fuzzy theory 1

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


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


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

fuzzy theory 1

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