U.P.B. Sci. Bull., Series A, Vol. 77, Iss. 1, 2015 ISSN 1223-7027 TEMPERED INTERVAL-VALUED FUZZY HYPERGRAPHS Muhammad Akram 1 , Noura Omair Alshehri 2 In this article, we first present concepts and properties of interval-valued fuzzy hypergraphs. Then we introduce the notion of A =[μ - ,μ + ]-tempered interval-valued fuzzy hypergraphs and investigate some of their properties. Keywords: interval-valued fuzzy hypergraphs, A-tempered interval-valued fuzzy hyper- graph. MSC2010: 05C99. 1. Introduction Zadeh [20] introduced the notion of interval-valued fuzzy sets as extensions of Zadeh’s fuzzy set theory [19] for representing vagueness and uncertainty. Interval-valued fuzzy set theory reflects the uncertainty by the length of the interval membership degree [μ 1 ,μ 2 ]. In intuitionistic fuzzy set theory for every membership degree (μ 1 ,μ 2 ), the value π =1 − μ 1 − μ 2 denotes a measure of non-determinacy (or undecidedness). Interval-valued fuzzy sets provide a more adequate description of vagueness than traditional fuzzy sets. It is therefore important to use interval-valued fuzzy sets in applications, such as fuzzy control. One of the computationally most intensive parts of fuzzy control is defuzzification [13]. Since interval- valued fuzzy sets are widely studied and used, we describe briefly the work of Gorzalczany on approximate reasoning [8, 9], Roy and Biswas on medical diagnosis [17], Turksen on multivalued logic [18] and Mendel on intelligent control [13]. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert system. Kaufmann’s initial definition of a fuzzy hypergraph [10] was based on Zadeh’s fuzzy relations [19]. Lee-kwang et al. [11] generalized and redefined the concept of fuzzy hypergraphs whose basic idea was given by Kaufmann [10]. Further the concept of fuzzy hypergraphs was discussed in [7]. The concepts and applications of intuitionistic fuzzy hypergraphs are discussed in [3, 15]. Chen [5] introduced the concept of interval-valued fuzzy hypergraphs. In this article, we introduce the notion of A =[μ - ,μ + ]−tempered interval-valued fuzzy hypergraphs and investigate some of their properties. 2. Preliminaries In this section, we review some elementary concepts whose understanding is necessary fully benefit from this paper. A hypergraph is a pair H * =(V,E * ), where V is a finite set of nodes (vertices) and E * is a set 1 Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan, E-mail: m.akram@pucit.edu.pk 2 Department of Mathematics, Faculty of Sciences(Girls), King Abdulaziz University, Jeddah, Saudi Arabia, E-mail: nalshehrie@kau.edu.sa 39