Document Type : Research Paper


Interdisciplinary Centre for Computational Modelling, Faculty of Mathematics and Natural Sciences, University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Poland


In this paper, the problem of measuring the degree of inclusion and similarity measure for two   interval-valued intuitionistic  fuzzy sets is considered. We propose inclusion and similarity measure by using  order on interval-valued intuitionistic fuzzy sets connected with lexicographical order. Moreover, some properties of inclusion and similarity measure and some correlation, between them and aggregations are examined. Finally, we have included example of ranking problem in car showrooms.


[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87{96.
[2] K. T. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst., 64
(1994), 159{174.
[3] K. T. Atanassov, Intuitionistic Fuzzy Sets. Theory and Applications, Physica-Verlag, Heidelberg/
New York, 1999.
[4] U. Bodenhofer, B. De Baets and J. Fodor, A compendium of fuzzy weak orders: Representa-
tions and constructions, Fuzzy Sets Syst., 158 (2007), 811{829.
[5] K. Bosteels and E. E. Kerre, On a re
exivity-preserving family of cardinality-based fuzzy
comparison measures, Inform. Sci., 179 (2009), 2342{2352.
[6] H. Bustince, J. Fernandez, R. Mesiar, J. Montero and R. Orduna, Overlap functions, Nonlinear
Anal.: Theory Methods Appl., 72 (2010), 1488{1499.
[7] H. Bustince, M. Pagola, R. Mesiar, E. Hullermeier and F. Herrera, Grouping, overlaps, and
generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans.
Fuzzy Syst., 20(3) (2012), 405{415.
[8] H. Bustince, J. Fernandez, A. Kolesarova and R. Mesiar, Generation of linear orders for
intervals by means of aggregation functions, Fuzzy Sets Syst., 220 (2013), 69-77.
[9] T. Calvo, A. Kolesarova, M. Komornikova and R. Mesiar, Aggregation operators: properties,
classes and construction methods, In T. Calvo, G. Mayor, and R. Mesiar (Eds.), Physica-
Verlag, New York, Aggregation Operators. Studies in Fuzziness and Soft Computing, 97
(2002), 3-104.
[10] B. De Baets and R. Mesiar, Triangular norms on product lattices, Fuzzy Sets Syst., 104
(1999), 61{76.
[11] B. De Baets, H. De Meyer and H. Naessens, A class of rational cardinality-based similarity
measures, J. Comp. Appl. Math., 132 (2001), 51{69.
[12] B. De Baets and H. De Meyer, Transitivity frameworks for reciprocal relations:cycle-
transitivity versus FG-transitivity, Fuzzy Sets Syst., 152 (2005), 249{270.
[13] B. De Baets, S. Janssens and H. De Meyer, On the transitivity of a parametric family of
cardinality-based similarity measures, Int. J. Appr. Reason., 50 (2009), 104{116.
[14] M. De Cock and E. E. Kerre, Why fuzzy T-equivalence relations do not resolve the Poincar'e
paradox, and related issues, Fuzzy Sets Syst., 133 (2003), 181{192.
[15] L. De Miguel, H. Bustince, J. Fernandez, E. Indurain, A. Kolesarova and R. Mesiar, Con-
struction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets
with an application to decision making, Information Fusion, 27 (2016), 189-197.
[16] S. Freson, B. De Baets and H. De Meyer, Closing reciprocal relations w.r.t. stochastic tran-
sitivity, Fuzzy Sets Syst., 241 (2014), 2{26.
[17] B. Jayaram and R. Mesiar, I-Fuzzy equivalence relations and I-fuzzy partitions, Inf. Sci., 179
(2009), 1278{1297.
[18] D. F. Li, Toposis-based nonlinear-programming methodology for multiattribute decision mak-
ing with interval-valued intuitionistic fuzzy sets, IEEE Trans. Fuzzy Syst., 18 (2010), 299{311.
[19] X. D. Liu, S. H. Zheng and F. L. Xiong, Entropy and subsethood for general interval-valued
intuitionistic fuzzy sets, Lecture Notes Artif. Intell., 3613 (2005), 42{52.
[20] N. Madrid, A. Burusco, H. Bustince, J. Fernandez and I. Per lieva, Upper bounding overlaps
by groupings, Fuzzy Sets Syst., 264 (2015), 76{99.
[21] S. Ovchinnikov, Numerical representation of transitive fuzzy relations, Fuzzy Sets Syst., 126
(2002), 225{232.
[22] D. G. Park, Y. C. Kwun, J. H. Park and I. Y. Park, Correlation coecient of interval-
valued intuitionistic fuzzy sets and its application to multiple attribute group decision making
problems, Math. Comput. Modell., 50 (2009), 1279{1293.
[23] Z.  Switalski, General transitivity conditions for fuzzy reciprocal preference matrices, Fuzzy
Sets Syst., 137 (2003), 85{100.
[24] L. A. Zadeh, Fuzzy sets, Information Contr., 8 (1965), 338 { 353.
[25] W. Y. Zeng and P. Guo, Normalized distance, similarity measure, inclusion measure and
entropy of interval-valued fuzzy sets and their relationship, Inf. Sci., 178 (2008), 1334{1342.
[26] H. Y. Zhang and W. X. Zhang, Hybrid monotonic inclusion measure and its use in measuring
similarity and distance between fuzzy sets, Fuzzy Sets Syst., 160 (2009), 107{118.
[27] Q. Zhang, H. Xing, F. Liu and J. Ye, P. Tang, Some new entropy measures for interval-
valued intuitionistic fuzzy sets based on distances and their relationships with similarity and
inclusion measures, Inf. Sci., 283 (2014), 55{69.