Step 1:

Let $x$ litres of oil be supplied from depot $A$ to petrol pump $D$ and $y$ litres of oil be supplied from depot $A$ to petrol pump $E$,then $7000-(x+y)\geq 0$ (i.e) $x+y\leq 7000$

The requirements of oil at petrol pumps $D,E$ and $F$ ar $4500l,3000l$ and $3500l$

The quantity of oil transported from depot $B$ to petrol pump $D,E$ and $F$ are $(4500-x),(3000-y)$ and $3500-[7000-(x+y)]$litres respectively.

(i.e)$4500-x\geq 0$

$\Rightarrow x\leq 4500$

$3000-y\geq 0$

$\Rightarrow y\leq 3000$

$3500-[7000-(x+y)]\geq 0$

$\Rightarrow x+y\geq 3500$

Step 2:

The cost of transportation per km for 10litre oil is Rs1.

Hence the cost of transportation/km/litre=Rs$\large\frac{1}{10}=$Rs 0.1

$\Rightarrow Z=0.7x+0.6y+0.3[7000-(x+y)]+0.3(4500-x)+0.4(3000-y)+0.2[(x+y)-3500]$

On simplifying we get,

$Z=0.3x+0.1y+3950$

Step 3:

The graph can be plotted according to the inequalities above and the feasible area is the shaded portion ABECD

Now the corner points are $A(500,3000),B(3500,0),E(4500,0),C(4500,2500)$ and $D(4000,3000)$

Step 4:

Now let us calculate the values of the objective function as follows :

At the points $(x,y)$ the value of the objective function subjected to $Z=0.3x+0.1y+3950$

At $A(500,3000)$ the value of the objective function subjected to $Z=0.3\times 500+0.1\times 3000+3950=4400$

At $B(3500,0)$ the value of the objective function subjected to $Z=0.3\times 3500+0+3950=5000$

At $E(4500,0)$ the value of the objective function subjected to $Z=0.3\times 4500+0+3950=5300$

At $C(4500,2500)$ the value of the objective function subjected to $Z=0.3\times 4500+0.1\times 2500+3950=5550$

At $C(4000,3000)$ the value of the objective function subjected to $Z=0.3\times 4000+0.1\times 3000+3950=5450$

Step 5:

It is clear that the minimum transport charges are Rs4400 at $A(500,3000)$.

When $x=500$ and $y=3000$,

500litres,3000 litres and 3500 litres of oil should be transported from depot A to petrol pumps $D,E$ and $F$ and 4000litres,0 litre of oil be transported from depot B to petrol pumps D,E and F with minimum cost of transportation of Rs4400.