  The higher dimensional Frobenius problem was introduced by a preceding paper
[Fan, Rao and Zhang, Higher dimensional Frobenius problem: maximal saturated
cones, growth function and rigidity, Preprint 2014]. %the higher dimensional
Frobenius problem was introduced and a directional growth function was studied.
  In this paper, we investigate the Lipschitz equivalence of dust-like
self-similar sets in $\mathbb R^d$. For any self-similar set, we associate with
it a higher dimensional Frobenius problem, and we show that the directional
growth function of the associate higher dimensional Frobenius problem is a
Lipschitz invariant.
  As an application, we solve the Lipschitz equivalence problem when two
dust-like self-similar sets $E$ and $F$ have coplanar ratios, by showing that
they are Lipschitz equivalent if and only if the contraction vector of the
$p$-th iteration of $E$ is a permutation of that of the $q$-th iteration of $F$
for some $p, q\geq 1$. This partially answers a question raised by Falconer and
Marsh [On the Lipschitz equivalence of Cantor sets, \emph{Mathematika,}
\textbf{39} (1992), 223--233].
