Question 1066772: A chain saw requires 4 hours of assembly and a wood chipper 6 hours. A maximum of 48 hours of assembly time is available. The profit is $160 on a chain saw an $230 on a chipper. How many of each should be assembled for maximum profit?
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website! .
A chain saw requires 4 hours of assembly and a wood chipper 6 hours. A maximum of 48 hours of assembly time is available.
The profit is $160 on a chain saw an $230 on a chipper. How many of each should be assembled for maximum profit?
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It is a simplest problem on the method of linear programming.
You are asked to find the number of chain saws X and the wood chippers Y to maximize the profit
P(X,Y) = $160*X + $230*Y
under the restriction
4X + 6Y <= 48.
In other words, you need to find the maximum of the objective function (profit)
P(X,Y) =160*X + 230*Y
over the triangle area in the first quadrant X >= 0, Y >= 0 restricted by the straight line
4X + 6Y <= 48.
(see the figure below).
Plot y = 
The method of linear programming says:
1) Take the vertices of this quadrilateral
(x1,Y1) = (0,8) (red line Y-intercept)
(x2,Y2) = (12,0) (red line X-intercept)
(x3,Y3) = (0,0) (the origin of the coordinate system)
2) Calculate the objective function at these points
R(X1,Y1) = 160*0 + 230*8 = 1840;
R(X2,Y2) = 160*12 + 230*0 = 1920;
R(X3,Y3) = 160*0 + 230*0 = 0.
3) Then select one of these point where the objective function is maximal - In our case this point is (X2,Y2) = (12,0).
4) This point gives your optimal solution X = 12 chain saws and Y = 0 wood chippers.
If they follow this optimal solution, then profit under the given restriction will be MAXIMAL, $1920.
Solved.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! x = number of chain saws.
y = number of wood chippers.
4 hours to assemble chain saw.
6 hours to assemble wood chipper.
total hours of assembly less than or equal to 48.
4x + 6y <= 48
profit on chain saw is 160.
profit on wood chipper is 230.
objective function is profit = 160x + 230y.
graph the constraint and then evaluate the objective function at the corner points of the region of feasibility.
in the following graph, the region of feasibility is the area of the graph that is not shaded.
the corner points of the region of feasibility, and the value of the objective function at those corner points are:
x,y,value of objective function
0,8,1840
0,0,0
12,0,1920
the maximum profit is made when x = 12 and y = 0.
12 chain saws are made.
the total profit is 12 * 160 = 1920.
the total number of assembly hours are 12 * 4 = 48.
all the constraints are met, which are that the total number of hours of assembly has to be less than or equal to 48.
note that the graph inequalities are opposite that required.
that allows the region of feasibility to NOT be shaded, which makes it easier to see.
4x + 6y <= 48 becomes 4x + 6y >= 48.
x>= 0 becomes x <= 0.
y >= 0 becomes y <= 0.
x >= 0 and y >= 0 are implied constraints because the number of hours can't be negative.
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