CHAPTER 3

Modelling Methodology

Aims and objectives

In the previous chapter, 12 modelling problems were presented. Many alternatives could have been included. They were all presented at the beginner level and the intention was to give a student modeller an idea of what mathematical modelling is about. In this chapter, we attempt to lay out guidelines for doing modelling. A methodology is explained and an example is then developed to show how and why the methodology is useful in practice.

3.1 Introduction

Each of the 12 examples from chapter 2 has an element of real-life use to motivate the investigation. It could be claimed that when mathematics is used to solve problems then all the problems are ‘real life’ to some extent, so the question arises: what is so special about mathematical modelling? The terminology used to promote the 12 examples mentioned ‘problem solving’, ‘investigation’, ‘model’, ‘simulation’ as well as ‘real-life situation’. All these terms need clarifying or defining and this will now be tackled here. It is convenient to list some of the questions to be addressed:

1.What exactly is mathematical modelling?

2.When is the best time to start a mathematical modelling course (as a student)?

3.What new skills are acquired from modelling that cannot be learned from conventional mathematics courses?

4.Is there any difference between ‘investigations’, ‘problems’, ‘models’ and ‘simulations’ in mathematical modelling?

5.Is there some general methodology for doing modelling that we should be aware of (perhaps for future use in the professional arena)?

Before giving answers to these questions, we shall review the content of the 12 examples from chapter 2. The important features in each example are identified here:

Example 2.1 Data fitting

Collect data and analyse relations between the sets of data with the purpose of interpolation and prediction of further results.

Software: spreadsheet analysis using EXCEL or equivalent. Example 2.2

Windscreen wipers

A problem-solving scene using simple geometry of circles. Occurrence of simplifying assumptions. Variable list drawn up. Comparative outcome over a variety of car products is possible.

Software: none. Example 2.3

Traffic lights

An everyday situation that can be observed and the actual behaviour recorded. Car performance based on kinematics and driver reaction have to be estimated. Simplifying assumptions made. Physical units required. Paves way to many other car modelling problems.

Software: none, though results can be put on a spreadsheet. Example 2.4

Price wars

A simple business problem. Conceptual relations between variables required. Parameter fixing difficulty. Strategic outcome.

Software: spreadsheet useful in display of results and graphs. Example 2.5

Evacuation

An important organisational problem. Conceptual behaviour to be assumed. Good for beginners since the outcome can be easily tested and the results matter.

Software: MODELMAKER package used to analyse large situations. Example 2.6

Corridors and corners

Problem solving using simple trigonometry. Needs calculus to obtain the solution. Can be validated onsite as the ‘removal man's dilemma’.

Software: none. Example 2.7 Fish harvest

A population problem from the age-stratification angle. Parameter selection issues. Steady-state outcome required. Opportunity for a matrix model.

Software: EXCEL spreadsheet. Example 2.8

Student loan finance

A topical problem of finance. Introduces compound interest and use of the geometric series in analysis. Tabular outcome ideal for spreadsheet display.

Software: EXCEL spreadsheet. Example 2.9

Car exhaust bay

Random effects included. Idea of a (discrete) simulation model introduced to give a stochastic as compared with a deterministic model. The beginning of some important modelling problems where the outcome is measured statistically after many simulation runs.

Software: EXCEL spreadsheet. Example 2.10

Fixtures

An organisational problem. No physical concepts or units required. No data required in particular so a good problem for beginners. More of a mathematical puzzle. Outcome important nevertheless.

Software: none, though spreadsheet helps. Example 2.11

Snow plough

A physical model required. Relations between variables are drawn up and units are chosen. Kinematics of snowplough motion are formulated mathematically.

Software: none specifically, though DERIVE useful. Example 2.12

Random walk

Fungal growth behaviour analysed by conversion to a random walk situation. Interest centres on the density of growth away from a number of source points. A simulation model created, which is run many times to produce statistical results of the fungal spread.

Software: MATLAB.

It can be seen from this list and the efforts shown in chapter 2 that some of the examples require a direct answer to a direct set problem. In others there is more than one answer owing to the nature of the Example or the formulation presented. Also some may seem relatively easy and ‘closed’ such as Examples 2.6 or 2.10, whereas others can be extended and are ‘open’ in the sense that we may dispute part of the formulation or want to develop more complicated relations between certain variables (perhaps in Examples 2.4, 2.5 or 2.11).

These considerations partly deal with question 4 above: a so-called problem often has a specific correct answer while a model will be more general and speculative. Very often a model is a formula containing symbols that represent our variables generally, without particular numeric values being substituted (see equations (2.4), (2.12), (2.23), (2.25) and many others). Different models can be formulated for the same initial situation and different answers obtained. It is not the case that one formulation will be ‘correct’ and all the others wrong, although some may be more useful than the rest. Essentially model building will require creativity including making modifying assumptions on the way. A model can and should be re-visited after use to see if it can be improved or corrected. Reading chapter 1 in this book sets out the whole scene in detail.

Other issues have begun to emerge from working through the examples in chapter 2. Keeping to the model building theme, it seems to be a good plan to list all the variables and factors at the start that affect the particular situation giving them symbols and units if possible. Assumptions need to be identified and carefully weighed for effect and then the mathematical formulation is hopefully carried out. We then apply data to our model and consider the results for their worth. We try to interpret the results in context to see if they stand up to reality. It may be realised that if the outcome is actually wanted by someone (and we hope that it is) then the person will probably not be a mathematician. They will perhaps be a planner, engineer, sales person, fire officer, football league secretary etc., so we shall probably have to present the results to a non-expert who may not want too much mathematical detail. (How to present orally and report in writing on the findings of a modelling activity are discussed in chapter 10.)

Returning to the questions given at the start of this section, brief answers are:

1.Translating a real situation into a mathematical form with the purpose of solving the mathematics to provide useful answers to the real situation.

2.No later than the second year within an appropriate mathematics degree programme (so that simple calculus, algebra, statistics and IT have been covered).

3.The essence of learning how to apply mathematics in the real world through careful understanding of the situations under consideration. This

is an important issue: setting up one's own model is quite different from using an already approved textbook formula.

Overlap always appears in the terminology, but there are normally some differences – see section 3.2 below. Yes. A structured approach (probably through team work) is always more likely to lead to success rather than haphazard undirected individual work. Read on to section 3.3 below.

3.2 Definitions and terminology

A number of different terms are commonly used to help to classify models and it will be useful to explain them here. The term mathematical modelling itself has been described in chapter 1; so there is no need to cover the same ground again. Perhaps a reminder is appropriate that the word ‘mathematics’ is used ‘generically’ to include statistics, operational research and computing as well as conventional mathematics.

The term problem solving was mentioned in chapter 2, suggesting a slightly different activity from modelling. Without being too rigid in our definitions, this term is often reserved for those situations where a definite well-defined ‘problem’ has to be tackled and a particular answer is required. Expressing the problem in mathematical terms will not lead to much argument over alternatives. Examples 2.6 and 2.8 come into this category.

The term simulation can often be used synonymously with ‘modelling’, but some writers prefer to save it for use when the situation being modelled is very complex or more especially when there are random effects to be considered. Note in passing that The Oxford English Dictionary definition of ‘to simulate’ is ‘to assume the mere appearance of’, ‘to feign’, ‘to counterfeit’ or ‘to pretend’. This is not what we want at all since it suggests some sort of confidence trick! To explain what is meant, consider the following two examples.

Example 3.1

A health centre is to be set up in a small town to contain, under one roof, a team of both doctors and pharmacists. The local area health authority want to know whether there is likely to be any improvement in efficiency of this new system compared with that currently in use, where doctors all act individually and pharmacists work in separate chemist's shops. In particular, the local authority need to judge the efficiency before going ahead with building work.

The operation of the new centre can be tested by constructing a model of the situation using statistics and mathematics. Data would have to be collected

about doctors’ current consultancy times, number of doctors and pharmacists to be available and so on. The data would then be used to construct a model that simulates the real system. Repeated operation of the model would be carried out to provide output on the times patients spent at the centre, the times a doctor was idle and so on. The results would be averaged so that items of information such as ‘mean patient time at the centre’ and ‘mean number of patients treated per day’ could be calculated. These results would then be compared with similar calculations for the current system so that a decision could