A NEW METHOD FOR RANKING IN AREAS OF TWO GENERALIZED TRAPEZOIDAL FUZZY NUMBERS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

A NEW METHOD FOR RANKING IN AREAS OF TWO GENERALIZED TRAPEZOIDAL FUZZY NUMBERS Salim Rezvani1 1

Department of Mathematics, Imam Khomaini Mritime University of Nowshahr, Nowshahr, Iran salim_rezvani@yahoo.com

ABSTRACT In this paper, we want proposed a new method for ranking in areas of two generalized trapezoidal fuzzy numbers. A simpler and easier approach is proposed for the ranking of generalized trapezoidal fuzzy numbers. For the confirmation this results, we compared with different existing approaches.

2010 AMS CLASSIFICATION: 47S20, 03E72.

KEYWORDS Generalized Trapezoidal Fuzzy Numbers, Ranking Method.

1. INTRODUCTION Zadeh [1] introduced the concept of fuzzy set theory to meet those problems. Dubois and Prade defined any of the fuzzy numbers as a fuzzy subset of the real line [2]. Ranking fuzzy numbers were first proposed by Jain [3] for decision making in fuzzy situations by representing the illdefined quantity as a fuzzy set. Bortolan and Degani [4] reviewed some of these ranking methods [2,3,5,6] for ranking fuzzy subsets. Chen [9] presented ranking with maximizing set and minimizing set. [10] and Wang and Lee [11] also used the centroid concept in developing their ranking index. Chen and Chen [12] presented a method for ranking generalized trapezoidal fuzzy numbers. Abbasbandy [13] introduced a new approach for ranking of fuzzy numbers based on the left and right spreads at some levels of trapezoidal fuzzy numbers. Chen and Chen [14] presented a method based on ranking generalized fuzzy numbers with different heights and different spreads. Babu [21] proposed a ranking Generalized Fuzzy Numbers using centroid of centroids.

Also, Wen [18] proposed a new modified similarity measure of generalized fuzzy numbers.

Moreover, Rezvani [5-8] proposed a method for ranking in fuzzy numbers.

In this paper, we want proposed a new method for ranking in areas of two generalized trapezoidal fuzzy numbers. A simpler and easier approach is proposed for the ranking of generalized trapezoidal fuzzy numbers. For the confirmation this results, we compared with different existing approaches.

DOI : 10.5121/ijfls.2013.3102

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

2. PRELIMINARIES Generally, a generalized fuzzy number A is described as any fuzzy subset of the real line R, whose membership function µ A satisfies the following conditions,

µA is a continuous mapping from R to the closed interval [0,1] , ii) µA ( x ) = 0 , −∞ < x ≤ a , iii) µA ( x ) = L (x ) is strictly increasing on [a , b ] , i)

iv) µA ( x ) = w , b ≤ x ≤ c , v) µA ( x ) = R (x ) is strictly increasing on [c , d ] , vi) µA ( x ) = 0 , d ≤ x < ∞ Where 0 < w ≤ 1 and a,b,c and d are real numbers. We call this type of generalized fuzzy number a trapezoidal fuzzy number, and it is denoted by

A = ( a , b , c , d ;w )

.

(1 )

A = (a , b , c , d ;w ) is a fuzzy set of the real line R whose membership function µA (x ) is defined as

µA

 w  w (x ) =  w  0 

x − a b − a d − x d −c

if

a ≤ x ≤ b

if

b ≤ x ≤ c

if

c ≤ x ≤ d

(2)

o t h e rw i s e

Let L−1 and R −1 be the inverse function of functions L and R respectively, then the graded mean h-level value of

A = (a , b , c , d ;w ) Is

h [ L −1 ( h ) + R −1 ( h )] 2 Therefore, the graded mean integration representation of generalized trapezoidal fuzzy number A with grade w is w

G (A ) =

∫ 0

 h [L h  

−1

(h ) + R 2

−1

( h )]  dh / 

w

∫

h dh

(* )

0

Where h lies between 0 and w, 0 < w ≤ 1 . Here, 18

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

L (a ) = w

x −a b −a

if

a ≤ x ≤b

R (a ) = w

d −x d −c

if

c ≤x ≤d

And

Then L −1 ( h ) = a + (b − a ) h , 0 ≤ h ≤ 1 And R −1 ( h ) = d + (d − c ) h , 0 ≤ h ≤ 1 So h [ L −1 ( h ) + R −1 ( h )] (a + d ) + (b − a − d + c ) h = 2 2 Now, using the (*) the graded mean integration representation of A is w

w

 (a + d ) + (b − a − d + c ) h  G (A ) = ∫ h   dh / ∫ h dh 2  0 0 1 1  (a + d )h (b − a − d + c )h 2   (a + d ) (b − a − d + c )  1 = ∫ + dh / h dh =  +  ∫  / 2 2 2 6  4  0  0

=

(a + d ) (b − a − d + c ) a + 2b + 2c + d + = .

2 3 6

3. PROPOSED APPROACH Jiang Wen [18] proposed the concept of the method to calculate the degree of similarity between generalized fuzzy numbers. In this method, the horizontal center-of-gravity, the perimeter, the height and the area of the two fuzzy numbers are considered.

Suppose that A1 = (a1 , b1 , c1 , d 1 ;w 1 ) and A 2 = (a2 , b 2 , c 2 , d 2 ;w 2 ) be the generalized trapezoidal fuzzy numbers. Where 0 ≤ a1 ≤ b1 ≤ c1 ≤ d 1 ≤ 1 and 0 ≤ a2 ≤ b 2 ≤ c 2 ≤ d 2 ≤ 1 . Then the degree of similarity S ( A1 , A 2 ) between the generalized trapezoidal fuzzy numbers A1 and A 2 is calculated as follows:

S (A1 , A 2 ) = [1 − x ∗A1 , x ∗A2 ] × [1 − w A1 −w A2 ] ×

min(P (A1 ), P (A 2 )) + min(A (A1 ), A (A 2 )) max(P (A1 ), P (A 2 )) + max(A (A1 ), A (A 2 ))

(3)

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

Where x ∗ A1 and x ∗ A 2 are the horizontal center-of-gravity of the generalized trapezoidal fuzzy numbers A1 and A 2 is calculated as follows:

x

y

∗ A1

∗ A1

=

y

∗ A1

(c 1 + b 1 ) + ( d 1 + a 1 )(w 2w

 w A 1 ( dc 11 −−ba11 + 2 )   6 =   w A1  2

A1

− y

∗ A1

)

(4)

A1

if

a 1 ≠ d 1 an d

0 <w

A1

≤1 (5)

if

a 1 = d 1 an d

0 <w

A1

≤1

P (A1 ) and P (A 2 ) are the perimeters of two generalized trapezoidal fuzzy numbers which are calculated as follows: P (A1 ) =

( a1 − b 1 ) 2 + w

2

P (A 2 ) =

(a 2 − b 2 ) 2 + w

A1

+ 2

A2

+

(d 1 − c 1 ) 2 + w

2

A1

(c 2 − d 2 ) 2 + w

+ (c 1 − b 1 ) + (d 1 − a1 )

(6)

2

(7)

A2

+ (c 2 − b 2 ) + (d 2 − a 2 )

A (A1 ) and A (A 2 ) are the areas of two generalized trapezoidal fuzzy numbers which are calculated as follows:

1 w 2 1 A (A 2 ) = w 2 A (A 1) =

A1

A

2

(c 1 − b 1 + d 1 − a 1 )

(8 )

(c 2 − b 2 + d

(9 )

2

− a2 )

The larger the value of S ( A1 , A 2 ) , the more the similarity measure between two generalized trapezoidal fuzzy numbers A1 and A 2 .

Theorem 3.1.

Let

A1 = (a1 , b1 , c1 , d 1 ;w 1 ) and A 2 = (a2 , b 2 , c 2 , d 2 ;w 2 ) be the generalized

trapezoidal fuzzy numbers. Where 0 ≤ a1 ≤ b1 ≤ c1 ≤ d 1 ≤ 1 and 0 ≤ a2 ≤ b 2 ≤ c 2 ≤ d 2 ≤ 1 , and

A (A1 ) and A (A 2 ) are the areas of two generalized trapezoidal fuzzy numbers. Then i)

If A (A1 ) < A (A 2 ) , then A1 < A 2 ,

ii) If A (A1 ) > A ( A 2 ) , then A1 > A 2 , iii) If A (A1 )

A (A 2 ) , then A1

A2 .

4. RESULTS Example 4.1. Let A = (0.2, 0.4, 0.6, 0.8;0.35) generalized trapezoidal fuzzy number, then

and

B = (0.1, 0.2, 0.3, 0.4;7)

be two 20

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (0.35)(0.6 − 0.4 + 0.8 − 0.2) = 0.14 , 2 2 And

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (0.7)(0.3 − 0.2 + 0.4 − 0.1) = 0.14 , 2 2 So

A (A )

A (B ) ⇒ A

B.

Example 4.2. Let A = (0.1, 0.2, 0.4, 0.5;1) and B = (0.1, 0.3, 0.3, 0.5;1) be two generalized trapezoidal fuzzy number, then

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (1)(0.4 − 0.2 + 0.5 − 0.1) = 0.3 , 2 2 And

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (1)(0.3 − 0.3 + 0.5 − 0.1) = 0.2 , 2 2 So

A (A ) > A (B ) ⇒ A > B .

Example 4.3. Let A = (0.1, 0.2, 0.4,.5;1) and B = (1,1,1,1;1) be two generalized trapezoidal fuzzy number, then

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (1)(0.4 − 0.2 + 0.5 − 0.1) = 0.3 , 2 2 And

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (1)(1 − 1 + 1 − 1) = 0 , 2 2 So

A (A ) > A (B ) ⇒ A > B .

Example 4.4. Let A = ( −0.5, −0.3, −0.3, −0.1;1) generalized trapezoidal fuzzy number, then

and

B = (0.1, 0.3, 0.3, 0.5;1) be two

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (1)(−0.3 + 0.3 − 0.1 + 0.5) = 0.2 , 2 2 21

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

And

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (1)(0.3 − 0.3 + 0.5 − 0.1) = 0.2 , 2 2 So

A (A )

A (B ) ⇒ A

B.

Example 4.5. Let A = (0.3, 0.5, 0.5,1;1) and B = (0.1, 0.6, 0.6, 0.8;1) be two generalized trapezoidal fuzzy number, then

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (1)(0.5 − 0.5 + 1 − 0.3) = 0.35 , 2 2 And

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (1)(0.6 − 0.6 + 0.8 − 0.1) = 0.35 , 2 2 So

A (A )

A (B ) ⇒ A

B.

4.6. Let A = (0, 0.4, 0.6, 0.8;1) and B = (0.2, 0.5, 0.5, 0.9;1) C = (0.1, 0.6, 0.7, 0.8;1) be two generalized trapezoidal fuzzy number, then

Example

and

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (1)(0.6 − 0.4 + 0.8 − 0) = 0.5 , 2 2 and

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (1)(0.5 − 0.5 + 0.9 − 0.2) = 0.35 , 2 2 and

1 1 A (C ) = w A3 (c 3 − b3 + d 3 − a3 ) = (1)(0.7 − 0.6 + 0.8 − 0.1) = 0.4 2 2 So

A (A ) > A (C ) > A ( B ) ⇒ A > C > B .

Example 4.7. Let A = (0.1, 0.2, 0.4, 0.5;1) and B = ( −2, 0, 0, 2;1) be two generalized trapezoidal fuzzy number, then 22

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

1 1 A (A ) = w A1 (c1 − b1 + d 1 − a1 ) = (1)(0.4 − 0.2 + 0.5 − 0.1) = 0.3 , 2 2 And

1 1 A (B ) = w A 2 (c 2 − b 2 + d 2 − a2 ) = (1)(0 − 0 + 2 + 2) = 2 , 2 2 So

A ( A ) < A (B ) ⇒ A < B .

It is clear from Table 1. that the results of the proposed approach are same as obtained by using the existing approach.

Table 1: A comparison of the ranking results for different approaches Approaches

Example Example Example Example Example Example Example 1 2 3 4 5 6 7

Cheng [14]

A<B

A~B

Error

A~B

A>B

A<B<C

Error

Chu [15]

A<B

A<B

A<B

A<B

A>B

A<B<C

Error

Chen [10]

A<B

A<B

A<B

A<B

A>B

A<C<B

A>B

Abbasbandy [11]

Error

A~B

A<B

A~B

A<B

A<B<C

A>B

Chen [12]

A<B

A<B

A<B

A<B

A>B

A<B<C

A>B

Kumar [18]

A>B

A~B

A<B

A<B

A>B

A<B<C

A>B

Singh [17]

A<B

A<B

A<B

A<B

A>B

A<B<C

A>B

Rezvani 2012

A<B

A>B

A>B

A~B

A~B

A>C>B

A<B

A~B

A>B

A>B

A~B

A~B

A>C>B

A<B

Proposed approach

5. CONCLUSIONS A simpler and easier approach is proposed for the ranking of generalized trapezoidal fuzzy numbers. In this paper, we want proposed a new method for ranking in areas of two generalized trapezoidal fuzzy numbers.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No1, January 2013

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[11]

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Pushpinder Singh et al, (2010) Ranking of Generalized Trapezoidal Fuzzy Numbers Based on Rank, Mode, Divergence and Spread, Turkish Journal of Fuzzy Systems (eISSN: 13091190), Vol.1, No.2, pp. 141-152.

Kumar, A., Singh P., Kaur A., Kaur, P., (2010) RM approach for ranking of generalized trapezoidal fuzzy numbers. Fuzzy Information and Engineering, 2 (1), 37-47.

Babu et al, Ranking Generalized Fuzzy Numbers using centroid of centroids, International Journal of Fuzzy Logic Systems (IJFLS) Vol.2, No.3, July 2012.

Author Salim Rezvani, Department of Mathematics, Imam Khomaini Mritime University of Nowshahr, Nowshahr, Iran

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