Question

In a certain manufacturing firm, the production consists of a machining process that takes raw materials and converts them into unassembled parts. These parts are then sent to one of two divisions for assembly into the final product – Division I for product A and Division II for product B.

Product A requires 40 units of raw material and ten hours of machine processing time, whereas product B requires 80 units of raw material and four hours of machine processing time. During the period, 800 units of raw material and 80 hours of machine processing time are available. The capacities of the two assembly divisions during the period are 6 and 9 units for Divisions I and II respectively.

The profit contributions per unit are ₹200 for A and ₹120 for B. Formulate the problem in the linear programming framework and determine the optimal level of output for the two products using graphical method. Also, find the maximum profit to be obtained during this period.

Answer 1

Required Answer ✏

This implies that

x2+2ax=4x-4a-13

or

x2+2ax-4x+4a+133D0

or

x2+(2a-4)x+(4a+13)=0

Since the equation has just one solution instead of the usual two distinct solutions, then the two solutions must be same i.e. discriminant = 0.

Hence we get that :-

(2a-4)2=4.1.(4a+13)

or4a2-16a+16=16a+52

or

4a2-32a-36=0

or

a2-8a-9=0

or

(a-9)(a+1)=0

So the values of a are -1 and 9.

Answer 2

This implies that

x2+2ax=4x-4a-13

or

x2+2ax-4x+4a+133D0

or

x2+(2a-4)x+(4a+13)=0

Since the equation has just one solution instead of the usual two distinct solutions, then the two solutions must be same i.e. discriminant = 0.

Hence we get that :-

(2a-4)2=4.1.(4a+13)

or4a2-16a+16=16a+52

or

4a2-32a-36=0

or

a2-8a-9=0

or

(a-9)(a+1)=0

So the values of a are -1 and 9.