Svetlana V. Asmuss

Extremal problems of approximation theory in fuzzy context

Abstract The problem of approximation of a fuzzy subset of a normed space is considered. We study the error of approximation, which in this case is characterized by an L -fuzzy number. In order to do this we define the supremum of an L -fuzzy set of real numbers as well as the supremum and the infimum of a crisp set of L -fuzzy numbers. The introduced concepts allow us to investigate the best approximation and the optimal linear approximation. In particular, we consider approximation of a fuzzy subset in the space L p m of differentiable functions in the L q -metric. We prove the fuzzy counterparts of duality theorems, which in crisp case allows effectively to solve extremal problems of the classical approximation theory.

Approximation properties of higher degree F-transforms based on B-splines

The paper deals with the F-transform with polynomial components with respect to a generalized fuzzy partition given by B-splines. We investigate approximation properties of the inverse F-transform in this case and prove that using B-splines allows us to improve the quality of approximation of smooth functions.

A choice of bilevel linear programming solving parameters: factoraggregation approach

Our paper deals with the problem of choosing correct parameters for the bilevel linear program- ming solving algorithm proposed by M. Sakawa and I. Nishizaki. We suggest an approach based on fac- toraggregation, which is a specially designed general aggregation operator. The idea of factoraggregation arises from factorization by the equivalence relation generated by the upper level objective function. We prove several important properties of the factorag- gregation result regarding the analysis of param- eters in order to find an optimal solution for the problem. We illustrate the proposed method with some numerical and graphical examples, in particu- lar we consider a modification of the mixed produc- tion planning problem. operator. The idea of factoraggregation is based on factorization by an equivalence relation. We show that all properties of the definition of a general ag- gregation operator such as the boundary conditions and the monotonicity hold for the factoraggrega- tion operator. In the third section we show how factoraggregation can be applied for solving BLPP. This section is based on the interactive method of solution of bilevel linear programming problems in- troduced by M. Sakawa and I. Nishizaki (7),(8) and involving some parameters for the upper and lower level objectives. We illustrate with some numeri- cal and graphical examples showing how factorag- gregation is applied to the analysis of the choice of the parameters for solving BLPP. Several impor- tant properties of the result of factoraggregation are proved, these properties help us in the process of choosing the solving parameters. In the final section we illustrate the factoraggregation approach with the analysis of solving parameters for one particu- lar problem called the mixed production planning problem. We modify the problem described in (3) by considering new objective functions: we maxi- mize the profit of a production company with the higher priority and minimize the volume of environ- mentally damaging products and the dependence of external suppliers. We give numerical values for the parameters of the problem and describe how the analysis of solving parameters could be performed. 2. BLPP fuzzy solution approach

An Analysis of Bilevel Linear Programming Solving Parameters Based on Factoraggregation Approach

We introduce the notion of factoraggregation,which is a special construction of general aggregation operators, and apply it for an analysis of optimal solution parameters for bilevel linear programming problems. The aggregation observes lower level objective functions considering the classes of equivalence generated by an objective function on the upper level. The proposed method is illustrated with numerical and graphical examples.

On central algorithms of approximation under fuzzy information

We consider the problem of approximation of an operator by information described by n real characteristics in the case when this information is fuzzy. We develop the well-known idea of an optimal error method of approximation for this case. It is a method whose error is the infimum of the errors of all methods for a given problem characterized by fuzzy numbers in this case. We generalize the concept of central algorithms, which are always optimal error algorithms and in the crisp case are useful both in practice and in theory. In order to do this we define the centre of an L-fuzzy subset of a normed space. The introduced concepts allow us to describe optimal methods of approximation for linear problems using balanced fuzzy information.

Upper and lower generalized factoraggregations based on fuzzy equivalence relation

We develop the concept of a general factoraggre-gation operator introduced by the authors on the basis of an equivalence relation and applied in two recent papers for analysis of bilevel linear programming solving parameters. In the paper this concept is generalized by using a fuzzy equivalence relation instead of the crisp one. By using a left-continuous t-norm and its residuum we define and investigate two modifications of such generalized construction: upper and lower generalized factoraggregations. These generalized factoraggregations can be used for construction of extensional fuzzy sets.

Continuous and discrete higher-degree F-transforms based on B-splines

The paper deals with the continuous and discrete higher-degree fuzzy transforms (F-transforms with polynomial components) with respect to a generalized fuzzy partition given by B-splines. We investigate properties of the direct and inverse F-transforms in these cases and prove that using B-splines allows us to improve the quality of approximation of smooth functions.

General aggregation operators based on a fuzzy equivalence relation in the context of approximate systems

Our paper deals with special constructions of general aggregation operators, which are based on a fuzzy equivalence relation and provide upper and lower approximations of the pointwise extension of an ordinary aggregation operator. We consider properties of these approximations and explore their role in the context of extensional fuzzy sets with respect to the corresponding equivalence relation. We consider also upper and lower approximations of a t-norm extension of an ordinary aggregation operator. Finally, we describe an approximate system, considering the lattice of all general aggregation operators and the lattice of all fuzzy equivalence relations.

On Extensional Fuzzy Sets Generated by Factoraggregation

We develop the concept of a general factoraggregation operator introduced by the authors on the basis of an equivalence relation and applied in two recent papers for analysis of bilevel linear programming solving parameters. In the paper this concept is generalized by using a fuzzy equivalence relation instead of the crisp one. We show how the generalized factoraggregation can be used for construction of extensional fuzzy sets and consider approximations of arbitrary fuzzy sets by extensional ones.

Higher Degree F-transforms Based on B-splines of Two Variables

The paper deals with the higher degree fuzzy transforms (F-transforms with polynomial components) for functions of two variables in the case when two-dimensional generalized fuzzy partition is given by B-splines of two variables. We investigate properties of the direct and inverse F-transform in this case and prove that using B-splines as basic functions of fuzzy partition allows us to improve the quality of approximation.