Consider the transportation of minerals from two mines to four plants. The allowable amount of minerals taken from Mine $1$ is $15$ tons and from mine $2$ is $13$ tons. The capacity of each plant is for $9,6,7,$ and $9$ tons is for plant $1$ to $4$ respectively. The unit transportation costs are given as

$\backslash$table$[[\backslash$table$[[$Mines$/],[$Plants$]],$Plant $1,$Plant $2,$Plant $3,$Plant $4,\backslash$table$[[$Mine$\text{'}$s$],[$Supply$]]],[$Mine $1,45,17,21,30,15],[$Mine $2,14,18,19,31,13],[$Plants$,,,,,],[$Capacity$,9,6,7,9,]]$

A$.$ Is this a standard linear transportation problem? If not convert it to a standard problem

B$.$ Use north west corner method or least cost method to find an initial basic feasible solution

C$.$ Use the $u-v$ method and basic feasible solution from part $b$ to find the optimal solution

D$.$ What is the optimal total cost of transportation

E$.$ Explain the meaning of any transportation from the dummy mine to a given plant