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conventional classification 
  
level gentle moderate steep 
  
  
  
  
false 
  
— 
10 30 60 slope (96) 
(a) 
     
   
fuzzy classification 
moderate 
  
  
  
5 15 25 35 55 65 slope (76) 
(b) 
Figure 1: An example of the conventional and fuzzy classifi- 
cation of ground slope. 
classification a location with slope 696 is assigned a d.o.b. of 
0.6 for level, 0.1 for gentle, O for moderate, and 0 for steep. 
Individual locations of the region under study may by classi- 
fied in a similar way based on the rest of criteria posed by 
decision makers. For the constraints of the residential site se- 
lection example (Section 3), the following lexical values could 
be considered: 
e development: [vacant, semi-developed, developed] 
e soil moisture: [dry, moderate, wet, water] 
e ground slope: [level, gentle, moderate, steep] 
e nearness to highways: [close, near, moderate, far, far 
away] 
e aspect: [north, east, south, west] 
For decision criteria which combine more than one layer and 
lexical value (e.g., level ground and dry land) an overall mea- 
sure should be computed and assigned to individual loca- 
tions. This measure is derived from the consideration of 
d.o.bs on two or more layers. For a fuzzy set A € X with 
d.o.b. ua(x) € [0,1], the overall measure can be provided by 
an energy function, which is given by the following formula 
[Gupta et al., 1988]: 
e(A) = > Elpa(z)] , where E : pa[0, 1] = [0,1] 
xEX 
One such function commonly used is: 
e(A) 2 Y ui (a) 
rex 
Where q a positive integer?. 
For instance, if there is a requirement to highlight the most 
level and dry locations of the region under study the overall 
measure (energy function) is given by: 
  
Susing this equation (e.g., for q — 2: quadratic measure), big weight 
values (d.o.bs) are amplified, while small values are nearly eliminated. 
833 
e(level.and.dry) — Bars (x) - Hievei() 
for each individual location z. 
Reasoning based on lexical values involves the local oper- 
ations of classification, overlay and search and fuzzy logic 
methodologies should be incorporated into them as follows: 
e Fuzzy classification operations, i.e., assignment of the 
d.o.b. for a lexical value to individual locations on a 
layer. The d.o.b. is derived by applying the appropriate 
transformation function. 
e Fuzzy overlay operations, i.e., computation and assign- 
ment of an overall measure“ to each individual location, 
which is derived from the consideration of d.o.bs on two 
or more layers. The overall measure is also expressed 
in the fuzzy domain [0,1]. 
e Fuzzy search operations, i.e., retrieval of information 
based on a pre-defined threshold value for the overall 
measure (d.o.bs) assigned to individual locations on a 
layer. 
The procedure of residential site selection, based on a set 
of constraints expressed in lexical terms, i.e., vacant area, 
dry soil, level land, near to highway, and south aspect, may 
consist of the following set of operations’: 
e vacant(a.o.bs) = Local(fuzzy classification) of develop- 
ment 
© dfY(d.o.bs) = Local(fuzzy classification) of moisture 
e level(d.o.bs) = Local(fuzzy classification) of s/ope 
© Neäf(dobs) = Local(fuzzy classification) of road- 
proximity 
e south(a.o.bs) = Local(fuzzy classification) of aspect 
e good-sites(dobs) = Local(fuzzy overlay) of 
vacant(q.obs) and drY(dobs) and level gos) and 
Near(q.o.bs) and south(q oos) 
e best-sites — Local(fuzzy search) of good-sites(d.0.bs) 
Obviously, in this scheme, contrary to traditional logic, rea- 
soning is based on a “late and flexible classification”, and 
consequently the problems presented in the previous section 
are overcome. 
6 CONCLUSION 
Fuzzy logic methodologies appear to be instrumental in the 
design of efficient tools for spatial decision-making. The con- 
tribution of the paper can be summarized as follows: 
e After a brief classification of operations available in 
current GIS packages, it is shown how a sequence of 
them may compose one or more procedures to support 
the spatial decision-making process. 
  
The notion of the measure of information is a well-established concept in 
communication theory and is based on the probabilistic approach. Two met- 
rics that are used extensively for measuring the ambiguity in cognitive infor- 
mation are: the energy metric and the entropy metric [Gupta et al., 1988]. 
"the pointer d.o.bs characterizes layers whose individual locations are 
assigned the d.o.bs for a lexical value; this lexical value is identical to the 
name of the layer. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996