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: ffqiu@utdallas.edu

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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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

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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

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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.

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