Mathematics in real life applications

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 it is taken for granted.
Linear programming basically entails the use of math to quantify, define, and resolve real-world problems: problems that seemed too daunting to quantify prior to the middle of the 20th century. 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 maximum working hours for pilots and flight crew can be input alongside the different types of aircraft in the fleet, the flight schedules for each day, and also ground crew needs. Airlines with many different types of aircraft of different sizes need to know how many crew each flight needs, with fluctuating personnel needs throughout the day. Also, linear programming can help optimize human resources costs by reducing the numbers of personnel on overnight stops requiring hotel stays or reducing redundancies or overscheduling (“Applications of Linear Programming,” n.d.). Another example is with large stadiums or conference centers, where personnel needs vary widely throughout the year, or in the postal and delivery services, which require temporary workers during holiday peak times.
Financial Management
Financial planners, hedge fund managers, and investment bankers cannot do their jobs properly without linear programming. In fact, any financial manager in any firm needs to use linear programming if the company is interested in maximizing returns on investment while mitigating risks (Chand, n.d.). Investment firms need linear programming for complex equations that can help them to “allocate assets” into various investments (“Applications of Linear Programming,” p. 1). While linear programming cannot be used for forecasting, the astute investment advisor can combine linear programming methods with other mathematical models to help predict when to move assets or shuffle around investments to maximize profits strategically. Linear programming can also be used to strategically allocate venture capital into new firms, showing investors whether and when to take a risk. Financial managers working for firms can use linear programming to help them to make choices regarding the company’s desire to open a new branch. Wu (1997) offers a case study showing how real estate developers and urban planners use linear programming to make key decisions regarding land use and development, taking into account multiple factors such as the variable costs of construction, differential costs of licensing permits, and whatever other factors are relevant. Along with other types of mathematical modeling, linear programming is indispensible in financial management.
Logistics and Inventory Management
One of the most common applications of linear programming is in inventory management and logistics, and the most famous of all companies known for its optimization strategies is Amazon. Using linear programming in creative ways, Amazon is able to strategically allocate warehousing and transportation resources to get a stunning array of products from around the world delivered to even the world’s most remote areas. Amazon and other retailers use linear programming to determine which warehouses in which areas have capacity, which are low on inventory, how to schedule shipping, and how to account for fluctuations in costs too. Linear programming can even help Amazon determine when and where a new warehouse might be needed.
Discussion
There are no real cons of linear programming, but it is not a panacea that fixes all problems. Linear programming is good for expressing ranges or zones of optimization, not absolute values or objectives. However, in the world of business there are rarely any certainties. Prices of goods, services, communication, and transportation are variable. The great thing about linear programming is that it can incorporate both fixed and variable costs and offer best case scenarios. Linear programming provides decision makers with options and choices based on quantifiable inputs and outputs. Linear programming might not be useful when making decisions involving unknown variables, like trying to predict consumer behavior. No mathematical modeling can account for the truly unpredictable variables in human life.
Conclusion
From its initial use in the military sector to its use in global commerce, linear programming has radically transformed the way almost all businesses and organizations operate. The public and private sectors need linear programming to optimize resources, whether financial or human, showing how to fix problems or prevent problems from happening in the first place. Transportation and communication infrastructure also depend on linear programming. Learning about the history and evolution of linear programming shows how concepts in math are never stagnant; they evolve over time to meet the needs of human beings. As abstract as mathematics can seem while learning it, the complex equations and algorithms have concrete, real-world applications that are in use behind the scenes, embedded in the software programs we use every day.





References

“Applications of Linear Programming,” (n.d.). http://homepages.rpi.edu/~mitchj/handouts/lp/lp.pdf
Chand, S. (n.d.). Applications of linear programming for solving business problems. http://www.yourarticlelibrary.com/linear-programming/applications-of-linear-programming-for-solving-business-problems-economics/28947
Dantzig, G.B. (2002). Linear programming. Operations Research 50(1): 42-47
Lewis, C. (2008). Linear programming: theory and applications. https://www.whitman.edu/Documents/Academics/Mathematics/lewis.pdf
Overton, M.L. (1997). Linear programming. https://cs.nyu.edu/overton/g22_lp/encyc/article_web.html
Wu, M.Y. (1989). Application of linear programming — a case study. Land Development Studies 6(3): 201-216.