  It was conjectured by McKernan and Shokurov that for any Fano contraction
$f:X \to Z$ of relative dimension $r$ with $X$ being $\epsilon$-lc, there is a
positive $\delta$ depending only on $r,\epsilon$ such that $Z$ is $\delta$-lc
and the multiplicity of the fiber of $f$ over a codimension one point of $Z$ is
bounded above by $1/\delta$. Recently, this conjecture was confirmed by Birkar.
In this paper, we give an explicit value for $\delta$ in terms of $\epsilon,r$
in the toric case, which belongs to $O(\epsilon^{2^r})$ as $\epsilon \to 0$.
When $r=2$, the order $O(\epsilon^4)$ is optimal by an example given by Alexeev
and Borisov.
