Question

Solve the question from Chapter - LINEAR PROGRAMMING

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 protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 per kilogram and rice costs Rs. 4 per kilogram.

Answer 1

80x+100y≥88
40x+30y≥36
Z=5x+4y
Solve these equations and you would get the answer. Since the price is given per kg you can either convert price in grams or milligrams or convert both equations in kg.

Answer 2

Answer:

Solution :

Let the cereal contains x kg of bran and y kg of rice. Clearly x ≥ 0 and y ≥ 0.

According to the hypothesis the linear programming problem is :

Minimize Z = 5x + 4y

Subject to the constraints :

(80/1000) x + (100/1000) y ≥ (88/1000)

Or, 20 x + 25 y ≥ 22 ----------------- (1)

And (40/1000) x + (30/1000) y ≥ (36/1000)

Or, 20 x + 15 y ≥ 18 ------------------ (2)

And also x ≥ 0, y ≥0 ------------------- (3)

Now the lines 20 x + 25 y = 22 and 20x + 15 y = 18 are drawn.

These lines meet at E (0.6, 0.4). The feasible region is shaded and it is an unbounded region with vertices A (1.1, 0), E (0.6, 0.4) and D (0, 1.2).

Corner Point Z = 5 x + 4 y

A (1.1, 0) 5 × 1.1 + 4 × 0 = 5.5

E (0.6, 0.4) 5 × 0.6 + 4 × 0.4 = 4.6 ← Minimum

D (0, 1.2) 5 × 0 + 4 × 1.2 = 4.8

Therefore, minimum cost of producing this cereal is Rs4.60 per kg. [Ans.]