Good morning ! I would like to implement this constraint using CP-SAT (see image below).

x_i,j is a boolean variable, a and b are given. The problem is that I don't know how to implement the indicator function in a constraint. I tried with int() function but it cannot be applied with a BoundedLinearExpression. Here is my code:

for i in range(20):
  for j in range(30):
    x[(i,j)] = model.NewBoolVar('variable%ii%ij' % (i,j))
model.Add(a <= sum( int( sum(x[(i,j)] for j in range(30)) >= 5 ) for i in range(20) ) <= b)

Any idea on how to solve it ?

  • 1
    $\begingroup$ This is a use-case for or-tools docs: Channeling constraints. As mentioned in the docs, this is highly linked to the concept of half/full reification in constraint-programming. (You might also want to decompose this into multiple expressions -> a new result-vector for all geq-expression results followed by two inequalities). $\endgroup$
    – sascha
    Mar 31, 2023 at 0:40
  • 1
    $\begingroup$ You can use AddLinearConstraint to have both LHS and RHS. $\endgroup$ Mar 31, 2023 at 5:53
  • $\begingroup$ Hello @sascha and @laurent-perron, thanks for your response. I think I made the result-vector : b = [model.NewBoolVar('b%i' % (i)) for i in range(20)] for i in range(20): model.Add( sum(x[(i,j)] for j in range(30) ) >= 5).OnlyEnforceIf(b[i]) So now I have all my indicator functions, but I struggle a bit for the second sum. Is it simply this following line that I need to add ? model.AddLinearConstraint(sum(b[i] for i in range(20)), a, b) $\endgroup$
    – Arthursbr
    Mar 31, 2023 at 10:16
  • 1
    $\begingroup$ Welcome to OR.SE Arthursbr. Please edit your question and rather than the image, use MathJax to make your question searchable. $\endgroup$
    – EhsanK
    Mar 31, 2023 at 13:04

1 Answer 1


If I understand you are trying this.
For each $i$, sum across $j$. Then if all the sums across $j \ge 5$ for all $i$ (basically the $\pi$, the AND logic), then whole sum will be between constants $a,b$.
I can linearize it as below (OR-Tools will do the same thing)
Introduce set of new binary variables $y_i$. Add linear constraints
$ \sum_j x_{i,j} \le 5+My_i$
$ 5- \sum_j x_{i,j} \le M(1-y_i)$
where could be appropriate big number, like upper bound of sum of $x$ values possible.

Then you'd need another binary variable $\delta$ and below constraints
$\delta \le y_i \quad \forall i$
$ \sum_iy_i -I+1 \le \delta$ where $I$ is total number of your $i$s

Then the below constraint
$ a-M(1-\delta) \le \sum_i \sum_j x_{i,j} \le b + M(1- \delta)$

  • $\begingroup$ I think it is supposed to be indicator $\mathbb{I}$, not product $\prod$. $\endgroup$
    – RobPratt
    Mar 31, 2023 at 13:31
  • 1
    $\begingroup$ This seems to assume, CP-SATs native mode of inference is based on LP/MILP technology which is questionable. CP-SAT contains most features of a MILP-solver and indeed does offer automatic linearization based on global/high-level constraints, but this is optional (can be turned off; defaults are conservative imho). It's hard to predict if cp-sat successfully progresses on a given problem better because of search/propagation or linearization, but the API alone is CP-based and reification/channeling-based approaches are imho the native approach (solver can decide how to exploit it internally). $\endgroup$
    – sascha
    Mar 31, 2023 at 15:26

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