  In this paper, we introduce and analyze a new $LU$-factorization technique
for square matrices over idempotent semifields. In particular, more emphasis is
put on "max-plus" algebra here, but the work is extended to other idempotent
semifields as well. We first determine the conditions under which a square
matrix has $LU$ factors. Next, using this technique, we propose a method for
solving square linear systems of equations whose system matrices are
$LU$-factorizable. We also give conditions for an $LU$-factorizable system to
have solutions. This work is an extension of similar techniques over fields.
Maple procedures for this $LU$-factorization are also included.
