Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm Meimei Xia a , Zeshui Xu a,b,⇑ , Bin Zhu a a School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China b Institute of Sciences, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China article info Article history: Received 18 November 2011 Received in revised form 19 January 2012 Accepted 7 February 2012 Available online 15 February 2012 Keywords: Multi-criteria decision making Archimedean t-conorm Archimedean t-norm Intuitionistic fuzzy set Aggregation operator abstract Archimedean t-conorm and t-norm are generalizations of a lot of other t-conorms and t-norms, such as Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms or others, and some of them have been applied to intuitionistic fuzzy set, which contains three functions: the membership function, the non- membership function and the hesitancy function describing uncertainty and fuzziness more objectively. Recently, Beliakov et al. [3] constructed some operations about intuitionistic fuzzy sets based on Archi- medean t-conorm and t-norm, from which an aggregation principle is proposed for intuitionistic fuzzy information. In this paper, we propose some other operations on intuitionistic fuzzy sets, study their properties and relationships, and based on which, we study the properties of the aggregation principle proposed by Beliakov et al. [3], and give some specific intuitionistic fuzzy aggregation operators, which can be considered as the extensions of the known ones. In the end, we develop an approach for multi-cri- teria decision making under intuitionistic fuzzy environment, and illustrate an example to show the behavior of the proposed operators. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction An intuitionistic fuzzy set (IFS) [1] A on a fixed set X is defined as A ={hx, l A (x), v A (x)ijx 2 X} with the condition that 0 6 l A (x) + v A (x) 6 1, l A (x) and v A (x) P 0. We can find that an IFS is con- structed by two information functions, which not only describe the membership degree l A (x) of x 2 X in A, but also describe the non-membership degree v A (x) of x 2 X in A. Moreover, the hesitancy information of x 2 X in A can be denoted by p A (x)=1  l A (x)  v A (x) which is called the hesitant index, and therefore IFS can describe the uncertainty and fuzziness more objectively than the usual fuzzy set (FS) [40]. For convenience, the pair a =(l a , v a ) is called an intuition- istic fuzzy number (IFN) [29], where l a , v a P 0 and l a + v a 6 1. Since it was introduced, IFS has attracted more and more atten- tions from researchers and has been used to deal with many prob- lems, especially the multi-criteria decision making problem which can be roughly described as: Let y ={y 1 , y 2 , ... , y m } be the set of alternatives, c ={c 1 , c 2 , ... , c n } be the set of criteria, the degree that alternative y i satisfies to the criterion c j can be denoted as l ij , the degree that the alternative y i does not satisfy the criterion c j can be denoted as v ij , then the performance of the alternative y i under the criteria c j can be described as an IFN a ij =(l ij , v ij ) with the condition that 0 6 l ij , v ij 6 1 and l ij + v ij 6 1. When all the perfor- mances of the alternatives are provided, the intuitionistic fuzzy decision matrix D =(a ij ) nn = ((l ij , v ij )) mn is constructed. Up to now, many methods have been proposed to deal with the multi- criteria decision making under intuitionistic fuzzy environment, which fall into two groups, the first group is to calculate the rela- tive values of the alternatives and the second group is to calculate the actual aggregation values of the alternatives. By comparing these two types of methods for obtaining the ranking of the alter- natives, the second one can reflect the actual results of the alterna- tives more objectively, while the first one can only obtain the relative results of the alternatives to the ideal alternative or others. To calculate the relative values of the alternatives, many classical methods have been extended to intuitionistic fuzzy environment, such as the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method [5,15,19,21], the gray relational analysis (GRA) method [11,26,43], the Elimination et Choice Translating Reality (ELECTRE) method [28], the Vlse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) method [12,23], the maximizing deviation method [25,31] and the entropy method [9,28,37,38] and so on. To calculate the actual aggregation values of the alternatives, a lot of aggregation operators have been developed. Xu [29] pro- posed some operational laws for IFNs based on Algebraic t-conorm and t-norm, and developed the intuitionistic fuzzy weighed aver- aging operator, the intuitionistic fuzzy ordered weighted averaging operator and the intuitionistic fuzzy hybrid averaging operator, 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2012.02.004 ⇑ Corresponding author at: School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, China. E-mail addresses: meimxia@163.com (M. Xia), xu_zshui@263.net (Z. Xu), zhubinThomas@gmail.com (B. Zhu). Knowledge-Based Systems 31 (2012) 78–88 Contents lists available at SciVerse ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys