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Advance Topics in Mathematical Methods ME71

Fuzzy sets,System andModelling

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Advance Topics in Mathematical Methods ME71

Fuzzy sets were introduced by Zadeh in 1965 to represent/manipulate

data and information possessing nonstatistical uncertainties.L.A.Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.

There are two main characteristics of fuzzy systems that give them

better performance for specific applications.

Fuzzy systems are suitable for uncertain or approximate reasoning,

especially for the system with a mathematical model that is difficult

to derive.

Fuzzy logic allows decision making with estimated values under

incomplete or uncertain information.

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Advance Topics in Mathematical Methods ME71

Classical sets or crisp set

A = {12, 24, 36, 48, }

Notation: A = {x | x = 12n, n is a natural number}

A = {cities adjoining Hyderabad}

A = {Mahbubnagar, Medak, Nalgonda, Rangareddy}

x

xxA if0

if1)(

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Advance Topics in Mathematical Methods ME71

Classical sets vs Fuzzy set

A = {cities near from Hyderabad}

A = {Hyderabad, Adilabad, Khammam, Karimnagar, Mahbubnagar,

Medak, Nalgonda, Nizamabad, Rangareddy, Warangal, Bidar,

Gulbarga,.., Mumbai, Pune, Bangalore,., Delhi,Islamabad, Kabul, Kathmandu, Singapore. Rome, London,

Paris.}

Is the above information precise? : No it is Fuzzynot clear,

distinct, or precise; blurred

Definition of fuzzy logic : A form of knowledge representation suitable for

notions that cannot be defined precisely, but which depend upon their

contexts.

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Advance Topics in Mathematical Methods ME71

Fuzzy set : example 1

A = {cities near from Hyderabad}

What is near:A distance less the D kilometer

D = 100; if we are interested in adjoining cities

D = 200/300/400; if we are interested in cities in APD = 300/400/500..; if we are interested in cities in AP or Karnataka

D= 3000/4000/if we are interested in cities in India AP or Karnataka

Information is not precise and is context based

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Advance Topics in Mathematical Methods ME71

Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:

1. If 0 D 100; very near ( depending upon road condition and traffic one can reach within 2 hr)

2. If 50< D 150; near ( depending upon road condition and traffic one can reach within 3hrs)

3. If 100< D 200; far ( depending upon road condition and traffic u can reach within 6hrs)

4. If D> 200; very far

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Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:

1. If 0 D 100; very near ( depending upon road condition and traffic one can reach within 2 hr)

100 200 300

0

1

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Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:

2. If 50< D 150; near ( depending upon road condition and traffic one can reach within 3hrs)

100 200 300

0

1

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Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:

3. If 100< D 200; far ( depending upon road condition and traffic u can reach within 6hrs)

100 200 300

0

1

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Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:

4. If D> 200; very far

100 200 300

0

1

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Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:1. If 0 D 100; very near ( depending upon road condition and traffic one can reach within 2 hr)

2. If 50< D 150; near ( depending upon road condition and traffic one can reach within 3hrs)

3. If 100< D 200; far ( depending upon road condition and traffic u can reach within 6hrs)

4. If D> 200; very far

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

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Fuzzy set :

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

Distance (x)

Mem

bership(x)

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Fuzzy set :

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

Distance (x)

Mem

bership(x)

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Fuzzy set :

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

Distance (x)

Mem

bership(x)

Imp. Note: + sign stands for the union of membership grades; /

stands for a marker and does not imply division.

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Fuzzy set :

100 200 300

0.25

1

100 200 300

1

100 200 300

0.75

1

X=75

0

1

Distance (x)

Mem

bership(x)

A(x=75) = 0.25/very near + 0.75/near + 0.0 far + 0.0/very far

A(x=300) = 0.00/very near + 0.00/near + 0.0 far + 1.0/very far

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Fuzzy set : Example 2

150 210170 180 190 200160

Height, cmDegreeofMembership

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degreeof

Membership

Short Average Tall

170

1.0

0.0

0.2

0.4

0.6

0.8

Fuzzy Sets

CrispSets

Short Average Tall

Negnevitsky, Pearson Education, 2005

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Fuzzy set :

A fuzzy set A can be denoted as

If x is discrete

If x is continuous

Example 2. The membership function of the fuzzy set of real numbers close

to 1, is can be defined as

A x xA

x X

i i

i

( ) /

A x xA

X

( ) /

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Triangular Fuzzy Number :

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Trapezoidal Fuzzy Number :

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Operations on crisp sets

Intersection Union

Complement

Not A

A

Containment

AA

B

BA AA B

Negnevitsky, Pearson Education, 2005

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Operations on fuzzy sets

Intersection:The intersection of A and B is defined

as

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Operations on fuzzy sets

Intersection:The union of A and B is defined as

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Advance Topics in Mathematical Methods ME71

Operations on fuzzy sets

Negnevitsky, Pearson Education, 2005

Complement

0x

1

(x)

0x

1

Containment

0x

1

0x

1

AB

Not A

A

Intersection

0x

1

0x

AB

Union

0

1

AB

AB

0x

1

0x

1

B

A

B

A

(x)

(x) (x)

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Advance Topics in Mathematical Methods ME71

Operations on fuzzy sets

Negnevitsky, Pearson Education, 2005

Complement

0x

1

(x)

0x

1

Containment

0x

1

0x

1

AB

Not A

A

Intersection

0x

1

0x

AB

Union

0

1

AB

AB

0x

1

0x

1

B

A

B

A

(x)

(x) (x)

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Advance Topics in Mathematical Methods ME71

Operations on fuzzy sets

Important: Operations on fuzzy sets are also fuzzy, i.e., union

of intersection may