Document Type : Research Paper


Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou 363000, China


In this paper, the notions of $(T,S)$-composition matrix and
$(T,S)$-interval-valued intuitionistic fuzzy equivalence matrix are
introduced where $(T,S)$ is a dual pair of triangular module. They
are the generalization of composition matrix and interval-valued
intuitionistic fuzzy equivalence matrix. Furthermore, their
properties and characterizations are presented. Then a new method
based on $tilde{alpha}-$matrix for clustering is developed.
Finally, an example is given to demonstrate our method.


[1] K. Atanassov,Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
[2] K. Atanassov,Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Sys-tems,
64 (1994), 159-174.
[3] K. Atanassov,On intuitionistic fuzzy negations and De Morgan Laws, Proceedings of
Eleventh International Conference, IPMU 2006, Paris, July 2-7, (2006), 2399-2404.
[4] K. Atanassov and G. Gargov,Interval-valued intuitionistic fuzzy sets
, Fuzzy Sets and Sys-tems,31 (1989), 343-349.
[5] R. A. Borzooei and Y. B. Jun,Intuitionistic fuzzy hyper BCK-ideals of hyper BCK-algebras
,Iranian Journal of Fuzzy Systems,1 (2004), 65-78.
[6] H. Bustince and P. Burillo,Correlation of interval-valued intuitionistic fuzzy sets, Fuzzy Sets
and Systems,74 (1995), 237-244.
[7] Q. Chen, Z. S. Xu, S. S. Liu and X. H. Yu,A method based on interval-valued intuitionistic
fuzzy entropy for multiple attribute decision making, Information: An International Interdis-
ciplinary Journal,13 (2010), 67-77.
[8] D. Coker,Fuzzy rough sets are intuitionistic L-fuzzy sets, Fuzzy Sets and Systems, 96
[9] G. Deschrijver,Arithmetic operators in interval-valued fuzzy set theory, Information Sciences,
 177 (2007), 2906-2924.
[10] J. Garcia and S. E. Rodabaugh, Order-theoretic, topological, categorical redundancies of
interval-valued sets, grey sets, vague sets, interval-valued intuitionistic sets, intuitionistic
fuzzy sets and topologies , Fuzzy Sets and Systems, 156 (2005), 445-484.
[11] J. Goguen,L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18(1967),145-174.
 [12] H. L. Huang and F. G. Shi,L-fuzzy numbers and their properties, Information Sciences,
178(2008), 1141-1151.
[13] W. L. Hung and J. W. Wu,Correlation of intuitionistic fuzzy sets by centroid method
,Information Sciences,144 (2002), 219-225.
[14] Y. Jiang, Y. Tang, J. Wang and S. Tang,Reasoning within intuitionistic fuzzy rough descrip-
tion logics, Information Sciences, 189 (2009), 2362-2378.
[15] A. Khan, Y. B. Jun and M. Shabir,Ordered semigroups characterized by their intuitionistic
fuzzy BI-ideals, Iranian Journal of Fuzzy Systems, 7(2010), 55-69.
[16] M. L. Lin and H. L. Huang,(T,S)-based intuitionistic fuzzy composite matrix and its appli-cation
, International Journal of Applied Mathematics and Statistics, 23 (2011), 54-63.
[17] T. K. Mondal and S. K. Samanta,Topology of interval-valued intuitionistic fuzzy sets, Fuzzy
Sets and Systems,119 (2001), 483-494.
[18] A. Narayanan, S. Vijayabalaji and N. Thillaigovindan,Intuitionistic fuzzy bounded linear
operators, Iranian Journal of Fuzzy Systems, 4 (2007), 89-101.
[19] D. G. Park, Y. C. Kwun, J. H. Park and et al.,Correlation coecient of interval-valued intu-
itionistic fuzzy sets and its application to multiple attribute group decision making problems
 ,Mathematical and Computer Modelling, 50 (2009), 1279-1293.
[20] L. Torkzadeh, M. Abbasi and M. M. Zahedi,Some results of intuitionistic fuzzy weak dual
hyper K-ideals, Iranian Journal of Fuzzy Systems, 5(2008), 65-78.
[21] P. Z. Wang,Fuzzy set theory and its application, Shanghai Science and Technology Press,
Shanghai, 1983.
 [22] Z. Wang, K. W. Li and W. Wang, An approach to multiattribute decision making with
interval-valued intuitionistic fuzzy assessments and incomplete weights, Information Sciences,179
 (2009), 3026-3040.
[23] G. Wei,Some induced geometric aggregation operators with intuitionistic fuzzy information
and their application to group decision making , Applied Soft Computing, 10 (2010), 423-431.
[24] Z. S. Xu,A method based on distance measure for interval-valued intuitionistic fuzzy group
decision making, Information Sciences, 180 (2010), 181-190.
[25] Z. S. Xu,Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences,
 180 (2010), 726-736.
[26] Z. S. Xu,A deviation-based approach to intuitionistic fuzzy multiple attribute group decision
making, Group Decision and Negotiation, 19 (2010), 57-76.
[27] Z. S. Xu,On correlation measures of intuitionistic fuzzy sets, Lecture Notes in Computer Science,
 4224 (2006), 16-24.
[28] Z. S. Xu,Intuitionistic fuzzy hierarchical clustering algorithms, Journal of Systems Engineer-
ing and Electronics,20 (2009), 90-97.
[29] Z. S. Xu and X. Cai,Incomplete interval-valued intuitionistic preference relations
, Interna-tional Journal of General Systems, 38 (2009), 871-886.
[30] Z. S. Xu and X. Q. Cai, Nonlinear optimization models for multiple attribute group decision
making with intuitionistic fuzzy information, International Journal of Intelligent Systems, 25 (2010),
[31] Z. S. Xu and J. Chen,Approach to group decision making based on interval-valued intuition-
istic judgment matrices, Systems Engineering-Theory and Practice, 27(4) (2007), 126-133.
[32] Z. S. Xu, J. Chen and J. Wu,Clustering algorithm for intuitionistic fuzzy sets, InformationSciences,
 178 (2008), 3775-3790.
[33] Z. S. Xu and J. J. Wu,Intuitionistic fuzzy c-means clustering algorithms, Journal of Systems
Engineering and Electronics, 21 (2010), 580-590.
[34] Z. S. Xu and R. R. Yager,Dynamic intuitionistic fuzzy multi-attribute decision making,
International Journal of Approximate Reasoning,48 (2008), 246-262.
[35] Z. S. Xu and R. R. Yager,Intuitionistic and interval-valued intutionistic fuzzy preference
relations and their measures of similarity for the evaluation of agreement within a group
 ,Fuzzy Optimization and Decision Making,8 (2009), 123-139.
[36] J. Ye and Multicriteria,Fuzzy decision-making method based on a novel accuracy function
under interval-valued intuitionistic fuzzy environment, Expert Systems with Applications,
36(2009), 6899-6902.
[37] J. Ye and Multicriteria,Fuzzy decision-making method using entropy weights-based correla-
tion coecients of interval-valued intuitionistic fuzzy sets, Applied Mathematical Modelling,
 34(2010), 3864-3870.
[38] L. A. Zadeh,Fuzzy sets, Information Control, 8 (1965), 338-353.
[39] H. M. Zhang, Z. S. Xu and Q. Chen,On clustering approach to intuitionistic fuzzy sets
,Control and Decision,22 (2007), 882-888.
[40] H. Y. Zhang, W. X. Zhang and W. Z. Wu,On characterization of generalized interval-
valued fuzzy rough sets on two universes of discourse, International Journal of Approximate
Reasoning,51 (2009), 56-70.
[41] L. Zhou, W. Z. Wu and W. X. Zhang,On characterization of intuitionistic fuzzy rough sets
based on intuitionistic fuzzy implicators, Information Sciences, 179 (2009), 883-898.