# Linear Programming: Definition,Formula and Examples

Linear programming is a mathematical method that allows you to make the best decisions when you have limited resources. Imagine you are trying to pack the most delicious items in your picnic basket with a handful of ingredients. In this case, Linear programming will help you to figure out the perfect combination. This concept is widely used in businesses to maximize profits, in factories to optimize production, and even in transportation to find the most effective routes. In short, it is a tool that assists you with the best possible outcome given your constraints. To understand this in more detail, you can read about it through the article below.

## Definition of Linear Programming

Linear programming is a mathematical tool used to figure out the best way to use limited resources to achieve the best possible outcome. Imagine you’re running a factory that makes two types of products, and you want to figure out how many of each to make to earn the most money without running out of materials or time. This is where linear programming comes in handy.

It helps you consider all the factors involved, like the amount of materials you have, the time it takes to make each product, and the profit you make from each one. Then, it finds the perfect combination of products to maximize your profit.

In the business world, linear programming is used for everything from scheduling production to optimizing transportation routes. It’s a powerful tool that helps businesses make informed decisions and improve efficiency.

Also Read: Basics of Prime Numbers

## What are the Components of Linear Programming?

Below are the four basic components of linear programming, you can take a look at the given data.

• Decision Variables
• Constraints
• Data
• Objective Functions

Also Read: Properties of Isosceles Triangle

## What are the Methods of Linear Programming?

Linear programming offers several techniques to solve problems involving optimization with constraints. Let’s explore two primary methods:

### Graphical Method

This method is perfect for problems with just two variables. It involves visualizing the constraints as lines on a graph. The area where these lines overlap is called the feasible region. The optimal solution (maximum or minimum value) lies at a corner point of this region.

Example: A factory produces two products, A and B. Each product requires different amounts of labor and material. Given the available resources and profit margins, the goal is to determine the production quantities of A and B to maximize profit.

1. Plot the constraints on a graph.
2. Identify the feasible region.
3. Calculate the objective function (profit) at each corner point.
4. The corner point with the highest (or lowest) value represents the optimal solution.

### Simplex Method

For problems with more than two variables, the graphical method becomes impractical. The simplex method is an algebraic approach that improves the solution until an optimal value is reached.

Steps:

1. Convert inequality constraints into equations by introducing slack variables.
2. Create an initial simplex table.
3. Identify the pivot column and pivot row.
4. Perform row operations to make the pivot element 1 and other elements in the pivot column zero.
5. Repeat steps 3 and 4 until all values in the bottom row (except the last one) are non-negative.
6. The final tableau provides the optimal solution.

Example: A company produces three products with different resource requirements and profit contributions. The goal is to determine the production quantities to maximize overall profit.

While the simplex method can be performed manually, it becomes complex for larger problems. Computer software tools are often used to automate the calculations.

Apart from these two methods, other techniques and software tools are available for solving linear programming problems, and handling different problems.

## Why is Linear Programming Important?

Linear programming is helpful in almost every aspect of the daily life. For example, businesses use linear programming to solve much bigger problems. Take the instance that a factory might use it to decide how many of each product to make to earn the most money without wasting materials or time. Or a delivery company could use it to find the fastest and most efficient routes for their trucks. In crux, linear programming helps people and businesses make better decisions by using their resources wisely.

## Practical Problems on Linear Programming

Linear programming is like solving a problem where you have limited resources and want to make the best possible outcome. To understand the concept of linear programming better, let’s see some examples:

You run a cookie factory. You make two kinds of cookies: chocolate chip and oatmeal raisin. Each cookie needs a certain amount of flour, sugar, and chocolate chips (for chocolate chip cookies). You have a limited supply of these ingredients. How many of each kind of cookie should you make to earn the most money without running out of ingredients?

### Example 2: The Delivery Driver

You’re a delivery driver with a truck that can only hold a certain weight. You have to deliver packages of different sizes and weights to different addresses. How should you plan your route to deliver all the packages while carrying the least amount of weight at any given time?To solve each and every practical problem like the above mentioned, linear programming comes to your rescue.

## FAQs

What is Linear Programming?

Linear programming is a mathematical method used to determine the best way to spot limited resources to achieve a maximum or minimum outcome.

When is Linear Programming Used?

Linear programming is used in various fields to optimize processes and make decisions.

How Does Linear Programming Work?

Linear programming involves defining an objective function and constraints.

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