FUZZY SETS

Membership Value Assignment

• There are possible more ways to assignmembership values or function to fuzzyvariables than there are to assign probabilitydensity functions to random variables [Duboisand Prade, 1980]

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Membership Value Assignment

• Intuition• Inference• Rank ordering• Angular fuzzy sets• Neural networks• Genetic algorithms• Inductive reasoning• Soft partitioning

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Intutition

• Derived from the capacity of humans todevelop membership functions through theirown innate intelligence and understanding.

• Involves contextual and semantic knowledgeabout an issue; it can also involve linguistictruth values about this knowledge.

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Types of Membership Functions

• The most commonly used in practice are– Triangles– Trapezoids– Bell curves– Gaussian, and– Sigmoidal

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Triangular MF

Specified by three parameters {a,b,c} as follows:

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

triangle(x : a,b,c) =

0 x < a

(x − a) (b − a) a ≤ x ≤ b

(c − x) (c − b) b ≤ x ≤ c

0 x > c

a

b

c

Trapezoidal MF

Specified by four parameters {a,b,c,d} as follows:

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

trapezoidal(x : a,b,c,d) =

0 x < a

(x − a) (b − a) a ≤ x < b

1 b ≤ x < c

(d − x) (d − c) c ≤ x ≤ d

0 x ≥ d

a

b c

d

Gaussian MF

Specified by two parameters {m,σ} as follows:

Where m and σ denote the center and width of the function, respectivelyA small σ will generate a “thin”MF, while a big σ will lead to a “flat”MF.

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

gaussian(x : m,σ ) = exp −(x − m)2

σ 2

m

σ

Bell-shaped MF

Specified by three parameters {a,b,c} as follows:

Where the parameter b is usually positive and we can adjust c and a to vary thecenter and width of the function and then use b to control the slopes.

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

bell(x : a,b,c) =1

1 + x − ca

2b

c

Bell-shaped MF

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

bell(x : a,b,c) =1

1 + x − ca

2b

c

Sigmoidal MF

Specified by two parameters {a, c} as follows:

Where c is the center of the function and a control the slope.

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Sigmoidal(x : a,c) =1

1+ e− a(x − c)

Sigmoidal MF

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Sigmoidal(x : a,c) =1

1+ e− a(x− c)

Hedges: a modifier to a fuzzy set

• Hedge modifies the meaning of the originalset to create a compound fuzzy set

– Example:• Very (Concentration)• More or Less(Dilation)

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Hedges: Very & MoreOrLess

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Very :

µveryA x( ) = µA x( )[ ]2

MoreorLess :

µMoreOrLessA x( ) = µA x( )

Hedges: Very

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Hedges: VeryVeryVery (Extreme)

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Inference

• Use knowledge to perform deductivereasoning, i.e . we wish to deduce or infer aconclusion, given a body of facts andknowledge.

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Inference : Example• In the identification of a triangle

– Let A, B, C be the inner angles of a triangle• Where A ≥ B≥C

– Let U be the universe of triangles, i.e.,• U = {(A,B,C) | A≥B≥C≥0; A+B+C = 180˚}

– Let ‘s define a number of geometric shapes• I Approximate isosceles triangle• R Approximate right triangle• IR Approximate isosceles and right triangle• E Approximate equilateral triangle• T Other triangles

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Inference : Example• We can infer membership values for all of

these triangle types through the method ofinference, because we possess knowledgeabout geometry that helps us to make themembership assignments.

• For Isosceles,µi (A,B,C) = 1- 1/60* min(A-B,B-C)

– If A=B OR B=C THEN µi (A,B,C) = 1;– If A=120˚,B=60˚, and C =0˚ THEN µi (A,B,C) = 0.

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Inference : Example• For right triangle,

µR (A,B,C) = 1- 1/90* |A-90˚|– If A=90˚ THEN µi (A,B,C) = 1;– If A=180˚ THEN µi (A,B,C) = 0.

• For isosceles and right triangle– IR = min (I, R)

µIR (A,B,C) = min[µI (A,B,C), µR (A,B,C)] = 1 - max[1/60min(A-B, B-C), 1/90|A-90|]

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Inference : Example• For equilateral triangle

µE (A,B,C) = 1 - 1/180* (A-C)– When A = B = C then µE (A,B,C) = 1,

A = 180 then µE (A,B,C) = 0

• For all other triangles– T = (I.R.E)’ = I’.R’.E’

= min {1 - µI (A,B,C) , 1 - µR (A,B,C) , 1 - µE (A,B,C)

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Inference : Example

– Define a specific triangle:• A = 85˚ ≥ B = 50˚ ≥ C = 45˚

µR = 0.94 µI = 0.916 µIR = 0.916 µE = 0. 7

µT = 0.05

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Rank ordering

• Assessing preferences by a single individual, acommittee, a poll, and other opinion methodscan be used to assign membership values to afuzzy variable.

• Preference is determined by pairwisecomparisons, and these determine theordering of the membership.

Fuzzy Logic with Engineering Applications: Timothy J. Ross

Rank ordering: Example

Fuzzy Logic with Engineering Applications: Timothy J. Ross