Solution Verified by Toppr

Let's assume that number of toys of type A be X and number of toys of type B be Y.

Since, toy A need 12 minutes and toy B need 6 minutes of machine I time. Also machine I is available for maximum 6 hours ( 360 minutes ).

So, 12X+6Y≤360

⇒2X+Y≤60   ...(1)

Since, toy A need 18 minutes and toy B need 0 minutes of machine II time. Also machine I is available for maximum 6 hours ( 360 minutes ).

So, 18X+0Y≤360

⇒X≤20   ...(2)

Since, toy A need 6 minutes and toy B need 9 minutes of machine III time. Also machine III is available for maximum 6 hours ( 360 minutes ).

So, 6X+9Y≤360

⇒2X+3Y≤120   ...(3)

Since, count of toys can't be negative.

∴X≥0 and Y≥0   ...(4)

Now, profit on toy A is 7.50 Rs and profit on toy B is 5 Rs

So, total profit (Z)=7.5X+5Y

We have to maximize the total profit of manufacturer.

After plotting all the constraints give by equation (1), (2), (3) and (4), we get the feasible region as shown in the image.

Corner points	Value of Z=7.5X+5Y
A (0, 40)	200
B (15, 30)	262.50 (Maximum)
C (20, 20)	250
D (20, 0)	150

Hence maximum profit that manufacturer can earn is 262.50 Rs when 15 toys of type A and 30 toys of type B manufactured.

A producer of helical gears manufactures two different quality rated versions of the same product. Both the gears require three machines designated 1, 2, 3 for their fabrication. Gear A uses three hours, one hour and one hour respectively of machining time. Gear A yields a profit of $50 per unit. Gear B uses one hour, one hour and two hours respectively of machining time. Gear B yields a profit of $70 per unit. Machine 1 is available 24 hours a day, but due to other product line usage, Machine 2 is only available 10 hours a day and Machine 3 is available 16 hours a day. As a senior engineer, it is your job to determine the most profitable mix of production level of both gears A and B. a) Using the method of Linear Programming, develop the merit function, constraints and bounds for this problem. b) Determine the most profitable number of gears of each type to manufacture as well as the profit. (25 pts.)