Question

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 board feet of quality redwood. Each bench that Outdoor Furniture produces requires 4 labor-hours and 10 board feet of redwood; each picnic table takes 6 labor-hours and 35 board 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 Outdoor Furniture produce in order to obtain the largest possible profit? The optimum solution is: Number of benches produced = (round your response to the nearest whole number). Number of tables produced = (round your response to the nearest whole number). Optimal solution value = $ (round your response to the nearest whole number).

Answer 1

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

Number of benches = 262

Number of tables = 25

Optimal solution value = $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

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