What's a basic solution, and how do we find them?
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I've just started learning linear programming, and for some reason, have run into a question about something that isn't mentioned in the first chapter (and we're supposed to answer these questions based on the first chapter).
What is a "basic" solution? It is only mentioned once (so far) in the book, in the following manner:
$$text{"The solutions we obtain by setting the nonbasic variables to zero are called basic feasible solutions"}$$
But the question now asks; "indicate EACH basic solution, and determine which are feasible and which are infeasible", and I don't see how the above quote says anything about several basic solutions. Surely there's only one way we can set all variables to zero, and so only one basic solution?
For example, what are the basic solutions for something as simple as; $$max 2x + y \ s.t. x + y le 3 \ x le 2 $$
where both variables are nonnegative.
linear-programming
asked Feb 2 '15 at 13:22
Arda Daniels
1612
add a comment |
up vote
3
down vote
favorite
I've just started learning linear programming, and for some reason, have run into a question about something that isn't mentioned in the first chapter (and we're supposed to answer these questions based on the first chapter).
What is a "basic" solution? It is only mentioned once (so far) in the book, in the following manner:
$$text{"The solutions we obtain by setting the nonbasic variables to zero are called basic feasible solutions"}$$
But the question now asks; "indicate EACH basic solution, and determine which are feasible and which are infeasible", and I don't see how the above quote says anything about several basic solutions. Surely there's only one way we can set all variables to zero, and so only one basic solution?
For example, what are the basic solutions for something as simple as; $$max 2x + y \ s.t. x + y le 3 \ x le 2 $$
where both variables are nonnegative.
linear-programming
asked Feb 2 '15 at 13:22
Arda Daniels
1612
You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $;(0,0),,,,(2,0),,,,(2,1),,,,(0,3);$ , so you probably have to evaluate the target function $;f(x,y)=2x+y;$ on each of these points and choose the max/min value.
– Timbuc
Feb 2 '15 at 13:43
BTW, what book are you reading on this?
– Timbuc
Feb 2 '15 at 13:50
Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix.
– Arda Daniels
Feb 2 '15 at 13:53
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I've just started learning linear programming, and for some reason, have run into a question about something that isn't mentioned in the first chapter (and we're supposed to answer these questions based on the first chapter).
What is a "basic" solution? It is only mentioned once (so far) in the book, in the following manner:
$$text{"The solutions we obtain by setting the nonbasic variables to zero are called basic feasible solutions"}$$
But the question now asks; "indicate EACH basic solution, and determine which are feasible and which are infeasible", and I don't see how the above quote says anything about several basic solutions. Surely there's only one way we can set all variables to zero, and so only one basic solution?
For example, what are the basic solutions for something as simple as; $$max 2x + y \ s.t. x + y le 3 \ x le 2 $$
where both variables are nonnegative.
linear-programming
asked Feb 2 '15 at 13:22
Arda Daniels
1612
I've just started learning linear programming, and for some reason, have run into a question about something that isn't mentioned in the first chapter (and we're supposed to answer these questions based on the first chapter).
What is a "basic" solution? It is only mentioned once (so far) in the book, in the following manner:
$$text{"The solutions we obtain by setting the nonbasic variables to zero are called basic feasible solutions"}$$
But the question now asks; "indicate EACH basic solution, and determine which are feasible and which are infeasible", and I don't see how the above quote says anything about several basic solutions. Surely there's only one way we can set all variables to zero, and so only one basic solution?
For example, what are the basic solutions for something as simple as; $$max 2x + y \ s.t. x + y le 3 \ x le 2 $$
where both variables are nonnegative.
linear-programming
linear-programming
asked Feb 2 '15 at 13:22
Arda Daniels
1612
asked Feb 2 '15 at 13:22
Arda Daniels
1612
asked Feb 2 '15 at 13:22
Arda Daniels
1612
asked Feb 2 '15 at 13:22
Arda Daniels
1612
asked Feb 2 '15 at 13:22
Arda Daniels
1612
1612
You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $;(0,0),,,,(2,0),,,,(2,1),,,,(0,3);$ , so you probably have to evaluate the target function $;f(x,y)=2x+y;$ on each of these points and choose the max/min value.
– Timbuc
Feb 2 '15 at 13:43
BTW, what book are you reading on this?
– Timbuc
Feb 2 '15 at 13:50
Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix.
– Arda Daniels
Feb 2 '15 at 13:53
add a comment |
You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $;(0,0),,,,(2,0),,,,(2,1),,,,(0,3);$ , so you probably have to evaluate the target function $;f(x,y)=2x+y;$ on each of these points and choose the max/min value.
– Timbuc
Feb 2 '15 at 13:43
BTW, what book are you reading on this?
– Timbuc
Feb 2 '15 at 13:50
Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix.
– Arda Daniels
Feb 2 '15 at 13:53
You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $;(0,0),,,,(2,0),,,,(2,1),,,,(0,3);$ , so you probably have to evaluate the target function $;f(x,y)=2x+y;$ on each of these points and choose the max/min value.
– Timbuc
Feb 2 '15 at 13:43
You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $;(0,0),,,,(2,0),,,,(2,1),,,,(0,3);$ , so you probably have to evaluate the target function $;f(x,y)=2x+y;$ on each of these points and choose the max/min value.
– Timbuc
Feb 2 '15 at 13:43
BTW, what book are you reading on this?
– Timbuc
Feb 2 '15 at 13:50
BTW, what book are you reading on this?
– Timbuc
Feb 2 '15 at 13:50
Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix.
– Arda Daniels
Feb 2 '15 at 13:53
Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix.
– Arda Daniels
Feb 2 '15 at 13:53
add a comment |
1 Answer
1
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oldest
votes
up vote
0
down vote
Rewrite the inequalities as
$$x+y+a=3\x+b=2\xgeq0,ygeq0,ageq0,bgeq0$$
A basic solution has any two of $a,b,x,y$ equal to zero. The other two variables are forced by the two equations.
The feasible basic solutions have the other two variables positive or zero.
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Rewrite the inequalities as
$$x+y+a=3\x+b=2\xgeq0,ygeq0,ageq0,bgeq0$$
A basic solution has any two of $a,b,x,y$ equal to zero. The other two variables are forced by the two equations.
The feasible basic solutions have the other two variables positive or zero.
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
add a comment |
up vote
0
down vote
Rewrite the inequalities as
$$x+y+a=3\x+b=2\xgeq0,ygeq0,ageq0,bgeq0$$
A basic solution has any two of $a,b,x,y$ equal to zero. The other two variables are forced by the two equations.
The feasible basic solutions have the other two variables positive or zero.
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
add a comment |
up vote
0
down vote
up vote
0
down vote
Rewrite the inequalities as
$$x+y+a=3\x+b=2\xgeq0,ygeq0,ageq0,bgeq0$$
A basic solution has any two of $a,b,x,y$ equal to zero. The other two variables are forced by the two equations.
The feasible basic solutions have the other two variables positive or zero.
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
Rewrite the inequalities as
$$x+y+a=3\x+b=2\xgeq0,ygeq0,ageq0,bgeq0$$
A basic solution has any two of $a,b,x,y$ equal to zero. The other two variables are forced by the two equations.
The feasible basic solutions have the other two variables positive or zero.
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
answered Feb 2 '15 at 13:26
Empy2
33.3k12261
33.3k12261
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
add a comment |
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
– Arda Daniels
Feb 2 '15 at 13:37
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
That means there are only five basic solutions, not six
– Empy2
Feb 2 '15 at 14:57
add a comment |
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You should read definitions. Perhaps "basic solution" is one on the vertices of the domain, which in this case are $;(0,0),,,,(2,0),,,,(2,1),,,,(0,3);$ , so you probably have to evaluate the target function $;f(x,y)=2x+y;$ on each of these points and choose the max/min value.
– Timbuc
Feb 2 '15 at 13:43
BTW, what book are you reading on this?
– Timbuc
Feb 2 '15 at 13:50
Vanderbel, Linear Programming. It's also not using matrices and linear algebra (but again, have only read the first two chapters), so I can't get much help from reading online, where apparently "basic solution" has something to do with linearly independent columns in a matrix.
– Arda Daniels
Feb 2 '15 at 13:53