The Research On The Unique Solution Of Fuzzy Relation Equations With Max-product Composition And Strong Regularity Of Matrices

Along with depth research on practical problems,many fuzzy relation equations (FRE) are applied to solve many practical problems, such as decision-making theory,fuzzy con-trol. There have been some results on the unique solution of fuzzy relation equations with max-min composition and the regularity of fuzzy relation matrices.But,theories on other composition operators have not been seen in literatures.In this paper,we prove the:necessary and sufficient condition under which unique so-lution of FRE with max-product composition.To do so,the structures of maximal solutions are investigated.Furthermore, strong regularity of fuzzy relation matrix is considered.The main content is organized as following:In the second chapter, firstly, the necessary and sufficient condition for fuzzy relation equations with max-product composition having unique solution is proposed. Two rules for simplifying the problem are introduced. An algorithm for solving the unique solu-tion of fuzzy relation equations with max-product compositions is given. The numerical experiments are presented to illustrate the effectiveness of the algorithm.In the third chapter, we give a sufficient condition and two necessary condition on the regularity of matrices. Meanwhile, three different special matrix are induced and two different conclusions on the regularity are obtained.In the fourth chapter, an algorithm for checking the strong regularity of a matrix is given.Firstly, according to the classification of the elements of matrix.we get an algorithm for checking the pseudo-regularity of a matrix.After that,we check the strong regularity of a matrix from the pseudo-regularity by solving a system of the linear inequality.