CHAPTER 9

Example Models

Aims and objectives

In this chapter we illustrate the application of the techniques discussed in previous chapters to a variety of example problems.

9.1 Introduction

In this chapter we look at a selection of models from a variety of backgrounds. There is no common theme and the models are not developed in detail. The aim of this chapter is to illustrate and complement the work of the previous chapters and it gives us an opportunity to put into practice some of the principles discussed.

As far as is practical, we have followed the methodology of chapter 3 in order to emphasise again that the underlying modelling process involves the same stages even when the individual problems vary widely in context. You should adopt this practice, or one similar to it, in your own modelling efforts with the qualification that the methodology is to be regarded as a helpful framework rather than a compulsory strait-jacket. For reference an outline of the methodology is repeated here.

Context.

Problem statement, objective, given …, find ….

Formulate a mathematical model, list factors and assumptions. Obtain the mathematical solution.

Interpret the mathematical solution, validate the model. Using the model, further thoughts.

The examples given in this chapter are not all complete; in fact, there are many questions left unanswered. You should read each modelling development critically. Try out your own ideas on these models and improve on them if you can.

9.2 Driving speeds

Context

A firm is carrying out a cost-cutting exercise and requires your help with an investigation into how it can reduce its transport costs. The firm employs a

number of drivers who cover a substantial amount of mileage every day. There has recently been a large increase in their fuel costs and drivers can achieve a higher rate of miles per gallon from their vehicles by driving at a lower speed. This, however, increases journey times and the cost of the drivers’ time.

Problem statement

Can you develop a model which, given the relevant information, could give advice on the optimum

driving speed to keep costs to a minimum?

Formulate a mathematical model

This problem involves journeys, vehicles and drivers.

Factors concerning the journey

Distance travelled

Speed

Cost

Factors concerning the vehicle

Fuel cost

Fuel consumption rate

Factors concerning the driver

Cost

We now list our variables and parameters as shown in Table 9.1.

Table 9.1

Note that we have listed fuel consumption as a variable because it depends on the driving speed. Note also that we regard the driving speed as a variable in the

sense that any value can be chosen for it, while we regard the wage rate and fuel cost as given. We will in fact assume that the journey is done at a constant speed s.

Rewritten problem statement

Given values of d, f and w and given the relationship between g and s find the value of s which makes C a minimum.

Assumptions

1.The journey is travelled at a constant speed.

2.Drivers’ pay is directly proportional to time.

3.The possible effect of speed on vehicle maintenance cost is ignored.

4.The vehicles deliver 40 mpg at 30 mph decreasing steadily with increasing speed to 20 mpg at 70 mph.

Obtain the mathematical solution

First we need a model for the relationship between g and s. The simplest model is shown in Figure 9.1. The equations relating g and s are:

The journey time is d/s so the cost of the driver's time is dw/s.

The cost of the fuel is df/g, so the total cost is:

Figure 9.1

Differentiating, we have:

and we see that for 30 < s < 70 d C /d s can = 0 if (55 − 0.5 s ) 2 = 0.5 fs 2 / w which gives:

So the optimum speed depends on the ratio f/w.

Figure 9.2 shows this dependence in graphical form bearing in mind that we are measuring f and w in our chosen units.

Our formula for the optimum driving speed applies for f/w between about 0.1 and 3.5 (which give optimum speeds of 70 and 30 mph respectively).