Resolution of a System of the Max-Product Fuzzy Relation Equations Using $$ B\circ B^t $$-Factorization

Abstract

In this paper, the Cholesky-factorization is extended to the square fuzzy symmetric matrix with respect to the max-product composition operator called \( B\circ B^t \)-factorization. Equivalently, we will find two fuzzy triangular matrices B and \(B^t\) for a fuzzy square symmetric matrix A such that \(A =B\circ B^t\) , where “\(\circ \)” is the max-product composition. An algorithm is presented to find the matrix B. Furthermore, some necessary and sufficient conditions are proposed for the existence and uniqueness of the \( B\circ B^t \)-factorization for a given fuzzy square symmetric matrix A. An algorithm is also proposed to find the solution set of a square system of Fuzzy Relation Equations (FRE) using the \( B\circ B^t \)-factorization.