[OR-Tools][CP-SAT] Implement an indicator function in a constraint

Question

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 ?

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)$