Linear
Programming: An Example & Sections: Optimizing linear systems, Setting up word problems
First I'll solve the fourth and fifth constraints for easier graphing: The feasibility region looks like this: From the graph, I can see which lines cross to form the corners, so I know which lines to pair up in order to verify the coordinates. I'll start at the "top" of the shaded area and work my way clockwise around the edges:
Now I'll plug each corner point into the optimization equation, z = –0.4x + 3.2y: (1, 6): z =
–0.4(1) + 3.2(6) = –0.4 + 19.2 = 18.8
Then the maximum is 18.8 at (1, 6) and the minimum is –2 at (5, 0). Given the inequalities, linearprogramming exercise are pretty straightforward, if sometimes a bit long. The hard part is usually the word problems, where you have to figure out what the inequalities are. So I'll show how to set up some typical linearprogramming word problems.
If gasoline is selling for $1.90 per gallon and fuel oil sells for $1.50/gal, how much of each should be produced in order to maximize revenue? The question asks for the number of gallons which should be produced, so I should let my variables stand for "gallons produced". x:
gallons of gasoline produced Since this is a "real world" problem, I know that I can't have negative production levels, so the variables can't be negative. This gives me my first two constraints: namely, x > 0 and y > 0. Since I have to have at least two gallons
of gas for every gallon of oil, then For graphing, of course, I'll use the more manageable form "y < ( ^{1}/_{2} )x". The winter demand says that y
> 3,000,000; note that
this constraint eliminates the need for the "y
> 0" constraint. The
gas demand says that x
< 6,400,000. I need to maximize revenue R, so the optimization equation is R = 1.9x + 1.5y. Then the model for this word problem is as follows: R = 1.9x +
1.5y, subject to:
Using a scale that counts by millions (so "y = 3" on the graph means "y is three million"), the above system graphs as follows: Taking a closer look, I can see the feasibility region a little better: When you test the corner points at (6.4m, 3.2m), (6.4m, 3m), and (6m, 3m), you should get a maximal solution of R = $16.96m at (x, y) = (6.4m, 3.2m). << Previous Top  1  2  3  4  5  Return to Index Next >>



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