An improved algorithm on the content of realizable fuzzy matrices
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 maxmin 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: 701706, 1999).
algorithm content realizable fuzzy matrix relation