The minimum value of $\left|Z\right|+\left|Z-3\right|$ occurs at points on the line segment joining $O$ and $A$
Similarly, the minimum value of $\left|Z+i\right|+\left|Z-3-2i\right|$ occurs at a point on the line segment joining $B$ and $C$.
$\Rightarrow$ Minimum value of $\left|Z\right|+\left|Z-3\right|+\left|Z+i\right|+\left|Z-3-2i\right|$ occurs at a point on the intersection of $OA$ and $BC$

Hence, the minimum value occurs at $Z=1$