Ranking Algorithm for Symmetric Octagonal Fuzzy Numbers

ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci.

2014, Vol.5(3): Pg.310-318

An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

Ranking Algorithm for Symmetric Octagonal Fuzzy Numbers

Dhanalakshmi V. and Felbin C. Kennedy Research Scholar,

Stella Maris College (Autonomous), Chennai, INDIA.

Research Guide, Associate Professor,

Stella Maris College (Autonomous), Chennai, INDIA.

(Received on: June 27, 2014) ABSTRACT

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, we propose a ranking algorithm for symmetric octagonal fuzzy numbers, which results in equality only when the numbers coincide. Also the ranking algorithm is verified for some reasonable properties of ordering of fuzzy numbers

Keywords: Symmetric Octagonal fuzzy number, ranking, mode, deviation, spread.

1. INTRODUCTION

Ranking fuzzy numbers is an important task in analyzing fuzzy information in optimization, data mining, decision making and related areas. Unlike real numbers, fuzzy numbers have no natural order; also no ordering technique is conducive to all. Thus significant contributions have been made in ranking fuzzy numbers 1–7,10,12,13

. All the ranking procedures in literature defines A ~ B ~

f , B

A ~ ~

p and A ~ B ~

≈ , here A ~ B ~

≈ implies the

two fuzzy numbers are equivalent, not

necessarily be equal. That is, two different

fuzzy numbers depending on the ranking

method are infered as equivalent. But in

many applications the two fuzzy numbers

need to be shown different. In this paper, we

have proposed a ranking algorithm for

symmetric octagonal fuzzy number, in

which we obtain the equality only when the

two symmetric octagonal fuzzy numbers

coincide. Some of the reasonable properties

for ranking fuzzy numbers proposed in ¹¹ are

verified for this algorithm.

311 Dhanalakshmi V., et al., J. Comp. & Math. Sci. Vol.5 (3), 310-318 (2014)

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

After the introduction, this paper is arranged as follows. In Section 2, the concept of fuzzy number and octagonal fuzzy numbers are recalled. Section 3 introduces the ranking algorithm for symmetric octagonal fuzzy numbers and the properties of the proposed algorithm are discussed in Section 4. The concluding remarks are presented in Section 5.

2. SYMMETRIC OCTAGONAL FUZZY NUMBERS

For the sake of completeness we recall the required definitions and results.

Definition 2.1 [8]The characteristic function A ~

µ of a crisp set A ⊆ X assigns a value either 0 or 1 to each member in X . This function can be generalized to a function

A ~

µ such that the value assigned to the element of the universal set X fall within a specified range i.e. µ _A ^~ : X → [ 0,1]. ^The assigned value indicates the membership grade of the element in the set A . The function µ _A ^~ is called the membership function and the set

} :

)) ( , {(

~ =

~ x x X

x

A µ _A ∈ is called a fuzzy

set.

Definition 2.2 [8]A fuzzy set A ~

,defined on the universal set of real numberR, is said to be a fuzzy number if its membership function has the following characteristics:

i. A ~

is convex i.e.

0,1]

[ , ,

)) ( ), ( ( min ) ) (1 (

2 1

~ 2

~ 1 2

~ 1

∈

∀

∈

∀

≥

− +

λ

µ µ

λ λ

µ

R x x

x x

x

x _{A A}

A

ii. A ~

is normal i.e. ∃ x ₀ ∈ Rsuch that µ _A ^~ ( x ₀ ) = 1

iii. µ _A ^~ is piecewise continuous Definition 2.3[9]A fuzzy number A ~

is said to be a generalized octagonal fuzzy number denoted by

) ,

; , , , , , , , (

~ =

8 7 6 5 4 3 2

1 a a a a a a a k w

a A

where a ₁ , a ₂ , a ₃ , a ₄ , a ₅ , a ₆ , a ₇ , a ₈ , k , w are real numbers such that

8 7 6 5 4 3 2

1 a a a a a a a

a ≤ ≤ ≤ ≤ ≤ ≤ ≤

and 0 < k < w and its membership function

A ~

µ is given by

 

 

 

 





 

 

 





≥

≤

− ≤

− ≤ ≤

≤

− ≤

− +

− ≤ ≤

≤

− ≤

− +

− ≤ ≤

≤

− ≤

− ≤

8 8 7

7 8

8

7 6

6 5

5 6

6 5

5 4

4 3

3 4

3 4

3 2

2 1

1 2

1

1

~

0

) ( ) (

) ( ) (

0

= ) (

a x

a x a a

a a k x

a x a k

a x a a

a

x a w a x k

a x a w

a x a a

a

a x w x a k

a x a k

a x a a

a a k x

a x

A x µ

Definition 2.5A fuzzy number A ~

is said to be a symmetric octagonal fuzzy number denoted by ~ = ( , , , , ; , )

w k t s r a a

A _{L U}

where a _L ≤ a _U and r, s, t are real numbers

such that its representation as generalized

octagonal fuzzy number is

).

,

; ,

, , ,

(

w k t s r a t s a

t a t s a t s r a

U U

L L

L

+ + + + +

−

−

−

−

−

−

Figure 1: Graphical representation of the symmetric octagonal fuzzy number

Definition 2.6 The α -cut of the symmetric octagonal fuzzy number

) ,

; , , , , (

~ =

w k t s r a a

A _{L U} given in

Definition 2.5 is



 



( , ) (

( , )

= ( ] , [

~ =

2 1

A A

A A A

A

A L R

R R L

L

α α

α α α

α where

k A r k t s a A

k A r k t s a A

R U L L

+ − + +

− −

−

−

α α α

α

( ,

= ) (

( ,

= ) (

1 1

Definition 2.7 Let

) ,

; , , , , (

~ =

1 1 1 1

1 s t k w

r a a

A _{L U}

,

; , , , , (

~ =

2 2 2 2

2 s t k w

r b b

B _{L U}

symmetric octagonal fuzzy numbers, then their sum is defined as

, , ,

, a _L a _U a _U + t

Figure 1: Graphical representation of the symmetric octagonal fuzzy number (2 , 5 , 1 , 2 ,

cut of the symmetric given in

∈

∈ ,1]

( ] )

] 0, [ ] )

2 1

k k

R R

α α

α α

k w t w a A

k w t w a A

R U L L

− + −

−

− − α α α

α

= )

,

= )

2 2

) and

2 ) be two symmetric octagonal fuzzy numbers, then

) ) , min(

, ) , min(

, , ,

(

~ =

~

2 1 2

1

1 2 1

w w k

k

s r r b a b a B

A + _L + _{L U} + _U +

3. RANKING OF SYMMETRIC OCTAGONAL FUZZY NUMBERS Definition 3.1 For any symmetric octagonal fuzzy number ~ = ( , ,

a a

A _{L U}

mode, divergence & spread at w are defined as :

i. 2

= (

~ )

( w a ^L

A Mode

w − +

ii. ~ ) = (

( A w a Divergence

w −

iii. Spread _{k w} A ~ ) = w t

, (

iv. 2

= (

~ )

( k a ^L

A Mode

k − +

v. (2

=

~ )

( k a

A rightmode k −

2 ,1)

; 1 3 ,

; , _{1 2}

2 t t

s +

+

3. RANKING OF SYMMETRIC OCTAGONAL FUZZY NUMBERS

For any symmetric octagonal ) ,

; , ,

, r s t k w w and k level

2

U ) + a

) 2t a a _U − _L +

U ) a

2

)

2 t s

a _U + +

313 Dhanalakshmi V., et al., J. Comp. & Math. Sci. Vol.5 (3), 310-318 (2014)

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

vi. 2

) 2

= (2

~ )

( k a t s

A leftmode

k − ^L − −

vii.

) 2 2 2 (

=

~ ) ( 0

t s r a a k

A Divergence

L

U − + + +

−

viii. Spread _k A ~ ) = k r

0, (

Given any two symmetric octagonal fuzzy numbers we present here an algorithm to compare them.

ALGORITHM

Let ~ = ( , , , , ; , )

1 1 1 1

1 s t k w

r a a

A _{L U} and

) ,

; , , , , (

~ =

2 2 2 2

2 s t k w

r b b

B _{L U} be two

symmetric octagonal fuzzy numbers, then use the following steps to compare A ~

and B ~

Step1: Find ~ )

(A Mode

w − and

~ ) (B Mode w − Case(i): If

~ ) (

>

~ ) (

B Mode w

A Mode w

−

− then A ~ B ~

f

Case(ii): If ~ )

(

<

~ )

( A w Mode B

Mode

w − −

then A ~ B ~ p

Case(iii): If

~ ) (

=

~ ) (

B Mode w

A Mode w

−

− then go to

Step 2

Step2: Find ~ )

(A Divergence

w − ^and

~ ) (B Divergence w −

If ~ ) 0

( ≥

− Mode A

w then

Case(i): If ~ ) >

( A Divergence w −

~ ) (B Divergence

w − ^{then A ~} f ^{B ~}

Case(ii): If ~ ) <

( A Divergence w −

~ ) (B Divergence

w − ^{then A ~} p ^{B ~}

Case(iii): If ~ ) =

( A Divergence w −

~ ) (B Divergence

w − then go to Step 3

else

Case(i): If ~ ) <

( A Divergence w −

~ ) (B Divergence

w − ^{then A ~} f ^{B ~}

Case(ii) : If ~ ) >

( A Divergence w −

~ ) (B Divergence

w − ^{then A ~} p ^{B ~}

Case(iii): If ~ ) =

( A Divergence w −

~ ) (B Divergence

w − then go to Step 3

Step 3: Find ~ )

, ( A Spread _{k w} and

~ )

, ( B Spread _{k w}

If ~ ) 0

( ≥

− Mode A

w then

Case(i): If ~ ) <

, ( A

Spread _{k w} ~ )

, ( B Spread _{k w} then A ~ B ~

f

Case(ii): If ~ ) >

, ( A Spread _{k w}

~ )

, ( B

Spread _{k w} then A ~ B ~ p Case(iii): If ~ ) =

, ( A Spread _{k w}

~ )

, ( B

Spread _{k w} then go to Step 4 else

Case(i): If ~ ) >

, ( A

Spread _{k w} ~ )

, ( B Spread _{k w} then A ~ B ~

f

Case(ii): If ~ ) <

, ( A

Spread _{k w} ~ )

, ( B Spread _{k w} then A ~ B ~

p

Case(iii): If ~ ) =

, ( A

Spread _{k w}

~ )

, ( B

Spread _{k w} then go to Step 4 Step 4: Find w ₁ and w ₂

Case (i): If w ₁ > w ₂ then A ~ B ~ f Case (ii): If w ₁ < w ₂ then A ~ B ~

p

Case (iii): If w ₁ = w ₂ then go to Step 6

Step 5: Find ~ ) (A Mode

k − and

~ ) (B Mode k −

Case(i): If ~ )

(

>

~ )

( A k Mode B

Mode

k − −

then A ~ B ~ f

Case(ii): If ~ )

(

<

~ )

( A k Mode B

Mode

k − −

then A ~ B ~ p

Case(iii): If ~ )

(

=

~ )

( A k Mode B

Mode

k − −

then go to Step 6

Step 6: Find ~ )

( A rightmode

k − and

~ ) (B rightmode k −

Case(i): If

~ ) (

>

~ ) (

B rightmode k

A rightmode k

−

− then A ~ B ~

f

Case(ii): If

~ ) (

<

~ ) (

B rightmode k

A rightmode k

−

− then A ~ B ~

p

Case(iii): If

~ ) (

=

~ ) (

B rightmode k

A rightmode k

−

− then go to

Step 7

Step7: Find ~ )

(A leftmode

k − and

~ ) (B leftmode k −

Case(i): If

~ ) (

>

~ ) (

B leftmode k

A leftmode k

−

− then A ~ B ~

f

Case(ii): If

~ ) (

<

~ ) (

B leftmode k

A leftmode k

−

− then A ~ B ~

p

Case(iii): If

~ ) (

=

~ ) (

B leftmode k

A leftmode k

−

− then go to

Step 8

Step 8: Find ~ )

(

0 − Divergence A ^and

~ ) ( 0 − Divergence B

Case(i): If ~ ) >

( 0 − Divergence A

~ ) (

0 − Divergence B ^{then A ~} f ^{B ~} Case(ii): If ~ ) <

( 0 − Divergence A

~ ) (

0 − Divergence B ^{then A ~} p ^{B ~}

Case(iii): If ~ ) =

( 0 − Divergence A

~ ) (

0 − Divergence B then go to Step 9 Step 9: Find k ₁ and k ₂

Case (i): If k ₁ > k ₂ then A ~ B ~ f Case (ii): If k ₁ < k ₂ then A ~ B ~

p Case (iii): If k ₁ = k ₂ then A ~ B ~

≈ 4. PROPERTIES OF THE RANKING ALGORITHM

Proposition 4.1: If equality holds in all the nine steps of the above algorithm, then the two symmetric octagonal fuzzy numbers are equal.

Proof:

Let ~ = ( , , , , ; , )

1 1 1 1

1 s t k w

r a a

A _{L U} and

) ,

; , , , , (

~ =

2 2 2 2

2 s t k w

r b b

B _{L U} be two

symmetric octagonal fuzzy numbers, such

that all the nine steps of the above algorithm

315 Dhanalakshmi V., et al., J. Comp. & Math. Sci. Vol.5 (3), 310-318 (2014)

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

leads equality. To prove that A ~

and B ~ are equal, i.e. to prove that

. ,

, , , , ,

2 1 2 1

2 1 2 1 2 1

w w k k

t t s s r r b a b

a _{L L U U}

=

=

=

=

=

=

=

From step 4,

2

1 = w

w ₍₁₎

From step 9,

2 1 = k

k ₍₂₎

From step 3 and equation (1),

2 1 = t

t ₍₃₎

From step 1 and equation (1),

U L U L

U L U

L

b b a a

b b w a a w

B Mode w A Mode w

+

= +

⇒

= +

⇒ +

−

−

2 ) (

2 ) (

~ ) (

=

~ ) (

2 1

(4) From step 2 and equation (1) and (3),

L U L U

L U L

U

b b a a

t b b w t a a w

B Divergence w

A Divergence w

−

=

−

⇒

+

−

= +

−

⇒

−

−

) 2 (

) 2 (

~ ) (

=

~ ) (

2 2

1

1 (5)

From equations (4) and (5),

L L U

U b a b

a = , =

(6) From step 6 and equations (3) and (6),

2 1

2 2 2

1 1 1

2

) 2 (2 2

) 2 (2

~ ) (

=

~ ) (

s s

s t b k s t a k

B rightmode k

A rightmode k

U U

=

⇒

+

= + +

⇒ +

−

−

(7)

From step 8 and equations (3), (6) and (7),

2 1

2 2 2 2

1 1 1 1

) 2 2 2 (

) 2 2 2 (

~ ) ( 0

=

~ ) ( 0

r r

t s r b b k

t s r a a k

B Divergence A

Divergence

L U

L U

=

⇒

+ + +

−

=

+ + +

−

⇒

−

−

Hence the proof.

Remark 4.2: In the above theorem, we have proved that equality, not just the equivalence of symmetric octagonal fuzzy numbers is obtained by the proposed algorithm.

Proposition 4.3: Let ) ,

; , , , , (

~ =

1 1 1 1

1 s t k w r

a a

A _{L U} ,

) ,

; , , , , (

~ =

2 2 2 2

2 s t k w

r b b

B _{L U} and

) ,

; , , , , (

~ =

3 3 3 3

3 s t k w

r c c

C _{L U} be three

symmetric octagonal fuzzy numbers such that A ~ B ~

f and ~ ~ ,

B f C ^then A ~ C ~ f ^. Proof:

Case (i):

Let the order be determined in step 1, i.e B

A ~ ~

f by ~ )

(

>

~ )

( A w Mode B

Mode

w − −

and B ~ C ~ f ^by

~ ) (

>

~ )

( B w Mode C

Mode

w − − _,

then

~ ) (

>

~ ) (

~ ) (

>

~ ) (

>

~ ) (

C Mode w A Mode w

C Mode w B Mode w A Mode w

−

−

⇒

−

−

− C A ~ ~

f

⇒ Case (ii):

Suppose A ~ B ~

f by step 1 and B ~ C ~ f _in step i say i=2,3,...,9

i.e. ~ )

(

>

~ )

( A w Mode B

Mode

w − − _and

~ ) (

~ )

( B w Mode C

Mode

w − = − _{and the}

inequality happens in the later steps, then

~ ) (

>

~ ) (

~ ) (

~ ) (

>

~ ) (

C Mode w A Mode w

C Mode w B Mode w A Mode w

−

−

⇒

−

=

−

−

C A ~ ~

f

⇒

Similarly, A ~ C ~ f ^if

~ ) (

~ )

( A w Mode B

Mode

w − = − ^and

~ ) (

>

~ )

( B w Mode C

Mode

w − −

Case (iii):

Similar to case (ii), it can be proved that C

A ~ ~

f _if ^{A ~} f ^{B ~} by step i and B ~ C ~ f _in step j where j=i+1,...,9or B ~ C ~

f by step i and A ~ B ~

f in step j where j=i+1,...,9.

Thus in all possibilities, A ~ B ~

f and B ~ C ~ f C

A ~ ~ f

⇒ _. Hence the proof.

Proposition 4.4: Let A ~

and B ~

be two symmetric octagonal fuzzy numbers with same height and

~ ), supp(

sup

~ ) supp(

inf A > B then ~ ~ .

A f B

Proof:

Let ~ = ( , , , , ; , )

1 1 1 1

1 s t k w

r a a

A _{L U} and

) ,

; , , , , (

~ =

2 2 2 2

2 s t k w

r b b

B _{L U}

be the two given symmetric octagonal fuzzy numbers.

Since they have the same height, w ¹ = w ² .

U U

L L

U L

b t s r b t s r a a

t s r b t s r a

B A

>

+ + +

>

−

−

−

>

⇒

+ + +

>

−

−

−

⇒

>

2 2 2 1

1 1

2 2 2 1

1 1

~ ) supp(

sup

~ ) supp(

inf

U

L b

a >

⇒ ₍₈₎

L U L

U a b b

a > > >

⇒ , by definition of A ~

and B ~

L

U b

a >

⇒ ₍₉₎

Adding (8) and (9),

U L L

U a b b

a + > +

~ .

~

~ ) (

>

~ ) ( B A

B Mode w

A Mode w

f

⇒

−

−

⇒

Hence the proof.

Proposition 4.5: Let A ~ , B ~

and C ~

be three symmetric octagonal fuzzy numbers with same k and w then A ~ B ~

f

~ .

~ ~

~ C B C

A + +

⇒ f

Proof: Let ~ = ( , , , , ; , )

1 1 1 1

1 s t k w r

a a

A _{L U} ,

) ,

; , , , , (

~ =

2 2 2 2

2 s t k w

r b b

B _{L U} and

) ,

; , , , , (

~ =

3 3 3 3

3 s t k w

r c c

C _{L U} be the three

given symmetric octagonal fuzzy numbers.

By hypothesis, k ₁ = k ₂ = k ₃

3 2

and w 1 = w = w ^{Suppose A ~} f ^{B ~} happens in step 1, then

~ ) (

~ )

( A w Mode B Mode

w − > −

317 Dhanalakshmi V., et al., J. Comp. & Math. Sci. Vol.5 (3), 310-318 (2014)

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

~ .

~ ~

~

~ ) ( ~

~ ) ( ~

2

) )(

, min(

2

) )(

, min(

sides both adding

, 2

) (

2 ) (

3 2

3 1

2 1

C B C A

C B Mode w

C A Mode w

c c b b w w

c c a a w w

c c

c c b b c c a a

b b a a

b b w a a w

U L U L

U L U L U L

U L U L U L U L

U L U L

U L U

L

+ +

⇒

+

−

>

+

−

⇒

+ +

> +

+ +

⇒ +

+

+ + +

>

+ + +

⇒

+

>

+

⇒

> +

⇒ +

f Suppose A ~ B ~

f happens in step 2, then

~ ) (

~ )

( A w Mode B

Mode

w − = − _and

~ ) (

~ )

( A w Divergence B Divergence

w − > −

~ ) (

~ )

( A w Mode B

Mode

w − = −

~ ) ( ~

~ ) ( ~

2 ) (

2 )

( ₂

1

C B Mode w

C A Mode w

c c b b c c a a

b b a a

b b w a a w

U L U L U L U L

U L U L

U L U

L

+

−

= +

−

⇒

+ + +

= + + +

⇒

+

= +

⇒

= +

⇒ +

and

~ ) (

~ )

( A w Divergence B Divergence

w − > −

~ .

~ ~

~

~ ) ( ~

~ ) ( ~

2 2

2 2

2 2

3 2

3 1

2 1

C B C A

C B Divergence w

C A Divergence w

t c c t b b

t c c t a a

t b b t a a

L U L

U

L U L

U

L U L

U

+ +

⇒

+

−

>

+

−

⇒

+

− + +

−

>

+

− + +

−

⇒

+

−

>

+

−

⇒

f

In a similar manner, the result can be proved for all cases.

Remark 4.6: The above Proposition may not hold for arbitrary k and w, for

Let ,1)

2

; 1 1 , 1 , 1 , 4 ,

~ = (1

A ,

2 ) , 1 4

; 1 1 , 1 , 1 , 4 ,

~ = (2

B and )

2 , 1 4

; 1 1 , 1 , 1 , 2 ,

~ = (1

C ,

then A ~ B ~ f ^as

~ ) 4 (

> 5 2 ) 5

( A ~ w Mode B

Mode

w − = = − ^,

but )

2 , 1 4

; 1 2 , 2 , 2 , 6 ,

~ (2

~ + C =

A and

2 ) , 1 4

; 1 2 , 2 , 2 , 6 ,

~ (3

~ + C = B

~ .

~

~

~

~ ) ( ~ 4

2 9

~ ) ( ~

C B C A

C B Mode w C

A Mode w

+ +

⇒

+

−

=

<

= +

−

⇒

p

5. CONCLUSION

In this paper, an algorithm for pairwise comparison of symmetric octagonal fuzzy numbers is introduced. A strong property of uniqueness of a symmetric octagonal fuzzy numbers is established in Proposition 4.1. Also some of the reasonable properties of ranking fuzzy numbers were verified.

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