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Formulating these logical constraint in an ILP

Operations Research Asked on January 18, 2021

I have these two constraints :

$z leq My$

$t leq M’y $

where $z$ and $t$ are two integer variables $ z, tgeq 0$, $y$ is a binary variable, and $M$, $M’$ are two big numbers.

So basically these constraints ensure that if $y = 1$ then $z, t leq M , M’$ respectively, otherwise $z,t = 0$.

However these constraints won’t give me a positive value for $t$ if $z > 0$.

My question is : how to connect variables $z$ and $t$ to ensure that if $z > 0$ then $t > 0$ .

The logical constraint that I want to write is as following:

if $y= 1$ then $z >0$ and $t>0$ .

Thank you.

One Answer

Let $epsilon > 0$ be a tolerance for what you consider positive. Now impose linear constraints $z ge epsilon y$ and $t ge epsilon y$. Because $z$ and $t$ are integer variables, you can take $epsilon=1$.

Correct answer by RobPratt on January 18, 2021

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