## Google OR-Tools

Here is a basic example of using $OR-Tools$ in $Python$ to solve a simple linear programming problem.

$OR-Tools$ is a powerful optimization library provided by Google, and it can be used to solve a wide range of problems, including linear programming, mixed-integer programming, constraint programming, and more.

## Example: Linear Programming with OR-Tools

This example demonstrates solving a linear programming problem where we want to maximize the objective function $3x + 4y$ subject to some constraints.

1 | from ortools.linear_solver import pywraplp |

## Explanation

**Solver:**We create a solver instance using`pywraplp.Solver.CreateSolver('SCIP')`

. $SCIP$ is a powerful mixed-integer programming solver, and $OR-Tools$ uses it as one of its backends.**Variables:**We define two variables,`x`

and`y`

, both with a lower bound of`0`

and an upper bound of infinity.**Constraints:**We add three constraints:- $(2x + 3y \leq 12)$
- $(4x + y \leq 14)$
- $(3x - y \geq 0)$

**Objective:**We want to maximize the function $3x + 4y$.**Solve:**The solver solves the problem, and we check if an optimal solution was found.**Result:**If a solution is found, it prints the optimal objective value and the values of`x`

and`y`

.

## Output

1 | Solution: |

Letâ€™s break down the result of the optimization problem using $OR-Tools$:

### Objective Value:

**Objective value = 17.0**: This is the maximum value of the objective function`3x + 4y`

given the constraints.

The solver found that this is the highest value that can be achieved without violating any of the constraints.

### Variable Values:

**x = 2.9999999999999996**: This is the optimal value of the variable`x`

that maximizes the objective function.

Due to floating-point precision in computational mathematics, this value is extremely close to 3 (but not exactly 3).**y = 2.0000000000000018**: Similarly, this is the optimal value of the variable`y`

. This value is extremely close to 2.

### Interpretation:

**Floating-Point Precision:**The values`2.9999999999999996`

for`x`

and`2.0000000000000018`

for`y`

are due to the way computers handle floating-point arithmetic. In practice, these values can be considered as`x = 3`

and`y = 2`

.**Objective Function Calculation:**Given the optimal values of`x`

and`y`

, we can calculate the objective function:

$$

3x + 4y = 3(3) + 4(2) = 9 + 8 = 17

$$

This confirms that the objective value of`17.0`

is indeed the maximum value that can be achieved under the given constraints.

### Summary:

The solver has determined that to achieve the maximum value of `17`

for the objective function `3x + 4y`

, the values of `x`

and `y`

should be approximately `3`

and `2`

, respectively.

The slight deviations from exact integers are due to the limitations of floating-point representation in computers.

## Running the Code

To run this code, ensure you have installed the $OR-Tools$ package.

You can install it using pip:

1 | pip install ortools |

This example should give you a good starting point for working with $OR-Tools$ in $Python$.