Question

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and I hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he produced each day so as to maximise the profit if he operates his machines for at the most 12 hours a day? 

(i) By suitable defining the variables write the objective function of the problem. 

(ii) Formulate the problem as a linear programming problem(LPP)

(iii) Solve the LPP graphically and and the number of packages of nuts and bolts to be manufactured.

Answer 1

(i) Let x be the number of packages of nuts

produced and y be the number of packages of bolts produced. Then; Maximise profit is; Z = 17. 5x + 7y

(ii) Time constraint for Machine A; x + 3y < 12 Time constraint for Machine B; 3x + y < 12 Therefore; Maximise; Z = 17.5x + 7y, x + 3y < 12, 3x + y < 12, x < 0, y < 0

(iii) The shaded region OABC is the visible region. Here the region is bounded. The corner points are 0(0,0), A (4, 0) B(3, 3), C(0, 4).

Given; Z = 17.5x + 7y

Corner points	Value of Z
O	Z =17.5(0) +7(0) = 0
A	Z =17.5(4)+ 7(0) = 70
B	Z= 17.5(3)+ 7(3) = 73.5
C	Z= 17.5(0)+7(4) = 28

Since maximum value of Z occurs at B, the solution is 

Z = 17.5(3) + 7(3) 

= 73.5.