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Carlo and Anita make mailboxes and toys in their craft shop near Lincoln.
Each mailbox requires 3 hours of work from Carlo and 4 hours from Anita.
Each toy requires 2 hours of work from Carlo and 4 hours from Anita.
Carlo cannot work more than 18 hours per week and
Anita cannot work more than 32 hours per week.
If each mailbox sells for $11 and each toy sells for $12,
then how many of each should they make to maximize their revenue? What is their maximum revenue?
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I re-formatted the input to provide the maximal readability (!!)
The question is: how many mailboxes (X) and how many toys (Y) should be produced to maximize the revenue $11*X + $12*Y
under these restrictions:
3X + 2Y <= 18 (1) (Carlo restricted by 18 hours per week) and
4X + 4Y <= 32 (2) (Anita restricted by 32 hours per week).
In other words, you must maximize the objective function (revenue)
R(X,Y) = 11X + 12Y
over the domain on the plot below, which is a quadrilateral in the first quadrant (X >= 0, Y >= 0) restricted
by the red and the green lines.
Plots y = (red) and y = (green)
The method of linear programming says:
1) Take the vertices of this quadrilateral
(x1,Y1) = (0,8) (green line Y-intercept)
(x2,Y2) = (6,0) (red line X-intercept)
(x3,Y3) = (2,6) (intersection point of the straight lines Y = and Y = )
2) Calculate the objective function at these points
R(X1,Y1) = 11*0 + 12*8 = 96;
R(X2,Y2) = 11*6 + 12*0 = 66;
R(X3,Y3) = 11*2 + 12*6 = 94.
3) Then select one of these point where the objective function is maximal - In our case this point is (X1,Y1) = (0,8)
4) This point gives your optimal solution X = 0 mailboxes and Y = 8 toys.
If they follow this optimal solution, their weekly revenue will be MAXIMAL, $96.
Solved.