Solving Linear Problems Using Pulp

This guide explains how to use Pulp, a Python library for linear programming, to solve optimization problems. Pulp is particularly suitable for linear programming (LP) problems and provides an easy-to-use interface for defining variables, constraints, and objective functions.

Installation

Before using Pulp, make sure you have Cython installed, then install Pulp itself:

  1. Install Cython: If you haven’t installed Cython yet, run:

    pip install cython

  2. Install Pulp: Install Pulp using pip:

    pip install pulp

Example Problem

In this example, we’ll solve a linear programming problem from S. Vedula’s book (Problem 2.2.6) using Pulp.

Objective: Maximize 7x + 5y

Constraints:

  1. 2x + 3y <= 12
  2. 3x + y >= 6
  3. 5x + 3y <=> 15

Step-by-Step Explanation

Step 1: Import Pulp

Import the Pulp module using the alias pl:

import pulp as pl

Step 2: Create a Linear Programming Problem

Create an LP problem named ‘Example’ to maximize the objective function:

model = pl.LpProblem('Example', pl.LpMaximize)

Step 3: Define Decision Variables

Define the decision variables x and y with lower bounds of 0 and upper bounds of 100:

x = pl.LpVariable('x', 0, 100)
y = pl.LpVariable('y', 0, 100)

Step 4: Add Constraints

Add the constraints to the model using the += operator:

model += 2*x + 3*y <= 12
model += 3*x + y >= 6
model += 5*x + 3*y <= 15

Step 5: Set the Objective Function

Set the objective function to maximize (7x + 5y). Notice that you don’t need to explicitly specify model += for the objective function; Pulp recognizes it based on the lack of bounds:

model += 7*x + 5*y  # Pulp recognizes this as the objective function to maximize

Step 6: Solve the Model

Solve the LP problem using the solve() method:

status = model.solve()

Step 7: Retrieve and Print the Solution

Retrieve the optimal values of x and y using pl.value() and print them:

x_value = pl.value(x)
y_value = pl.value(y)
print(f'Optimal values: x = {x_value}, y = {y_value}')

Complete Code

Here is the complete code with all the steps:

import pulp as pl

# Create an LP problem named 'Example' to maximize
model = pl.LpProblem('Example', pl.LpMaximize)

# Define decision variables with bounds
x = pl.LpVariable('x', 0, 100)
y = pl.LpVariable('y', 0, 100)

# Add constraints
model += 2*x + 3*y <= 12
model += 3*x + y >= 6
model += 5*x + 3*y <= 15

# Set the objective function to maximize (implicitly)
model += 7*x + 5*y  # Pulp recognizes this as the objective function to maximize

# Solve the model
status = model.solve()

# Retrieve and print the optimal values of x and y
x_value = pl.value(x)
y_value = pl.value(y)
print(f'Optimal values: x = {x_value}, y = {y_value}')

Summary

This script demonstrates how to use Pulp to define and solve a linear programming problem. Pulp simplifies the process of modeling LP problems by providing an intuitive syntax for defining variables, constraints, and objectives, making it a valuable tool for optimization tasks in various domains.