What's a basic solution, and how do we find them?
$begingroup$
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 DanielsArda Daniels
2112
$endgroup$
add a comment |
$begingroup$
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 DanielsArda Daniels
2112
$endgroup$
$begingroup$
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.
$endgroup$
– Timbuc
Feb 2 '15 at 13:43
$begingroup$
BTW, what book are you reading on this?
$endgroup$
– Timbuc
Feb 2 '15 at 13:50
$begingroup$
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.
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:53
add a comment |
$begingroup$
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 DanielsArda Daniels
2112
$endgroup$
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 DanielsArda Daniels
2112
asked Feb 2 '15 at 13:22
Arda DanielsArda Daniels
2112
asked Feb 2 '15 at 13:22
Arda DanielsArda Daniels
2112
asked Feb 2 '15 at 13:22
Arda DanielsArda Daniels
2112
asked Feb 2 '15 at 13:22
Arda DanielsArda Daniels
2112
2112
$begingroup$
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.
$endgroup$
– Timbuc
Feb 2 '15 at 13:43
$begingroup$
BTW, what book are you reading on this?
$endgroup$
– Timbuc
Feb 2 '15 at 13:50
$begingroup$
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.
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:53
add a comment |
$begingroup$
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.
$endgroup$
– Timbuc
Feb 2 '15 at 13:43
$begingroup$
BTW, what book are you reading on this?
$endgroup$
– Timbuc
Feb 2 '15 at 13:50
$begingroup$
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.
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:53
$begingroup$
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.
$endgroup$
– Timbuc
Feb 2 '15 at 13:43
$begingroup$
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.
$endgroup$
– Timbuc
Feb 2 '15 at 13:43
$begingroup$
BTW, what book are you reading on this?
$endgroup$
– Timbuc
Feb 2 '15 at 13:50
$begingroup$
BTW, what book are you reading on this?
$endgroup$
– Timbuc
Feb 2 '15 at 13:50
$begingroup$
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.
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:53
$begingroup$
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.
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:53
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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
Empy2Empy2
33.5k12261
$endgroup$
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1130269%2fwhats-a-basic-solution-and-how-do-we-find-them%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
Empy2Empy2
33.5k12261
$endgroup$
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
add a comment |
$begingroup$
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
Empy2Empy2
33.5k12261
$endgroup$
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
add a comment |
$begingroup$
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
Empy2Empy2
33.5k12261
$endgroup$
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
Empy2Empy2
33.5k12261
answered Feb 2 '15 at 13:26
Empy2Empy2
33.5k12261
answered Feb 2 '15 at 13:26
Empy2Empy2
33.5k12261
answered Feb 2 '15 at 13:26
Empy2Empy2
33.5k12261
33.5k12261
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
add a comment |
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
So set x and b equal to zero. The second equality gives $0 + 0= 2$....So what does that mean?
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:37
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
$begingroup$
That means there are only five basic solutions, not six
$endgroup$
– Empy2
Feb 2 '15 at 14:57
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1130269%2fwhats-a-basic-solution-and-how-do-we-find-them%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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.
$endgroup$
– Timbuc
Feb 2 '15 at 13:43
$begingroup$
BTW, what book are you reading on this?
$endgroup$
– Timbuc
Feb 2 '15 at 13:50
$begingroup$
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.
$endgroup$
– Arda Daniels
Feb 2 '15 at 13:53