L3 Fuzzy11

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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