# Ranking triangular interval-valued fuzzy numbers based on the relative preference relation

Document Type: Original Manuscript

Author

Department of Shipping and Transportation Management, National Penghu University of Science and Technology, Penghu 880, Taiwan, Republic of China

Abstract

In this paper, we first use a fuzzy preference relation with a membership function representing preference degree for
comparing two interval-valued fuzzy numbers and then utilize a relative preference relation improved from the fuzzy
preference relation to rank a set of interval-valued fuzzy numbers. Since the fuzzy preference relation is a total ordering
relation that satisfies reciprocal and transitive laws on interval-valued fuzzy numbers, the relative preference relation is
also a total ordering relation. Practically, the fuzzy preference relation is more reasonable on ranking interval-valued
fuzzy numbers than defuzzification because defuzzification does not present preference degree between fuzzy numbers
and loses messages. However, fuzzy pair-wise comparison for the fuzzy preference relation is more complex and difficult
than defuzzification. To resolve fuzzy pair-wise comparison tie, the relative preference relation takes the strengths of
defuzzification and the fuzzy preference relation into consideration. The relative preference relation expresses preference
degrees of interval-valued fuzzy numbers over average as the fuzzy preference relation does, and ranks fuzzy numbers
by relative crisp values as defuzzification does. In fact, the application of relative preference relation was shown in
traditional fuzzy numbers, such as triangular and trapezoidal fuzzy numbers, for previous approaches. In this paper, we
extend and utilize the relative preference relation on interval-valued fuzzy numbers, especially for triangular intervalvalued fuzzy numbers. Obviously, interval-valued fuzzy numbers based on the relative preference relation are easily and quickly ranked, and able to reserve fuzzy information.

Keywords

### References

[1] G. Bortolan, R. Degani, A review of some methods for ranking fuzzy numbers, Fuzzy Sets and Systems, 15 (1985),
1-19.
[2] C. T. Chen,
Extensions to the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems,
114 (2000), 1-9.
[3] L. H. Chen, H. W. Lu,
An approximate approach for ranking fuzzy numbers based on left and right fuzzy dominance,
Computers and Mathematics with Applications,
41 (2001), 1589-1602.
[4] L. H. Chen, H. W. Lu,
The preference order of fuzzy numbers, Computers and Mathematics with Applications, 44
(2002), 1455-1465.
[5] C. H. Cheng,
A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95 (1998),
307-317.
[6] F. Choobineh, H. Li,
An index for ordering fuzzy numbers, Fuzzy Sets and Systems, 54 (1993), 287-294.
[7] T. C. Chu, C. T. Tsao,
Ranking fuzzy numbers with an area between the centroid point and the original point,
Computers and Mathematics with Applications,
43 (2002), 111-117.
[8] G. Dias,
Ranking alternatives using fuzzy numbers: A computational approach, Fuzzy Sets and Systems, 56 (1993),
247-252.
Operations on fuzzy numbers, The International Journal of Systems Sciences, 9 (1978), 631-
626.
[10] S. S. Epp,
Discrete Mathematics with Applications, Wadsworth, California, 1990.
[11] P. Fortemps and M. Roubens,
Ranking and defuzzification methods based on area compensation, Fuzzy Sets and
Systems,
82(1996), 319-330.
[12] M. B. Gorzalczany,
A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets
and Systems,
21 (1987), 1-17.
[13] P. Grzegoezewski,
Mertics and orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97 (1998), 83-94.
[14] R. Jain,
Decision-making in the presence of fuzzy variables, IEEE Transactions on Systems, Man, and Cybernetics,
6 (1976), 698-703.
[15] R. Jain,
A procedure for multi-aspect decision making using fuzzy sets, The International Journal of Systems
Sciences,
8 (1978), 1-7.
[16] A. Kauffman, M. M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold,
New York, 1991.
[17] M. S. Kuo, G. S. Liang,
A soft computing method of performance evaluation with MCDM based on interval-valued
fuzzy numbers
, Applied Soft Computing, 12 (2012), 476-485.
[18] C. S. Lee, C. C. Chung, H. S. Lee, G. Y. Gan, M. T. Chou,
An interval-valued fuzzy number approach for supplier
selection
, Marine Science and Technology, 24 (2016), 384-389.
[19] H. S. Lee,
A fuzzy multi-criteria decision making model for the selection of the distribution center, Lecture Notes
in Computer Science,
3612 (2005), 1290-1299.
[20] H. S. Lee,
On fuzzy preference relation in group decision making, International Journal of Computer Mathematics,
82 (2005), 133-140.
[21] K. M. Lee, C. H. Cho, H. Lee-Kwang,
Ranking fuzzy values with satisfaction function, Fuzzy Sets and Systems, 64
(1994), 295-311.
[22] E. S. Lee, R. J. Li,
Comparison of fuzzy numbers based on the probability measure of fuzzy events, Computers and
Mathematics with Applications,
15 (1988), 887-896.
[23] R. J. Li,
Fuzzy method in group decision making, Computers and Mathematics with Applications, 38 (1999),
91-101.
[24] L. Mikhailov,
Deriving priorities from fuzzy pairwise comparison judgements, Fuzzy Sets and Systems, 134 (2003),
365-385.
[25] S. Murakami, S. Maeda, S. Imamura,
Fuzzy decision analysis on the development of centralized regional energy
control system
, In: Proceedings of the IFAC Symposium on Fuzzy Information, Knowledge Representation and
Decision Analysis, (1983), 363-368.
[26] A. Raj, D. N. Kumar,
Ranking alternatives with fuzzy weights using maximizing set and minimizing set, Fuzzy
Sets and Systems,
105 (1999), 365-375.
[27] I. Requena, M. Delgado, J. I. Verdagay,
Automatic ranking of fuzzy numbers with the criterion of decision-maker
learnt by an artificial neural network
, Fuzzy sets and Systems, 64 (1994), 1-19.
[28] T. L. Saaty,
Decision making with the analytic hierarchy process, Services Sciences, 1 (2008), 83-98.
[29] A. Sanayei, S. Farid Mousavi, A. Yazdankhah,
Group decision making process for supplier selection with VIKOR
under fuzzy environment
, Expert Systems with Applications, 37 (2010), 24-30.
[30] H. C. Tang,
Inconsistent property of Lee and Li fuzzy ranking method, Computers and Mathematics with Applications, 45 (2003), 709-713.
[31] Y. J. Wang,
Ranking triangle and trapezoidal fuzzy numbers based on the relative preference relation, Applied
Mathematical Modelling,
39 (2015), 586-599.
[32] J. Wu, F. Chiclana,
A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on
novel attitudinal expected score and accuracy functions
, Applied Soft Computing, 22 (2014), 272-286.
[33] J. S. Yao, F. T. Lin,
Constructing a fuzzy flow-shop sequencing model based on statistical data, International Journal
of Approximate Reasoning,
29 (2002), 215-234.
[34] Y. Yuan,
Criteria for evaluating fuzzy ranking methods, Fuzzy Sets and Systems, 44 (1991), 139-157.
[35] C. Yue,
A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group
decision-making
, Computers and Industrial Engineering, 102 (2016), 233-245.
[36] L. A. Zadeh, Fuzzy sets, Information and Control,
8 (1965), 338-353.