Partial decision ordering for linear programs
$begingroup$
I'm very new to linear programming, so please bear with me:
I have a problem where I want to maximize the amount of money I can return for $d$ products to a group of members that are split into 3 groups arbitrarily per product. The amount I can give per product has an upper bound, $b in mathbb{R}^d$, and each grouping has an amount of money I'm wanting to return a percentage on. The formulation I have so far looks like this:
Maximize: $$c^Tx$$
Subject to:
$$
c_1x_1 + c_2x_2 + c_3x_3 le b_1 \
vdots \
c_{3d-2}x_{3d-2} + c_{3d-1}x_{3d-1} + c_{3d}x_{3d} le b_d \
$$
There are additional bounds on the $x$'s to define a range of values that they can take. Is there a way to constrain only some of the $x$'s such that:
$$
x_1 le x_2 le x_3 \
vdots \
x_{3d-2} le x_{3d-1} le x_{3d} \
$$
so that there's some kind of partial ordering due to the products? How would I go about formalizing this?
optimization linear-programming
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
$endgroup$
add a comment |
$begingroup$
I'm very new to linear programming, so please bear with me:
I have a problem where I want to maximize the amount of money I can return for $d$ products to a group of members that are split into 3 groups arbitrarily per product. The amount I can give per product has an upper bound, $b in mathbb{R}^d$, and each grouping has an amount of money I'm wanting to return a percentage on. The formulation I have so far looks like this:
Maximize: $$c^Tx$$
Subject to:
$$
c_1x_1 + c_2x_2 + c_3x_3 le b_1 \
vdots \
c_{3d-2}x_{3d-2} + c_{3d-1}x_{3d-1} + c_{3d}x_{3d} le b_d \
$$
There are additional bounds on the $x$'s to define a range of values that they can take. Is there a way to constrain only some of the $x$'s such that:
$$
x_1 le x_2 le x_3 \
vdots \
x_{3d-2} le x_{3d-1} le x_{3d} \
$$
so that there's some kind of partial ordering due to the products? How would I go about formalizing this?
optimization linear-programming
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
$endgroup$
1
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
$endgroup$
– LinAlg
Jan 6 at 19:59
$begingroup$
This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
$endgroup$
– Brad Flynn
Jan 6 at 22:14
$begingroup$
Glad that's what you needed. I have added my comment as an answer for you to accept.
$endgroup$
– LinAlg
Jan 6 at 23:41
add a comment |
$begingroup$
I'm very new to linear programming, so please bear with me:
I have a problem where I want to maximize the amount of money I can return for $d$ products to a group of members that are split into 3 groups arbitrarily per product. The amount I can give per product has an upper bound, $b in mathbb{R}^d$, and each grouping has an amount of money I'm wanting to return a percentage on. The formulation I have so far looks like this:
Maximize: $$c^Tx$$
Subject to:
$$
c_1x_1 + c_2x_2 + c_3x_3 le b_1 \
vdots \
c_{3d-2}x_{3d-2} + c_{3d-1}x_{3d-1} + c_{3d}x_{3d} le b_d \
$$
There are additional bounds on the $x$'s to define a range of values that they can take. Is there a way to constrain only some of the $x$'s such that:
$$
x_1 le x_2 le x_3 \
vdots \
x_{3d-2} le x_{3d-1} le x_{3d} \
$$
so that there's some kind of partial ordering due to the products? How would I go about formalizing this?
optimization linear-programming
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
$endgroup$
I'm very new to linear programming, so please bear with me:
I have a problem where I want to maximize the amount of money I can return for $d$ products to a group of members that are split into 3 groups arbitrarily per product. The amount I can give per product has an upper bound, $b in mathbb{R}^d$, and each grouping has an amount of money I'm wanting to return a percentage on. The formulation I have so far looks like this:
Maximize: $$c^Tx$$
Subject to:
$$
c_1x_1 + c_2x_2 + c_3x_3 le b_1 \
vdots \
c_{3d-2}x_{3d-2} + c_{3d-1}x_{3d-1} + c_{3d}x_{3d} le b_d \
$$
There are additional bounds on the $x$'s to define a range of values that they can take. Is there a way to constrain only some of the $x$'s such that:
$$
x_1 le x_2 le x_3 \
vdots \
x_{3d-2} le x_{3d-1} le x_{3d} \
$$
so that there's some kind of partial ordering due to the products? How would I go about formalizing this?
optimization linear-programming
optimization linear-programming
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
edited Jan 6 at 2:28
Brad Flynn
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
asked Jan 5 at 11:36
Brad FlynnBrad Flynn
446
446
1
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
$endgroup$
– LinAlg
Jan 6 at 19:59
$begingroup$
This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
$endgroup$
– Brad Flynn
Jan 6 at 22:14
$begingroup$
Glad that's what you needed. I have added my comment as an answer for you to accept.
$endgroup$
– LinAlg
Jan 6 at 23:41
add a comment |
1
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
$endgroup$
– LinAlg
Jan 6 at 19:59
$begingroup$
This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
$endgroup$
– Brad Flynn
Jan 6 at 22:14
$begingroup$
Glad that's what you needed. I have added my comment as an answer for you to accept.
$endgroup$
– LinAlg
Jan 6 at 23:41
1
1
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
$endgroup$
– LinAlg
Jan 6 at 19:59
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
$endgroup$
– LinAlg
Jan 6 at 19:59
$begingroup$
This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
$endgroup$
– Brad Flynn
Jan 6 at 22:14
$begingroup$
This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
$endgroup$
– Brad Flynn
Jan 6 at 22:14
$begingroup$
Glad that's what you needed. I have added my comment as an answer for you to accept.
$endgroup$
– LinAlg
Jan 6 at 23:41
$begingroup$
Glad that's what you needed. I have added my comment as an answer for you to accept.
$endgroup$
– LinAlg
Jan 6 at 23:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1−x_2leq 0$, $x_2-x_3leq 0$, etc. Or, more formally:
$$x_i - x_j leq 0 quad forall (i,j)in S$$
where $S$ is the set of pairs $(i,j)$ such that $x_i leq x_j$.
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
$endgroup$
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%2f3062632%2fpartial-decision-ordering-for-linear-programs%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
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1−x_2leq 0$, $x_2-x_3leq 0$, etc. Or, more formally:
$$x_i - x_j leq 0 quad forall (i,j)in S$$
where $S$ is the set of pairs $(i,j)$ such that $x_i leq x_j$.
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
$endgroup$
add a comment |
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1−x_2leq 0$, $x_2-x_3leq 0$, etc. Or, more formally:
$$x_i - x_j leq 0 quad forall (i,j)in S$$
where $S$ is the set of pairs $(i,j)$ such that $x_i leq x_j$.
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
$endgroup$
add a comment |
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1−x_2leq 0$, $x_2-x_3leq 0$, etc. Or, more formally:
$$x_i - x_j leq 0 quad forall (i,j)in S$$
where $S$ is the set of pairs $(i,j)$ such that $x_i leq x_j$.
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
$endgroup$
Your constraint already look formal. You can rewrite them to $x_1−x_2leq 0$, $x_2-x_3leq 0$, etc. Or, more formally:
$$x_i - x_j leq 0 quad forall (i,j)in S$$
where $S$ is the set of pairs $(i,j)$ such that $x_i leq x_j$.
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
answered Jan 6 at 23:41
LinAlgLinAlg
10.1k1521
10.1k1521
add a comment |
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%2f3062632%2fpartial-decision-ordering-for-linear-programs%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
1
$begingroup$
Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
$endgroup$
– LinAlg
Jan 6 at 19:59
$begingroup$
This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
$endgroup$
– Brad Flynn
Jan 6 at 22:14
$begingroup$
Glad that's what you needed. I have added my comment as an answer for you to accept.
$endgroup$
– LinAlg
Jan 6 at 23:41