Probabilistic Transmission System Planning (Li/Transmission System Planning) || Appendix B: Elements of Fuzzy Mathematics

321

B.1 FUZZY SETS

B.1.1 Defi nition of Fuzzy Set

Let U be a traditional set and its member be denoted by x . A fuzzy set A on U is defi ned as a set of ordered pairs and is expressed by

A x x x UA= ( ) ∈{ }, ( ) |μ (B.1)

where μ A ( x ), whose value varies from 0 to 1, is called the membership function of A . If U is a discrete set with n members, A can be represented as

Ax

x

x

x

x

xA A n

n

A i

ii

n

= + + ==

∑μ μ μ( ) ( ) ( )1

1 1

� (B.2)

where + or Σ indicates the union of the members in A and μ A ( x i ) is the membership grade of x i . If U is a continuous set, A can be represented as

Ax

xA

U

= ∫ μ ( ) (B.3)

APPENDIX B

ELEMENTS OF FUZZY MATHEMATICS

Probabilistic Transmission System Planning, by Wenyuan LiCopyright © 2011 Institute of Electrical and Electronics Engineers

bapp02.indd 321bapp02.indd 321 1/20/2011 10:23:29 AM1/20/2011 10:23:29 AM

322 ELEMENTS OF FUZZY MATHEMATICS

where ∫ indicates the union of the members in A . Note that the horizontal bar in Equations (B.2) and (B.3) is not a quotient but a delimiter.

An α cutset of A , denoted by A α , is defi ned as

A x U xAα μ α α= ∈ ≥ ∈{ }| ( ) , [ , ]0 1 (B.4)

For any fuzzy set A , the membership function can be expressed using its α cutset by

μ α μα

αA Ax x x U( ) sup min ( , ( ) )[ , ]

= ∈∈ 0 1

(B.5)

In particular, A is probabilistic fuzzy set if μ A ( x ) is a random variable defi ned on a probabilistic space.

B.1.2 Operations of Fuzzy Sets

Assume that A and B are two fuzzy sets. A new fuzzy set C can be obtained using the following operations:

Intersection C = A ∩ B :

μ μ μC A Bx x x( ) min( ( ), ( ))= (B.6)

Union C = A ∪ B :

μ μ μC A Bx x x( ) max( ( ), ( ))= (B.7)

Complement C A= :

μ μC Ax x( ) ( )= −1 (B.8)

Algebraic product C = A • B :

μ μ μC A Bx x x( ) ( ) ( )= ⋅ (B.9)

The majority of relation laws of crisp sets hold for fuzzy sets. These include

Commutativity A ∪ B = B ∪ A A ∩ B = B ∩ A Associativity ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) Distributivity A ∪ ( B ∩ C ) = ( A ∪ B ) ∩

( A ∪ C ) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪

( A ∩ C ) Absorption A ∪ (A ∩ B ) = A A ∩ (A ∪ B ) = A

DeMorgan ’ s law A B A B∩ ∪= A B A B∪ ∩=


B.2 FUZZY NUMBERS 323

However, it should be noted that the following two laws are not the same as those for crisp sets (where W represents the universal set and ϕ represents the empty set):

A A W∪ ≠ A A∩ ≠ φ

B.2 FUZZY NUMBERS

B.2.1 Defi nition of Fuzzy Number

A fuzzy number is a special type of fuzzy set. A fuzzy set A , which is defi ned on the real number space R , is convex if the following inequality holds (where x 1 , x 2 ∈ U , λ ∈ [0,1]):

μ λ λ μ μA A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21+ − ≥ (B.10)

A fuzzy number is defi ned as a convex, normalized fuzzy set with a piecewise continu-ous membership function. Under this defi nition, it is clear that the α cutset A α of a fuzzy number A is an interval: A α = [ a l ( α ), a u ( α )] with a l ( α ) ≤ a u ( α ).

It is obvious that a l ( α ) and a u ( α ) are both a monotonically decreasing function of α . Therefore, operations of fuzzy numbers can be performed using calculations of intervals.

B.2.2 Arithmetic Operation Rules of Fuzzy Numbers

For two given fuzzy numbers A α = [ a l ( α ), a u ( α )] and B α = [ b l ( α ), b u ( α )], the following operation rules apply.

B.2.2.1 Addition.

( ) [ ( ) ( ), ( ) ( )]A B a b a bl l u u+ = + +α α α α α (B.11)

B.2.2.2 Subtraction.

( ) [ ( ) ( ), ( ) ( )]A B a b a bl u u l− = − −α α α α α (B.12)

B.2.2.3 Multiplication.

( ) [min( ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )AB a b a b a b a bl l u l l u u uα α α α α α α α α= ⋅ ⋅ ⋅ ⋅ ,,

max( ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )]a b a b a b a bl l u l l u u uα α α α α α α α⋅ ⋅ ⋅ ⋅ (B.13)

If A and B are defi ned on positive monotonic real number space, (B.13) becomes

( ) [ ( ) ( ), ( ) ( )]AB a b a bl l u uα α α α α= ⋅ ⋅ (B.14)



In particular, if H is a positive constant number, we have

( ) [ ( ), ( )]HA Ha Hal uα α α= (B.15)

B.2.2.4 Division.

A

Ba a

b bl u

u l

⎛⎝⎜

⎞⎠⎟ = ⋅ ⎡

⎣⎢⎤⎦⎥α

α αα α

[ ( ), ( )]( )

,( )

1 1 (B.16)

where b 1 ( α ) ≠ 0 and b u ( α ) ≠ 0. Otherwise, one or both ends of the interval are extended to ∞ .

B.2.2.5 Maximum and Minimum Operations.

( ) [ ( ) ( ), ( ) ( )]A B a b a bl l u u∨ = ∨ ∨α α α α α (B.17)

( ) [ ( ) ( ), ( ) ( )]A B a b a bl l u u∧ = ∧ ∧α α α α α (B.18)

where ∨ or ∧ denotes fi nding maximum or minimum.

Note that when the arithmetic operation rules are used, the fuzzy numbers in the opera-tion must be independent of each other.

B.2.3 Functional Operation of Fuzzy Numbers

If C = f ( A , B ) is a monotonic function of A and B on the real number space R, then its α cutset is calculated by

C f a b f a b f a b f al l u l l u uα α α α α α α= [min (( ( ), ( )), ( ( ), ( )), ( ( ), ( )), ( (αα αα α α α α

), ( )) ,

max (( ( ), ( )), ( ( ), ( )), ( ( ), (

b

f a b f a b f a bu

l l u l l u

{ }αα α α)), ( ( ), ( )) ]f a bu u{ } (B.19)

It is apparent that this is a calculation to fi nd the minimum and maximum in all possible combinations. A similar rule can be extended to a function containing more fuzzy variables. In this situation, the number of combinations can be very large. If the fuzzy variables in a function are not independent, the arithmetic operation rules given in Section B.2.2 are invalid and the general rule given in Equation (B.19) must be applied. For example, if a fuzzy number occurs in more than one term in a functional expres-sion, the repeated use of the arithmetic operation rules is generally incorrect because the independence condition is violated.

Theoretically, a general function of fuzzy numbers is still a fuzzy set, but there is no guarantee that it is a fuzzy number. However, the arithmetic operations of indepen-dent fuzzy numbers on the positive real number space result in a fuzzy number.


B.3 TWO TYPICAL FUZZY NUMBERS IN ENGINEERING APPLICATIONS 325

B.3 TWO TYPICAL FUZZY NUMBERS IN ENGINEERING APPLICATIONS

B.3.1 Triangular Fuzzy Number

The triangular fuzzy number, which is often denoted by A = ( a 1 , a 2 , a 3 ), is defi ned by the following membership function:

μA x

x a a a

a x a a

a x a

a x a( )

( ) /( )

( ) /( )=− −− −

⎧⎨⎪

⎩⎪

≤ ≤≤ ≤

1 2 1

3 3 2

1 2

2 3

0

if

if

iif orx a x a≤ ≥1 3

(B.20)

This membership function is shown in Figure B.1 . The α cutset A α of the triangular fuzzy number is calculated by

A a a a a a aα α α= + − − −[ ( ), ( )]1 2 1 3 3 2 (B.21)

For two triangular fuzzy numbers A = ( a 1 , a 2 , a 3 ) and B = ( b 1 , b 2 , b 3 ), it can be shown that their sum and difference are still triangular fuzzy numbers:

A B a b a b a b+ = + + +( , , )1 1 2 2 3 3 (B.22)

A B a b a b a b− = − − −( , , )1 3 2 2 3 1 (B.23)

However, their product AB is no longer a triangular fuzzy number.

B.3.2 Trapezoidal Fuzzy Number

The trapezoidal fuzzy number, which is often denoted by A = ( a 1 , a 2 , a 3 , a 4 ), is defi ned by the following membership function:

Figure B.1. Triangular fuzzy number.

a3a1

1.0

0.0

µA(x)

x a2



μA x

x a

a aa x a

a x a

a x

a aa x a

x a

( ) =

−−

≤ ≤

≤ ≤−−

≤ ≤

≤

1

2 11 2

2 3

4

4 33 4

1

0

if

if

if

if 11 4or x a≥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

(B.24)

This membership function is shown in Figure B.2 . The α cutset A α of the trapezoidal fuzzy number is calculated by

A a a a a a aα α α= + − − −[ ( ), ( )]1 2 1 4 4 3 (B.25)

For two trapezoidal fuzzy numbers A = ( a 1 , a 2 , a 3 , a 4 ) and B = ( b 1 , b 2 , b 3 , b 4 ), it can be shown that their sum and difference are still trapezoidal fuzzy numbers:

A B a b a b a b a b+ = + + + +( , , , )1 1 2 2 3 3 4 4 (B.26)

A B a b a b a b a b− = − − − −( , , , )1 4 2 3 3 2 4 1 (B.27)

Similarly, the product AB is no longer a trapezoidal fuzzy number.

B.4 FUZZY RELATIONS

B.4.1 Basic Concepts

If the relation between two sets X and Y is vague and cannot be described by “ yes ” or “ no, ” it is a fuzzy relation. Such a fuzzy relation is a fuzzy set, which is denoted by �R X Y( , ). The membership function of �R X Y( , ) is expressed by μ �R i jx y( , ), where x i and y j are the elements of X and Y , respectively. The fuzzy relation between the two sets X and Y can be expressed using the following matrix:

Figure B.2. Trapezoidal fuzzy number.

a4a1

1.0

0.0

µA(x)

x a2 a3


B.4 FUZZY RELATIONS 327

μ μ μμ μ μ� � �

� � �

�R R R n

R R R

x y x y x y

x y x y x

( , ) ( , ) ( , )

( , ) ( , ) ( ,1 1 1 2 1

2 1 2 2 2 yy

x y x y x y

n

R m R m R m n

)

( , ) ( , ) ( , )

� � � �

� � �μ μ μ1 2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Specifi cally, �R X X( , ) is the fuzzy relation between the elements in a set X and can be expressed using an n × n square matrix �R where n is the number of elements.

B.4.1.1 Refl exivity. A fuzzy relation �R X X( , ) is refl exive if and only if

μ �R i i ix x x X( , ) = ∀ ∈1 (B.28)

This corresponds to a fuzzy relation matrix �R in which each diagonal element equals to 1.

B.4.1.2 Symmetry. A fuzzy relation �R X X( , ) is symmetric if and only if

μ μ� �R i j R j i i jx x x x x x X( , ) ( , ) ,= ∀ ∈ (B.29)

This corresponds to a symmetric fuzzy relation matrix �R.

B.4.1.3. Resemblance. If a fuzzy relation �R X X( , ) is both refl exive and sym-metric, it is called a resemblance relation.

B.4.1.4 Transitivity. A fuzzy relation �R X X( , ) is transitive if and only if

μ μ μ� � �R i kx

R i j R j k i j kx x x x x x x x x Xj

( , ) sup min( ( , ), ( , )) , ,≥ ∀ ∈ (B.30)

This corresponds to � � � �R R R⊂ , where the symbol of � � �R R denotes the self - multiplication of the relation matrix �R [see Equation (B.34) below].

B.4.1.5 Equivalence. If a resemblance fuzzy relation �R X X( , ) satisfi es transi-tivity, it is an equivalence relation.

B.4.2 Operations of Fuzzy Matrices

A fuzzy matrix can be denoted by A = [ a ij ]. For the two fuzzy matrices with the same dimension A = [ a ij ] and B = [ b ij ], the following operation rules apply:

Intersection C = [ c ij ] = A ∩ B :

c a b a bij ij ij ij ij= = ∧min[ , ] (B.31)



Union C = [ c ij ] = A ∪ B :

c a b a bij ij ij ij ij= = ∨max[ , ] (B.32)

Complement C c Aij= =[ ] :

c aij ij= −[ ]1 (B.33)

Product C A B= � :

C a b a bijk

ik kjk

ik kj= = ∨ ∧max min[ , ] [ ] (B.34)

Equation (B.34) signifi es that each element in the fuzzy matrix C is calculated using a rule similar to that for multiplication of crisp matrices, except that the product of two elements is replaced by taking the minimum one and the addition of two products is replaced by taking the maximum one. Note that in general, A B B A� �≠ even if A and B are both square matrices of the same size.

A square fuzzy relation matrix �R represents the fi rst - order relation. � � � �R R R2 = represents the second - order fuzzy relation. Similarly, the following equation represents the n th - order fuzzy relation:

� � � � � �� R R R R Rn

n

=self-multiplications

If a fuzzy set contains n elements and �R is a resemblance relation matrix on the set, then the ( n − 1)th - order fuzzy relation matrix �Rn−1 not only maintains refl exivity and symmetry but also has transitivity. In other words, �Rn−1 obtained from self - multiplications of a resemblance relation matrix is an equivalence relation matrix and satisfi es

� � � � �R R R Rn n n n m− + += = = =1 1 (B.35)

where m is any positive integer. The equivalence relation matrix can be used for fuzzy clustering.


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Probabilistic Transmission System Planning (Li/Transmission System Planning) || Appendix B: Elements of Fuzzy Mathematics