In a transportation problem $3$ sources with capacities $30,40$ and $20$ units respectively, can cater to the demands of $4$ destinations with demands $40,20,30$ and $20$ units respectively. The unit transportation cost matrix is given in the following table. Write the LPP formulation of this transportation problem. Carry out one iteration of identifying a Basic Feasible Solution for the problem using North$-$West Corner rule, Least Cost method and Vogel$$s Approximation method respectively.

Destinations

P

Q

R

S

Sources

A

$4$

$6$

$7$

$8$

B

$5$

$8$

$7$

$9$

C

$7$

$7$

$6$

$5$