Question

Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 per kg and rice costs Rs. 4 per kg.

Answer 1

The above information can be expressed using the following table:

Let the amount of Bran and Rice required be ‘x’ and ‘y’ kgs respectively.

Cost of Bran = 5x

Cost of Rice = 4y

Cost of the cereal = 5x + 4y

Now,

⟹ 80x + 100y ≥ 88

i.e. the minimum requirement of protein in the cereal, from Bran and Rice combined, is 88g, each of which have 80g and 100g of proteins respectively.

⟹ 40x + 30y ≥ 36

i.e. the minimum requirement of iron in the cereal, from Bran and Rice combined, is 36mg, each of which have 40mg and 30mg of iron.

Hence, mathematical formulation of LPP is as follows:

Find ‘x’ and ‘y’ that:

Minimises Z = 5x + 4y

Subject to the following constraints:

(i) 80x + 100y ≥ 88

(ii) 40x + 30y ≥ 36

(iii) 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(0.6, 0.4)

Let us consider 5x + 4y ≤ 4.6

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

The minimum cost of the cereal is Rs. 4.6