Three key requirements for successful mathematical modelling

A. Introduction

The Oxford dictionary defines a ‘model’ as “A simplified description, especially a mathematical one, of a system or process, to assist calculations and predictions.”

As the functional head of commercial planning and analysis, every day revolves around models: to forecast revenues, analyze route performance, determine pricing, the quantity of meals to be uplifted, and the quantity of seats to sell in advance, amongst others. Some models are simple, quick and dirty. Others are more involved, detailed, and complex. Some may model the market, others may model systems, and some may model processes. Whatever the model, there are just three key requirements to be able to successfully build one.

B. Background

Since my department started out as a single member department, the requirements for successful modelling were implicit, and never did the need arise to externalise these basic requirements. As I started interviewing candidates for my team, externalisation became a necessity, to both explain to, and evaluate candidates.

C. The three requirements

1. Qualitatively establishing a relationship between quantities

Let us take an example of the effect of capacity on demand. Does an increase in capacity (such as the number of airline seats deployed between two cities) increase demand? Or does demand decrease? Or are the two quantities not related at all?

This requirement to establish a relationship relies on communication skills, and structured thinking. Communication skills are necessary to read up on literature, to ask questions, or to interact with others who may be more knowledgeable in the matter. Structured thinking is necessary to break down the problem into components, and arrange the components in the right sequence.

For example, an increase in capacity, while keeping airfares the same, may lead to low occupancy in airplanes. To boost occupancy in airplanes, airfares may be lowered. Lowering airfares may stimulate demand, which may in turn lead to a higher demand for air travel.

Incorrectly structuring the thought process may be as bad as, (for example), an increase in capacity leading to higher demand, due to which airfares start to fall. This is counter intuitive, as a higher demand usually leads to higher market prices!

Good observation skills, common sense, and to some extent a relevant educational background also help.

2. Quantitatively establishing a relationship between quantities

Once a qualitative relationship is established, it is necessary to establish a quantitative relationship. Continuing with the same example, how much an increase in capacity results in how much of an increase in demand? How much must the prices be lowered to stimulate demand? Are there other factors which influence this demand? If so, what are they? How much of an effect do each of these other quantities have on demand?

A qualitative relationship answers the ‘how?’, while a quantitative relationship answers the ‘how much?’. Does a 10% increase in capacity require a 5% fall in prices to raise demand by 15%?

Good data driven analytical skills come into the picture, to work with various quantities, and identify strong correlations. A sufficiently strong command over mathematics helps, in arriving at a mathematical relationship between various quantities. An eye for detail also helps.

3. Bringing the quantitative relationship(s) to life

Strictly speaking, the quantitative relationship between quantities is the model. The model could be approximate, or exact, depending on the desired tolerance in error. In some cases, a 5% error may be too insignificant, but in other cases, a 0.1% error may be too much. That is beyond the scope of this article.

For the sake of efficiency, and usability, it may be very necessary to realise the quantitative relationship through a tool, such as Microsoft Excel. The intent of realising the relationship on a tool is to allow for faster, more efficient, and more powerful computations (especially for complex models). Further, the model may be shared with end users, for whom the relationship (or the workings) is not as important as the result.

Bringing a model to life requires hard skills, such as proficiency on either a tool such as Excel or any suitable programming language. It also requires a sense of aesthetics, to make the input and output visually appealing. Design for usability also plays a role, such as the ease with which the model may be used by a user who may not be technically savvy.

Documentation and an organized approach are also important. When revisiting the model after months, to effect changes, most of how the model was implemented may be forgotten. Good documentation, and an organized, well-structured implementation will allow for both quick debugging and quick changes at a later date.

D. Conclusion

Modelling a system, process, or a phenomenon is as much an art as it is science. It does require a lot of quiet time, to allow for good research and development. Just as junk data in results in junk data out, a junk model also results in junk data out. Invest good time while creating a model.

About the author: Vasuki Prasad heads commercial planning and analysis at an India based full service airline. He is passionate about processes, automation, analysis, training, and strategic decision making, though not necessarily in that order. Sworn to the world of aviation, he loves airplanes, and flies for fun (including behind the controls) at every possible opportunity. Loves photography. Has two US patents, designed aircraft LED lights, designed and built flight simulators, blogged, and consulted. All views expressed are the author’s, and do not represent those of the employer.