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Consider the following trivial, theoretical model:

$$ \max x+2y+3z \qquad s.t. $$ $$ x \leq b_x $$ $$ y \leq b_y $$ $$ z \leq b_z $$ $$ x+y+z = 1 $$ $$ x,y,z \in \{0, 1\} $$

and $b_x$, $b_y$ and $b_z$ integer, non-negative parameters. Whenever the parameter $b_z \geq 1$ the trivial solution would be $z=1, x=y=0$. Let us assume I have hypotethical reasons to "prefer" $x$ over $z$, and $z$ over $y$; say, when $b_x$, $b_y$ and $b_z$ are greater/equal 1, return $x=1, y=z=0$ as solution. In other words, I would maximize the objective function under a "preference constraint" on variables. Is it possible to express such a constraint as a MIP/MILP?

For example, imposing $z \leq x$ and $y \leq z$ would do the trick when both $b_x$ and $b_z$ are greater/equal 1, but when $b_x = 0$, $b_z \geq 1$ and $b_y \geq 1$ it would return $x=y=z=0$, whereas the optimal solution for me according to my preference would be $z=1, x=y=0$. Similarly, with $b_x = b_z = 0$ and $b_y\geq 1$ the optimal solution from solving the problem would be again $x=y=z=0$ whereas what I want according to my preference would be $y=1, x=z=0$. So my attempt $z \leq x$ and $y \leq z$ clearly does not work.

I am looking for a generalizable solution for any number of variables and any preference, not just for a solution to this specific case.

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    $\begingroup$ Why not express the preferences in the objective function? $\endgroup$
    – RobPratt
    Mar 2, 2023 at 23:04
  • $\begingroup$ @RobPratt you mean changing the objective completely or adding a term to the current objective? I want to maximize the objective $x+2y+3z$ (let's say it represents a profit) under the constraint that, when possible, $x$ should be selected before $z$. $\endgroup$
    – Libra
    Mar 3, 2023 at 7:42
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    $\begingroup$ Then it sounds like you have a primary objective of maximizing preference and a secondary objective of maximizing profit. You might consider the lexicographic approach to this multiobjective problem. $\endgroup$
    – RobPratt
    Mar 3, 2023 at 13:37

1 Answer 1

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How is this following constraint
Either $b_z + y \le b_x+z +b_y\le x + b_z +yb_y$ Or let's break it
$b_x+z \le x + b_z$
$b_z + y \le z +b_y $

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  • $\begingroup$ It does not seems to work. When $b_x = b_z = 0$ and $b_y = m$ it gives $0+y \leq 0 + 0 + m \leq 0 + 0$, that is $y\leq m \leq 0$. I am looking for a general solution, not just a solution to this specific example. Thanks. $\endgroup$
    – Libra
    Mar 2, 2023 at 22:57
  • $\begingroup$ Ok let me break it up then $\endgroup$ Mar 2, 2023 at 23:02

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