On the Efficiency of Alternative Aggregations in Region-Building Problems

It is clear that, whenever sub-areas are aggregated into regions, the set of regions obtained is only one of a large number of alternative groupings. To illustrate this, equations are derived which give the number of different ways n counties may be grouped to form k regions, k ≤ n, in two limiting situations: the totally unconstrained case, in which each county may be grouped with any other county; and the maximally constrained case, when each county can only be grouped with its physically contiguous neighbours. A measure of the efficiency, co, of any regional grouping is given, and the problem of finding the regional grouping with the ‘best’ value of co in the set of alternative grouping patterns is discussed.