Figure 9.4

9.4 Shopping trips

Context

Where do people do most of their shopping? If we look at shopping trips made by the residential population of an area over a period of time, can we develop a model which fits the behaviour? Such a model could be used to predict the consequences of proposed redevelopments or new transport policies.

Problem statement

Given physical and economic data relating to shopping expenditure, find the shopping pattern.

Suppose that we divide a region into a number of ‘zones’, each of which has its residential population as well as shopping facilities.

Table 9.3 gives the distance in miles between the centres of three zones of a region and also the average shopping trip distances within each zone.

We are avoiding the question of what exactly we mean by a ‘zone’ and its ‘centre’. In a real situation, there will obviously be a number of options and we would be guided by geographical and economic factors.

Table 9.3

At present the resident population figures are

the number of shops in each zone are

Formulate a mathematical model

Some of the factors involved are population, distances, number of shops, types of shop, transport facilities, travelling costs and shopping expenditure.

What causes people to choose to shop in a particular zone? Attractions are the number of shops and ease of access. Disincentives are distances, travelling costs and travelling time.

The factors involved in formulating a model are given in Table 9.4.

Our objective is to formulate a model relating x ij (the number of shopping trips made from zone i to zone j in a specified period of time) with N j, d ij and P i. Alternatively, we could have defined x ij in

terms of the amount of money spent on shopping during that period. Assumptions

1.x ij is proportional to P i. (The more people live in zone i, the more trips there will be from zone i. )

Table 9.4

2.

x ij is proportional to N j. (The more shops there are in zone j, the more shoppers will be attracted there.)

xij decreases as dij increases. (Distance will put people off.) The mathematical consequences of assumptions 1 and 2 are easy to formulate but what form could we use for assumption 3? Some possibilities are as follows. xij – dij where α is a parameter xij exp(− βdij) where β is a parameter

Models such as this have been developed and used successfully.

Generally, it is found that the effect of distance is quite pronounced and that x ij decreases with d ij more sharply than both the linear model (a) and the inverse model (b). The inverse square model (c) seems to be about right and by analogy with the inverse square law of Newton's mechanics this is often known as the gravity model. Many researchers have constructed useful models using the more general forms (d) and (e). These have the advantage of a parameter so that the model can be calibrated to some particular application. Usually, it is found that a value of α between 1.5 and 3 gives the best fit.

In this example, we shall use the inverse square model (c). Combining this with assumptions 1 and 2, we have

To change the proportionality into an equation, we must put in some constant k so that x ij = kP i N j / d ij 2 .

Obtain the mathematical solution

The most practical way to proceed with the model is to calculate the quantities P i N j / d ij 2 for all i, j

and to find the sum . Remembering that our objective was to predict the shopping pattern, we can do this in terms of percentages by dividing all the P i N j / d ij 2 by S. This is done for the current example in Tables 9.5 and 9.6.

The values P i N j / d ij 2 are as tabulated and . Dividing by S, we find the ratios given in Table 9.6. These can be converted into actual numbers of shopping trips by multiplying by the constant k. The shopping pattern, however, can be seen in Table 9.5 without using the k value.

Interpret the mathematical solution

Our model predicts that, in the area covered by our model, about 7% of the shopping will be done in zone A, 23% in zone B and 70% in zone C.

We could be more flexible than this by allowing the ‘constant’ k to be different for different parts of the table. For example, if the three zones differed significantly in average per capita income, we would expect the more affluent zones to spend more on shopping. We could use different k values, k 1 , k 2 and k 3 , for the three lines in the table to reflect different ‘spending powers’. To fit the model, we would need data on average incomes.

Note that, instead of distances between the zones, we could substitute travel times or costs. Instead of the number of shops as the attraction, we could substitute the number of different shops, the size of the shops (measured by floor space), the amount of car parking space, or some combination of these.

Table 9.5

Table 9.6

Using the model

1.A proposed new shopping centre in zone A will mean an additional 20 shops in that zone. What effect will this have on the shopping pattern?

2.A proposed new motorway will shorten the effective distance between zones A and B to 5 miles. How will this affect the shopping pattern?