Decomposition And Resolution Of Fuzzy Relation Equations Based On Boolean Implications

The resolution problem of fuzzy relation equations (FRES) is a very important subject in fuzzy sets and systems, the majority of fuzzy inference systems can be implemented by using FERS. It is well known that there are different types of FRES corresponding to different fuzzy relation composite operations. At present, both sup -t composition FRE and inf -θ composition FRE (where θ is an implication) have been mainly studied. The material forms of compositions are presented in the following: Let A = (aij)n×m be a fuzzy input, R = (rij)m×k be a fuzzy relation, then a fuzzy output B = (bij)n×k is given by the following formula:The sup - t composition of A and R is the following:the inf-θ composition of A and R is the following:The concept of max-min composition FRES and their solutions originated by Sanchez in 1976 have been developed by quite a few researchers and have also provided the mathematical foundation and an effective methodology for the application of fuzzy set theory to various fields. In 1985, Miyakoshi and shimbo have investigated the two types of relational compositions based on a i-norm and its residual operator called jR-implication (i.e. T-composition and at-composition) and have given the solutions of the two equations.This works aims to study the type of inf - 9 FRES, where 6 is any Bloolean implication. Based upon fl-implications, S-implications and QL-implications, we discuss the resolution problem of the following two types of FRES:(I) Given A = (aij)n×m,B - (6ij)n×k, determine X = (xij)m×k, satisfying:(II) Given R = (rij}m×k,B = (bij)n×k, determine X = (zIJ)n×m, satisfying:Especially, the general decomposition methods of the two types of FRES are introduced in this paper. By decomposing the matrices of the two FRES, the elements of the two FRES are presented by the form of vectors, which is greatly convenient for the resolution of the two FRES. Some new solvability criteria based on the notations of 'solution matrices' and 'mean solution matrices' are suggested. When the solution sets of the two FERS are nonempty we show that Eq.(I) has a unique minimal-solution and finite maximal-solutions, and that Eq.(II) has a unique maximal-solution and finite minimal-solutions. On the other hand, the complete solution sets of the two FRES are determineded by their minimal-solution and maximal-solution sets, and we give effective methods of calculating their complete solution sets. Finally, we provide some numerical examples to illustrate the methodologies proposed in the paper.