# An Introduction to Linear Programming

## What is Linear Programming?

Linear programming is a mathematical technique used to optimize a certain objective while abiding by a set of constraints. The objective and constraints are formulated as linear equations or inequalities, hence the name linear programming. The goal is to find the optimal values for a set of decision variables that satisfies the constraints and maximizes (or minimizes) the objective function. Uncover more information about the subject by checking out this recommended external website. www.Analyticsvidhya.com.

## The History of Linear Programming

The origins of linear programming can be traced back to the late 19th century when the concept of optimization was first introduced in mathematics. However, the formal development of linear programming as a mathematical technique began in the 1940s with the works of George Dantzig, Leonid Kantorovich, and Tjalling Koopmans. Linear programming gained popularity during World War II due to its application in solving military logistics problems.

## Applications of Linear Programming

Linear programming has a wide range of applications in various fields such as economics, finance, engineering, transportation, and resource allocation. Some common examples of linear programming applications are:

• Optimizing production and inventory management in manufacturing
• Optimizing investment portfolios in finance
• Optimizing transportation routes in logistics
• Optimizing workforce scheduling in healthcare
• ## The Components of Linear Programming

A linear programming problem consists of the following components:

• Decision variables: These are the variables that represent the values that need to be optimized. For example, in a production optimization problem, decision variables may include the number of products to be produced.
• Objective function: This is the function that needs to be optimized. For example, in a profit maximization problem, the objective function may be the total revenue minus the total cost.
• Constraints: These are the limitations or restrictions on the decision variables that must be satisfied. Constraints may be expressed as linear equations or inequalities. For example, in a production optimization problem, constraints may include the availability of resources such as labor, raw materials, and machinery.
• ## The Simplex Method

The simplex method is an algorithm used to solve linear programming problems. The simplex method starts with an initial feasible solution and systematically searches for a better solution by moving along the edges of the feasible region. The process continues until an optimal solution is found or it is determined that no further improvement can be made.

## Conclusion

Linear programming is a powerful mathematical technique used to solve a wide range of optimization problems. It has numerous applications in various fields and has been instrumental in improving efficiency and reducing costs. The simplex method is the most commonly used algorithm for solving linear programming problems, but other methods such as the interior point method and the branch-and-bound method are also widely used. If you want to know more about the subject covered, linear programming examples, check out the carefully selected external content to complement your reading and enrich your knowledge of the topic.

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