the use of Zadeh's fuzzy calculus to model impression, and an axiomatic system is suggested towards this end. This is seen as an attempt at extending Savage's axioms of subjective probability to produce "approximate probabilities." It is argued that the method proposed by Watson et al. for comparing decision options is unsatisfactory, and several alternative methods are developed. Some computational anomalies are pointed out which severely limit the potential of this methodology. It is suggested that, for individual decisionmaking, fuzzy decision analysis should be viewed as an automatic sensitivity analysis, but that fuzzy sets may be useful with another interpretation for group decisionmaking. The conclusions are that the methodology has too many limitations to be of use for isolated decisions but that it may have a value for often repeated generic decisions.

Fuzzy sets versus probability

Abstract: One of the aims of the theory of fuzzy sets is the development of a methodology for the formulation and solution of problems which are too complex or ill-defined to be susceptible to analysis by conventional techniques. Because of its unorthodoxy, it has been and will continue to be controversial for some time. Eventually, though, the theory of fuzzy sets is likely to be recognized as a natural development in the evolution of scientific thinking. In retrospect, the skepticism about its usefulness will be viewed as a manifestation of the human attachment to tradition andre sistance to innovation.

Duality in Fuzzy Linear Programming

Abstract: In classical duality theory of linear programming the saddlepoint of the Lagrangian is the solution of the max-min problem as well as of the min-max problem. In using the theory of fuzzy sets these problems are interpreted in a new sense — leading to a pair of “fuzzy dual” optimization problems. An economic interpretation is given and properties of the fuzzy dual problems are derived.

Lattice fuzzy logics

Abstract: Although the characterizing membership functions of fuzzy sets normally have as their range the interval [0, 1], it is possible for the range to be a partially ordered set. The use of lattices for this set is explored. Various forms of restricted infinite lattice are considered. Rose's logical operator for logics whose truth values form lattices are reviewed. A basis for lattice fuzzy logics, using Rose's operators, is discussed and a particular infinite lattice is proposed for use in characterizing lattice fuzzy sets. Some of the concepts are used to extend edge detection techniques in image processing from grey scale to colour images.

Finite linearly ordered fuzzy sets with applications to decisions

Abstract: We introduce S-fuzzy sets as fuzzy sets whose membership grades lie in a finite linearly ordered set. We extend the basic operations from fuzzy sets to these sets. We introduce the shift operation on S fuzzy sets. We then use this new operation to present a method for multiple objective decision making with ordinal information on both membership grades and importances of the objectives.

Fuzzy sets, probabilities, and decision

Abstract: The purpose of this article is to show how fuzzy sets can be used to help make decisions in a probabilistic environment. We first show that in the face of uncertainty a one-dimensional decision becomes a multidimensional one. We then apply fuzzy multiobjective techniques to solve this. We extend this to the case where we have more than one consideration. We then examine two stage decisions.

Fuzzy economic spaces

Abstract: Traditional spatial economic analysis is limited to the description of precise spaces. To say that an economic space is precise means: (1) that this space has, or else, has not given constituent characteristics and (2) that the agents located there prefer, or else, do not prefer one possible action to another. Proposition (1) implies that an economic space is perfectly delimited and that it can be clearly partitioned into homogeneous subspaces. Proposition (2) implies that economic agents undertake exact economic calculations and optimize, under rigid constraints of resource limitation, objective functions whose arguments are clearly defined. Thus, traditional spatial economic analysis is based on a binary logic: presence or absence of the space's characteristics, preference or non-preference of agents with respect to possible actions. This logic supposes the principle of the excluded middle. However, the real world is imprecise. The observed economic spaces (areas of influence, regions, attraction zones, market areas, etc.) have "more or less" the given characteristics; instead of having frontiers, they have ill-chiselled limits; they partially overlap one another and they do not allow themselves to be subdivided without ambiguity. Likewise, economic agents pursue vague objectives, sometimes incompatible or contradictory, and they appraise imperfectly the constraints which limit their resources. The analyst who admits that the lights and shades of expression "modify everything" and are, at the same time essential, intends to retain them in full. But he must go beyond the usual literary comments which are often juxtaposited to scientific analysis and whose purpose is to relativize the conclusions, in other words, to contest implicitly the results. He is bound to give a formalized expression of these nuances and gradations of the real world and he ought to reconcile the imprecision inherent in the latter with the precision of the mathematical model being used. 0.3. It is true that n-ary logics have been in use for some time now: POST (1921), LUKASIEWICZ (1937), MOISIL (1940). But, it is with the recent development of the theory of fuzzy subsets that the elaboration of a spatial economic study, perfectly rigorous and fully formalized, has become possible. This theory, presented for the first time by ZADEH (1965) is making great strides and penetrates every branch of mathematics. A few primers are now available. Since 1974, the Institut de Mathematiques Economiques of the University of Dijon, associated with the Centre National de la Recherche Scientifique (France), devotes an important part of its researches to the theory of fuzzy subsets and its applications, especially its applications to spatial economic analysis. In the Institute, research has followed four directions : 0.4.1. : Firstly, it was absolutely necessary to rigorously formulate the axiomatic framework of the theory of fuzzy subsets, to clearly distinguish between the latter and probability calculus, to assemble the principal mathematical results to present the concepts and theorems which are useful to economics and more importantly to spatial economic analysis and to resolve a number of algorithmic problems. 0.4.2. : Next, many types of fuzzy economic spaces were studied: attraction zones for sale-points, areas of fuzzy spatial interactions, fuzzy regions, french fuzzy regions defined by a fuzzy numeric taxonomy, fuzzy regional dynamic systems, fuzzy interregional relations, fuzzy hierarchy of a system of central places and fuzzy urban spaces. 0.4.3. : Then, analyses of fuzzy spatial behaviours of the consumer and of the producer led to a reformulation of the theories of partial equilibria which prepares the way for that of the theory of general spatial equilibrium and of the optimum [ under study ] . 0.4.4. : Finally, various contributions have been made to general economics: fuzzy multicriterion analysis, fuzzy decision theory and fuzzy econometrics. 0.5. The aim of the present study is not to summarize the totality of these works. The time has come for the presentation, with all the rigour called for in this new and, for some people, unwonted field, of the scientific foundations of the theory of fuzzy economic spaces in the course of elaboration. This reconsideration of the foundations of the theory should answer two series of questions: 0.5.1. : On what axiomatic framework is the description of economic universes based? Has it at its disposal specific and novel mathematical instruments, sufficiently pertinent and sophisticated ? 0.5.2. : Can the description of fuzzy spatial behaviours of economic agents rely on a coherent and an appropriate type of economic calculation? On what theory of value is a fuzzy economic calculation based? 0.6. The above set of questions command the plan which will be followed: 1 - Fuzzy economic universes; 2 - Fuzzy spatial behaviours. 0.7. Remark: In order to avoid any ambiguity in the notation of mathematical terms, ordinary concepts (non-fuzzy) are underlined, whereas fuzzy concepts are not. For instance, A C E is read: A is a fuzzy subset of the ordinary reference set E. Furthermore: g(x) designates an ordinary function, whereas f(x) defines a fuzzy function. Likewise: [a1,a2 ] designates a non-fuzzy interval, whereas [t1 ,t2 ] represents a fuzzy interval. For lack of space, the results of numerous theorems are cited without demonstrations, but the complete references indicate in what books and articles these demonstrations can be found.

Fuzzy sets and non-stochastic linear partial information

Abstract: We extend the fuzzy set (FS)-membership measures for finite universal sets in Zadeh's sense by the passage from points in the n-dimensional unit cube to points in n-dimensional polyhedra of the cube.

Identification of fuzzy sets with a class of canonically induced random sets

Abstract: Any random subset of a space X clearly determines the membership function of a fuzzy subset through its one point coverages. This paper shows, conversely, that any fuzzy subset A of X can always be identified with, in general, many random subsets S(A) of X with respect to one point coverages i.e., simultaneously, for all x ? X, ?A(x) = Pr(x?S(A)). In a related manner, it is shown that any fuzzy set can be uniformly closely approximated with respect to one point coverages by a random set having a finite number of outcomes. In particular, the canonical mapping SU, defined by A ? SU (A) =df ?A -1 ([U,1]), where A is any fuzzy subset of X and U is any uniformly distributed r.v. over [0, 1], produces such an identification. Moreover, SU is an isomorphism from the collection of all fuzzy subsets onto a proper subcollection of all random subsets of X, with respect to many of the basic fuzzy set operations and corresponding ordinary set operations among random sets. In addition, SU, among all possible mappings from the class of all fuzzy subsets of X into the class of all random subsets of X which preserve one point coverages, induces both the maximal lower probability measure and the minimal upper probability measure in Dempster's sense on P (X). Applications of the results to fuzzy attribute reasoning are presented, emphasizing the close connection between fuzzy and random confidence sets.

Probability, Vague Statements and Fuzzy Sets

Abstract: The relationship between vague statements and fuzzy sets is examined. It is shown that the probability of vague statements may be defined in a manner analogous to that discussed in Reichenbach's logic of weight.

From absolute to probable and fuzzy in decision‐making

Abstract: General Systems Theory postulates the existence of many general theories that serve to describe isomorphisms across systems. The theory of Fuzzy Sets can be considered as one particular general theory which describes the phenomenon of ambiguity across all systems displaying this property and its consequences. Fuzzy Set Theory is a mathematical development that holds great promise in becoming the metalanguage of ambiguity, in a way parallel to Statistics and Probability Theory which represent the metalanguage of uncertainty. Fuzzy Sets appear particularly well suited to model ambiguity in the context of the systems paradigm which has been offered as a counterpart to the traditional science paradigm. A decision model is used to discuss the differences between these two paradigms and to show the role which Fuzzy Sets can play in resolving some of the epistemological problems in the domain of the social sciences.

Characterization of Fuzzy Measures by Classical Measures

Abstract: In this paper we first recall L. A. Zadeh’s8 probability of a fuzzy event and the axiomatic definition of fuzzy probability measures given in ref. 5. It is shown that these fuzzy probability measures can be characterized uniquely either by a probability measure and a Markoff-kernel or by a probability measure on the product space measuring the area below the membership function of the fuzzy event.