A factory manufactures two types of screws A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 min on the automatic and 6 min on hand operated machines to manufacture a package of screws A, while it takes 6 min on automatic and 3 min on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 h on any day. The manufacturer can sell a package of screws A at a profit on Rs. 7 and screws B at a profit of Rs.10. Assuming that he can sell all the screws he manufactured, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.
Let the manufacturer produces x package of screws A and y package of screws B, We construct the following table:
Type of screwNumberofTime on automaticTime on handProfitpackagesmachine(in min)machine(in min)(in Rs.)Ax4x6x7xBy6y3y10yTotalx+y4x+6y6x+3y7x+10yAvailability4×60=2404×60=240
The profits on a package of screws A is Rs. 7 and on the package of screws B is Rs. 10.
Our problem is to maximize Z=7x+10y..........(i)
Subject to constraints 4x+6y≤240⇔2x+3y≤120.........(ii)
6x+3y≤240⇔2x+y≤80.........(iii)
and x≥0, y≥0..........(iv)
Firstly, draw the graph of the line 2x + 3y = 120
x060y400
Putting (0, 0) in the inequality
2x+3y≤120, we have 2×0+3×0≤120
⇒0≤120 (which is true)
So, the half plane is towards the origin. Secondly, draw the graph of the line 2x+y=80
x400y080
Putting (0, 0) in the inequality 2x+y≤80, we have 2×0+0≤80⇒0≤80 (which is true)
So, the half plane is towards the origin. Since, x, y ≥ 0
So, the feasible region lies in the first quadrant.
On solving equations 2x+3y=120 and 2x+y=80, we get B(30, 20).
The corner points of the feasible region are O(0, 0), A(40, 0), B(30, 20) and C(0, 40). the values of Z at these points are as follows:
∴ Feasible region is OABCO.
Corner pointZ=7x+10yO(0, 0)0A(40, 0)280B(30, 20)410→MaximumC(0,40)400
The maximum value of Z is Rs. 410 at B(30, 20)
Thus, the factory should produce 30 packages of screws A and 20 packages of screws B to get the maximum profit of Rs. 410.