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.

Question

Philoid · Accepted Answer

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.