Question

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F₁ and F₂ are available. Food F₁ costs Rs 4 per unit food and F₂ costs Rs 6 per unit. One unit of food F₁ contains 3 units of vitamin A and 4 units of minerals. One unit of food F₂ contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum costs for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

Answer 1

Let x units of food F₁ and y units of food F₂ be in the diet

Total cost Z = 4x + 6y

Then the LPP is

Minimize Z = 4x + 6

Subject to the constraints

3x + 6y ≥ 80

4x + 3y ≥ 100

x, y ≥ 0

The feasible region is unbounded

As the feasible region is unbounded, 104 may or may not be the minimum value of 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 points with 2x + 3y < 52 Therefore minimum cost of the mixture will be 104.