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Model Tanımlamak için Dizileri Kullanma

Önceki bölümde, yalnızca birkaç değişkenle MIP'nin nasıl çözülebileceği kısıtlanmasıdır. Daha büyük problemlerde döngüler üzerinden değişken ve kısıtları tanımlamak için oldukça kolaydır. İlgili içeriği oluşturmak için kullanılan bunu gösteriyor.

Örnek

Bu örnekte aşağıdaki sorunu çözeceğiz.

Aşağıdaki kısıtlamalara tabi olarak 7x₁ + 8x₂ + 2x₃ + 9x₄ + 6x₅ boyutlarını en üst düzeye çıkarın:

5 x₁ + 7 x₂ + 9 x₃ + 2 x₄ + 1 x₅ &leq; 250
18 x₁ + 4 x₂ - 9 x₃ + 10 x₄ + 12 x₅ &leq; 285
4 x₁ + 7 x₂ + 3 x₃ + 8 x₄ + 5 x₅ &leq; 211
5 x₁ + 13 x₂ + 16 x₃ + 3 x₄ - 7 x₅ &leq; 315

burada x₁, x₂, ..., x₅ negatif olmayan değerlerdir tam sayılar.

Aşağıdaki bölümlerde bu sorunu çözen programlar sunulmaktadır. Programlar önceki MIP örneğindeki ile aynı yöntemleri kullanın, ancak bu örnekte, bunları bir döngüdeki dizi değerlerine uygulayabilirsiniz.

Çözücüyü açıklama

Herhangi bir MIP programında, doğrusal çözücü sarmalayıcıyı içe aktararak ve aşağıdaki gibi MIP çözücüyü tanımlama önceki MIP örneği.

Verileri oluşturma

Aşağıdaki kod, örnekteki verileri içeren diziler oluşturur: kısıtlar ve hedef fonksiyon için değişken katsayılar ve ve kısıtlamalara dönüştürmenizi sağlar.

Python

def create_data_model():
    """Stores the data for the problem."""
    data = {}
    data["constraint_coeffs"] = [
        [5, 7, 9, 2, 1],
        [18, 4, -9, 10, 12],
        [4, 7, 3, 8, 5],
        [5, 13, 16, 3, -7],
    ]
    data["bounds"] = [250, 285, 211, 315]
    data["obj_coeffs"] = [7, 8, 2, 9, 6]
    data["num_vars"] = 5
    data["num_constraints"] = 4
    return data

C++

struct DataModel {
  const std::vector<std::vector<double>> constraint_coeffs{
      {5, 7, 9, 2, 1},
      {18, 4, -9, 10, 12},
      {4, 7, 3, 8, 5},
      {5, 13, 16, 3, -7},
  };
  const std::vector<double> bounds{250, 285, 211, 315};
  const std::vector<double> obj_coeffs{7, 8, 2, 9, 6};
  const int num_vars = 5;
  const int num_constraints = 4;
};

Java

static class DataModel {
  public final double[][] constraintCoeffs = {
      {5, 7, 9, 2, 1},
      {18, 4, -9, 10, 12},
      {4, 7, 3, 8, 5},
      {5, 13, 16, 3, -7},
  };
  public final double[] bounds = {250, 285, 211, 315};
  public final double[] objCoeffs = {7, 8, 2, 9, 6};
  public final int numVars = 5;
  public final int numConstraints = 4;
}

C#

class DataModel
{
    public double[,] ConstraintCoeffs = {
        { 5, 7, 9, 2, 1 },
        { 18, 4, -9, 10, 12 },
        { 4, 7, 3, 8, 5 },
        { 5, 13, 16, 3, -7 },
    };
    public double[] Bounds = { 250, 285, 211, 315 };
    public double[] ObjCoeffs = { 7, 8, 2, 9, 6 };
    public int NumVars = 5;
    public int NumConstraints = 4;
}

Verileri örneklendirme

Aşağıdaki kod, veri modelini örneklendirir.

Python

data = create_data_model()

C++

DataModel data;

Java

final DataModel data = new DataModel();

C#

DataModel data = new DataModel();

Çözücüyü örneklendirme

Aşağıdaki kod çözücüyü örneklendirir.

Python

# Create the mip solver with the SCIP backend.
solver = pywraplp.Solver.CreateSolver("SCIP")
if not solver:
    return

C++

// Create the mip solver with the SCIP backend.
std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP"));
if (!solver) {
  LOG(WARNING) << "SCIP solver unavailable.";
  return;
}

Java

// Create the linear solver with the SCIP backend.
MPSolver solver = MPSolver.createSolver("SCIP");
if (solver == null) {
  System.out.println("Could not create solver SCIP");
  return;
}

C#

// Create the linear solver with the SCIP backend.
Solver solver = Solver.CreateSolver("SCIP");
if (solver is null)
{
    return;
}

Değişkenleri tanımlayın

Aşağıdaki kod, döngüdeki örnek için değişkenleri tanımlar. Büyük için bu, değişkenleri tek tek tanımlamaktan daha kolaydır, örneğin önceki örneği inceleyin.

Python

infinity = solver.infinity()
x = {}
for j in range(data["num_vars"]):
    x[j] = solver.IntVar(0, infinity, "x[%i]" % j)
print("Number of variables =", solver.NumVariables())

C++

const double infinity = solver->infinity();
// x[j] is an array of non-negative, integer variables.
std::vector<const MPVariable*> x(data.num_vars);
for (int j = 0; j < data.num_vars; ++j) {
  x[j] = solver->MakeIntVar(0.0, infinity, "");
}
LOG(INFO) << "Number of variables = " << solver->NumVariables();

Java

double infinity = java.lang.Double.POSITIVE_INFINITY;
MPVariable[] x = new MPVariable[data.numVars];
for (int j = 0; j < data.numVars; ++j) {
  x[j] = solver.makeIntVar(0.0, infinity, "");
}
System.out.println("Number of variables = " + solver.numVariables());

C#

Variable[] x = new Variable[data.NumVars];
for (int j = 0; j < data.NumVars; j++)
{
    x[j] = solver.MakeIntVar(0.0, double.PositiveInfinity, $"x_{j}");
}
Console.WriteLine("Number of variables = " + solver.NumVariables());

Kısıtlamaları tanımlama

Aşağıdaki kod, MakeRowConstraint (veya kodlama diline bağlı olarak bazı varyantlar). İlgili içeriği oluşturmak için kullanılan yöntemin ilk iki bağımsız değişkeni, kısıtlayın. Kısıtlamanın adı olan üçüncü bağımsız değişken isteğe bağlıdır.

Her kısıt için değişkenlerin katsayılarını tanımlamak için yöntem SetCoefficient. Yöntem, değişkenin katsayısını i kısıtlamasında, dizinin [i][j] girişi olacak x[j] constraint_coeffs.

Python

for i in range(data["num_constraints"]):
    constraint = solver.RowConstraint(0, data["bounds"][i], "")
    for j in range(data["num_vars"]):
        constraint.SetCoefficient(x[j], data["constraint_coeffs"][i][j])
print("Number of constraints =", solver.NumConstraints())
# In Python, you can also set the constraints as follows.
# for i in range(data['num_constraints']):
#  constraint_expr = \
# [data['constraint_coeffs'][i][j] * x[j] for j in range(data['num_vars'])]
#  solver.Add(sum(constraint_expr) <= data['bounds'][i])

C++

// Create the constraints.
for (int i = 0; i < data.num_constraints; ++i) {
  MPConstraint* constraint = solver->MakeRowConstraint(0, data.bounds[i], "");
  for (int j = 0; j < data.num_vars; ++j) {
    constraint->SetCoefficient(x[j], data.constraint_coeffs[i][j]);
  }
}
LOG(INFO) << "Number of constraints = " << solver->NumConstraints();

Java

// Create the constraints.
for (int i = 0; i < data.numConstraints; ++i) {
  MPConstraint constraint = solver.makeConstraint(0, data.bounds[i], "");
  for (int j = 0; j < data.numVars; ++j) {
    constraint.setCoefficient(x[j], data.constraintCoeffs[i][j]);
  }
}
System.out.println("Number of constraints = " + solver.numConstraints());

C#

for (int i = 0; i < data.NumConstraints; ++i)
{
    Constraint constraint = solver.MakeConstraint(0, data.Bounds[i], "");
    for (int j = 0; j < data.NumVars; ++j)
    {
        constraint.SetCoefficient(x[j], data.ConstraintCoeffs[i, j]);
    }
}
Console.WriteLine("Number of constraints = " + solver.NumConstraints());

Hedefi tanımlama

Aşağıdaki kod, bu örnek için hedef işlevini tanımlar. İlgili içeriği oluşturmak için kullanılan SetCoefficient yöntemi hedefin katsayılarını atar SetMaximization bunu bir maksimizasyon problemi olarak tanımlar.

Python

objective = solver.Objective()
for j in range(data["num_vars"]):
    objective.SetCoefficient(x[j], data["obj_coeffs"][j])
objective.SetMaximization()
# In Python, you can also set the objective as follows.
# obj_expr = [data['obj_coeffs'][j] * x[j] for j in range(data['num_vars'])]
# solver.Maximize(solver.Sum(obj_expr))

C++

// Create the objective function.
MPObjective* const objective = solver->MutableObjective();
for (int j = 0; j < data.num_vars; ++j) {
  objective->SetCoefficient(x[j], data.obj_coeffs[j]);
}
objective->SetMaximization();

Java

MPObjective objective = solver.objective();
for (int j = 0; j < data.numVars; ++j) {
  objective.setCoefficient(x[j], data.objCoeffs[j]);
}
objective.setMaximization();

C#

Objective objective = solver.Objective();
for (int j = 0; j < data.NumVars; ++j)
{
    objective.SetCoefficient(x[j], data.ObjCoeffs[j]);
}
objective.SetMaximization();

Çözücüyü çağırın

Aşağıdaki kod çözücüyü çağırır.

Python

print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()

C++

const MPSolver::ResultStatus result_status = solver->Solve();

Java

final MPSolver.ResultStatus resultStatus = solver.solve();

C#

Solver.ResultStatus resultStatus = solver.Solve();

Çözümü gösterin

Çözüm aşağıdaki kodda gösterilir.

Python

if status == pywraplp.Solver.OPTIMAL:
    print("Objective value =", solver.Objective().Value())
    for j in range(data["num_vars"]):
        print(x[j].name(), " = ", x[j].solution_value())
    print()
    print(f"Problem solved in {solver.wall_time():d} milliseconds")
    print(f"Problem solved in {solver.iterations():d} iterations")
    print(f"Problem solved in {solver.nodes():d} branch-and-bound nodes")
else:
    print("The problem does not have an optimal solution.")

C++

// Check that the problem has an optimal solution.
if (result_status != MPSolver::OPTIMAL) {
  LOG(FATAL) << "The problem does not have an optimal solution.";
}
LOG(INFO) << "Solution:";
LOG(INFO) << "Optimal objective value = " << objective->Value();

for (int j = 0; j < data.num_vars; ++j) {
  LOG(INFO) << "x[" << j << "] = " << x[j]->solution_value();
}

Java

// Check that the problem has an optimal solution.
if (resultStatus == MPSolver.ResultStatus.OPTIMAL) {
  System.out.println("Objective value = " + objective.value());
  for (int j = 0; j < data.numVars; ++j) {
    System.out.println("x[" + j + "] = " + x[j].solutionValue());
  }
  System.out.println();
  System.out.println("Problem solved in " + solver.wallTime() + " milliseconds");
  System.out.println("Problem solved in " + solver.iterations() + " iterations");
  System.out.println("Problem solved in " + solver.nodes() + " branch-and-bound nodes");
} else {
  System.err.println("The problem does not have an optimal solution.");
}

C#

// Check that the problem has an optimal solution.
if (resultStatus != Solver.ResultStatus.OPTIMAL)
{
    Console.WriteLine("The problem does not have an optimal solution!");
    return;
}

Console.WriteLine("Solution:");
Console.WriteLine("Optimal objective value = " + solver.Objective().Value());

for (int j = 0; j < data.NumVars; ++j)
{
    Console.WriteLine("x[" + j + "] = " + x[j].SolutionValue());
}

Sorunun çözümü burada açıklanmıştır.

Number of variables = 5
Number of constraints = 4
Objective value = 260.0
x[0]  =  10.0
x[1]  =  16.0
x[2]  =  4.0
x[3]  =  4.0
x[4]  =  3.0

Problem solved in 29.000000 milliseconds
Problem solved in 315 iterations
Problem solved in 13 branch-and-bound nodes

Programları tamamlama

Programların tamamını burada bulabilirsiniz.

Python

from ortools.linear_solver import pywraplp


def create_data_model():
    """Stores the data for the problem."""
    data = {}
    data["constraint_coeffs"] = [
        [5, 7, 9, 2, 1],
        [18, 4, -9, 10, 12],
        [4, 7, 3, 8, 5],
        [5, 13, 16, 3, -7],
    ]
    data["bounds"] = [250, 285, 211, 315]
    data["obj_coeffs"] = [7, 8, 2, 9, 6]
    data["num_vars"] = 5
    data["num_constraints"] = 4
    return data



def main():
    data = create_data_model()
    # Create the mip solver with the SCIP backend.
    solver = pywraplp.Solver.CreateSolver("SCIP")
    if not solver:
        return

    infinity = solver.infinity()
    x = {}
    for j in range(data["num_vars"]):
        x[j] = solver.IntVar(0, infinity, "x[%i]" % j)
    print("Number of variables =", solver.NumVariables())

    for i in range(data["num_constraints"]):
        constraint = solver.RowConstraint(0, data["bounds"][i], "")
        for j in range(data["num_vars"]):
            constraint.SetCoefficient(x[j], data["constraint_coeffs"][i][j])
    print("Number of constraints =", solver.NumConstraints())
    # In Python, you can also set the constraints as follows.
    # for i in range(data['num_constraints']):
    #  constraint_expr = \
    # [data['constraint_coeffs'][i][j] * x[j] for j in range(data['num_vars'])]
    #  solver.Add(sum(constraint_expr) <= data['bounds'][i])

    objective = solver.Objective()
    for j in range(data["num_vars"]):
        objective.SetCoefficient(x[j], data["obj_coeffs"][j])
    objective.SetMaximization()
    # In Python, you can also set the objective as follows.
    # obj_expr = [data['obj_coeffs'][j] * x[j] for j in range(data['num_vars'])]
    # solver.Maximize(solver.Sum(obj_expr))

    print(f"Solving with {solver.SolverVersion()}")
    status = solver.Solve()

    if status == pywraplp.Solver.OPTIMAL:
        print("Objective value =", solver.Objective().Value())
        for j in range(data["num_vars"]):
            print(x[j].name(), " = ", x[j].solution_value())
        print()
        print(f"Problem solved in {solver.wall_time():d} milliseconds")
        print(f"Problem solved in {solver.iterations():d} iterations")
        print(f"Problem solved in {solver.nodes():d} branch-and-bound nodes")
    else:
        print("The problem does not have an optimal solution.")


if __name__ == "__main__":
    main()

C++

#include <memory>
#include <vector>

#include "ortools/linear_solver/linear_solver.h"

namespace operations_research {
struct DataModel {
  const std::vector<std::vector<double>> constraint_coeffs{
      {5, 7, 9, 2, 1},
      {18, 4, -9, 10, 12},
      {4, 7, 3, 8, 5},
      {5, 13, 16, 3, -7},
  };
  const std::vector<double> bounds{250, 285, 211, 315};
  const std::vector<double> obj_coeffs{7, 8, 2, 9, 6};
  const int num_vars = 5;
  const int num_constraints = 4;
};

void MipVarArray() {
  DataModel data;

  // Create the mip solver with the SCIP backend.
  std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP"));
  if (!solver) {
    LOG(WARNING) << "SCIP solver unavailable.";
    return;
  }

  const double infinity = solver->infinity();
  // x[j] is an array of non-negative, integer variables.
  std::vector<const MPVariable*> x(data.num_vars);
  for (int j = 0; j < data.num_vars; ++j) {
    x[j] = solver->MakeIntVar(0.0, infinity, "");
  }
  LOG(INFO) << "Number of variables = " << solver->NumVariables();

  // Create the constraints.
  for (int i = 0; i < data.num_constraints; ++i) {
    MPConstraint* constraint = solver->MakeRowConstraint(0, data.bounds[i], "");
    for (int j = 0; j < data.num_vars; ++j) {
      constraint->SetCoefficient(x[j], data.constraint_coeffs[i][j]);
    }
  }
  LOG(INFO) << "Number of constraints = " << solver->NumConstraints();

  // Create the objective function.
  MPObjective* const objective = solver->MutableObjective();
  for (int j = 0; j < data.num_vars; ++j) {
    objective->SetCoefficient(x[j], data.obj_coeffs[j]);
  }
  objective->SetMaximization();

  const MPSolver::ResultStatus result_status = solver->Solve();

  // Check that the problem has an optimal solution.
  if (result_status != MPSolver::OPTIMAL) {
    LOG(FATAL) << "The problem does not have an optimal solution.";
  }
  LOG(INFO) << "Solution:";
  LOG(INFO) << "Optimal objective value = " << objective->Value();

  for (int j = 0; j < data.num_vars; ++j) {
    LOG(INFO) << "x[" << j << "] = " << x[j]->solution_value();
  }
}
}  // namespace operations_research

int main(int argc, char** argv) {
  operations_research::MipVarArray();
  return EXIT_SUCCESS;
}

Java

package com.google.ortools.linearsolver.samples;
import com.google.ortools.Loader;
import com.google.ortools.linearsolver.MPConstraint;
import com.google.ortools.linearsolver.MPObjective;
import com.google.ortools.linearsolver.MPSolver;
import com.google.ortools.linearsolver.MPVariable;

/** MIP example with a variable array. */
public class MipVarArray {
  static class DataModel {
    public final double[][] constraintCoeffs = {
        {5, 7, 9, 2, 1},
        {18, 4, -9, 10, 12},
        {4, 7, 3, 8, 5},
        {5, 13, 16, 3, -7},
    };
    public final double[] bounds = {250, 285, 211, 315};
    public final double[] objCoeffs = {7, 8, 2, 9, 6};
    public final int numVars = 5;
    public final int numConstraints = 4;
  }

  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    final DataModel data = new DataModel();

    // Create the linear solver with the SCIP backend.
    MPSolver solver = MPSolver.createSolver("SCIP");
    if (solver == null) {
      System.out.println("Could not create solver SCIP");
      return;
    }

    double infinity = java.lang.Double.POSITIVE_INFINITY;
    MPVariable[] x = new MPVariable[data.numVars];
    for (int j = 0; j < data.numVars; ++j) {
      x[j] = solver.makeIntVar(0.0, infinity, "");
    }
    System.out.println("Number of variables = " + solver.numVariables());

    // Create the constraints.
    for (int i = 0; i < data.numConstraints; ++i) {
      MPConstraint constraint = solver.makeConstraint(0, data.bounds[i], "");
      for (int j = 0; j < data.numVars; ++j) {
        constraint.setCoefficient(x[j], data.constraintCoeffs[i][j]);
      }
    }
    System.out.println("Number of constraints = " + solver.numConstraints());

    MPObjective objective = solver.objective();
    for (int j = 0; j < data.numVars; ++j) {
      objective.setCoefficient(x[j], data.objCoeffs[j]);
    }
    objective.setMaximization();

    final MPSolver.ResultStatus resultStatus = solver.solve();

    // Check that the problem has an optimal solution.
    if (resultStatus == MPSolver.ResultStatus.OPTIMAL) {
      System.out.println("Objective value = " + objective.value());
      for (int j = 0; j < data.numVars; ++j) {
        System.out.println("x[" + j + "] = " + x[j].solutionValue());
      }
      System.out.println();
      System.out.println("Problem solved in " + solver.wallTime() + " milliseconds");
      System.out.println("Problem solved in " + solver.iterations() + " iterations");
      System.out.println("Problem solved in " + solver.nodes() + " branch-and-bound nodes");
    } else {
      System.err.println("The problem does not have an optimal solution.");
    }
  }

  private MipVarArray() {}
}

C#

using System;
using Google.OrTools.LinearSolver;

public class MipVarArray
{
    class DataModel
    {
        public double[,] ConstraintCoeffs = {
            { 5, 7, 9, 2, 1 },
            { 18, 4, -9, 10, 12 },
            { 4, 7, 3, 8, 5 },
            { 5, 13, 16, 3, -7 },
        };
        public double[] Bounds = { 250, 285, 211, 315 };
        public double[] ObjCoeffs = { 7, 8, 2, 9, 6 };
        public int NumVars = 5;
        public int NumConstraints = 4;
    }
    public static void Main()
    {
        DataModel data = new DataModel();

        // Create the linear solver with the SCIP backend.
        Solver solver = Solver.CreateSolver("SCIP");
        if (solver is null)
        {
            return;
        }

        Variable[] x = new Variable[data.NumVars];
        for (int j = 0; j < data.NumVars; j++)
        {
            x[j] = solver.MakeIntVar(0.0, double.PositiveInfinity, $"x_{j}");
        }
        Console.WriteLine("Number of variables = " + solver.NumVariables());

        for (int i = 0; i < data.NumConstraints; ++i)
        {
            Constraint constraint = solver.MakeConstraint(0, data.Bounds[i], "");
            for (int j = 0; j < data.NumVars; ++j)
            {
                constraint.SetCoefficient(x[j], data.ConstraintCoeffs[i, j]);
            }
        }
        Console.WriteLine("Number of constraints = " + solver.NumConstraints());

        Objective objective = solver.Objective();
        for (int j = 0; j < data.NumVars; ++j)
        {
            objective.SetCoefficient(x[j], data.ObjCoeffs[j]);
        }
        objective.SetMaximization();

        Solver.ResultStatus resultStatus = solver.Solve();

        // Check that the problem has an optimal solution.
        if (resultStatus != Solver.ResultStatus.OPTIMAL)
        {
            Console.WriteLine("The problem does not have an optimal solution!");
            return;
        }

        Console.WriteLine("Solution:");
        Console.WriteLine("Optimal objective value = " + solver.Objective().Value());

        for (int j = 0; j < data.NumVars; ++j)
        {
            Console.WriteLine("x[" + j + "] = " + x[j].SolutionValue());
        }

        Console.WriteLine("\nAdvanced usage:");
        Console.WriteLine("Problem solved in " + solver.WallTime() + " milliseconds");
        Console.WriteLine("Problem solved in " + solver.Iterations() + " iterations");
        Console.WriteLine("Problem solved in " + solver.Nodes() + " branch-and-bound nodes");
    }
}