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I am trying to model the following constraint using binary variables $a$, $b$, $c$ and $d$:

$a$ will be true if at least one of $b$,$c$, or $d$ is true.

I know how to do $a$ if $b$ but I am having a very hard time introducing the "at least one of" part.

What I tried but realized didn't work is $a \leq b+c+d$ because this does mean $a$ can only be true if one of he others is true but it does not mean $a$ will be true if one of the others is true.

UPDATE: I just solved it! I did $a\geq b$,$a\geq c$,$a\geq d$. To show if any are $1$ (true) then $a$ must be as well.

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2 Answers 2

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HINT

  1. Learn how to define $x = b \text{ or } c$ .
  2. Similarly, define $y = x \text{ or } d$.
  3. Use the technique you mentioned you know to force $a$ if $y$.

UPDATE

What does $x \ge b$ enforce in the relationship between $x$ and $b$? What about $x \ge b$ and $x \ge c$ simultaneously?

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  • $\begingroup$ Here is my take at this, please let me know if my logic is correct: $x \leq b+c$ and $y \leq x+d$ and $a=y$ where all the variables are binary $\endgroup$ Commented May 18, 2017 at 20:43
  • $\begingroup$ @mhlzzz if you just want $b \text{ or } c \text{ or } d = 1 \implies a = 1$ then it's much simpler $\endgroup$ Commented May 18, 2017 at 20:46
  • $\begingroup$ @mhlzzz your solution does not force anything, in fact, if both $b,c$ are true, $x$ can be anything. See another update $\endgroup$ Commented May 18, 2017 at 20:48
  • $\begingroup$ Regarding your update, what I gather is that $x=1$ if $b=1$ or $c=1$ but if $b=0$ and $c=0$ then $x$ can be $0$ or $1$ $\endgroup$ Commented May 18, 2017 at 20:52
  • $\begingroup$ @mhlzzz yes, i figured from your own answer you figured ot where i was leading you. $\endgroup$ Commented May 19, 2017 at 17:06
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I did $a\geq b$,$a\geq c$,$a\geq d$. To show if any of $b$, $c$, or $d$ are $1$ (true) then $a$ must be as well.

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