A Note On Advanced Normalization Algorithms And Their Applications

Abstract—The max-plus semiring is a semiring where the
addition operation is replaced with maximum operation and the
multiplication operation is replaced with the usual addition.
In this paper we suggest ‘Advanced normalization method’
and its algorithms to solve the max linear system and its
applications are discussed. We use these algorithms to determine the existence, uniqueness and the principal solution of
a max linear system. An important aspect of the advanced
normalization method is finding the free variables and the
leading variables of any solvable max linear system. We have
also found the generalized principal solution of the max linear
system with some special matrices such as row-wise arithmetic
matrix, column-wise arithmetic matrix, ϕ- diagonal matrix,
Jc×d(l) matrix, arithmetic circulant matrix. The generalized
principal solution of the max linear system had been found in
both cases, T
Jx = u with u ̸= 0 and T
Jx = 0. We have
given some important results on the principal solution of the
max linear system.