Question

A chemical company produces two compounds, A and B. The following table gives the units of ingredients, C and D per kg of compounds A and B as well as minimum requirements of C and D and costs per kg of A and B. Find the quantities of A and B which would give a supply of C and D at a minimum cost.

	Compound		Minimum requirement
	A	B
Ingredient C	1	2	80
Ingredient D	3	1	75
Cost(in Rs) per kg	4	6

Answer 1

Let required quantity of compound A and B are x and y kg.

Since, cost of one kg of compound A and B are Rs 4 and Rs 6 per kg. So,

Cost of x kg of compound A and y kg of compound B are Rs 4x and Rs 6 respectively.

Let Z be the total cost of compounds, so,

Z = 4x + 6y

Since, compound A and B contain 1 and 2 units of ingredient C per kg respectively, So x kg of compound A and y kg of compound B contain x and 2y units of ingredient C respectively but minimum requirement of ingredient C is 80 units, so,

x + 2y ≥ 80 {first constraint}

Since, compound A and B contain 3 and 1 units of ingredient D per kg respectively,

So x kg of compound A and y kg of compound B contain 3x and y units of ingredient D respectively but minimum requirement of ingredient C is 75 units, so,

3x + y ≥ 75 {second constraint}

Hence, mathematical formulation of LPP is,

Min Z = 4x + 6y

Subject to constraints,

x + 2y ≥ 80

3x + y ≥ 75

x, y ≥ 0 [Since production can not be less than zero]

Region x + 2y ≥ 80: line x + 2y = 80 meets axes at A(80, 0), B(0, 40) respectively.

Region not containing origin represents x + 2y ≥ 80 as (0, 0) does not satisfy x + 2y ≥ 80.

Region 3x + y ≥ 75: line 3x + y = 75 meets axes at C(25, 0), D(0, 75) respectively.

Region not containing origin represents 3x + y ≥ 75 as (0, 0) does not satisfy 3x + y ≥ 75.

Region x, y ≥ 0: it represents first quadrant.

The corner points are D(0, 75), E(14, 33), A(80, 0).

The values at Z at these corner points are as follows:

Corner Point Z = 4x + 6y

D 450

E 254

A 320

The minimum value of Z is 254 which is attained at E(14, 33).

Thus, the minimum cost is Rs 254 obtained when 14 units of compound A

and 33 units compound B are produced.