The following sections will get you started with ORTools with C++:
 What is an optimization problem?
 Running a C++ program
 Solving an optimization problem in C++
 More C++ examples
 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. (Sometimes you'll be satisfied with finding any feasible solution; ORTools can do that as well.)
Here's a typical optimization problem. 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 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" can be either a maximum or a minimum.)  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 weight to trucks, this would impose a constraint on the solutions. A feasible solution is one that satisfies all the given constraints for the problem, without necessarily being optimal.
The first step in solving an optimization problem is identifying the objective and constraints.
Running a C++ program
This section shows how to run a C++ program that solves a very simple optimization problem: find the maximum value of the objective function x + y, subject to the constraints 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2.
The program is shown below. At this point, you don't need to understand the program in detail—we're just seeing how to run the code.
#include "ortools/linear_solver/linear_solver.h" #include "ortools/linear_solver/linear_solver.pb.h" namespace operations_research { void RunTest( MPSolver::OptimizationProblemType optimization_problem_type) { MPSolver solver("Glop", optimization_problem_type); // Create the variables x and y. MPVariable* const x = solver.MakeNumVar(0.0, 1, "x"); MPVariable* const y = solver.MakeNumVar(0.0, 2, "y"); // Create the objective function, x + y. MPObjective* const objective = solver.MutableObjective(); objective>SetCoefficient(x, 1); objective>SetCoefficient(y, 1); objective>SetMaximization(); // Call the solver and display the results. solver.Solve(); printf("\nSolution:"); printf("\nx = %.1f", x>solution_value()); printf("\ny = %.1f", y>solution_value()); } void RunExample() { RunTest(MPSolver::GLOP_LINEAR_PROGRAMMING); } } int main(int argc, char** argv) { operations_research::RunExample(); return 0; }
You can run the C++ example as follows:
 Copy and paste the code above into new file, and save it as
program.cc
.  Open a command window at the
top level of the directory where you installed ORTools, and enter:
make run SOURCE=relative/path/to/program.cc
whererelative/path/to/
is the path to the directory where you saved the program.
The program returns the values of x and y that maximize the objective function:
Solution: x = 1.0 y = 2.0
To just compile the program without running it, enter:
make build SOURCE=relative/path/to/program.cc
Compiling in opt mode
To compile in O3 mode:
make DEBUG='O3' all
Running the C++ executable
When you compile a C++ program, the executable is created in the bin
directory. You can run the executable for the example program as follows:
cd bin
program
If you make changes to the program, you'll need to recompile it as shown above.
Solving an optimization problem in C++
Next, we give an example of an optimization problem, and show how to set up and solve it in C++.
A 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.
C++ program
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");
The complete program
The complete program is 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; }
You can run the program as shown in Running a C++ program above.
Optimal solution
The 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.
More C++ examples
ORTools includes a number of C++ example programs that illustrate how to solve various types
of optimization problems. The examples are in the examples/cpp
subdirectory of
the directory where you installed ORTools.
You can run any C++ example as follows:
make run SOURCE=examples/cpp/example.cc
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
 Constraint 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
Constraint optimization
Constraint optimization, or constraint programming (CP), identifies feasible solutions out of a very large set of candidates, where the problem can be modeled in terms of arbitrary constraints. CP is based on feasibility (finding a feasible solution) rather than optimization (finding an optimal solution) and focuses on the constraints and variables rather than the objective function. However, CP can be used to solve optimization problems, simply by comparing the values of the objective function for all feasible solutions.
Learn more about constraint 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.