Question

Smita is a diet conscious house wife, wishes to ensure certain minimum intake of vitamins A, B and C for the family. The minimum daily needs of vitamins A, B, and C for the family are 30, 20, and 16 units respectively. For the supply of the minimum vitamin requirements Smita relies on 2 types of foods F₁ and F₂. F₁ provides 7, 5 and 2 units of A, B, C vitamins per 10 grams and F₂ provides 2, 4 and 8 units of A, B and C vitamins per 10 grams. F₁ costs ₹ 3 and F₂ costs ₹ 2 per 10 grams. How many grams of each F₁ and F₂ should buy every day to keep her food bill minimum

Answer 1

Let food F₁ be x grams and food F₂ be y grams.

Since x and y cannot be negative, x ≥ 0, y ≥ 0.

F₁ costs ₹ 3 and F₂ costs ₹ 2 per 10 grams.

∴ Total cost = Z = 3x + 2y

We construct a table with constraints of vitamins A, B and C as follows:

Vitamins/Food	F1	F2	Minimum requirement
A	7	2	30
B	5	4	20
C	2	8	16

From the table, the constraints are

7x + 2y ≥ 30

5x + 4y ≥ 20

2x + 8y ≥ 16

∴ Given problem can be formulated as follows:

Minimize Z = 3x + 2y

Subject to 7x + 2y ≥ 30

5x + 4y ≥ 20,

2x + 8y ≥ 16, x ≥ 0, y ≥ 0

To draw the feasible region, construct table as follows:

Inequality	7x + 2y ≥ 30	5x + 4y ≥ 20	2x + 8y ≥ 16
Corresponding equation (of line)	7x + 2y = 30	5x + 4y = 20	2x + 8y = 16
Intersection of line with X-axis	`(30/7, 0)`	(4, 0)	(8, 0)
Intersection of line with Y-axis	(0, 15)	(0, 5)	(0, 2)
Region	Non-Origin side	Non-Origin side	Non-Origin side

Shaded portion XABCY is the feasible region, whose vertices are A(8, 0), B and C(0, 15),

B is the point of intersection of the lines 7x + 2y = 30 and 2x + 8y = 16

Solving the above equations, we get

x = 4, y = 1

∴ B ≡ (4, 1)

Here, the objective function is

Z = 3x + 2y

Z at A (8, 0) = 3(8) + 2(0) = 24

Z at B (4, 1) = 3(4) + 2(1)

= 12 + 2

= 14

Z at C (0, 15) = 3(0) + 2(15)

= 30

∴ Z has minimum value 14 at x = 4 and y = 1.

∴ Smita should buy 4 grams of food F₁ and 1 gram of food F₂ every day to keep her food bill minimum.