Question 1131675: Each coffee table produced by Yeehaw Designs nets the firm a profit of $9. Each
bookcase yields a $12 profit. Yeehaw Designs is small and its resources limited.
During any given production period (of one week), 10 gallons of varnish and 12
lengths of high-quality red wood are available. Each coffee table requires approx. 1
gallon of varnish and 1 length of redwood. Each bookcase takes 1 gallon of varnish
and 2 lengths of redwood.
Formulate Yeehaw’s production-mix decision as a linear programming problem and
solve. How many tables and bookcases should be produced each week? What will
be the maximum profit be?
Answer by greenestamps(13208)   (Show Source):
You can put this solution on YOUR website!
Let x = number of coffee tables
let y = number of bookcases
varnish constraint: x+y <=10
wood constraint: x+2y <= 12
and of course x >=0 and y >= 0
Graph the constraint boundary lines to determine the feasibility region:
The corners of the feasibility region are (0,0), (0,6), (8,2), and (10,0).
Evaluate the objective function at each corner:
(0,0): 9(0)+12(0) = 0
(0,6): 9(0)+12(6) = 72
(8,2): 8(9)+2(12) = 96
(10,0): 10(9)+0(12) = 90
The maximum profit is $96, when 8 coffee tables and 2 bookcases are built.
Note as a refinement of this process, you can determine the corner of the feasibility region with the maximum profit without evaluating the objective function at every corner. The corner with the maximum profit can be found by comparing the slopes of the constraint lines and the objective function.
In this problem, the constraint lines have slopes of -1 and -.5; the objective function has a slope of -.75. Because the slope of the objective function is between the slopes of the two constraint lines, the maximum profit will be at the intersection of the two constraint lines.
In more complex linear programming problems, where the feasibility region has several corners, this can save a lot of time in finding the answer.
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