Vol.:(0123456789) 1 3 Int. J. Mach. Learn. & Cyber. DOI 10.1007/s13042-016-0632-x ORIGINAL ARTICLE Exponential operational laws and new aggregation operators of intuitionistic Fuzzy information based on Archimedean T-conorm and T-norm Xue Luo 1  · Zeshui Xu 1  · Xunjie Gou 1   Received: 21 June 2016 / Accepted: 27 December 2016 © Springer-Verlag Berlin Heidelberg 2017 1 Introduction Most of the existing mathematical tools for formal mod- eling, reasoning and computing are crisp, deterministic and precise in nature, which are not capable of dealing with the problems involving uncertainty, imprecision or fuzziness. Fuzzy set (FS) [1], characterized by the membership func- tion, is suitable to deal with those uncertain or fuzzy prob- lems. Later, Atanassov extended the FS to intuitionistic fuzzy set (IFS) [2, 3]. The IFS is constructed by three func- tions, i.e., the membership function, the non-membership function, and the indeterminacy function, and thus, the IFS can describe uncertainty and fuzziness more comprehen- sively than the FS. Atanassov [4] and De et al. [5] intro- duced some basic operational laws of IFSs. For simplicity, Xu and Yager [6–8] defned the concept of intuitionis- tic fuzzy number (IFN) and gave some operational laws of IFNs, such as “intersection”, “union”, “supplement”, “power” and so on. Besides, Gou et al. [9] presented the exponential operational law of IFNs, which is an efective supplement for the calculations of IFNs. Based on these operational laws of IFNs, lots of intui- tionistic fuzzy aggregation operators have been developed, such as the intuitionistic fuzzy weighted averaging (IFWA) operator [7], the intuitionistic fuzzy weighted geomet- ric (IFWG) operator [6], the intuitionistic fuzzy ordered weighted averaging (IFOWA) [8] operator, and the intui- tionistic fuzzy weighted exponential aggregation (IFWEA) operator [9], etc. With the advantage in depicting uncertain and fuzzy information, IFSs and IFNs have been widely applied in many practical areas of modern life, including aggrega- tion techniques [6–16], distance measures [17–19], corre- lation measures [20–23], intuitionistic preference relations [24–26], dynamic decision making [27–29], intuitionistic Abstract Atanassov extended the fuzzy set to intuition- istic fuzzy set (IFS) whose basic components are intuition- istic fuzzy numbers (IFNs). IFSs and IFNs can depict the fuzzy characteristics of the objects comprehensively, and lots of operational laws have been introduced to facilitate the use of IFSs and IFNs for solving the practical problems under intuitionistic fuzzy environments. As a supplement of the existing operational laws, we defne the exponen- tial operational laws of IFSs and IFNs based on Archime- dean t-conorm and t-norm (EOL-IFS-A and EOL-IFN-A), which can be considered as the more general forms of the original exponential operational law. After that, we study the properties of the EOL-IFS-A and EOL-IFN-A. Then, we develop an approach for multiple criteria decision mak- ing with intuitionistic fuzzy information. Finally, we give an example to illustrate the application of the developed approach, and make a detailed comparison with the exist- ing method so as to show the advantages of our approach. Keywords Intuitionistic fuzzy set · Intuitionistic fuzzy number · Exponential operational law · Archimedean t-conorm and t-norm · Multiple criteria decision making * Zeshui Xu xuzeshui@263.net Xue Luo 961586293@qq.com Xunjie Gou gouxunjie@qq.com 1 Business School, Sichuan University, Chengdu 610064, China