Restricciones de canal

Una restricción de canal vincula variables dentro de un modelo. Se usan cuando expresar una relación complicada entre variables, como "si este satisface una condición, fuerza otra variable a un valor particular".

Por lo general, la canalización se implementa usando restricciones lineales rematizadas: una. la restricción implica otro (a → b), pero no necesariamente de la otra forma alrededor de (a ← b).

Expresiones si-entonces-si no

Supongamos que deseas implementar lo siguiente: "Si x es menor que 5, establece y. en 0. De lo contrario, establece y en 10-x". Puedes hacerlo creando una evaluación variable booleana b que sea verdadera si x es mayor o igual que 5, y false; de lo contrario,

b implica y == 10 - x

no(b) implica y == 0

Estas se implementan con el método OnlyEnforceIf, como se muestra a continuación.

Python

#!/usr/bin/env python3
# Copyright 2010-2024 Google LLC
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

"""Link integer constraints together."""


from ortools.sat.python import cp_model


class VarArraySolutionPrinter(cp_model.CpSolverSolutionCallback):
    """Print intermediate solutions."""

    def __init__(self, variables: list[cp_model.IntVar]):
        cp_model.CpSolverSolutionCallback.__init__(self)
        self.__variables = variables

    def on_solution_callback(self) -> None:
        for v in self.__variables:
            print(f"{v}={self.value(v)}", end=" ")
        print()


def channeling_sample_sat():
    """Demonstrates how to link integer constraints together."""

    # Create the CP-SAT model.
    model = cp_model.CpModel()

    # Declare our two primary variables.
    x = model.new_int_var(0, 10, "x")
    y = model.new_int_var(0, 10, "y")

    # Declare our intermediate boolean variable.
    b = model.new_bool_var("b")

    # Implement b == (x >= 5).
    model.add(x >= 5).only_enforce_if(b)
    model.add(x < 5).only_enforce_if(~b)

    # Create our two half-reified constraints.
    # First, b implies (y == 10 - x).
    model.add(y == 10 - x).only_enforce_if(b)
    # Second, not(b) implies y == 0.
    model.add(y == 0).only_enforce_if(~b)

    # Search for x values in increasing order.
    model.add_decision_strategy([x], cp_model.CHOOSE_FIRST, cp_model.SELECT_MIN_VALUE)

    # Create a solver and solve with a fixed search.
    solver = cp_model.CpSolver()

    # Force the solver to follow the decision strategy exactly.
    solver.parameters.search_branching = cp_model.FIXED_SEARCH
    # Enumerate all solutions.
    solver.parameters.enumerate_all_solutions = True

    # Search and print out all solutions.
    solution_printer = VarArraySolutionPrinter([x, y, b])
    solver.solve(model, solution_printer)


channeling_sample_sat()

C++

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#include <stdlib.h>

#include "absl/types/span.h"
#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"
#include "ortools/sat/model.h"
#include "ortools/sat/sat_parameters.pb.h"

namespace operations_research {
namespace sat {

void ChannelingSampleSat() {
  // Create the CP-SAT model.
  CpModelBuilder cp_model;

  // Declare our two primary variables.
  const IntVar x = cp_model.NewIntVar({0, 10});
  const IntVar y = cp_model.NewIntVar({0, 10});

  // Declare our intermediate boolean variable.
  const BoolVar b = cp_model.NewBoolVar();

  // Implement b == (x >= 5).
  cp_model.AddGreaterOrEqual(x, 5).OnlyEnforceIf(b);
  cp_model.AddLessThan(x, 5).OnlyEnforceIf(~b);

  // Create our two half-reified constraints.
  // First, b implies (y == 10 - x).
  cp_model.AddEquality(x + y, 10).OnlyEnforceIf(b);
  // Second, not(b) implies y == 0.
  cp_model.AddEquality(y, 0).OnlyEnforceIf(~b);

  // Search for x values in increasing order.
  cp_model.AddDecisionStrategy({x}, DecisionStrategyProto::CHOOSE_FIRST,
                               DecisionStrategyProto::SELECT_MIN_VALUE);

  // Create a solver and solve with a fixed search.
  Model model;
  SatParameters parameters;
  parameters.set_search_branching(SatParameters::FIXED_SEARCH);
  parameters.set_enumerate_all_solutions(true);
  model.Add(NewSatParameters(parameters));
  model.Add(NewFeasibleSolutionObserver([&](const CpSolverResponse& r) {
    LOG(INFO) << "x=" << SolutionIntegerValue(r, x)
              << " y=" << SolutionIntegerValue(r, y)
              << " b=" << SolutionBooleanValue(r, b);
  }));
  SolveCpModel(cp_model.Build(), &model);
}

}  // namespace sat
}  // namespace operations_research

int main() {
  operations_research::sat::ChannelingSampleSat();

  return EXIT_SUCCESS;
}

Java

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

package com.google.ortools.sat.samples;

import com.google.ortools.Loader;
import com.google.ortools.sat.BoolVar;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverSolutionCallback;
import com.google.ortools.sat.DecisionStrategyProto;
import com.google.ortools.sat.IntVar;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.SatParameters;

/** Link integer constraints together. */
public class ChannelingSampleSat {
  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    // Create the CP-SAT model.
    CpModel model = new CpModel();

    // Declare our two primary variables.
    IntVar[] vars = new IntVar[] {model.newIntVar(0, 10, "x"), model.newIntVar(0, 10, "y")};

    // Declare our intermediate boolean variable.
    BoolVar b = model.newBoolVar("b");

    // Implement b == (x >= 5).
    model.addGreaterOrEqual(vars[0], 5).onlyEnforceIf(b);
    model.addLessOrEqual(vars[0], 4).onlyEnforceIf(b.not());

    // Create our two half-reified constraints.
    // First, b implies (y == 10 - x).
    model.addEquality(LinearExpr.sum(vars), 10).onlyEnforceIf(b);
    // Second, not(b) implies y == 0.
    model.addEquality(vars[1], 0).onlyEnforceIf(b.not());

    // Search for x values in increasing order.
    model.addDecisionStrategy(new IntVar[] {vars[0]},
        DecisionStrategyProto.VariableSelectionStrategy.CHOOSE_FIRST,
        DecisionStrategyProto.DomainReductionStrategy.SELECT_MIN_VALUE);

    // Create the solver.
    CpSolver solver = new CpSolver();

    // Force the solver to follow the decision strategy exactly.
    solver.getParameters().setSearchBranching(SatParameters.SearchBranching.FIXED_SEARCH);
    // Tell the solver to enumerate all solutions.
    solver.getParameters().setEnumerateAllSolutions(true);

    // Solve the problem with the printer callback.
    solver.solve(model, new CpSolverSolutionCallback() {
      public CpSolverSolutionCallback init(IntVar[] variables) {
        variableArray = variables;
        return this;
      }

      @Override
      public void onSolutionCallback() {
        for (IntVar v : variableArray) {
          System.out.printf("%s=%d ", v.getName(), value(v));
        }
        System.out.println();
      }

      private IntVar[] variableArray;
    }.init(new IntVar[] {vars[0], vars[1], b}));
  }
}

C#

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

using System;
using Google.OrTools.Sat;
using Google.OrTools.Util;

public class VarArraySolutionPrinter : CpSolverSolutionCallback
{
    public VarArraySolutionPrinter(IntVar[] variables)
    {
        variables_ = variables;
    }

    public override void OnSolutionCallback()
    {
        {
            foreach (IntVar v in variables_)
            {
                Console.Write(String.Format("{0}={1} ", v.ToString(), Value(v)));
            }
            Console.WriteLine();
        }
    }

    private IntVar[] variables_;
}

public class ChannelingSampleSat
{
    static void Main()
    {
        // Create the CP-SAT model.
        CpModel model = new CpModel();

        // Declare our two primary variables.
        IntVar x = model.NewIntVar(0, 10, "x");
        IntVar y = model.NewIntVar(0, 10, "y");

        // Declare our intermediate boolean variable.
        BoolVar b = model.NewBoolVar("b");

        // Implement b == (x >= 5).
        model.Add(x >= 5).OnlyEnforceIf(b);
        model.Add(x < 5).OnlyEnforceIf(b.Not());

        // Create our two half-reified constraints.
        // First, b implies (y == 10 - x).
        model.Add(y == 10 - x).OnlyEnforceIf(b);
        // Second, not(b) implies y == 0.
        model.Add(y == 0).OnlyEnforceIf(b.Not());

        // Search for x values in increasing order.
        model.AddDecisionStrategy(new IntVar[] { x }, DecisionStrategyProto.Types.VariableSelectionStrategy.ChooseFirst,
                                  DecisionStrategyProto.Types.DomainReductionStrategy.SelectMinValue);

        // Create the solver.
        CpSolver solver = new CpSolver();

        // Force solver to follow the decision strategy exactly.
        // Tell the solver to search for all solutions.
        solver.StringParameters = "search_branching:FIXED_SEARCH, enumerate_all_solutions:true";

        VarArraySolutionPrinter cb = new VarArraySolutionPrinter(new IntVar[] { x, y, b });
        solver.Solve(model, cb);
    }
}

Esto muestra lo siguiente:

x=0 y=0 b=0
x=1 y=0 b=0
x=2 y=0 b=0
x=3 y=0 b=0
x=4 y=0 b=0
x=5 y=5 b=1
x=6 y=4 b=1
x=7 y=3 b=1
x=8 y=2 b=1
x=9 y=1 b=1
x=10 y=0 b=1

Un problema de empaquetado en contenedores

Como otro ejemplo de restricción de canal, considera un problema de empaquetado en contenedores en la que una parte del modelo calcula la carga de cada discretización, mientras que otra maximiza la cantidad de discretizaciones bajo un umbral determinado. Para implementar esto, puedes channel en un conjunto de variables booleanas, en las que cada una indica si están por debajo del umbral.

Para que sea más concreto, digamos que tienes 10 contenedores de capacidad de 100, y en los contenedores. Quieres maximizar la cantidad de discretizaciones que puede aceptar una carga de emergencia de tamaño 20.

Para ello, debes maximizar la cantidad de discretizaciones que tienen una carga menor por encima de 80. En el siguiente código, la canalización se usa para vincular load y slack. variables juntas:

Python

#!/usr/bin/env python3
# Copyright 2010-2024 Google LLC
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

"""Solves a binpacking problem using the CP-SAT solver."""


from ortools.sat.python import cp_model


def binpacking_problem_sat():
    """Solves a bin-packing problem using the CP-SAT solver."""
    # Data.
    bin_capacity = 100
    slack_capacity = 20
    num_bins = 5
    all_bins = range(num_bins)

    items = [(20, 6), (15, 6), (30, 4), (45, 3)]
    num_items = len(items)
    all_items = range(num_items)

    # Model.
    model = cp_model.CpModel()

    # Main variables.
    x = {}
    for i in all_items:
        num_copies = items[i][1]
        for b in all_bins:
            x[(i, b)] = model.new_int_var(0, num_copies, f"x[{i},{b}]")

    # Load variables.
    load = [model.new_int_var(0, bin_capacity, f"load[{b}]") for b in all_bins]

    # Slack variables.
    slacks = [model.new_bool_var(f"slack[{b}]") for b in all_bins]

    # Links load and x.
    for b in all_bins:
        model.add(load[b] == sum(x[(i, b)] * items[i][0] for i in all_items))

    # Place all items.
    for i in all_items:
        model.add(sum(x[(i, b)] for b in all_bins) == items[i][1])

    # Links load and slack through an equivalence relation.
    safe_capacity = bin_capacity - slack_capacity
    for b in all_bins:
        # slack[b] => load[b] <= safe_capacity.
        model.add(load[b] <= safe_capacity).only_enforce_if(slacks[b])
        # not(slack[b]) => load[b] > safe_capacity.
        model.add(load[b] > safe_capacity).only_enforce_if(~slacks[b])

    # Maximize sum of slacks.
    model.maximize(sum(slacks))

    # Solves and prints out the solution.
    solver = cp_model.CpSolver()
    status = solver.solve(model)
    print(f"solve status: {solver.status_name(status)}")
    if status == cp_model.OPTIMAL:
        print(f"Optimal objective value: {solver.objective_value}")
    print("Statistics")
    print(f"  - conflicts : {solver.num_conflicts}")
    print(f"  - branches  : {solver.num_branches}")
    print(f"  - wall time : {solver.wall_time}s")


binpacking_problem_sat()

C++

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#include <stdlib.h>

#include <vector>

#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"

namespace operations_research {
namespace sat {

void BinpackingProblemSat() {
  // Data.
  const int kBinCapacity = 100;
  const int kSlackCapacity = 20;
  const int kNumBins = 5;

  const std::vector<std::vector<int>> items = {
      {20, 6}, {15, 6}, {30, 4}, {45, 3}};
  const int num_items = items.size();

  // Model.
  CpModelBuilder cp_model;

  // Main variables.
  std::vector<std::vector<IntVar>> x(num_items);
  for (int i = 0; i < num_items; ++i) {
    const int num_copies = items[i][1];
    for (int b = 0; b < kNumBins; ++b) {
      x[i].push_back(cp_model.NewIntVar({0, num_copies}));
    }
  }

  // Load variables.
  std::vector<IntVar> load(kNumBins);
  for (int b = 0; b < kNumBins; ++b) {
    load[b] = cp_model.NewIntVar({0, kBinCapacity});
  }

  // Slack variables.
  std::vector<BoolVar> slacks(kNumBins);
  for (int b = 0; b < kNumBins; ++b) {
    slacks[b] = cp_model.NewBoolVar();
  }

  // Links load and x.
  for (int b = 0; b < kNumBins; ++b) {
    LinearExpr expr;
    for (int i = 0; i < num_items; ++i) {
      expr += x[i][b] * items[i][0];
    }
    cp_model.AddEquality(expr, load[b]);
  }

  // Place all items.
  for (int i = 0; i < num_items; ++i) {
    cp_model.AddEquality(LinearExpr::Sum(x[i]), items[i][1]);
  }

  // Links load and slack through an equivalence relation.
  const int safe_capacity = kBinCapacity - kSlackCapacity;
  for (int b = 0; b < kNumBins; ++b) {
    // slack[b] => load[b] <= safe_capacity.
    cp_model.AddLessOrEqual(load[b], safe_capacity).OnlyEnforceIf(slacks[b]);
    // not(slack[b]) => load[b] > safe_capacity.
    cp_model.AddGreaterThan(load[b], safe_capacity).OnlyEnforceIf(~slacks[b]);
  }

  // Maximize sum of slacks.
  cp_model.Maximize(LinearExpr::Sum(slacks));

  // Solving part.
  const CpSolverResponse response = Solve(cp_model.Build());
  LOG(INFO) << CpSolverResponseStats(response);
}

}  // namespace sat
}  // namespace operations_research

int main() {
  operations_research::sat::BinpackingProblemSat();

  return EXIT_SUCCESS;
}

Java

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

package com.google.ortools.sat.samples;

import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.IntVar;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;

/** Solves a bin packing problem with the CP-SAT solver. */
public class BinPackingProblemSat {
  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    // Data.
    int binCapacity = 100;
    int slackCapacity = 20;
    int numBins = 5;

    int[][] items = new int[][] {{20, 6}, {15, 6}, {30, 4}, {45, 3}};
    int numItems = items.length;

    // Model.
    CpModel model = new CpModel();

    // Main variables.
    IntVar[][] x = new IntVar[numItems][numBins];
    for (int i = 0; i < numItems; ++i) {
      int numCopies = items[i][1];
      for (int b = 0; b < numBins; ++b) {
        x[i][b] = model.newIntVar(0, numCopies, "x_" + i + "_" + b);
      }
    }

    // Load variables.
    IntVar[] load = new IntVar[numBins];
    for (int b = 0; b < numBins; ++b) {
      load[b] = model.newIntVar(0, binCapacity, "load_" + b);
    }

    // Slack variables.
    Literal[] slacks = new Literal[numBins];
    for (int b = 0; b < numBins; ++b) {
      slacks[b] = model.newBoolVar("slack_" + b);
    }

    // Links load and x.
    for (int b = 0; b < numBins; ++b) {
      LinearExprBuilder expr = LinearExpr.newBuilder();
      for (int i = 0; i < numItems; ++i) {
        expr.addTerm(x[i][b], items[i][0]);
      }
      model.addEquality(expr, load[b]);
    }

    // Place all items.
    for (int i = 0; i < numItems; ++i) {
      LinearExprBuilder expr = LinearExpr.newBuilder();
      for (int b = 0; b < numBins; ++b) {
        expr.add(x[i][b]);
      }
      model.addEquality(expr, items[i][1]);
    }

    // Links load and slack.
    int safeCapacity = binCapacity - slackCapacity;
    for (int b = 0; b < numBins; ++b) {
      //  slack[b] => load[b] <= safeCapacity.
      model.addLessOrEqual(load[b], safeCapacity).onlyEnforceIf(slacks[b]);
      // not(slack[b]) => load[b] > safeCapacity.
      model.addGreaterOrEqual(load[b], safeCapacity + 1).onlyEnforceIf(slacks[b].not());
    }

    // Maximize sum of slacks.
    model.maximize(LinearExpr.sum(slacks));

    // Solves and prints out the solution.
    CpSolver solver = new CpSolver();
    CpSolverStatus status = solver.solve(model);
    System.out.println("Solve status: " + status);
    if (status == CpSolverStatus.OPTIMAL) {
      System.out.printf("Optimal objective value: %f%n", solver.objectiveValue());
      for (int b = 0; b < numBins; ++b) {
        System.out.printf("load_%d = %d%n", b, solver.value(load[b]));
        for (int i = 0; i < numItems; ++i) {
          System.out.printf("  item_%d_%d = %d%n", i, b, solver.value(x[i][b]));
        }
      }
    }
    System.out.println("Statistics");
    System.out.println("  - conflicts : " + solver.numConflicts());
    System.out.println("  - branches  : " + solver.numBranches());
    System.out.println("  - wall time : " + solver.wallTime() + " s");
  }
}

C#

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

using System;
using Google.OrTools.Sat;

public class BinPackingProblemSat
{
    static void Main()
    {
        // Data.
        int bin_capacity = 100;
        int slack_capacity = 20;
        int num_bins = 5;

        int[,] items = new int[,] { { 20, 6 }, { 15, 6 }, { 30, 4 }, { 45, 3 } };
        int num_items = items.GetLength(0);

        // Model.
        CpModel model = new CpModel();

        // Main variables.
        IntVar[,] x = new IntVar[num_items, num_bins];
        for (int i = 0; i < num_items; ++i)
        {
            int num_copies = items[i, 1];
            for (int b = 0; b < num_bins; ++b)
            {
                x[i, b] = model.NewIntVar(0, num_copies, String.Format("x_{0}_{1}", i, b));
            }
        }

        // Load variables.
        IntVar[] load = new IntVar[num_bins];
        for (int b = 0; b < num_bins; ++b)
        {
            load[b] = model.NewIntVar(0, bin_capacity, String.Format("load_{0}", b));
        }

        // Slack variables.
        BoolVar[] slacks = new BoolVar[num_bins];
        for (int b = 0; b < num_bins; ++b)
        {
            slacks[b] = model.NewBoolVar(String.Format("slack_{0}", b));
        }

        // Links load and x.
        int[] sizes = new int[num_items];
        for (int i = 0; i < num_items; ++i)
        {
            sizes[i] = items[i, 0];
        }
        for (int b = 0; b < num_bins; ++b)
        {
            IntVar[] tmp = new IntVar[num_items];
            for (int i = 0; i < num_items; ++i)
            {
                tmp[i] = x[i, b];
            }
            model.Add(load[b] == LinearExpr.WeightedSum(tmp, sizes));
        }

        // Place all items.
        for (int i = 0; i < num_items; ++i)
        {
            IntVar[] tmp = new IntVar[num_bins];
            for (int b = 0; b < num_bins; ++b)
            {
                tmp[b] = x[i, b];
            }
            model.Add(LinearExpr.Sum(tmp) == items[i, 1]);
        }

        // Links load and slack.
        int safe_capacity = bin_capacity - slack_capacity;
        for (int b = 0; b < num_bins; ++b)
        {
            //  slack[b] => load[b] <= safe_capacity.
            model.Add(load[b] <= safe_capacity).OnlyEnforceIf(slacks[b]);
            // not(slack[b]) => load[b] > safe_capacity.
            model.Add(load[b] > safe_capacity).OnlyEnforceIf(slacks[b].Not());
        }

        // Maximize sum of slacks.
        model.Maximize(LinearExpr.Sum(slacks));

        // Solves and prints out the solution.
        CpSolver solver = new CpSolver();
        CpSolverStatus status = solver.Solve(model);
        Console.WriteLine(String.Format("Solve status: {0}", status));
        if (status == CpSolverStatus.Optimal)
        {
            Console.WriteLine(String.Format("Optimal objective value: {0}", solver.ObjectiveValue));
            for (int b = 0; b < num_bins; ++b)
            {
                Console.WriteLine(String.Format("load_{0} = {1}", b, solver.Value(load[b])));
                for (int i = 0; i < num_items; ++i)
                {
                    Console.WriteLine(string.Format("  item_{0}_{1} = {2}", i, b, solver.Value(x[i, b])));
                }
            }
        }
        Console.WriteLine("Statistics");
        Console.WriteLine(String.Format("  - conflicts : {0}", solver.NumConflicts()));
        Console.WriteLine(String.Format("  - branches  : {0}", solver.NumBranches()));
        Console.WriteLine(String.Format("  - wall time : {0} s", solver.WallTime()));
    }
}