How do I convert this into a linear programming problem?
A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of $50 and each acre of barley yields a profit of $70. To sow the crop, two machines, a tractor and tiller, are rented. The tractor is available for 200 hours, and the tiller is available for 100 hours, Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. How many acres of each crop should be planted to maximize the farmer's profit?
I'll try to use simplex method, afterwards.
linear-algebra linear-programming
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
asked Mar 29 '15 at 22:48
randev
63
add a comment |
A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of $50 and each acre of barley yields a profit of $70. To sow the crop, two machines, a tractor and tiller, are rented. The tractor is available for 200 hours, and the tiller is available for 100 hours, Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. How many acres of each crop should be planted to maximize the farmer's profit?
I'll try to use simplex method, afterwards.
linear-algebra linear-programming
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
asked Mar 29 '15 at 22:48
randev
63
add a comment |
A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of $50 and each acre of barley yields a profit of $70. To sow the crop, two machines, a tractor and tiller, are rented. The tractor is available for 200 hours, and the tiller is available for 100 hours, Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. How many acres of each crop should be planted to maximize the farmer's profit?
I'll try to use simplex method, afterwards.
linear-algebra linear-programming
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
asked Mar 29 '15 at 22:48
randev
63
A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of $50 and each acre of barley yields a profit of $70. To sow the crop, two machines, a tractor and tiller, are rented. The tractor is available for 200 hours, and the tiller is available for 100 hours, Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. How many acres of each crop should be planted to maximize the farmer's profit?
I'll try to use simplex method, afterwards.
linear-algebra linear-programming
linear-algebra linear-programming
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
asked Mar 29 '15 at 22:48
randev
63
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
asked Mar 29 '15 at 22:48
randev
63
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
edited Mar 29 '15 at 22:51
George V. Williams
4,50321746
4,50321746
asked Mar 29 '15 at 22:48
randev
63
asked Mar 29 '15 at 22:48
randev
63
asked Mar 29 '15 at 22:48
randev
63
63
add a comment |
add a comment |
1 Answer
1
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oldest
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The first step is to describe what your variables are: $x$ which could be the number of acres of Barley sown, and $y$ which could be the number of acres of wheat.
Now, what is the profit function: $P = 70x + 50y$ (70 dollars per acre of barley + 50 dollars per acre of wheat).
Now, what are the constraints: certainly $xgeq 0$ and $ygeq 0$. What about not overusing the tiller?
$$2x + 1y leq 100$$ 2 hours on tiller for each acre of barley, 1 hour for each acre of wheat... not more than 100 hours available.
Similarly, what about not overusing the tractor?
$$3x + 4y leq 200$$ 3 hours on tractor for each acre of barley, 4 hours on the tractor for each acre of wheat... not more than 200 hours total on the tractor.
Condensing all those constraints you have:
Maximize $P=70x + 50y$
Subject to
begin{align*}
2x + 1y &leq 100 \
3x + 4y &leq 200 \
x, y &geq 0
end{align*}
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
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oldest
votes
active
oldest
votes
The first step is to describe what your variables are: $x$ which could be the number of acres of Barley sown, and $y$ which could be the number of acres of wheat.
Now, what is the profit function: $P = 70x + 50y$ (70 dollars per acre of barley + 50 dollars per acre of wheat).
Now, what are the constraints: certainly $xgeq 0$ and $ygeq 0$. What about not overusing the tiller?
$$2x + 1y leq 100$$ 2 hours on tiller for each acre of barley, 1 hour for each acre of wheat... not more than 100 hours available.
Similarly, what about not overusing the tractor?
$$3x + 4y leq 200$$ 3 hours on tractor for each acre of barley, 4 hours on the tractor for each acre of wheat... not more than 200 hours total on the tractor.
Condensing all those constraints you have:
Maximize $P=70x + 50y$
Subject to
begin{align*}
2x + 1y &leq 100 \
3x + 4y &leq 200 \
x, y &geq 0
end{align*}
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
add a comment |
The first step is to describe what your variables are: $x$ which could be the number of acres of Barley sown, and $y$ which could be the number of acres of wheat.
Now, what is the profit function: $P = 70x + 50y$ (70 dollars per acre of barley + 50 dollars per acre of wheat).
Now, what are the constraints: certainly $xgeq 0$ and $ygeq 0$. What about not overusing the tiller?
$$2x + 1y leq 100$$ 2 hours on tiller for each acre of barley, 1 hour for each acre of wheat... not more than 100 hours available.
Similarly, what about not overusing the tractor?
$$3x + 4y leq 200$$ 3 hours on tractor for each acre of barley, 4 hours on the tractor for each acre of wheat... not more than 200 hours total on the tractor.
Condensing all those constraints you have:
Maximize $P=70x + 50y$
Subject to
begin{align*}
2x + 1y &leq 100 \
3x + 4y &leq 200 \
x, y &geq 0
end{align*}
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
add a comment |
The first step is to describe what your variables are: $x$ which could be the number of acres of Barley sown, and $y$ which could be the number of acres of wheat.
Now, what is the profit function: $P = 70x + 50y$ (70 dollars per acre of barley + 50 dollars per acre of wheat).
Now, what are the constraints: certainly $xgeq 0$ and $ygeq 0$. What about not overusing the tiller?
$$2x + 1y leq 100$$ 2 hours on tiller for each acre of barley, 1 hour for each acre of wheat... not more than 100 hours available.
Similarly, what about not overusing the tractor?
$$3x + 4y leq 200$$ 3 hours on tractor for each acre of barley, 4 hours on the tractor for each acre of wheat... not more than 200 hours total on the tractor.
Condensing all those constraints you have:
Maximize $P=70x + 50y$
Subject to
begin{align*}
2x + 1y &leq 100 \
3x + 4y &leq 200 \
x, y &geq 0
end{align*}
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
The first step is to describe what your variables are: $x$ which could be the number of acres of Barley sown, and $y$ which could be the number of acres of wheat.
Now, what is the profit function: $P = 70x + 50y$ (70 dollars per acre of barley + 50 dollars per acre of wheat).
Now, what are the constraints: certainly $xgeq 0$ and $ygeq 0$. What about not overusing the tiller?
$$2x + 1y leq 100$$ 2 hours on tiller for each acre of barley, 1 hour for each acre of wheat... not more than 100 hours available.
Similarly, what about not overusing the tractor?
$$3x + 4y leq 200$$ 3 hours on tractor for each acre of barley, 4 hours on the tractor for each acre of wheat... not more than 200 hours total on the tractor.
Condensing all those constraints you have:
Maximize $P=70x + 50y$
Subject to
begin{align*}
2x + 1y &leq 100 \
3x + 4y &leq 200 \
x, y &geq 0
end{align*}
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
answered Mar 29 '15 at 22:58
TravisJ
6,35831730
6,35831730
add a comment |
add a comment |
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