Modeling

The principles of mathematical modeling can be applied to almost any area of life. The Editors of Encyclopedia Britannica sum this up quite well in the following statement: “Essentially, anything in the physical or biological world, whether natural or involving technology and human intervention, is subject to analysis by mathematical models if it can be described in terms of mathematical expressions” [20]. Mathematical models can be used to model the patterns of traffic or industrial processes [20]. They can also be used to model the transmission of messages, changes in the atmosphere, and the distribution of stress in physical structures [20]. They are also seen in business situations with optimization and minimization, and they can be used in really any area of science, often alongside of or in place of experimentation [8]. These are just a few of the many ways mathematical models can be utilized in the real world.

In this section, we will begin by discussing a subsection of modeling known as linear programming. Linear programming is one method that can be used for certain optimization and minimization problems so it is especially useful in the business world. We will discuss a couple of the many methods of linear programming. We will start with the graphing method, which is helpful in visualizing optimization and minimization problems involving two variables. However, more complex methods are needed for more than two variables. This is where we will introduce the simplex method and work through an example demonstrating the general process.

Next, we will move on to the type of modeling that involves plotting data points and finding a polynomial that passes through every point. We will start with Lagrangian polynomials and go through an example of how to find one. However, because the predictions of Lagrangian polynomials become less reliable as the number of data points increases, we will look at two other methods, linear splines and cubic splines. In each of these sections, we will go through an example of how to find these splines.

We will then look at Markov chains and how they can be used to predict long-term behavior, and then we will move on to decision trees. These are visual representations that are created to organize the information given for certain optimization and minimization problems. We will then work through a problem that we solve using a decision tree and bit of probability theory.

Finally, we will discuss game theory which examines how individuals or groups of individuals interact in different scenarios when each is trying to maximize their payoffs.

Mathematical modeling is also connected to other branches of mathematics. Here are a few examples that we will see. When working through the simplex method, we will see the concepts of matrices and elementary row operations. When we work through cubic splines, we will see that one of the main concepts from calculus, derivatives, can be used to find the cubic splines. We will also need to use matrix multiplication and inverses of matrices, topics covered in linear algebra, to find the cubic splines. As previously mentioned, there will also be a little probability theory in the section on decision trees. Now, without further ado, let’s begin our observation of the world through the lenses of mathematical models.