Find the starting solution in the following transportation problem by a$)$ north$-$west corner method, b$)$ least cost method, c$)$ Vogel$$s approximation method. Also, obtain optimum solution by using the best starting solution:

D$1$

D$2$

D$3$

D$4$

Supply

S$1$

$3$

$7$

$6$

$4$

$5$

S$2$

$2$

$4$

$3$

$2$

$2$

S$3$

$3$

$3$

$8$

$5$

$3$

Demand

$3$

$3$

$2$

$2$