An improved algorithm on the content of realizable fuzzy matrices

Abstract

An $n{\times}n$ fuzzy matrix A is called realizable if there exists an $n{\times}t$ fuzzy matrix B such that $A=B\odot B^{T},$ where $\odot$ is the max–min composition. Let $r(A)={min}\{p:A=B\odot B^{T}, B\in L^{n\times p}\}.$ Then $r(A)$ is called the content of A. Since 1982, how to calculate r(A) for a given $n{\times}n$ realizable fuzzy matrix A was a focus problem, many researchers have made a lot of research work. X. P. Wang in 1999 gave an algorithm to find the fuzzy matrix B and calculate r(A) within $[r(A)]^{n^{2}}$ steps. Therefore, to find a simpler algorithm is a problem what we have to consider. This paper makes use of the symmetry of the realizable fuzzy matrix A to simplify the algorithm of content $r(A)$ based on the work of Wang (Chin Ann Math A 6: 701–706, 1999).