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IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 11, November 2015.

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Applications Of Fuzzy Number Mathematics Dr. S. Chandrasekaran

Head & Associate Professor of Mathematics

E.Tamilmani Research Scholar , Khadir Mohideen College , Adirampattinam.

ABSTRACT Fuzzy sets have been introduced by Lotfi.A.Zadeh(1965)[16] and

Dieter Klaua(1965)[7]. Fuzzy set theory permits the gradual assessment of

the membership of elements in a set which is described in the interval [0, 1].

It can be used in a wide range of domains where information is incomplete

and imprecise. Interval arithmetic was first suggested by Dwyer[7] in

1951,by means of Zadeh’s extension principle[15,16], the usual Arithmetic

operations on real numbers can be extended to the ones defined on Fuzzy

numbers. D.Dubois and H.Prade[3] in 1978 has defined any of the fuzzy

numbers as a fuzzy subset of the real line[4,5,6,8]. A fuzzy number is a

quantity whose values are imprecise, rather than exact as is the case with

single-valued numbers. Among the various shapes of fuzzy numbers,

Triangular fuzzy number and Trapezoidal fuzzy number are the most

commonly used membership function(Dubois and

Prade[3],1980,Zimmermann[17], 1996) In this paper a new operation of

Decagonal fuzzy numbers has been introduced with its basic membership

function followed by the properties of its arithmetic operations of fuzzy

numbers[1,2,3,9,13]. In few cases Triangular or Trapezoidal is not applicable

to solve the problem if it has ten different points; hence we make use of this

new operation of Decagonal fuzzy number to solve in such cases. Key words : Fuzzy number, Addition, Subtraction, Multiplication and division

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HEXAGONAL FUZZY NUMBERS

A fuzzy number Ac H is a hexagonal fuzzy number denoted by Ac H

(a1, a2, a3, a4, a5, a6) where a1, a2, a3, a4, a5, a6 are real numbers and its

membership function µAc H

(x) is given below.

0 for x < a1

12fff x@a1

a2 @a1

ffffffffffffffff g

for a1 ≤ x ≤ a2

12fff+ 1

2fff x@a2

a3 @a2

fffffffffffffffff g

for a2 ≤ x ≤ a3

µAc H

(x) = 1 for a3 ≤ x ≤ a4

1@12fff x@a4

a5 @a4

fffffffffffffffff g

for a4 ≤ x ≤ a5

12fff a6 @x

a6 @a5

fffffffffffffffff g

for a5 ≤ x ≤ a6

0 for x> a6

Figure 1 Graphical representation of a normal hexagonal fuzzy number for x ∈[0, 1]

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Decagonal fuzzy number :

A fuzzy number Ac D is a decagonal fuzzy number denoted by Ac D

(a1, a2, a3, a4, a5 a6, a7, a8, a9, a10) where a1, a2, a3, a4, a5 a6, a7, a8, a9, a10

are real numbers and its membership function µAc D

xa given below.

14fff x@a1

a2 @a1

ffffffffffffffff g

a1 ≤ x ≤ a2

14fff+ 1

4fff x@a2

a3 @a2

fffffffffffffffff g

a2 ≤ x ≤ a3

12fff+ 1

4fff x@a3

a4 @a3

fffffffffffffffff g

a3 ≤ x ≤ a4

34fff+ 1

4fff x@a4

a5 @a4

fffffffffffffffff g

a4 ≤ x ≤ a5

µAc D

xa = 1 a5 ≤ x ≤ a6

1@14fff x@a6

a7 @a6

fffffffffffffffff g

a6 ≤ x ≤ a7

34fff@1

4fff x@a7

a8 @a7

fffffffffffffffff g

a7 ≤ x ≤ a8

12fff@1

4fff x@a8

a9 @a8

fffffffffffffffff g

a8 ≤ x ≤ a9

14fff a10 @x

a10 @a9

fffffffffffffffffff g

a9 ≤ x ≤ a10

0 Otherwise

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Fig – 2 Graphical representation of a normal decagonal fuzzy number for

x ∈ [0, 1]

ALPHA CUT :

The classical set Ac α called alpha cut set is the set of elements whose

degree of membership is the set of elements whose degree of membership in

Ac D = (a1, a2, a3, a4, a5 a6, a7, a8, a9, a10) is no less than α it is defined as,

Aα = x 2 X /µAc D

≥ αT U

Dc α = P1 (α), P2 (α) for α ∈ [0, 0.25)

Q1 (α), Q2 (α) for α ∈ [0.25, 0.5)

R1 (α), R2 (α) for α ∈ [0.5, 0.75)

S1 (α), S2 (α) for α ∈ [0.75, 1)

α Cut Operations

If the crisp interval by α cut operations interval Aα shall be obtained

as follows for all α ∈ [0, 1].

Consider P1 (x) = α,

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(i.e) 14fff x@a1

a2 @a1

ffffffffffffffff g

a2 @a1

ffffffffffffffff g

x – a1 = 4α (a2 – a1)

x = 4 α (a2 – a1) + a1

(i.e) P1 (α) = 4 α (a2 - a1) + a1

Similarly from P2 (x) = α ,

(i.e) 14fff a10 @x

a10 @a9

a10 @xa10 @a9

a10 – x = 4α (a10 – a9)

-x = 4α (a10 – a9) – a10

x = - 4α (a10 – a9) + a10

(i.e) P2 (α) = - 4 α (a10 – a9) + a10

This implies

[P1 (α), P2 (α)] = [ 4 α (a2 – a1) + a1, - 4 α (a10 – a9) + a10]

Consider Q1 (x) = α,

14fff+ 1

4fff x@a2

a3 @a2

` affffffffffffffffffff = α,

14fff x@a2

a3 @a2

` affffffffffffffffffff = α @14fff

a3 @a2

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a3 @a2

` affffffffffffffffffff = 4 α @1` a

(x – a2) = (4α -1) (a3 – a2)

(x-a2) = 4 α (a3) – 4 α (a2) – a3 + a2

= 4 α (a3 – a2) – a3 + a2

x = 4 α (a3 – a2) – a3 + a2 + a2

x = 4 α (a3 – a2) – a3 + 2a2

x = 4 α (a3 – a2) – 2a2 – a3

Q1 (α)= 4 α (a3 – a2) + 2a2 – a3

Similarly from

Q2 (x) = α

(i.e) 12fff@1

4fff x@a8

a9 @a8

` affffffffffffffffffff = α

@14fff x@a8

a9 @a8

` affffffffffffffffffff= α @12fff

- (x – a8) = (4α - 2) (a9 – a8)

- x + a8 = (4α - 2) (a9 – a8)

- x = 4 α (a9) – 4 α (a8) – 2 a9 + 2 a8 – a8

- x = 4 α (a9 – a8) – 2a9 + a8

x = -4 α (a9 – a8) + 2a9 – a8

(i.e) Q2 (α) = - 4 α (a9 – a8) + 2a9 – a8

This implies

[Q1 (α), Q2 (α)] = [4 α (a3 – a2)–a3 + 2a2, -4α (a9 –a8)+2a9 –

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Consider R1 (x) = α

(i.e) 12fff+ 1

4fff x@a3

a4 @a3

` affffffffffffffffffff= α

14fff x@a3

a4 @a3

` affffffffffffffffffff = 4α @2` a

(x - a3) = (4α - 2) (a4 – a3)

(x – a3) = 4 α (a4) – 4α (a3) – 2a4 + 2a3

x = 4α (a4 – a3) – 2a4 + 2a3 + a3

x = 4α (a4 – a3) – 2a4 + 3a3

x = 4α (a4 – a3) + 3a3 – 2a4.

(i.e) R1 (α) = 4α (a4 – a3) + 3a3 – 2a4

Similarly from,

R2 (x) = α

(i.e) 34fff@1

4fff x@a7

a8 @a7

` affffffffffffffffffff= α

@14fff x@a7

a8 @a7

` affffffffffffffffffff= 4α @3` a

- (x – a7) = (4α - 3) (a8 – a7)

- (x – a7) = 4α (a8) – 4 α (a7) – 3a8 + 3a7

- x + a7 = 4α (a8 – a7) – 3a8 + 3a7 155

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- x = 4α (a8 – a7) – 3a8 + 3a7 – a7

x = - 4α (a8 – a7) + 3a8 – 2a7

x = - 4α (a8 – a7) – 2a7 + 3a8

(i.e) R2 (α) = - 4α (a8 – a7) – 2a7 + 3a8

This implies

[R1, (α), R2 (α)] = [4α (a4 – a3) + 3a3 – 2a4,

- 4α (a8 – a7) – 2a7 + 3a8]

Consider,

S1 (x) = α

(i.e) 34fff+ 1

4fff x@a4

a5 @a4

14fff x@a4

a5 @a4

` affffffffffffffffffff= α @34fff

a5 @a4

ffffffffffffffff= 4α @3` a

x – a4 = (4α - 3) (a5 – a4)

x – a4 = 4α (a5) - 4α (a4) – 3a5 + 3a4

x – a4 = 4α (a5 – a4) – 3a5 + 3a4

x = 4α (a5 – a4) – 3a5 + 3a4 + a4

x = 4α (a5 – a4) – 3a5 + 4a4

(i.e) S1 (α) = 4 α (a5 – a4) + 4a4 – 3a5

Similarly from,

S2 (x) = α

(i.e) 1@14fff x@a6

a7 @a6

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@14fff x@a6

a7 @a6

` affffffffffffffffffff = α @1

a7 @a6

` affffffffffffffffffff= α @1` a

- (x – a6) = (4α - 4) (a7 – a6)

- (x – a6) = (4α) (a7) - 4α (a6) – 4a7 + 4a6

- x + a6 = 4α (a7 – a6) – 4a7 + 4a6

- x = 4 α (a7 – a6 ) – 4a7 + 4a6 – a6

- x = 4α (a7 – a6) – 4a7 + 3a6

x = - 4α (a7 – a6) + 4a7 – 3a6

(i.e) S2 (α) = - 4α (a7 – a6) + 4a7 – 3a6

This implies

[S1 (α), S2 (α)] = [4α (a5 – a4) + 4a4 – 3a5,

- 4α (a7 – a6) + 4a7 – 3a6]

[4α (a2 – a1) + a1, - 4α(a10 – a9) + a10] for α ∈ [0, 0.25)

[4α (a3 – a2) + 2a2 – a3, - 4α (a9 – a8) + 2a9 – a8]

for α∈[0.25, 0.5)

Aα = [4α (a4 – a3) + 3a3 – 2a4, - 4α (a8 – a7) + 3a8 – 2a7]

for α ∈ [0.5, 0.75)

[4α (a5 – a4) + 4a4 – 3a5, -4α (a7 – a6) + 4a7 – 3a6]

for α ∈ [0.75, 1]

Operations of decagonal fuzzy numbers : [1, 2, 3, 4] 157

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Following are the three operations that can be performed on decagonal

fuzzy numbers. Suppose

Let Ac D = (a1, a2, a3, a4, a5 a6, a7, a8, a9, a10)

and Bc D = (b1, b2, b3, b4, b5 b6, b7, b8, b9, b10)

Addition :

Ac D (+) Bc D = (a1 + b1, a2 + b2 , a3 + b3, a4 + b4, a5 + b5, a6 + b6, a7 +

a8 + b8, a9 + b9, a10 + b10).

Subtraction :

Ac D (-) Bc D = (a1 - b1, a2 - b2 , a3 - b3, a4 - b4, a5 – b5, a6 - b6, a7 -

a8 - b8, a9 - b9, a10 - b10).

Multiplication :

Ac D (*) Bc D = (a1 * b1, a2 * b2 , a3 * b3, a4 * b4, a5 * b5, a6 * b6, a7 * b7,

a8 * b8, a9 * b9, a10 * b10).

Division :

Ac D÷ Bc D = (a1 ÷ b1, a2 ÷ b2 , a3 ÷ b3, a4 ÷ b4, a5 ÷ b5, a6 ÷ b6, a7 ÷ b7,

a8 ÷ b8, a9 ÷ b9, a10 ÷ b10).

Example : 1

Let Ac D = (3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0)

and Bc D = (4.8, 5.1, 5.4, 5.7, 6.0, 6.3, 6.6, 6.9, 7.2, 7.5)

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Ac D (+) Bc D = (8.1, 8.7, 9.3, 9.9, 10.5, 11.1, 11.7, 12.3, 12.9, 13.5)

Figure -1

A New Operation for addition, subtraction, Multiplication and division

on Decagonal fuzzy number.

α cut of a normal decagonal fuzzy number. The α cut of a normal

decagonal fuzzy number Ac D = (a1, a2, a3, a4, a5 a6, a7, a8, a9, a10) given by

the definition

(i.e) W = 1 for all α ∈ [0, 1] is

[4α (a2 – a1) + a1, - 4α (a10 – a9) + a10] for α ∈[0, 0.25)

[4α(a3 – a2)+2a2 – a3, - 4 α(a9 – a8) + 2a9 – a8] for α∈[0.25,

Aα = [4α(a4 – a3)+3a3 – 2a4, - 4α(a8 – a7)+3a8 – 2a7] for α∈[0.5,

[4α(a5 – a4)+ 4a4 – 3a5, - 4α(a7 – a6) + 4a7 – 3a6] for α∈[0.75,

Ac D + Bc D

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Addition of two decagonal fuzzy numbers :

and Bc D = (b1, b2, b3, b4, b5 b6, b7, b8, b9, b10) be two

decagonal fuzzy numbers for all α ∈ [0, 1].

Let us add the alpha cuts Aα and Bα of Ac D and Bc D using interval

arithmetic.

[4α (a2 – a1) + a1, - 4α (a10 – a9) + a10] +

[4α (b2 – b1) + b1, - 4α (b10 – b9) + b10] for α ∈ [0, 0.25)

[4α(a3 – a2) + 2a2 – a3, - 4α (a9 – a8) + 2a9 – a8] +

[4α (b3 – b2) + 2b2 – b3, - 4α (b9 – b8) +2b9 – b8]

for α ∈ [0.25, 0.5)

Aα + Bα = [4α (a4 – a3) + 3a3 – 2a4 , - 4α (a8 – a7) + 3a8 – 2a7] +

[4α (b4 – b3) + 3b3 – 2b4, - 4α (b8 – b7) + 3b8 – 2b7]

for α ∈ [0.5, 0.75)

[4α (a5 – a4) + 4a4 – 3a5, - 4α (a7 – a6) + 4a7 – 3a6]

+ [4α (b5 – b4) + 4b4 – 3b5, - 4α (b7 – b6) + 4b7 – 3b6]

for α ∈ [0.75, 1]

Addition Operation :

1. Aα = [4α (a2 – a1) + a1, - 4α (a10 – a9) + a10]

Bα = [4α (b2 – b1) + b1, - 4α (b10 – b9) + b10]

a1 = 3.3, a2 = 3.6, a3 = 3.9, a4 = 4.2, a5 = 4.5

a6 = 4.8, a7 = 5.1, a8 = 5.4, a9 = 5.7, a10 = 6.0

b1 = 4.8, b2 = 5.1, b3 = 5.4, b4 = 5.7, b5 = 6.0

b6 = 6.3, b7 = 6.6, b8 = 6.9, b9 = 7.2, b10 = 7.5

Aα = [4α (3.6 – 3.3) + 3.3, - 4 α (6.0 – 5.7) + 6.0]

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= [4α (0.3) + 3.3, - 4α (0.3) + 6.0]

Aα = [1.2 α + 3.3, -1.2 α + 6.0]

Bα = [4α (5.1 - 4.8) + 4.8, - 4α (7.5 – 7.2) + 7.5]

= [4α (0.3)+4.8, - 4α (0.3) + 7.5]

Bα = [1.2α + 4.8, - 1.2α+ 6.0]

For α ∈ [0, 0.25)

Aα = [1.2α + 3.3, - 1.2α + 6.0]

Bα = [1.2α + 4.8, - 1.2α + 7.5]

Aα + Bα = [2.4 α + 8.1, - 2.4 α + 13.5]

2. Aα = [4α (a3 – a2) + 2a2, - a3, - 4α (a9 – a8) +2a9 – a8]

Bα = [4α (b3 – b2) + 2b2 - b3 - 4α (b9 – b8) + 2b9 – b8]

Aα = [4α (3.9 – 3.6) + 2 (3.6) – 3.9,

- 4α (5.7 – 5.4) + 2 (5.7) – 5.4]

= [4α (0.3) + 7.2 – 3.9, - 4α (0.3) + 11.4 – 5.4]

Aα = [1.2 α+ 3.3, - 1.2 α + 6.0]

Bα = [4α(5.4 – 5.1)+2 (5.1) – 5.4, - 4α(7.2 – 6.9)+2 (7.2) - 6.9]

= [4α (0.3) + 4.8, - 4α (0.3) + 7.5]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

For α ∈ [0.25, 0.5)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα + Bα = [2.4 α+ 8.1, - 2.4 α + 13.5]

3. Aα = [4α (a4 – a3) + 3a3 – 2a4, - 4α (a8 – a7) + 3a8 - 2a7]

Bα = [4α (b4 – b3) + 3b3 – 2b4, - 4α (b8 – b7) + 3b8 – 2b7] 161

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Aα = [4α (4.2 – 3.9) + 3 (3.9) – 2 (4.2),

- 4α (5.4 - 5.1) + 3 (5.4) – 2(5.1)].

= [4α (0.3) + 3 (3.9) – 2 (4.2), - 4α (0.3) + 3 (5.4) – 2 (5.1)]

Aα = 1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [4α (6.0 – 5.7) + 3 (5.4) – 2 (5.7),

- 4α (6.9 – 6.6) + 3 (6.9) – 2 (6.6)]

= [4α (0.3) + 3 (5.4) – 2(5.7) , - 4α (0.3) + 3 (6.9) – 2 (6.6)]

Bα = [1.2 α + 4.8, - 1.2α + 7.5]

For α ∈ [0.5, 0.75)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8 – 1.2 α + 7.5]

Aα + Bα = [2.4 α + 8.1, - 2.4 α + 13.5]

4. Aα = [4α (a5 – a4) + 4a4 – 3a5, - 4α (a7 – a6) + 4a7 - 3a6]

Bα = [4α (b5 – b4) + 4b4 – 3b5, - 4α (b7 – b6) + 4b7 – 3b6]

Aα = [4α (4.5 – 4.2) + 4 (4.2) – 3 (4.5) ,

- 4 α (5.1 – 4.8) + 4 (5.1) – 3 (4.8)]

= [4α (0.3) + 4 (4.2) – 3 (4.5), - 4α (0.3) + 4 (5.1) – 3 (4.8)]

Aα = 1.2 α + 3.3, - 1.2 α + 6.0

Bα = [4α (6.0 – 5.7) + 4 (5.7) – 3 (6.0)]

- 4α (6.6 – 6.3) + 4 (6.6) – 3 (6.3)]

= [4 α (0.3)+ 4 (5.7) – 3 (6.0), - 4α (0.3) + 4 (6.6) – 3 (6.3)]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

For α ∈ [0.75, 1]

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5] 162

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Aα + Bα = [2.4 α + 8.1, - 2.4 α+ 13.5]

To verify this new addition operation with ordinary addition operation :

Ac D = (3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0)

Bc D = (4.8, 5.1, 5.4, 5.7, 6.0, 6.3, 6.6, 6.9, 7.2, 7.5)

For α ∈ [0, 0.25)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

For α ∈ [0.25, 0.5)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα + Bα = [2.4 α + 8.1, - 2.4 α + 13.5]

For α ∈ [0.5 , 0.75)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα + Bα = [2.4 α + 8.1, - 2.4 α + 13.5]

For α ∈ [0.75, 1]

Aα = [1.2 α + 3.3, - 1.2 α+ 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα + Bα = [2.4 α + 8.1, - 2.4 α + 13.5]

As for α ∈ [0, 0.25), α ∈ [0.25, 0.5)

α ∈ [0.5, 0.75), α ∈ [0.75, 1]

Arithmetic intervals are same.

Aα + Bα = [2.4 α + 8.1, - 2.4 α + 13.5] for α ∈ [0, 1] 163

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When α = 0,

⇒ A0 + B0 = [2.4 (0) + 8.1, - 2.4 (0) + 13.5]

A0 + B0 = [8.1, 13.5]

α = 0.25,

⇒ A0.25 + B0.25 = [2.4 (0.25) + 8.1, - 2.4 (0.25) + 13.5]

A0.25 + B0.25 = [8.7, 12.9]

α = 0.5,

⇒ A0.5 + B0.5 = [2.4 (0.5) + 8.1, - 2.4 (0.5) + 13.5]

A0.5 + B0.5 = [9.3, 12.3]

α = 0.75,

⇒ A0.75 + B0.75 = [2.4 (0.75) + 8.1, - 2.4 (0.75) + 13.5]

A0.75 + B0.75 = [9.9, 11.7]

α = 1,

⇒ A1 + B1 = [2.4 (1) + 8.1, - 2.4 (1) + 13.5]

A1 + B1 = [10.5, 11.1]

Hence Aα + Bα = [8.1, 8.7, 9.3, 9.9, 10.5, 11.1, 11.7, 12.3, 12.9, 13.5]

Hence all the points coincides with the sum of the two decagonal

fuzzy number.

Therefore addition of two α - cuts lies within the interval.

Subtraction of two decagonal fuzzy numbers :

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Let us subtract the alpha cuts Aα and Bα of Ac D and Bc D using interval

arithmetic.

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[4α (a2 – a1) + a1, - 4α (a10 – a9) + a10] -

[4α (b2 – b1) + b1, - 4α (b10 – b9) + b10] for α ∈ [0, 0.25)

[4α(a3 – a2) + 2a2 – a3, - 4α (a9 – a8) + 2a9 – a8] -

[4α (b3 – b2) + 2b2 – b3, - 4α (b9 – b8) +2b9 – b8]

for α ∈ [0.25, 0.5)

Aα - Bα = [4α (a4 – a3) + 3a3 – 2a4 , - 4α (a8 – a7) + 3a8 – 2a7] -

[4α (b4 – b3) + 3b3 – 2b4, - 4α (b8 – b7) + 3b8 – 2b7]

for α ∈ [0.5, 0.75)

[4α (a5 – a4) + 4a4 – 3a5, - 4α (a7 – a6) + 4a7 – 3a6] –

[4α (b5 – b4) + 4b4 – 3b5, - 4α (b7 – b6) + 4 b7 – 3b6]

for α ∈ [0.75, 1]

To verify this new Subtraction operation with ordinary subtraction

operation :

Ac D = (3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0)

Bc D = (4.8, 5.1, 5.4, 5.7, 6.0, 6.3, 6.6, 6.9, 7.2, 7.5)

For α ∈ [0, 0.25)

Aα = [4α (a2 – a1) + a1, - 4 α (a10 – a9) + a10]

Bα = [4α (b2 – b1) + b1, - 4α (b10 – b9) + b10]

Aα = [4α (3.6 – 3.3) + 3.3, - 4α (6.1 – 5.7) + 6.0]

Aα = [4α (0.3) + 3.3, - 1.2 α - 6.0]

Aα = [1.2α + 3.3, - 4α (0.3) + 6.0]

Bα = [4α (5.1 – 4.8) + 4.8, - 4α (7.5 – 7.2) + 7.5]

Bα = [4α (0.3) + 4.8, - 4α (0.3)+ 7.5]

Bα = [1.2α + 4.8, - 1.2α + 7.5]

For α ∈ [0, 0.25) 166

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Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα - Bα = [- 1.5, - 1.5]

For α ∈ [0.25, 0.5)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα - Bα = [- 1.5, - 1.5]

For α ∈ [0.5, 0.75)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα - Bα = [- 1.5, - 1.5]

For α ∈ [0.75, 1]

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα - Bα = [- 1.5, - 1.5]

As For α ∈ [0, 0.25), α ∈ [0.25, 0.5), α ∈ [0.5, 0.75)

α ∈ [0.75, 1] arithmetic intervals are

Therefore Aα - Bα = [2.4 α + 8.1, - 2.4 α + 13.5] for α ∈ [0, 1]

when α = 0, A0 – B0 = [ - 1.5, - 1.5]

when α = 0.25, A0.25 – B0.25 = [ - 1.5, - 1.5]

when α = 0.75, A0.75 – B0.75 = [ - 1.5, - 1.5]

when α = 1, A1 – B1 = [ - 1.5, - 1.5]

Hence Aα - Bα = [- 1.5, - 1.5, - 1.5, - 1.5, - 1.5, - 1.5, - 1.5, - 1.5, - 1.5, -

1.5,] 167

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Hence all the points coincides with the difference of the two

decagonal fuzzy number.

Therefore subtraction of two α - cuts lies within the intervals.

Multiplication of two decagonal fuzzy number :

Let us multiply the alpha cuts Aα and Bα of Ac D and Bc D using

interval arithmetic.

[4α (a2 – a1) + a1, - 4α (a10 – a9) + a10] *

[4α (b2 – b1) + b1, - 4α (b10 – b9) + b10] for α ∈ [0, 0.25)

[4α(a3 – a2) + 2a2 – a3, - 4α (a9 – a8) + 2a9 – a8] *

[4α (b3 – b2) + 2b2 – b3, - 4α (b9 – b8) +2b9 – b8]

for α ∈ [0.25, 0.5)

Aα * Bα = [4α (a4 – a3) + 3a3 – 2a4 , - 4α (a8 – a7) + 3a8 – 2a7] *

[4α (b4 – b3) + 3b3 – 2b4, - 4α (b8 – b7) + 3b8 – 2b7]

for α ∈ [0.5, 0.75)

[4α (a5 – a4) + 4a4 – 3a5, - 4α (a7 – a6) + 4a7 – 3a6] *

[4α (b5 – b4) + 4b5 – 3b5, - 4α (b7 – b6) + 4b7 – 3b6]

for α ∈ [0.75, 1]

To verify the new multiplication operation with ordinary multiplication

operation :

Ac D = (3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0)

Bc D = (4.8, 5.1, 5.4, 5.7, 6.0, 6.3, 6.6, 6.9, 7.2, 7.5) 168

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For α ∈ [0, 0.25)

Aα = [4α (a2 – a1) + a1, - 4 α (a10 – a9) + a10]

Bα = [4α (b2 – b1) + b1, - 4α (b10 – b9) + b10]

Aα = [4α (3.6 – 3.3) + 3.3, - 4α (6.1 – 5.7) + 6.0]

Aα = [4α (0.3) + 3.3, - 4 α (0.3) + 6.0]

Aα = [1.2α + 3.3, - 1.2α + 6.0]

Bα = [4α (5.1 – 4.8) + 4.8, - 4α (7.5 – 7.2) + 7.5]

Bα = [4α (0.3) + 4.8, - 4α (0.3)+ 7.5]

Bα = [1.2α + 4.8, - 1.2α + 7.5]

For α ∈ [0, 0.25)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα * Bα = [1.44 α + 15.84, 1.44 α + 45]

For α ∈ [0.25, 0.5)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα * Bα = [1.44 α + 15.84, 1.44 α + 45]

For α ∈ [0.5, 0.75)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα * Bα = [1.44 α + 15.84, 1.44 α + 45]

For α ∈ [0.75, 1]

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα * Bα = [1.44 α + 15.84, 1.44 α + 45] 169

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As For α ∈ [0, 0.25), α ∈ [0.25, 0.5), α ∈ [0.5, 0.75)

and α ∈ [0.75, 1] arithmetic intervals are same

Therefore Aα * Bα = [1.44 α + 15.84, 1.44 α + 45] for α ∈ [0, 1]

when α = 0 ⇒

A0 * B0 = [1.44 (0) + 15.84, 1.44 (0) + 45]

A0 * B0 = [15.84, 45]

when α = 0.25 ⇒

A0.25 * B0.25 = [1.44 (0.25) + 15.84, 1.44 (0.25) + 45]

A0.25 * B0.25 = [18.36, 41.04]

when α = 0.5 ⇒

A0.5 * B0.5 = [1.44 (0.5) + 15.84, 1.44 (0.5) + 45]

A0.5 * B0.5 = [21.06, 37.26]

when α = 0.75 ⇒

A0.75 * B0.75 = [1.44 (0.75) + 15.84, 1.44 (0.75) + 45]

A0.75 * B0.75 = [23.94, 33.66]

when α = 1 ⇒

A1 * B1 = [1.44 (1) + 15.84, 1.44 (1) + 45]

A1 * B1 = [27, 30.43]

Hence Aα * Bα = [15.84, 18.36, 21.06, 23.94, 27, 30.43, 33.66, 37.26,

41.04, 45]

Hence all the points coincides with the multiply of two decagonal

fuzzy number.

Therefore multiplication of two α - cuts lies within the intervals.

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Symmetric image :

If Ac D = (a1, a2, a3, a4, a5 a6, a7, a8, a9, a10) is the decagonal fuzzy

number then - Ac D = (- a10, - a9, - a8, - a7, - a6, - a5, - a4, - a3, - a2, - a1)

which in the symmetric image of Ac D is also an decagonal fuzzy number.

Example :

If Ac D = (3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0)

- Ac D = (- 6.0, - 5.7, - 5.4, - 5.1, - 4.8, - 4.5, - 4.2, - 3.9, -3.6, -3.3)

which is again an decagonal fuzzy number.

Dividing of two decagonal fuzzy number :

Let us divided the alpha cuts Aα and Bα of Ac D and Bc D using interval

arithmetic.

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[4α (a2 – a1) + a1, - 4α (a10 – a9) + a10] ÷

[4α (b2 – b1) + b1, - 4α (b10 – b9) + b10] for α ∈ [0, 0.25)

[4α(a3 – a2) + 2a2 – a3, - 4α (a9 – a8) + 2a9 – a8] ÷

[4α (b3 – b2) + 2b2 – b3, - 4α (b9 – b8) +2b9 – b8]

for α ∈ [0.25, 0.5)

Aα ÷ Bα = [4α (a4 – a3) + 3a3 – 2a4 , - 4α (a8 – a7) + 3a8 – 2a7] ÷

[4α (b4 – b3) + 3b3 – 2b4, - 4α (b8 – b7) + 3b8 – 2b7]

for α ∈ [0.5, 0.75)

[4α (a5 – a4) + 4a4 – 3a5, - 4α (a7 – a6) + 4a7 – 3a6] ÷

[4α (b5 – b4) + 4b4 – 3b5, - 4α (b7 – b6) + 4b7 – 3b6]

for α ∈ [0.75, 1]

To verify this new division operation with ordinary division operation :

Ac D = (3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0)

Bc D = (4.8, 5.1, 5.4, 5.7, 6.0, 6.3, 6.6, 6.9, 7.2, 7.5)

For α ∈ [0, 0.25)

Aα = [4α (a2 – a1) + a1, - 4 α (a10 – a9) + a10]

Bα = [4α (b2 – b1) + b1, - 4α (b10 – b9) + b10]

Aα = [4α (3.6 – 3.3) + 3.3, - 4α (6.0 – 5.7) + 6.0]

Aα = [4α (0.3) + 3.3, - 4 α (0.3) + 6.0]

Aα = [1.2α + 3.3, - 1.2α + 6.0]

Bα = [4α (5.1 – 4.8) + 4.8, - 4α (7.5 – 7.2) + 7.5]

Bα = [4α (0.3) + 4.8, - 4α (0.3)+ 7.5]

Bα = [1.2α + 4.8, - 1.2α + 7.5]

For α ∈ [0, 0.25) 172

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Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

Aα ÷ Bα = 1.2α + 3.31.2α + 4.8ffffffffffffffffffffffffff, @1.2α + 6.0

@1.2α + 7.5fffffffffffffffffffffffffffffffF G

For α ∈ [0.25, 0.5)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

For α ∈ [0.5, 0.75)

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

For α ∈ [0.75, 1]

Aα = [1.2 α + 3.3, - 1.2 α + 6.0]

Bα = [1.2 α + 4.8, - 1.2 α + 7.5]

As for α ∈ [0, 0.25), α ∈ [0.25, 0.5), α ∈ [0.5, 0.75)

and α ∈ [0.75, 1] arithmetic intervals are same

When α= 0

⇒ A0 ÷ B0 = 1.2 0

a+ 3.3

1.2 0` a

+ 4.8fffffffffffffffffffffffffffff,

@1.2 0a

+ 6.0@1.2 0

` a+ 7.5

fffffffffffffffffffffffffffffffffffH

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= 3.34.8ffffffff, 6.0

7.5ffffffffF G

A0 ÷ B0 = 0.6875, 0.8

When α= 0.25

⇒ A0.25 ÷ B0.25 = 1.2 0.25

a+ 3.3

1.2 0.25` a

+ 4.8ffffffffffffffffffffffffffffffffffff,

@1.2 0.25a

+ 6.0@1.2 0.25

` a+ 7.5

ffffffffffffffffffffffffffffffffffffffffffH

= 3.65.1ffffffff, 5.7

7.2ffffffffF G

A0.25 ÷ B0.25 = [0.705, 0.79]

When α= 0.5

⇒ A0.5 ÷ B0.5 = 1.2 0.5

a+ 3.3

1.2 0.5` a

+ 4.8fffffffffffffffffffffffffffffffff,

@1.2 0.5a

+ 6.0@1.2 0.5

` a+ 7.5

fffffffffffffffffffffffffffffffffffffffH

= 3.95.4ffffffff, 5.4

6.9ffffffffF G

A0.5 ÷ B0.5 = [0.722, 0.782]

When α= 0.75

⇒ A0.75 ÷ B0.75 = 1.2 0.75

a+ 3.3

1.2 0.75` a

+ 4.8ffffffffffffffffffffffffffffffffffff,

@1.2 0.75a

+ 6.0@1.2 0.75

` a+ 7.5

ffffffffffffffffffffffffffffffffffffffffffH

= 4.25.7ffffffff , 5.1

6.6ffffffffF G

A0.75 ÷ B0.75 = [0.736, 0.772]

When α= 1

⇒ A1 ÷ B1 = 1.2 1

a+ 3.3

1.2 1` a

+ 4.8ffffffffffffffffffffffffffff,

@1.2 1a

+ 6.0@1.2 1

` a+ 7.5

ffffffffffffffffffffffffffffffffffH

= 4.560ffffffff, 4.8

6.3ffffffffF G

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A1 ÷ B1 = [0.75, 0.761]

A1 ÷ B1 = [0.6875, 0.705, 0.722, 0.736, 0.75, 0.761, 0.772, 0.782, 0.79, 0.8]

Hence all the points coincides with the divide of the two decagonal

fuzzy number. Therefore division of two α - cuts lies within the interval.

CONCLUSIONS

In this paper decagonal Fuzzy number has been newly introduced and

the alpha cut operations of arithmetic function principles using addition,

subtraction multiplication and division has been fully modified with some

conditions and has been explained with numerical examples. In a particular

case of the growth rate in bacteria which consists of ten points is difficult to

solve using trapezoidal or triangular fuzzy numbers, therefore decagonal

fuzzy numbers plays a vital role in solving the problem. It also helps us to

solve many optimization problems in future which has ten parameters as in

the above case.

REFERENCES:

[1] Abhinav Bansal (2011) Trapezoidal Fuzzy numbers

(a,b,c,d):Arithmetic behavior. International Journal of Physical and

Mathematical Sciences, ISSN-2010-1791.

[2] Bansal,A.,(2010)Some non linear arithmetic operations on triangular

fuzzy numbers(m,B, a). Advances in fuzzy mathematics, 5,147-156.

[3] Dubois.D and Prade.H,(1978) Operations on fuzzy numbers,

,International Journal of Systems Science, vol.9, no.6.,pp.613-626.

[4] Dwyer.,(1965), P.S. Fuzzy sets. Information and Control,

No.8:338–353. 175

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ISSN 2348 – 7968

[5] Fuller.R and Majlender.P.,(2003), On weighted possibilistic mean and

variance of fuzzy numbers, Fuzzy Sets and Systems, vol.136,

pp.363-374

[6] Heilpern.S.,(1997), Representation and application of fuzzy numbers,

Fuzzy sets and Systems, vol.91, no.2, pp.259-268.

[7] Klaua.D.,(1965) ,Über einen Ansatz zur mehrwertigen Mengenlehre.

Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876

[8] Klir.G.J., (2000), Fuzzy Sets: An Overview of Fundamentals,

Applications, and Personal views. Beijing Normal University Press,

pp.44-49.

[9] Klir., (1997) Fuzzy arithmetic with requisite constraints, Fuzzy Sets

System, vol. 91, ,pp. 165– 175.

[10] Kauffmann,A.,(1980) Gupta,M., Introduction to Fuzzy Arithmetic

:Theory and Applications,Van Nostrand Reinhold, New York.

[11] Malini.S.U,Felbin.C.Kennedy.,(2013), An approach for solving Fuzzy

Transportation using Octagonal Fuzzy numbers,Applied Mathematical

Sciences,no.54,2661-2673

[12] Nasseri.H(2008) Positive and non-negative, International

Mathematical Forum,3,1777-1780.

[13] Rezvani .S.,(2011).,Multiplication Operation on Trapezoidal Fuzzy

numbers, Journal of Physical Sciences,Vol no-15,17-26

[14] Yager.R.,(1979) On Solving Fuzzy Mathematical relationships,

Information control,41,29-55.

[15] Zadeh,L.A.,(1965) Fuzzy Sets, Information and Control.,No.8

pp.338-353.

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