A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 of calories. Two foods A and B, are available at the cost of ₹ 4 and ₹ 3 per unit respectively. If one unit of A contains 200 units of vitamin, 1 unit of mineral and 40 calories and one unit of food B contains 100 units of vitamin, 2 units of minerals and 40 calories, find what combination of foods should be used to have the least cost?

Question

Philoid · Accepted Answer

The above information can be expressed in the form of the following table:

Let the quantity of the foods be ‘x’ and ‘y’ respectively.

Cost of food A = 4x

Cost of food B = 3y

Total cost of the combination = 4x + 3y

Now,

⟹ 200x + 100y ≥ 4000

i.e. the minimum requirement of vitamins from the two foods should be 4000.

⟹ x + 2y ≥ 50

i.e. the minimum requirement of minerals from the two foods should be 50.

⟹ 40x + 40y ≥ 1400

i.e. the minimum requirement of calories from the two foods should be 1400

Hence, mathematical formulation of LPP is as follows:

Find ‘x’ and ‘y’ which

Minimize Z = 4x + 3y

Subject to the following constraints:

(i) 200x + 100y ≥ 4000

i.e. 2x + y ≥ 40

(ii) x + 2y ≥ 50

(iii) 40x + 40y ≥ 1400

i.e. x + y ≥ 35

(iv) x,y ≥ 0 (∵ quantity cant be negative)

The feasible region is unbounded.

The corner points of the feasible region is as follows:

Z is smallest at B(5,30)

Let us consider 4x + 3y ≤ 110

As it has no intersection with the feasible region, the smallest value is the minimum value.

The minimum cost of foods is ₹110