Soft Computing

, Volume 22, Issue 22, pp 7435–7444 | Cite as

Operations and aggregation method of neutrosophic cubic numbers for multiple attribute decision-making



A neutrosophic cubic set (NCS) consists of both an interval neutrosophic set and a single-valued neutrosophic set. Hence, NCSs are very suitable for expressing the hybrid information of both the interval neutrosophic set (related to the uncertain part of the information) and the single-valued neutrosophic set (related to the certain part) simultaneously. However, there is little research on operations and applications of NCNs in existing literature. First, therefore, this paper proposes the operations of neutrosophic cubic numbers (NCNs), which are basic elements in NCSs. Second, we define the score, accuracy, and certainty functions, along with a ranking method for NCNs. Third, we develop a neutrosophic cubic number weighted arithmetic averaging (NCNWAA) operator and a neutrosophic cubic number weighted geometric averaging (NCNWGA) operator in order to aggregate NCN information and indicate their properties. Then, a multiple attribute decision-making method based on the NCNWAA or NCNWGA operator and the ranking method of NCNs is established under a NCN environment. Finally, an illustrative example about the selection problem of investment alternatives is provided in order to demonstrate the application and feasibility of the developed approach.


Neutrosophic cubic set Neutrosophic cubic number Ranking method Neutrosophic cubic number weighted arithmetic averaging (NCNWAA) operator Neutrosophic cubic number weighted geometric averaging (NCNWGA) operator Multiple attribute decision-making 



This paper is supported by the National Natural Science Foundation of China (No. 71471172).

Compliance with ethical standards

Conflict of interest

The author declares that I have no conflict of interest regarding the publication of this paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Information EngineeringShaoxing UniversityShaoxingPeople’s Republic of China

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