On the composition of intuitionistic fuzzy relations

Fuzzy relations are able to model vagueness, in the sense that they provide the degree to which two objects are related to each other. However, they cannot model uncertainty: there is no means to attribute reliability information to the membership degrees. Intuitionistic fuzzy sets, as defined by Atanassov (Instuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999), give us a way to incorporate uncertainty in an additional degree. Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a cartesian product of universes. One of the main concepts in relational calculus is the composition of two relations. Burillo and Bustince (Fuzzy Sets and Systems 78 (1996) 293; Soft Comput. 2 (1995) 5) have extended the sup-T composition of fuzzy relations to a composition of intuitionistic fuzzy relations. In this paper, we present an intuitionistic fuzzy version of the triangular compositions of Bandler and Kohout (in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, p. 341) and the variants of these compositions given by De Baets and Kerre (Adv. Electron. Electron Phys. 89 (1994) 255). Some properties of these compositions are investigated: containment, convertibility, monotonicity, interaction with union and intersection. (C) 2002 Elsevier Science B.V. All rights reserved.