# Most constraining constraint CP-SAT

I am using CP-SAT to minimize a problem similar to Flexible Job Shop Scheduling (https://github.com/google/or-tools/blob/stable/examples/python/flexible_job_shop_sat.py), with a minimization objective.

I was wondering if OR-tools provided feedback on 'most constraining constraint'. For example;

Minimize 3x + 5y --> assuming integers for x and y

Constraint1: x, y ≥ 1

Constraint2: x + y ≥ 3

Constraint3: x, y nonnegative

Then constraint1 makes it so that the minimal value is 8. However, by constraint2, the minimum value would be 11. Would the CP-SAT model be able to report that this constraint would yield the most objective value improvement if relaxed?

• In practice, you might want to consider relaxing constraints without removing them (e.g., change the right hand side of constraint2 from 3 to 2). In LP models, you can get at this through dual values ("shadow prices"). In MIP and CP-SAT models, I think you are limited to trial and error.
– prubin
Commented Jul 19 at 15:35

\begin{aligned} \min\> & 3x+5y \\ & \delta_1 = 1 \implies x \ge 1 \\ & \delta_1 = 1 \implies y \ge 1 \\ & \delta_2 = 1 \implies x+y \ge 3 \\ & \delta_3 = 1 \implies x \ge 0 \\ & \delta_3 = 1 \implies y \ge 0 \\ & \sum_k \delta_k = 2 \\ & \delta_k \in \{0,1\} \end{aligned}
The equations to relax are identified by $$\delta_k=0$$ in the solution.