Linear Programming with OR-Tools in Python

This guide will help you understand how to solve a simple linear programming problem using the OR-Tools library in Python. Linear programming helps us find the best outcome (such as maximum profit or lowest cost) under a given set of constraints.

Step-by-Step Explanation

Install OR-Tools Library First, you need to install the OR-Tools library. This library contains tools for optimization, including linear programming.
```
pip install ortools
```
Import the OR-Tools Linear Solver We import the necessary module from OR-Tools:
```
from ortools.linear_solver import pywraplp
```
Create a Solver We create a solver object using the pywraplp.Solver.CreateSolver method. Here, we use “GLOP”, which stands for Google’s Linear Optimization Package.
```
solver = pywraplp.Solver.CreateSolver("GLOP")
```
Define Variables We define the variables we want to solve for. In this case, we have two variables, x and y. We also set their lower and upper bounds (0 to 100).
```
x = solver.NumVar(0, 100, 'x')
y = solver.NumVar(0, 100, 'y')
```

Add Constraints We add the constraints to our problem. Constraints are conditions that our solution must satisfy.

solver.Add(2*x + 3*y >= 15)  # First constraint
solver.Add(3*x + 5*y <= 60)  # Second constraint
solver.Add(x + y == 18)      # Third constraint

Define the Objective Function We define our objective function, which is what we want to maximize (or minimize). Here, we aim to maximize the expression 5*x + 8*y.
```
solver.Maximize(5*x + 8*y)
```
Solve the Problem We call the Solve method to find the optimal solution.
```
results = solver.Solve()
```
Print the Results We print the values of x and y that give the optimal solution.
```
print(f"x: {x.solution_value()}")
print(f"y: {y.solution_value()}")
```

9. Check for Optimal Solution We check if the solver found an optimal solution and print a confirmation message.

if results == pywraplp.Solver.OPTIMAL:
    print("Optimal Found")

Complete Code

Here is the complete code with all the steps explained:

# First: pip install ortools
from ortools.linear_solver import pywraplp

# Create the solver
solver = pywraplp.Solver.CreateSolver("GLOP")

# Define the variables with their bounds
x = solver.NumVar(0, 100, 'x')
y = solver.NumVar(0, 100, 'y')

# Add the constraints
solver.Add(2 * x + 3 * y >= 15)  # First constraint
solver.Add(3 * x + 5 * y <= 60)  # Second constraint
solver.Add(x + y == 18)          # Third constraint

# Define the objective function to maximize
solver.Maximize(5 * x + 8 * y)

# Solve the problem
results = solver.Solve()

# Print the results
print(f"x: {x.solution_value()}")
print(f"y: {y.solution_value()}")

# Check if an optimal solution was found
if results == pywraplp.Solver.OPTIMAL:
    print("Optimal Found")