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I have 6 binary variables $a_i$ for $i$ from 0 to 5. I would like to model the next if-else statement using only MILP constraints

  if $(a_0+a_1+a_2)\mod 2=1$ then $(a_3+a_4+a_5) \mod 2 = 0$

I tried constraint (a_3+a_4+a_5) mod 2 = (a_0+a_1+a_2)mod 2+1, but this constraint is also taking the case when $(a_0+a_1+a_2)\mod 2 = 0$. And I need to take in consideration only the case $(a_0+a_1+a_2)\mod 2 = 1$.

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1 Answer 1

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One way to do this is to introduce three auxiliary variables: $x$ and $y$ are non-negative integers and $z$ is binary. The binary variable $z$ should equal 1 if and only if $a_0+a_1+a_2 \mbox{ mod } 2=1 $.

Then you may add the following constraints \begin{align} &a_0+a_2+a_2=2x+z\\ &a_3+a_4+a_5=2y+(1-z)\\ &x,y\in\mathbb{N}_0,\ z\in\{0,1\} \end{align} The first constraint says that $a_0+a_1+a_2 \mbox{ mod } 2=1 $ if and only if $z=1$ and the other constraint says tha $a_3+a_4+a_5 \mbox{ mod } 2=0 $ if and only if $z=1$.


Edit: Based on the comment by @RobPratt my first answer was a bit too restrictive. In stead of adding one binary variable $z$, introduce two binary variables $z_1$ and $z_2$ and add the following constraints \begin{align} &a_0+a_2+a_2=2x+z_1\\ &a_3+a_4+a_5=2y+z_2\\ &z_1+z_2\leq 1\\ &x,y\in\mathbb{N}_0,\ z_i\in\{0,1\}, i=1,2 \end{align}

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    $\begingroup$ What you proposed is a bit too strong, but you can repair it by replacing the $z$ In the first constraint with $z_1$, replacing the $1-z$ In the second constraint with $z_2$, and imposing $z_1 +z_2\le 1$. $\endgroup$
    – RobPratt
    Nov 3, 2021 at 12:33
  • $\begingroup$ Good point, I will add the edit $\endgroup$
    – Sune
    Nov 3, 2021 at 13:54

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