ORIGINAL PAPER
Application of Fuzzy Set Theory to Rock EngineeringClassification Systems: An Illustration of the Rock MassExcavability Index
Jafar Khademi Hamidi Kourosh Shahriar Bahram Rezai Hadi Bejari
Received: 14 June 2008 / Accepted: 8 January 2009 / Published online: 7 February 2009
Springer-Verlag 2009
Abstract The characterization of rock masses is one of the
integral aspects of rock engineering. Over the years, many
classification systems have been developed for character-
ization and design purposes in mining and civil engineering
practices. However, the strength and weak points of such
rating-based classifications have always been questionable.
Such classification systems assign quantifiable values to
predefined classified geotechnical parameters of rock mass.
This results in subjective uncertainties, leading to the misuse
of such classifications in practical applications. Fuzzy set
theory is an effective tool to overcome such uncertainties by
using membership functions and an inference system. This
study illustrates the potential application of fuzzy set theory
in assisting engineers in the rock engineering decision pro-
cesses for which subjectivity plays an important role. So, the
basic principles of fuzzy set theory are described and then it
was applied to rock mass excavability (RME) classification
to verify the applicability of fuzzy rock engineering classi-
fications. It was concluded that fuzzy set theory has an
acceptable reliability to be employed for all rock engineering
classification systems.
Keywords Fuzzy set theory Rock engineeringclassification Rock mass excavability (RME) TBM performance
1 Introduction
Since the earliest days of history, man has searched for a
way to describe the properties of rock. For this, he has
always classified and characterized rocks based on features
such as color, shape, weight, hardness, etc. Unlike the
recently developed (multiple-parameter) rock engineering
classifications, they considered only one feature of rock for
its description. Nowadays, rock engineering classification
systems form the backbone of the empirical design
approach and are widely employed in civil and mining
engineering practices. Rock mass classifications, as an
example, have recently been quite popular and are mostly
being used for the preliminary design and planning pur-
poses of a project. According to Bieniawski (1989), a rock
mass classification scheme is intended to classify the rock
masses, provide a basis for estimating deformation and
strength properties, supply quantitative data for support
estimation, and present a platform for communication
between the exploration, design, and construction groups.
So, the role of classification is generally to obtain a
better overview of a phenomenon or set of data in order to
understand them or to take different actions concerning
them. With this task for rock mass classifications, classi-
fication is defined as the arrangement of objects into
groups on the basis of their relationship (Bieniawski
1989). According to Stille and Palmstrom (2003), the use
of the term classification in various ways has led to
confusion when the rules and roles of classification are
discussed. They declared that the meaning of classification
is different from what is usually used in rock engineering
and design.
As briefly mentioned by Stille and Palmstrom (2003),
the requirements to build up such a system to be able to
adequately solve rock engineering problems include:
J. Khademi Hamidi (&) K. Shahriar B. RezaiDepartment of Mining, Metallurgical and Petroleum
Engineering, Amirkabir University of Technology,
Hafez 424, P.O. Box 15875-4413, Tehran, Iran
e-mail: [email protected]; [email protected]
H. Bejari
Department of Mining, Petroleum and Geophysics Engineering,
Shahrood University of Technology, Shahrood, Iran
123
Rock Mech Rock Eng (2010) 43:335350
DOI 10.1007/s00603-009-0029-1
Use of a supervised classification adapted to the
specific project
The reliability of the classes to handle the given rock
engineering problem must be estimated
The classes must be exhaustive and mutually exclusive
(i.e., every object has to belong to a class and no object
can belong to more than one class)
Establish the principles of the division into classes
based on suitable indicators
The indicators should be related to the different tools
used for the design
The principles of division into classes must be flexible,
so that additional indicators can be incorporated
The principles of division into classes have to be
updated to take account of experiences gained during
the construction
The uncertainties or quality of the indicators must be
established so that the probability of mis-classification
can be estimated
The system should be practical and robust, and give an
economic and safe design
In practice, none of the existing classification systems
fulfil the requirements mentioned above for a true classi-
fication system for rock engineering problems. This may be
due to the fact which was addressed by Williamson and
Kuhn (1988): no classification system can be devised that
deals with all the characteristic of all possible rock material
or rock masses and/or by Riedmuller and Schubert
(1999): complex properties of a rock mass can not suffi-
ciently be described by a single number. So, rock mass
classification systems are to group rocks in such a way that
those parameters which are of the most universal concern
are clearly dealt with.
Rock mass classification schemes have been developing
for over 100 years since Ritter (1879) attempted to for-
malize an empirical approach to tunnel design, in
particular, for determining support requirements. Probably,
the first successful attempt of classifying rock masses for
engineering purposes was the rock-load concept, which
was introduced by Terzaghi (1946). He considered the
structural discontinuities of the rock masses and classified
them qualitatively into nine categories, including: (1) hard
and intact; (2) hard, stratified, and schistose; (3) massive to
moderately jointed; (4) moderately blocky and seamy; (5)
very blocky and seamy; (6) completely crushed but
chemically intact; (7) squeezing rock at moderate depth;
(8) squeezing rock at great depth; and (9) swelling rock.
Lauffer (1958) proposed that the stand-up time for an
unsupported tunnel span is related to the quality of the rock
mass in which the active span is excavated. He classified
tunnel rocks into seven groups according to the stand-up
time concept. Lauffers original classification has since
been modified by a number of authors, notably Pacher et al.
(1974), and now forms part of the general tunneling approach
known as the new Austrian tunneling method (NATM).
Deere et al. (1967) introduced the rock quality desig-
nation (RQD) index for design and characterization
afterwards. The RQD index was developed to provide a
quantitative estimate of rock mass quality from drill logs.
Afterwards, other rock mass classifications (mostly multi-
ple-parameter) have been introduced by other researchers.
Of the existing classification systems, RSR by Wickham
et al. (1972), RMR by Bieniawski (1973, 1974, 1976, 1979,
1989), Q-system by Barton et al. (1974), GSI by Hoek and
Brown (1997), and RMi by Palmstrom (1995) are the most
commonly used in mining and civil fields of application.
It has been experienced repeatedly that, when used
correctly, a rock mass classification can be a powerful tool
in designs. In fact, on many projects, the classification
approach serves as the only practical basis for the design of
complex underground structures.
However, due to the complex nature of the rock masses,
the rock mass classification systems always include some
uncertainties, leading to difficulty in the determination of
rock mass parameters and related ratings used by the sys-
tems as definite values. To minimize the uncertainties,
engineering judgment is commonly used by experienced
engineers.
These challenging uncertainties in rock engineering and
design have always been addressed by different researchers.
Karl Terzaghi in his latest years stated that, the geotech-
nical engineer should apply theory and experimentation but
temper them by putting them into the context of the uncer-
tainty of nature (Palmstrom and Broch 2006). According to
Brekke and Howard (1972), rock masses are so variable in
nature that the chances of ever finding a common set of
parameters and a common set of constitutive equations valid
for all rock masses is quite remote. Nguyen (1985) referred to
the important role of subjective judgment which was nor-
mally prominent in the assessment of the stability of
underground excavations, and, in general, many decision-
making processes in mining geomechanics. Bieniawski
(1989) stated that, unlike other engineering materials, rock
presents the designer with unique problems by being a
complex material varying widely in its properties. Goodman
(1995) declared that, when the materials are natural rock, the
only thing known with certainty is that this material will
never be known with certainty. Alvarez Grima (2000)
expressed that, in comparison to many other civil engineer-
ing situations, the uncertainties in underground rock
engineering are high. He, therefore, called rock engineering
classifications complex and ill-defined systems. Swart
et al. (2005) indicated that the rock engineering challenge is
to convince management to minimize the uncertainty by
336 J. Khademi Hamidi et al.
123
spending money on geotechnical investigations and by col-
lecting more geotechnical data.
Generally speaking, most geosciences suffer from insuf-
ficient data. On the other hand, in most engineering design
and characterization works, the value of the required vari-
ables changes frequently in short intervals. These are the
reasons why the ideas of some experts should be taken into
consideration during the design process in most geo-related
practices. In other words, the subjective judgment of the
engineer during characterizing a rock mass or designing a
tunnel support system is risky to accept. These all impose an
uncertainty and imprecision in such engineering fields.
According to Zadeh (2006), uncertainty is an unavoidable
attribute of information. By using the rules of probability,
scientists were capable of dealing with such uncertainties in
information. With fuzzy set theory coming into existence, it
is done better by fuzzy logic. Zadeh (2008), in answering the
question is there a need for fuzzy logic?, believes that,
today, close to four decades after its conception, fuzzy logic
is a precise logic of imprecision and approximate reasoning,
which shows itself to be more effective than an attempt at the
formalization/mechanization of human reasoning capabili-
ties. Fuzzy sets theory, as a soft computing technique, has
established itself as a new methodology for dealing with any
sort of ambiguity and uncertainty. Soft computing, as
introduced by Zadeh (1992), includes approaches to human
reasoning, which try to make use of the human tolerance for
incompleteness, uncertainty, imprecision, vagueness, and
fuzziness in decision-making problems (Jang et al. 1997).
With the spread of fuzzy set applications in many areas
of engineering that are sufficiently modeled by conven-
tional deterministic and probabilistic analyses, it can also
be recognized that there are also many classes of problems
in rock engineering that are suitably receptive to fuzzy set
applications. As was stated by Nguyen (1985), the key to
such application lies in the inevitability of subjective
uncertainty being involved in many decision-making pro-
cesses, whereas the main advantage of fuzzy set application
is the incorporation of expert knowledge.
This paper proposes the application of fuzzy set theory
in assisting engineers in the rock engineering decision
processes for which subjectivity plays an important role.
Also, as an example, the applicability of fuzzy set theory to
the newly developed rock mass excavability (RME) index
for tunneling technique selection is illustrated.
2 Application Potential of Fuzzy Set Theory to Rock
Engineering Classification Systems
Classification systems have recently become quite popular
and are widely employed in rock engineering. This may be
due to the following reasons (Singh and Goel 1999):
They provide better communication between geolo-
gists, designers, contractors, and engineers
Engineers observations, experience, and judgment are
correlated and consolidated more effectively by a
quantitative classification system
Engineers prefer numbers in place of descriptions,
hence, a quantitative classification system has consid-
erable application in an overall assessment of rock
quality
A classification approach helps in the organization of
knowledge
Despite their widespread use, the currently used clas-
sification systems have some deficiencies in practical
applications. The most common disadvantages are its
subjective uncertainties resulting from the linguistic input
value of some parameters, low resolution, fixed weight-
ing, sharp class boundaries, etc. Figure 1 illustrates the
procedures for the measurement and calculation of the
RQD. The relationship between RQD and the engineering
quality of rock mass as proposed by Deere (1968) is
given in Table 1. A close examination of the table reveals
that there are some uncertainties on data that are close to
the range boundaries of rock classes. For example, it is
not clear whether a rock having an RQD index of 50%
will be included in Class 2 or 3, leading to subjective
decision-making. The other limitation of this classification
is the decisive length of 10-cm (4-in) core pieces in the
determination of the RQD. For example, suppose a
borehole is drilled in a rock mass with a joint spacing of
9 and 11 cm. The RQD values will be 0 and 100%,
respectively, if other conditions have no contribution to
the formation of core pieces. The RQD is employed
Fig. 1 Procedure for the measurement and calculation of the rockquality designation (RQD) (after Deere 1989)
Application of Fuzzy Set Theory to Rock Engineering Classification Systems 337
123
during the determination of some other rock mass clas-
sifications, such as RMR and Q.
Sometimes, the problems arise from the rating on each
input parameter being a fixed numerical score for a given
rock class interval. This causes the engineer to apply the
same numerical scores in the regions close to the lower and
upper boundaries of a given class. Table 2 illustrates the
RMR system measured for two rock masses. Upon
assigning such ratings, each input parameter from the
tables were given for the calculation of the RMR
(Bieniawski 1989), and a situation is reached where the
same rock mass class and average stand-up time is attrib-
uted for both rock masses. However, from the point of view
of an experienced field engineer, it is expected that the
quality of Rock mass 2 is much more than that of Rock
mass 1.
Another deficiency of such a classification scheme is the
existence of sharp transitions between two adjacent classes.
For example, in Table 3, the determining RMR values
between Rock mass class I and Rock mass class II is 81 and
80, respectively. Consequently, for the rating difference of
only 1 (i.e. 81 - 80), the average stand-up time of 10 years
for a 15-m span is determined for a tunnel roof having a
rock mass rating of 81, while an average stand-up time of
6 months for an 8-m span is determined for a tunnel roof
having a rock mass rating of 80. Such a rating procedure
employing sharp transitions between classes exhibits
uncertainties in the assessment of rock mass classes and
related design parameters, as the transitions between rock
classes are not so sharp but gradational in the field. In such
cases, it is imperative that an engineering judgment be
made for a final decision on subsequent design parameters,
such as the support system.
The above-mentioned uncertainties will be encountered
in the practical application of such rock engineering clas-
sification systems. It deserves mention that, although
careful consideration has been given to the precise wording
for each category and to the relative weights assigned to
each combination of involved parameters in rock
Table 1 Correlation between the rock quality designation (RQD) androck mass quality (Deere 1968)
Class no. RQD (%) Rock quality
1 \25 Very poor2 2550 Poor
3 5075 Fair
4 7590 Good
5 90100 Excellent
Table 2 Comparison between the two different rock masses in terms of RMR according to Bieniawski (1989)
RMR input parameters Rock mass properties Ratings
Rock mass 1 Rock mass 2 Rock mass 1 Rock mass 2
Point load index (MPa) 4 7 12 12
RQD (%) 51 74 13 13
Spacing of discontinuities (m) 0.22 0.59 10 10
Condition of discontinuities Rough and slightly weathered,
wall rock surface separation
\1 mm
Rough and slightly weathered,
wall rock surface separation
\1 mm
25 25
Inflow per 10 m tunnel length (l/min) 24 11 7 7
Joint orientation adjustment (for tunnel) Very favorable Very favorable 0 0
RMR 67 67
Table 3 Design parameters and engineering properties of rock mass (Bieniawski 1989)
Properties of rock mass Rock mass rating (rock class)
10081 (I) 8061 (II) 6041 (III) 4021 (IV) \20 (V)
Classification of rock mass Very good Good Fair Poor Very poor
Average stand-up time 10 years for
15-m span
6 months for
8-m span
1 week for
5-m span
10 h for
2.5-m span
30 min for
1-m span
Cohesion of rock mass (MPa) [0.4 0.30.4 0.20.3 0.10.2 \0.1Angle of internal friction
of rock mass
[45 3545 2535 1525 15
338 J. Khademi Hamidi et al.
123
engineering classification systems, their use involves some
subjectivity. For instance, as was addressed by Nguyen
(1985) in the RMR system, the ratings for criteria on dis-
continuities and groundwater conditions are largely
subjective. Hence, good experiences and sound judgment is
required to successfully employ these rock engineering
classifications. However, this may be considered as a
serious problem, particularly for young engineers with
limited experience.
Over the years, fuzzy set theory has shown itself to be an
appropriate alternative for engineering judgment to cope
with the uncertainties encountered in decision-making
processes. Up to now, many researchers have studied the
potential application of fuzzy set theory to rock engineer-
ing classification systems. The first attempt for the
application of fuzzy set theory in rock engineering classi-
fication systems after fuzzy theory came into existence was
carried out by Nguyen (1985) and Nguyen and Ashworth
(1985) in rock mass classification based on the minmax
aggregation operation proposed by Bellman and Zadeh
(1970) for multi-criteria decision modeling. This approach,
which aims at the selection of the most likely rock mass
class, was used in the RMR and Q classification systems.
The subjective judgment in the decision-making process
for the design and characterization of geomechanical- and
geotechnical-related projects was addressed as the main
reason for the application of fuzzy theory in mining and
civil engineering fields. Juang and Lee (1990) applied
fuzzy set theory to the RMR system by aggregating the
individual fuzzy ratings of different criteria into an overall
classification rating. Habibagahi and Katebi (1996) also
modified the RMR classification system using fuzzy set
theory. In their model, each of the numerical RMR criteria
(UCS, RQD, and JS) is fuzzified by five trapezoidal
membership functions defined over the universal domain of
the criterion in question. Gokay (1998) applied the fuzzy
logic concept to the weightings of the Q classification
system proposed by Barton et al. (1974). Sonmez et al.
(2003) applied fuzzy set theory to the geological strength
index (GSI), which is used as an input parameter in the
HoekBrown failure criterion to handle the uncertainties
involved in the characterization of rock masses.
A rock mass classification approach was made by
Aydin (2004) based on the concept of partial fuzzy sets
representing the variable importance of each parameter in
the universal domain of rock mass quality. Use of the
partial set concept was shown to be capable of expressing
the variability of rock quality conditions due to all types
of non-random uncertainties or fuzziness in a rock mass
classification process. Iphar and Goktan (2006) applied
fuzzy sets to the diggability index rating method for
surface mine equipment selection. The diggability index
rating method devised by Scoble and Muftuoglu (1984)
defines seven rock excavation classes based on four
geotechnical parameters, namely, uniaxial compressive
strength, bedding spacing, joint spacing, and weathering.
More recently, Khademi Hamidi et al. (2007a, b) applied
fuzzy set theory to RME classification. This classification
system is discussed in detail in Sect. 4.
Fuzzy set theory has also been used for the construc-
tion of prediction models in engineering geological and
rock mechanics problems. Sakurai and Shimizu (1987)
employed fuzzy set theory for the assessment of rock
slope stability. They proposed a classification for evalu-
ating the stability of slopes on the basis of the fuzzified
factor of safety (FOS), defined as a trapezoidal member-
ship function. They also consider that, in general, many
failed slopes fall into the fair class. Hence, they sug-
gested engineering judgment while interpreting the FOS
fuzzy set. Ghose and Dutta (1987) developed a classifi-
cation model to assess the cavability of a coal mines roof
using fuzzy set theory. Alvarez Grima and Babuska
(1999) employed a fuzzy model to predict the uniaxial
compressive strength of various rocks, where it was
concluded that the fuzzy model performed better than the
conventional multilinear regression model. Li and Tso
(1999) used a fuzzy classification method to classify the
tool wear states so as to facilitate defective tool replace-
ment at the proper time in drilling. A tool wear estimation
model was also developed by Yao et al. (1999) using
fuzzy logic and a neural network approach by utilizing
data obtained from cutting tests. Wu et al. (1999) sug-
gested a fuzzy probability model to describe the damage
threshold of a rock mass under explosive loads. Finol
et al. (2001) developed a fuzzy model for the prediction
of petrophysical rock parameters. Using a fuzzy rule-
based expert system, Klose (2002) constructed a predic-
tion model for short-range seismicity by interpreting those
seismic images. Gokceoglu (2002) developed a fuzzy
triangular chart to predict the uniaxial compressive
strength of the Ankara agglomerates from their petro-
graphic composition. A comparative study was carried out
by Kayabasi et al. (2003) for estimating the deformation
modulus of rock masses through fuzzy and multiple
regression models. They constructed three prediction
models, namely, simple regression, multiple regression,
and fuzzy inference system (FIS), for the indirect esti-
mation of the modulus of deformation of rock masses and
reported that FIS provided more reliable results than those
of the others. Wei et al. (2003) proposed a fuzzy ranking
model to predict the sawability of granites with the use of
petrographic analyses and mechanical property testing.
Lee et al. (2003) developed a fuzzy model to estimate
rock mass properties, including deformation modulus,
cohesion, and friction angle. Gokceoglu and Zorlu (2004)
developed a fuzzy model to predict the uniaxial
Application of Fuzzy Set Theory to Rock Engineering Classification Systems 339
123
compressive strength and the modulus of elasticity of a
problematic rock. The model includes four inputs,
namely, P-wave velocity, block punch index, point load
index, and tensile strength, and two outputs, namely,
uniaxial compressive strength and the modulus of elas-
ticity. The comparative study of fuzzy and multiple
regression predictive models showed that the prediction
performances of the fuzzy model are higher than those of
multiple regression equations. Chen and Liu (2007) and
Liu and Chen (2007) combined the analytic hierarchy
process (AHP) and the fuzzy Delphi method (FDM) for
assessing the ratings of rock mass quality for the cases of
tunnel and rock slope stability analyses. They considered
the rock mass classification as a group decision problem
and applied fuzzy logic theory as the criterion to calculate
the weighting of factors. The results of this analysis
showed that this model can provide a more quantitative
measure of rock mass quality and, hence, minimize
judgmental bias. Recently, Acaroglu et al. (2008) con-
structed a fuzzy logic model to predict the specific energy
requirement for TBM performance prediction. The model
includes six rock- and machine-related input parameters,
namely, uniaxial compressive strength, Brazilian tensile
strength, disc dimensions such as disc diameter and tip
width, and cutting geometry such as spacing and
penetration.
Fuzzy set theory has also been used for the performance
prediction of tunnel and trench excavation machines (Den
Hartog et al. 1997; Deketh et al. 1998; Alvarez Grima and
Verhoef 1999; Alvarez Grima 2000; Alvarez Grima et al.
2000).
The above-mentioned literature indicates that fuzzy set
theory is about to establish itself as a reliable new meth-
odology for dealing with any sort of ambiguity and
uncertainty which can be inevitably found in engineering
geology and rock mechanics-related projects.
This paper discusses the applicability of fuzzy set theory
to rock engineering classification systems, with particular
illustration of the RME index, which was newly developed
based on a rating system similar to other classification
systems. So, first, the basic principles of fuzzy set theory
are described.
3 Fuzzy Set Theory
Fuzzy theory started with the concept of fuzziness and its
expression in the form of fuzzy sets was introduced by
Zadeh (1965). Fuzzy set theory provides the means for
representing uncertainty using set theory. A fuzzy set is an
extension of the concept of a crisp set. A crisp set only
allows full membership or no membership to every element
of a universe of discourse, whereas a fuzzy set allows for
partial membership. The membership or non-membership
of an element x in the crisp set A is represented by the
characteristic function of lA, defined by:
lA x 1 if x 2 A0 if x 62 A
Fuzzy sets generalize this concept to partial membership
by extending the range of variability of the characteristic
function from the two-point set {0, 1} to the whole interval
[0, 1]:
lA : U ! 0; 1 where U refers to the universe of discourse defined for a
specific problem. If U is a finite set U = {x1, x2,, xn},then a fuzzy set A in this universe U can be represented by
listing each element and its degree of membership in the
set A as:
A lA x1 =x1; lA x2 =x2; . . .; lA xn =xnf gAccording to the International Society for Rock
Mechanics (ISRM), rocks with uniaxial compressive
strength between 50 and 100 MPa belong to the Hard
Rock class. Figure 2 compares two crisp and fuzzy models
of the hard rock set. As can be followed from Fig. 2, the
belonging of a rock to the fuzzy set of hard rock is quite
different from that of the classical one (crisp set).
3.1 Membership Function
An element of the variable can be a member of the fuzzy
set through a membership function that can take values in
the range from 0 to 1. Membership functions (MF) can
either be chosen by the user arbitrarily, based on the users
experience (MF chosen by two users could be different
depending upon their experiences, perspectives, etc.) or can
also be designed using machine learning methods (e.g.,
artificial neural networks, genetic algorithms, etc.).
There are different shapes of membership functions;
triangular, trapezoidal, piecewise-linear, Gaussian, bell-
shaped, etc. In this study, triangular and trapezoidal
membership functions are used. Triangular and trapezoidal
MFs are shown in Fig. 3.
In Fig. 3, points a, b, and c in the triangular MF repre-
sent the x coordinates of the three vertices of lA(x) in afuzzy set A (a: lower boundary and c: upper boundary
8800 MMPPaa
7700 MMPPaa9955 MMPPaa5555 MMPPaa
4400 MMPPaa
111100 MMPPaa
112200 MMPPaa
3300 MMPPaa
8800 MMPPaa7700 MMPPaa
9955 MMPPaa5555 MMPPaa
4400 MMPPaa
111100 MMPPaa
112200 MMPPaa
3300 MMPPaa
Fig. 2 Crisp (left) and fuzzy (right) models of the hard rock set
340 J. Khademi Hamidi et al.
123
where the membership degree is zero, b: the center where
membership degree is 1).
The belonging of an element to a definite set in the
method of fuzzy membership models (gradual membership
degree) gives the fuzzy sets flexibility in modeling com-
monly used linguistic expressions, such as the uniaxial
compressive strength of rock is high or low water
inflow, which are frequently used in rock engineering
classification systems.
3.2 Fuzzy ifthen Rules
Like most classical expert systems, fuzzy logic has an
expert and implication logic behind it. This expert system
is constructed by using ifthen rules. The fuzzy rules
provide a system for describing complex (uncertain, vague)
systems by relating input and output parameters using
linguistic variables. A fuzzy ifthen rule assumes the form
if x is A then y is B, where A and B are linguistic values
defined by fuzzy sets on universes of discourse X and Y,
respectively. Often x is A is called the antecedent or
premise, while y is B is called the consequence or
conclusion. Examples of fuzzy ifthen rules are widespread
in daily linguistic expressions in rock engineering designs,
such as if quartz content is high, then disc cutter life is
low.
Each rule in a fuzzy model is a relation such as
Ri = (X 9 Y ? [0, 1]), which is calculated by using thefollowing equation (Alvarez Grima 2000):
lRi x; y I lAi x ; lBi y where lRi(x, y) is the R relations membership degree ofrule i according to x and y inputs, lAi(x) and lBi(y) are themembership degrees of x and y inputs, respectively, and I
denotes the AND or OR operator.
Most rule-based systems involve more than one rule.
The process of obtaining the overall consequent (conclu-
sion) from the individual consequents contributed by each
rule in the rule base is known as the aggregation of rules. In
determining an aggregation strategy, two simple extreme
cases exist, namely, conjunctive and disjunctive system of
rules by using AND and OR connectives, respectively
(Ross 1995).
3.3 Fuzzy Inference System (FIS)
Fuzzy inference is the process of formulating an input
fuzzy set map to an output fuzzy set using fuzzy logic. In
fact, the core section of a fuzzy system is the FIS part,
which combines the facts obtained from the fuzzification
with the rule base and conducts the fuzzy reasoning
process.
Generally, the basic structure of a FIS consists of three
conceptual components, which are rule base, database, and
reasoning mechanism. A rule base contains a selection of
fuzzy rules and a database defines the membership func-
tions used in the fuzzy rules. A reasoning mechanism
performs the fuzzy reasoning based on the rules and given
facts to derive a reasonable output or conclusion.
There are several FISs that have been employed in
various applications, such as the Mamdani fuzzy model,
TakagiSugenoKang (TSK) fuzzy model, Tsukamoto
fuzzy model, and Singleton fuzzy model. Among different
FISs, the Mamdani algorithm is one of the most used fuzzy
models to apply in complex engineering geological prob-
lems, since most geological processes are defined with
linguistic variables or simple vague predicates. The
Mamdani FIS was proposed by Mamdani to control a
steam engine and boiler combination by a set of linguistic
control rules obtained from experienced human operators
(Mamdani and Assilian 1975).
The general ifthen rule structure of the Mamdani
algorithm is given in the following equation:
if x1 is Ai1 and x2 is Ai2 and . . .xr is Air then y is Bifor i 1; 2; . . .; k
where k is the number of rules, xi is the input variable
(antecedent variable), Air and Bi are linguistic terms or
fuzzy sets which are defined by the membership functions
Air(xr) and Bi, and y is the output variable (consequent
variable).
Figure 4 is an illustration of a two-rule Mamdani FIS
which derives the overall output z when subjected to two
crisp inputs x and y (Jang et al. 1997).
As shown in Fig. 4, the fuzzy output is the aggregation
(max) of the two truncated fuzzy sets. The outputs are
obtained after defuzzification by using the centroid of area
(COA) method.
>
/GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 599 /MonoImageMinResolutionPolicy /Warning /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False
/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice