6 DETERMINATION OF WEIGHTS
Given the consideration of factors as fuzzy sets
and the nature of the aggregation process, the
criterion weights of weighted linear combination
clearly represent trade-off weights – that is,
expressions of the manner in which they will trade
with other factors when aggregated in multi-criteria
evaluation. Rao et al (1991) have suggested that a
logical process for the development of such weights
is the procedure of pairwise comparisons
developed by Saaty (1977). In this process each
factor is rated for its importance relative to every
other factor using a 9-point reciprocal scale (i.e. if
7 represents substantially more important, 1/7
would indicate substantially less important). This
leads to a n x n matrix of ratings (where n is the
number of factors being considered). Saaty (1977)
has shown that the principal eigenvector of this
matrix represents a best fit set of weights. Figure 5,
for example, illustrates this rating scale along with
a completed comparison matrix and the best fit
weights produced. Eastman et al (1993) have
implemented this procedure in a raster GIS with a
modification that also allows the degree of
consistency to be evaluated as well as the location
of inconsistencies to allow for an orderly reevaluation.
The process is thus an iterative one that
converges on a consistent set of consensus weights.
A problem still exists, however, in how these
weights should be applied in the context of the
ordered weighted average discussed above. It seems
clear that these weights will have full effect with the
weighted linear combination operator (where full
trade-off exists), and that they should have no effect
when no trade-off is in effect (i.e. with the minimum
and maximum operators). It seems logical,
therefore, that their effect should be graded between
these extremes as the degree of trade-off is
manipulated with the ordered weighted average
process. However, the logic for this gradation has
not been established. In their implementation of the
ordered weighted average for GIS, Eastman and
Jiang (1996) have used a measure of relative