The Meaning of Application of Principles in Real Life

The realm of mathematics have a variety of implications on many real word activities that take place in today’s society. From the construction of a buildings to the usage of models in stocks and investment, math has a very effective role in the productivity and results of certain activities in the world. As such, the purpose of this essay is to focus on the implications of mathematics in the realm of business; to be more specific, the usage of data and calculations to gain the greatest output of products through the least amount of labor and resources, thereby creating larger profits while reserving resources, that can later be used for further expansion in the business realm. In order to further explore this area of interest, it is important to first be exposed to the ideas of the Pareto’s Principle, as well as the usage of Optimal Control Theory in an effort to capitalize on the inputs of a system that creates the greatest outputs. The Pareto’s Principle is term that specifies an uneven relationship between the inputs and outputs of a certain system. Accordingly, the principle states that 20% of the actual inputs used to create a certain result, is actually responsible for 80% of the outputs that are produced. In other words, this principle is representative of the imbalance that is prevalent between the inputs and outputs of a system; more energy and expenditure may not necessarily create a proportionate amount of greater results. A concocted example to provide further visualization of this statement would be that 20% of a business's employees are actually responsible for 80% of the corporation’s profits.

As a quick note, it is important to realize that because the inputs and outputs of a system, such as a business, has different units, the percentage of these factors do not have to actually add up to 100%. For example, 83% of the outputs can be the result of 25% of the inputs-- the sum of these percentages is 108% and this is still valid. The core principle is that an emphasis must be placed on certain inputs in order to maximize the amount of desired results that can be achieved. Now with this in mind, it is practical to only assume that businesses should focus all of their resources on the inputs that are responsible for the majority of their outputs, in order to further create a proportionate amount of profits through the expenditure of energy in the right areas. However, this is easier said than done, as this emphasis should be done without causing harm to any of the other functions of the business, such as communications, networking, and other departments of the corporation. This state that most businesses are always trying to achieve (it is a broad fact that businesses are always looking for methods to gain larger profits and more expansion at the least expense possible) is known as Pareto efficiency, or Pareto optimality. This is, in essence, an economic condition in which resources are apportioned in the most effective manner possible so as to gain the most desireable results; it is evident that this state does not imply the equal distribution of resources, but rather the concentration of energy in the right areas to maximize the utility of employees and energy in the world of business. Now, Pareto efficiency is achieved once individuals are maximizing their work’s value; this is when the production of materials, or outputs, has been maximized completely with an optimal amount of input that does not waste any extra energy or money.

However, in the effort to attain this state of Pareto efficiency, one must make Pareto improvements. Pareto improvements are, in essence, improvements in the optimization of resource allocation and work that do not put other inputs at risk. For example, it is not ideal to move one employee of a production line to another, if it causes a decrease in the rate of efficiency of the former. As such, for further clarification, Pareto efficiency is also that state in which the amount of outputs is maximized with an optimal amount of inputs, without causing detrimental effects to other factors of the system. This implies that the variables that are being inputted into the system, like the number of employees, working hours, and money that is put into industrial activities in a factory, may not be functioning at their full potential, but rather are functioning on a level that allows for maximum efficiency while not harming the rate of production that any other variables have in play, as well. The consequences of the reallocation of resources in order to attain a state of Pareto efficiency will me the restructuring of processes in the work environment, as well as changes in the use of materials, equipment, and employers to commit to completing certain tasks. As such, the aim of this paper is to discover how to use the variables associated with the business industry in order to achieve this state of efficiency, as well as how to control them in order to keep these inputs functioning at a steady rate so as to achieve constant, high profits.

In order to discover this system, the principles of Control Theory must be implemented. Now, the solving of this problem with the control theory in mind consists of a two-step process: the adequate description of the behavior of the system in a manner that is mathematically precise (through either differential equations or integrals) and the purpose of the control and the environment must also be specified, mathematically, as well. The description of the behavior of the system will be used to measure the effect of any control action, in this case the maximization of the necessary inputs, in any possible circumstance, and then the optimization of the actual number of choices available to obtain the desired outputs will be based off of the information provided by the description of the control action and the environment. From this point on, further principles that fall under the umbrellas of the Control Theory, like the Pontryagin Maximum Principle, which is used to find the best possible control for taking a system from one state to another in the presence of input controls, as well as the use of dynamic programming methods, like the Hamilton-Jacobi-Bellman equation, which is a partial differential equation whose solution will ultimately be a function that gives the minimum cost for a given system, will be used to further narrow down the inputs that will allow for the achievement of a Pareto efficient-mathematical model.