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.
