Some notes
2. The locational framework 1. Districting
Although many factors contribute to locational decisions, one which is always important is the subject of patient accessibility. How facilities are spaced in an urban region, however, clearly depends on the prevailing costs of travel. If these costs are high, it can be argued that locations will be chosen so that their catchment areas avoid extremes of distance;
if they are low, then other factors will operate that give more· weight to the size of the service population, so enabling a more cost-effective pattern of services.
An illustration of this point is showrt in fig. 1, which shows the complications in cities caused by a variable population density. In the top half, an evenly populated city is partitioned into five equal-spaced sub-regions each served by an imaginary hospital (L1 to L5). It is important to notice that the population P serviced by each facility is the same while the dividing line d between each sub-region is co-terminous with the points of maximum travel (MC) and the total distance (TC) of the contained population from each facility. In the bottom half of fig. 1, in which a centre to periphery decline in density is shown, this property vanishes. Holding constant the locations, we note that (i) the population influenced by each centre decreases from left to right, (ii) the total distance of travel also declines because fewer people are travelling;
and (iii) the intersections of the divides ( d, d', d') under each criterion,
A theory of health care facility location in cities. Some notes 417
418 L.D. Mayhew
as the population density. Only in a uniformly populated region could the sizes associated with all three be the same.
We can make a case for each type of districting: P- districted facilities could be built to similar specifications to obtain the best economic returns to scale and other advantages of uniformity. The increased distance of travel at low densities, however, could be effective in reducing unit consumption, so introducing a measure of spatial
inequality. TC- districted facilities take accessibility into account, but to lesser degree than MC- districted facilities for which the maximum distance in every sub-region is the same. Difficulties will be experienced with of these types of organization, however, in providing an economic mix of services at very low population densities.
2.2. Hierarchies and transformations
The exact pattern of densities varies between cities and between times. The recurrence of certain mathematical urban density functions (Clark, 1951), however, provides one common link. Similarities in the organization and functioning of health care facilities provide another (Shigan, Kitsul, 1980). The suggestion is, therefore, that the arrangement of facilities in different cities at the same time, or the same city at different times can be regarded as transformations of one another. The specification of the transformations is the ultimate goal of a dynamic spatial theory. The problem is difficult because it involves combinations of discrete and continuous processes (location decisions and population dynamics) plus uncertainty with regard to consumer behaviour.
Nevertheless, certain transformations are easy to produce, and are relevant to the discussion.
In fig. 2 the districting principle is extended to the plane. Embedded in the geometry is a hierarchy of five levels organized on well-known lines (Christaller, 1933; Dietrich, 1977) for supplying services at varying intensity. In the hierarchy, there exists a centrally located facility, which in addition to supplying high order services throughout the region, also subsumes the functions provided by layers lower in the hierarchy.
Facilities in lower layers are more numerous, but they attract patients from more limited areas. At the lowest level, a facility serves only the immediate locality, providing only those low order services that are in general demand and that are used most frequently. Finally, some facilities bordering the region share services in an unspecified way with neighbouring regions: their sub-regions are truncated by the urban perimeter.
Fig. 2 has two parts. In (b) a system is shown in which each level serves equal populations. The distortion of districts into curvilinear polygons is inevitable under such a transformation. This is hence P- districting: (a), on the other hand, is based on the MC principle.
The grid super-imposed on (a) demonstrates how the system must bend to get from (b) to (a) either in time or between cities.
A theory of health care facility location in cities. Some notes
(B)
10 kma POPULATION NORMALLY DISTRIBUTED
(A)
POPULATION UNIFORMLY DISTRIBUTED
Figure 2 Hierarchies and transformations: a) MC-districting; b) P-districting assuming a normally distributed population
419
420 L.D. Mayhew
The transforming function in fig. 1 is an equation fitted to the 1971 distribution of population in London. It is
D(r) = 92.767 exp(-0.00354r2) (1)
where D is the density in persons per hectare and r is the distance in kilometers from the city centre. The procedure for obtaining a
P-transform, if cities are radially symmetric, is to set equal the integral of the density function in the region of interest to the integral over an image region, and then solve for r. That is
~ ~
<l>(p) pdpdAR'
~ ~
D(r) rdrd8R
where R and R' are the region and image region respectively. For MC-districting the solution is always the identity transformation.
Equation 2 can be more generally written, to include non-radially symmetric cases as well. That is
~ ~<I>(p,A)plJldrd8
=~ ~D(r,8)
rdrd8R' R
where I JI is the Jacobian determinant
± J = ap aA ar a8
aA ap
-ar a8
For fig. 2b above, let <I> (p, A) = constant and D (r) = A exp (- br).
Then, if A = 8, J simplifies to 8p/ar, and the required equation is
r = [ ]
1 . 2 I
- b
log(l - p(p))where p (p) is the proportion of the population out to p in the uniformly populated image region.
(2)
(3)
(4)
(5)
As the city develops along a time path, growing in population and area, the existing health care facilities cannot move with it, because they are fixed in position for the duration of their functioning. Services will approximate the theoretical change partly by the development of new facilities, but mostly by the shuffiing of resources between existing sites.
A theory of health care facility location in cities. Some notes 421
3. The impact of time on facility behaviour at particular locations