Fuzzy numbers and fuzzy arithmetic - DCA - Unicamp

An Overview of Fuzzy Numbersand

Fuzzy Arithmetic

Fernando GomideUnicamp-FEEC-DCA

DcaFeecUnicampGomide

Purpose

To provide a tutorial review of operations withuncertain quantities from the point of view ofthe theory of fuzzy sets.


Outline

1- Introduction

2- Fuzzy Numbers and Arithmetic

3- New Approaches for Fuzzy Arithmetic

4- Analysis and Discussion

5- Conclusions


1- Introduction

• Exact values (e.g. parameters) are rare in practice

• Reason: incomplete or imprecise information

• How to model e.g. imprecise system parameters ?

• How to compute with imprecise parameters ?

• Fuzzy numbers and fuzzy arithmetic provide an answer

– L. Zadeh, Calculus of fuzzy restrictions, in L. Zadeh, K. Fu, K. TanakaM. Shimura (Eds.), Fuzzy Sets and Their Applications to Cognitiveand Decision Processes, Academic Press, NY, 1975, pp. 1-39.

– D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and ApplicationsAcademic Press, NY, 1980.


• Issues with standard fuzzy arithmetic:

• Overestimation: accumulation of fuzziness skews membership functions

• Shape preservation: membership function forms are not preserved

• Properties: commutative, associative, sub-distributive

• Consequences for system modeling and applications:

• Analysis, validation and interpretation of imprecise models more complex

• Overestimation and shape preserving mean non-intuitive results

• Mathematically speaking: fuzzy arithmetic is well developed


2- Fuzzy Numbers and Arithmetic

• Fuzzy quantity A: R → [0,1] is a fuzzy set A of R such that:

1- A is a normal fuzzy set

2-The support { x: A(x) > 0 } of A is bounded

3-The α-cuts of A are closed intervals

• Fuzzy Number: If A(x)=1 for exactly one x

• Fuzzy Interval: Otherwise


General form of a fuzzy quantity

φ≠

∈∈∈

=

],[

otherwise0],[)(],[1],[)(

)(

cb

dcxxgcbxbaxxf

xAA

A

xa b c d

A(x)

AfAg

1


Examples of fuzzy quantities

1 1

2.5 2.5

1 1

2.2

2.2 3.0

3.0 2.2 3.0

Real number2.5

Fuzzy numberabout 2.5

Real interval[2.2, 3.0]

Fuzzy intervalaround [2.2, 3.0]

2.5


Major approaches for fuzzy arithmetic

α

A

Aαx

I- Extension of interval arithmetic

• α-cuts Aα of fuzzy sets

• representation theorem

)()(]1,0[

xAxA

AA

αα

∈α

α

α=

= U

α

A

Aα

β

γ

Aγ

Aβ

xDcaFeecUnicampGomide

II- Extension Principle

A

Bf

X

Y

)(sup)()(

)()(::

)(/xAyB

AfB

YFXFfYXf

xfyx ==

=

→→


I- Extension of Interval arithmetic

• If * is any of the arithmetic operations: +, - , . , / , then:

[a, b] * [d, e] = { v * w | a ≤ v ≤ b, d ≤ w ≤ e} except when 0∈[d, e]

• Arithmetic operations with closed intervals are:

],[0)]/,/,/,/max(),/,/,/,/[min(]/1,/1].[,[],/[],[

)],,,max(),,,,[min(],].[,[

],[],[],[

],[],[],[

edebdbeadaebdbeadadebaedba

bebdaeadbebdaeadedba

dbeaedba

ebdaedba

∉==

=

−−=−

++=+


]0,0[],1,1[],,[],,[],,[:

]2,2[]5.0,2/[]1,1[

]2,2[]5.0,2].[1,1[

]4,1[]3,1[]5,2[

]8,3[]3,1[]5,2[

212121 =====•

−=−−−

−=−−−

−=−

=+

•

01ccCbbBaaALet

Examples


vityDistributithenandIf

utivitySubdistrib

Identity

ityAssociativ

ityCommutativ

Properties

CABACBACcBbcb

CABACBA

AAAAAA

CBACBACBACBA

ABBAABBA

..).(0.5

..).(4

..3

)..()..()()(2

..1

+=+∈∀∈∀≥−

+⊆+−

==+=+=−

=++=++−

=+=+−

•

1100


Fuzzy arithmetic

]1,0(0/

)(

]1,0(},),(|{)(

]1,0[

∈α∀∉=∗•

•

∗=∗

∈α×∈∗=∗=∗

α

∈αα

ααααα

B

BABA

BAyxyxBABA

assumewhen

functionsmembershipcontinuousassuming

U


Example

]25,12[]23,12[

332/)5(312/)1(

5;10)(

312/)3(112/)1(3;10

)(

α−+α=α−−α=

≤<−≤<−

>≤=

≤<−≤<−+>−≤

=

αα BA

xxxx

xxxB

xxxxxx

xA

-1 1 3 840

1A B


-1 1 3 840

1A B

A+ Bα=0.5

-6 -4 -2 3-1-5

1 A BA-B

α=0.5

0 5DcaFeecUnicampGomide

0 1 2-1-2

A B

A/B

5-5

1

α=0.5

0 1 2-1-2

A BA.B

5-5 15

1

α=0.5


II- Extension Principle

RzyBxAxBA

RzyBxAxBA

yxz

yxz

∈∀=∗

∈∀=∗

∗=

∗=

,)](t)([sup))((

)],(),([minsup))((

t is a t-norm


Example

A B

A+Btm

A+Btd

1 2 30 R 210 3 4

1 3 5 7 2 3 5 60 0

R

R R


3- New Approaches for Fuzzy Arithmetic

I - Requisite constraints (Klir, 1997)

II - Discrete fuzzy arithmetic (Hanss, 1999, 2000)

III - Specialized approaches

- Kreinovich and Pedrycz, 2001

- Filev and Yager, 1997


I - Requisite constraints

0 1-1-3

B

5-5 73

B/B

B/ B ≠ 10 1-1-3

B

5-5

- B

73

B - B

B - B ≠ 0


choices x in Aα and y in Bαare not constrained by the equality x = y

• constraints: supplementary information not containedin operands

• constraints: result from the meaning of operandsrather than the operands themselves

• constraints: are not taken into account by standardfuzzy arithmetic

Greater imprecision than justifiable in all computationsthat involve the requisite equality constraint (Klir, 1997)


• Fuzzy interval arithmetic with requisite constraint

U]1,0[

)()(

]1,0(,)(),(|{)(

∈α

α

αααα

∗=∗

∈αℜ∩×∈∗=∗

RR

R

BABA

BAyxyxBA

• Extension principle with requisite constraint

)},()](t)({[sup)()(

)],(),(),([minsup)()(

yxyBxAxBA

yxyBxAxBA

yxz

R

yxz

R

ℜ∧=∗

ℜ=∗

∗=

∗=


0 1-1-3

B

5-5 73

B/B

B/ B = 1

0}|{)( =∈−=− αα BbbbBB E

B - B = 0

0 1-1-3

B

5-5

- B

73

B - B

1}0,|/{)/(

=∉∈= ααα BBbbbBB R


4- Analysis and Discussion

• Overestimation

-1 1 3 640

1A

A+ A

2-2


• Shape preservation

0 1 2-1-2

A BA.B

5-5 15

1

• Properties: still to be proved for all three approaches


5- Conclusions

• Fuzzy quantities is important in many fields

• Mathematically OK but not always intuitive

• Requisite constraints seem to be a key

• Heuristics and approximate reasoning as a link


References

1-D. Dubois and H. Prade, Fuzzy Numbers: An Overview. In: J. Bezdek (ed.) Analysis ofFuzzy Information, CRC Press, Boca Raton, 1988, vol. 2, pp. 3-39.

2-G. Klir, Fuzzy Arithmetic with Requisite Constraints. Fuzzy sets and systems, 91 (1997) 165-175.

3-M. Hanss, On Implementation of Fuzzy Arithmetical Operations for Engineering Problems.Proceedings of NAFIPS 1999 (1999) 462-466, New York.

4- M. Hanss, A Nearly Strict Fuzzy Arithmetic for Solving Problems with Uncertainties.Proceedings of NAFIPS 2000 (2000), 439-443, Atlanta.

5-V. Kreinovich and W. Pedrycz, How to Make Sure that "≈100"+1 is ≈100 in Fuzzy Arithmetic:Solution and its (Inevitable) Drawbacks. Proceedings of the FUZZ-IEEE (2001), 1653-1658,Melbourne.

6-D. Filev and R. Yager, Operations on Fuzzy Numbers via Fuzzy Reasoning. Fuzzy sets andSystems, 91 (1997) 137-142.


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Fuzzy numbers and fuzzy arithmetic - DCA - Unicamp

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