siti zaiton - fuzzy1

8/13/2019 Siti Zaiton - Fuzzy1

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FUZZY LOGIC & FUZZY SETS

Siti Zaiton Mohd Hashim, PhD


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Fuzzy logicFuzzy logic

Introduction:Introduction: what is fuzzy thinking?what is fuzzy thinking?

Fuzzy setsFuzzy sets

Linguistic variables and hedgesLinguistic variables and hedges

Operations of fuzzy setsOperations of fuzzy sets

Fuzzy rulesFuzzy rules

SummarySummary


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What is fuzzy thinking?What is fuzzy thinking?

Experts rely onExperts rely on common sensecommon sense when they solvewhen they solveproblems.problems.

How can we represent expert knowledge thatHow can we represent expert knowledge that

uses vague and ambiguous terms in a computer?uses vague and ambiguous terms in a computer?

Fuzzy logic is not logic that is fuzzy, but logic thatFuzzy logic is not logic that is fuzzy, but logic that

is used to describe fuzziness. Fuzzy logic is theis used to describe fuzziness. Fuzzy logic is the

theory of fuzzy sets, sets that calibrate vagueness.theory of fuzzy sets, sets that calibrate vagueness.

Fuzzy logic is based on the idea that all thingsFuzzy logic is based on the idea that all thingsadmit of degrees. Temperature, height, speed,admit of degrees. Temperature, height, speed,

distance, beautydistance, beauty all come on a sliding scale. Theall come on a sliding scale. The

motor is runningmotor is running really hotreally hot. Tom is a. Tom is a very tallvery tall guy.guy.


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Boolean logic uses sharp distinctions. It forces usBoolean logic uses sharp distinctions. It forces us

to draw lines between members of a class and nonto draw lines between members of a class and non--

members. For instance, we may say, Tom is tallmembers. For instance, we may say, Tom is tall

because his height is 181 cm. If we drew a line atbecause his height is 181 cm. If we drew a line at

180 cm, we would find that David, who is 179 cm,180 cm, we would find that David, who is 179 cm,

is short. Is David really a short man or we haveis short. Is David really a short man or we have

just drawn an arbitrary line in the sand?just drawn an arbitrary line in the sand?

Fuzzy logic reflects how people think. It attemptsFuzzy logic reflects how people think. It attemptsto model our sense of words, our decision makingto model our sense of words, our decision making

and our common sense. As a result, it is leading toand our common sense. As a result, it is leading to

new, more human, intelligent systems.new, more human, intelligent systems.


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In 1965In 1965 LotfiLotfi ZadehZadeh, published his famous paper, published his famous paper

Fuzzy sets.Fuzzy sets.

ZadehZadeh extended the work on possibility theory intoextended the work on possibility theory into

a formal system of mathematical logic, anda formal system of mathematical logic, and

introduced a new concept for applying naturalintroduced a new concept for applying natural

language terms.language terms.

This new logic for representing and manipulatingThis new logic for representing and manipulatingfuzzy terms was calledfuzzy terms was called fuzzy logicfuzzy logic, and, and ZadehZadeh

became thebecame the Master/FatherMaster/Father ofoffuzzy logicfuzzy logic..


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Why fuzzy?Why fuzzy?

AsAs ZadehZadeh said, the term is concrete, immediate andsaid, the term is concrete, immediate and

descriptive.descriptive. However, many people in the WestHowever, many people in the West

were repelled by the wordwere repelled by the wordfuzzyfuzzy, because it is, because it isusually used in a negative sense.usually used in a negative sense.

Why logic?Why logic?

Fuzziness rests on fuzzy set theory, and fuzzy logicFuzziness rests on fuzzy set theory, and fuzzy logicis just a small part of that theory.is just a small part of that theory. ZadehZadeh used theused the

term fuzzy logic in a broader sense.term fuzzy logic in a broader sense.


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Fuzzy logic is a set of mathematical principlesFuzzy logic is a set of mathematical principles

for knowledge representation based on degreesfor knowledge representation based on degrees

of membershipof membership rather than on crisp membership ofrather than on crisp membership of

classical binary logicclassical binary logic..

Unlike twoUnlike two--valued Boolean logic, fuzzy logic isvalued Boolean logic, fuzzy logic is

multimulti--valuedvalued. It deals with. It deals with degrees ofdegrees of

membershipmembership andand degrees of truthdegrees of truth..

What is fuzzy logic?What is fuzzy logic?


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Range of logical values inRange of logical values in

Boolean and fuzzy logicBoolean and fuzzy logic

(a) Boolean Logic. (b) Multi-valued Logic.

0 1 10 0.2 0.4 0.6 0.8 100 1 10

Fuzzy logic uses the continuum of logical values between 0Fuzzy logic uses the continuum of logical values between 0

(completely false) and 1 (completely true).(completely false) and 1 (completely true).Instead of just black and white, it employs the spectrum ofInstead of just black and white, it employs the spectrum of

colours, accepting that things can be partly true and partlycolours, accepting that things can be partly true and partly

false at the same time.false at the same time.


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Fuzzy setsFuzzy sets

The concept of aThe concept of a setset is fundamental to mathematics.is fundamental to mathematics.

Crisp set theory is governed by a logic that uses one ofCrisp set theory is governed by a logic that uses one of

only two values: true or false.only two values: true or false.

This logic cannot represent vague concepts, andThis logic cannot represent vague concepts, and

therefore fails to give the answers on the paradoxes.therefore fails to give the answers on the paradoxes.

In fuzzy set theory: an element is with a certain degreeIn fuzzy set theory: an element is with a certain degree

of membership.of membership.

Thus, a proposition is not either true or false, butThus, a proposition is not either true or false, butmay be partly true (or partly false) to any degree.may be partly true (or partly false) to any degree.

This degree is usually taken as a real number in theThis degree is usually taken as a real number in the

interval [0,1].interval [0,1].


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The classical example in fuzzy sets isThe classical example in fuzzy sets is tall mentall men. The. Theelements of the fuzzy set tall men are all men,elements of the fuzzy set tall men are all men,

but their degrees of membership depend on theirbut their degrees of membership depend on their

height.height.D e g re e o f M e m b e rs h ip

F u zzy

M a r k

J o h n

T o m

B o b

B i ll

1

1

1

00

1 . 0 0

1 . 0 0

0 . 9 8

0 . 8 2

0 . 7 8

Pe t e r

S t e v e n

M ik eD a v i d

C h r i s

C ris p

1

0

0

0

0

0 . 2 4

0 . 1 5

0 . 0 6

0 . 0 1

0 . 0 0

N a m e H e ig h t, c m

2 0 5

1 9 8

1 8 1

1 6 7

1 5 5

1 5 2

1 5 8

1 7 21 7 9

2 0 8


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D e g re e o f M e m b e rs h ip

F u zzy

M a r k

J o h n

T o m

B o b

B i ll

1

1

1

0

0

1 . 0 0

1 . 0 0

0 . 9 8

0 . 8 2

0 . 7 8

Pe t e r

S t e v e n

M ik e

D a v i d

C h r i s

C ris p

1

0

0

0

0

0 . 2 4

0 . 1 5

0 . 0 6

0 . 0 1

0 . 0 0

N a m e H e ig h t, c m

2 0 5

1 9 8

1 8 1

1 6 7

1 5 5

1 5 2

1 5 8

1 7 2

1 7 9

2 0 8

Crisp set asks the question: Is the man tall?Crisp set asks the question: Is the man tall?

Tall men are above 180, and not tall men are below 180.Tall men are above 180, and not tall men are below 180.

Fuzzy set asks the question: How tall is the man?Fuzzy set asks the question: How tall is the man?

The tall is partial membership in the fuzzy set, Tom is 0.82The tall is partial membership in the fuzzy set, Tom is 0.82

tall.tall.


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150 210170 180 190 200160

Height, cmDegree ofMembership

Tall Men

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degree ofMembership

170

1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

Fuzzy Sets

Crisp Sets

Crisp and fuzzy sets of Crisp and fuzzy sets of tall mentall men


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TheThexx--axis represents theaxis represents the universe of discourseuniverse of discourse the range ofthe range of

all possible values applicable to a chosen variable.all possible values applicable to a chosen variable.

The variable is the mans height. According to thisThe variable is the mans height. According to this

representation, the universe of mens heights consists of all tallrepresentation, the universe of mens heights consists of all tallmen.men.

TheTheyy--axis represents theaxis represents the membership value of the fuzzy setmembership value of the fuzzy set..

The fuzzy set of The fuzzy set of tall mentall men maps height values into maps height values into

corresponding membership values.corresponding membership values.

Fuzzy sets of Fuzzy sets of tall mentall men


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LetLetXXbe thebe the universe of discourseuniverse of discourse and its elements be denotedand its elements be denoted

asasxx. In the classical set theory,. In the classical set theory, crisp setcrisp setAA ofofXXis defined asis defined as

functionfunctionffAA((xx)) called the characteristic function ofcalled the characteristic function ofAA

ffAA((xx))::XX {{00,, 11},}, wherewhere

Ax

AxxfA

if0,

if1,)(

This set maps universeThis set maps universeXXto a set of two elements. For anyto a set of two elements. For any

elementelementxx of universeof universeXX, characteristic function, characteristic functionffAA((xx) is equal to 1) is equal to 1ififxx is an element of setis an element of setAA, and is equal to 0 if, and is equal to 0 ifxx is not an elementis not an element

ofofAA..

Crisp set definitionCrisp set definition


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A fuzzy set is a set with fuzzy boundaries.A fuzzy set is a set with fuzzy boundaries.

In the fuzzy theory, fuzzy setIn the fuzzy theory, fuzzy setAA of universeof universeXXis definedis defined

by functionby function AA((xx) called the) called the membership functionmembership function of setof set

AA

AA((xx))::XX [[00,, 11],], wherewhere AA((xx)) == 11 ififxx isis totallytotally ininAA;;

AA((xx)) == 00 ififxx isis notnot ininAA;;

0


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AA((xx))::XX [[00,, 11],], wherewhere AA((xx)) == 11 ififxx isis totallytotally ininAA;;

AA((xx)) == 00 ififxx isis notnot ininAA;;

0


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How to represent a fuzzy set inHow to represent a fuzzy set in

a computer?a computer?

First, determine the membership functions.First, determine the membership functions. In In tall mentall men example, the fuzzy sets of example, the fuzzy sets of talltall,,

shortshortandand averageaverage men, can be obtained.men, can be obtained.

The universe of discourseThe universe of discourse the mens heightsthe mens heights consists of three sets:consists of three sets: shortshort,, averageaverage andand tall mentall men..

As shown in the following figure, a man who isAs shown in the following figure, a man who is

184 cm tall is a member of the184 cm tall is a member of the average menaverage men setset

with a degree of membership of 0.1, and at thewith a degree of membership of 0.1, and at thesame time, he is also a member of thesame time, he is also a member of the tall mentall men

set with a degree of 0.4.set with a degree of 0.4.


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Crisp and fuzzy sets of short, averageCrisp and fuzzy sets of short, average

and tall menand tall men

150 210170 180 190 200160

Heig ht , cmDegree ofMembership

Tall Men

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degree ofMembership

Short Average ShortTall

170

1.0

0.0

0.2

0.4

0.6

0.8

Fuzzy Sets

Crisp Sets

Short Average

Tall

Tall


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Representation of crisp andRepresentation of crisp and

fuzzy subsetsfuzzy subsets

Fuzzy SubsetA

Fuzziness

1

0Crisp SubsetA Fuzziness

x

X(x)

Typical functions that can be used to represent a fuzzyTypical functions that can be used to represent a fuzzyset areset are sigmoid, gaussiansigmoid, gaussian andandpipi. However, these. However, these

functions increase the time of computation. Therefore,functions increase the time of computation. Therefore,

in practice, most applications usein practice, most applications use linear fit functionslinear fit functions..


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Representation of crisp andRepresentation of crisp and

fuzzy subsetsfuzzy subsets

The above figure can be represented as aThe above figure can be represented as a fitfit--vectorvector::

tall mentall men = (0/180, 0.5/185, 1/190)= (0/180, 0.5/185, 1/190)

average menaverage men = (0/165, 1/175, 0/185)= (0/165, 1/175, 0/185)

short menshort men = (1/160, 0.5/165, 0/170)= (1/160, 0.5/165, 0/170)


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Linguistic variablesLinguistic variables

At the root of fuzzy set theory lies the idea ofAt the root of fuzzy set theory lies the idea of

linguistic variables.linguistic variables.

A linguistic variable is a fuzzy variable.A linguistic variable is a fuzzy variable.

For example, the statement John is tall impliesFor example, the statement John is tall impliesthat the linguistic variablethat the linguistic variableJohnJohn takes thetakes the

linguistic valuelinguistic value talltall..


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In fuzzy expert systems, linguistic variables are usedIn fuzzy expert systems, linguistic variables are used

in fuzzy rules. For example:in fuzzy rules. For example:

IFIF wind is strongwind is strong

THENTHEN sailingsailing isis goodgood

IFIF project_durationproject_duration isis longlong

THENTHEN completion_riskcompletion_risk isis highhigh

IFIF speedspeed isis slowslow

THENTHEN stopping_distancestopping_distance isis shortshort



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The range of possible values of a linguistic variableThe range of possible values of a linguistic variable

represents the universe of discourse of that variable.represents the universe of discourse of that variable.

For example, the universe of discourse of theFor example, the universe of discourse of the

linguistic variablelinguistic variable speedspeedmight have the rangemight have the rangebetween 0 and 220 km/h and may include suchbetween 0 and 220 km/h and may include such

fuzzy subsets asfuzzy subsets as very slowvery slow,, slowslow,, mediummedium,,fastfast, and, and

very fastvery fast..



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A linguistic variable carries with it the concept ofA linguistic variable carries with it the concept of

fuzzy set qualifiers, calledfuzzy set qualifiers, called hedgeshedges..

Hedges are terms that modify the shape of fuzzyHedges are terms that modify the shape of fuzzy

sets. They include adverbs such assets. They include adverbs such as veryvery,,somewhatsomewhat,, quitequite,,more or lessmore or less andandslightlyslightly..

Hedges in Fuzzy LogicHedges in Fuzzy Logic


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Fuzzy sets with the hedgeFuzzy sets with the hedge veryvery

Short

Very Tall

ShortTall

Degree ofMembership

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160 170

Height, cm

Average

TallVery Short Very Tall


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Representation of hedges inRepresentation of hedges in

fuzzy logicfuzzy logic

Hedge MathematicalExpression

A little

Slightly

Very

Extremely

Hedge MathematicalExpression

Graphical Representation

[A(x)]1.3

[A(x)]1.7

[A

(x)]2

[A(x)]3


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Representation of hedges inRepresentation of hedges in

fuzzy logic (continued)fuzzy logic (continued)

Hedge MathematicalExpressionHedge MathematicalExpression Graphical Representation

Very very

More or less

Indeed

Somewhat

2 [A(x)]2

A(x)

A(x)

if 0 A0.5

if 0.5 < A1

1 2 [1 A(x)]2

[A(x)]4


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Operations of fuzzy setsOperations of fuzzy sets

The classical set theory developed in the late 19thThe classical set theory developed in the late 19thcentury by Georg Cantor describes how crisp sets cancentury by Georg Cantor describes how crisp sets can

interact. These interactions are calledinteract. These interactions are called operationsoperations..


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Cantors setsCantors sets

Intersection Union

Complement

Not A

A

Containment

AA

B

BA BAA B


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CrispCrisp SetsSets:: WhoWho doesdoes notnot belongbelong toto thethe set?set?FuzzyFuzzy SetsSets:: HowHow muchmuch dodo elementselements notnot belongbelong toto thethe set?set?

The complement of a set is an opposite of this set.The complement of a set is an opposite of this set.

For example, if we have the set ofFor example, if we have the set of tall mentall men, its complement, its complement

is the set ofis the set ofNOT tall menNOT tall men. When we remove the tall men. When we remove the tall men

set from the universe of discourse, we obtain theset from the universe of discourse, we obtain thecomplement.complement.

IfIfAA is the fuzzy set, its complementis the fuzzy set, its complement AA can be found ascan be found as

follows:follows:

AA((xx)) == 11 AA((xx))

QuestionQuestion:: IfIf thethe fuzzyfuzzy setset ofoftalltall menmen isis asas follows,follows, whatwhat isis itsits

complement?complement?

talltall menmen == ((00//180180,, 00..2525//182182..55,, 00..55//185185,, 00..7575//187187..55,, 11//190190))

ComplementComplement


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CrispCrisp SetsSets:: WhichWhich setssets belongbelong toto whichwhich otherother sets?sets?FuzzyFuzzy SetsSets:: WhichWhich setssets belongbelong toto otherother sets?sets?

A set can contain other sets. The smaller set is called theA set can contain other sets. The smaller set is called the

subsetsubset..

For example, the set ofFor example, the set of tall mentall men contains all tall men;contains all tall men; veryvery

tall mentall men is a subset ofis a subset of tall mentall men. However, the. However, the tall mentall men set isset isjust a subset of the set ofjust a subset of the set of menmen..

In crisp sets, all elements of a subset entirely belong to a largerIn crisp sets, all elements of a subset entirely belong to a larger

set.set.

In fuzzy sets, however, each element can belong less to theIn fuzzy sets, however, each element can belong less to the

subset than to the larger set. Elements of the fuzzy subset havesubset than to the larger set. Elements of the fuzzy subset havesmaller memberships in it than in the larger set.smaller memberships in it than in the larger set.

QuestionQuestion: If the set of: If the set of tall mentall men is as follows, which setsis as follows, which sets

belong tobelong to very tall menvery tall men ?

talltall menmen == ((00//180180,, 00..2525//182182..55,, 00..55//185185,, 00..7575//187187..55,, 11//190190

ContainmentContainment


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CrispCrisp SetsSets:: WhichWhich elementelement belongsbelongs toto bothboth sets?sets?FuzzyFuzzy SetsSets::HowHow muchmuch ofof thethe elementelement isis inin bothboth sets?sets?

In classical set theory, an intersection between two setsIn classical set theory, an intersection between two sets

contains the elements shared by these sets.contains the elements shared by these sets.

For example, the intersection of the set ofFor example, the intersection of the set of tall mentall men and theand the

set ofset offat menfat men is the area where these sets overlap.is the area where these sets overlap.

In fuzzy sets, an element may partly belong to both sets withIn fuzzy sets, an element may partly belong to both sets with

different memberships. A fuzzy intersection is the lowerdifferent memberships. A fuzzy intersection is the lower

membership in both sets of each element.membership in both sets of each element.

The fuzzy intersection of two fuzzy setsThe fuzzy intersection of two fuzzy setsAA andandBB ononuniverse of discourseuniverse of discourseXX::

AABB((xx) =) =minmin [[AA((xx),), BB((xx)] =)] = AA((xx)) BB((xx)),,wherewherexxXX

IntersectionIntersection


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QuestionQuestion: Consider, the set of: Consider, the set of talltall andand average manaverage man are asare asfollows, what is the intersection of these two setsfollows, what is the intersection of these two sets?

tall men =tall men = (0/165, 0/175, 0/180, 0.25/182.5, 0.5/185, 1/190)(0/165, 0/175, 0/180, 0.25/182.5, 0.5/185, 1/190)

Average men =Average men = ((0/165, 1/175, 0.5/180, 0.25/182.5, 0/185,0/165, 1/175, 0.5/180, 0.25/182.5, 0/185,

0/1900/190))

IntersectionIntersection


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CrispCrisp SetsSets:: WhichWhich elementelement belongsbelongs toto eithereither set?set?Fuzzy SetsFuzzy Sets::How much of the element is in either set?How much of the element is in either set?

The union of two crisp sets consists of every element that fallsThe union of two crisp sets consists of every element that falls

into either set.into either set.

For example, the union ofFor example, the union of tall mentall men andandfat menfat men contains allcontains all

men who are tallmen who are tall OROR fat.fat. In fuzzy sets, the union is the reverse of the intersection. That is,In fuzzy sets, the union is the reverse of the intersection. That is,

the union is the largest membership value of the element inthe union is the largest membership value of the element in

either set.either set.

The fuzzy operation for forming the union of two fuzzy setsThe fuzzy operation for forming the union of two fuzzy sets

AA andandBB on universeon universeXXcan be given as:can be given as:

AABB((xx) =) =maxmax [[AA((xx),), BB((xx)] =)] = AA((xx)) BB((xx)),,wherewherexxXX

UnionUnion


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QuestionQuestion: Consider, the set of: Consider, the set of talltall andand average manaverage man are asare asfollows, what is the union of these two setsfollows, what is the union of these two sets?

tall men =tall men = (0/165, 0/175, 0/180, 0.25/182.5, 0.5/185, 1/190)(0/165, 0/175, 0/180, 0.25/182.5, 0.5/185, 1/190)

Average men =Average men = ((0/165, 1/175, 0.5/180, 0.25/182.5, 0/185,0/165, 1/175, 0.5/180, 0.25/182.5, 0/185,

0/1900/190))

UnionUnion


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Fuzzy Logical OperationsFuzzy Logical Operations

FuzzyFuzzy logicallogical reasoningreasoning isis aa supersetsuperset ofof standardstandard BooleanBoolean logiclogic.. InIn otherother words,words, ifif wewe keepkeep thethe fuzzyfuzzy valuesvalues atat theirtheir extremesextremes ofof

11 (completely(completely true),true), andand 00 (completely(completely false),false), standardstandard logicallogical

operationsoperations willwill holdhold..

The standard logical operations:


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InIn FuzzyFuzzy logiclogic thethe truthtruth ofof anyany statementstatement cancan bebe realreal numbersnumbersbetweenbetween 00 andand 11..

QuestionQuestion:: HowHow willwill thesethese truthtruth tablestables bebe altered?altered? WhatWhat

functionfunction willwill preservepreserve thethe resultsresults ofof thethe truthtruth tablestables andand alsoalso

extendextend toto allall realreal numbersnumbers betweenbetween 00 andand 11??


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TheThe truthtruth tablestables isis unchangedunchanged byby thisthis substitutionsubstitution::


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Fuzzy sets andFuzzy sets and

Logical OperationsLogical Operations


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In 1973,In 1973, Lotfi ZadehLotfi Zadehpublished his second mostpublished his second mostinfluential paper. This paper outlined a new approachinfluential paper. This paper outlined a new approach

to analysis of complex systems, in which Zadehto analysis of complex systems, in which Zadeh

suggested capturing human knowledge in fuzzy rules.suggested capturing human knowledge in fuzzy rules.


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What is a fuzzy rule?What is a fuzzy rule?

A fuzzy rule can be defined as a conditionalA fuzzy rule can be defined as a conditionalstatement in the form:statement in the form:

IFIF xx isisAA

THENTHENyy isisBB

wherewherexx andandyy are linguistic variables; andare linguistic variables; andAA andandBB

are linguistic values determined by fuzzy sets on theare linguistic values determined by fuzzy sets on the

universe of discoursesuniverse of discoursesXXandand YY, respectively., respectively.


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What is the difference betweenWhat is the difference between

classical and fuzzy rules?classical and fuzzy rules?

A classical IFA classical IF--THEN rule uses binary logic, forTHEN rule uses binary logic, for

example,example,

Rule: 1Rule: 1

IFIF speed speed isis >> 100100THENTHEN stopping_distancestopping_distance isis longlong

Rule: 2Rule: 2

IFIF speed speed isis


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We can also represent the stopping distance rules in aWe can also represent the stopping distance rules in afuzzy form:fuzzy form:

RuleRule:: 11

IFIF speed speed isis fastfast

THEN stopping_distance is longTHEN stopping_distance is long

Rule: 2Rule: 2

IFIF speed speed isis slowslow

THEN stopping_distance is shortTHEN stopping_distance is short

In fuzzy rules, the linguistic variableIn fuzzy rules, the linguistic variable speedspeedalso hasalso has

the range (the universe of discourse) between 0 andthe range (the universe of discourse) between 0 and

220 km/h, but this range includes fuzzy sets, such as220 km/h, but this range includes fuzzy sets, such as

slowslow,, mediummedium andandfastfast. The universe of discourse of. The universe of discourse ofthe linguistic variablethe linguistic variable stopping_distancestopping_distance can becan be

between 0 and 300 m and may include such fuzzybetween 0 and 300 m and may include such fuzzy

sets assets as shortshort,, mediummedium andand longlong..


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Fuzzy rules relate fuzzy sets.Fuzzy rules relate fuzzy sets.

In a fuzzy system, all rules fire to some extent,In a fuzzy system, all rules fire to some extent,

or in other words they fire partially.or in other words they fire partially.

If the antecedent is true to some degree ofIf the antecedent is true to some degree ofmembership, then the consequent is also true tomembership, then the consequent is also true to

that same degreethat same degree.

IF speed is fast THEN stopping_distance is longIF speed is fast THEN stopping_distance is long

IF speed is slow THEN stopping_distance is shortIF speed is slow THEN stopping_distance is short

antecedent consequent


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Tall men Heavy men

180

Degree of

Mem bership

1.0

0.0

0.2

0.4

0.6

0.8

Height , cm

190 200 70 80 100160

Weight, kg

120

Degree of

Mem bersh ip

1.0

0.0

0.2

0.4

0.6

0.8

Fuzzy sets ofFuzzy sets oftalltallandandheavyheavy menmen

These fuzzy sets provide the basis for a weight estimationThese fuzzy sets provide the basis for a weight estimation

model. Consider the model that is based on a relationshipmodel. Consider the model that is based on a relationship

between a mans height and his weight:between a mans height and his weight:

IFIF heightheight isistalltall

THENTHEN weightweight isisheavyheavy


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The value of the output or a truth membership gradeThe value of the output or a truth membership gradeof the rule consequent can be estimated directly from aof the rule consequent can be estimated directly from a

corresponding truth membership grade in thecorresponding truth membership grade in the

antecedent. This form of fuzzy inference uses aantecedent. This form of fuzzy inference uses a

method calledmethod called monotonic selectionmonotonic selection..

Tall menHeavy men

180

Degree ofMembership

1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

190 200 70 80 100160

Weight, kg

120

Degree ofMembership

1.0

0.0

0.2

0.4

0.6

0.8

Monotonic Selection ofMonotonic Selection of

Fuzzy inferenceFuzzy inference


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A fuzzy rule can have multiple antecedents, forA fuzzy rule can have multiple antecedents, forexample:example:

IFIF project_durationproject_duration isis longlong

ANDAND project_staffingproject_staffing isis largelargeANDAND project_fundingproject_funding isis inadequateinadequate

THENTHEN riskrisk isis highhigh

IFIF serviceservice isis excellentexcellentOROR food food isis deliciousdelicious

THENTHEN tiptip isis generousgenerous

Multiple antecedents ofMultiple antecedents of



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The consequent of a fuzzy rule can also includeThe consequent of a fuzzy rule can also includemultiple parts, for instance:multiple parts, for instance:

IFIF temperaturetemperature isis hothot

THENTHEN hot_waterhot_water isis reducedreduced;;

cold_watercold_water isis increasedincreased

Multiple consequents ofMultiple consequents of


siti zaiton - fuzzy1

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