Question

A diet is to contain at least $80$ units of vitamin $A$ and $100$ units of minerals. Two foods $F_1$ and $F_2$ are available. Food $F_1$ costs Rs.$4$ per unit food and $F_2$s costs Rs.$6$ per unit. One unit of food $F_1$ contains $3$ units of vitamin $A$ and $4$ units of minerals. One unit of food $F_2$ contains $6$ units of vitamin $A$ and $3$ units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the mineral nutritional requirements?

Answer 1

Let the diet contain $x$ units of food $F_1$ and $y$ units of food $F_2$.

Therefore, $x\ge 0$ and $y\ge 0$
The given information can be compiled in a table as follows

	Vitamin A(units)	Mineral (units) Cost Per unit	(Rs)
Food $F_1\left(x\right)$	$3$	$4$ $4$
Food $F_2\left(y\right)$	$6$	$3$ $6$
Requirement	$80$	$100$

The cost of food $F_1$ is Rs.$4$ per unit and of food $F_2$ is Rs.$6$ per unit. Therefore, the constraints are
$3x+6y\ge 80$
$4x+3y\ge 100$
$x,y\ge 0$
Total cost of the diet, $Z=4x+6y$
The mathematical formulation of the given problem is
Minimise $Z=4x+6y......\left(1\right)$
subject the constraints
$3x+6y\ge \mathrm{80......}\left(2\right)$
$4x+3y\ge \mathrm{100.....}\left(3\right)$
$x,y\ge \mathrm{0.......}\left(4\right)$
The feasible region determined by the constraints is as given.
It can be seen that the feasible region is unbounded
The corner points are $A(\frac{80}{3},0),B(24,\frac{4}{3})$ and $C(0,\frac{100}{3})$
The values of $Z$ at these corner points are as follows.

Corner point	$Z=4x+6y$
$A(\frac{80}{3},0)$	$\frac{320}{3}=106.67$
$B(24,\frac{4}{3})$	$104$	$\to$ Minimum
$C(0,\frac{100}{3})$	$200$

As the feasible region is unbounded, therefore, $104$ may or may not be the minimum value $Z$
For this, we draw a graph of the inequality, $4x+6y<104$ or $2x+3y<52$, 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 $2x+3y<52$
Therefore, the minimum cost of the mixture will be Rs.$104$.