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soft_constraints_sat

<table align="left"> <td> <a href="https://colab.research.google.com/github/google/or-tools/blob/main/examples/notebook/sat/soft_constraints_sat.ipynb">Run in Google Colab</a> </td> <td> <a href="https://github.com/google/or-tools/blob/main/ortools/sat/samples/soft_constraints_sat.py">View source on GitHub</a> </td> </table>

First, you must install ortools package in this colab.

python

%pip install ortools

The sample shows multiple ways to model soft constraints in CP-SAT.

python

from ortools.sat.python import cp_model



def infeasible_model() -> None:
    """Base model that is infeasible."""
    # Creates the model.
    model = cp_model.CpModel()

    # Creates the variables.
    x = model.new_int_var(0, 10, "x")
    y = model.new_int_var(0, 10, "y")
    z = model.new_int_var(0, 10, "z")

    # Creates the constraints.
    model.add(x > y)
    model.add(y > z)
    model.add(z > x)

    # Creates a solver and solves.
    solver = cp_model.CpSolver()
    status = solver.solve(model)

    # Print solution.
    print(f"  Status = {solver.status_name(status)}")


def model_with_enforcement_literals() -> None:
    """Adds fixed costs to violated constraints."""
    # Creates the model.
    model = cp_model.CpModel()

    # Creates the variables.
    x = model.new_int_var(0, 10, "x")
    y = model.new_int_var(0, 10, "y")
    z = model.new_int_var(0, 10, "z")
    a = model.new_bool_var("a")
    b = model.new_bool_var("b")

    # Creates the constraints. Adds enforcement literals to the first two
    # constraints, we assume the third constraint is always enforced.
    model.add(x > y).only_enforce_if(a)
    model.add(y > z).only_enforce_if(b)
    model.add(z > x)

    # Adds an objective to maximize the number of enforced constraints.
    model.maximize(a + 2 * b)

    # Creates a solver and solves.
    solver = cp_model.CpSolver()
    status = solver.solve(model)

    # Print solution.
    print(f"  Status = {solver.status_name(status)}")
    if status == cp_model.OPTIMAL:
        print(f"  Objective value = {solver.objective_value}")
        print(f"  Value of x = {solver.value(x)}")
        print(f"  Value of y = {solver.value(y)}")
        print(f"  Value of z = {solver.value(z)}")
        print(f"  Value of a = {solver.boolean_value(a)}")
        print(f"  Value of b = {solver.boolean_value(b)}")


def model_with_linear_violations() -> None:
    """Adds fixed costs to violated constraints."""
    # Creates the model.
    model = cp_model.CpModel()

    # Creates the variables.
    x = model.new_int_var(0, 10, "x")
    y = model.new_int_var(0, 10, "y")
    z = model.new_int_var(0, 10, "z")
    a = model.new_int_var(0, 10, "a")
    b = model.new_int_var(0, 10, "b")

    # Creates the constraints. Adds enforcement literals to the first two
    # constraints, we assume the third constraint is always enforced.
    model.add(x > y - a)
    model.add(y > z - b)
    model.add(z > x)

    # Adds an objective to minimize the added slacks.
    model.minimize(a + 2 * b)

    # Creates a solver and solves.
    solver = cp_model.CpSolver()
    status = solver.solve(model)

    # Print solution.
    print(f"  Status = {solver.status_name(status)}")
    if status == cp_model.OPTIMAL:
        print(f"  Objective value = {solver.objective_value}")
        print(f"  Value of x = {solver.value(x)}")
        print(f"  Value of y = {solver.value(y)}")
        print(f"  Value of z = {solver.value(z)}")
        print(f"  Value of a = {solver.value(a)}")
        print(f"  Value of b = {solver.value(b)}")


def model_with_quadratic_violations() -> None:
    """Adds fixed costs to violated constraints."""
    # Creates the model.
    model = cp_model.CpModel()

    # Creates the variables.
    x = model.new_int_var(0, 10, "x")
    y = model.new_int_var(0, 10, "y")
    z = model.new_int_var(0, 10, "z")
    a = model.new_int_var(0, 10, "a")
    b = model.new_int_var(0, 10, "b")
    square_a = model.new_int_var(0, 100, "square_a")
    square_b = model.new_int_var(0, 100, "square_b")

    # Creates the constraints. Adds enforcement literals to the first two
    # constraints, we assume the third constraint is always enforced.
    model.add(x > y - a)
    model.add(y > z - b)
    model.add(z > x)

    model.add_multiplication_equality(square_a, a, a)
    model.add_multiplication_equality(square_b, b, b)

    # Adds an objective to minimize the added slacks.
    model.minimize(square_a + 2 * square_b)

    # Creates a solver and solves.
    solver = cp_model.CpSolver()
    status = solver.solve(model)

    # Print solution.
    print(f"  Status = {solver.status_name(status)}")
    if status == cp_model.OPTIMAL:
        print(f"  Objective value = {solver.objective_value}")
        print(f"  Value of x = {solver.value(x)}")
        print(f"  Value of y = {solver.value(y)}")
        print(f"  Value of z = {solver.value(z)}")
        print(f"  Value of a = {solver.value(a)}")
        print(f"  Value of b = {solver.value(b)}")


def main() -> None:
    print("Infeasible model:")
    infeasible_model()
    print("Model with enforcement literals:")
    model_with_enforcement_literals()
    print("Model with linear violations:")
    model_with_linear_violations()
    print("Model with quadratic violations:")
    model_with_quadratic_violations()


main()