ORIGINAL PAPER Distance measures for cubic Pythagorean fuzzy sets and its applications to multicriteria decision making Pranjal Talukdar 1 • Palash Dutta 1 Received: 22 February 2019 / Accepted: 1 July 2019 Ó Springer Nature Switzerland AG 2019 Abstract The main objective of this paper is to develop a sophisticated mathematical expression which can carry much more information than the general intuitionistic fuzzy set (IFS), interval-valued intuitionistic fuzzy set (IVIFS) and cubic intuitionistic fuzzy set (CIFS). CIFS is one of the powerful tools to handle uncertainty in complex situation. It is the simultaneous consideration of both the IVIFS and IFS. As in many real life situation, interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) are more capable than IVIFS and IFS to represent the vagueness or ill-defined information; therefore, it motivates us to enhance the capability of CIFS in complex decision-making problems. This paper presents a novel notion of cubic Pythagorean fuzzy set (CPFS) incorporating IVPFS and PFS simultaneously, to encounter uncertainty in a more specific manner. Furthermore, a family of distance measures for CPFSs is defined and applications of the proposed distance measures are shown in medical decision-making problem. Keywords Cubic intuitionistic fuzzy sets  Cubic Pythagorean fuzzy sets  Distance measure  Medical diagnosis 1 Introduction Multicriteria decision-making process is the technique of finding the best alternative among a set of alternatives under some criteria. Researchers have developed a lot of methods for multicriteria decision-making problems under the environment of crisp set and fuzzy set. To deal with the multicriteria decision-making process under uncertain environments, researchers prefer fuzzy set rather than crisp set. Thus, researchers have been giving their efforts to develop the multicriteria decision-making process under fuzzy environments. In 1986, Atanassov developed the theory of intuitionistic fuzzy set (IFS), which is the extension of Zadeh’s (1965) Fuzzy Set Theory (FST). IFS allows to assign each element a membership degree, a non- membership degree and a hesitation degree, whereas FST only assigns to each element a membership degree. In FST, the non-membership degree is just the compliment of the membership degree. Consequently, IFS has been treated as a more effective tool in dealing with the uncertainty than the ordinary FST since its appearance. But sometimes, in reality, it may not be possible to identify the appropriate value for membership and non-membership degree of an element in a set. In such cases, a range of values may prove to be more appropriate measurement to accommodate the information or vagueness. After the successful implemen- tation of IFSs in different branches, Atanassov and Gargov (1989) introduced the notion of IVIFSs, where both the membership and the non-membership degrees are repre- sented by intervals instead of single numbers. Thus, IVIFS is the generalisation of IFS. During the last decades, researchers have successfully studied the decision-making process, medical diagnosis, pattern recognition, artificial intelligence, fuzzy optimisation, etc., under the environ- ments of FST (Chen and Wang 1995; Chen and Tanuwi- jaya 2011; Chen and Chuan 2011; Chen et al. 2016a, b, c; Garg 2018c; Lee and Chen 2008; Chen and Huang 2003), IFS (Garg and Kumar 2019; Jamkhaneh and Garg 2018; Chen and Chang 2015; Chen et al. 2016a, b, c; Dutta and Talukdar 2018; Talukdar and Dutta 2019; Liu and Chen & Pranjal Talukdar pranjaltalukdar70@gmail.com Palash Dutta palash.dtt@gmail.com 1 Department of Mathematics, Dibrugarh University, Dibrugarh 786004, India 123 Granular Computing https://doi.org/10.1007/s41066-019-00185-3