5.1 Fuzzy numbers | unit 5 fuzzy arithmetic

Unit – 5

Fuzzy Arithmetic

5.1 Fuzzy numbers

A fuzzy number is a generalization of a regular, real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function.

Set A: R

To qualify as fuzzy number, a fuzzy set A on R must have following properties:

1. Set A must be normal Fuzzy set;

2. of A must be closed interval foe every in (0,1]

3. Support and strong of A must be bounded.

Every fuzzy number is a Convex fuzzy set, the inverse is not necessarily true.

Example 1: It is necessary to compare 2 sensors based upon their detection levels & gain settings the tails of gain settings & sensor detection level with a obtained item being monitored providing typical membership values to reprocess the detection levels for each sensor is given in tales.

Find following membership function:

a) b) c)

Solution:

Gain setting	Detection level of sensor 1(D1)	Detection level sensor 2(D2)
0	0	0
10	0.2	0.35
20	0.35	0.25
30	0.65	0.8
40	0.85	0.95
50	1	1

= max {}

b) = min {}

c) = 1-

5.2 Fuzzy cardinality

Definition: If aA is a fuzzy set defined on universal set X then its fuzzy cardinality is denoted and defined as

Where is membership grade from given fuzzy set A and is a scalar cardinality (total number of elements present in crisp set) of corresponding

Example 1: Find the fuzzy cardinality of A defined as follows:

Solution: Here X= {0,1,2,3,4,5} s and its scalar cardinality are

	{0,2,4,5}
	{0,1,2,3,4,5}
	{2,5}
	{0,2,3,4,5}
	{2,4,5}
	{5}

Thus,

Note: Fuzzy cardinality of Fuzzy set is a fuzzy set

Example 2: Find fuzzy cardinality of fuzzy set A and B whose membership function are given as follows , B(x)= where x

Solution:

X	0	1	2	3	4
,	0	0.5	0.67	0.75	0.8
B(x)=	1	0.9	0.8	0.7	0.6

	{0,1,2,3,4}
	{1,2,3,4}
	{2,3,4}
	{3,4}
	{4}

	{0}
	{0,1}
	{0,1,2}
	{0,1,2,3}
	{0,1,2,3,4}

Thus,

5.3 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:

Arithmetic Operations on Intervals: Arithmetic operations on fuzzy intervals satisfy following useful properties:

Properties:

Commutativity (+,..) A*B = B*A

Associativity (+,..) (A*B) *C = A*(B*C)

Identity A = 0+A = A+0, A=1.A=A.1

Distributivity A.(B+C) = A.B + A.C

Sub-distributivity A.(B+C)

A.B + A.C

Inclusion monotonicity (all). If

0,1 are included in -,/ operations between same fuzzy intervals respectively. 0

and 1

Examples Illustrating interval-valued arithmetic operations:

[-1,3] + [1,5] = [-1+1, 3+5] = [0,8]

[-1,3] – [1,5] = [-1-5, 3-1] = [-6,2]

[-1,3]. [1,5] = [min(-1,-5,3,15), max(-1/1, -1/5, 3/1, 3/5)] = [-5,15]

[-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]

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

Applying the extension principle to arithmetic operations, we have

Fuzzy Addition:

Fuzzy Subtraction

Fuzzy Multiplication

Fuzzy Division

Example 1:

Their

Step 1: Find and

Step 2: Use decomposition theorem which is given as and arithmetic operation on closed interval

Step 3: Find membership function.

Example 2: The membership function of two fuzzy numbers A and B are given as follows

A(x)	=	-1 < x	B(x)	=
	=			=
	= 0			= 0	otherwise

Find A+B and A-B

Solution:

Step 1: Find and

For -1 < x

By def. of ,

A(x)

For

= [2-1, 3-2 for all

For

By def. of cut,

B(x)

For 3 < x

B(x)

= [2+1, 5-2 for all

Step 2: By decomposition theorem & arithmetic operation on closed interval we find as follows.

Step 3: Now find membership function

Let 4

Let 8-4

Thus, (A+B) (x) = ; 0 < x

= 4

= 0; Otherwise

To find A-B

Step 4: By decomposition theorem and arithmetic operation on closed interval we find follows

= [

= [4 for all

Step 5: Now find membership function to define fuzzy numbers A-B

Let 4

Let 2-4

Thus,

5.4 Solutions of Fuzzy equations of type A + X = B&A.X=B

Algorithm:

Step 1: Find

Step 2: Use decomposition theorem which is given as

Let

A+X = B

Taking alpha cut on both side

Step 3: Find membership function

Example 1: The membership function of two fuzzy numbers A and B are

A(x)	= x-1	1 < x	B(x)	=
	=			=
	= 0			= 0	otherwise

Solve A+X =B

Solution:

Let

A+X = B

Taking alpha cut on both side

Check


0.1	12.8	56.9
0.6	26.8	46.4
0.8	32.4	42.2

Thus, hence solution exists.

Let

Example 2: The membership function of two fuzzy numbers A and B are

A(x)	= x-3	3 < x	B(x)	=
	=			=
	= 0			= 0	otherwise

Solve A.X =B

Solution:

Let

A.X = B

Taking alpha cut on both side

Check


0.3	4.3636	6.0426
0.7	4.7568	5.4884
0.9	4.9231	5.1707

Thus, hence solution exists.

Let

Reference

Advance Engineering Mathematics by Erwin Kreyszig (WileyIndia).

Mathematical Methods of Science and Engineering, by Kanti B. Datta(Cengage Learning)

Advanced Engineering Mathematics, 3e, by Jack Goldberg (Oxford University Press).

Engineering Mathematics by V. Sundaram (Vikas Publication).

Higher Engineering Mathematics, by B. S. Grewal (Khanna Publication Delhi).

Higher Engineering Mathematics, by B. V. Ramana (Tata McGraw-Hill).

Advanced Engineering Mathematics, by H. K. Das (S. Chand Publication).

Fuzzy Sets and Fuzzy Logic: Theory and Applications, by George J. Klir and Bo Yuan (Prentice Hall of India Private Limited).

Applied Mathematics by Navneet D. Sangle (CengagePublication)

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