The following sections introduce you to solving optimization problems using ORTools:
 What is an optimization problem?
 Setting up and solving an optimization problem
 Identifying the type of problem you wish to solve
What is an optimization problem?
The goal of optimization is to find the best solution to a problem out of a large set of possible solutions. Here's an example. Suppose that a shipping company delivers packages to its customers using a fleet of trucks. Every day, the company must assign packages to trucks, and then a choose a route for each truck to deliver its packages. Each possible assignment of packages and routes has a cost, based on the total travel distance for the trucks, and possibly other factors as well. The problem is to choose the assignments of packages and routes that has the least cost.
Like all optimization problems, this problem has the following elements:
 The objective — the quantity you want to optimize.
In the example above, the objective is to
minimize cost.
To set up an optimization problem, you need to define
a function that calculates the value of the objective for any
possible solution. This is called the objective function.
In the preceding example, the objective function would calculate the total cost
of any assignment of packages and routes.
An optimal solution is one for which the value of the objective function is the best. ("Best" will be either a maximum or a minimum, depending on the problem statement.)  The constraints — restrictions on the set of possible solutions, based on the specific requirements of the problem. For example, if the shipping company can't assign packages above a given size to certain trucks, due to space limitations, this would impose a constraint on the solutions. A feasible solution is one that satisfies all the given constraints for the problem.
Setting up and solving an optimization problem
Next, we give an example of an optimization problem, and show how to set up and solve it in each of the supported languages.
A simple linear optimization example
One of the oldest and most widelyused areas of optimization is linear optimization (or linear programming), in which the objective function and the constraints can be written as linear expressions. Here's a simple example of this type of problem.
 Maximize 3x + 4y subject to the following constraints:

x + 2y ≤ 14 3x – y ≥ 0 x – y ≤ 2
The objective function in this example is f(x, y) = 3x + 4y. Both the objective function and the constraints are given by linear expressions, which makes this a linear problem.
The constraints define the feasible region, which is the triangle shown below, including its interior.
Main steps in solving the problem
For each language, the basic steps for setting up and solving a problem are the same:
 Create the variables.
 Define the constraints.
 Define the objective function.
 Declare the solver — the method that implements an algorithm for finding the optimal solution.
 Invoke the solver and display the results.
The following sections show how to implement these steps in each of the supported languages to solve the linear optimization problem:
Finding information in the C++ Reference pages
You can find detailed information about ORTools solvers and methods in the C++ Reference pages. Because ORTools is written in C++, the reference pages describe these methods in terms of their C++ code. If you are writing in Python, Java, or C#, the names of the corresponding methods in those languages are, for the most part, the same as the C++ names (with a few minor differences). So you can find useful information in the Reference pages no matter which language you are working in.
C++ example
Let's start by showing how to set up the problem in C++. We've added links from the methods used in this example to their reference pages, in case you want to learn about them in more detail. Here are the steps for setting up and solving the problem in C++:
 Create the variables using the method
MakeNumVar.
MPVariable* const x = solver.MakeNumVar(0.0, infinity, "x"); MPVariable* const y = solver.MakeNumVar(0.0, infinity, "y");
 Define the constraints using the methods
MakeRowConstraint and
SetCoefficient.
// x + 2y <= 14. MPConstraint* const c0 = solver.MakeRowConstraint(infinity, 14.0); c0>SetCoefficient(x, 1); c0>SetCoefficient(y, 2); // 3x  y >= 0. MPConstraint* const c1 = solver.MakeRowConstraint(0.0, infinity); c1>SetCoefficient(x, 3); c1>SetCoefficient(y, 1); // x  y <= 2. MPConstraint* const c2 = solver.MakeRowConstraint(infinity, 2.0); c2>SetCoefficient(x, 1); c2>SetCoefficient(y, 1);
For example, for the first inequality, $$x + 2y \leq 14$$ the constraint is defined as follows:
MakeRowConstraint(infinity, 14)
creates an inequality constraint in which the left side is less than or equal to 14. 
c0>SetCoefficient(x, 1);
sets the coefficient of x to 1. 
c0>SetCoefficient(y, 2);
sets the coefficient of y to 2.

 Define the objective function. The method
SetCoefficient sets the coefficients of the function. The method
SetMaximization makes this a maximization problem.
// Objective function: 3x + 4y. MPObjective* const objective = solver.MutableObjective(); objective>SetCoefficient(x, 3); objective>SetCoefficient(y, 4); objective>SetMaximization();
 Declare the solver. In this example, we use the
ORTools linear solver wrapper to invoke Glop, Google's linear optimizer.
The following code declares the solver.
void RunLinearExample( MPSolver::OptimizationProblemType optimization_problem_type) { MPSolver solver("LinearExample", optimization_problem_type);
When the program calls the solver byRunLinearExample(MPSolver::GLOP_LINEAR_PROGRAMMING);
the argumentGLOP_LINEAR_PROGRAMMING
, which tells the solver to use Glop, is passed to the solver through theOptimizationProblemType
method.  Invoke the solver and display the results.
printf("\nNumber of variables = %d", solver.NumVariables()); printf("\nNumber of constraints = %d", solver.NumConstraints()); solver.Solve(); // The value of each variable in the solution. printf("\nSolution:"); printf("\nx = %.1f", x>solution_value()); printf("\ny = %.1f", y>solution_value()); // The objective value of the solution. printf("\nOptimal objective value = %.1f", objective>Value()); printf("\n");
Python example
Here are the steps for setting up and solving the problem in Python:
 Create the variables.
x = solver.NumVar(solver.infinity(), solver.infinity(), 'x') y = solver.NumVar(solver.infinity(), solver.infinity(), 'y')
Note:NumVar
is the Python name for the C++ method MakeNumVar.  Define the constraints.
# Constraint 1: x + 2y <= 14. constraint1 = solver.Constraint(solver.infinity(), 14) constraint1.SetCoefficient(x, 1) constraint1.SetCoefficient(y, 2) # Constraint 2: 3x  y >= 0. constraint2 = solver.Constraint(0, solver.infinity()) constraint2.SetCoefficient(x, 3) constraint2.SetCoefficient(y, 1) # Constraint 3: x  y <= 2. constraint3 = solver.Constraint(solver.infinity(), 2) constraint3.SetCoefficient(x, 1) constraint3.SetCoefficient(y, 1)
Note:Constraint
is the Python name for the C++ method MakeRowConstraint  Define the objective function.
# Objective function: 3x + 4y. objective = solver.Objective() objective.SetCoefficient(x, 3) objective.SetCoefficient(y, 4) objective.SetMaximization()
 Declare the solver.
pywraplp
is a Python wrapper for the underlying C++ solver. As in the C++ program, the argumentGLOP_LINEAR_PROGRAMMING
tells the solver to use Glop.from ortools.linear_solver import pywraplp def main(): # Instantiate a Glop solver, naming it LinearExample. solver = pywraplp.Solver('LinearExample', pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)
 Invoke the solver and display the results.
# Solve the system. solver.Solve() opt_solution = 3 * x.solution_value() + 4 * y.solution_value() print('Number of variables =', solver.NumVariables()) print('Number of constraints =', solver.NumConstraints()) # The value of each variable in the solution. print('Solution:') print('x = ', x.solution_value()) print('y = ', y.solution_value()) # The objective value of the solution. print('Optimal objective value =', opt_solution)
Java example
Here are the steps for setting up and solving the problem in Java:
 Create the variables.
double infinity = MPSolver.infinity(); // x and y are continuous nonnegative variables. MPVariable x = solver.makeNumVar(0.0, infinity, "x"); MPVariable y = solver.makeNumVar(0.0, infinity, "y");
 Define the constraints.
// x + 2y <= 14. MPConstraint c0 = solver.makeConstraint(infinity, 14.0); c0.setCoefficient(x, 1); c0.setCoefficient(y, 2); // 3x  y >= 0. MPConstraint c1 = solver.makeConstraint(0.0, infinity); c1.setCoefficient(x, 3); c1.setCoefficient(y, 1); // x  y <= 2. MPConstraint c2 = solver.makeConstraint(infinity, 2.0); c2.setCoefficient(x, 1); c2.setCoefficient(y, 1);
Note:MakeConstraint
is the Java name for the C++ method MakeRowConstraint.  Define the objective function.
// Maximize 3 * x + 4 * y. MPObjective objective = solver.objective(); objective.setCoefficient(x, 3); objective.setCoefficient(y, 4); objective.setMaximization();
 Declare the solver. In this example, we use the
ORTools linear solver wrapper to invoke Glop, Google's linear optimizer.
The following code creates the solver.
private static MPSolver createSolver (String solverType) { return new MPSolver("LinearExample", MPSolver.OptimizationProblemType.valueOf(solverType)); }
When the program calls the solver byrunLinearExample("GLOP_LINEAR_PROGRAMMING");
the argumentGLOP_LINEAR_PROGRAMMING
, which tells the solver to use Glop, is passed to the solver through theOptimizationProblemType
method.  Invoke the solver and display the results.
System.out.println("Number of variables = " + solver.numVariables()); System.out.println("Number of constraints = " + solver.numConstraints()); solver.solve(); // The value of each variable in the solution. System.out.println("Solution"); System.out.println("x = " + x.solutionValue()); System.out.println("y = " + y.solutionValue()); // The objective value of the solution. System.out.println("Optimal objective value = " + solver.objective().value());
C# example
Here are the steps for setting up and solving the problem in C#:
 Create the variables.
Variable x = solver.MakeNumVar(0.0, double.PositiveInfinity, "x"); Variable y = solver.MakeNumVar(0.0, double.PositiveInfinity, "y");
 Define the constraints.
// x + 2y <= 14. Constraint c0 = solver.MakeConstraint(double.NegativeInfinity, 14.0); c0.SetCoefficient(x, 1); c0.SetCoefficient(y, 2); // 3x  y >= 0. Constraint c1 = solver.MakeConstraint(0.0, double.PositiveInfinity); c1.SetCoefficient(x, 3); c1.SetCoefficient(y, 1); // x  y <= 2. Constraint c2 = solver.MakeConstraint(double.NegativeInfinity, 2.0); c2.SetCoefficient(x, 1); c2.SetCoefficient(y, 1);
Note:MakeConstraint
is the C# name for the C++ method MakeRowConstraint.  Define the objective function.
// Objective function: 3x + 4y. Objective objective = solver.Objective(); objective.SetCoefficient(x, 3); objective.SetCoefficient(y, 4); objective.SetMaximization();
 Declare the solver. In this example, we use the
ORTools linear solver wrapper to invoke Glop, Google's linear optimizer.
The following code creates the solver.
private static void RunLinearExample(String solverType) { Solver solver = Solver.CreateSolver("LinearExample", solverType);
When the program calls the solver byRunLinearExample("GLOP_LINEAR_PROGRAMMING");
the argumentGLOP_LINEAR_PROGRAMMING
, which tells the solver to use Glop, is passed to the solver through theOptimizationProblemType
method.  Invoke the solver and display the results.
Console.WriteLine("Number of variables = " + solver.NumVariables()); Console.WriteLine("Number of constraints = " + solver.NumConstraints()); solver.Solve(); // The value of each variable in the solution. Console.WriteLine("Solution:"); Console.WriteLine("x = " + x.SolutionValue()); Console.WriteLine("y = " + y.SolutionValue()); // The objective value of the solution. Console.WriteLine("Optimal objective value = " + solver.Objective().Value());
Optimal solution
Each program returns the optimal solution to the problem, as shown below.
Number of variables = 2 Number of constraints = 3 Solution: x = 6.0 y = 4.0 Optimal objective value = 34.0
Here is a graph showing the solution:
The dashed green line is defined by setting the objective function equal to its optimal value of 34. Any line whose equation has the form 3x + 4y = c is parallel to the dashed line, and 34 is the largest value of c for which the line intersects the feasible region.
If you think about the geometry in the above graph, in any linear optimization problem at least one vertex of the feasible region must be an optimal solution. As a result, you can find an optimal solution by traversing the vertices of the feasible region until there is no more improvement in the objective function. This is the idea behind simplex algorithm, the most widelyused method for solving linear optimization problems.
To learn more about solving linear optimization problems, see The Glop linear solver.
Complete programs in all the languages
The complete programs in all four languages are shown below.
#include "ortools/linear_solver/linear_solver.h" #include "ortools/linear_solver/linear_solver.pb.h" namespace operations_research { void RunLinearExample( MPSolver::OptimizationProblemType optimization_problem_type) { MPSolver solver("LinearExample", optimization_problem_type); const double infinity = solver.infinity(); // x and y are nonnegative variables. MPVariable* const x = solver.MakeNumVar(0.0, infinity, "x"); MPVariable* const y = solver.MakeNumVar(0.0, infinity, "y"); // Objective function: 3x + 4y. MPObjective* const objective = solver.MutableObjective(); objective>SetCoefficient(x, 3); objective>SetCoefficient(y, 4); objective>SetMaximization(); // x + 2y <= 14. MPConstraint* const c0 = solver.MakeRowConstraint(infinity, 14.0); c0>SetCoefficient(x, 1); c0>SetCoefficient(y, 2); // 3x  y >= 0. MPConstraint* const c1 = solver.MakeRowConstraint(0.0, infinity); c1>SetCoefficient(x, 3); c1>SetCoefficient(y, 1); // x  y <= 2. MPConstraint* const c2 = solver.MakeRowConstraint(infinity, 2.0); c2>SetCoefficient(x, 1); c2>SetCoefficient(y, 1); printf("\nNumber of variables = %d", solver.NumVariables()); printf("\nNumber of constraints = %d", solver.NumConstraints()); solver.Solve(); // The value of each variable in the solution. printf("\nSolution:"); printf("\nx = %.1f", x>solution_value()); printf("\ny = %.1f", y>solution_value()); // The objective value of the solution. printf("\nOptimal objective value = %.1f", objective>Value()); printf("\n"); } void RunExample() { RunLinearExample(MPSolver::GLOP_LINEAR_PROGRAMMING); } } // namespace operations_research int main(int argc, char** argv) { operations_research::RunExample(); return 0; }
from __future__ import print_function from ortools.linear_solver import pywraplp def main(): # Instantiate a Glop solver, naming it LinearExample. solver = pywraplp.Solver('LinearExample', pywraplp.Solver.GLOP_LINEAR_PROGRAMMING) # Create the two variables and let them take on any value. x = solver.NumVar(solver.infinity(), solver.infinity(), 'x') y = solver.NumVar(solver.infinity(), solver.infinity(), 'y') # Constraint 1: x + 2y <= 14. constraint1 = solver.Constraint(solver.infinity(), 14) constraint1.SetCoefficient(x, 1) constraint1.SetCoefficient(y, 2) # Constraint 2: 3x  y >= 0. constraint2 = solver.Constraint(0, solver.infinity()) constraint2.SetCoefficient(x, 3) constraint2.SetCoefficient(y, 1) # Constraint 3: x  y <= 2. constraint3 = solver.Constraint(solver.infinity(), 2) constraint3.SetCoefficient(x, 1) constraint3.SetCoefficient(y, 1) # Objective function: 3x + 4y. objective = solver.Objective() objective.SetCoefficient(x, 3) objective.SetCoefficient(y, 4) objective.SetMaximization() # Solve the system. solver.Solve() opt_solution = 3 * x.solution_value() + 4 * y.solution_value() print('Number of variables =', solver.NumVariables()) print('Number of constraints =', solver.NumConstraints()) # The value of each variable in the solution. print('Solution:') print('x = ', x.solution_value()) print('y = ', y.solution_value()) # The objective value of the solution. print('Optimal objective value =', opt_solution) if __name__ == '__main__': main()
import com.google.ortools.linearsolver.MPConstraint; import com.google.ortools.linearsolver.MPObjective; import com.google.ortools.linearsolver.MPSolver; import com.google.ortools.linearsolver.MPVariable; public class LinearExample { static { System.loadLibrary("jniortools"); } private static MPSolver createSolver (String solverType) { return new MPSolver("LinearExample", MPSolver.OptimizationProblemType.valueOf(solverType)); } private static void runLinearExample(String solverType) { MPSolver solver = createSolver(solverType); double infinity = MPSolver.infinity(); // x and y are continuous nonnegative variables. MPVariable x = solver.makeNumVar(0.0, infinity, "x"); MPVariable y = solver.makeNumVar(0.0, infinity, "y"); // Maximize 3 * x + 4 * y. MPObjective objective = solver.objective(); objective.setCoefficient(x, 3); objective.setCoefficient(y, 4); objective.setMaximization(); // x + 2y <= 14. MPConstraint c0 = solver.makeConstraint(infinity, 14.0); c0.setCoefficient(x, 1); c0.setCoefficient(y, 2); // 3x  y >= 0. MPConstraint c1 = solver.makeConstraint(0.0, infinity); c1.setCoefficient(x, 3); c1.setCoefficient(y, 1); // x  y <= 2. MPConstraint c2 = solver.makeConstraint(infinity, 2.0); c2.setCoefficient(x, 1); c2.setCoefficient(y, 1); System.out.println("Number of variables = " + solver.numVariables()); System.out.println("Number of constraints = " + solver.numConstraints()); solver.solve(); // The value of each variable in the solution. System.out.println("Solution"); System.out.println("x = " + x.solutionValue()); System.out.println("y = " + y.solutionValue()); // The objective value of the solution. System.out.println("Optimal objective value = " + solver.objective().value()); } public static void main(String[] args) throws Exception { runLinearExample("GLOP_LINEAR_PROGRAMMING"); } }
using System; using Google.OrTools.LinearSolver; public class LinearExample { private static void RunLinearExample(String solverType) { Solver solver = Solver.CreateSolver("LinearExample", solverType); // x and y are continuous nonnegative variables. Variable x = solver.MakeNumVar(0.0, double.PositiveInfinity, "x"); Variable y = solver.MakeNumVar(0.0, double.PositiveInfinity, "y"); // Objective function: 3x + 4y. Objective objective = solver.Objective(); objective.SetCoefficient(x, 3); objective.SetCoefficient(y, 4); objective.SetMaximization(); // x + 2y <= 14. Constraint c0 = solver.MakeConstraint(double.NegativeInfinity, 14.0); c0.SetCoefficient(x, 1); c0.SetCoefficient(y, 2); // 3x  y >= 0. Constraint c1 = solver.MakeConstraint(0.0, double.PositiveInfinity); c1.SetCoefficient(x, 3); c1.SetCoefficient(y, 1); // x  y <= 2. Constraint c2 = solver.MakeConstraint(double.NegativeInfinity, 2.0); c2.SetCoefficient(x, 1); c2.SetCoefficient(y, 1); Console.WriteLine("Number of variables = " + solver.NumVariables()); Console.WriteLine("Number of constraints = " + solver.NumConstraints()); solver.Solve(); // The value of each variable in the solution. Console.WriteLine("Solution:"); Console.WriteLine("x = " + x.SolutionValue()); Console.WriteLine("y = " + y.SolutionValue()); // The objective value of the solution. Console.WriteLine("Optimal objective value = " + solver.Objective().Value()); } static void Main() { RunLinearExample("GLOP_LINEAR_PROGRAMMING"); } }
Identifying the type of problem you wish to solve
There are many different types of optimization problems in the world. For each type of problem, there are different approaches and algorithms for finding an optimal solution. Before you can start writing a program to solve an optimization problem, you need to identify what type of problem you are dealing with, and then choose an appropriate solver — an algorithm for finding an optimal solution.
Below you will find a brief overview of the types of problems that ORTools solves, and links to the sections in this guide that explain how to solve each problem type.
 Linear optimization
 Mixedinteger optimization
 Bin packing
 Network flows
 Assignment
 Scheduling
 Routing
Linear optimization
As you learned in the previous section, a linear optimization problem is one in which the objective function and the constraints linear expressions in the variables. The primary solver in ORTools for this type of problem is the linear optimization solver, which is actually a wrapper for several different libraries for linear and mixedinteger optimization, including thirdparty libraries.
Learn more about linear optimization
Mixedinteger optimization
A mixed integer optimization problem is one in which some or all of the variables are required to be integers. An example is the assignment problem, in which a group of workers needs be assigned to a set of tasks. For each worker and task, you define a variable whose value is 1 if the given worker is assigned to the given task, and 0 otherwise. In this case, the variables can only take on the values 0 or 1.
Learn more about mixedinteger optimization
Bin packing
Bin packing is the problem of packing a set of objects of different sizes into containers with different capacities. The goal is to pack as many of the objects as possible, subject to the capacities of the containers. A special case of this is the knapsack problem, in which there is just one container.
Network flows
Many optimization problems can be represented by a directed graph consisting of nodes and directed arcs between them. For example, transportation problems, in which goods are shipped across a railway network, can be represented by a graph in which the arcs are rail lines and the nodes are distribution centers. In the maximum flow problem, each arc has a maximum capacity that can be transported across it. The problem is to assign the amount of goods to be shipped across each arc so that the total quantity being transported is as large as possible.
Learn more about network flows
Assignment
Assignment problems involve assigning a group of agents (say, workers or machines) to a set of tasks, where there is a fixed cost for assigning each agent to a specific task. The problem is to find the assignment with the least total cost. Assignment problems are actually a special case of network flow problems.
Scheduling
Scheduling problems involve assigning resources to perform a set of tasks at specific times. An important example is the job shop problem, in which multiple jobs are processed on several machines. Each job consists of a sequence of tasks, which must be performed in a given order, and each task must be processed on a specific machine. The problem is to assign a schedule so that all jobs are completed in as short an interval of time as possible.
Routing
Routing problems involve finding the optimal routes for a fleet of vehicles to traverse a network, defined by a directed graph. The problem of assigning packages to delivery trucks, described in What is an optimization problem?, is one example of a routing problem. Another is the traveling salesman problem.