Intuitionistic fuzzy hypergraphs with applications Muhammad Akram a,⇑ , Wieslaw A. Dudek b a Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore 54000, Pakistan b Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland article info Article history: Received 27 October 2011 Received in revised form 26 February 2012 Accepted 23 June 2012 Available online 6 July 2012 Keywords: Intuitionistic fuzzy hypergraph Strength of edge (class) Intuitionistic fuzzy partition Dual intuitionistic fuzzy hypergraph Clustering problem abstract Hypergraphs are considered a useful tool for modeling system architectures and data struc- tures and to represent a partition, covering and clustering in the area of circuit design. In this paper, we apply the concept of intuitionistic fuzzy set theory to generalize results con- cerning hypergraphs. For each intuitionistic fuzzy structure defined, we use cut-level sets to define an associated sequence of crisp structures. We determine what properties of the sequence of crisp structures characterize a given property of the intuitionistic fuzzy struc- ture. We also present applications of intuitionistic fuzzy hypergraphs. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Presently, science and technology is featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing ele- ments of uncertainty. A large number of these models is based on an extension of the ordinary set theory, namely, fuzzy sets. The notion of fuzzy sets was introduced by Zadeh [32] as a method of representing uncertainty and vagueness. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines, including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multi- agent systems, pattern recognition, robotics, computer networks, expert systems, decision making and automata theory. In 1986, Atanassov [6] introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Atanassov added in the definition of fuzzy set a new component which determines the degree of nonmembership. Fuzzy sets give the degree of membership of an element in a given set (the nonmembership of degree equals one minus the degree of mem- bership), while intuitionistic fuzzy sets give both a degree of membership and a degree of nonmembership, which are more- or-less independent from each other; the only requirement is that the sum of these two degrees is not greater than 1. Intui- tionistic fuzzy sets are higher order fuzzy sets. Application of higher order fuzzy sets makes the solution-procedure more complex, but if the complexity on computation-time, computation-volume or memory-space are not the matter of concern then a better result could be achieved. Graph theory has numerous applications to problems in computer science, electrical engineering, system analysis, oper- ations research, economics, networking routing, and transportation. However, in many cases, some aspects of a graph-the- oretic problem may be uncertain. For example, the vehicle travel time or vehicle capacity on a road network may not be known exactly. In such cases, it is natural to deal with the uncertainty using the methods of fuzzy sets and fuzzy logic. 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.06.024 ⇑ Corresponding author. Fax: +92 99212505. E-mail addresses: makrammath@yahoo.com, m.akram@pucit.edu.pk (M. Akram), wieslaw.dudek@pwr.wroc.pl (W.A. Dudek). Information Sciences 218 (2013) 182–193 Contents lists available at SciVerse ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins