Question

Mechanical Design 1, please answer ASAP and clear. Thank You

3) 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.)

Answer 1

Let x unts cf gearA md Y units af gear B gre momutactured. Then merit fun ction, Rofit P= 50X+toy ( machine s avalloble 24har aday) (machine 2 avaltoblefor 1ohr aday) (madhne a, 16 h/day) constraint: 3x+Y=24 자2렇 드16 bounds: x20 using graphical pmethad: draw all 3 constain te on 1-y PAme (Notice that constzumta crre all straightun) Co-ordinaesof corner pointe f feas ible reglon: 26 24 A= (0,8) B is intersecton of constauint 2 cnd 3(on saluing) x+y=10 나정-16 y= 6 X=4 B= (416) cin intersecHon of cons, 12 3XFY-24 3=3 CE (713) D= (80) 2. feasible region we knouo that objective functHodo colll have may value atone of 4. 14 K6 12 18 these corner polnts. P(A) = 50x0+ 8x70= 560 PUB) = 50 x4+ tox 6= 620 P() = 50x7+7ox3 = 5G0 5ox8+Toy o= 400 PLO = lerly profit 18 mayimum at manimum profit = $ 620 no, of units ef ger A= 4 dmd that of gear B36 B= (4,6)