Who's who in fuzzy sets

In this paper some fuzzy relation equations provided with one solution on a finite set are characterized: we consider fuzzy relation equations H ° Q = T with Q?F(XxY) and card X ? cardY. After recalling the definition of equivalent fuzzy relation equations, we introduce the definition of ρ-equivalent ones, which allows us to constrain our research without loss of generality to fuzzy relation equations where T does not have zero-components.     

Convex-cone-based comparisons of and difference evaluations for fuzzy sets

Abstract

This paper focuses on how to compare two fuzzy sets and, from the viewpoint of set optimization, proposes eight types of fuzzy-set relations based on a convex cone as new comparison criteria of fuzzy sets. Then, difference evaluation functions for fuzzy sets are introduced. Under suitable assumptions of certain compactness and stability of fuzzy sets, we show that these functions correspond well to the fuzzy-set relations. In addition, through transforming these functions stepwise, we deal with numerical calculation methods of them in particular cases. Consequently, we can judge whether each fuzzy-set relation holds or not for given two fuzzy sets with the aid of computers.     

Extracting compact and information lossless sets of fuzzy association rules

Applying classical association rule extraction framework on fuzzy datasets leads to an unmanageably highly sized association rule sets. Moreover, the discretization operation leads to information loss and constitutes a hamper towards an efficient exploitation of the mined knowledge. To overcome such a drawback, this paper proposes the extraction and the exploitation of compact and informative generic basis of fuzzy association rules. The presented approach relies on the extension, within the fuzzy context, of the notion of closure and Galois connection, that we introduce in this paper. In order to select without loss of information a generic subset of all fuzzy association rules, we define three fuzzy generic basis from which remaining (redundant) FARs are generated. This generic basis constitutes a compact nucleus of fuzzy association rules, from which it is possible to informatively derive all the remaining rules. In order to ensure a sound and complete derivation process, we introduce an axiomatic system allowing the complete derivation of all the redundant rules. The results obtained from experiments carried out on benchmark datasets are very encouraging. They highlight a very important reduction of the number of the extracted fuzzy association rules without information loss.     

Convex fuzzy sets

This paper gathers some elementary known results about convex fuzzy sets and completes the theory, introducing the necessary concepts. Using a representation theorem for fuzzy subspaces it gives separation theorems for convex fuzzy sets in the proper setting.     

A fuzzy ordering on multi-dimensional fuzzy sets induced from convex cones

A fuzzy ordering for fuzzy sets on is presented by a fuzzy relation on which is induced by closed convex cones. The suitability of the fuzzy order is discussed using the axioms A₁–A₇ in (Fuzzy Sets and Systems 118 (2001) 375). For fuzzy sets on which are incomparable with respect to the fuzzy order, a method to evaluate the degree of satisfaction regarding the fuzzy order is presented by using a subsethood degree. Approximation by discrete cases is discussed for numerical calculation on the degree of the fuzzy order. Numerical examples are also given to illustrate our idea.     

The Arrow paradox with fuzzy preferences

This note considers a new factorization of a fuzzy weak binary preference relation into its asymmetric and symmetric parts. Arrow’s General Possibility Theorem is then examined within the resulting framework of vague individual and social preferences. The outcome of this exercise is compared with some earlier results available in the literature on the Arrow paradox with fuzzy preferences.     

Group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development

The purpose of this study is not only to build a group decision making structure model of risk in software development but also to propose two algorithms to tackle the rate of aggregative risk in a fuzzy environment by fuzzy sets theory during any phase of the life cycle. While evaluating the rate of aggregative risk, one may adjust or improve the weights or grades of the factors until she/he can accept it. Moreover, our result will be more objective and unbiased since it is generated by a group of evaluators.     

Parallel processing for fuzzy sets operations

Binary operations with fuzzy sets using parallel computing are investigated. A parallel algorithm for merging pairs is presented with a special functionality of the comparators.     

A computer program for fuzzy reasoning

An interactive computer program is described which implements the procedure proposed in “A Formal System for Fuzzy Reasoning” [1]. The problem in question is that of deciding what conclusions may be drawn in the presence of (posibly conflicting) evidence provided, generally with associated partial degrees of belief, by several sources of differing reliability. In using the program, each piece of evidence is entered as a sentence (using the terms NOT, AND, OR, IMPLIES as necessary), with an associated ‘degree of belief’ and ‘weight’; followed by a tentative conclusion. The system returns the degree(s) of belief and weight(s) which may rationally be attached to the conclusion. Copies of the program, written in FORTRAN IV (870 lines) have been lodged with the program libraries CUBE, DECUS, and SHARE, or may be obtained by writing to the author.     

Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

Axiomatics for fuzzy rough sets

A fuzzy T-rough set consists of a set X and a T-similarity relation R on X, where T is a lower semi-continuous triangular norm. We generalize the Farinas-Prade definition for the upper approximation operator of a fuzzy T-rough set (X, R); given originally for the special case T = Min, to the case of arbitrary T. We propose a new definition for the lower approximation operator of (X,R). Our definition satisfies the two important identities and , as well as a number of other interesting properties. We provide axiomatics to fully characterize those upper and lower approximations.     

Groups operating on fuzzy sets

In crisp environment the notions of normal subgroup and group operating on a set are well known due to many applications. In this paper we study extensions of these classical notions to the larger universe of fuzzy sets. We obtain a characterization of operations of a fuzzy group on a fuzzy set in terms of homomorphisms of crisp groups.     

The role of fuzzy sets in civil engineering

Civil Engineering is an art, the practise of which requires the use of scientific knowledge as a basic tool. The engineer has to use considerable judgement in setting up and interpreting his scientific calculations and in making decisions based upon incomplete information. Because a very high safety level is required in civil engineering structures the uncertainty associated with the application of scientific calculations is very crudely and conservatively estimated using traditional methods. Current reliability theory enables a discussion of random parameter uncertainty but in civil engineering, system uncertainty and the possibility of human error is extremely important and must be included in any reliability calculations.The potential role of fuzzy sets in analysing system and human uncertainty is discussed in the paper and two numerical examples are given of the calculation of system and random uncertainty using probability measures of fuzzy sets.     

Evaluation of fuzzy regression models by fuzzy neural network

In this paper, a novel hybrid method based on fuzzy neural network for approximate fuzzy coefficients (parameters) of fuzzy linear and nonlinear regression models with fuzzy output and crisp inputs, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate parameters, a simple algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.     

Perturbation of fuzzy sets and fuzzy reasoning based on normalized Minkowski distances

In this paper, we extend the concept of the perturbation of fuzzy sets based on normalized Minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets. These operations are induced by triangular norms and conorms. Furthermore, we discuss the perturbation of fuzzy reasoning.