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
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11]
[12] [13] [14]
[15]
[16] [17] [18] [19]
[20] [21]
L. A. Zadeh, Fuzzy set ,(1965) Information and Control, vol.8,no.3, pp.338-353. D. Dubois and H. Prade, (1987) The mean value of a fuzzy number, Fuzzy Sets and Systems, vol. 24, no. 3, pp. 279-300. R. Jain, (1976) Decision making in the presence of fuzzy variables, IEEE Transactions on Systems, Man and Cybernetics, vol. 6, no. 10, pp. 698-703. G. Bortolan and R. Degani, (1985) A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, vol. 15, no. 1, pp. 119. S. Rezvani, (2010) Graded Mean Representation Method with Triangular Fuzzy Number, World Applied Sciences Journal 11 (7): 871-876. S. Rezvani, (2011) Multiplication Operation on Trapezoidal Fuzzy Numbers, Journal of Physical Sciences, Vol. 15, 17-26. S. Rezvani, (2012) A New Method for Ranking in Perimeters of two Generalized Trapezoidal Fuzzy Numbers, International Journal of Applied Operational Research, Vol. 2, No. 3, pp. 83-90. S. Rezvani, Ranking method of trapezoidal intuitionistic fuzzy numbers Annals of Fuzzy Mathematics and Informatics, accepted, 2012. S.-H. Chen, (1985) Ranking fuzzy numbers withmaximizing set and minimizing set, Fuzzy Sets and Systems, vol. 17, no. 2, pp. 113 129. C. Liang, J. Wu and J. Zhang, (2006) Ranking indices and rules for fuzzy numbers based on gravity center point, Paper presented at the 6th world Congress on Intelligent Control and Automation, Dalian, China, pp.21-23 Y. J.Wang and H. S.Lee, (2008) The revised method of ranking fuzzy numbers with an area between the centroid and original points, Computers and Mathematics with Applications, vol. 55, pp.20332042. S. j. Chen and S. M. Chen, (2007) Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers, Applied Intelligence, vol. 26, pp. 1-11. S. Abbasbandy and T. Hajjari, (2009) A new approach for ranking of trapezoidal fuzzy numbers, Computers and Mathematics with Applications, vol. 57, pp. 413-419. S.M Chen and J. H. Chen, (2009) Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads, Expert Systems with Applications, vol. 36. pp. 68336842. Shan-Hou Chen and Guo-Chin Li, (2000) Representation, Ranking, ang Distance of Fuzzy Number eith Exponential Membership Function Using Graded mean Integration method, Tamsui Oxford journal of Mathematical Sciences 16, 123-131. Cheng, CH., (1998) A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95 (3), 307-317. Chu, TC., Tsao, CT., (2002) Ranking fuzzy numbers with an area between the centroid point and original point. Computers and Mathematics with Applications, 43 (1-2), 111-117. Jiang Wen et al, (2011) A modified similarity measure of generalized fuzzy numbers, Procedia Engineering 15. 2773- 2777. 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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