Question

An oil company has two depots $A$ and $B$ with capacities of $7000L$ and $4000L$ respectively. The company is to supply oil to three petrol pumps, $D,E$ and $F$ whose requirements are $4500L,3000L$ and $3500L$ respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:

Distance in (km.)
From/ To	$A$	$B$
$D$	$7$	$3$
$E$	$6$	$4$
$F$	$3$	$2$

Assuming that the transportation cost of $10$ litres of oil is $Re.1$ per km, how should the delivery be scheduled in order that the transportation cost is minimum?
What is the minimum cost?

Answer 1

Let's assume that depot A supplies $X$ L oil to petrol pump D and $Y$ L oil to petrol pump E.

So, depot A will supply remaining oil $7000-X-Y$ L to petrol pump F.

Also, depot B will supply $4500-X$ L oil to petrol pump D, $3000-Y$ L oil to petrol pump E and $3500-(7000-X-Y)=X+Y-350$ L oil to petrol pump F.

Since cost of transportation of 10 L oils for 1 km is 1 Rs.

So, per L oils for 1 km, cost = $\frac{1}{10}=0.1Rs$

Now total transportation cost is $Z=0.7\times X+0.6\times Y+0.3\times (7000-X-Y)+0.3\times (4500-X)+0.4\times (3000-Y)+0.2\times (X+Y-3500)$

$\Rightarrow Z=0.7X+0.6Y+2100-0.3X-0.3Y+1350-0.3X+1200-0.4Y+0.2X+0.2Y-700$

$\Rightarrow Z=0.3X+0.1Y+3950...\left(1\right)$

Now, Since depot A can supply maximum 4500 L to petrol pump D and 3000 L to petrol pump E and have maximum 7000 L capacity to supply.

$\therefore X+Y\le 7000,X\le 4500andY\le 3000...\left(2\right)$

Also, if depot A supplies all 3500 L to petrol pump F, then remaining 3500 L will be supplied to petrol pump D and E.

So, $X+Y\ge 3500...\left(3\right)$

Since, X and Y is amount of oil. It can never be negative.

So, $X\ge 0,Y\ge 0...\left(4\right)$

We have to minimise transportation cost given by equation (1).

After plotting all the constraints given by equation (2), (3) and (4), we got the feasible region as shown in the image.

Corner points	Value of $Z=0.3X+0.1Y+3950$
A (500, 3000)	4400 (Minimum)
B (3500, 0)	5000
C (4500, 0)	5300
D (4500, 2500)	5550
E (4000, 3000)	5450

Hence minimum transportation cost is $4400Rs$. Transporter needs to supply as follows:

Depot A to petrol pump D $=X=500$ L

Depot A to petrol pump E $=Y=3000$ L

Depot A to petrol pump F $=7000-X-Y=3500$ L

Depot B to petrol pump D $=4500-X=4000$ L

Depot B to petrol pump E $=3000-Y=0$ L

Depot B to petrol pump F $=X+Y-3500=0$ L