Question 127434: The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 feet of good quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood: each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoors Furniture produce to obtain the largest possible profit? Use graphical Linear programming approach.
Found 2 solutions by ankor@dixie-net.com, stanbon:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 feet of good quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood: each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoors Furniture produce to obtain the largest possible profit? Use graphical Linear programming approach.
:
Let x = no. of picnic tables
Let y = no. of benches
:
The labor constraint:
6x + 4y <= 1200
4y <= 1200 - 6x; divide equation by 4
y <= 300 - 1.5x
:
Plot above equation for x = 0 and x = 60
x | y
-------
0 | 300
60 | 210
;
:
The material constraint
35x + 10y <= 3500
10y <= 3500 - 35x
y <= 350 - 3.5x; divide equation by 10
:
Plot the above equation for x = 0 and x = 60
x | y
-------
0 | 350
60 | 140
:
Graph will look like this
The area of feasibility is at or below the two graphs, which ever is lowest
Area is bounded by coordinates; 0,0; 0,300; 100,0; and an integer values of 25, 262
Find the profit using each:
Tables + Benches
20(0) + 9(300) = $2700
20(100) + 9(0) = $2000
20(25) + 9(262) = $2858;
:
25 tables and 262 benches will yield max profit;
utilizes:
6(25) + 4(262) = 1190 hrs of labor
35(25) + 10(262) = 3495 ft of red-wood
:
This is the general idea. Check my math here, a lot of chance for mistakes
Answer by stanbon(75887)  (Show Source):
You can put this solution on YOUR website! During the next production cycle, 1,200 hours of labor are available under a union agreement.
The firm also has a stock of 3,500 feet of good quality redwood.
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Let # of benches produced be "b"; Let # of picnic tables produced be "P".
Each bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood:
each picnic table takes 6 labor hours and 35 feet of redwood.
------------
Labor Inequality: 4b+6p <= 1200
Redwood Inequality: 10b+35p <= 3500
Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each.
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Objective Function: Profit = 9b+20p
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How many benches and tables should Outdoors Furniture produce to obtain the largest possible profit? Use graphical Linear programming approach.
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INEQUALITIES:
Labor: p <= (-2/3)b + 200
Redwood: p <= (-2/7)b + 100
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Graph the solution sets of both inequalities:
-------------------------
Determine the vertices of the solution set:
(0,100), (262.5,25) (300,0)
----------------------
Check each vertex pair in the objective function to see which pair
yields the maximum profit.
Profit = 9b+20p
(0,100) yields: 2000
(262.5,25) yields: 9*262.5+20*25=2862.50
(300,0) yields 9*300 = 2700
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Conclusion: maximum comes with 263 benches and 25 picnic tables.
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Cheers,
Stan H.
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