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.

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.


17.jpg


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.


17