Linear programming is a branch of mathematics and statistics that allows researchers to determine solutions to problems of optimization. Linear programming problems are distinctive in that they are clearly defined in terms of an objective function, constraints and linearity. The characteristics of linear programming make it an extremely useful field that has found use in applied fields ranging from logistics to industrial planning.

## Optimization

All linear programming problems are problems of optimization. This means that the true purpose behind solving a linear programming problem is to either maximize or minimize some value. Thus, linear programming problems are often found in economics, business, advertising and many other fields that value efficiency and resource conservation. Examples of items that can be optimized are profit, resource acquisition, free time and utility.

## Linearity

As the name hints, linear programming problems all have the trait of being linear. However, this trait of linearity can be misleading, as linearity only refers to variables being to the first power (and therefore excluding power functions, square roots and other non-linear functions). Linearity does not, however, mean that the functions of a linear programming problem are only of one variable. In short, linearity in linear programming problems allows the variables to relate to each other as coordinates on a line, excluding other shapes and curves.

## Objective Function

All linear programming problems have a function called the “objective function.” The objective function is written in terms of the variables that can be changed at will (e.g., time spent on a job, units produced and so on). The objective function is the one that the solver of a linear programming problem wishes to maximize or minimize. The result of a linear programming problem will be given in terms of the objective function. The objective function is written with the capital letter “Z” in most linear programming problems.

## Constraints

All linear programming problems have constraints on the variables inside the objective function. These constraints take the form of inequalities (e.g., “b < 3” where b may represent the units of books written by an author per month). These inequalities define how the objective function can be maximized or minimized, as together they determine the “domain” in which an organization can make decisions about resources.

#### References

- “An Introduction to Linear Programming and Game Theory”; Paul Thie and G. E. Keough; 2008