Question

A farmer mixes two brands $P$ and $Q$ of cattle feed. Brand $P$, costing Rs.$250$ per bag contains $3$ units of nutritional element $A$, $2.5$ units of element $B$ and $2$ units of element $C$. Brand $Q$ costing Rs.$200$ per bag contains $1.5$ units of nutritional elements $A$, $11.25$ units of element $B$ and $3$ units of element $C$. The minimum requirements of nutrients $A,B$ and $C$ are $18$ units, $45$ units and $24$ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag> What is minimum cost of the mixture per bag?

Answer 1

Let the farmer mix $x$ bags of brand $P$ and $y$ bags of brand $Q$.
The given information can be compiled in a table as follows.

	Vitamin A (units/kg)	Vitamin B (units/kg)	Vitamin C (units/kg)
Food P	$3$	$5$	$60$
Food Q	$4$	$2$	$80$
Requirement (units/kg)	$8$	$11$

The given problem can be formulated as follows
Minimise $z=250x+200y.........\left(1\right)$
subject to the constraints
$3x+1.5y\ge \mathrm{18........}\left(2\right)$
$2.5x+11.25y\ge \mathrm{45......}\left(3\right)$
$2x+3y\ge \mathrm{24........}\left(4\right)$
$x,y\ge \mathrm{0..............}\left(5\right)$
The feasible region determined by the system of constraints is as shown
The corner points of the feasible region are $A(18,0),B(9,2),C(3,6)$ and $D(0,12)$
The values of $z$ at these corner points are as follows

Corner point	$x=250x+200y$
$A(18,0)$	$4500$
$B(9,2)$	$2650$
$C(3,6)$	$1950$	$\to$ Minimum
$D(0,12)$	$2400$

As the feasible region is unbounded, therefore, $1950$ may or may not be the minimum value of $z$.
For this, we draw a graph of the inequality, $250x+200y<1950$ or $5x+4y<39$ and check whether the resulting half plane has points in common with the feasible region or not.
IT can be seen that the feasible region has no common point with $5x+4y<39$
Therefore, the minimum value of $z$ is $2000$ at $(3,6)$
Thus, $3$ bags of brand $P$ and $6$ bags of brand $Q$ should be used in the mixture to minimize the cost to Rs.$1950$