Partial decision ordering for linear programs












1












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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
















1












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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














1












1








1





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








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














  • 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










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 $x_i leq x_j$.






share|cite|improve this 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 $x_i leq x_j$.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      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 $x_i leq x_j$.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        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 $x_i leq x_j$.






        share|cite|improve this answer









        $endgroup$



        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 $x_i leq x_j$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 6 at 23:41









        LinAlgLinAlg

        10k1521




        10k1521






























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