## PuLP in Python

*PuLP* is a Python library used for **linear programming (LP)** and **mixed-integer linear programming (MILP)**.

It allows you to define and solve optimization problems where you want to minimize or maximize a **linear objective** function subject to **linear constraints**.

## Installation

First, you need to install *PuLP*. You can do this using pip:

1 | pip install pulp |

## Basic Usage of PuLP

Here’s a step-by-step guide to solving a simple **linear programming problem** with *PuLP*.

### Example Problem:

Suppose you are a factory manager and you want to determine the optimal number of products $A$ and $B$ to produce to maximize your profit.

Each product requires a certain amount of resources (time, materials) and yields a certain profit.

**Objective**: Maximize profit.**Constraints**:- You have $100$ units of material.
- You have $80$ hours of labor.
- Product $A$ requires $4$ units of material and $2$ hours of labor.
- Product $B$ requires $3$ units of material and $5$ hours of labor.

**Profit**:- Product $A$ gives $20 profit per unit.
- Product $B$ gives $25 profit per unit.

### Step-by-Step Implementation

1 | import pulp |

## Explanation of the Code:

**Define the Problem**: We create a linear programming problem with the objective to maximize profit.**Decision Variables**: We define the decision variables`x`

and`y`

for the number of products A and B to produce.

These variables are continuous and non-negative.**Objective Function**: We define the objective function to maximize, which is the total profit:`20*x + 25*y`

.**Constraints**: We add constraints based on the available resources:- Material constraint:
`4*x + 3*y <= 100`

- Labor constraint:
`2*x + 5*y <= 80`

- Material constraint:
**Solve the Problem**: We solve the linear programming problem using PuLP’s`solve`

method.**Display the Results**: We print the optimal values of`x`

and`y`

, and the maximum profit.

## Interpretation of Output:

### Output

1 | Status: Optimal |

### Interpretation

- The optimal solution is to produce $18.5$ units of product A and $8.57$ units of product B, which yields a maximum profit of $585.71.

This is a basic example of using *PuLP* in Python.

The library is powerful and can handle more complex constraints, variables, and objectives, including mixed-integer programming.