**5. Conclusions**

According to the intuitionistic fuzzy cosine similarity measures and cotangent similarity measures, based on q-rung orthopair fuzzy sets (q-ROFSs), we proposed another form of ten similarity measures by considering the function of membership degree, nonmembership degree and indeterminacy membership degree in q-ROFSs. In addition, we utilized our presented ten similarity measures and ten weighted similarity measures between q-ROFSs to deal MADM problems, including pattern recognition and scheme selection. Finally, two numerical examples and some comparative analysis

were provided to illustrate the scientific and effective of the similarity measures for pattern recognition and scheme selection. By utilizing our developed ten similarity measures, we can deal with MADM problems regarding pattern recognition and scheme selection. When comparing our developed ten similarity measures with the q-rung orthopair fuzzy weighted average (q-ROFWA) operator and q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator, our proposed methods can be applied in scheme selection and pattern recognition applications as the q-ROFWA and q-ROFWG operators can be only used to select best alternatives. Moreover, q-ROFWA and q-ROFWG operators have the limitation of considering the interrelationship between fused arguments; our proposed methods can overcome this disadvantage and derive more accuracy and reasonable decision-making results. In the future, works concerning q-ROFSs could focus on dealing with other kinds of decision-making problems such as: staff selection, investment selection, machine selection, project selection, manufacturing systems, etc. [52–59].

**Author Contributions:** All the authors conceived and worked together to achieve this work; P.W. and J.W. compiled the computing program by Excel and analyzed the data; J.W. and G.W. wrote the paper. Finally, all the authors have read and approved the final manuscript.

**Funding:** The work was supported by the National Natural Science Foundation of China under Grant No. 61174149 and 71571128, the Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China (Grant No. 14YJCZH091, 15XJA630006), and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).

**Conflicts of Interest:** The authors declare no conflict of interest.
