  Max-plus algebra is a kind of idempotent semiring over
$\mathbb{R}_{\max}:=\mathbb{R}\cup\{-\infty\}$ with two operations $\oplus :=
\max$ and $\otimes := +$.In this paper, we introduce a new model of a walk on
one dimensional lattice on $\mathbb{Z}$, as an analogue of the quantum walk,
over the max-plus algebra and we call it max-plus walk. In the conventional
quantum walk, the summation of the $\ell^2$-norm of the states over all the
positions is a conserved quantity. In contrast, the summation of eigenvalues of
state decision matrices is a conserved quantity in the max-plus walk.Moreover,
spectral analysis on the total time evolution operator is also given.
