Question

Vitamin A and B are found in two different foods F1 and F2. One unit of food F1 contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F2 contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F1 and F2 cost Rs.50 and 25 respectively. The minimum daily requirement for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any things in excess of the daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F1 and F2 at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.

Answer 1

These types of problems are considered under the linear programming problems. In this we are given units of food F1 and F2 at different costs respectively. By calculating the equations we can calculate the optimum mixture of food at the minimum cost required of vitamin A and B.

Complete step-by-step answer:
Let us assume x units for food F1 and y units of food for F2.
As in the question it was given that food gets mixed.
So, we can clearly say that $x \ge 0$ and $y \ge 0$ .
As mentioned in the question that one unit of the food F1​ contains 2 units of the vitamin A and one unit of the food F2​ contains 4 units of the vitamin A.
Therefore, x and y units of food F1​and food F2​contains 2x and 4y units of vitamin A respectively.
It is clearly mentioned in the question that the minimum daily requirements for a person of vitamin A should be 40 units.
Let’s convert this statement into an equation for further solving, that is $2x + 4y \ge 40$ .
As mentioned in the question that one unit of the food F1​ contains 3 units of the vitamin B and one unit of the food F2​ contains 2 units of the vitamin B.
Therefore, x and y units of food F1 and food F2​contains 3x and 2y units of vitamin B respectively
It is clearly mentioned in the question that the minimum daily requirements for a person of vitamin A should be 50 units.
Let’s convert this statement into an equation for further solving, that is $3x + 2y \ge 50$
So a unit of food F1 will cost us Rs.50 whereas a unit of food F2 will ​cost us Rs.25.
Which means, x and y units of food F1​ and food F2​ will cost Rs.50x and Rs.25y respectively.
Let us use T to denote the total cost.
Then, $T = Rs\left( {50x + 25y} \right)$
Hence, the required LPP here is
Minimize $T = 50x + 25y$
As per these equations.
$\Rightarrow$ $2x + 4y \ge 40$
$\Rightarrow$ $3x + 2y \ge 50$
$\Rightarrow$ $x \ge 0$
$\Rightarrow$ $\;y \ge 0$

Note: Here, in the above question we solved by the costs of one unit of food F1 and F2 are Rs50 and Rs25 respectively, So, by using this cost and units we calculated different equations for different values.