Linear Programming Overview

Linear Programming Overview

Linear programming is a mathematical optimization method used to maximize or minimize a linear objective function under given constraints. It's especially useful when dealing with problems involving multiple variables where each variable can take on only certain values. These problems often arise in economics, business management, engineering, and other fields of science and technology.

At its core, linear programming involves setting up a model which consists of variables representing decision quantities, equations defining relationships between these quantities and a goal defined by one or more functions whose value needs to be optimized. This set of equations can be solved using graphical methods, substitution methods, or mathematical techniques such as the simplex algorithm.

Some key components of linear programming include:

Objective Function

The first step towards solving any linear program is identifying what we want our solution to achieve—this is known as the objective function. For example, if you're trying to make sure your production line has enough materials while keeping costs low, this might mean minimizing expenses. In contrast, if you want to maximize profit from selling goods produced by this line, it would involve maximizing revenue generated through sales.

Constraints

Constraints define the boundaries within which decisions must occur. They represent restrictions based on factors like limited resources or physical laws governing systems. Each constraint represents an inequality relationship among the variables involved in the problem. For instance, if there's a limit imposed on how many employees can work simultaneously during peak hours due to space limitations, then worker capacity becomes a hard constraint.

Decision Variables

These are unknown parameters that require resolution before arriving at optimum solutions. Their number may vary depending upon specific scenarios being tackled - they could range anywhere between two and dozens. A common approach here is assigning integer values to ensure accuracy in calculations.

Understanding these concepts allows us to apply linear programming successfully across various domains including transportation logistics planning for airline scheduling, healthcare resource allocation, agricultural commodity pricing, and even sports team selection.