# Get Started with OR-Tools for C++

The following sections will get you started with OR-Tools with C++:

## 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; OR-Tools 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:

1. Copy and paste the code above into new file, and save it as program.cc.
2. Open a command window at the top level of the directory where you installed OR-Tools, and enter:
make run SOURCE=relative/path/to/program.cc
where relative/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 re-compile 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 widely-used 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(xy) = 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 OR-Tools 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 by
RunLinearExample(MPSolver::GLOP_LINEAR_PROGRAMMING);
the argument GLOP_LINEAR_PROGRAMMING, which tells the solver to use Glop, is passed to the solver through the OptimizationProblemType 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 non-negative 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 widely-used method for solving linear optimization problems.

To learn more about solving linear optimization problems, see The Glop linear solver.

## More C++ examples

OR-Tools 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 OR-Tools.

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 OR-Tools solves, and links to the sections in this guide that explain how to solve each problem type.

### 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 OR-Tools for this type of problem is the linear optimization solver, which is actually a wrapper for several different libraries for linear and mixed-integer optimization, including third-party libraries.

### 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.

### Mixed-integer 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.

### 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.

### 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.