Mixing
Represent point (20,10) as a convex combination of the LP’s bfs.
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I am having difficulty understanding how you represent a specific point and write it as a convex combination of the linear programming bfs.
According to an example found in operations research textbook, it gives an example where you represent a point $G=$ $(20,10)$ is within the feasible region of four extreme points $(0,40)$ , $(0,0)$ , $(30,0)$ , and $(20,20)$.
Given the contraints:
$max$ $z$ = $4$ $x_3$ + $3$$x_2$
$st.$ $x_1$ + $x_2$ $ leq 40$
$2x_1$ + $x_2$ $ leq 60$
$x_1$ , $x_2$ $ geq 0$
From this I can see that the feasible region is bounded, and that there is some optimal solution. I also know that point $G$ is a combination of $F$ and $H$, and a combination of $E$ and $C$. Therefore point $G$ is convex combination of $FEC$. However, i don't know how they got this result where:
point $G$ can be written as $1/6F + 5/6H $ , where $H= (24, 12)$ (Comes from intersection between line segments $FH$ and $EC$). They say that $H$ may be written as $.6E + .4C $. By putting these two relationships together, they write point $G$ as $1/6F + 5/6 (.6E +.4C) = 1/6F + 1/2E + 1/3C $
Based on the functions they've written I can't see where are these fractions coming from?
linear-programming
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
$endgroup$
add a comment |
$begingroup$
I am having difficulty understanding how you represent a specific point and write it as a convex combination of the linear programming bfs.
According to an example found in operations research textbook, it gives an example where you represent a point $G=$ $(20,10)$ is within the feasible region of four extreme points $(0,40)$ , $(0,0)$ , $(30,0)$ , and $(20,20)$.
Given the contraints:
$max$ $z$ = $4$ $x_3$ + $3$$x_2$
$st.$ $x_1$ + $x_2$ $ leq 40$
$2x_1$ + $x_2$ $ leq 60$
$x_1$ , $x_2$ $ geq 0$
From this I can see that the feasible region is bounded, and that there is some optimal solution. I also know that point $G$ is a combination of $F$ and $H$, and a combination of $E$ and $C$. Therefore point $G$ is convex combination of $FEC$. However, i don't know how they got this result where:
point $G$ can be written as $1/6F + 5/6H $ , where $H= (24, 12)$ (Comes from intersection between line segments $FH$ and $EC$). They say that $H$ may be written as $.6E + .4C $. By putting these two relationships together, they write point $G$ as $1/6F + 5/6 (.6E +.4C) = 1/6F + 1/2E + 1/3C $
Based on the functions they've written I can't see where are these fractions coming from?
linear-programming
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
$endgroup$
$begingroup$
can you label the points and also tell us "which values" can't you see where they are coming from?
$endgroup$
– Siong Thye Goh
Jan 24 at 7:12
$begingroup$
It's added in graph and explanation link.
$endgroup$
– mathrookie3
Jan 24 at 14:27
add a comment |
$begingroup$
I am having difficulty understanding how you represent a specific point and write it as a convex combination of the linear programming bfs.
According to an example found in operations research textbook, it gives an example where you represent a point $G=$ $(20,10)$ is within the feasible region of four extreme points $(0,40)$ , $(0,0)$ , $(30,0)$ , and $(20,20)$.
Given the contraints:
$max$ $z$ = $4$ $x_3$ + $3$$x_2$
$st.$ $x_1$ + $x_2$ $ leq 40$
$2x_1$ + $x_2$ $ leq 60$
$x_1$ , $x_2$ $ geq 0$
From this I can see that the feasible region is bounded, and that there is some optimal solution. I also know that point $G$ is a combination of $F$ and $H$, and a combination of $E$ and $C$. Therefore point $G$ is convex combination of $FEC$. However, i don't know how they got this result where:
point $G$ can be written as $1/6F + 5/6H $ , where $H= (24, 12)$ (Comes from intersection between line segments $FH$ and $EC$). They say that $H$ may be written as $.6E + .4C $. By putting these two relationships together, they write point $G$ as $1/6F + 5/6 (.6E +.4C) = 1/6F + 1/2E + 1/3C $
Based on the functions they've written I can't see where are these fractions coming from?
linear-programming
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
$endgroup$
I am having difficulty understanding how you represent a specific point and write it as a convex combination of the linear programming bfs.
According to an example found in operations research textbook, it gives an example where you represent a point $G=$ $(20,10)$ is within the feasible region of four extreme points $(0,40)$ , $(0,0)$ , $(30,0)$ , and $(20,20)$.
Given the contraints:
$max$ $z$ = $4$ $x_3$ + $3$$x_2$
$st.$ $x_1$ + $x_2$ $ leq 40$
$2x_1$ + $x_2$ $ leq 60$
$x_1$ , $x_2$ $ geq 0$
From this I can see that the feasible region is bounded, and that there is some optimal solution. I also know that point $G$ is a combination of $F$ and $H$, and a combination of $E$ and $C$. Therefore point $G$ is convex combination of $FEC$. However, i don't know how they got this result where:
point $G$ can be written as $1/6F + 5/6H $ , where $H= (24, 12)$ (Comes from intersection between line segments $FH$ and $EC$). They say that $H$ may be written as $.6E + .4C $. By putting these two relationships together, they write point $G$ as $1/6F + 5/6 (.6E +.4C) = 1/6F + 1/2E + 1/3C $
Based on the functions they've written I can't see where are these fractions coming from?
linear-programming
linear-programming
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
edited Jan 24 at 15:38
Siong Thye Goh
103k1468119
103k1468119
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
asked Jan 24 at 3:50
mathrookie3mathrookie3
84
84
$begingroup$
can you label the points and also tell us "which values" can't you see where they are coming from?
$endgroup$
– Siong Thye Goh
Jan 24 at 7:12
$begingroup$
It's added in graph and explanation link.
$endgroup$
– mathrookie3
Jan 24 at 14:27
add a comment |
$begingroup$
can you label the points and also tell us "which values" can't you see where they are coming from?
$endgroup$
– Siong Thye Goh
Jan 24 at 7:12
$begingroup$
It's added in graph and explanation link.
$endgroup$
– mathrookie3
Jan 24 at 14:27
$begingroup$
can you label the points and also tell us "which values" can't you see where they are coming from?
$endgroup$
– Siong Thye Goh
Jan 24 at 7:12
$begingroup$
can you label the points and also tell us "which values" can't you see where they are coming from?
$endgroup$
– Siong Thye Goh
Jan 24 at 7:12
$begingroup$
It's added in graph and explanation link.
$endgroup$
– mathrookie3
Jan 24 at 14:27
$begingroup$
It's added in graph and explanation link.
$endgroup$
– mathrookie3
Jan 24 at 14:27
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
We have $F=(0,0)$ and $H$ is the intersection of the line between $FG$ and $EC$. We can solve for the intersection and find out that $H=(24,12).$
and we can see that $$G=(20,10)=frac56(24,12)=frac16(0,0)+frac56(24,12)=frac16F+frac56H$$
Now, we just have to figure out where is $H$ located along the line $EC$.
Here $E=(20,20)$ and $C=(30,0).$
From the $y$ coordinate, $12=frac35 cdot 20$
We have $$H=frac35E+frac25$$
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
$endgroup$
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
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active
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$begingroup$
We have $F=(0,0)$ and $H$ is the intersection of the line between $FG$ and $EC$. We can solve for the intersection and find out that $H=(24,12).$
and we can see that $$G=(20,10)=frac56(24,12)=frac16(0,0)+frac56(24,12)=frac16F+frac56H$$
Now, we just have to figure out where is $H$ located along the line $EC$.
Here $E=(20,20)$ and $C=(30,0).$
From the $y$ coordinate, $12=frac35 cdot 20$
We have $$H=frac35E+frac25$$
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
$endgroup$
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
add a comment |
$begingroup$
We have $F=(0,0)$ and $H$ is the intersection of the line between $FG$ and $EC$. We can solve for the intersection and find out that $H=(24,12).$
and we can see that $$G=(20,10)=frac56(24,12)=frac16(0,0)+frac56(24,12)=frac16F+frac56H$$
Now, we just have to figure out where is $H$ located along the line $EC$.
Here $E=(20,20)$ and $C=(30,0).$
From the $y$ coordinate, $12=frac35 cdot 20$
We have $$H=frac35E+frac25$$
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
$endgroup$
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
add a comment |
$begingroup$
We have $F=(0,0)$ and $H$ is the intersection of the line between $FG$ and $EC$. We can solve for the intersection and find out that $H=(24,12).$
and we can see that $$G=(20,10)=frac56(24,12)=frac16(0,0)+frac56(24,12)=frac16F+frac56H$$
Now, we just have to figure out where is $H$ located along the line $EC$.
Here $E=(20,20)$ and $C=(30,0).$
From the $y$ coordinate, $12=frac35 cdot 20$
We have $$H=frac35E+frac25$$
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
$endgroup$
We have $F=(0,0)$ and $H$ is the intersection of the line between $FG$ and $EC$. We can solve for the intersection and find out that $H=(24,12).$
and we can see that $$G=(20,10)=frac56(24,12)=frac16(0,0)+frac56(24,12)=frac16F+frac56H$$
Now, we just have to figure out where is $H$ located along the line $EC$.
Here $E=(20,20)$ and $C=(30,0).$
From the $y$ coordinate, $12=frac35 cdot 20$
We have $$H=frac35E+frac25$$
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
answered Jan 24 at 15:38
Siong Thye GohSiong Thye Goh
103k1468119
103k1468119
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
add a comment |
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
$begingroup$
Thank you, I understand the example now
$endgroup$
– mathrookie3
Jan 25 at 0:17
add a comment |
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$begingroup$
can you label the points and also tell us "which values" can't you see where they are coming from?
$endgroup$
– Siong Thye Goh
Jan 24 at 7:12
$begingroup$
It's added in graph and explanation link.
$endgroup$
– mathrookie3
Jan 24 at 14:27