**1. Introduction**

As an important branch of multiple attribute decision-making (MADM) domains, the similarity measures have been regarded as very useful tools to determine the degree of similarity between two objects. In the previous research literature, an increasing number of researchers pay attention to similarity measures between fuzzy sets (FSs) due to their broad applications in a variety of fields, for instance, pattern recognition, scheme selection, machine learning, image processing, and decision-making, many theories and applications of similarity measures between fuzzy sets (FSs) have been presented and investigated for the past few years. Atanassov [1,2] presented the definition of intuitionistic fuzzy set (IFS), which is an extension form of fuzzy set (FS). Each element contained in IFS was depicted by an ordered pair including the degree of membership μ and non-membership *v*, and the sum of them is limited to 1. Since IFS theory was proposed, a variety of similarity measures between intuitionistic fuzzy sets (IFSs) have been studied in the document [3–6]. Based on IFS and theories of similarity measures, Li and Cheng [7] presented appropriate similarity measure and gave a numerical example of pattern recognition problems to illustrate the e ffective of this method. Besides, Mitchell [8] improved Li and Cheng's similarity measures to deal with MADM. According to the extension of the Hamming distance (HD) of fuzzy sets (FSs), Park et al. [9] computed the distance between IFSs based on Hamming distance (HD) and proposed some similarity measures to solve MADM problems [10]. According to the Hausdor ff distance, Torra and Narukawa [11] defined some new similarity measures between IFSs. Based on geometric aggregation operators, Xia and Xu [12] proposed the intuitionistic fuzzy geometric distance and intuitionistic fuzzy similarity measures to deal with MADM problems. Ye [13] initially developed the intuitionistic fuzzy cosine similarity measure based on cosine function. Kuo-ChenHung [14] defined the likelihood-based measurement of IFSs for the medical diagnosis and bacteria classification problems. Shi and Ye [15] further modified

the cosine similarity measure of IFSs. Based on the cotangent function, Tian [16] presented the intuitionistic fuzzy cotangent similarity measure between IFSs for medical diagnosis. To contain more fuzzy information, Rajarajeswari and Uma [17] further defined the cotangent similarity measure which considered the function of membership degree, non-membership degree, and indeterminacy membership degrees described in IFSs. In addition, Szmidt [18] introduced distances between IFSs and introduced a family of similarity measures which considered the function of membership degree, non-membership degree, and indeterminacy membership degree in IFSs. Ye [19] developed two new cosine similarity measures and weighted cosine similarity measures based on cosine function and the fuzzy information denoted by the function of membership degree, non-membership degree, and indeterminacy membership degree described in intuitionistic fuzzy sets (IFSs). Wei [20] proposed some picture fuzzy similarity measures and applied them in MADM problems. Le Hoang and Pham Hong [21] defined the intuitionistic vector similarity measures for medical diagnosis. Wei and Wei [22] introduced some Pythagorean fuzzy similarity measures based on cosine function and applied them in pattern recognition and medical diagnosis.

More recently years, Pythagorean fuzzy set (PFS) [23] has emerged to describe the indeterminacy and complexity of the evaluation information. Similar to IFS, the PFS also consisted of the function of membership μ and non-membership *v*; the sum of squares of μ and *v* is restricted to 1, thus it is clear that the PFS is more widespread than the IFS and can express more decision-making information. For instance, the membership is given as 0.6 and the non-membership is given as 0.8, therefore it is obvious that this problem is only valid for PFS. In other words, all the intuitionistic fuzzy decision-making problems are the special case of Pythagorean fuzzy decision-making problems, which means that PFS can more e fficiently deal with MADM problems. In previous literatures, some researching works have been studied by a large amount of investigators [24–28]. Zhang and Xu [29] defined the Pythagorean fuzzy TOPSIS model to deal with the MADM problems. Peng and Yang [30] primarily proposed two Pythagorean fuzzy operations including the division and subtraction operations to better understand PFS. Reformat and Yager [31] handled the collaborative-based recommender system with Pythagorean fuzzy information. Garg [32] defined some new Pythagorean fuzzy aggregation operators including Pythagorean fuzzy Einstein weighted averaging (PFEWA) operator, Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA) operator, generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA) operator, and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA) operator. Zeng, et al. [33] utilized the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator to study Pythagorean fuzzy MADM issues. Ren, et al. [34] built the Pythagorean fuzzy TODIM model. Wei and Lu [35] developed some new Maclaurin symmetric mean (MSM) [36] operator based on Pythagorean fuzzy environment. Wei and Wei [22] defined ten cosine similarity measures under Pythagorean fuzzy environment. Liang, et al. [37] investigated some Bonferroni mean operators with Pythagorean fuzzy information. Liang, et al. [38] presented Pythagorean fuzzy Bonferroni mean aggregation operators based on geometric averaging (GA) operations. Combined the PFSs [39–41] and dual hesitant fuzzy sets (DHFSs) [42], Zhao et al. [43] introduced the definition of the dual hesitant Pythagorean fuzzy sets (DHPFSs) and proposed some dual hesitant Pythagorean fuzzy Hamacher aggregation operators.

In spite of this, to express more decision information, Yager [44] initially defined the q-rung orthopair fuzzy sets (q-ROFSs), in which the sum of the *qth* power of the membership and non-membership is less or equal to 1, that is to say, μ*q* + *vq* ≤ 1. Obviously, q-ROFS are more general for the IFS, and PFSs are special issues of it. Liu and Wang [45] developed the q-rung orthopair fuzzy weighted averaging (q-ROFWA) operator and the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator. Wei, et al. [46] proposed some q-rung orthopair fuzzy MSM operators, including q-rung orthopair fuzzy MSM (q-ROFMSM) operator, q-rung orthopair fuzzy weighted MSM (q-ROFWMSM) operator, q-rung orthopair fuzzy dual MSM (q-ROFDMSM) operator, and q-rung orthopair fuzzy weighted DMSM (q-ROFWDMSM) operator. Wei, et al. [47] defined some

q-rung orthopair fuzzy Heronian mean operators. Yang and Pang [48] presented some new partitioned Bonferroni mean operators under q-rung orthopair fuzzy environment. Liu and Liu [49] provided some power Bonferroni mean operators with linguistic q-rung orthopair fuzzy information. Xu, et al. [50], given the concept of q-rung dual hesitant orthopair fuzzy set (q-RDHOFS), proposed some q-rung dual hesitant orthopair fuzzy Heronian mean operators. Lei and Xu [51] gave some methods for MAGDM with q-rung interval-valued orthopair fuzzy information for green supplier selection.

Although the intuitionistic fuzzy set (IFS) [1,2] and Pythagorean fuzzy set (PFS) [23,39] have been applied in some decision-making areas, for some special cases, such as when the membership degree and non-membership degree are given as 0.7 and 0.8, it is clear that both IFS and PFS theories cannot satisfy this situation. The q-rung orthopair fuzzy set (q-ROFS) is also denoted by the degree of membership and non-membership whose *q-th* power sum of them is restricted to 1. Obviously, the q-ROFS is more general than the q-ROFS and can express more fuzzy information. In other words, the q-ROFS can deal with the MADM problems which IFS cannot, and it is clear that IFS is a part of the q-ROFS, which indicates q-ROFS can be more e ffective and powerful to deal with fuzzy and uncertain decision-making problems. Thus, to solve such issues, based on the cosine functions and cotangent functions, we shall propose the concept of q-rung orthopair fuzzy cosine similarity measures and q-rung orthopair fuzzy cotangent similarity measures under q-rung orthopair fuzzy environment in this paper, which is a new extension of the similarity measure of IFSs.

To do this, the rest of this article is structured as follows. In the next section, we briefly review some fundamental theories of intuitionistic fuzzy set (IFS) and some intuitionistic fuzzy similarity measures. Some q-rung orthopair fuzzy cosine similarity measures, q-rung orthopair fuzzy weighted cosine similarity measures, q-rung orthopair fuzzy cotangent similarity measures, and q-rung orthopair fuzzy weighted cotangent similarity measures are developed in Section 3. All the above-mentioned similarity measures for q-ROFSs are used to pattern recognition and scheme selection in Section 4. Section 5 concludes the paper with some remarks.
