Extension Principle — Concepts

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# Extension Principle — Concepts - PowerPoint PPT Presentation

Extension Principle — Concepts. To generalize crisp mathematical concepts to fuzzy sets . Extension Principle.

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Extension Principle — Concepts
• To generalize crisp mathematical concepts to fuzzy sets.

Extension Principle

Extension Principle
• Let X be a cartesian product of universes X=X1…Xr, and be r fuzzy sets in X1,…,Xr, respectively. f is a mapping from X to a universe Y, y=f(x1,…,xr), Then the extension principle allows us to define a fuzzy set in Y by

where

Extension Principle

Example 1

f(x)=x2

Extension Principle

Fuzzy Numbers
• To qualify as a fuzzy number, a fuzzy set on R must possess at least the following three properties:
• must be a normal fuzzy set
• must be a closed interval for every α(0,1](convex)
• the support of , must be bounded

Extension Principle

Positive (negative) fuzzy number
• A fuzzy number is called positive (negative) if its membership function is such that

Extension Principle

Increasing (Decreasing) Operation
• A binary operation  in R is called increasing (decreasing) if

for x1>y1 and x2>y2

x1x2>y1y2(x1x2<y1y2)

Extension Principle

Example 2
• f(x,y)=x+y is an increasing operation
• f(x,y)=x•y is an increasing operation on R+
• f(x,y)=-(x+y) is an decreasing operation

Extension Principle

Notation of fuzzy numbers’ algebraic operations
• If the normal algebraic operations +,-,*,/ are extended to operations on fuzzy numbers they shall be denoted by 

Extension Principle

Theorem 1
• If and are fuzzy numbers whose membership functions are continuous and surjectivefromR to [0,1] and  is a continuous increasing (decreasing) binary operation, then is a fuzzy number whose membership function is continuous and surjective from R to [0,1].

Extension Principle

Theorem 2
• If , F(R) (set of real fuzzy number) with and continuous membership functions, then by application of the extension principle for the binary operation : R R→R the membership function of the fuzzy number is given by

Extension Principle

Special Extended Operations
• If f:X→Y, X=X1 the extension principle reduces for all F(R) to

Extension Principle

Example 31
• For f(x)=-x the opposite of a fuzzy number is given with , where
• If f(x)=1/x, then the inverse of a fuzzy number is given with

, where

Extension Principle

Example 32
• For λR\{0} and f(x)=λx then the scalar multiplication o a fuzzy number is given by , where

Extension Principle

• Since addition is an increasing operation→ extended addition  of fuzzy numbers that

is a fuzzy number — that is

Extension Principle

Properties of 
• ( )( )
•  is commutative
•  is associative
• 0RF(R) is the neutral element for , that is , 0= , F(R)
• For  there does not exist an inverse element, that is,

Extension Principle

Extended Product
• Since multiplication is an increasing operation on R+ and a decreasing operation on R-, the product of positive fuzzy numbers or of negative fuzzy numbers results in a positive fuzzy number.
• Let be a positive and a negative fuzzy number then is also negative and results in a negative fuzzy number.

Extension Principle

(

)

(

)

=

1=

1=

Properties of
• is commutative
• is associative
• , 1RF(R) is the neutral element for , that is , ,F(R)
• For there does not exist an inverse element, that is,

Extension Principle

Theorem 3
• If is either a positive or a negative fuzzy number, and and are both either positive or negative fuzzy numbers then

Extension Principle

Extended Subtraction
• Since subtraction is neither an increasing nor a decreasing operation,
• is written as ( )

Extension Principle

Extended Division
• Division is also neither an increasing nor a decreasing operation. If and are strictly positive fuzzy numbers then

The same is true if and are strictly negative.

Extension Principle

={(2,0.3),(3,0.3),(4,0.7),(6,1),(8,0.2),(9,0.4),(12,0.2)}

Note
• Extended operations on the basis of min-max can’t directly applied to “fuzzy numbers” with discrete supports.
• Example
• Let ={(1,0.3),(2,1),(3,0.4)}, ={(2,0.7),(3,1),(4,0.2)} then

No longer be convex → not fuzzy number

Extension Principle

Extended Operations for LR-Representation of Fuzzy Sets
• Extended operations with fuzzy numbers involve rather extensive computations as long as no restrictions are put on the type of membership functions allowed.
• LR-representation of fuzzy sets increases computational efficiency without limiting the generality beyond acceptable limits.

Extension Principle

Definition of L (and R) type
• Map R+→[0,1], decreasing, shape functions if
• L(0)=1
• L(x)<1, for x>0
• L(x)>0 for x<1
• L(1)=0 or [L(x)>0, x and L(+∞)=0]

Extension Principle

Definition of LR-type fuzzy number1
• A fuzzy number is of LR-type if there exist reference functions L(for left). R(for right), and scalars α>0, β>0 with

Extension Principle

Definition of LR-type fuzzy number2
• m; called the mean value of , is a real number
• α,β called the left and right spreads, respectively.
• is denoted by (m,α,β)LR

Extension Principle

Example 4
• Let L(x)=1/(1+x2), R(x)=1/(1+2|x|), α=2, β=3, m=5 then

Extension Principle

Fuzzy Interval
• A fuzzy interval is of LR-type if there exist shape functions L and R and four parameters , α, β and the membership function of is

The fuzzy interval is denoted by

Extension Principle

Different type of fuzzy interval
• is a real crisp number for mR→

=(m,m,0,0)LR L, R

• If is a crisp interval, →

=(a,b,0,0)LRL, R

• If is a “trapezoidal fuzzy number”→ L(x)=R(x)=max(0,1-x)

Extension Principle

Theorem 4
• Let , be two fuzzy numbers of LR-type: =(m,α,β)LR, =(n,γ,δ)LR Then
• (m,α, β)LR(n, γ,δ)LR=(m+n, α+γ, β+δ)LR
• -(m, α, β)LR=(-m, β, α)LR
• (m, α, β)LR (n, γ, δ)LR=(m-n, α+δ, β+γ)LR

Extension Principle

Example 5
• L(x)=R(x)=1/(1+x2)
• =(1,0.5,0.8)LR
• =(2,0.6,0.2)LR
•  =(3,1.1,1)LR
• =(-1,0.7,1.4)LR

Extension Principle

Theorem 5
• Let , be fuzzy numbers →

(m, α, β)LR (n, γ, δ)LR≈(mn,mγ+nα,mδ+nβ)LR for , positive

• (m, α, β)LR (n, γ, δ)LR≈(mn,nα-mδ,nβ-mγ)LR for positive, negative
• (m, α, β)LR (n, γ, δ)LR ≈(mn,-nβ-mδ,-nα-mγ)LR for , negative

Extension Principle