Question

An oil company has tow 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	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 $Rs.1$ per km, how should the delivery be scheduled in order that the transportation cost is minimum.

Answer 1

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 $7000-(x+y)\ge 0(i.e)x+y\le 7000$

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

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

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

$\Rightarrow x\le 4500$

$3000-y\ge 0$

$\Rightarrow$ $y\le 3000$

$3500-[7000-(x+y\left)\right]\ge 0$

$\Rightarrow x+y\ge 3500$

Step 2:

The cost of transportation per km for $10litre$ oil is $Rs.1.$

Hence the cost of transportation/km/litre$=Rs110=Rs0.1$

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

On simplifying we get,

$Z=0.3x+0.1y+3950$

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 $Rs.4400$ at $A(500,3000).$

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

$500L$,$3000L$ and $3500L$ of oil should be transported from depot $A$ to petrol pumps $D,E$ and $F$ and $4000L$,$0L$ of oil be transported from depot $B$ to petrol pumps $D,E$ and $F$ with minimum cost of transportation of $Rs.4400.$