Question

A diet of a sick person must contain at least 48 units of vitamin A and 64 units of vitamin B. Two foods F ₁ and F₂ are available. Food F₁ costs Rs. 6 per unit and food F₂ costs Rs. 10 per unit. One unit of food F₁ contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F₂ contains 8 units of vitamin A and 12 units of vitamin B.Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutritional requirements.

Answer 1

Let x and y be two different types of food.
Thus, our objective function is minimise the cost
Z = 6x + 10y, subject to the constraints,
6x + 8y ≥48
7x + 12y ≥ 64
Plotting the above lines in a graph, we have,

Thus, the region above ABC is unbounded.
Let us check the value of the function at the corner points A, B and C

Corner point	Value of Z = 6x + 10y
(0,6)	Z = 0 + 10 x 6 = 60
(4,3)	Z = 6 x 4 + 10 x 3 = 54
(64/7,0)	Z = 6 x 64/7+ 10 x 0 = 54.85

Minimum of the function is at 4, 3
Minimum cost of the optimum diet is Rs. 54