I am formulating a MILP and one of the constraints is 

$\min\{\max\{0,y-x+a\},b\} \leq c$.

with decision variables $x, y \geq 0$ and $a,b,c$ as constants.

How would I ideally introduce auxiliary variables and formulate constraints to get rid of the nonlinearity of the $\min$ and $\max$ functions?