Halaman ini diterjemahkan oleh Cloud Translation API.

Batasan penyaluran

Batasan saluran menautkan variabel di dalam model. {i>Low-fidelity<i} digunakan saat Anda ingin menyatakan hubungan yang rumit antar variabel, seperti "jika variabel memenuhi kondisi, memaksa variabel lain ke nilai tertentu".

Penyaluran biasanya diimplementasikan menggunakan batasan linear half-reified: satu batasan menyiratkan hal lain (a → b), tetapi bukan berarti sebaliknya di sekitar (a ← b).

Ekspresi If-Kemudian-Else

Misalnya Anda ingin menerapkan hal berikut: "Jika x kurang dari 5, tetapkan y menjadi 0. Jika tidak, tetapkan y ke 10-x". Anda dapat melakukannya dengan membuat perantara variabel boolean b yang bernilai benar jika x lebih besar dari atau sama dengan 5, dan false jika tidak:

b menyiratkan y == 10 - x

not(b) menyiratkan y == 0

Hal ini diimplementasikan menggunakan metode OnlyEnforceIf seperti yang ditunjukkan di bawah ini.

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);
    }
}

Hal ini akan menampilkan hal berikut:

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

Masalah saat {i>bin-packing<i}

Sebagai contoh lain dari kendala saluran, pertimbangkan masalah pengemasan bin di di mana satu bagian model menghitung beban setiap bin, sementara bagian lainnya memaksimalkan jumlah bin dalam nilai minimum tertentu. Untuk menerapkannya, Anda dapat channel pemuatan setiap bin ke dalam kumpulan variabel boolean, yang masing-masing menunjukkan apakah berada di bawah ambang batas.

Untuk membuatnya lebih konkret, katakanlah Anda memiliki 10 {i>bin<i} dengan kapasitas 100, dan barang-barang untuk dimasukkan ke dalam tempat sampah. Anda ingin memaksimalkan jumlah {i>bin<i} yang dapat menerima satu beban darurat berukuran 20.

Untuk melakukannya, Anda perlu memaksimalkan jumlah {i>bin<i} yang memiliki beban lebih sedikit dari 80. Pada kode di bawah ini, channeling digunakan untuk menautkan load dan slack variabel bersama-sama:

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()));
    }
}

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