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P-Indeterminate Vector Similarity Measures of Orthopair Neutrosophic Number Sets and Their Decision-Making Method with Indeterminate Degrees

Mailing Zhao1, Jun Ye1,2,*
1 Department of Electrical Engineering and Automation, Shaoxing University, Shaoxing, 312000, China
2 School of Civil and Environmental Engineering, Ningbo University, Ningbo, 315211, China
* Corresponding Author: Jun Ye. Email:
(This article belongs to this Special Issue: Advances in Neutrosophic and Plithogenic Sets for Engineering and Sciences: Theory, Models, and Applications (ANPSESTMA))

Computer Modeling in Engineering & Sciences 2021, 128(3), 1219-1230.

Received 04 April 2021; Accepted 07 May 2021; Issue published 11 August 2021


In the complexity and indeterminacy of decision making (DM) environments, orthopair neutrosophic number set (ONNS) presented by Ye et al. can be described by the truth and falsity indeterminacy degrees. Then, ONNS demonstrates its advantages in the indeterminate information expression, aggregations, and DM problems with some indeterminate ranges. However, the existing research lacks some similarity measures between ONNSs. They are indispensable mathematical tools and play a crucial role in DM, pattern recognition, and clustering analysis. Thus, it is necessary to propose some similarity measures between ONNSs to supplement the gap. To solve the issue, this study firstly proposes the p-indeterminate cosine measure, p-indeterminate Dice measure, p-indeterminate Jaccard measure of ONNSs (i.e., the three parameterized indeterminate vector similarity measures of ONNSs) in vector space. Then, a DM method based on the parameterized indeterminate vector similarity measures of ONNSs is developed to solve indeterminate multiple attribute DM problems by choosing different indeterminate degrees of the parameter p, such as the small indeterminate degree (p = 0) or the moderate indeterminate degree (p = 0.5) or the big indeterminate degree (p = 1). Lastly, an actual DM example on choosing a suitable logistics supplier is provided to demonstrate the flexibility and practicability of the developed DM approach in indeterminate DM problems. By comparison with existing relative DM methods, the superiority of this study is that the established DM approach indicates its flexibility and suitability depending on decision makers’ indeterminate degrees (decision risks) in ONNS setting.


Orthopair neutrosophic number set; p-indeterminate vector similarity measure; p-indeterminate cosine measure; p-indeterminate Dice measure; p-indeterminate Jaccard measure; decision making

Cite This Article

Zhao, M., Ye, J. (2021). P-Indeterminate Vector Similarity Measures of Orthopair Neutrosophic Number Sets and Their Decision-Making Method with Indeterminate Degrees. CMES-Computer Modeling in Engineering & Sciences, 128(3), 1219–1230.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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