whose attribute values satisfy absolutely a constraint posed
by decision makers. In addition, decision makers are obliged
to express their constraints through arithmetical terms and
mathematical symbols (e.g., slope < 10%), since they are
not allowed to use natural language lexical terms (e.g., level
land). Finally, there in no factor available for the ordering of
qualified locations derived from a sequence of GIS operations.
These problems caused by the standard logical foundation
can also be distinguished by examining the simplified, though
representative, approach (process) to spatial decision mak-
ing in a real world problem, as given in the previous section.
Specifically, the employment of a sequence of standard GIS
operations to support the residential site selection is accom-
panied with all problems of an “early and sharp classification" .
First, the overall decision is made in steps which drastically
and sharply reduce the intermediate results. Any constraint
is accompanied with an absolute threshold value and no ex-
ception is allowed. For instance, if the threshold for a level
land is slope = 10%, a location with slope equal to 9.9%
is characterized as level, while a second location with slope
equal to 10.1% is characterized as steep. Moreover, for de-
cisions based on multiple criteria, it is usually the case an
entity (i.e., an individual location) which satisfies quite well
the majority of constraints and is marginally rejected in one
of them to be selected as valid by decision makers. However,
based on boolean logic, a location with slope 10.1% will be
rejected, even if it satisfies quite well all other constraints
posed by decision makers. Finally, the effect of classical set
theory is that the selection result is flat in the sense that
there is no overall ordering of the valid entities as regard to
the degree they fulfill the set of constraints. For instance,
dry-level layer (Section 3) highlights all locations which sat-
isfy the constraints: dry land and level ground, however there
is no distinction between a location with moisture = 10% and
slope = 3% and another with moisture = 20% and slope =
7%.
These impediments, and all those stated in literature
[Aubert et al., 1994, Burrough, 1989, Davidson et al., 1994,
Kollias et al., 1991, Leung et al., 1993], call for a more gen-
eral and sound logical foundation for GISs as offered by the
concept of fuzzy logic.
5 SPATIAL DECISION MAKING AND FUZZY LOGIC
Fuzzy logic methodologies [Zadeh, 1988] may provide a
scheme for the representation and manipulation of the un-
certainty which is related to the classification of individual
locations according to their attribute values. Instead of nu-
merical values real world entities and measurements are as-
signed lexical values. For instance, “the site is far away from
the highway". This statement has uncertainty features. The
uncertainty is related to the perception of distance between
the site and the road network. The perception of distance
may be formed by the objective distance measurement to the
nearest highway (e.g., 20 km) and the perceptual and cogni-
tive background of the observer. The concept of the uncer-
tainty represents the degree to which an object belongs to a
set. This measure is referred to as degree of belief (d.o.b.)
[Gupta et al., 1988, Zadeh, 1968]. The d.o.b. is usually nor-
malized in the interval [0,1], termed as fuzzy domain.
Lexical values assigned to physical entities correspond to a
^ also referred to as grade of membership
range of physical values (e.g., far away — distance € [15 km,
oo)). The transformation of physical values to fuzzy values
(i.e., values in the fuzzy domain) is accomplished through the
employment of transformation functions of the form f : R +
[0,1].
This procedure is called fuzzification and fuzzy values are
measures of the d.o.b. that the corresponding physical value
belongs to the set denoted by the lexical value.
An important issue for decision making is reasoning based on
lexical values assigned to physical entities. According to the
scheme proposed in [Vazirgiannis et al., 1994] a set of lexical
values should be assumed to classify entities and measure-
ments in categories. Each lexical value corresponds to a range
of physical values, while transformation functions are provided
to map physical values to fuzzy values. There is one transfor-
mation function associated to each lexical value. Hence the
number of transformation functions is equal to the number
of lexical values assumed. Several transformation functions
are exploited [Gupta et al., 1988]. For the purposes of this
study, the following simple linear transformation functions are
assumed:
e Linear increasing: It is used in the cases where a
straight-forward mapping of physical values to the
fuzzy domain is needed. The linear increasing func-
tion is represented by the equation:
T --CO
Mni.
C1 — Co
Vz € [co, ci]
e Linear decreasing: lt is represented by the equation:
LD(z) = S +1, Vz € [co, c1]
Co — C1
e Triangle: The set of physical values is divided into k
parts: [co, c1], [c1,¢2], ..., [ck=1,ck]. The transforma-
tion function of the physical values to the fuzzy domain
are:
TRı(z)= SIS , Vx € [co,ci]
Co 77 C1
2(x — c;) Ci Ci+1
TRa(z)- ——— —- , Va E [ci, ——]
Ci+1 — Ci 2
2(ci — i t ci
TRılz) = 20 0 rl vei
Ci+1 — Ci 2
TRs(x) = — , vs € [cr-1,0x]
€1 —'C0
Consider the classification of individual locations on a layer
based on the slope values of the ground (physical values).
Four lexical values are used: [level, gentle, moderate, steep).
The transformation functions are linear decreasing and in-
creasing for the first and last lexical values respectively, and
triangle for the rest of them. Figure 1 illustrates the con-
ventional (Fig.1a) and fuzzy classification (Fig.1b) of slope
values. Notice that the conventional way to classify slope in-
volves discrete classes with specific ranges, while fuzzy classi-
fication captures the gradual transition between classes (lex-
ical values), providing a better way to categorize imprecise
concepts such as gentle and steep land. Based on the fuzzy
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
