Linear programming is a method of economic and business analysis that relies on matrix algebra and other mathematics techniques to achieve the highest level of satisfaction – maximum profits, for example – subject to a set of known constraints. The challenge of maximizing satisfaction within a set of limits makes linear programming an ideal tool of analysis for economics, which studies the ways in which households, businesses and societies allocate limited resources to achieve needs and wants.
Russian mathematician and economist Leonid Kantorovich first developed linear programming problems in the late 1930s. His early work represented a great advance in the understanding of allocating resources to achieve optimal outcomes. Kantorovich later won the Nobel Prize in economics. American mathematician George B. Dantzig applied techniques of linear programming to his work as a statistician for the Pentagon during World War II. He later developed an algorithm known as the simplex method for solving linear programming problems. Dantzig’s 1963 book, “Linear Programming and Extensions,” is considered a classic in the field.
Although computers provide tremendous help with the complex mathematics used in linear programming, the word “programming” does not refer to a set of instructions to a computer. In this context, the word refers to operations planning.
In economic analysis, linear programming has broad applications in industrial management and operations. Managers of industry want to maximize their companies’ profits or minimize production costs, but recognize the existence of constraints. For example, managers of an automobile production plant want to produce the number of vehicles that will maximize factory profits, but they recognize that finite resources such as materials, labor and production equipment present constraints on the level of output the factory can achieve. Linear programming helps managers optimize output, subject to these and other constraints.
Any linear programming problem in economic analysis contains the following three elements: an objective function, constraints and a set of variables. The objective function is a mathematical expression of the goal you want to achieve. For example, as a manager of a company, you want to maximize your company’s profit. Your constraint is a mathematical expression of the limits that exist on achieving that goal of maximum profit. These might include a finite work force and limited materials for manufacturing your product. The variables represent factors that can be adjusted, such as the daily rate of output or costs of production. Linear programming involves combining the relevant variables that provide the highest value of the objective function, subject to existing constraints.
In economic analysis, linear programming facilitates the search for an optimization solution involving large numbers of variables and constraints. Some large-scale optimization issues can involve hundreds of thousands of variables and thousands of constraints. Linear programming, combined with the power of computer programs, can solve these problems in practical amounts of time.