A company manufactures commercial heating system components and domestic furnaces at its factories in Monaca, PA and Hamburg, NY. At the Monaca plant, no more than 1000 units per day can be produced, and the number of commercial components cannot exceed 100 more than half the number of domestic furnaces. At the Hamburg plant, no more than 850 units per day can be produced. The profit on each commercial component is $400 at the Monaca plant and $390 at the Hamburg plant. The profit on each domestic furnace is $200 at the Monaca plant and $215 at the Hamburg plant. If there is a rush order for 540 commercial components and 740 domestic furnaces, how many of each should be produced at each plant in order to maximize profits? Find the maximum profit. Note: Assume that exactly enough units are produced to fill the rush order.
A)Maximum profit is $365,750 with 210 heating comproments produced at Monaca and 330 at Hamburg and 520 domestic furnces produced at Monaca and 220 at Hamburg.
B)Maximum profit is $355,250 with 330 heating comproments produced at Monaca and 210 at Hamburg and 220 domestic furnces produced at Monaca and 520 at Hamburg.
C)Maximum profit is $368,500 with 210 heating comproments produced at Monaca and 330 at Hamburg and 220 domestic furnces produced at Monaca and 520 at Hamburg.
D)Maximum profit is $344,500 with 330 heating comproments produced at Monaca and 220 at Hamburg and 520 domestic furnces produced at Monaca and 220 at Hamburg.
E)Maximum profit is $364,000 with 210 heating comproments produced at Monaca and 330 at Hamburg and 220 domestic furnces produced at Monaca and 520 at Hamburg.

Question

A company manufactures commercial heating system components and domestic furnaces at its factories in Monaca, PA and Hamburg, NY. At the Monaca plant, no more than 1000 units per day can be produced, and the number of commercial components cannot exceed 100 more than half the number of domestic furnaces. At the Hamburg plant, no more than 850 units per day can be produced. The profit on each commercial component is $400 at the Monaca plant and $390 at the Hamburg plant. The profit on each domestic furnace is $200 at the Monaca plant and $215 at the Hamburg plant. If there is a rush order for 540 commercial components and 740 domestic furnaces, how many of each should be produced at each plant in order to maximize profits? Find the maximum profit. Note: Assume that exactly enough units are produced to fill the rush order. ​
A)Maximum profit is $365,750 with 210 heating comproments produced at Monaca and 330 at Hamburg and 520 domestic furnces produced at Monaca and 220 at Hamburg.
B)Maximum profit is $355,250 with 330 heating comproments produced at Monaca and 210 at Hamburg and 220 domestic furnces produced at Monaca and 520 at Hamburg.
C)Maximum profit is $368,500 with 210 heating comproments produced at Monaca and 330 at Hamburg and 220 domestic furnces produced at Monaca and 520 at Hamburg.
D)Maximum profit is $344,500 with 330 heating comproments produced at Monaca and 220 at Hamburg and 520 domestic furnces produced at Monaca and 220 at Hamburg.
E)Maximum profit is $364,000 with 210 heating comproments produced at Monaca and 330 at Hamburg and 220 domestic furnces produced at Monaca and 520 at Hamburg.

Quizplus · Accepted Answer

Let x be the number of domestic furnaces produced at Monaca, y be the number of domestic furnaces produced at Hamburg, z be the number of commercial components produced at Monaca, and w be the number of commercial components produced at Hamburg. 
The objective function to be maximized is: 
f(x,y,z,w) = 200x + 215y + 400z + 390w 
subject to the following constraints: 
z + w = 540 (rush order for commercial components) 
x + y = 740 (rush order for domestic furnaces) 
z ≤ (1/2)(y) + 100 (Monaca plant constraint) 
z + w ≤ 1000 (Monaca plant constraint) 
w ≤ 850 (Hamburg plant constraint) 
We can use a linear programming tool to solve this problem, or we can solve it graphically. The feasible region is a polygon with vertices at: 
(0,740), (340,400), (440,300), (540,200), (440,0), (0,0)

The objective function is maximized at point C (210 domestic furnaces produced at Monaca, 520 domestic furnaces produced at Hamburg, 330 commercial components produced at Hamburg, and 210 commercial components produced at Monaca), where the maximum profit is $368,500. Therefore, the best choice is (C).

A Company Manufactures Commercial Heating System Components and Domestic Furnaces