I need to model the following statement:
if $\sum\limits_{i=1}^N X_i=k$ then $Y=1$ else $Y=0$
$X_i$'s are binary variables
$k$ is an integer between $0$ and $N$
$Y$ is a binary variable
Thank you in advance.
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In case $N$ is a constant, then add the following two inequalities to your MILP: $$NY \leq k $$ $$N-1+Y \geq k$$
As $Y$ is a binary variable, if $k\lt N$ then $Y$ must be $0$ to fulfill both inequalities. And when $k=N$ then $Y$ must be $1$ to fulfill both inequalities
Maksim
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Thank you so much for your proposition. but when k=N (k<>0), with Y=0 both inequalities are true and with Y=1, the second inequality is False. Thanks again. – Sou Mar 26 '19 at 14:51
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My apologies, I corrected the inequalities. – Maksim Mar 26 '19 at 16:10
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Thank you Maksim for your solution. – Sou Mar 26 '19 at 17:20
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HI, Is it possible to model: if X>k then Y=0, if X<k then Y=0, if X=k then Y=1. Any help is appreciated. – Sou Mar 27 '19 at 12:41
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Is $X$ a constant ? – Maksim Mar 27 '19 at 12:49
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X is a decision variable between 0 and N, Y is a binary decision variable and k is an integer (constant). – Sou Mar 27 '19 at 12:58
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Is $X$ bounded, i.e. are there some constants $a,b$ such that $a\leq X\leq b$? – Maksim Mar 27 '19 at 13:25
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Yes, $$0\leq X\leq N$$. – Sou Mar 27 '19 at 13:27
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added a separate answer to your 2nd Q – Maksim Mar 27 '19 at 17:28
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To your question in the comments: $N$ positive constant, $k$ positive integer constant with $k\leq N$, $X$ decision variable with $0\leq X \leq N$.
Add 3 binary variables $r_1, r_2,r_3$ together with the following constraints: $$ \eqalign{ kr_1 & \leq X \\ X & \leq (1-r_1)(k-1)+r_1N \\ (k+1)(1-r_2) & \leq X\\ X & \leq kr_2+(1-r_2)N\\ 2r_3 & \leq r_1 +r_2 \\ r_1 +r_2 & \leq 1+ r_3 }$$
Then $r_3 = 1$ iff $X=k$, else $r_3 = 0$
Maksim
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Thank you Maksim again for your response. I noticed that in case X>k or X<k it works perfectly, but when X=k, $$r_1$$ could be 0/1 and $$r_2$$ could be 0/1 and if they both are not 1 and 1 $$r_3$$ should be 0 and it is false. – Sou Mar 27 '19 at 18:04
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Yes, it works in all cases!! Thank you so much. If possible, I would like to know if you find these solution just by intuition (experience) or there is a method that helps (I am relatively new and sometimes I get stuck!) . Thkx again. – Sou Mar 27 '19 at 19:20