The value of

$\frac{4 \sqrt{12}}{12 \sqrt{27}}$

$=\frac{4 \sqrt{2 \times 2 \times 3}}{12 \sqrt{3 \times 3 \times 3}}$

$=\frac{4 \times 2 \sqrt{3}}{12 \times 3 \sqrt{3}}$

$=\frac{2}{9}$

Hence, the correct answer is option (b).