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Guided by

Dr. A G KeskarProfessor

Department of Electronics EngineeringVNIT, Nagpur

Visvesvaraya National Institute Of Technology, Nagpur

Presentation on Fuzzy Arithmetic

Presented By:

LOKESH GAHANE (MT15CMN007)

MOHIT CHIMANKAR (MT15CMN008)

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Contents

• Fuzzy Numbers

• Linguistic Variables

• Arithmetic Operations On Fuzzy Intervals

• Arithmetic Operations On Fuzzy Numbers

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4.1 Fuzzy Numbers• Set A : R [0,1]• To qualify as fuzzy number, a fuzzy set A on R must have following

properties:1. Set A must be Normal Fuzzy set;2. α-cut of A must be closed interval for every α in (0,1];3. Support and strong α-cut of A must be bounded. Every Fuzzy number is a Convex fuzzy set, The Inverse is not

necessarily true.

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Examples of fuzzy quantities : Real number Fuzzy number/interval

4.1 Fuzzy Numbers

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4.2 Linguistic Variables

• It is fully described by name of the variable, linguistic values, base variable range, various rules.

• A numerical variables takes numerical values: Age = 65

• A linguistic variables takes linguistic values: Age is old

• All such linguistic values set is a fuzzy set.

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Example: Linguistic variable age :

4.2 Linguistic Variables

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• Appearance = {beautiful, pretty, cute, handsome, attractive, not beautiful, very pretty, very very handsome, more or less pretty, quite pretty, quite handsome, fairly handsome, not very attractive..}

• Truth = {true, not true, very true, completely true, more or less true, fairly true, essentially true, false, very false...}

• Age = {young, not young, very young, middle aged, not middle aged, old, not old, very old, more or less old, not very young and not very old...}

4.2 Linguistic VariablesExample: Linguistic variables :

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4.3 Arithmetic Operations On Intervals

• Fuzzy arithmetic is based on two properties :

– Each fuzzy set and thus each fuzzy number is uniquely be represented by its α-cuts;

– α-cut of A must be closed intervals of real numbers for all α in (0,1].

Using above properties we define arithmetic operations on fuzzy numbers in terms of arithmetic operations on their α cuts.

Later operations are subject to interval analysis

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• [a, b]*[d, e]={f*g|a ≤ f ≤ b, d ≤ g ≤ e}1. Addition [a, b]+[d, e]=[a+d, b+e];2. Subtraction [a, b]-[d, e]=[a-e, b-d];3.Multiplication [a, b].[d, e]=[min(ad,ae,bd,be), max(ad,ae,bd,be)];4. Division* [a, b]/[d, e]=[a, b].[1/e, 1/d] =[min(a/d,a/e,b/d,b/e), max(a/d,a/e,b/d,b/e)].5. Inverse Interval* [d,e] inverse =[min(1/d,1/e),max(1/d,1/e)] * 0 [d,e] i.e excluding the case d=0 or e=0.∉

4.3 Arithmetic Operations On IntervalsLet * denotes any of the four arithmetic operations on closed intervals :

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1. [-1,3]+[1, 5] = [-1+1, 3+5] = [0, 8]2 . [-1,3]-[1, 5] = [-1-5, 3-1] = [-6,2]3. [-1,3].[1, 5] = [min(-1,-5,3,15),max(-1,-5,3,15)] =[-5,15]4. [-1,3]/[1, 5] = [-1, 3].[1/1, 1/5]

= [min(-1/1,-1/5,3/1,3/5),max(-1/1,-1/5,3/1,3/5)] = [-1/1,3/1] = [-1,3]5. [-2,7]inverse = [min(1/-2,1/7),max(1/-2,1/7)] = [-1/2,1/7]

4.3 Arithmetic Operations On Intervals

Examples illustrating interval-valued arithmetic operations :

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Arithmetic Operations On IntervalsA=[-1,3] and B=[1,5]

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Arithmetic Operations On Intervals

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Arithmetic Operations On Intervals

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Properties:1. Commutativity (+, .) A*B = B*A2. Associativity (+, .) (A*B)*C = A*(B*C)3. Identity A = 0+A = A+0 , A = 1.A = A.14. Distributivity A.(B+C) = A.B+A.C5. Subdistributivity A.(B+C)⊆ A.B+A.C6. Inclusion monotonicity (all). If A ⊆ E & B ⊆ F then A*B ⊆

E*F7. 0, 1 are included in -,/ operations between same fuzzy

intervals respectively. 0 A-A and 1 A/A∈ ∈

4.3 Arithmetic Operations On IntervalsArithmetic operations on Fuzzy intervals satisfy following useful properties :

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4.4 Arithmetic Operations On Fuzzy Numbers

Moving from intervals we can define arithmetic on fuzzy numbers based on principles of Interval Arithmetic

Let A and B denote fuzzy numbers and let * denote any of the four basic arithmetic operations. Then, we define a fuzzy set on R, A*B, by defining its alpha-cut as:

4.4 Arithmetic Operations On Fuzzy Numbers

•

(A + B)(x)

(A . B)(x)

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4.5 Lattice Of Fuzzy Numbers

4.5 Lattice Of Fuzzy Numbers

4.5 Lattice Of Fuzzy Numbers

• How to find MIN?• Solution• How to find MAX?• Solution

4.5 Lattice Of Fuzzy Numbers

4.5 Lattice Of Fuzzy Numbers

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4.6 Fuzzy Equations

ABX

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4.6 Fuzzy Equations

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0.6

α = 0.4

0.8

0.2

0

A B

10 20 30

αa1 = 4αa2 = 16 αb2 = 26

A(x) , B(x)

x

α b1 – αa1 ≤ α b2 – α a2

αb1 = 14

Fig 5

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1

0.6

α = 0.4

β = 0.8

0.2

0

A B

10 20 30

αa1 = 4αa2 = 16 αb2 = 26

βa1 = 8 βa2 = 12 βb1 = 18 βb2 = 22

αb1 = 14

A(x) , B(x)

x

α b1 – αa1 ≤ β b1 – βa1 ≤ β b2 – βa2 ≤ α b2 – α a2

Fig 6

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4.6 Fuzzy Equations

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4.6 Fuzzy Equations

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4.6 Fuzzy Equations

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4.6 Fuzzy Equations

THANK YOU.