Union Of Fuzzy Sets Research Articles

Set operations of fuzzy sets using gradual elements

The conventional set operations of fuzzy sets are based on the membership functions using the $$\max $$ and $$\min $$ functions. In this paper, we shall consider the set operations of fuzzy sets based on the concepts of gradual sets and gradual elements. When the fuzzy sets can be formulated as consisting of gradual elements like the usual set consisting of usual elements, the intersection and union of fuzzy sets can be defined as the same way as the intersection and union of usual sets. In this case, the set operations of fuzzy sets will be similar to the set operations of crisp sets.

Longitudinal measures of fuzzy poverty: a focus on Czechia, Hungary and Poland after the crisis

The main scope of this paper consists in presenting an original contribution for the estimation of measures of monetary and multi-dimensional poverty in a longitudinal context; we also provide empirical on how this could be very useful in explaining the different recovery in three Visegrad Countries from the 2008 crisis. The traditional approach of poverty, defined by the dichotomy (poor, non-poor) is replaced by a defining based on the concept “a matter of degree”, which could briefly be defined as the position of an individual in the income distribution, or in any other variable’s distribution taken into account in a multidimensional approach. Another original contribution of the paper consists in proposing rules for the intersection and union of fuzzy sets, which we demonstrate to be appropriate for the study of poverty and deprivation, especially when the approach is applied to a time sequence in a longitudinal perspective; in particular, we propose the new ‘fuzzy “At-persistent-risk-of-poverty rate”, which can be seen as the fuzzy alter ego of the well-known Eurostat Laeken indicator.

AN ABSTRACT APPROACH TO CONSEQUENCE RELATIONS

AbstractWe generalise the Blok–Jónsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and Jónsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariablyaggregatedvia set-theoretical union. Our approach is more general in that nonidempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus,deductive relationsare defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic.

Optimizing of Linear Problems Subjected to Sugeno - Weber FRI

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno-Weber family of t-norms is considered as fuzzy composition. SugenoWeber family of t-norms and t-conorms is one of the most applied one in various fuzzy modeling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for socalled λ –fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Sugeno-Weber composition. A necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Sugeno-Weber fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.

Optimizing of Linear Problems Subjected to SugenoWeber FRI

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno-Weber family of t-norms is considered as fuzzy composition. SugenoWeber family of t-norms and t-conorms is one of the most applied one in various fuzzy modeling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for socalled λ –fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Sugeno-Weber composition. A necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Sugeno-Weber fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.

Factorization of fuzzy preferences

In the ordinary framework, the factorization of a weak preference relation into a strict preference relation and an indifference relation is unique. However, in fuzzy set theory, the intersection and the union of fuzzy sets can be represented different ways. Furthermore, some equivalent properties in the ordinary case have generalizations in the fuzzy framework that may be not equivalent. For these reasons there exist in the literature several factorizations of a fuzzy weak preference relation. In this paper we obtain and characterize different factorizations of fuzzy weak preference relations by means of two courses of action which are equivalent in the ordinary framework: axioms and definitions of strict preference and indifference.

Łukasiewicz Operations in Fuzzy Set and Many-Valued Representations of Quantum Logics

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered.

Subjective analysis of system reliability and its analyzer

Subjectivity, which is related to fuzziness, is inherent in system reliability analysis. Therefore, it is necessary to construct a system reliability analysis model into which subjectivity can be incorporated. This paper describes a model of subjective analysis of system reliability. In the model, information about system component reliability and human reliability is expressed by natural language or by numerical values. The information is translated into a fuzzy set that is defined on the unit interval. This fuzzy set is considered as a subjective reliability measure. The model applies a fault tree analysis and uses parametrized “and” and “or” operations of the fuzzy sets. These operations can cover the range of the reliability estimate from the most pessimistic estimate through the most optimistic one and reflect analyst's subjectivity towards the analyzed system. The model also considers dependence between subsystems and possibility of subsystem failure leading to a system accident. Analysis results are expressed by the use of natural language expressions. In this model various results can be obtained even if reliability of the same system is analyzed since analyst's subjectivity is incorporated into the analysis. This model considers two kinds of mutual agreements that are an intersection and a union of fuzzy sets, and the weighted average of fuzzy sets. This paper also proposes a prototype of an analyzer based on the present model in order to show the feasibility of the model. The analyzer is implemented on a workstation using the C language and the X-windows system. This analyzer is explained with the help of an example. Subjectivity can be incorporated into the system reliability analysis by the present model, and the subjective reliability analysis can be performed by the use of this analyzer. Finally, this paper compares the present method and the probabilistic method and emphasizes the usefulness of the present consideration.

Decomposition of properties and definition of fuzzy set

A mathematical model of fuzzy set is suggested in this paper. It is based on the understanding of a fuzzy set as a collection of objects showing some common property. This property itself is defined as decomposable into elementary properties. The degree to which an object shows a decomposable property, depends on the size, or more generally, importance of the collection of its elementary properties, that is evaluated by means of a set function that we refer to as pseudomeasure. With each pseudomeasure on a class of collections of elementary properties we associate a pseudomeasure that is complementary to it, and with each decomposable property, a complementary decomposable property interpreted as not showing the original property. We consider products of decomposable properties corresponding to the traditional operations of intersection and union of fuzzy sets. The model of a fuzzy set introduced leads to formulating compositions of decomposable properties (and associated fuzzy sets) that are not confined only to those based on operations of intersection and union, although they include them.

Operations on fuzzy sets—an axiomatic approach

We introduce intersection and union of fuzzy sets, constructed by means of operations on the unit interval which may depend on the considered element of the universe of discourse. A similar approach for complementation was given by Lowen [7]. The main result is that element-dependent operations are not compatible with the preservation of the intersection and union under the inverse image of a function. Finally we consider fuzzy σ-algebras built with these set-theoretic operations.

Applications of fuzzy set concepts to behavioral sciences

Because people do not seem to use hard-edged categories in everyday thought, fuzzy set theory has some intuitively appealing formalisms which appear suitable for modelling natural cognitive schema. This paper presents tools for applying fuzzy set concepts to the social and behavioral sciences, and examples of their uses. The following issues are addressed: (1) Definitions of fuzziness and its measurement; (2) The measurement of intercategory overlap or synonymy; (3) Fuzzy set inclusion and overlap as the basis for a new hierarchical clustering method; and (4) Fuzzy set union and intersection as the basis for regional-interpretive models in multidimensional scaling and fuzzy taxonomies. Most of the examples for this paper are taken from empirical research into individuals' cognitive representations of helping behavior.

Latent connectives in human decision making

The interpretation of a decision as the intersection of fuzzy sets, computed by applying either the minimum or the product operator to the membership functions of the fuzzy sets concerned implies that there is no compensation between low and high degrees of membership. If, on the other hand, a decision is defined to be the union of fuzzy sets, represented by the maximum or algebraic sum of the degrees of membership, full compensation is assumed. Managerial decisions hardly even represent any of these extremes. The aggregation of subjective categories in the framework of human decisions or evaluations almost always shows some degree of compensation. This indicates that human beings partially are using non-verbal aggregation procedures which do not correspond to the verbal and logical connectives ‘and’ and ‘or’. The results of our experiments support the hypothesis that people often use compensatory procedures. Several well-known operators are tested. However, they do not predict our data very well. Therefore a new class of operators is suggested which varies with respect to a parameter of compensation. Our data do confirm this concept.