**1. Introduction**

As a generalization concept of fuzzy set (FS) introduced by Zadeh [1], the definition of intuitionistic fuzzy set (IFS) was initiated by Atanassov [2] for dealing with vague and uncertain information, which elaborately describe uncertain information by membership degree, non-membership degree and hesitancy degree. In [3], Gau and Buehrer presented the definition of vague set. In [4], Bustince and Burillo have showed that the notion of IFSs and vague sets coincide with each other. In order to deal with indeterminate and inconsistent information, Smarandache [5] proposed a neutrosophic set (NS). In the NS, indeterminacy-membership *IA*(*x*) is independent, thus making the NS more flexible and the most suitable for solving some decision-making problems related to the use of incomplete and imprecise information, uncertainties, predictions and so on. Zhang [6,7] studied algebraic and lattice structure for neutrosophic sets.

The conception of similarity measure for IFSs is one of the most important subjects for degree of similarity between objects in IFS theory. Chen [8] proposed the similarity measure based on a vague set for the first time. Hong [9] introduced a new similarity measure based on vague set and overcame some drawbacks of Chen's similarity measure. Szmidt and Kacprzyk [10] extend Hamming distance and Euclidean distance to construct intuitionistic fuzzy similarity measure. However, Wang and Xin [11] implied that Szmidt and Kacprzyk's distance measure [10] were ineffective in some situations. Grzegorzewski [12] extended some novel similarity measures for IFSs based on Hausdorff distance. Chen [13] pointed out some defects of Grzegorzewski's similarity measure and show some counter examples. On the other hand, some studies defined new similarity measures for IFSs, rather than extending the well-known distance measures. Li and Cheng [14] presented a new similarity measure between IFSs and applied it to pattern recognition. Mitchell [15] indicated that similarity measure of Li and Cheng [14] had some counter-intuitive cases and modified that similarity measure based on a statistical perspective. Furthermore, Liang and Shi [16] presented some counter instances to indicate that the similarity measure of Li and Cheng [14] was not suitable for some situations, and proposed several new similarity measures for IFS. Ye [17] conducted a similarity comparative study of existing similarity measures for IFSs and proposed a cosine similarity measure and weighted cosine similarity measure. Xu [18] acquainted a sequence of similarity measures for IFSs and applied

to solve multiple attribute decision-making problems. Boran et al. [19] proposed a new general type of similarity measures for IFSs with two parameters, expressing *Lp*-norm and give its relation with existing similarity measures. Zhang and Yu [20] presented a new distance measure based on interval comparison, where the IFSs were respectively transformed into the symmetric triangular fuzzy numbers. Comparison with the widely used methods indicated that the proposed method contained more information, with much less loss of information. Luo and Zhao [21] proposed a new distance measure for IFSs, which is based on a matrix norm and a strictly increasing (or decreasing) binary function, and applied it to solve pattern recognition problems.

As the development of IFSs, Atanassov introduced interval-valued intuitionistic fuzzy set (IVIFS) [22], which the membership degree, non-membership degree and hesitancy degree are represented by subinterval of [0, 1]. It therefore can represent the dynamic character of features accurately. Due to the advantages of IVIFSs in practical application, various similarity measures based on IVIFSs were studied extensively by many researchers from different angles and applied to many areas such as medical diagnosis, pattern recognition problem and so on. Liu [23] proposed a set of axiomatic definitions for entropy measures between IVIFSs, which extends Szmidt and Kacprzyk's axioms formulated for entropy between IFSs. Xu [24] generalized some formulas of similarity measures of IFSs to IVIFSs. Wei [25] proposed an new similarity measure for IVIFSs, and also applied to solve problems on pattern recognitions, multi-criteria fuzzy decision-making and medical diagnosis. Singh [26] introduced a new cosine similarity measure for IVIFSs and applied to pattern recognition. Khalaf [27] advanced a new approach for medical diagnosis by IVIFSs, which is generalized by the application of IFS theory. Dhivya [28] presented a new similarity measure for IVIFSs based on the mid points of transformed triangular fuzzy numbers.

However, there are some drawbacks in some existing similarity measures for IVIFSs, most of which ge<sup>t</sup> counterintuitive results in some situations and they cannot ge<sup>t</sup> correct classification results for dealing with the pattern recognition problems and medical diagnosis problems. For example, letting *A* = < [0.20, 0.30], [0.40, 0.60] >, *B*1 = < [0.30, 0.40], [0.40, 0.60] > and *B*2 = < [0.30, 0.40], [0.30, 0.50] > be IVIFSs, we can compute the similarity measures between *A* and *Bi* (*i* = 1, 2) by Formulas (1), (2) and (4) (see Section 3). Obviously, we have the result *B*1 =*B*2 because the membership degree of *B*1 is identical to that of *B*2, and the non-membership degree of *B*1 is not identical to that of *B*2. Therefore, we should obtain *Si*(*<sup>A</sup>*, *<sup>B</sup>*1) = *Si*(*<sup>A</sup>*, *<sup>B</sup>*2)(*<sup>i</sup>* = 1, <sup>2</sup>). However, we can obtain that *<sup>S</sup>*1(*<sup>A</sup>*, *<sup>B</sup>*1) = *<sup>S</sup>*1(*<sup>A</sup>*, *<sup>B</sup>*2) = *<sup>S</sup>*2(*<sup>A</sup>*, *<sup>B</sup>*1) = *<sup>S</sup>*2(*<sup>A</sup>*, *<sup>B</sup>*2) = 0.9 by the Formulas (1) and (2) (for *p* = 1), which is not reasonable. Meanwhile, we can ge<sup>t</sup> *SD*(*<sup>A</sup>*, *<sup>B</sup>*1) = 1 by Formula (4), which does not satisfy the second axiom of the definition for similarity measure. Therefore, we need to develop a new similarity measure to overcome these drawbacks.

The rest of the paper is organized as follows: Section 2 reviews some necessary definitions related to IVIFS. In Section 3, some existing similarity measures are reviewed. In Section 4, a novel similarity measure is introduced. The geometric interpretation of the new similarity measure and the explanation of parameters are briefly given in Section 5. Applications in pattern recognition and medical diagnosis are presented in Section 6. The conclusions for this paper are given in the last section.
