**7. Conclusions**

In this study, we propose the concept of *INLMSM*, which can not only adapt to the cognitive situation of decision maker, but also provide convenience for decision making. We introduce the basic concept of *INLMSM* and its generalized form, give some operators based on *INLMSM*, and introduce the theory of weight to investigate *WINLMSM* and *WINLGMSM*. Afterwards, we put forward the *INLMSM* operator, the *INLGMSM* operator, the *WINLMSM* operator, and the *WINLGMSM* operator. In addition, we proved these operators. In addition, we introduce the MADM methods with *INLMSM* in detail and illustrate their usefulness and effectiveness by showing examples. Finally, we compare other methods to demonstrate our approach. From this paper, we can see that *WINLGMSM* is more practical and flexible in application and *INLMSM* can express fuzzy information more conveniently. In further study, we can use the *INLMSM* operator to solve practical problems and pattern recognition. We should develop other aggregation operators for future research.

**Acknowledgments:** This paper is supported by the National Natural Science Foundation of China (Nos. 71771140 and 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project (Nos. 17BGLJ04,16CGLJ31 and 16CKJJ27), the Natural Science Foundation of Shandong Province (Nos. ZR2013FM017, ZR2017MG007), and the Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (2015Z057).

**Author Contributions:** Yushui Geng proposed extended MSM operators for INLNs; Xingang Wang designed the experiments; Xuemei Li and Kun Yu analyzed the data, and wrote the paper; Peide Liu proposed the ideas of this paper.

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

**Compliance with Ethical Standards:** (1) Disclosure of potential conflicts of interest. We declare that we have no commercial or associative interests that represent a conflict of interest in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work. (2) Research involving human participants and/or animals. This article does not contain any studies with human participants or animals performed by any of the authors.
