Question

A dietician wishes to mix two kinds ·of food X· and Y in such a way that the  mixture contains at least 10 units of vitamin A, 12 units of vitamin B arid 8 units of vitamin C. The vitamin contents of one kg food is given below:

Food	Vitamin A	Vitamin.B	Vitamin C
X	1 unit	2 unit	3 unit
Y	2 unit	2 unit	1 unit

Orie kg of food X costs Rs 24 and one kg of food Y costs Rs 36. Using Linear Programming, find the least cost of the total mixture. which will contain the required vitamins.

Answer 1

Let one type of food be x and another type be y

Food	Vitamin A	Vitamin B	Vitamin C
X	1 units	2 units	3 units
Y	2 units	2 units	1 units

∴ According to the given condition

Minimize z = 24x + 36y

Subject to

`x + 2y >= 10`

`x + 2y >= 12 or x + y >= 6`

`x + y >=  8`

X + 2y = 10	x + y = 6	3x + 1y = 8
`x/10 + y/5 = 1`	x + y = 6	`x/(8/3) + y/8 = 1`

 

Corner points	Objective function z= 24x + 36y
A(10,0)	`z= 24xx 10 + 36 xx 0 = 240`
B(2,4)	`z = 24 xx 2 + 36 xx 4 = 192`
C(1,5)	`z = 24 xx 1 + 36 xx 5 = 204`
D(0,8)	`z = 0 xx 24 + 36 xx 8 = 288`

Minimum value of z is 192 at B (2, 4)