Abstract. Type-2 fuzzy sets are the generalization of ordinary fuzzy sets which consider secondary membership function and primary membership function. In this paper a new approach is proposed for the ranking of triangular type-2 fuzzy sets by considering both secondary membership function and uncertainty in primary membership function. It is shown that the proposed approach satisfies reasonable properties for the ordering of fuzzy quantities. Comparison with existing method is also presented. For the illustration of the proposed approach some numerical examples are given. Application of proposed ranking function in multicriteria decision making problem is also presented.

Received: July 22, 2016

Revised: August 20, 2016

Published: October 1, 2016

AMS Subject Classification: 05C72

Key Words and Phrases: fuzzy sets, type-2 fuzzy sets, ranking functions, multicriteria decision making problems
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Bibliography

1

S.J. Chen, S.M. Chen, Fuzzy risk analysis based on the ranking of generalizedtrapezoidal fuzzy numbers, Appl. Intelligence, 26 (2007), 1-11. doi: 10.1007/s10489-006-0003-5

2

S.M. Chen, J.H. Chen, Fuzzy risk analysis based on ranking generalized fuzzynumbers with dierent heights and dierent spreads, Expert Syst. Appl., 36 (2009), 6833-6842. doi: 10.1016/j.eswa.2008.08.015

3

S.M. Chen, K. Sanguansat, Analyzing fuzzy risk based on a new fuzzy rankingmethod between generalized fuzzy numbers, Expert Syst. Appl., 38 (2011), 2163-2171. doi: 10.1016/j.eswa.2010.08.002

4

S.M. Chen, M.W. Yang , L.W. Lee, S.W. Yang, Fuzzy multiple attributes groupdecision-making based on ranking interval type-2 fuzzy sets, Expert Syst. Appl., 39 (2012), 5295-5308. doi: 10.1016/j.eswa.2011.11.008

5

C.H. Cheng, A new approach for ranking fuzzy numbers by distance method,Fuzzy Sets and Systems, 95 (1998), 307-317. doi: 10.1016/S0165-0114(96)00272-2

6

D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications,Academic Press, 1980.

7

A. Gupta, A. Kumar, M.K. Sharma, Application of tabu search for solvingthe bi-objective warehouse problem in a fuzzy environment, Iran. J.Fuzzy Syst., 9 (2012), 1-19.

8

E. Hisdal, The If THEN ELSE statement and interval-valued fuzzy sets ofhigher type, Internat. J. Man-Mach. Stud., 15 (1981), 385-455. doi: 10.1016/S0020-7373(81)80051-X

9

R. Jain, Decision-making in the presence of fuzzy variables, IEEE Trans. Systems Man Cybernet., 6 (1976), 698-703. doi: 10.1109/TSMC.1976.4309421

10

R.I. John, Type 2 fuzzy sets: an appraisal of theory and applications, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 6 (1998), 563-576. doi: 10.1142/S0218488598000434

11

R.I. John, Embedded interval valued type-2 fuzzy sets, Proc. 2002 IEEE Int. Conf. Fuzzy Systems, 2 (2002), 1316-1320.

12

R.I. John, C. Czarnecki, A type-2 adaptive fuzzy inferencing system, Proc. 1998 Int. Conf. Systems, Man Cybernet., (1998) 2068-2073.

13

A. Kumar, P. Singh, A. Kaur, P. Kaur, RM approach for ranking of generalizedtrapezoidal fuzzy numbers. Fuzzy Inf. Engin., 1 (2010), 37-47. doi: 10.1007/s12543-010-0036-7

14

A. Kumar, P. Singh, A. Kaur, P. Kaur, RM approach for ranking of - type generalized fuzzy numbers, Soft Computing, 15 (2011), 1373-1381. doi: 10.1007/s00500-010-0676-x

15

A. Kumar, P. Singh, A. Kaur, P. Kaur, A new approach for ranking nonnormalp-norm trapezoidal fuzzy numbers, Comput. Math.Appl., 61 (2011), 881-887. doi: 10.1016/j.camwa.2010.12.036

16

A. Kumar, P. Singh, P. Kaur, A. Kaur, A new approach for ranking of -type generalized fuzzy numbers, Expert Syst. Appl., 38 (2011), 10906-10910. doi: 10.1016/j.eswa.2011.02.131

17

H.C. Kwang, J.H. Lee, A method for ranking fuzzy numbers and its applicationto decision-making. IEEE Trans. Fuzzy Systems, 7 (1999), 677-685. doi: 10.1109/91.811235

18

T.S. Liou, M.J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems, 50 (1992) 247-255. doi: 10.1016/0165-0114(92)90223-Q

19

J.M. Mendel, Uncertainty, fuzzy logic and signal processing, Signal Process, 80 (2000), 913-933. doi: 10.1016/S0165-1684(00)00011-6

20

J.M, Mendel, R.I.B. John, Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Systems, 10 (2002), 117-127. doi: 10.1109/91.995115

21

J.M. Mendel, H. Wu, New results about the centroid of an interval type-2 fuzzyset, including the centroid of a fuzzy granule, Inform. Sci., 177 (2007),360-377. doi: 10.1016/j.ins.2006.03.003

22

H.B. Mitchell, Ranking type-2 fuzzy numbers, IEEE Trans. Fuzzy Systems, 14 (2006), 287-294. doi: 10.1109/TFUZZ.2005.864078

23

M. Mizumoto, K. Tanaka, Some properties of fuzzy sets of type-2, Informationand Control, 31 (1976), 312-340. doi: 10.1016/S0019-9958(76)80011-3

24

M. Mizumoto, K. Tanaka, Fuzzy sets of type-2 under algebraic product andalgebraic sum, Fuzzy Sets and Systems, 5 (1976), 277-290. doi: 10.1016/0165-0114(81)90056-7

25

M. Modarres, S. Sadi-Nezhad, Ranking fuzzy numbers by preference ratio,Fuzzy Sets and Systems, 118 (2001), 429-436. doi: 10.1016/S0165-0114(98)00427-8

26

N. Ramli, D. Mohamad, A comparative analysis of centroid methods in rankingfuzzy numbers, Eur. J. Sci. Res., 28 (2009), 492-501.

27

P. Singh, A new approach for the ranking of fuzzy sets with different heights,J. Appl. Res. Technol., 10 (2012), 941-949.

28

P. Singh, A novel method for ranking generalized fuzzy numbers, J. Inform. Sci. Engin., 31 (2015), 1373-1385.

29

I.B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, 20 (1986), 191-210. doi: 10.1016/0165-0114(86)90077-1

30

I.B. Turksen, Type-2 representation and reasoning for CWW, Fuzzy Sets and Systems, 127 (2002), 17-36. doi: 10.1016/S0165-0114(01)00150-6

31

X. Wang, E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems, 118 (2001), 375-385. doi: 10.1016/S0165-0114(99)00062-7

32

D. Wu, J.M. Mendel, Uncertainty measures for interval type-2 fuzzy sets, Inform. Sci., 177 (2007), 5378-5393. doi: 10.1016/j.ins.2007.07.012

33

D. Wu, J.M. Mendel, A comparative study of ranking methods, similarity measuresand uncertainty measures for interval type-2 fuzzy sets, Inform. Sci., 179 (2009), 1169-1192. doi: 10.1016/j.ins.2008.12.010

34

R.R. Yager, Fuzzy subsets of type II in decisions, J. Cybernet., 10(1980), 137-159. doi: 10.1080/01969728008927629

35

L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353.

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DOI: 10.12732/ijpam.v109i3.12 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 109
Issue: 3
Pages: 631 - 649

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This work is licensed under the Creative Commons Attribution International License (CC BY).

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