Using Arrays to Define a Model

The previous section showed how to solve a MIP with just a few variables and constraints, which are defined individually. For larger problems, it's more convenient to define the variables and constraints by looping over arrays. The next example illustrates this.

Example

In this example we'll solve the following problem.

Maximize 7x1 + 8x2 + 2x3 + 9x4 + 6x5 subject to the following constraints:
5x1 + 7x2 + 9x3 + 2x4 + 1x5 250
18x1 + 4x2 - 9x3 + 10x4 + 12x5 285
4x1 + 7x2 + 3x3 + 8x4 + 5x5 211
5x1 + 13x2 + 16x3 + 3x4 - 7x5 315

where x1, x2, ..., x5 are non-negative integers.

The following sections present programs that solve this problem. The programs use the same methods as the previous MIP example, but in this case apply them to array values in a loop.

Declare the solver

In any MIP program, you start by importing the linear solver wrapper and declaring the MIP solver, as shown in the previous MIP example.

Create the data

The following code creates arrays containing the data for the example: the variable coefficients for the constraints and objective function, and bounds for the constraints.

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;
}

Define the variables

The following code defines the variables for the example in a loop. For large problems, this is easier than defining the variables individually, as in the previous example.

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());

Define the constraints

The following code creates the constraints for the example, using the method MakeRowConstraint (or some variant, depending on the coding language). The first two arguments to the method are the lower and upper bounds for the constraint. The third argument, a name for the constraint, is optional.

For each constraint, you define the coefficients of the variables using the method SetCoefficient. The method assigns the coefficient of the variable x[j] in constraint i to be the [i][j] entry of the array 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());

Define the objective

The following code defines the objective function for the example. The SetCoefficient method assigns the coefficients for the objective, while SetMaximization defines this as a maximization problem.

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();

Call the solver

The following code calls the solver.

Python

status = solver.Solve()

C++

const MPSolver::ResultStatus result_status = solver.Solve();

Java

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

C#

Solver.ResultStatus resultStatus = solver.Solve();

Display the solution

The following code displays the solution.

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('Problem solved in %f milliseconds' % solver.wall_time())
  print('Problem solved in %d iterations' % solver.iterations())
  print('Problem solved in %d branch-and-bound nodes' % solver.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());
}

Here is the solution to the problem.

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

Complete programs

Here are the complete programs.

Python

from __future__ import print_function
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')
  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))

  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('Problem solved in %f milliseconds' % solver.wall_time())
    print('Problem solved in %d iterations' % solver.iterations())
    print('Problem solved in %d branch-and-bound nodes' % solver.nodes())
  else:
    print('The problem does not have an optimal solution.')


if __name__ == '__main__':
  main()
 

C++

#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.
  MPSolver solver("simple_mip_program",
                  MPSolver::SCIP_MIXED_INTEGER_PROGRAMMING);
  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

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 {
    System.loadLibrary("jniortools");
  }
  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 {
    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");
    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");
  }
}