Abstract
The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system.
Highlights
The max-Łukasiewicz algebra, is one of the so-called max-T fuzzy algebras, which are defined for various triangular norms T.A max-T fuzzy algebra works with variables in the unit interval I = h0, 1i and uses the binary operations of maximum and a t-norm, T, instead of the conventional operations of addition and multiplication
The main result of this paper is description of a recognition algorithm for the parametric solvability problem
Existence of a bounded solution to a one-sided linear system in max-Łuk algebra has been considered in dependence on a given linear parameter factor on the fixed side of the system
Summary
The max-Łukasiewicz algebra (max-Łuk algebra, for short), is one of the so-called max-T fuzzy algebras, which are defined for various triangular norms T. The eigenvectors in a max-T algebra, for various triangular norms T, have applications in fuzzy set theory. Such eigenvectors have been studied in [5,7,22]. The Łukasiewicz conjunction can be used, for example, in describing backup of data on a computer, the maximal capacity of an oil tank or lump payment in finances Such applications often lead to systems of max-Łuk linear equations. The aim of this paper is to present an algorithm for recognizing solvability of a given one-sided max-Łuk linear system with bounded variables, in dependence of a linear parameter factor on the right side, see (9) and (10) for an exact formulation This problem has not yet been studied in the parametrized version. Discussion, comparison of the results with other papers, as well as future developments, are given in Conclusions
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