Showing papers on "Fuzzy number published in 1977"
TL;DR: A method to deal with multiple-alternative decision problems under uncertainty by considering each of these variables as fuzzy quantities, characterized by appropriate membership functions of fuzzy sets induced by mappings is proposed.
737 citations
01 Jul 1977-International Journal of Human-computer Studies \/ International Journal of Man-machine Studies
TL;DR: This paper presents a model for solving multi-objective decision problems when the objectives are of varying degrees of importance by assigning to each objective a power indicative of its importance and then raising each fuzzy set to its appropriate power.
Abstract: One of the most useful aspects of fuzzy set theory is its ability to represent multio-bjective decision problems involving vague or fuzzy objectives. This paper presents a model for solving multi-objective decision problems when the objectives are of varying degrees of importance. This is done by assigning to each objective a power indicative of its importance and then raising each fuzzy set to its appropriate power. These powers are obtained by getting the eigenvector of the maximum eigenvalue of a matrix of paired comparisons of the objectives.
452 citations
Journal Article•
TL;DR: A linguistic control algorithm is synthesized, capable of dealing with a continuously reproduced decisionmaking situation, and a fuzzy set theoretic representation of these instructions, called a fuzzy logic controller, was tried as an answer to the control modeling problem, which gave very satisfactory results.
Abstract: The system is a traffic junction and the problem of its control is considered as a classical example of nonprogrammed decisionmaking, i.e. , decisionmaking characterized by the lack of well-specified analytical means for coping with a particular problem. Thus a linguistic control algorithm is synthesized, capable of dealing with a continuously reproduced decisionmaking situation. The starting point is an adequate (though qualitative) knowledge of the system and a protocol of control instructions used by a human operator. A fuzzy set theoretic representation of these instructions which we call "a fuzzy logic controller" was tried as an answer to the control modeling problem, which gave very satisfactory results. The work done on the construction of the model of the system and the implementation of the fuzzy logic controller is presented.
440 citations
01 Oct 1977
TL;DR: It is shown that the use of a fuzzy logic controller results in a better performance than a conventional effective vehicle-actuated controller.
Abstract: Work done on the implementation of a fuzzy logic controller in a single intersection of two one-way streets is presented. The model of the intersection is described and validated, and the use of the theory of fuzzy sets in constructing a controller based on linguistic control instructions is introduced. The results obtained from the implementation of the fuzzy logic controller are tabulated against those corresponding to a conventional effective vehicle-actuated controller. With the performance criterion being the average delay of vehicles, it is shown that the use of a fuzzy logic controller results in a better performance.
380 citations
TL;DR: The qualitative terms of the alternatives are represented quantitatively using fuzzy sots and then the fuzzy optimal alternative is computed and gives the relative merit of each alternative and thus helps in decision making.
Abstract: In many situations one has to find the best alternative on the basis of many aspects which have varying degrees of importance. The problem is more complex if the ratings of the alternatives and the importance of the aspects are expressed using qualitative variables like good, fair, very important, etc. This paper presents a method for decision making in such situations. The qualitative terms are represented quantitatively using fuzzy sots and then the fuzzy optimal alternative is computed. The fuzzy set representing the optimal alternative gives the relative merit of each alternative and thus helps in decision making.
253 citations
245 citations
TL;DR: In this paper, some sufficient conditions for convergence under “max(min)” products of the powers of a square fuzzy matrix and of a fuzzy state process are established.
229 citations
TL;DR: Techniques for processing simple fuzzy queries expressed in the relational query language SEQUEL are introduced and the feasibility of implementing such techniques in a real environment is studied.
Abstract: This paper is concerned with techniques for fuzzy query processing in a database system. By a fuzzy query we mean a query which uses imprecise or fuzzy predicates (e.g. AGE = “VERY YOUNG”, SALARY = “MORE OR LESS HIGH”, YEAR-OF-EMPLOYMENT = “RECENT”, SALARY ⪢ 20,000, etc.). As a basis for fuzzy query processing, a fuzzy retrieval system based on the theory of fuzzy sets and linguistic variables is introduced. In our system model, the first step in processing fuzzy queries consists of assigning meaning to fuzzy terms (linguistic values), of a term-set, used for the formulation of a query. The meaning of a fuzzy term is defined as a fuzzy set in a universe of discourse which contains the numerical values of a domain of a relation in the system database. The fuzzy retrieval system developed is a high level model for the techniques which may be used in a database system. The feasibility of implementing such techniques in a real environment is studied. Specifically, within this context, techniques for processing simple fuzzy queries expressed in the relational query language SEQUEL are introduced.
189 citations
176 citations
TL;DR: The notions of initial and final fuzzy topologies are introduced and it is shown that from a categorical point of view they are the right concepts to generalize the topological ones.
TL;DR: Using fuzzy measures and fuzzy integrals, a mathematical model of learning is presented which is able to learn through fuzzy information and is compared with an ordinary Bayesian learning model.
Abstract: Using fuzzy measures and fuzzy integrals, the paper presents a mathematical model of learning which is able to learn through fuzzy information. The characteristics of the model are studied theoretically and in numerical examples, where the model is compared with an ordinary Bayesian learning model. The problem of seeking an extremum of multimodel objective function is given as an example.
01 Dec 1977
TL;DR: The use of fuzzy logic is proposed for the method of failure diagnosis under fuzzy environment and the results obtained show that the present method would be applicable as a methodology of the more general diagnosis.
Abstract: The use of fuzzy logic is proposed for the method of failure diagnosis under fuzzy environment. The distinctive points are the description of the relationship between causes and symptoms by a set of fuzzy relational inequalities and the use of the solutions of the inverse problem of these relations. Recently, the method to solve the inverse problem of fuzzy relational equation has been proposed by E. Sanchez et. al,. A new algorithm for the inverse problem is developed With the consideration on the existence condition of the solution. The proposed scheme of diagnosis is illustrated by considering the diagnosis of car troubles. The results obtained from some numerical examples show that the present method would be applicable as a methodology of the more general diagnosis.
TL;DR: A fuzzy equilibrium solution is introduced, which can provide a base for an agreement between the players in analysing games with fuzzy sets of strategies of the players.
Abstract: Two solution concepts for a FMP problem are suggested. The first one makes use of level sets of the fuzzy set of feasible alternatives. The second solution is based on the concept of Pareto maximum in vector optimization. It is shown that both solutions are equivalent in a sense that they give the same fuzzy value of a function maximized. It is suggested that if a decision‐maker is to choose a single element, then his choice must be based not only on the membership value of this element in the solution fuzzy set but also on the corresponding value of the function maximized. In this respect the situation is similar to that typical for vector optimization. The approach suggested in this paper is further used for analysing games with fuzzy sets of strategies of the players. A fuzzy equilibrium solution is introduced, which can provide a base for an agreement between the players.
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TL;DR: A type of fuzzy set called a level fuzzy set is defined in the paper, definitions of basic operations performed on level fuzzy sets are given, and properties of these operations are presented.
Abstract: In practical applications of fuzzy set theory, e.g., in pattern recognition, information retrieval, etc., the reader is not advised-for the reasons of the amount of the considered object universal set and the capacity of computer memory involved-to consider fuzzy sets defined in the whole universal set, when it is sufficient-from the point of view of a given process description exactness-to take into consideration fuzzy sets determined in the universal subsets. Such a type of fuzzy set called a level fuzzy set is defined in the paper, definitions of basic operations performed on level fuzzy sets are given, and properties of these operations are presented.
TL;DR: A fuzzy set is approximated by an ordinary set using the Chebyshev norm, and a set is said to approximate a fuzzy set if the norm of a difference of its characteristic functions is smaller than a given number.
Abstract: By fuzzy optimization we here mean optimization in a fuzzy environment, i.e., optimization with fuzzy constraints. Such a problem can be reduced to a family of ordinary optimization problems by using the representation theorem which states that a fuzzy set is a family of ordinary sets. Since it is difficult to work with a family of sets, in this paper a fuzzy set is approximated by an ordinary set. The Chebyshev norm is introduced into the set of all fuzzy sets, and a set is said to approximate a fuzzy set if the norm of a difference of its characteristic functions is smaller than a given number.
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TL;DR: In this paper, the authors define a category of fuzzy complemented spaces and give necessary and sufficient conditions for subcategories to have properties analogous to those of the category of sets and functions.
Abstract: In [1] Bellman and Giertz showed that under reasonable restrictions the generalized operations of union and intersection for fuzzy sets have to be l.u.b, and g.l.b, in IX endowed with the usual order, where I denotes the unit interval and X is an arbitrary set. It is the purpose of this paper to characterize those operations which are acceptable as complementation. We demand that the operation would be idempotent, orderreversing and that it would be a productmap. We see that these conditions are minimal and not sufficient to guarantee the preservation of important properties of complementation with regard to sets and functions. We then define a category of fuzzy complemented spaces and give necessary and sufficient conditions for subcategories to have properties analogous to those of the category of sets and functions.
01 Dec 1977
TL;DR: The general formulation of optimization under elastic constraints is proposed based upon the notion of maximizing and minimizing sets and it is argued that the extension problem of possibility measures to fuzzy sets is justified.
Abstract: Transformations of fuzzy sets are discussed using the relation between random sets and fuzzy sets. We propose the general formulation of optimization under elastic constraints. This formulation is based upon the notion of maximizing and minimizing sets. Using this formulation we justify the extension problem of possibility measures to fuzzy sets.
01 Dec 1977
TL;DR: A flat, convex set of fuzzy relations is utilizes to provide a physically realistic and mathematically tractable model for small group decision theory.
Abstract: This paper utilizes a flat, convex set of fuzzy relations to provide a physically realistic and mathematically tractable model for small group decision theory. Measures of individual preference, group consensus, and distance to consensus are defined, and a number of theoretical results about them are given. Various decision-oriented goals are characterized, and the theory is illustrated numerically using data from a small group communication experiment.
01 Dec 1977
TL;DR: A rigorous mathematical theory is developed for general sample sets in order to allow for future studies of an statistical nature relating the classification of a general set with that of its finite samples.
Abstract: The problem of clustering a sample set on the basis of a fuzzy resemblance structure is considered. A rigorous mathematical theory is developed for general (i.e., non-necessary finite) sample sets in order to allow for future studies of an statistical nature relating the classification of a general set with that of its finite samples. The theory is presented in two major steps. The first step deals with the formalization of the concept of cluster. Unlike previous approaches which defined clusters as the fuzzy sets obtained as a result of the application of an algorithmic procedure (frequently based on an optimization goal) to the sample set, in this work clusters are characterized as fuzzy subsets which must satisfy certain conditions (independent of the classification goals being pursued) in order to qualify as potential taxonomical elements. The qualifying conditions are expressed as fuzzy relational equations derived extending simple relational equations in the conventional set domain. The problems in making such transition are discussed. Conditions for existence of fuzzy classifications are derived in terms of a class of fuzzy resemblance relations, called "likeness" relations. These conditions are shown to be much less restrictive than the corresponding conditions for the conventional case. In addition, the resulting cluster families provide richer representations of the underlying taxonomical structures. The problem of likeness relation derivation is analyzed from several viewpoints. The second step in this theoretical development deals with the problem of optimal representation of a sample set as a union of clusters. Using again the procedure of extending conventional set formulations, it is shown that approaches based on the minimization of functionals defined over all possible cluster representations, which satisfy certain desirable properties, must necessarily optimize one member of a uniparametric functional family. It is also shown that the extension of conventional concepts to the fuzzy domain provides a generalization of the concept of a number of clusters (which is no longer required to be an integer) and of "prototype element" of a cluster.
TL;DR: The modified version of the definition of fuzzy consensus is used to find the set of all fuzzy prime implicants of a fuzzy switching function as defined previously.
01 Dec 1977
TL;DR: In this article, it was shown that the internal model principle is valid for regulators (plant, controller and exosystem) working in fuzzy, environments, and that this principle is also valid for control systems.
Abstract: In the setting of modern algebra, it is shown that the internal model principle is valid for regulators (plant, controller and exosystem) working in fuzzy, environments.
TL;DR: The principal advantages of this technique are the new concept of direct simplification via essential fuzzy prime implicants (without generation of all of the fuzzy primeimplicants), and the fact that the algorithm is the first one to be suitable for efficient computer implementation.
Abstract: This paper is concerned with the study of simplification of fuzzy switching functions. A novel algorithm for generating all fuzzy prime implicants is introduced, followed by a new method of simplification of fuzzy switching functions. This algorithm is then reduced to a simple algorithm that produces only those fuzzy prime implicants that are essential. There areonly two other valid techniques for the minimization of fuzzy switching functions in the literature,(1,2) and those methods are not very suitable for computerized application. Thus, the principal advantages of this technique are the new concept of direct simplification via essential fuzzy prime implicants (without generation of all of the fuzzy prime implicants), and the fact that the algorithm is the first one to be suitable for efficient computer implementation.
01 Dec 1977
TL;DR: The main purpose of this method is to describe and illustrate a formal procedure for constructing the graphic presentation of the hierarchical arrangement given the necessary information concerning the relation of each element to each other element.
Abstract: On the basis of fuzzy sets theory, we propose a method for structuring hierarchy for the several complex problems, and call it Fuzzy Structural Modeling (FSM) method. An important requirement for structural modeling of complex systems is that the necessary data is acquired and organized into a form such that a structural model can be developed. The main purpose of this method is to describe and illustrate a formal procedure for constructing the graphic presentation of the hierarchical arrangement given the necessary information concerning the relation of each element to each other element. The procedure permits an automatic development of the graphic structure that portrays the hierarchy.
TL;DR: Certain fundamental questions pertaining to the nature of automatic control are raised and the impact of fuzzy concepts on their resolution is discussed.
Abstract: Certain fundamental questions pertaining to the nature of automatic control are raised and the impact of fuzzy concepts on their resolution is discussed.
01 Dec 1977
TL;DR: A method for finding the optimal alternative in the presence of both, the fuzziness and the uncertainty, is presented to deal with those situations in which the ratings of the alternatives are known imprecisely and the state of the system is known with uncertainty.
Abstract: The concept of fuzzy sets is being increasingly utilized to deal with illdefined terms and variables. Mathods have been proposed to find the optimal alternative in presence of fuzzy variables. In this paper a method for finding the optimal alternative in the presence of both, the fuzziness and the uncertainty, is presented. This method deals with those situations in which the ratings of the alternatives are known imprecisely and the state of the system is known with uncertainty. The impreciseness of the ratings is represented using fuzzy sets and the expected fuzzy rating for each alternative is calculated. The existing methods for finding the optimal alternative on the basis of the fuzzy ratings may then be employed for finding the optimal alternative from the expected ratings. This method is extended to those situations in which the optimal alternative is to be determined on the basis of more than one aspects having varying amount, known imprecisely, of importance.
TL;DR: The application of fuzzy models to hazard detection in binary systems is discussed, and this leads to methods of detection for multiple zero and one hazards.
Abstract: This paper presents some theoretical considerations for the detection of hazards in combinational switching systems. The application of fuzzy models to hazard detection in binary systems is discussed, and this leads to methods of detection for multiple zero and one hazards. Finally, a method for theorem proving applicable to the detection method is presented.