Introduction Maximizing profit, minimizing loss, optimizing resources: these are the buzzwords of business. Before the Second World War, though, businesses would use only basic mathematical equations, estimations, and even intuition to maximize profit, minimize loss, and optimize resources. The underlying principles of linear programming have been around a long time; these are not revolutionary algorithmic concepts. Yet the industrial age and its models and methods of mass production meant increasing demands for calculations that would help solve complex operational and financial challenges. Linear programming allows for the introduction of several decision variables into the equation, allowing specialists in a wide range of fields to help companies improve their overall operations, with the ultimate goal of making calculated decisions based on math instead of guesswork. In this paper, I will discuss multiple methods for applying linear equations to the real world. For example, I will show how linear equations are used in human resources and personnel management in firms with fluctuating needs. Next, I will show how linear programming is used to maximize investment portfolios for businesses and individuals. Finally, I will show how linear programming is used in novel and creative ways, particularly in inventory management and logistics.

Background

Linear programming evolved around the Second World War, when the American military used basic mathematical functions to plan military deployments in ways that optimized manpower, resources, and time constraints (Lewis, 2008, p. 4). After the war, Air Force officer George Dantzig developed the first branded optimization algorithm known as Simplex, with the goal of providing “an efficient algorithm for solving programming problems that had linear structures,” (Lewis, 2008, p. 4). Reflecting on his work, Dantzig (2002) states that linear programming evolved as “part of a great revolutionary development which has given mankind the ability to state general goals and to lay out a path of detailed decisions to take in order to ‘best’ achieve its goals when faced with practical situations of great complexity,” (p. 42). Simply put, linear programming has become so ubiquitous since the days of simplex that...

The evolution of computing had a huge role to play in using linear programming in the real world. As Dantzig (2002) puts it, there are three components to linear programming: the mathematical models, the algorithms, and the technological tools. In addition to its role in optimization more generally, linear programming represents one of the earliest functions of computing. Linear programming is therefore one of the most important developments in math within the past century. Although Dantzig (2002) never won the Nobel Prize for the Simplex method, two other mathematicians would. In 1975, mathematician Leonid Kantorovich of the former Soviet Union and American economist Tjalling Koopmans were awarded a dual Nobel Prize in economics, for their “contributions to the theory of optimal allocation of resources, in which linear programming played a key role,” (Overton, 1997, p. 1). Since then, organizations in every imaginable sector from the American military to Amazon rely on linear programming.
Personnel Management

Businesses around the world now depend on linear programming methods and models for all aspects of human resources and personnel management. Linear programming is used at every stage of business planning to optimize human resources, not just in terms of how many personnel to hire at any given point in time, but also which types of personnel, what skills they might need, how much to pay them, what departments they are needed in most at any given time, and how to respond to differential needs throughout the day, week, month, or year. As Chand (n.d.) point out, linear programming “enables the personnel manager to solve problems relating to recruitment, selection, training, and deployment of manpower to different departments of the firm,” (p. 1). Personnel managers in large firms likely cannot remember a time when they did not have linear programming tools to aid them.

One specific example of an industry that relies heavily on linear programming for personnel management is in aviation. Variables like the number of…