Mathematical Modelling

A few days before, we had a test on mathematical modelling. Those interested could write it. It was announced that eight of us would be selected on the basis of the test to attend a workshop on mathematical modelling. I wrote the test, but did not get selected because I didn’t know many mathematical concepts. But then, they selected a few others, including me, whose approach to the problem in the test was logical. I got in through that, and I was part of the audience in the workshop.

Today, we had the awesome workshop on mathematical modelling in our school. The session was very interesting. We had experts in the field speaking to us, and explaining what it all really meant.

This post has everything that I learnt from the experts today. Let’s start!

What is mathematical modelling?

To answer this question, we need to know what a mathematical model is. A mathematical model is converting and understanding a physical, normal problem/situation using  mathematical concepts, and then projecting them mathematically to find a viable solution. The process of making a mathematical model is called as mathematical modelling.

Why is mathematical modelling important?

In the next 10 years, the biggest and the fanciest subject is going to be Big Data. And jobs such as a Data Scientist and a whole other new professions are going to come into existence. Big Data revolves chiefly around making mathematical models, analysing the models, projecting them in the right directions, and applying the models in the real-life situation. So, mathematical modelling becomes highly important.

Apart from the futuristic perspective, let us look at the concept of mathematical modelling logically. When we consider a problem involving huge numbers, like the population of a country or the world, and we are trying to solve a problem, it is impossible to use a system of trial-and-error. We will not be able to deal with all the elements individually. At the same time, we need to have a fairly accurate solution. Wrong solutions can be destructive, and can lead to a disaster! This is where mathematical modelling becomes useful.

For instance, Uber and Ola cabs have grown rapidly right now. The number of users of their different services have shot up phenomenally. They wouldn't be able to manage this boost successfully if not for mathematical modelling in their strategies, which can forewarn them about such boosts and also give viable solutions to tackle the situation in the best way.

Another example- the Duckworth-Lewis method. Most of us watch cricket. We all know that when it is bad weather and the match gets cancelled abruptly, Duckworth-Lewis method is adopted to finalise the winner. This method is purely a mathematical model. Without even the match happening, the result is determined by considering all aspects of the match before it got cancelled, and mathematically projecting the scores to see the winner.

So, such is the importance of mathematical models and their role in our day-to-day lives that it cannot be ignored.

Understanding a problem

A good understanding of the problem presented is necessary before approaching the problem. Many times, the problem may not be of a subject that we know about. So, the first step in understanding a problem is researching about it. Reading up a lot about the issue can give us a fair idea of what it is all about, and also how we can form a solution to the problem. The second step is defining variables. We need to have a complete understanding of all the factors, dependent or independent, affecting the problem that we have taken up. Otherwise, we will not be able to fully comprehend the situation and can make costly mistakes in our mathematical model. The third important thing is to make certain assumptions. We need to be aware of all the factors, but we also must be able to imagine an ideal situation and make some assumptions. We have all the right to make any logically correct assumptions that we want, but it is essential that we always remember that it was something that we assumed.

Approaching a problem and creating a mathematical model

A systematic approach is required to handle a problem and create a mathematical model of it, so as to reduce the errors and make it as accurate as possible.

Most of the time the problem has various other sub-problems combined. This makes it complex. Therefore, we have to decide on the problem that we are going to tackle, before starting with the model itself. So, the first thing to do when we have a problem is set a GOAL. This will keep us in the right track all through our model.

The process of making a mathematical model can be broadly divided into four steps:

1. Building (collecting data, researching, listing out factors, reading up, pooling all ideas)
2. Working (arranging the data collected in an organised manner, converting the data into graphs/equations/etc., studying how the factors affect the data, considering hidden factors, making a clear study of the problem and the solution proposed)
3. Reviewing(analysing the solution, finding loopholes, relating the study to the actual problem)
4. Reflecting(imagining and projecting the mathematical model in real-life and considering its viability)

In the third step, more often than not, we will have a confusion after we arrive at a solution. Are we doing the right model? Is this what we wanted? Is the model logical? Does the solution even make sense? There is no proper ‘question’ as such for us to go back and check. This is where setting a GOAL comes handy. At any point, when we have a doubt about the solution relating to the problem, we can go back to the GOAL that we have written clearly and check if the ideas match. If it does, then we are on the right track. If it doesn’t we just have to go back in our steps to find where we had deviated from the actual problem at hand.

Sticking to this broad outline methodology can help making a mathematical model easier.

Comparing our data with external evidences

It is important to check if the data we have collected is right by confirming with other external sources. For example, let us assume that we collect data on the number of people affected by influenza in a particular locality over a period of one month. We can, more or less, confirm if our data is right by approaching the hospitals in the locality and matching their data, of influenza affected patients, with ours. This is how we confirm our data before proceeding with the model.

Hidden factors

Most of the times, a problem is not presented as soon as it is recognised. So, there is a significant time period when the problem has been neglected. On top of this, the problem could have been there for a long time before it was even recognised! Now, the situation becomes more complex because the happenings during these times are not for sure. This is an example of a hidden factor affecting a problem. A problem can have more than one hidden factor. Right assumptions have to be made to bring out the best solution.

Improvising the model

The model is complete. But that is not enough. There would be a lot of improvisations to do. We will have to consider leftover factors, and try to reduce the assumptions that we have made. Also, it would be necessary for us to review our methods, and consider the limitations of the model. A mathematical model can never be totally right. So, as long as we have time, we should go back and check on our methods and assumptions, again and again and again.

Presenting the solution

Let us assume that we have made a successful mathematical model of a situation and have arrived at a brilliant solution for the problem. The actual success lies in how we present it. We may understand the mathematical equations that convey the solution. But the people for whom we are doing it need not be mathematicians, and need not understand our model equations. The whole purpose of the model will be defeated if we cannot make them understand. So, just like we translated the real-life problem to mathematics, we have to convert it back and project it in the real-life situation. We have to discuss the solution’s  implications in real-life. Technology can help in this. Videos and images aid this presentation. Therefore, mathematical modelling is not just about mathematics. It is an interdisciplinary topic, and has to be dealt in the same way. A mathematician alone cannot make a model. Professionals from various fields have to get together to create a successful one.

At the end of the presentation, we must be able to justify the existence of the model solution that we have created. We must be able to explain convincingly to others why our model is better than any other. And when people are convinced, and the model brings in a change for the better, the mathematical model’s role is fulfilled.

TIPS from the experts

1. concentrate on modelling one solution rather than trying to compare many
2. induce your own creativity
3. make a simple model and then improvise
4. you don’t have to use advanced and fancy mathematical concepts
5. there are no completely right or wrong answers
6. use technology to project
7. use diagrams and pictures in your model solution

All these are the points that I learned in the workshop. It was very fascinating. We were also given a simple problem to try our hand at mathematical modelling.

After all, the best way to learn about something is by doing it yourself!

The workshop is not yet over. We have one more day of it. The solutions for the problem will be discussed then. I will share the problem along with interesting solutions in a separate post. Look out for it!