Question

Vitamins 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 requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any thing in excess of 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

Let x and y units of food F₁ and food F₂ were mixed.
Clearly, x ≥ 0 and y ≥ 0

One unit of food F₁ contains 2 units of vitamin A and one unit of of food F₂ contains 4 units of vitamin A. Therefore, x and y units of food F₁ and food F₂ respectively contains 2x and 4y units of vitamin A.

It is given that the minimum daily requirements for a person of vitamin A is 40 units.
Hence, 2x+ 4y ≥ 40

One unit of food F₁ contains 3 units of vitamin B and one unit of food F₂ contains 2 units of of vitamin B. Therefore, x and y units of F₁ and F₂ respectively contains 3x and 2y units of vitamin B.

It is given that the minimum daily requirements for a person of vitamin B is 50 units.
Hence, 3x+ 2y ≥ 50

One unit of food F₁ and food F₂ cost Rs 50 and 25 respectively. Therefore, x and y units of food F₁ and food F₂ costs Rs 50x and Rs 25y respectively.
​Let Z denote the total cost

Then, Z = Rs (50x + 25y)

​Hence, the required LPP is

Minimize Z = 50x + 25y

subject to

2x+ 4y ≥ 40
3x+ 2y ≥ 50
x ≥ 0,y ≥ 0