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Question

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.

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Solution

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+6y2402x+3y120.........(ii)

6x+3y2402x+y80.........(iii)

and x0, y0..........(iv)

Firstly, draw the graph of the line 2x + 3y = 120

x060y400

Putting (0, 0) in the inequality

2x+3y120, we have 2×0+3×0120

0120 (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+y80, we have 2×0+080080 (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)410MaximumC(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.


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