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Question 1189015: A small petroleum company owns two re neries. Re nery 1 costs $20,000 per day to operate, and it can produce
400 barrels of high-grade oil, 300 barrels of medium-grade oil, and 200 barrels of low-grade oil each day. Re nery
2 is newer and more modern. It costs $25,000 per day to operate, and it can produce 300 barrels of high-grade
oil, 400 barrels of medium-grade oil, and 500 barrels of low-grade oil each day. The company has orders totaling
25,000 barrels of high-grade oil, 27,000 barrels of medium-grade oil, and 30,000 barrels of low-grade oil. How
many days should it run each re nery to minimize its costs and still re ne enough oil to meet its orders?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your objective function is 20,000 * x + 25,000 * y = cost
this is what you want to minimize.
your constraint functions are:
400x * x + 300 * y >= 25,000
300x + 400y >= 27,000
200x + 500y >= 30,000
x >= 0
y >= 0
using the desmos.com calculator at https://www.desmos.com/calculator, you would do the following:
graph the opposite of the inequalities.
the feasible region is the area on the graph that is not shaded.
evaluate the objective function at each of the corner points.
the minimum cost will be at one of those corner pointas.
the graph is shown below.
the corner points are in (x,y) format.
you will find that the minimum cost will be at (25,50).
all the constraints must be met at that coordinate point as well.
the graph is shown below:
at (x,y) = (25,50), the cost is 25 * 20000 + 50 * 25000 = 1,750,000.
the number of days are 25 at refinery 1 and 50 at refinery 2.
the number of barrels of high grade oil are 400 * 25 + 300 * 50 = 25,000
the number of barrels of medium grade oil are 300 * 25 + 400 * 50 = 27,500
the number of barrels of low grade oil are 200 * 25 + 500 * 50 = 30,000.
i should have graphed x <= 0 and y <= 0.
in this case, it didn't matter, because those inequalities are already shaded.
let me know if you have any questions.
remember, with the desmos.com calculator, you graph the opposite of the inequalities.
this is necessary so that the feasible region is not shaded.
trying to graph the inequalities as they are will result in shading that overlaps other shading.
that makes identifying the feasible region more difficult.
if you we to manually create the graph, then shading the inequalities as they are makes more sense, because you can control the shading better.
it's also more work.
doing it by using the desmos.com software is much easier.
theo
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