modeling land suitability/capability using fuzzy...
TRANSCRIPT
Modeling land suitability/capability using fuzzy evaluation
Fang Qiu • Bryan Chastain • Yuhong Zhou •
Caiyun Zhang • Harini Sridharan
Published online: 20 September 2013
� Springer Science+Business Media Dordrecht 2013
Abstract Modeling the suitability of land to support
specific land uses is an important and common GIS
application. Three classic models, specifically pass/
fail screening, graduated screening and weighted
linear combination, are examined within a more
general framework defined by fuzzy logic theory.
The rationale underlying each model is explained
using the concepts of fuzzy intersections, fuzzy unions
and fuzzy averaging operations. These fuzzy imple-
mentations of the three classic models are then
operationalized and used to analyze the distribution
of kudzu in the conterminous United States. The fuzzy
models achieve better predictive accuracies than their
classic counterparts. By incorporating fuzzy suitabil-
ity membership of environment factors in the model-
ing process, these fuzzy models also produce more
informative fuzzy suitability maps. Through a defuzz-
ification process, these fuzzy maps can be converted
into conventional maps with clearly defined bound-
aries, suitable for use by individuals uncomfortable
with fuzzy results.
Keywords Suitability analysis � Capability
analysis � Fuzzy evaluation
Introduction
Geographic information systems (GISs) have been
widely used to support real world decision-making
processes that involve finding regions capable of
supporting certain land uses. For example, identifying
an area capable of supporting a certain agricultural
crop or locating a site‘ suitable for a landfill are tasks
often tackled with the assistance of GIS tools. Spatial
decision-making processes like these require the
assessment of alternative sites based on criteria that
are defined by a variety of environmental and/or
socioeconomic factors. The process of assessing these
factors involves comparing the actual conditions of the
alternative sites with desirable characteristics, and is
usually referred to as capability/suitability evaluation
(Stoms et al. 2002). While the two terms are often used
interchangeably, there exist some subtle differences
between suitability and capability. Suitability most
often refers to social-economical promise, whereas
capability usually indicates natural environmental
potential. However, since it has become common
practice to ignore these subtle differences, we have
adopted the norm and do not make any distinction
between the terms in this paper.
Suitability/capability evaluation is also known as
multi-criteria evaluation, site selection, or resource
allocation analysis (Eastman 1999; Malczewski 2000).
The general procedure used by the decision maker to
perform suitability analysis using GIS usually entails
(1) selecting important factors and defining evaluation
F. Qiu (&) � B. Chastain � Y. Zhou � C. Zhang �H. Sridharan
Geospatial Information Sciences, University of Texas at
Dallas, 800 W. Campbell Rd. MS. GR 32, Richardson,
TX 75080, USA
e-mail: [email protected]
123
GeoJournal (2014) 79:167–182
DOI 10.1007/s10708-013-9503-0
criteria, (2) comparing the attributes of the alternative
sites on the desirability criteria and generating com-
mensurate suitability values/ratings for each of the
factors, (3) aggregating ratings of individual factors
into a combined suitability map to identify suitable
area(s) (Malczewski 2002). The important factors
representing site characteristics under consideration
are usually available as attributes of vector or raster
layers in a GIS. These characteristics might include
such common measures as precipitation, temperature,
slope or aspect. They can also be derived from
measurements representing spatial relations among
land features, such as proximity to major roads,
distance from the previous distribution of a species,
or any other metric deemed appropriate to the analysis.
The ratings of the alternative sites based on individual
factors are generated either using a simple threshold or
by applying a transformation/standardization function
so that all factors will have commensurate values,
allowing for their subsequent aggregation. The aggre-
gation is often conducted either using traditional map
overlay or more recent fuzzy logic based approaches
(Hall et al. 1992).
In this paper, predicting the distribution of an
invasive vine species, Kudzu (Pueraria lobata), is
used as an example to demonstrate GIS based
suitability studies. Kudzu is a climbing, perennial
legume, characterized by broad, tri-foliate leaves and
woody stems. It was introduced to the United States
from Japan in 1876, first used as an ornamental
plant, and later popularized as food for cattle, and for
soil restoration and erosion control. However, its
prodigious and uncontrollable growth eventually
made it a nuisance, as it can completely engulf
trees, utility poles and buildings in its path. In 1970,
the US Department of Agriculture classified the vine
as a pest and efforts have been made to eradicate it
(Winberry and Jones 1973). The prediction of future
kudzu distributions and the mapping of areas
susceptible to future kudzu infestation based on
various physical characteristics are important to
these eradication efforts. Winberry (1996) indicated
that kudzu requires a certain annual precipitation, a
long growing season, a mild winter and proximity to
existing kudzu growth. It can grow on almost any
soil and is seldom bothered by insects or plant
disease, hindered only by insufficient moisture and
severe winters.
Traditional approaches
McHarg is considered to be a pioneer in using overlay
for suitability evaluation. His seminal work (McHarg
1969) superimposed individual transparent maps for
each factor to obtain an overall suitability map, a
technique regarded as a precursor of modern GIS
overlay. Currently, the use of various map overlay
operations in GIS for suitability evaluation has
become a common practice. Map overlay is often
conducted in a vector GIS to aggregate various map
layers of evaluation factors. The cartographic model-
ing environment of many GIS packages can also be
used to construct efficient models of suitability
evaluation using a wide variety of map algebra
operators and functions (Tomlin 1990). Map over-
lay-based approaches are often grouped into three
categories based on how the factors under consider-
ation are combined to evaluate alternative choices
(Eastman 1999; Malczewski 2004).
The first approach is referred to as pass/fail
screening, or alternatively, as Boolean Overlay (Mal-
czewski 2004). It is a simple method that treats
environmental factors as absolute limiting criteria. In
this approach, all factors are first converted to Boolean
(i.e. true/false) values of suitability using thresholds.
For example, in order to determine the distribution of
kudzu in the conterminous United States in 1996,
Winberry (1996) used two factors: annual precipita-
tion and annual frost-free days. Based on the previous
studies on kudzu distribution, it was determined that,
in 1970, the minimum annual precipitation needed for
kudzu was 1,500 mm and the minimum number of
frost-free days was 146. Therefore, land units with an
annual precipitation greater than or equal to 1,500 mm
were given a True value (or 1) representing suitable
areas. Areas with annual precipitation less than
1,500 mm were assigned a False value (or 0). Like-
wise, land units with annual frost-free days greater
than or equal to 146 were assigned a value of 1 while
areas with fewer than 146 frost-free days were
assigned False (or 0) values (Table 1). When the
criteria for precipitation and frost-free days were
combined using a Boolean intersection (AND), it was
then possible to derive a predicted kudzu distribution
map delineating the locations that met both thresholds
simultaneously. Applying Winbery’s thresholds to the
1996 annual precipitation and frost-free days data, we
168 GeoJournal (2014) 79:167–182
123
can first derive two Boolean variable surfaces, which
can then be combined using a GIS overlay operation to
predict the suitable areas for kudzu in 1996.
This pass/fail screening method is very simple to
comprehend and implement. It is also intuitively
appealing to decision makers, because many legal
requirements exist that are specified as clear-cut
boundaries (Malczewski 2004). The theoretical foun-
dation upon which map overlay suitability evaluation
approaches are built is traditional Boolean logic, thus
the term Boolean overlay. Boolean logic is based on
‘‘crisp sets’’ that allow only two-value/binary mem-
bership (i.e. True or False or {0, 1}). A ‘‘crisp set’’ has
to satisfy the principle of mutual exclusivity and
exhaustivity, meaning that an area can only be
‘‘suitable’’ or ‘‘unsuitable’’; it has to be one of the
two and cannot be both at the same time. Usually, a
threshold has to be determined in order to assign the
suitability membership as either true or false. In the
example of kudzu distribution, 1,500 mm was used to
classify suitable and unsuitable areas based on the
annual precipitation factor. Although there exist nat-
ural cut-offs for some attributes, such as some political
mandates, many geographic phenomena cannot be
easily distinguished with a clear-cut boundary. For
example, areas with 1,499 mm of annual precipitation
and areas with 1,501 mm do not differ significantly in
their rainfall magnitude. However, using the pass/fail
screening approach, these two areas have to be
assigned to two different categories, one as unsuitable
and the other as suitable, respectively. The principle
behind pass/fail screening excludes the possibility of
partial membership or between-class overlap.
The pass/fail method uses inference rules formu-
lated by Boolean intersection operators to combine
different factors. The rules may look like this, ‘‘if
factor 1 (e.g. precipitation) indicates suitability AND
factor 2 (e.g. frost-free days) indicates suitability
AND… factor n indicates suitability, then the land is
suitable’’ (Burrough and McDonnell 1998). In this
case, the land unit is suitable only if all of the
requirements are met. Conversely, if any one of these
requirements is not met, then the area is determined to
be unsuitable. A key disadvantage of this method is
that it does not allow for ranking of sites in terms of
suitability. The results are strictly binary, either
suitable or unsuitable. The lack of partial membership
in Boolean logic contributes to this weakness because
a crisp Boolean suitability set does not permit the
comparison of two land units to see if one is more
suitable. Fundamentally, crisp Boolean logic only
works well for non-continuous phenomena or those
that are easily categorized into discrete classifications
(Openshaw and Openshaw 1997). Continuous data
that do not straightforwardly fall into arbitrary cate-
gories require a more thought-out approach. Further-
more, it does not allow the good suitability of one
factor to compensate for the poor suitability of another
factor, a feature that is referred to as trade-off.
A second approach, called graduated screening, has
been widely used in agriculture land evaluation (FAO
1976, 1982; Hall et al. 1992). This approach first
converts the raw values for an environmental factor
into a numerical rating covering a predefined range
(e.g. 1–3, 0–100, or 0–250) which represents relative
suitability rankings. In this way, all factors will have a
commensurate measurement based on a transforma-
tion scheme or a standardization function so that they
can be combined subsequently. Returning to our
Kudzu example, areas with annual precipitation
between 3,001 and 4,500 mm can be assigned a high
rating of 3, while 1 designates the least suitable areas
with annual precipitation of less than 1,500 mm
(Table 2). Likewise, areas with a number of frost-free
days of less than 145 will be assigned a low rating of 1,
while the areas with more 201 frost-free days will have
high rating of 3. Once individual criteria are ranked in
this fashion, each land unit is assigned an overall
rating equal to the lowest or the highest rating recorded
in the unit using a minimum or a maximum function. If
the rating received by an area based on the number of
frost-free days is 2, and that based on the mean annual
precipitation is 3, the overall rating will be 2 is a
minimum function is used, or it will be 3 if a maximum
function is used.
Table 1 Pass/fail screening suitability analysis
Factor Pass/fail screening
Constraint
value
Factor
constraint (FC)
Number of
frost-free days
C146 1, 0 otherwise
Mean annual
precipitation
C1,500 mm/year 1, 0 otherwise
Equation Score ¼ FC1 � FC2 � . . . � FCi
A unit failing to meet the criteria for any factor is considered to
be unsuitable
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123
Graduated screening attempts to evade the two-
value problem by introducing multi-valued suitability
ratings. Instead of using binary suitability values,
these methods utilize ordinal measurements to support
the ranking of suitability among alternative sites.
However, the introduction of ordinal ratings does not
actually overcome the shortcomings suffered by the
pass/fail screening method. The conversion of the raw
values for an environmental factor into discrete
suitability ratings substitutes one clear-cut boundary
with many clear-cut boundaries. It is still not logical or
reasonable to assign an area with 1,499 mm of annual
precipitation to the suitability rating of 1 while an area
with 1,501 mm of annual precipitation is given a
suitability rating of 2 (Table 2). The ranking of
alternative sites permits the graduated screening
approach to identify which land units are more suitable
and which are less suitable. Additionally, if we apply a
threshold to the final ratings, we can also separate the
land units into suitable and unsuitable areas, which
generate a final result comparable to that of the pass/
fail approach.
The employment of minimum or maximum func-
tions to obtain the lowest or the highest values for the
overall rating in the graduated screening approach is
equivalent to the use of Boolean intersection (AND) or
union (OR) operators. It may result in too few or too
many candidates for suitability evaluation because
these Boolean logic operators are either overly exclu-
sive (with AND) or overly inclusive (with OR).
During the evaluation process, the graduated screen-
ing is actually based only on the rating of a single
factor, either the lowest or the highest one. Thus, like
the pass/fail screening, it does not allow trade-off
among different factors, which means a good rating of
a site for one factor will not compensate a poor rating
for another factor.
The last approach is weighted linear combination.
Like graduated screening, this method first transfers
the raw value of each factor to a suitability rating.
However, in this approach, some factors will be valued
as more important than others during the evaluation
process (Malczewski 2000). In this case, different
weights are assigned to each factor to represent their
relative importance. Typically, factor weights are
defined so that the sum of all the weights equals 1. The
final suitability rating is determined by the sum of all
the weighted ratings (Table 3). If the factors under
consideration are all of equal importance (i.e. all the
weights are equal), the final suitability rating becomes
the simple arithmetic mean of all ratings for the
factors. This method can also be combined with the
pass/fail screening to offer, not only the delineation of
suitability areas, but also the suitability rankings of
different suitable land units (Eastman et al. 1993;
Eastman et al. 1995). Allowing factors to be treated
differentially in importance is an advantage for the
weighted linear combination approach. However, the
implicit linearity and additive assumptions of
weighted linear combinations may not always be
valid (Stoms et al. 2002; Malczewski and Rinner
2005), and the determination of the weight for each
factor is sometimes difficult and subjective (Malczew-
ski 2000).
Table 2 Graduated screening suitability analysis
Factor Graduated screening
Value range Factor ratingi
Number of
frost-free days
0–145 days 1
146–200 days 2
201–365 days 3
Mean annual
precipitation
1–1,500 mm/year 1
1,501–3,000 mm/year 2
3,001–4,500 mm/year 3
Equation Score ¼ min or max FR1 � FR2 � . . . � FRIð Þ
Final score is the lowest or highest rating a unit received in
reviewing all factors
Table 3 Weight linear combination suitability analysis
Factor Weighted linear combination
Value range Factor
ratingi
Number of
frost-free days
0–145 days 1
146–200 days 2
201–365 days 3
Mean annual
precipitation
1–1,500 mm/year 1
1,501–3,000 mm 2
3,001–4,500 mm 3
Weights (w) Mean annual
precipitation
0.4
Number of
frost-free days
0.6
Equation Score ¼ R FR � w
Final Score is the sum of all ratings multiplied by the weight of
the associated factor
170 GeoJournal (2014) 79:167–182
123
Unlike pass/fail screening and graduated screening,
the weighted linear combination approach does allow
for trade-offs in decision-making, in that all factors
contribute to the final ratings of a land unit during the
combination stage and the poor rating of one factor can
be compensated by a better rating of another factor
(Eastman 1999). Both pass/fail screening and gradu-
ated screening can find their theoretic support within a
Boolean logic framework, which provides the ratio-
nales for these two approaches. However, it is not clear
which logic operator or inference rule is employed to
determine the final rating of a candidate with the
weighted linear combination approach.
Fuzzy evaluation approaches
The terms used to describe suitability criteria such as
‘‘kudzu requires certain minimum annual precipita-
tion, a long growing season, a mild winter and
proximity to existing kudzu growth’’ (Winberry
1996) are often very uncertain and imprecise. The
rigid thresholds for annual precipitation and frost free-
days used in the pass/fail screening to define suability
are unreasonable or unrealistic because such natural
cut-off points do not exist in reality. An alternative site
with an attribute slightly below the threshold should
not lead to absolute rejection. To deal with the
uncertainty and imprecision involved in suitability
evaluation where crisply defined boundaries are
difficult to define fuzzy logic based approaches are
often employed (Malczewski 2004, 2006).
Fuzzy logic is built upon the concept of fuzzy sets.
Fuzzy sets (Zadeh 1965) are an extension of crisp
Boolean sets that combine Lukasiewicz’s (1970) idea
of having grades of membership with a multi-valued
logic. Fuzzy sets allow partial membership within the
range of 0 and 1 (i.e. [0, 1]) to represent the extent to
which an entity belongs to a certain class. This implies
that crisp Boolean sets with membership in {0,1} are
actually contained within fuzzy sets as a special case.
The partial membership capability of fuzzy sets allows
for the representation of complex spatial relationships
(Malczewski 2004).
Fuzzy sets not only allow partial memberships, but
also multiple memberships to different classes. Hall
et al. (1992) used multidimensional Euclidean dis-
tance to ideal characteristics of different classes to
derive fuzzy membership grades for various suitability
categories. Their research revealed that fuzzy logic
results were more informative to decision makers
when compared with those from traditional Boolean
methods and could provide recommendations for
possible improvement of suitability. Fuzzy member-
ship functions play an important role in determining
fuzzy membership grades, which explicitly models the
uncertainty of suitability analysis in terms of vague-
ness, imprecision and lack of information (Malczew-
ski 2004). The many available fuzzy membership
functions also provide a wider range of standardiza-
tion functions to derive commensurate ratings for
individual criteria with stronger logical support. For
example, nonlinear scaling using piecewise linear or
sigmoidal functions is possible (Eastman 1999).
To address issues present in conventional suitabil-
ity screening, such as ambiguity of cut-off definitions
and difficulty of weight assignment, Malczewski
(2002) proposed a fuzzy screening approach that
introduced fuzzy linguistic variables in the suitability
evaluation to define both cut-off values and the
relative importance of attributes. A symbolic approach
was then used to aggregate qualitative linguistic
values. The uniqueness of this approach is its use of
negation of attribute preferences as cut-off values to
emphasize characteristics of important attributes and
deemphasize the low ratings of unimportant attributes.
By doing this, fuzzy screening is able to deal with
decision-maker preferences in conjunction with the
threshold values.
By introducing fuzzy measurement, Jiang and
Eastman (2000) attempted to deal with the trade-off
and standardization process problems of the Boolean
overlay and weighted linear combination approaches.
They adopted the ordered weighted average (OWA)
procedure by Yager (1988), a multi-criteria evaluation
method based on two sets of weights, the conventional
criteria weights and new ordered weights. The ordered
weights apply to the ranked criteria after the criteria
weights are utilized. The primary benefit to this
approach is its ability to control the degree of
ANDORness and trade-offs between different factors.
It can deliver either pure Boolean AND/OR analysis
with no trade-off, WLC analysis with full trade-off, or
any mixture in-between, thus providing a theoretical
link between Boolean overlay and WLC. Malczewski
and Rinner (2005) extended the OWA-based model by
incorporating linguistic qualifiers to simplify the
specification of the degree of trade-off and AND/
GeoJournal (2014) 79:167–182 171
123
ORness for the aggregation procedures with a single
value. The improved OWA implementation also
allowed an exploratory multi-criteria evaluation of
suitability. Boroushaki and Malczewski (2008) further
improved upon this model by combining OWA with
the analytic hierarchy process (AHP) by Saaty (1977).
By doing this, they were able to incorporate both
criteria weights and order weights into the fuzzy
suitability evaluation process.
Clearly, fuzzy methods provide new and interesting
approaches to conventional problems such as suitabil-
ity analysis. Researchers have also begun to use fuzzy
logic in other environmental applications, especially
those involving imprecise boundaries such as land-use
and land-cover image classification, and have com-
pared their results with those from crisp classifications
(Woodcock and Gopal 2000; Benz et al. 2004; Fritz
and See 2005). However, to date, the accuracies of
fuzzy and crisp models have not been explicitly
compared in the suitability evaluation literature, nor
has fuzzy logic been used in GIS-based kudzu
capability/suitability modeling.
Fuzzy logic based kudzu suitability analysis
The analysis and prediction of kudzu distribution in
the conterminous United States is used as an example
to demonstrate the potential of fuzzy logic within a
GIS-based capability/suitability modeling framework
to support and augment sophisticated environmental
decision-making processes. The specific objectives of
the paper are to (1) integrate the three classic
suitability models into a more generalized fuzzy logic
framework, similar to the work of Jiang and Eastman
(2000), but including additional alternative aggrega-
tion operators in the fuzzy logic based models; (2)
formalize the fuzzy suitability methods within a fuzzy
suitability evaluation expert system; (3) map the fuzzy
suitability for kudzu distribution in the United States
and introduce a model calibration procedure to
transform the continuous fuzzy suitability map into a
conventional predictive map with clear-cut bound-
aries; and (4) compare the results obtained from the
conventional crisp suitability evaluation approaches
against those of their fuzzy counterparts.
The introduction of fuzzy logic theory to the kudzu
suitability modeling process is implemented in a fuzzy
system, which is also referred to as a fuzzy expert
system because of the involvement of expert knowl-
edge in the construction of the system components. The
conceptual model of this fuzzy system, as shown in the
Fig. 1, consists of environmental factors, a fuzzification
module to derive fuzzy suitability membership, a fuzzy
inference engine, a fuzzy suitability map, a defuzzifi-
cation module, and a conventional suitability map. It is
a system in which at least some or all of the variables are
fuzzy sets (Klir and Yuan 1995). However, in most
cases, the factors under evaluation within a suitability
analysis are not initially fuzzy variables. Therefore, the
first task of a fuzzy system is to convert the measure-
ment of an environmental factor into an appropriate
fuzzy set, expressing the membership grades of each
land unit belonging to the suitability class. This step is
called fuzzification. The fuzzy membership grades thus
obtained for all of the environmental factors are then
combined by an inference engine based on certain fuzzy
rules. The result of this procedure is a fuzzy suitability
map defining the overall degree of suitability for kudzu
distribution across all land units. This map provides a
continuous suitability field and is more informative
than the conventional two-value suitability/unsuitabil-
ity map. The fuzzy suitability map may be converted to
a conventional suitability map, if desired, since many
analysts are still accustomed to conventional suitability
maps with clearly defined boundaries. Since this
conversion process is the inverse of the fuzzification
procedure it is referred to as defuzzification. Figure 1
also shows the interconnection among the data sets and
the operation modules. The methodology to support
each component of the fuzzy system is given in detail in
the following subsections.
Environmental factors
The environmental factors for suitability evaluation are
often chosen from the land characteristics and land use
requirements. Physiographical characteristics such as
temperature, rainfall, soil types, and slopes usually
significantly impact the distribution of a plant. How-
ever, because of the robust nature of kudzu it is less
likely to be affected by soil types and slopes and in most
kudzu studies only temperature and rainfall are selected
as environmental factors (Winberry 1996). Therefore,
in this study, two climatic variables, annual precipita-
tion that represents the rainfall factor and annual frost-
free days that characterizes the temperature factor, are
used in the models. These two variables are obtained
172 GeoJournal (2014) 79:167–182
123
through the interpolation of weather station measure-
ments in the conterminous United States. The spatial
variations of these two variables are shown in Fig. 2a
and b respectively. The dispersion of the kudzu species
is a contagious diffusion process, thus the proximity of a
land unit to the existing distribution of kudzu is also of
importance. In order to analyze the distribution of the
kudzu in 1996, the nearest distance to the kudzu
distribution in 1970 was calculated using a GIS-based
Euclidean distance operation. The continuous surface
for the nearest distance factor is displayed in Fig. 2c.
Fuzzification
As discussed above, fuzzification is the process of
converting the raw environmental measurements into
fuzzy membership grades on the basis of some expert-
defined fuzzy membership function for each suitability
factor. The fuzzification process is similar to the
conventional standardization procedure in that the
fuzzy membership grades are commensurate values in
the range between 0 and 1. There are two methods that
can be used to define fuzzy membership functions for
generating fuzzy membership grades (Burrough and
McDonnell 1998). One method is the fuzzy k-mean
approach, which determines the fuzzy membership
function for many classes based on large quantities of
training data. The fuzzy k-mean approach is often used
in complex systems where many factors are involved
and a multitude of training examples are available. The
other method is the semantic import (SI) approach,
which is often utilized when sufficient training data are
not available and the analyst has a good, general sense
of where to put the boundaries between classes, but has
difficulty with the precision associated with these
boundaries. Fortunately, previous studies have sug-
gested some general ideas about the environmental
requirements of kudzu distribution that are helpful to
define class boundaries (Winberry 1996). Thus, in an
application such as this that involves the suitability of
only one species, the SI approach is appropriate and was
chosen to determine the fuzzy membership functions of
the three environmental factors in this study.
Compared to the conventional standardization pro-
cess, a wider variety of suitability membership functions
can be employed for use with the SI approach to deriving
membership grades, including triangular, trapezoidal,
Gaussian, linear, piecewise linear, and sigmoidal func-
tions. The first three are symmetric functions, which are
often used to define linguistic variables with different
levels of magnitude, such as high, medium and low
temperatures. However, in the suitability analysis of
kudzu distribution, asymmetric, monotonically increas-
ing functions such as the last three are needed to represent
the fuzzy membership grades of a binary fuzzy concept
such as suitability versus unsuitability. In order to model
Fig. 1 Conceptual model
of the fuzzy suitability
systems
GeoJournal (2014) 79:167–182 173
123
the nonlinearity of the three environmental factors in this
study, piece-wise linear functions were chosen instead of
simple linear functions to define the fuzzy suitability
membership functions. The general form of the piece-
wise linear functions is given as follows:
MFðxÞ ¼0 for x� a
ðx� aÞ=ðb� aÞ for a\x\b
1 for x� b
8<
:ð1Þ
where MF (x) is the membership function for
measurement x. The parameters a and b are the
threshold points, beyond which, the users are very
confident that a land unit will be unsuitable or suitable
for the existence of kudzu species (a is less than b).
This agrees with the implication made earlier that
classic Boolean sets can be included in a fuzzy set as
special cases, because if a equals b, then the fuzzy set
defined collapses into a typical crisp Boolean set.
The definitions of the fuzzy membership function
for each of the three environmental factors are given
below, and the fuzzy membership maps derived based
on these functions are shown in Fig. 3.
where p represents annual precipitation (in millime-
ters),
where f represents the annual frost-free days
where d represents the distance (in meters) to the
kudzu distribution in 1970.
Fuzzy inference engine
Based on the discussion above, we know that the
Boolean logic intersection (AND) operator has been
used to combine the evaluation of the multiple factors
involved in the pass/fail screening method. In special
cases where the standardization process results in
ratings of 0 and 1, the minimum and maximum
functions used in the graduated screening are also
equivalent to the Boolean intersection and union
operators. However, the theorems that have been
employed to support the theoretical foundation for
weighted combination methods cannot be found in the
Boolean logic framework.
The application of the logical intersection
implies that the evaluation rule used is in the form
of ‘‘if an area is suitable based on precipitation,
AND the area is suitable based on frost-free days,
AND the area is suitable based on distance to the
previous kudzu distribution (that is, if the area is a
land unit that satisfies all suitability factors), then
this area is a suitable area for kudzu growth’’. This
inference rule can also be extended to the fuzzy set
MFðpÞ ¼0 for p� 500 mm
ðp� 500Þ=ð1; 700� 500Þ for 500 mm\p\1; 700 mm
1 for p� 1; 700 mm
8<
:ð2Þ
MFðf Þ ¼0 for f � 100 days
ðf � 100Þ=ð200� 100Þ for 100 days\f \200 days
1 for f � 200 days
8<
:ð3Þ
MFðdÞ ¼0 for d� 1; 000; 000 m
ð1; 000; 000� dÞ=ð1; 000; 000� 100; 000Þ for 100; 000 m\d\1; 000; 000 m
1 for d� 100; 000 m
8<
:ð4Þ
174 GeoJournal (2014) 79:167–182
123
if the fuzzy logic intersection is used to replace the
Boolean logic intersection. Unlike a Boolean inter-
section which has only one type of operator, a
fuzzy intersection has many different operator
versions. To decide which fuzzy operator to use,
it is instructive to review some of the popular fuzzy
logic operators.
Fuzzy intersection, also called t-norm, defines the
logical intersection of two fuzzy sets A and B. T-norms
are specified in general by a binary operator on the unit
interval (Klir and Yuan 1995); that is, a function of the
form
i : ½0; 1�X ½0; 1� ) ½0; 1� ð5Þ
such that
ðA \ BÞðxÞ ¼ iðAðxÞ;BðxÞÞ ð6Þ
for all x [ X, and X is the universal set.
Examples of some t-norms that are frequently used
as fuzzy intersection operators include (each defined
for all a,b [ [0,1]):
Standard intersection : iða; bÞ ¼ minða; bÞ ð7ÞAlgebraic product : iða; bÞ ¼ ab ð8ÞBounded difference : iða; bÞ ¼ maxð0; aþ b� 1Þ
ð9Þ
Drastic intersection : iminða; bÞ
¼a when b ¼ 1
b when a ¼ 1
0 otherwise
8<
:ð10Þ
There exists a partial order among these fuzzy
intersection operations, because
iminða; bÞ�maxð0; aþ b� 1Þ� ab�minða; bÞð11Þ
Fuzzy union, also called t-conorm or s-norm, defines the
logical union of two fuzzy sets A and B. The fuzzy union
operator is specified in general by a binary operator on
the unit interval; that is, a function of the form
Fig. 2 Environment factors for Kudzu suitability analysis
Fig. 3 Fuzzy suitability membership of the environment factor
GeoJournal (2014) 79:167–182 175
123
u : ½0; 1�X ½0; 1� ) ½0; 1� ð12Þ
such that
ðA [ BÞðxÞ ¼ uðAðxÞ;BðxÞÞ ð13Þ
for all x [ X, where X is the universal set.
Examples of some t-conorms that are frequently
used as fuzzy union operators include (each defined
for all a,b [ [0,1]).
Standard union : uða; bÞ ¼ maxða; bÞ ð14ÞAlgebraic sum : uða; bÞ ¼ aþ b� ab ð15ÞBounded sum : uða; bÞ ¼ minð1; aþ bÞ ð16Þ
Drastic union : uminða; bÞ ¼a when b ¼ 0
b when a ¼ 0
1 otherwise
8<
:
ð17Þ
The fuzzy union operation also has partial order
because
maxða; bÞ� aþ b� ab�minð1; aþ bÞ� umaxða; bÞð18Þ
Because min(a,b) B max(a,b), the partial orders of
fuzzy intersection and fuzzy union operations can be
connected to form a longer sequence of partial order
from imin (a,b) to umax (a,b). The fuzzy logic operation
that falls inside the interval between min(a,b) and
max(a,b) operators is called the aggregated operator.
Because it is a fuzzy logic operator sitting in between
standard fuzzy intersect (AND) and standard fuzzy
union operations (OR), it is often referred to as an
ANDOR operator, or an averaging operation. The
continuum of the partial order (Klir and Yuan 1995)
that covers the all the range from imin (a,b) to umax
(a,b) is shown as
iminða; bÞ. . .. . .minða; bÞ. . .. . .maxða; bÞ. . .. . .umax
ða; bÞ Intersection operator Averaging operatorj jðUnion operationÞ ð19Þ
One class of ANDOR operators that lies in range
from imin (a,b) to umax (a,b) is the linear weighted
combination operator, defined as
Hða1; a2; . . .; anÞ ¼Xn
i¼0
aiwi ð20Þ
wherePn
i¼1 wi ¼ 1 (Yager 1988).
The geometric mean is another class of fuzzy
averaging operator, which is defined as
Hða1; a2; . . .; anÞ ¼Yn
i¼0ai
� �1=n
ð21Þ
The application of a fuzzy logic operator to a classic
set should also generate an outcome that satisfies the
expectation of using a corresponding classic operator,
because the classic set is a special case of the fuzzy set.
For example, when the algebraic product fuzzy
intersection is applied, the equation used is exactly
the same as that in the pass/fail screening suitability
analysis. Therefore, without any modification in
operation, the pass/fail screening method can be easily
extended to incorporate fuzzy sets as a fuzzy pass/fail
screening method. The only difference is that the
operands become fuzzy suitability membership grades
that lie in the interval of [0, 1], rather than classic
suitability values of either 0 or 1 ({0.1}).
At the same time, it is obvious that the minimum or
maximum function utilized in the graduated screening
approach is actually the standard fuzzy intersection or
fuzzy union operator. When the underlying logical
operation supporting the graduated screening model is
fuzzy intersection (e.g. a minimum function), the pass/
fail screening method is then a special case of the
graduated screening approach. If the factor ratings are
standardized in the range between 0 and 1 instead of
exactly 0 or 1, they can be used to represent the fuzzy
suitability membership grade of the factors under
consideration. In this case, a classic graduated screen-
ing model can also be extended to a fuzzy graduated
screening model based on fuzzy suitability
membership.
As previously mentioned, the weights in the
weighted linear combination method fall between 0
and 1 and all the weights for different factors will sum
to 1. Also, the function used in the weighted linear
combination model is actually the fuzzy weighted
averaging (ANDOR) operator. As a result, the logical
basis for the linear combination model can now be
provided by operations within the fuzzy logic frame-
work. We can then extend the weighted linear
combination model to incorporate the fuzzy sets and
build a fuzzy weighted linear combination model if the
standardization process is conducted using member-
ship functions and results in values in the range
between 0 and 1. The fuzzy inference rule behind the
weighted linear combination method becomes, ‘‘if the
176 GeoJournal (2014) 79:167–182
123
precipitation factor indicates an area to be suitable,
ANDOR the frost-free days factor indicates an area as
suitable, ANDOR distance to the previous distribution
factor indicates that the area is suitable, then the area is
suitable for kudzu distribution’’, because the weighted
averaging operator is an ANDOR fuzzy averaging
operator.
Models based on a logical intersection (AND), such
as fuzzy pass/fail screening and fuzzy graduated
screening methods, tend to evaluate based on only
the worst rating received by a land unit. The fuzzy
intersection (AND) operation is therefore a pessimistic
or risk-averse approach to decision-making (Mal-
czewski and Rinner 2005) and may have higher
omission errors. On the contrary, if a logical union
(OR) is used for a fuzzy graduated screening model, it
will evaluate using only the best characteristics of all
the factors. The fuzzy intersection (AND) operation is
therefore an optimistic or risk-taking approach to
decision-making (Malczewski and Rinner 2005) and
may have higher commission errors. A model using a
fuzzy averaging operator (ANDOR) attempts to
achieve a balance between these two extremes by
taking many of the factor ratings into the consider-
ation. Both the weighted averaging operator and the
geometric averaging operator mentioned above cal-
culate certain means of the factor ratings. The fuzzy
averaging operator based models allow for the com-
pensation of a low rating on one factor by a high rating
on another factor, that is, trade-offs. Jiang and
Eastman (2000) employed a more general case of the
fuzzy weighted averaging operator, ordered weighted
average, by introducing an additional set of order
weights. In their system, the amount of ANDness and
ORness, and the trade-off among the factors can be
controlled by varying the amount of dispersion and
skew of the order weights. However, like factor
weights, the determination of appropriate order
weights is also a difficult and subjective process.
If the relative importance of the environmental
factors imposes no difference and the weights are not
easy to define, we can also use the geometric mean
fuzzy averaging (ANDOR) operator to create a fourth
fuzzy suitability model. As a matter of fact, the
calculation of the geometric mean operator is similar
to the algebraic product fuzzy intersection operator
used in the fuzzy pass/fail screening approach, except
that the output is the nth root of the product. The
advantage of the geometric mean operator is that it is
an idempotent operator, which means h (a, a, a, a) = a
(Klir and Yuan 1995). For example, given the fuzzy
suitability memberships for all three of the factors are
0.5, it is expected that the final suitability membership
should also be 0.5. The fuzzy pass/fail screening
model, if used, will end up with a final suitability
membership of 0.125 (0.5*0.5*0.5), which underesti-
mates the actual suitability of the land unit and may
cause confusion when the results are used to produce a
fuzzy suitability map. The geometric mean operator,
however, can produce a final suitability membership
of 0.5, which is the same as what one would expect.
The geometric averaging operator is therefore an
appropriate alternative when the relative importance
of the factors and/or order weights is not considered or
unavailable. For this reason, a geometric averaging
based fuzzy suitability model was also implemented in
this research to analyze the distribution of Kudzu in
the conterminous United States.
Fuzzy suitability map
Each of the fuzzy suitability models extended from
their classic counterparts discussed in the Fuzzy
Inference Engine subsection can then be used to
combine the fuzzy membership grades of all the
environmental factors obtained in the fuzzification
process. The result derived from the fuzzy inference
engine is a final fuzzy suitability map. The final
suitability map defines a continuous surface depicting
the suitability membership grades of all land units in
the study area.
Unlike a conventional suitability map that is
composed of two colors, a fuzzy suitability map needs
many color intensities so that the continuous variation
of suitability across the space can be represented.
Similar to conventional suitability mapping, a fuzzy
suitability map also encloses areas completely suitable
(membership grade equals 1) and completely unsuit-
able (membership grade equals 0), although the
coverage of both areas will be much less than their
counterparts on a conventional suitability map. The
fuzzy suitability map contains more information
because, not only are the completely suitable and
complete unsuitable areas shown, but also the transi-
tion areas that fall in between are presented with their
degree of suitability. The extra information provided
by a fuzzy suitability map is useful for estimating the
likelihood that a land unit will have kudzu growing
GeoJournal (2014) 79:167–182 177
123
there. By comparing membership suitability grades,
researchers are able to gain insight into where the
possible infestation of kudzu will likely happen next.
This insight allows for the estimation of where a cost-
benefit kudzu eradication/prevention effort should be
concentrated, information which is not available with
Boolean analysis.
Defuzzification and conventional suitability
mapping
The fuzzy suitability map by itself does not serve as a
prediction map of kudzu distribution. The actual
distribution of kudzu will occur in the areas with
sufficient suitability. Although clear-cut boundaries
separating kudzu and no-kudzu distribution are often
extremely difficult to define in the natural environment
because of the transitional characteristics of kudzu
spreading, many environmental analysts may still like
to have the conventional suitability map with clear-cut
boundaries since the vast majority of suitability maps
that have been made in the past bear such boundaries.
Therefore, there exists a desire to derive a conven-
tional prediction map for kudzu distribution from the
fuzzy suitability map. The reconverting of a fuzzy
suitability map to a classic suitability map is known as
defuzzification.
Defuzzification is the reverse of the fuzzification
process previously discussed. It converts a fuzzy
membership grade in the interval of [0, 1] to a classic
membership of either 0 or 1 according to a threshold
membership grade. Any membership grade that is
greater than this threshold will be clumped into 1,
otherwise it will be collapsed into 0. The application of
the defuzzification process to a fuzzy suitability map
results in a conventional suitability map with values of
1 indicating suitable areas and 0 representing unsuit-
able units.
The determination of the threshold is not arbitrary
and must rely on the expert knowledge about previous
kudzu distribution. For example, by comparing the
fuzzy suitability map produced for 1970 with the
actual distribution of kudzu in 1970, we can derive an
optimal threshold for the defuzzification process. Then
this optimal threshold can be used to predict the
distribution of kudzu in other years such as 1996. In
this research, the determination of such an optimal
threshold is obtained by adapting the method used in
calibrating GIS-based predictive models. By varying
the threshold for fuzzy membership from 0 to 1 with a
small increment (such as 0.01) in each step, the
predicted distribution is compared to the actual
distribution of kudzu. The total number of correct
and the total number of incorrect units are then
obtained and stored in a confusion table. The threshold
with the highest overall accuracy is chosen as the
optimal defuzzification threshold to be used to predict
the kudzu distribution in the future years.
It is important to keep in mind that the optimal
threshold obtained from one fuzzy suitability model
will be different from that of another model. It is not
appropriate to apply the optimal threshold derived
from a fuzzy pass/fail screening model to a fuzzy
suitability map generated by a fuzzy geometric
averaging model. This is because the same values of
environment factors may produce different final fuzzy
suitability grades if different models are employed.
Fig. 4 Fuzzy pass\fail screening suitability model
178 GeoJournal (2014) 79:167–182
123
Generally speaking, models utilizing fuzzy intersec-
tion operators tend to treat the environmental factors
as limiting criteria and are prone to underestimating a
final fuzzy suitability grade when compared with
models using fuzzy averaging or fuzzy union
operators.
Implementation and results
Suitability models have been successfully imple-
mented in both vector (Wang et al. 1990) and raster
(Eastman 1999) GIS systems. A raster-based GIS
system was chosen to implement the four fuzzy
suitability models in this application. All of the
environmental factors, such as precipitation, frost-free
days, and nearest Euclidean distance are continuous
data fields and can be easily represented in a raster data
model. However, with minor modification, these
models can be migrated to a vector-based GIS system
to model the land use suitability in an urban
environment, where more discrete geographic features
are involved. The resolution of the data set is
25,000 m, which allows for adequate spatial accuracy
with only moderate storage size requirements for the
data.
The fuzzy suitability maps from the four models
and the conventional suitability maps derived from the
defuzzification of these four fuzzy maps are displayed
in Figs. 4, 5, 6, and 7. Theoretically the results from
pass/fail screening and weighted linear combination
might be very different. However, a visual examina-
tion of the results in the defuzzified maps from these
models reveals that they are actually similar to each
other. The visual conformity of the defuzzified
suitability map with the actual distribution in 1996
demonstrates the predictive ability of fuzzy suitability.
The optimal thresholds of the four models that were
obtained through model calibration approach and were
Fig. 5 Fuzzy graduated screening suitability model
Fig. 6 Fuzzy weight linear combination suitability model
GeoJournal (2014) 79:167–182 179
123
then used to defuzzify the fuzzy suitability maps are
listed in the Table 4, along with their overall accuracy
of prediction. It is observed that the defuzzification
optimal thresholds for the four models are quite
different, as discussed above. Each of the four fuzzy
models yielded an overall accuracy above 94 %, while
most of classic suitability models have accuracy less
than 90 %, with only WLC obtaining the highest
accuracy of 93.23 % (not shown). These results
indicate that the fuzzy suitability models could
provide an improved prediction when compared to
their classic counterparts.
Among these four models, the best prediction (with
a 96.37 % accuracy), was achieved by the fuzzy
weighted linear combination model, while the least
accuracy (94.73 %) was produced by the fuzzy
graduated screening system. The models that utilized
fuzzy ANDOR averaging operator yielded better
predictions than those that employed fuzzy intersec-
tion and union operators, highlighting the advantage of
allowing tradeoff evaluation through fuzzy averaging
operator based models. The fuzzy weighted linear
combination model produced the best accuracy, even
better than the fuzzy geometric mean model. This may
be ascribed to the fact that the fuzzy weighted linear
combination model permits weights corresponding to
different levels of importance to be assigned to
different environmental factors, while the fuzzy
geometric mean model does not.
Conclusions
This research examined the three popular classic
suitability models within the theoretical framework
defined by fuzzy logic. Similar to Jiang and Eastman
(2000), we utilized the concepts of fuzzy intersection,
fuzzy union and fuzzy averaging to explain the
theoretical rationale underlying these three classic
suitability models. Unlike their research focus on
linking Boolean overlay with weighted linear combi-
nation using OWA with adjustable AND/ORness and
trade-offs, our study attempted to fuzzify these crisp
models through incorporating continuous suitability
membership grades for each environmental factor
using piece-wise linear functions. A new fuzzy
suitability evaluation approach based on the geometric
averaging operations was also proposed as an alterna-
tive when factor weights or order weights are not
available or are difficult to define. The fuzzy suitabil-
ity models were implemented within a fuzzy expert
system, which consists of environment factors, fuzz-
ification, fuzzy inference engine, fuzzy suitability
maps, defuzzification and conventional suitability
maps. The fuzzy suitability maps produced by the
fuzzy models are more informative than conventional
suitability maps because of the extra information
provided by the partial degree of suitability across
Fig. 7 Fuzzy geometric average suitability model
Table 4 The defuzzification threshold and the associated
overall accuracy of all the fuzzy suitability models
Fuzzy suitability
models
Defuzzification
threshold
Overall
accuracy
(%)
Pass/fail screening 0.45 96.32
Graduated screening 0.55 94.73
Weighted linear combination 0.70 97.23
Geometric average
aggregation
0.75 96.37
180 GeoJournal (2014) 79:167–182
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space. Through a defuzzification procedure based on
the model calibration procedure proposed in the study,
conventional suitability maps with clearly defined
boundaries were also derived. By doing this, we were
able to directly compare classic suitability approaches
with their fuzzy counterparts. The results from the
fuzzy models were demonstrated to be not only more
useful as indicated by others (Hall et al. 1992) in the
literature, but also were shown to be more accurate
than those of their crisp counterparts in all cases. The
geometric averaging based fuzzy model also outper-
formed the fuzzy models without trade-off, and
performed almost as well as the weighted linear
combination model.
The determination of the parameters (a and b) for
the membership functions and the factor weights is
currently based on expert knowledge from previous
experience. This is acceptable for a simple study such
as this but not for a suitability study that involves many
environmental factors and/or the suitability of many
land uses. A procedure that can fine-tune the param-
eters and weights automatically based on training data
provided by human experts is desirable and constitutes
the possibility for future research.
References
Benz, U. C., Hofmann, P., Willhauck, G., Lingenfelder, I., &
Heynen, M. (2004). Multi-resolution, object-oriented
fuzzy analysis of remote sensing data for GIS-ready
information. ISPRS Journal of Photogrammetry and
Remote Sensing, 58(3–4), 239–258.
Boroushaki, S., & Malczewski, J. (2008). Implementing an
extension of the analytical hierarchy process using ordered
weighted averaging operators with fuzzy quantifiers in
ArcGIS. Computers & Geosciences, 34(4), 399–410.
Burrough, P. A., & McDonnell, R. A. (1998). Fuzzy sets and
fuzzy geographical objects. In Principles of geographical
information systems (pp. 265–292). Oxford, New York:
Oxford University Press.
Eastman, J. R. (1999). Multi-criteria evaluation and GIS. In P.
A. Longley, M. F. Goodchild, D. J. Maguire, & D.
W. Rhind (Eds.), Geographical information systems (pp.
493–502). New York: Wiley.
Eastman, J. R., Jin, W., Kyem, P. A. K., & Toledano, J. (1995).
Raster procedures for multi-criteria/multi-objective deci-
sions. Photogrammetric Engineering and Remote Sensing,
61(5), 539–547.
Eastman, J. R., Kyem, P. A. K., Toledano, J., & Jin, W. (1993).
GIS and decision making. Geneva: UNITAR.
FAO. (1976). A framework for land evaluation. FAO Soils
Bulletin No. 32. Rome: UN Food and Agriculture
Organization.
FAO. (1982). Fourth meeting of the eastern African sub-com-
mittee for soil correlation and land evaluation. Arusha,
Tanzania: UN Food and Agriculture Organization.
Fritz, S., & See, L. (2005). Comparison of land cover maps using
fuzzy agreement. International Journal of Geographical
Information Science, 19(7), 787–807.
Hall, G. B., Wang, F., & Subaryono, J. (1992). Comparison of
Boolean and fuzzy classification methods in land suitabil-
ity analysis by using geographical information systems.
Environment and Planning A, 24(4), 497–516.
Jiang, H., & Eastman, J. R. (2000). Application of fuzzy mea-
sures in multi-criteria evaluation in GIS. International
Journal of Geographical Information Science, 14(2),
173–184.
Klir, G. J., & Yuan, B. (1995). Fuzzy sets and fuzzy logic: Theory
and applications. Upper Saddle River, NJ: Prentice Hall
PTR.
Lukasiewicz, J. (1970). In defence of logistic. In L. Borkowski
(Ed.), Selected works. London: North-Holland.
Malczewski, J. (2000). On the use of weighted linear combi-
nation method in GIS: Common and best practice
approaches. Transactions in GIS, 4(1), 5–22.
Malczewski, J. (2002). Fuzzy screening for land suitability
analysis. Geographical and Environmental Modelling,
6(1), 27–39.
Malczewski, J. (2004). GIS-based land-use suitability ana-
lysis: A critical overview. Progress in Planning, 62(1),
3–65.
Malczewski, J. (2006). GIS-based multicriteria decision
analysis: A survey of the literature. International
Journal of Geographical Information Science, 20(7),
703–726.
Malczewski, J., & Rinner, C. (2005). Exploring multicriteria
decision strategies in GIS with linguistic quantifiers: A
case study of residential quality evaluation. Journal of
Geographical Systems, 7(2), 249–268.
McHarg, I. (1969). Design with nature. New York: Doubleday.
Openshaw, S., & Openshaw, C. (1997). Fuzzy logic, fuzzy
systems and soft computing. In Artificial intelligence
in geography (pp. 268–308). Chichester, New York:
Wiley.
Saaty, T. J. (1977). A scaling method for priorities in hierar-
chical structures. Journal of Mathematical Psychology,
15(3), 234–281.
Stoms, D. M., McDonald, J. M., & Davis, F. W. (2002). Fuzzy
assessment of land suitability for scientific research
reserves. Environmental Management, 29(4), 545–558.
Tomlin, D. (1990). Geographic information systems and car-
tographic modeling. New York: Prentice Hall.
Wang, F., Hall, G. B., & Subaryono, J. (1990). Fuzzy infor-
mation representation and processing in conventional GIS
software: Database design and application. International
Journal of Geographical Information Science, 4(3),
261–283.
Winberry, J. J. (1996). Kudzu: The vine that almost ate the
south. In G. G. Bennett (Ed.), Snapshots of the Carolinas
(pp. 99–102). Washington, DC: Association of American
Geographers.
Winberry, J. J., & Jones, D. M. (1973). Rise and decline of the
‘‘miracle vine’’ kudzu in the southern landscape. South-
eastern Geographer, 13(3), 61–70.
GeoJournal (2014) 79:167–182 181
123
Woodcock, C. E., & Gopal, S. (2000). Fuzzy set theory and
thematic maps: Accuracy assessment and area estimation.
International Journal of Geographical Information Sci-
ence, 14(2), 153–172.
Yager, R. R. (1988). Ordered weighted averaging aggregation
operators in multi-criteria decision making. IEEE
Transactions on Systems, Man, and Cybernetics, 18(1),
183–190.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3),
338–353.
182 GeoJournal (2014) 79:167–182
123