How do you solve this problem?
You operate a small wooden toy company making two products: alphabet blocks and wooden trucks. Your profit is 30$per box of blocks and $40 per box of trucks. Producing a box of blocks requires one hour of woodworking and two hours of painting: producing a box of trucks takes three hours of woodworking, but only one hour of painting. You employ three woodworkers and two painters, each working 40 hours a week. How many boxes of blocks (B) and trucks (T) should you make each week to maximize profit? Solve graphically as a linear program and confirm analytically
I can't show you the graph, but here's the rest of it:
The problem is to maximise the value of 30B + 40T subject to the limits of resources.
Actually, we can maximise 3B + 4T, and then the maximum profit will be 10 times the maximum value of 3B + 4T,
The number of hours of woodworking required is B + 3T
Three woodworkers work 120 hours total in a week, so B + 3T ≤ 120
The number of hours of painting required is 2B + T,
and there are two painters, so 2B + T ≤ 80
Now with the B axis horizontal and the T axis vertical (you can do it the other way round if you like), draw the graph of B + 3T = 120. It goes through 120 on the B axis and 40 on the T axis.
Draw the graph of 2B + T = 80, through 40 on the B axis and 80 on the T axis.
Now draw the line 3B + 4T = 0
We want a line parallel to this, as far as possible across to the right while still having at least one point in the region enclosed by the B and T axes and the two lines we plotted above. The line we want is the line which passes through the intersection of those two lines, which is the point
B = 24 , T = 32
So those are the numbers which give maximum profit.
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