Catálogo de publicaciones - libros

This paper is a continuation of the paper [ 22 ]. It goes on investigating a kind of ‘fuzzy limits’ based on the notion of nearness relation. We will study limits of functional sequences.

A probability theory on IFS events has been constructed in [ 2 ], and axiomatically in [ 4 ]. Here we use a general system of axioms for probabilities and observables. Using analogous results of probability theory on MV-algebras (see [ 6 ]) a version of the weak law of large numbers can be proved.

There are several approaches to the cardinality of fuzzy sets. One group of them are constructive approaches. Following these approaches we get single numbers (scalar cardinalities) or convex fuzzy sets (fuzzy cardinalities) as cardinalities of fuzzy sets. Wygralak in [ 8 ] has presented an axiomatic theory of the scalar cardinality of finite fuzzy sets which contains as particular cases all standard concepts of the scalar cardinality. In our contribution we will present possible extensions to the case of IF sets.

The shortest path problem is an optimization problem in which the best path between two considered objects is searched for in accordance with an optimization criterion, which has to be minimized. In this paper this problem is investigated in the case when the distances between the nodes are fuzzy numbers. The problem is formulated as a linear optimization problem with fuzzy coefficients in the objective function. This problem is solved using crisp parametric two-criterial linear optimization. Special emphasis is given to the sensitivity of the solution with respect to the fuzzy objective function coefficients.

Practical tasks of map coloring in case of objects groups’ allocation, not connected by any binary relation, come to the problem of coloring of graph [ 1 ]. This task is closely connected to the calculation of internal stable sets of graphs, calculation of chromatic number and a chromatic class of the graph.

Fuzzy constrained optimization problems have been extensively studied since the seventies. In the linear case, the first approaches to solve the so-called fuzzy linear programming problem were made in [ 12 ] and [ 15 ]. Since then, important contributions solving different linear models have been done and these models have been recipients of a great dealt of work. In the nonlinear case the situation is quite different, as there is a wide variety of specific and both practical and theoretically relevant nonlinear problems, each having a different solution method. In the following we consider a Nonlinear Programming problem with fuzzy constraints. From a mathematical point of view the problem can be addressed as: 1 $$ \begin{gathered} Min f(x) \hfill \\ s.t.:g_j (x) \lesssim b_j , j = 1, \ldots ,m \hfill \\ x_i \in [l_i ,u_i ], i = 1, \ldots ,n, l_i \geqslant 0 \hfill \\ \end{gathered} $$ where x = ( x _1, . . ., x _n) ∈ ℜ^n is a n dimensional real-valued parameter vector, [ l _i, u _i] ⊂ ℜ, b _j ∈ ℜ, f ( x ), g _j ( x ) are arbitrary functions, and the symbol ≲ indicates a fuzzy constraint [ 15 ]. Here we will consider the following linear membership function related to each fuzzy constraint: 2 $$ \mu _j (x) = \left\{ \begin{gathered} 0 if g_j (x) \geqslant b_j + d_j \hfill \\ h\left( {\tfrac{{b_j + d_j - g_j (x)}} {{d_j }}} \right) if b_j \leqslant g_j (x) \leqslant b_j + d_j \hfill \\ 1 if g_j (x) \leqslant b_j \hfill \\ \end{gathered} \right. $$ which gives the accomplishment degree of g _j ( x ), and consequently of x , with respect to the j -th constraint (the decision maker can tolerate violations of each constraint up to the value b _j + d _j, j = 1, . . ., m ). We assume that the function h is a arbitrary function which allows to represent accurately the accomplishment degree.

Researchers proposed many reasoning methods. However, many of the methods are suitable neither for fuzzy control nor for fuzzy modeling. In the paper some possible reasoning methods are compared from this point of view. The most popular approach to fuzzy control and modeling is based on if ... then rules. Using this approach four general problems must be solved: what interpretations of sentence connectives “and” “or” and negation “not” may be used for if part what implication or other operation may be used for conclusion (then) part what interpretation to use for rule aggregator “also” what defuzzification procedure can be applied.

The generation of new, useful knowledge is the mission of scientists. In the data mining area researchers try to find interesting new knowledge about data [ 1 , 2 ]. Especially in text mining interesting knowledge is extracted from texts [ 3 ]. Automated theorem proving aims at finding automatically theorems and proofs [ 4 ]. Genetic programming is used for automatically generating new programs [ 5 ]. To find new and interesting knowledge automatically , a system should be “computational intelligent” as much as possible.

In this paper we investigate the dynamic systems, which are represented by recurrent Takagi-Sugeno rule bases that are widely used in many applications. The main question to be answered is under what conditions the recurrent rule base can reconstruct the chaotic bit series. We use for this purpose so-called ‘backward interval mapping’.

In this study, we discuss the use of Dempster- Shafer theory as a well-rounded algorithmic vehicle in the construction of fuzzy decision rules. The concept of fuzzy granulation realized via fuzzy clustering is aimed at the discretization of continuous attributes. Next we use Genetic Algorithms (GA) to find the best points of division for discretization of continuous attributes. The rules, generated using Fuzzy Dempster-Shafer model (FDS), were verified by GA methods. The natural crossover improved by random changes (mutation and selection) can help us to find the best set of rules.