A note on ranking generalized fuzzy numbers

Expert Systems with Applications 39 (2012) 6454–6457

Contents lists available at SciVerse ScienceDirect

Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A note on ranking generalized fuzzy numbers Peida Xu a,c,f, Xiaoyan Su a,d, Jiyi Wu f, Xiaohong Sun c, Yajuan Zhang b,e, Yong Deng b,⇑ a

School of Electronics and Information Technology, Shanghai Jiao Tong University, Shanghai, China College of Computer and Information Sciences, Southwest University, Chongqing 400715, China c College of Food Science and Technology, Shanghai Ocean University, Shanghai, China d Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Hunan, China e Key Subject Laboratory of National Defense for Radioactive Waste and Environmental Security, Southwest University of Science and Technology, Mianyang, China f Hangzhou Key Lab of E-Bussiness and Information Security, Hangzhou Normal University, Zhejiang, China b

a r t i c l e

i n f o

Keywords: Generalized fuzzy numbers Ranking fuzzy numbers Modification

a b s t r a c t Ranking fuzzy numbers plays an important role in decision making under uncertain environment. Recently, Chen and Sanguansat (2011) [Chen, S. M. & Sanguansat, K. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3), (pp. 2163–2171)] proposed a method for ranking generalized fuzzy numbers. It considers the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate ranking scores of the generalized fuzzy numbers. Chen and Sanguansat’s method (2011) can overcome the drawbacks of some existing methods for ranking generalized fuzzy numbers. However, in the situation when the score is zero, the results of the Chen and Sanguansat’s ranking method (2011) ranking method are unreasonable. The aim of this short note is to give a modification on Chen and Sanguansat’s method (2011) to make the method more reasonable. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Ranking generalized fuzzy numbers ranking plays an important role in many fields. In recent years, the methods for ranking generalized fuzzy numbers have been extensively researched and used to solve many problems (Cheng, 1998; Chen & Chen, 2009; Chu & Tsao, 2002; Deng & Liu, 2005; Deng, Zhu, & Liu, 2006; Yager, 1978). Chen and Sanguansat (2011) proposed a method for ranking generalized fuzzy numbers. The method considers the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate the ranking scores of the generalized fuzzy numbers. It can overcome many drawbacks of the existing methods. But we find that, in some cases, the result of Chen and Sanguansat’s method (2011) is unreasonable. Based on Chen and Sanguansat’s method (2011), this paper proposed a modified method. The remainder of this paper is organized as follows. In Section 2, we briefly review Chen and Sanguansat’s method (2011). In Section 3, we present a modified fuzzy ranking method and make a comparison of the calculation results of the modified method with the existing methods. In Section 4, we summarize and conclude.

⇑ Corresponding author. E-mail address: [email protected] (Y. Deng). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.12.062

2. Chen and Sanguansat’s method (2011) for ranking generalized fuzzy numbers In this section, we give a review of Chen and Sanguansat’s method (2011) for ranking generalized fuzzy numbers (Chen & Sanguansat, 2011). The method calculates the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate the ranking scores of the generalized fuzzy numbers. It can get ranking scores by considering many factors of the generalized fuzzy numbers. e1; A e2; . . . ; A e n to be Assume that there are n fuzzy numbers A e i ¼ ðai1 ; ai2 ; ai3 ; ai4 ; w Þ,1 < ai1 6 ai2 6 ai3 6 ai4 ranked, where A eA i < 1; we 2 ð0; 1 and 1 6 i 6 n. Chen and Sanguansat’s method Ai (2011) for ranking generalized fuzzy numbers is shown as follows: Step 1: Transform each generalized fuzzy number e i ¼ ðai1 ; ai2 ; ai3 ; ai4 ; w Þ into a standardized generalized A eA i e, fuzzy number A i

e ¼ A i

    ai1 ai2 ai3 ai4 ; ; ; ; we ¼ ai1 ; ai2 ; ai3 ; ai4 ; we Ai Ai k k k k

ð1Þ

where k = maxij(djaijje, 1), jaijj denotes the absolute value of aij and djaijje denotes the upper bound of jaijj, 1 6 i 6 n and 1 6 j 6 4.  Step 2: Calculate the areas Area iL and AreaiR on the negative side, respectively, which denote the trapezoidal areas from the membership function curve of the generalized fuzzy

6455

P. Xu et al. / Expert Systems with Applications 39 (2012) 6454–6457

number ð1; 1; 1; 1; we Þ to the membership function Ai

L

R

curves of f  and f  of the standardized generalized fuzzy eA eA i

i

e  , respectively, where number A i L

fe ¼ we Ai A i

feR ¼ we Ai A i

ðx  ai1 Þ    ; a 6 x < ai2 ; ðai2  ai1 Þ i1 ðx  ai4 Þ    ; a < x 6 ai4 ; ðai3  ai4 Þ i3

ð2Þ ð3Þ

and

ða þ 1Þ þ ðai2 þ 1Þ ¼ we  i1 Ai 2 ðai3 þ 1Þ þ ðai4 þ 1Þ  AreaiR ¼ we  Ai 2

AreaiL

ð4Þ ð5Þ

þ Then, calculate the areas Areaþ iL and AreaiR on the positive side, respectively, which denote the trapezoidal areas from the membership function curve of f L  and f R defined in Eqs. (2) and (3), respeceA eA i

Ai

ð1 

ai1 Þ

ai2 Þ

þ ð1  2 ð1  ai3 Þ þ ð1  ai4 Þ þ AreaiR ¼ we  Ai 2

AreaþiL ¼ we 

ð6Þ

Ai

ð7Þ

Step 3: Calculate the values XIe and XDe of each standardized Ai Ai e  , where XIe denotes the generalized fuzzy number A i Ai

e  that have a positive sum of the factors on the X-axis of A i  e influence on the ranking score of A , i.e., the ranking score i

increases as the values of these factors increase; XDe A i

e  that have denotes the sum of the factors on the X-axis of A i e  , i.e., the a negative influence on the ranking score of A i

ranking score decreases as the values of these factors increase, shown as follows:

XIe ¼ AreaiL þ AreaiR A

ð8Þ

i

XDe ¼ AreaþiL þ AreaþiR A

ð9Þ

i





e  of each standardized Step 4: Calculate the ranking score Score A i  e generalized fuzzy number A i shown as follows:

  1  XIeA  þ ð1Þ  XDeA  i i e ¼ Score A i XIe þ XDe þ ð1  we Þ A A Ai i

¼

i

XIe  XDe A A i

i

XIe þ XDe þ ð1  we Þ A A Ai i

Therefore, in some cases, the ranking results are unreasonable. Some examples will be presented in Section 3. 3. Modified method and examples In this section, we present a modified method for ranking generalized fuzzy numbers based on Chen and Sanguansat’s method (2011). Then we give some examples to compare the calculation results of the modified ranking method with some existing methods.

ð10Þ

3.1. Modified method e  on the X-axis As previously stated, if the center of gravity of A i   e   0. In some cases, the ranking results of Chen is 0, than Score A i

and Sanguansat’s method (2011) are unreasonable. To improve Chen and Sanguansat’s method (2011), we suggest adding Step 5. The modified method is shown as follows: e i into a standardStep 1: Transform each generalized fuzzy number A  e ized generalized fuzzy number A , as defined in Eq. (1). i

 þ þ Step 2: Calculate the areas Area iL ; AreaiR ; AreaiL and AreaiR of each e  , as defined in standardized generalized fuzzy number A i Eqs. (4)–(7). Step 3: Calculate the values XIe and XDe . of each standardized genAi Ai e  , as defined eralized fuzzy number A  in  Eqs. (8) and (9). i e  of each standardized Step 4: Calculate the ranking score Score A i

e  , as defined in Eq. (10). The largeneralized fuzzy number A   i e . e  , the better the ranking of A ger the value of Score A i n  i  e e Step 5: For generalized fuzzy number set: R ¼ A i jScore A i ¼ 0; 1 6 i 6 ng. If jRj > 1,where jRj denotes the cardinality of R, then rank the generalized fuzzy numbers in R according to their heights we . The larger the value of we , the better the Ai

Ai

ei. ranking of A

By adding Step 5, the modified method can deal with the situations that some generalized fuzzy numbers’ centers of gravity on the X-axis are 0. So the modified method can be an improvement of Chen and Sanguansat’s method (2011).

i

  e  2 ½1; 1, 1 6 i 6 n. The larger the value of where Score A i    e e  . Based on Eq. (10), we Score A i , the better the ranking of A i can see that XIe and XDe are used for weighting the maximal A A i

Chen and Sanguansat’s method (2011) has some properties (Chen & Sanguansat, 2011). Among them, Property 1 is: If e  ¼ ða1 ; a2 ; a3 ; a4 ; wÞ, where 1 6 a1 6 a2 6 a3 6 a4 6 1 and A i   e  ¼ 0 .It means that, if a1 + a2 + a3 + a4 = 0 or w = 0, then Score A i    e   0. e the center of gravity of A i on the X-axis is 0, than Score A i

i

tively, to the membership function curves of the generalized fuzzy number ð1; 1; 1; 1; we Þ, where

generalized fuzzy numbers, ð1  we Þ is added into the denominaAi     e   decreases when w decreases. tor in Eq. (10), i.e., Score A i eA i

i

and the minimal possible values of the universe of discourse of e  , 1 and 1, respectively. This means the larger the value of A i   e  is to 1; the larger the value XIe , the closer the value of Score A i Ai   e  is to 1. Moreover, beof XDe , the closer the value of Score A i A i

cause Chen and Sanguansat’s method (2011) considers the factor on the Y-axis as less influential to the ranking scores of

3.2. Numerical example In this section, we use four sets of generalized fuzzy numbers shown in Fig. 1 to make an experiment to compare the calculation results of the modified ranking method with Yager’s method(1978), Cheng’s method (1998), Chu and Tsao’s method (2002), Chen and Chen’s method (2009) and Chen and Sanguansat’s method (2011). The results are shown in Table 1. From Table 1, we can see that: e B e shown in Set 1 of Fig. 1, e and C (1) For the fuzzy numbers A; e which coincides e > A, e>C only the modified method gets B with the intuition of human beings.

6456

P. Xu et al. / Expert Systems with Applications 39 (2012) 6454–6457

Fig. 1. Four sets of fuzzy numbers.

Table 1 A comparison of the ranking results of the modified fuzzy ranking method with the existing methods.

Note: ‘‘⁄’’ denotes the method cannot calculate the ranking value of the fuzzy numbers; ‘‘ ’’denotes unreasonable results.

e and B e shown in Set 2 of Fig. 1, only (2) For the fuzzy numbers A e which coincides with the e > A, the modified method gets B intuition of human beings. e and B e shown in Set 3 of Fig. 1. Chen (3) For the fuzzy numbers A and Chen’s method (2009), Chen and Sanguansat’s method e which coincides e > A, (2011) and the modified method get B with the intuition of human beings. Under such circumstances, Chen and Sanguansat’s method (2011) and the modified method are the same. e and B e shown in Set 4 of Fig. 1, Yag(4) For the fuzzy numbers A e e er’s method(1978) gets B ¼ A, which does not coincide with the intuition of human beings due to the fact that the height e is less than the height of B e .The other methods get of A e which coincides with the intuition of human beings. e > A, B Under such circumstances, Chen and Sanguansat’s method (2011) and the modified method are the same. 4. Conclusions This note presents a modification on Chen and Sanguansat’s method (2011), to make the method more effective. The modified

method can deal with the situations that some generalized fuzzy numbers’ centers of gravity on the X-axis are 0. So the modified method can be an improvement of Chen and Sanguansat’s method (2011).

Acknowledgement The work is partially supported by National Natural Science Foundation of China, Grant Nos. 60874105 and 61174022, Program for New Century Excellent Talents in University, Grant No. NCET08-0345, Chongqing Natural Science Foundation, Grant No. CSCT, 2010BA2003, the Fundamental Research Funds for the Central Universities Grant No. XDJK2010C030 and XDJK2011D002, the Southwest University Scientific & Technological Innovation Fund for Postgraduates, Grant No. ky2011011,Doctor Funding of Southwest University Grant No. SWU110021, the Open Project Program of Hangzhou Key Lab of E-Business and Information Security, Hangzhou Normal University Grant No. HZEB201001, Key Subject Laboratory of National defense for Radioactive Waste and Environmental Security Grant No. 10zxnk08.

P. Xu et al. / Expert Systems with Applications 39 (2012) 6454–6457

References Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95(3), 307–317. Chu, T. C., & Tsao, C. T. (2002). Ranking fuzzy numbers with an area between the centroid point and original point. Computers and Mathematics with Applications, 43(1), 111–117. Chen, S. M., & Chen, J. H. (2009). Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications, 36(3), 6833–6842.

6457

Chen, S. M., & Sanguansat, K. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3), 2163–2171. Deng, Y., & Liu, Q. (2005). A TOPSIS-based centroid-index ranking method of fuzzy numbers and its application in decision-making. Cybernetics and Systems, 36(6), 581–595. Deng, Y., Zhu, Z. F., & Liu, Q. (2006). Ranking fuzzy numbers with an area method using radius of gyration. Computers & Mathematics with Applications, 51(6-7), 1127–1136. Yager, R.R. (1978). Ranking fuzzy subsets over the unit interval. In Proceeding of the 17th IEEE international conference on decision and control (pp. 1435–1437). San Diego, CA.