Partial decision ordering for linear programs
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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
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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
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1
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Your constraint already look formal. You can rewrite them to $x_1-x_2 leq 0$, x_2-x_3leq 0$, etc
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– LinAlg
Jan 6 at 19:59
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This is exactly what I needed! My mental block had to do with putting it in terms of a number, thank you!
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– Brad Flynn
Jan 6 at 22:14
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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
$endgroup$
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
edited Jan 6 at 2:28
Brad Flynn
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
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$begingroup$
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 $x_i leq x_j$.
$endgroup$
add a comment |
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$begingroup$
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 $x_i leq x_j$.
$endgroup$
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 $x_i leq x_j$.
$endgroup$
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 $x_i leq x_j$.
$endgroup$
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
10k1521
10k1521
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$begingroup$
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