Showing papers on "Fuzzy number published in 1988"

Fuzzy Sets, Uncertainty and Information

Abstract: (1990). Fuzzy Sets, Uncertainty, and Information. Journal of the Operational Research Society: Vol. 41, No. 9, pp. 884-886.

Possibility Theory: An Approach to Computerized Processing of Uncertainty

Abstract: 1. Measures of Possibility and Fuzzy Sets.- 1.1. Imprecision and Uncertainty.- 1.2. Traditional Models of Imprecision and Uncertainty.- 1.3. Confidence Measures.- 1.3.1. Measures of Possibility and of Necessity.- 1.3.2. Possibility and Probability.- 1.4. Fuzzy Sets.- 1.5. Elementary Fuzzy Set Operations.- 1.6. Practical Methods for Determining Membership Functions.- 1.6.1. Vague Categories as Perceived by an Individual.- 1.6.2. Fuzzy Sets Constructed from Statistical Data.- 1.6.3. Remarks on the Set of Degrees of Membership.- 1.7. Confidence Measures for a Fuzzy Event.- 1.8. Fuzzy Relations and Cartesian Products of Fuzzy Sets.- References.- 2. The Calculus of Fuzzy Quantities.- 2.1. Definitions and a Fundamental Principle.- 2.1.1. Fuzzy Quantities, Fuzzy Intervals, Fuzzy Numbers.- 2.1.2. The Extension Principle.- 2.2. Calculus of Fuzzy Quantities with Noninteractive Variables.- 2.2.1. Fundamental Result.- 2.2.2. Relation to Interval Analysis.- 2.2.3. Application to Standard Operations.- 2.2.4. The Problem of Equivalent Representations of a Function.- 2.3. Practical Calculation with Fuzzy Intervals.- 2.3.1. Parametric Representation of a Fuzzy Interval.- 2.3.2. Exact Practical Calculation with the Four Arithmetic Operations.- 2.3.3. Approximate Calculation of Functions of Fuzzy Intervals.- 2.4. Further Calculi of Fuzzy Quantities.- 2.4.1. "Pessimistic" Calculus of Fuzzy Quantities with Interactive Variables.- 2.4.2. "Optimistic" Calculus of Fuzzy Quantities with Noninteractive Variables.- 2.5. Illustrative Examples.- 2.5.1. Estimation of Resources in a Budget.- 2.5.2. Calculation of a PERT Analysis with Fuzzy Duration Estimates.- 2.5.3. A Problem in the Control of a Machine Tool.- Appendix: Computer Programs.- References.- 3. The Use of Fuzzy Sets for the Evaluation and Ranking of Objects.- 3.1. A Quantitative Approach to Multiaspect Choice.- 3.1.1. Basic Principles of the Approach.- 3.1.2. Fuzzy Set-Theoretic Operations.- 3.1.3. Application to the Combination of Criteria.- 3.1.4. Identification of Operators.- 3.1.5. Example.- 3.2. Comparison of Imprecise Evaluations.- 3.2.1. Comparison of a Real Number and a Fuzzy Interval.- 3.2.2. Comparison of Two Fuzzy Intervals.- 3.2.3. Ordering of n Fuzzy Intervals.- 3.2.4. Computer Implementation.- 3.2.5. Example.- Appendix: Computer Programs.- References.- 4. Models for Approximate Reasoning in Expert Systems.- 4.1. Remarks on Modeling Imprecision and Uncertainty.- 4.1.1. Credibility and Plausibility.- 4.1.2. Decomposable Measures.- 4.1.3. Vague Propositions.- 4.1.4. Evaluating the Truth Value of a Proposition.- 4.2. Reasoning from Uncertain Premises.- 4.2.1. Deductive Inference with Uncertain Premises.- 4.2.2. Complex Premises.- 4.2.3. Combining Degrees of Uncertainty Relative to the Same Proposition.- 4.3. Inference from Vague or Fuzzy Premises.- 4.3.1. Representation of the Rule "if X is A, then Y is B".- 4.3.2. "Generalized" Modus Ponens.- 4.3.3. Complex Premises.- 4.3.4. Combining Possibility Distributions.- 4.4. Brief Summary of Current Work and Systems.- 4.5. Example.- Appendix A..- Appendix B: Computer Programs.- References.- 5. Heuristic Search in an Imprecise Environment, and Fuzzy Programming.- 5.1. Heuristic Search in an Imprecise Environment.- 5.1.1. A and A* Algorithms.- 5.1.2. The Classical Traveling Salesman Problem (Reminder).- 5.1.3. Heuristic Search with Imprecise Evaluations.- 5.1.4. Heuristic Search with Fuzzy Values.- 5.2. An Example of Fuzzy Programming: Tracing the Execution of an Itinerary Specified in Imprecise Terms.- 5.2.1. Execution and Chaining of Instructions.- 5.2.2. Illustrative Example.- 5.2.3. Problems Arising in Fuzzy Programming.- 5.2.4. Concluding Remarks.- Appendix: Computer Programs.- A.1. Selection of "the Smallest" of N Fuzzy Numbers.- A.2. Tracing Imprecisely Specified Itineraries.- References.- 6. Handling of Incomplete or Uncertain Data and Vague Queries in Database Applications.- 6.1. Representation of Incomplete or Uncertain Data.- 6.1.1. Representing Data by Means of Possibility Distributions.- 6.1.2. Differences and Similarities with Other Fuzzy Approaches.- 6.1.3. Dependencies and Possibilistic Information.- 6.2. The Extended Relational Algebra and the Corresponding Query Language.- 6.2.1. Generalization of ?-Selection.- 6.2.2. Cartesian Product, ?-Join, and Projection.- 6.2.3. Union and Intersection-Redundancy.- 6.2.4. Queries Employing Other Operations.- 6.3. Example.- 6.3.1. Representation of Data.- 6.3.2. Examples of Queries.- 6.4. Conclusion.- Appendix: Computer Program.- A.1. Data Structures.- A.2. Representation of Queries.- A.3. Description of Implemeted Procedures.- References.

Fuzzy Mathematical Models in Engineering and Management Science

Abstract: (1990). Fuzzy Mathematical Models in Engineering and Management Science. Technometrics: Vol. 32, No. 2, pp. 238-238.

Comparison of fuzzy numbers based on the probability measure of fuzzy events

Abstract: Most approaches for ranking fuzzy numbers proposed in the literature are based on fuzzy sets theory only, and suffer from lack of discrimination and occasionally conflict with intuition. It is true that fuzzy numbers are frequently partial order and cannot be compared. However,this does not alleviate the need for comparison in practical applications. In this paper the order of fuzzy numbers are determined based on the concept of probability measure of fuzzy events due to Zadeh. It considers both the mean and dispersion of alternatives and gives two groups of indices based on the uniform and the proportional probability distributions. The approach is also extended to the comparison of random fuzzy numbers by means of a mean fuzzy number. It is shown that several comparison indices in the literature can be obtained based on the probability present measure approach. Finally some typical examples are used to compare the various different approaches. The different interpretations of the dispersion index under different physical situations are emphasized.

Fuzzy Petri nets for rule-based decisionmaking

Abstract: The technique of fuzzy reasoning by transformations of fuzzy truth state vectors by fuzzy matrices is extended to Petri nets. The result is a novel type of neural network in which the transition bars serve as the neutrons, and the nodes are conditions. Conditions may be conjuncted and disjuncted in a natural way to allow the firing of the neurons. The neuron fires to feed the implication truths into one or more consequent conditions when the MIN of the truth values of the antecedent conditions is greater than the neuron threshold. Disjunctions are also modeled in a natural way. Modifications are made to the usual Petri model to allow fuzzy rule-based reasoning by propositional logic. First, fuzzy values are allowed for rules and truths of conditions that appear in rules. Next, multiple copies, rather than the original, of the fuzzy truth tokens are passed along all arrows that depart a node or transition bar where the truth resides. An algorithm is presented for reasoning using these networks, as well as a simple example for exercising the algorithm. Abduction may be done analogously be reversing all arrows and propagating truth tokens backwards. >

Fuzzy functional dependencies and lossless join decomposition of fuzzy relational database systems

Abstract: This paper deals with the application of fuzzy logic in a relational database environment with the objective of capturing more meaning of the data It is shown that with suitable interpretations for the fuzzy membership functions, a fuzzy relational data model can be used to represent ambiguities in data values as well as impreciseness in the association among them Relational operators for fuzzy relations have been studied, and applicability of fuzzy logic in capturing integrity constraints has been investigated By introducing a fuzzy resemblance measure EQUAL for comparing domain values, the definition of classical functional dependency has been generalized to fuzzy functional dependency (ffd) The implication problem of ffds has been examined and a set of sound and complete inference axioms has been proposed Next, the problem of lossless join decomposition of fuzzy relations for a given set of fuzzy functional dependencies is investigated It is proved that with a suitable restriction on EQUAL, the design theory of a classical relational database with functional dependencies can be extended to fuzzy relations satisfying fuzzy functional dependencies

A new approach to handling fuzzy decision-making problems

Abstract: New techniques for handling fuzzy decision-making problems are introduced. Fuzzy production rules and fuzzy set theory are used for knowledge representation. In a classical production rule, the rule is executed if the pattern of its antecedent portion D/sub i/ perfectly matches the pattern of a set M of manifestations. However, in a fuzzy production rule, the rule is executed if the degree of matching is not less than a certain matching threshold value. By using a vector representation method, the antecedent portion of the fuzzy production rule and the set of manifestations can be represented by vectors of values and features, respectively. Then, a matching function can be used to measure the degree of similarity between the vectors, and the strength of confirmation calculation method can be used on the consequence d/sub i/ caused by M. An efficient algorithm to generate the maximum fuzzy cover of M to help the decision-maker make his decisions is proposed. >

Neighborhood systems and relational databases

Abstract: Queries in database can be classified roughly into two types: specific targets and fuzzy targets. Many queries are in effect fuzzy targets, however, because of lacking the supports, the user has been emulating them with specific targets by retiring a query repeatedly with minor changes. In this paper, we augment the relational database, with neighborhood systems, so the database can answer a fuzzy query. There have been many works to combine relational databases and fuzzy theory. Bucklles and Petry replaced attributes values by sets of values. Zemankova-Leech, Kandel, and Zviell used fuzzy logic. The formalism of present work is quite general, it allows numerical or nonnumerical measurements of fuzziness in relational databases. The fuzzy theory present here is quite different from the usual theory. Our basic assumption here is that: the data are not fuzzy, the queries are.Motro [Motr86] introduced the notion of distance into the relational databases. From that he can, then, define the notion of “close-ness” and develop goal queries. Though “distance” is a useful concept, yet very often the quantification of it is meaningless or extremely difficult. For example, “very close”, “very far” are meaningful concept of distance, yet there is no practical way to quantity them for all occasions. Our approach here is more direct, we define directly the meaning of “very close neighborhood”. Using the concept of neighborhoods is not very original, in fact, in the theory of topological spaces [Dugu66], mathematician has been using the “neighborhood system” to study the phenomena of “close-ness”. In the territory of fuzzy queries, the notion of “neighborhood” captures the essence of the qualitative information of “close-ness” better than the brute-force-quantified information (distance). A “fuzzy” neighborhood is a qualitative measure of fuzziness.On the surface, it seems a very complicated procedure to define a neighborhood for each value in the attribute. In fact, if we use the characteristic function (membership function) to define a subset, then the defining procedure is merely another type of distance function (non-measure distance or symbolic distance). Now, to define the neighborhood system one can simply re-entered the third column of the relation with linguistic values: “very close”, “close”, “far”. Note that there is a “greater than” relation among these linguistic values. In mathematical terms, they forms a lattice [Jaco60]. For technical reason, we require the values in third column be elements of a lattice. Note that real number is a lattice, so we get Motro's results back.

Fuzzy concepts in expert systems

Abstract: The authors present a comprehensive expert-system building tool, called System Z-II, that can deal with exact, fuzzy (or inexact), and combined reasoning, allowing fuzzy and normal terms to be freely mixed in the rules and facts of an expert system. This fully implemented tool has been used to build several expert systems in the fields of student curriculum advisement, medical diagnosis, psychoanalysis, and risk analysis. System Z-II is a rule-based system that uses fuzzy logic and fuzzy numbers for its inexact reasoning. It uses two basic inexact concepts, fuzziness and uncertainty, which are distinct from each other in the system. >

Recent convergence results for the fuzzy c-means clustering algorithms

Abstract: One of the main techniques embodied in many pattern recognition systems is cluster analysis — the identification of substructure in unlabeled data sets. The fuzzy c-means algorithms (FCM) have often been used to solve certain types of clustering problems. During the last two years several new local results concerning both numerical and stochastic convergence of FCM have been found. Numerical results describe how the algorithms behave when evaluated as optimization algorithms for finding minima of the corresponding family of fuzzy c-means functionals. Stochastic properties refer to the accuracy of minima of FCM functionals as approximations to parameters of statistical populations which are sometimes assumed to be associated with the data. The purpose of this paper is to collect the main global and local, numerical and stochastic, convergence results for FCM in a brief and unified way.

Spatial classification using fuzzy membership models

Abstract: In the usual statistical approach to spatial classification, it is assumed that each pixel belongs to precisely one of a small number of known groups. This framework is extended to include mixed-pixel data; then, only a proportion of each pixel belongs to each group. Two models based on multivariate Gaussian random fields are proposed to model this fuzzy membership process. The problems of predicting the group membership and estimating the parameters are discussed. Some simulations are presented to study the properties of this approach, and an example is given using Landsat remote-sensing data. >

Fuzzy cognitive maps, signal flow graphs, and qualitative circuit analysis

Abstract: Fuzzy cognitive maps (FCMs) represent a means of fuzzy causal knowledge processing, using the net rather than the traditional tree knowledge representation. The FCM approach allows various knowledge bases to be combined. Similarities between the FCMs and signal flow graphs (SFGs) are pointed out and the inference process used in FCMs is compared in parallel with a fixed point iterative solution of the equations describing the SFG. Then, applications to qualitative circuit analysis are discussed for a class of feedback amplifiers and general active RLC circuits, using a combination of the SFG and FCM concepts. Several examples are given. >

Combining Fuzzy Imprecision With Probabilistic Uncertainty in Decision Making

Abstract: Essay on the history of the development of many-valued logics and some related topics.- 1. Introductory Sections.- Uncertainty aversion and separated effects in decision making under uncertainty.- Essentials of decision making under generalized uncertainty.- Decision evaluation methods under uncertainty and imprecision.- 2. Basic Theoretical Issues.- Fuzzy random variables.- Fuzzy P-measures and their application in decision making.- Theory and applications of fuzzy statistics.- Confidence intervals for the parameters of a linguistic random variable.- On combining uncertainty measures.- On the combination of vague evidence of the probabilistic origin.- Fuzzy evaluation of communicators.- Uncertain associational relations: compatibility and transition relations in reasoning.- 3. Fuzzy Sets Involving Random Aspects.- Stochastic fuzzy sets: a survey.- Probabilistic sets - a survey.- 4. Decision - Making - Related Models Involving Fuzziness and Randomness.- Decision making based on fuzzy stochastic and statistical dominance.- Decision making in a probabilistic fuzzy environment.- Randomness and fuzziness in a linear programming problem.- Comparison of methodologies for multicriteria feasibility -constrained fuzzy and multiple - objective stochastic linear programming.- Fuzzy dynamic programming with stochastic systems.- Probabilistic - possibilistic approach to some statistical problems with fuzzy experimental observations.- Estimation of life-time with fuzzy prior information: application in reliability.- Questionnaires with fuzzy and probabilistic elements.- From fuzzy data to a single action - a simulation approach.- 5. Applications.- Probabilistic sets in classification and pattern recognition.- Fuzzy optimization of radiation protection and nuclear safety.- Application of fuzzy statistical decision making in countermeasures against great earthquakes.- From an oriental market to the European monetary system: some fuzzy - sers - related ideas.

An approach to the comparison of fuzzy subsets with an alpha -cut dependent index

Abstract: An approach to the comparison of fuzzy subsets or fuzzy numbers is proposed and studied. A function-type index J/sub ij/( alpha ), indicating a degree of dominance of a fuzzy subset A/sub i/ over the other A/sub j/, is derived from an alpha -cut of the difference of the two subsets compared. This index can be converted to a single-valued version F/sub ij//sup 0/ by way of a alpha -weighted averaging over alpha for a quick reference to the conclusion of comparison. Linguistic descriptions of the results of comparison and a fuzzy ordering relation are discussed. For the case of trapezoidal membership functions the shapes of J/sub ij/( alpha ) are classified and the formulas to calculate J/sub ij//sup 0/ are obtained. >

Project network analysis with fuzzy activity times

Abstract: Triangular fuzzy numbers are used to represent project activities whose duration times are uncertain and cannot be represented stochastically. A comprehensive path analysis method is devised using fuzzy arithmetic and a fuzzy number comparison method to determine fuzzy project completion time and the degrees of criticality of each network path. Possibility theory is then used to determine the possibilities of the project completion given the fuzzy project completion time. The existing composite method and the new comprehensive comparison method are compared and contrasted.

Probabilistic fuzzy model for dynamic systems

Abstract: We present a model, based on a fuzzy relation obtained from fuzzy referential sets on the input and output spaces, for predicting the behaviour of nonlinear dynamic systems. The model can be made to learn from experience, and the computing requirements are modest, making online application feasible. Some numerical results are compared with those of earlier models.

A new approach to connective generation in the framework of expert systems using fuzzy logic

Abstract: A technique for modeling uncertainty in expert systems is presented. Operators are defined using linguistic terms to avoid any numerical representation. These operators consider linguistic term set ordering, constraints that are the counterpart of properties fulfilled by triangular norms in fuzzy logic, and additional restrictions created by the expert's procedure to combining certainty. The method avoids the usual problems that arise in other treatments of certainty linguistic terms, where they are represented as fuzzy numbers or fuzzy truth labels. One of the most significant problems is the lack of consensus in the representation of each term by a group of experts, due to the necessity of representing the terms in a pseudonumerical scale. >

An event based fuzzy temporal logic

Abstract: A temporal logic is developed to deal with events that are uncertain with regard to their occurrence in a given interval of time. Events are represented as fuzzy sets with the membership function giving the possibility of occurrence of the event in a given interval of time. An axiomatization of the fuzzy event calculus is presented, and several of its properties are proved. The logic is simple but powerful; it can determine effectively the various temporal relations between uncertain events or their combinations. >

Numerical scaling of human judgement in pairwise-comparison methods for fuzzy multi-criteria decision analysis

Abstract: We are concerned with the multi-criteria problem of how to choose the best item from a finite set of alternatives. We face such a decision problem when we want to attain a goal, and when none of the alternative tools for attaining it seems to be perfect. In general, we consider the alternatives from various viewpoints, and we estimate how well they enable us to attain the goal. Briefly speaking, we estimate the relative importance of the criteria, and we express it in numerical weights. Similarly, we estimate the relative importance of the alternatives under each of the criteria separately. Lastly, we aggregate the weights to obtain a final score for each alternative. With this information, we are in a position to rank and rate the alternatives, and to select the best compromise.