How to basically solve Integer programming problems?
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I learnt to solve the task below Linear Programmming is employed. I have tried studying texts to better understand what methods to employ out of the following:
$(A) Gomory's-cut$
$(B) Mixed-Gomory's-cut $
$(C) Branch-and-Bound $
But I am finding it difficult to comprehend any of the three and all the example I see are some how more complex and non related.Is there a way to simplify the problem below given the constraint is just $a,b,c,d,e,fin 4,2 $
beginalign
a+b+c+d+e+f &= 18, tag1 \
b+d &= 4, tag2 \
e+f &= 8, tag3
endalign
find $a,b,c,d,e,f$ ?
abstract-algebra linear-programming integer-programming
asked Sep 1 at 9:08
LiNKeR
337
add a comment |Â
up vote
0
down vote
favorite
I learnt to solve the task below Linear Programmming is employed. I have tried studying texts to better understand what methods to employ out of the following:
$(A) Gomory's-cut$
$(B) Mixed-Gomory's-cut $
$(C) Branch-and-Bound $
But I am finding it difficult to comprehend any of the three and all the example I see are some how more complex and non related.Is there a way to simplify the problem below given the constraint is just $a,b,c,d,e,fin 4,2 $
beginalign
a+b+c+d+e+f &= 18, tag1 \
b+d &= 4, tag2 \
e+f &= 8, tag3
endalign
find $a,b,c,d,e,f$ ?
abstract-algebra linear-programming integer-programming
asked Sep 1 at 9:08
LiNKeR
337
1
we like to use the first letters of the alphabet for constants, and the last letters for variables
– LinAlg
Sep 1 at 13:15
Any integer program can be solved using (C) branch-and-bound (though it might be very slow—like, thousands of years slow). Any integer program can also be solved using (A) Gomory’s cut—if you add enough rounds of cuts, then you will find the optimal solution. This technique is ALSO very slow. As it turns out, if you combine (A) and (C) together, the result can be very fast. This is (very, very roughly) how IPs are solved today. Look up “branch and cutâ€Â.
– David M.
Sep 1 at 13:48
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I learnt to solve the task below Linear Programmming is employed. I have tried studying texts to better understand what methods to employ out of the following:
$(A) Gomory's-cut$
$(B) Mixed-Gomory's-cut $
$(C) Branch-and-Bound $
But I am finding it difficult to comprehend any of the three and all the example I see are some how more complex and non related.Is there a way to simplify the problem below given the constraint is just $a,b,c,d,e,fin 4,2 $
beginalign
a+b+c+d+e+f &= 18, tag1 \
b+d &= 4, tag2 \
e+f &= 8, tag3
endalign
find $a,b,c,d,e,f$ ?
abstract-algebra linear-programming integer-programming
asked Sep 1 at 9:08
LiNKeR
337
I learnt to solve the task below Linear Programmming is employed. I have tried studying texts to better understand what methods to employ out of the following:
$(A) Gomory's-cut$
$(B) Mixed-Gomory's-cut $
$(C) Branch-and-Bound $
But I am finding it difficult to comprehend any of the three and all the example I see are some how more complex and non related.Is there a way to simplify the problem below given the constraint is just $a,b,c,d,e,fin 4,2 $
beginalign
a+b+c+d+e+f &= 18, tag1 \
b+d &= 4, tag2 \
e+f &= 8, tag3
endalign
find $a,b,c,d,e,f$ ?
abstract-algebra linear-programming integer-programming
abstract-algebra linear-programming integer-programming
asked Sep 1 at 9:08
LiNKeR
337
asked Sep 1 at 9:08
LiNKeR
337
edited Sep 1 at 9:30
asked Sep 1 at 9:08
LiNKeR
337
asked Sep 1 at 9:08
LiNKeR
337
asked Sep 1 at 9:08
LiNKeR
337
337
1
we like to use the first letters of the alphabet for constants, and the last letters for variables
– LinAlg
Sep 1 at 13:15
Any integer program can be solved using (C) branch-and-bound (though it might be very slow—like, thousands of years slow). Any integer program can also be solved using (A) Gomory’s cut—if you add enough rounds of cuts, then you will find the optimal solution. This technique is ALSO very slow. As it turns out, if you combine (A) and (C) together, the result can be very fast. This is (very, very roughly) how IPs are solved today. Look up “branch and cutâ€Â.
– David M.
Sep 1 at 13:48
add a comment |Â
1
we like to use the first letters of the alphabet for constants, and the last letters for variables
– LinAlg
Sep 1 at 13:15
Any integer program can be solved using (C) branch-and-bound (though it might be very slow—like, thousands of years slow). Any integer program can also be solved using (A) Gomory’s cut—if you add enough rounds of cuts, then you will find the optimal solution. This technique is ALSO very slow. As it turns out, if you combine (A) and (C) together, the result can be very fast. This is (very, very roughly) how IPs are solved today. Look up “branch and cutâ€Â.
– David M.
Sep 1 at 13:48
1
1
we like to use the first letters of the alphabet for constants, and the last letters for variables
– LinAlg
Sep 1 at 13:15
we like to use the first letters of the alphabet for constants, and the last letters for variables
– LinAlg
Sep 1 at 13:15
Any integer program can be solved using (C) branch-and-bound (though it might be very slow—like, thousands of years slow). Any integer program can also be solved using (A) Gomory’s cut—if you add enough rounds of cuts, then you will find the optimal solution. This technique is ALSO very slow. As it turns out, if you combine (A) and (C) together, the result can be very fast. This is (very, very roughly) how IPs are solved today. Look up “branch and cutâ€Â.
– David M.
Sep 1 at 13:48
Any integer program can be solved using (C) branch-and-bound (though it might be very slow—like, thousands of years slow). Any integer program can also be solved using (A) Gomory’s cut—if you add enough rounds of cuts, then you will find the optimal solution. This technique is ALSO very slow. As it turns out, if you combine (A) and (C) together, the result can be very fast. This is (very, very roughly) how IPs are solved today. Look up “branch and cutâ€Â.
– David M.
Sep 1 at 13:48
add a comment |Â
2 Answers
2
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0
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If all the variable takes value from $2,4$.
The constraints reduces to
$$a+c=6$$
$$b+d=4 implies b=d=2$$
$$e+f=8 implies e=f=4$$
Hence $(a,c) = (4,2)$ or $(a,c)=(2,4)$.
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
1
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
add a comment |Â
up vote
0
down vote
In your case, there is no need to advanced methods, unless you want to practice how to use them. If you only want to solve this specific problem you can do this:
$a+b+c+d+e+f=18$
is
$a+(b+d)+c+(e+f) =18$
so
$a+4+c+8=18$
so
$a+c=6$
since $a,b,c,d,e,fin 4,2 $
you can easily determine integer values for $a,c$ from the last equation.
answered Sep 1 at 10:14
NoChance
3,35021221
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
If all the variable takes value from $2,4$.
The constraints reduces to
$$a+c=6$$
$$b+d=4 implies b=d=2$$
$$e+f=8 implies e=f=4$$
Hence $(a,c) = (4,2)$ or $(a,c)=(2,4)$.
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
1
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
add a comment |Â
up vote
0
down vote
If all the variable takes value from $2,4$.
The constraints reduces to
$$a+c=6$$
$$b+d=4 implies b=d=2$$
$$e+f=8 implies e=f=4$$
Hence $(a,c) = (4,2)$ or $(a,c)=(2,4)$.
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
1
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
add a comment |Â
up vote
0
down vote
up vote
0
down vote
If all the variable takes value from $2,4$.
The constraints reduces to
$$a+c=6$$
$$b+d=4 implies b=d=2$$
$$e+f=8 implies e=f=4$$
Hence $(a,c) = (4,2)$ or $(a,c)=(2,4)$.
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
If all the variable takes value from $2,4$.
The constraints reduces to
$$a+c=6$$
$$b+d=4 implies b=d=2$$
$$e+f=8 implies e=f=4$$
Hence $(a,c) = (4,2)$ or $(a,c)=(2,4)$.
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
answered Sep 1 at 9:16
Siong Thye Goh
81.9k1456104
81.9k1456104
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
1
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
add a comment |Â
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
1
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
How can an IP have "gaps" like the missing $3$ in $2,4$?
– Rodrigo de Azevedo
Sep 9 at 12:41
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
his constraint has gap isn't it? Did I misread his question?
– Siong Thye Goh
Sep 9 at 12:55
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
It has a gap. But is it still an IP? Is the feasible region of an IP the intersection of a convex polytope with $mathbb Z^n$? Or can it be the union of convex polytopes intersected with $mathbb Z^n$? Am I missing anything obvious?
– Rodrigo de Azevedo
Sep 9 at 12:59
1
1
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
well, that depends on the definiton of IP. For this particular question, we can divide all the variables by $2$ and the gap is gone.
– Siong Thye Goh
Sep 9 at 13:01
add a comment |Â
up vote
0
down vote
In your case, there is no need to advanced methods, unless you want to practice how to use them. If you only want to solve this specific problem you can do this:
$a+b+c+d+e+f=18$
is
$a+(b+d)+c+(e+f) =18$
so
$a+4+c+8=18$
so
$a+c=6$
since $a,b,c,d,e,fin 4,2 $
you can easily determine integer values for $a,c$ from the last equation.
answered Sep 1 at 10:14
NoChance
3,35021221
add a comment |Â
up vote
0
down vote
In your case, there is no need to advanced methods, unless you want to practice how to use them. If you only want to solve this specific problem you can do this:
$a+b+c+d+e+f=18$
is
$a+(b+d)+c+(e+f) =18$
so
$a+4+c+8=18$
so
$a+c=6$
since $a,b,c,d,e,fin 4,2 $
you can easily determine integer values for $a,c$ from the last equation.
answered Sep 1 at 10:14
NoChance
3,35021221
add a comment |Â
up vote
0
down vote
up vote
0
down vote
In your case, there is no need to advanced methods, unless you want to practice how to use them. If you only want to solve this specific problem you can do this:
$a+b+c+d+e+f=18$
is
$a+(b+d)+c+(e+f) =18$
so
$a+4+c+8=18$
so
$a+c=6$
since $a,b,c,d,e,fin 4,2 $
you can easily determine integer values for $a,c$ from the last equation.
answered Sep 1 at 10:14
NoChance
3,35021221
In your case, there is no need to advanced methods, unless you want to practice how to use them. If you only want to solve this specific problem you can do this:
$a+b+c+d+e+f=18$
is
$a+(b+d)+c+(e+f) =18$
so
$a+4+c+8=18$
so
$a+c=6$
since $a,b,c,d,e,fin 4,2 $
you can easily determine integer values for $a,c$ from the last equation.
answered Sep 1 at 10:14
NoChance
3,35021221
answered Sep 1 at 10:14
NoChance
3,35021221
answered Sep 1 at 10:14
NoChance
3,35021221
answered Sep 1 at 10:14
NoChance
3,35021221
3,35021221
add a comment |Â
add a comment |Â
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1
we like to use the first letters of the alphabet for constants, and the last letters for variables
– LinAlg
Sep 1 at 13:15
Any integer program can be solved using (C) branch-and-bound (though it might be very slow—like, thousands of years slow). Any integer program can also be solved using (A) Gomory’s cut—if you add enough rounds of cuts, then you will find the optimal solution. This technique is ALSO very slow. As it turns out, if you combine (A) and (C) together, the result can be very fast. This is (very, very roughly) how IPs are solved today. Look up “branch and cutâ€Â.
– David M.
Sep 1 at 13:48