Four different methods to model the impact of multiple opportunities were considered during the development of the accessibility calculation methodology. The four methods for combining all the raw accessibility indices for each site into a single value were: maximum, mean, sum and the sum of a harmonic series. The advantages and disadvantages of using each of these methods are presented in table 15.2.
The result of applying these equations to a real world example shows using the straight sum or the harmonic series yields results that are logically consistent with the concept of real world accessibility. That is, access to more instances of the same activity type increases the level of service that can be achieved from that site. The use of the mean provided erroneous results where suburbs with access to low numbers of facilities scored higher, and use of the maximum value reduced the accessibility to a simple map of distance to the closest activity and ignored multiple opportunities.
By analysing the slope of the equations used to model how the rate of return varies with each additional instance of an activity (figure 15.5), it can be seen that while both the sum and the harmonic series consider multiple opportunities, only the harmonic series recognises the diminishing value of subsequent opportunities.
Table 15.2 Summary of four different methods of accounting for multiple opportunities
Name 
Formulae 
Advantages 
Disadvantages 

Maximum 

· Only first destination counts · Measures best opportunity · Percentage measure of access · Simple to interpret, percentage of people who access activity. 
· Does not consider multiple opportunities. · Does not consider saturation of opportunity. 

Mean 

· All destinations count · Measures overall opportunity in travel range · Simple to interpret, percentage of people who find activity pool accessible.

· Distant opportunities dilute score of closer opportunities. · Does not consider saturation of opportunity. 

Sum 

· All destinations count · Measures overall opportunity in travel range · Closer opportunities not diluted · Can be used when sites are of unequal value. · Simple to interpret, measure of accessible activity equivalents. 
· Does not consider saturation of opportunities. 

Harmonic series 

· All destinations count · Measures overall opportunity in travel range · Closer opportunities not diluted · Additional opportunities have lesser value (saturation of opportunity) · Simple to interpret, measure of accessible activity equivalents. 
· The decay of value of successive opportunities cannot be fairly applied when the value of each instance of the activity is not equal. 
Note: the mean, sum and harmonic series all consider multiple opportunities, but only the harmonic series accounts for the concept of the saturation of opportunity.
The following points are noted regarding the four methods:
· Taking the average value only reports high values when close to a single instance of the activity (the purple values in the outer suburbs), providing an erroneous picture of accessibility. See figure 15.6 upper left quadrant.
· The maximum value method can be seen to reduce the measure of accessibility to a simple contour map of distance to nearest activity. See figure 15.6 lower left quadrant.
· Both the sum and the sum of a harmonic series of (λ) parameters illustrate access to more opportunities improves accessibility. This can be seen in the pink areas around the central city, as the centre of the city is the optimum place to minimise overall travel distance to the pool of primary health facilities distributed around the city. See figure 15.6 upper and lower right quadrants.
· The sum of a harmonic series of (λ) parameters also accounts for the saturation of opportunity. This is illustrated by the peak in the central city that is moderated, while the majority of the outer suburbs have midrange values (in many cases similar to the CBD). Highlighting the additional benefit gained by reaching the 4th, 5th, 6th doctor (for example) is of far less importance than the benefit provided by being able to choose between, say, the first three practitioners. See figure 15.6 upper right quadrant. A harmonic series is the same as a sum log function; which economic theory shows is the correct form for utility order competition.
If the origin and destination were colocated the travel distance would be zero and the raw accessibility index would yield a value of 1 (or 100%). With this assumption in mind, it is possible to compare the four methods that were considered for modelling the change in benefit provided by each additional instance of an activity as shown in figure 15.7.
For the maximum and average methods, there is no increase in benefit to be gained from accessing further instances of the activity. In fact with the average method, without the assumption of colocation, the distant values would lower the score of closer values.
The sum can be seen to weight each additional opportunity equally, while the harmonic series shows a decreasing benefit gained by each additional activity, thereby replicating the concept of a saturation of opportunities. It should be noted the harmonic series diverges as the number of opportunities increases but this rate of increase diminishes, ie the closest activity score is not scaled, while the second closest activity adds only half its score and the third adds only one third of its score and so on.
The above methods were compared with the modified isochronic measure proposed in the Austroads report on the development of accessibility measures (Espada and Luk 2009). The Austroads report assumed a saturation level of three opportunities. The modified isochronic measure accounts for the saturation of opportunities, but assumes that below the threshold value, all instances of an activity are of equal value and once the threshold is reached, additional instances provide no further benefit. In comparison, the harmonic series also accounts for the saturation of opportunities but factors in the concept that the greatest benefit comes from the closest activity, with a decreasing benefit gained by each successive opportunity.
The measurement of the ‘saturation of opportunities’ requires more in depth analysis where it may be possible to determine the actual values and the apportionment of value of each additional destination. This is an area of further research. Additionally it should also be noted that opportunity saturation probably changes over time as expectations grow for access to a larger range and choice of opportunities. Therefore saturation is a relative term. However the concept is still useful since an additional convenience store may provide very limited additional benefit as might the tenth and eleventh primary schools.
In the UK, the origin indicators publish the scale of the opportunity accessible from a location in terms of floor space, number of jobs or number of facilities (eg post offices). The value placed on the level of additional choice then becomes a matter for the user of the analysis results rather than a decision by the analyst. Until further research has been undertaken in New Zealand to ensure accurate representation of choices a similar approach to the UK could be adopted. Seven of the eight core+ activities are activities that are consumed and can be considered to be provided homogenously, and each instance of the activity is just as valuable as any other, ie the level of service provided to a consumer by Doctor A will be the same as the service provided by Doctors B and C, or that an equivalent range of items is provided by all supermarkets. These activities are named consumed activities.
Places of employment are considered differently because employment is supplied by an individual. An individual would not be qualified or able to fill all the employment opportunities within a city, therefore the level of service provided by these places of employment is not equal. This means the closest employment opportunity reached (in this instance in travel time), may not be realisable for a specific individual. This is a significant difference between the activities consumed and supplied by an individual.