What is a Mathematical Model?

This is part three in our series answering the question:  “With regards to what you do - scheduling, route and fleet optimization - what is artificial intelligence, and how does it compare to machine learning, or other technologies?”

The first two posts answer the questions “What is artificial intelligence?” and “What is machine learning?”.  In this post, we’ll look at the definition of a mathematical model.

What is a Mathematical Model?

Generally, “models describe our beliefs about how the world functions.  In mathematical modelling, we translate those beliefs into the language of mathematics.”

Stated another way, a mathematical model is “a representation in mathematical terms of the behavior of real devices and objects.”

Mathematical models are just quantitative ways to describe systems, things we see, and real-world problems.

There are many benefits to modelling a system mathematically, which include

  • being able to utilize computers to perform calculations regarding the system,

  • testing the effect of changes to a system, and

  • aiding decision making of managers and planners.

Again: a mathematical model is a representation of a given system (theoretical, or real-world) in mathematical terms.

How Mathematical Models Work

There are a variety of stages in building a mathematical model, and they often vary depending whether one is building a theoretical model, or a practical (real-world) model.

In general, the steps are as follows:

  1. Getting started: we need to define an objective, and then choose our level of detail and the scope of our model:

    • Level of detail - too low, and our model won’t be useful; too great, and the model will be too complex, or resource-intensive, to solve.

    • Scope - too small in scope, and our model will have limited application; too great, and again, the model will be too complex or resource-intensive to use.

  2. Construction of basic model framework: these are essentially our underlying assumptions about how a system operates.  In a real-world context, these might be specified by governing rules, regulations, in-house rules, etc.  This also includes the variables or objectives we have for the system.

  3. Choosing mathematical equations: once the system structure has been determined, it’s time to choose equations to represent the system.  There are a range of options here, but generally, this is where industry experience and familiarity with a particular type of problems becomes very important.  Potential sources for equations could be

    • previous literature,

    • experimental data, or

    • analogous domains or problems.

  4. Study and testing of the model: this typically takes two forms, with both a mathematical analysis of the behavior of the model for certain attributes, and then practical testing of the model:

    • In practical terms, this generally involves using some new input data to provide an output, and validating that output accurately describes the system we’re modeling.

    • In a scheduling context, this is the stage where some test schedules are produced, and they are checked for operational validity.

Applications of Mathematical Modeling

There are obviously a very large number of applications for mathematical models.  They form the basis of disciplines like engineering and physics: engineers use mathematical models to describe behavior of systems they design or are examining, while physicists attempt to use mathematical models to explain the universe.

You can look around you and see many applications where mathematical models were used - every product that you use has likely been designed using software powered by mathematical models (Illustrator, CAD software, etc.), or directly designed using them (cars, computers, etc.).

We will talk more about how we use mathematical modeling in a follow-up post, but let’s revisit the example we talked about previously - baseball.

If we want to predict where a ball will go once hit, we can do that using mathematical modeling.  If we know a ball’s initial direction and speed, it’s a relatively trivial problem to predict where it will land, depending on how precise is required, of course, and many first-year physics or engineering students will solve many similar projectile problems.

Assuming we know the ball’s initial location and velocity, it becomes relatively trivial to predict where it will land with good accuracy (certainly good enough to make a catch).

In our next post, we’ll examine the similarities and differences between mathematical modeling and machine learning, and their applicability to transportation and logistics - our domain.

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