A NEW OPERATION ON HEXAGONAL FUZZY NUMBER

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

A NEW OPERATION ON HEXAGONAL FUZZY NUMBER P. Rajarajeswari1 , A.Sahaya Sudha2 and R.Karthika3 1

Department of Mathematics, Chikkanna Government Arts College, Tirupur-641 602 p.rajarajeswari29@gmail.com 2

Department of Mathematics, Nirmala College for women, Coimbatore-641018 sudha.dass@yahoo.com

3

Department of Mathematics, Hindustan Institute of Technology, Coimbatore-641028. karthika526@gmail.com

ABSTRACT: The Fuzzy set Theory has been applied in many fields such as Management, Engineering etc. In this paper a new operation on Hexagonal Fuzzy number is defined where the methods of addition, subtraction, and multiplication has been modified with some conditions. The main aim of this paper is to introduce a new operation for addition, subtraction and multiplication of Hexagonal Fuzzy number on the basis of alpha cut sets of fuzzy numbers.

KEYWORDS: Fuzzy arithmetic, Hexagonal fuzzy numbers, Function principles

1. INTRODUCTION: 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 hexagonal 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 six different points; hence we make use of this new operation of hexagonal fuzzy number to solve in such cases.

2. PRELIMINARIES [15, 16]: 2.1 Definition: Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function : → [0,1]

and A(x) is interpreted as the degree of membership of element x in fuzzy A for each x ∈ X .

DOI : 10.5121/ijfls.2013.3302

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

The value zero is used to represent complete non-membership; the value one is used to represent complete membership and values in between are used to represent intermediate degrees of membership. The mapping A is also called the membership function of fuzzy set A

2.2 Definition: A Fuzzy number “A” is a convex normalized fuzzy set on the real line R such that: • •

There exist at least one xo ∈ R with µ A(xo) = 1 µ A(x) is piecewise continuous

2.3 Definition:

A fuzzy number is a triangular fuzzy number denoted by (a1, a2, a3) where a1, a2 and a3 are real numbers and its membership function is given below.

(x) =

2.4 Definition:

( ) ≤ ≤

( )

! ( ! ) ≤ ≤ " 0 #ℎ% &'(%

( )

)

A fuzzy set = (a1, a2, a3, a4) is said to be trapezoidal fuzzy number if its membership function is given by where a1 ≤ a2 ≤ a3 ≤ a4 0 < ( ) * * ( ) ≤ ≤

(x) = 1 ≤ ≤ " ) ( , ) " ≤ ≤ * * ( , !)

0 >

3. HEXAGONAL FUZZY NUMBERS:

A fuzzy number / is a hexagonal fuzzy number denoted by / (a1, a2, a3, a4, a5, a6) where a1, a2, a3, a4, a5, a6 are real numbers and its membership function µ (x) is given below.

0

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

µ (x) = 0

0 <

* 1 2 ≤ ≤ * * + 1 ! 2 ≤ ≤ " 1 " ≤ ≤ - )

, * 1 − 1 5 , 2 - ≤ ≤ 6 * 7 1 2 6 ≤ ≤ 8 * 7 5 0 > 8

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

3.2 Definition:

An Hexagonal fuzzy number denoted by / is defined as ; = ( P1(u) , Q1(v), Q2 (v), P2(u)) for u ∈ [0,0.5] and v ∈ [0.5,w] where, (i) P1(u) is a bounded left continuous non decreasing function over [0,0.5] (ii) Q1 (v) is a bounded left continuous non decreasing function over [0.5,w] (iii) Q2 (v) is a bounded continuous non increasing function over [w, 0.5] (iv) P2 (u) is a bounded left continuous non increasing function over [0.5,0] 3.2.1 Remark:

If w = 1, then the hexagonal fuzzy number is called a normal hexagonal fuzzy number. Here ; represents a fuzzy number in which “w” is the maximum membership value that a fuzzy number takes on whenever a normal fuzzy number is meant, the fuzzy number is shown by / for convenience.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

3.2.2 Remark:

Hexagonal fuzzy number / is the ordered quadruple P1(u) , Q1(v), Q2 (v), P2(u) for [0,0.5] and v ∈ [0.5,w] where, P1 (u) = 1

u∈

< 2

Q1 (v) = + 1

= 2 !

Q2 (v) = 1 − 1

= , 2 5 ,

P2 (u) = 1 7 2

< 7

5

3.2.3 Remark: Membership function µ

0

(x) are continuous functions.

3.3 Definition [12]:

A positive hexagonal fuzzy number / is denoted as / = (a1, a2, a3, a4, a5, a6) where all ai’s > 0 for all i= 1, 2, 3,4,5,6.

Example: A= (1, 2, 3, 5, 6, 7)

3.4 Definition:

A negative hexagonal fuzzy number / is denotes as / = (a1, a2, a3, a4, a5, a6) where all ai’s < 0 for all i= 1, 2, 3,4,5,6.

Example: / = (-8,-7,-6,-4,-3,-2)

Note:

A negative hexagonal fuzzy number can be written as the negative multiplication of a positive hexagonal fuzzy number.

Example: / = (-2,-4,-6,-8,-10,-12) Then / = -(2,4,6,8,10,12)

3.5 Definition:

Let / = (a1, a2, a3, a4, a5, a6) and > / = (b1, b2, b3, b4, b5, b6) be two hexagonal fuzzy number, If ̅/ is identically equal to >@/ only if a1 = b1, a2 = b2, a3 = b3, a4 = b4, a5 = b5, a6 = b6.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

4. ALPHA CUT:

The classical set A called alpha cut set is the set of elements whose degree of membership is the set of elements whose degree of membership in / = (a1, a2, a3, a4, a5, a6) is no less than, B it is defined as α = {x ∈ X/ µ C

=

0

(x) ≥ α }

[D (B) , D(B)] B ∈ [0,0.5))

[G (B) , G (B)] B ∈ [0.5,1]

4.1 α Cut Operations [18]:

If we get crisp Interval by α cut operations Interval α shall be obtained as follows for all α ∈ [0, 1], Consider G (x) = α , (i.e) + 1

2 !

=α

x = 2B( " − ) - " + 2

(i.e) G (B) = 2B( " − ) - " + 2 Similarly from G (x) = α , 1− 1

, 2 5 ,

=α

x = -2 B( 6 − - ) + 2 6 − -

(i.e) G (B) = -2 B( 6 − - ) + 2 6 − -

This implies

[G (B) , G (B)] = [2B( " − ) − " + 2 , −2 B( 6 − - ) + 2 6 − - ]

Consider D (x) = α ,

D (B) = x =2 B( − ) +

Similarly from D (x) = α , We get,

(ie) D (B) = -2 B( 8 − 6 ) + 8

This implies

[D (B) , D(B)] = [2 B( − ) + , −2 B( 8 − 6 ) + 8 ] 19

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

Hence

A = J

[2 B( − ) + , −2 B( 8 − 6 ) + 8 ] α ∈ [0,0.5)

) [2B( " − ) − " + 2 , −2 B( 6 − - ) + 2 6 − - ] α ∈ [0.5, w]

4.2 Operations of Hexagonal Fuzzy numbers [1, 2, 3, 17]:

Following are the three operations that can be performed on hexagonal fuzzy numbers, Suppose / = (a1, a2, a3, a4, a5, a6) and > / = (b1, b2, b3, b4, b5, b6) are two hexagonal fuzzy numbers then • • •

Addition : / (+)> / = (a1 + b1, a2 + b2 , a3 + b3 , a4 + b4 , a5 + b5 , a6 + b6 ) Subtraction: / (−)> / = (a1 - b1, a2 - b2 , a3 - b3 , a4 - b4 , a5 - b5 , a6 - b6 ) Multiplication: / (∗)> / = (a1 * b1, a2 * b2 , a3 * b3 , a4 * b4 , a5 * b5 , a6 * b6 )

Example 1:

Let / = (1,2,3,5,6,7) and > / = (2,4,6,8,10,12) be two fuzzy numbers then

/ + > / = (3, 6, 9, 13, 16, 19)

M N, B MN Figure 2: Hexagonal Fuzzy numberA

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

Figure 3:

Example 2:

M N+ B MN Hexagonal Fuzzy number A

Let / = (1,2,3,5,6,7) and > / = (2,4,6,10,12,14) be two fuzzy numbers

Then / - > / = (-1, -2, -3, -5, -6, -7)

Example 3:

M N- B MN Figure: 4 Hexagonal Fuzzy numbers A

Let / = (1,2,3,5,6,7) and> / = (2,4,6,8,10,12) be two fuzzy numbers Then / * > / = (2, 8, 18, 40, 60, 84)

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

Figure :5

MN* B MN Hexagonal fuzzy number A

4.3 A New operation for Addition, Subtraction & Multiplication on Hexagonal fuzzy number: 4.3.1 α Cut of a normal hexagonal fuzzy number:

The α Cut of a normal hexagonal fuzzy number / = (a1, a2, a3, a4, a5, a6) given by the definition (i.e.) w=1 for all B ∈ [0,1] is A

[2 B( − ) + , −2 B( 8 − 6 ) + 8 ] α ∈ [0,0.5)

) = P[2B( − ) − + 2 , −2 B( − ) + 2 − ] B ∈ [0.5,1] " " 6 6

4.3.2 Addition of two hexagonal fuzzy numbers:

Let / = (a1, a2, a3, a4, a5, a6) and > / = (b1, b2, b3, b4, b5, b6) be two hexagonal fuzzy numbers for all B ∈ [0,1]. Let us add the alpha cuts A and >A of / and > / using interval arithmetic.

A + >A =

[2 B( − ) + , −2 B( 8 − 6 ) + 8 ] + [2 B(T − T ) + T , −2 B(T8 − T6 ) + T8 ]

α ∈ [0,0.5) *

* 2B(T" − T ) − T" + 2T , ) −2 B(T6 − T- ) + V [2B( " − ) − " + 2 , −2 B( 6 − - ) + 2 6 − - ] + U 2 T6 − T- * * [0.5,1]

B ∈

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

Consider the example 1 such that

/ = (1,2,3,5,6,7) and > / = (2,4,6,8,10,12)

For B ∈ [0.5,1]

A = [2B + 1 , −2B + 7] >A = [4B + 2 , −4B + 12]

A + >A = [6B + 3 , −6B + 19]

For B ∈ [0,0.5)

A = [2B + 1 , −2B + 7]

>A = [4B + 2 , −4B + 12]

A + >A = [6B + 3 , −6B + 19]

Since for both B ∈ [0,0.5) & B ∈ [0.5,1] arithmetic intervals are same Therefore A + >A = [6B + 3 , −6B + 19] for all B ∈ [0,1] When B = 0 \ + >\ =[3,19]

Likewise for B = 0.5

\.6 + >\.6 =[6,16]

and for B = 1 + > =[9,13]

Hence A + >A = [3, 6, 9, 13, 16, 19] hence all the points coincides with the sum of the two hexagonal fuzzy number.

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

4.3.3 Subtraction of two hexagonal Fuzzy numbers:

Let / = (a1, a2, a3, a4, a5, a6) and > / = (b1, b2, b3, b4, b5, b6) be two hexagonal fuzzy numbers for all B ∈ [0,1]. Let us subtract the alpha cuts A and >A of / and > / using interval arithmetic.

A − >A = [2 B( − ) + , −2 B( 8 − 6 ) + 8 ] − [2 B(T − T ) + T , −2 B(T8 − T6 ) + T8 ]

α ∈ [0,0.5) *

* 2B(T" − T ) − T" + 2T , ) [2B( ) ) −

−

+ 2 , −2 B( −

+ 2 −

] − U −2 B(T6 − T- ) + V " " 6 6

2 T6 − T- * *

B ∈ [0.5,1]

Let / = (1,2,3,5,6,7) and> / = (2,4,6,10,12,14) Consider the example 2 such that For B ∈ [0.5,1]

A = [2B + 1 , −2B + 7] >A =[4B + 2 , −4B + 14] 23

A – >A = [−2B − 1 , 2B − 7]

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

For B ∈ [0,0.5)

A = [2B + 1 , −2B + 7] >A = [4B + 2 , −4B + 14]

A − >A = [−2B − 1 , 2B − 7]

Since for both B ∈ [0,0.5) & B ∈ [0.5,1] arithmetic intervals are same

Therefore A − >A = [−2B − 1 , 2B − 7] for all B ∈ [0,1] When B = 0 \ − >\ =[−1, −7] Likewise for B = 0.5 \.6 − >\.6 =[−3, −5] and for B = 1 − > =[−2, −6] Hence A − >A = [-1, -2, -3, -5, -6, -7] hence all the points coincides with the difference of the two hexagonal fuzzy number.

Therefore difference of two B-cuts lies within the interval 4.3.4 Multiplication of two hexagonal fuzzy numbers:

Let / = (a1, a2, a3, a4, a5, a6) and > / = (b1, b2, b3, b4, b5, b6) be two hexagonal fuzzy numbers for all B ∈ [0,1]. Let us multiply the alpha cuts A and >A of / and > / using interval arithmetic.

A ∗ >A = [2 B( − ) + , −2 B( 8 − 6 ) + 8 ] ∗ [2 B(T − T ) + T , −2 B(T8 − T6 ) + T8 ]

α ∈ [0,0.5) * *

2B(T" − T ) − T" + 2T , −2 B(T6 − T- ) + ) _ [2B( " − ) − " + 2 , −2 B( 6 − - ) + 2 6 − - ] ∗ ^ 2 T6 − T- * *

B ∈ [0.5,1]

/ = (1,2,3,5,6,7) and > / = (2,4,6,8,10,12) Consider the example 3 such that For B ∈ [0.5,1]

A = [2B + 1 , −2B + 7]

>A = [4B + 2 , −4B + 12]

A ∗ >A = [(2B + 1)(4B + 2), (−2B + 7)(−4B + 12)]

For B ∈ [0,0.5)

A = [2B + 1 , −2B + 7]

>A = [4B + 2 , −4B + 12]

A ∗ >A = [(2B + 1)(4B + 2), (−2B + 7)(−4B + 12)]

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Since for both B ∈ [0,0.5) & B ∈ [0.5,1] arithmetic intervals are same Therefore A ∗ >A =[(2B + 1)(4B + 2), (−2B + 7)(−4B + 12)] for all B ∈ [0,1] International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No3, July 2013

When B = 0

\ ∗ >\ =[2,84]

Likewise for B = 0.5 \.6 ∗ >\.6 =[8,60]

and for B = 1 ∗ > =[18,40]

Hence A ∗ >A = [2, 8, 18, 40, 60, 84] hence all the points lies within the interval which is the approximate value hence it coincides with the product of the two hexagonal fuzzy number.

Therefore multiplication of two B-cuts lies within the interval.

4.3.5 Symmetric Image:

If / = (a1, a2, a3, a4, a5, a6) is the hexagonal fuzzy number then − / = (-a6, - a5, -a4, - a3, -a2, -a1) which is the symmetric image of / is also an hexagonal fuzzy number Example: If / = (1, 2, 3, 5, 6, 7)

Then − / = (-7, -6, -5, -3, -2, -1) which is again an hexagonal fuzzy number.

5. CONCLUSIONS: In this paper Hexagonal Fuzzy number has been newly introduced and the alpha cut operations of arithmetic function principles using addition, subtraction and multiplication 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 six points is difficult to solve using trapezoidal or triangular fuzzy numbers, therfore hexagonal fuzzy numbers plays a vital role in solving the problem. It also helps us to solve many optimization problems in future which has six parameters as in the above case.

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