jmc

The upper diagonal line is the graph of $2x+y=4$. The lower diagonal line is the graph of $x+y=1$. The y-axis is the graph of $x=0$ and the x-axis is the graph of $y=0$. The shaded region includes the solutions to the system. The longest side is the upper diagonal side. The length of this side is $\sqrt{4^2+2^2}=2\sqrt{5}.$ Thus $a+b=2+5=\boxed{7}$.