Solutions Of Fuzzy Relation Equations Research Articles

Numerical analysis on fuzzy relation equations with various operators

Abstract In this study we investigate the properties of the approximate solutions of fuzzy relation equations for composite operators defined by max-min, Hamacher and its special case, disjunctive sum - conjunctive product. A quasi-Newton method is adopted to find the solutions. Apart from particular issues relevant to individual operators, comparisons among the various operators are made.

Equations of fuzzy relations defined on fuzzy subsets

Abstract In this paper some results of the theory of fuzzy relation equations are generalized, when we consider fuzzy relations defined on fuzzy sets instead of crisp sets. If the extension of classical definitions for composition of relations [12, 13] is not straightforward, we provide suitables counterexamples illustrating the necessity of change in definitions. We find the greatest (or the least) solutions of fuzzy relation equations with different types of compositions, like the sup- T , inf- T ′, inf-ϕ or sup-β compositions.

符号行列によるファジィ関係式の解法

ファジィ関係式は 1967 年に Sanchez によって提案され, その後多くの研究者によってファジィ関係式の解法に関する発表がなされた.しかし, 互いに包含関係にない解集合だけを求める方法は報告なされていない.本解法は塚本らの解法を基にするものである.塚本の解法では, 解を求めるための行列W^kのうち, 解が空集合となるWがあり, ある解集合が他の解集合に包含される場合があるることに対して, 大小を比較する符号を行列に導入し, 符号行列により包含関係にない解のみを得る方法を提案する.さらに, 解が存在しない場合に, 行列TとWにより簡単に判別できることを示す.

Multifactorial functions in fuzzy sets theory

The multifactorial function is a very important new concept which can be used to many aspects in fuzzy sets theory. It appeared firstly in [5] where it was used to define fuzzy perturbation function and where the stability for the solutions of fuzzy relation equations by using fuzzy perturbation was studied. In [6], by means of multifactorial functions, multifactorial fuzzy sets and the multifactorial degree of nearness, which are two new concepts too, were given, and they were used to multifactorial pattern recognition and clustering analysis with fuzzy characteristics. In fact, generally speaking, most of the mathematical models dependent on several factors should use multifactorial functions. Especially, fuzzy decision-making, fuzzy games, fuzzy programming and fuzzy linear programming with several objective functions should. Here we study properties of multifactorial functions, how to generate a new multifactorial function, and we introduce in detail its applications in fuzzy pattern recognition, fuzzy decision-making, fuzzy clustering, fuzzy games and fuzzy programming, etc.

Approximate solutions of fuzzy relational equations

A problem is studied of solving a system of fuzzy relational equations A i ∘ R = B i for known fuzzy sets A i and B i . It is pointed out that this is closely related to a task of discovering a structure of the data set ( A i , B i ), i = 1, 2,…, N, which plays an important role with respect to the solvability and identifiability property of the system of equations. Two main algorithms are proposed: their background is discussed with respect to solvablility of the system of equations. The relevant technical details are clarified in depth.

Fuzzy perturbation analysis, Part 2: Undirectional perturbation

In this paper, the stability of the solutions of fuzzy relation equations is discussed by the method of undirectional fuzzy perturbation. First, a general definition of ‘fuzzy perturbation’ is given. From this, the concept of fuzzy perturbation equation is introduced, and thus stability of the solutions is defined. Next, the degree of stability of the solutions, a kind of metric of the stability, is advanced. Three ordered quotient sets are induced by use of the degree of stability. These quotient sets have a well chained characteristic.

On measures of fuzziness of solutions of fuzzy relation equations with generalized connectives

By using a general class of fuzzy connectives of Yager [ Fuzzy Sets and Systems 4 (1980) , 235–242], Pedrycz [Fuzzy relational equations with generalized convectives and their applications, Fuzzy Sets and Systems 10 (1983) , 185–201] has shown that the classical fuzzy relation equations of Sanchez [ in “Fuzzy Automata and Decision Processes” (M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds.), pp. 221–234, North-Holland, Amsterdam, 1977 ] can be considered as a particular case of a more extensive class of fuzzy equations. For such types of equations, in this paper the solutions having the greatest energy measure and the smallest possible entropy measure of fuzziness are characterized.

Fuzzy relation equations and inequalities

This paper generalizes a method of solution of fuzzy relation equations introduced by Sanchez 1976 and developed by Pedrycz 1982. We find the greatest (or the least) solution of a composite L-fuzzy relation equation with arbitrary monotone composition.