Skip to main content
Springer Nature Link
Log in

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

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Ali M, Deli I, Smarandache F (2016) The theory of neutrosophic cubic sets and their applications in pattern recognition. J Intell Fuzzy Syst 30:1957–1963

    Article  Google Scholar 

  • Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96

    Article  Google Scholar 

  • Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349

    Article  MathSciNet  Google Scholar 

  • Banerjee D, Giri BC, Pramanik S, Smarandache F (2017) GRA for multi attribute decision making in neutrosophic cubic set environment. Neutrosophic Sets Syst 15:60–69

    Google Scholar 

  • Biswas P, Pramanik S, Giri BC (2016) TOPSIS method for multi-attribute group decision-making under single valued neutrosophic environment. Neural Comput Appl 27(3):727–737

    Article  Google Scholar 

  • Cheng HD, Guo Y (2008) A new neutrosophic approach to image thresholding. New Math Nat Comput 4(3):291–308

    Article  Google Scholar 

  • Guo Y, Cheng HD (2009) New neutrosophic approach to image segmentation. Pattern Recognit 42:587–595

    Article  Google Scholar 

  • Guo Y, Sengur A, Ye J (2014) A novel image thresholding algorithm based on neutrosophic similarity score. Measurement 58:175–186

    Article  Google Scholar 

  • Jun YB, Lee KJ (2010) Closed cubic ideals and cubic o-subalgebras in BCK/BCI-algebras. Appl Math Sci 4:3395–3402

    MathSciNet  MATH  Google Scholar 

  • Jun YB, Kim CS, Kang MS (2010) Cubic subalgebras and ideals of BCK/BCI-algebras. Far East J Math Sci (FJMS) 44:239–250

    MathSciNet  MATH  Google Scholar 

  • Jun YB, Kim CS, Kang JG (2011a) Cubic q-ideals of BCI-algebras. Ann Fuzzy Math Inform 1:25–34

    MathSciNet  MATH  Google Scholar 

  • Jun YB, Lee KJ, Kang MS (2011b) Cubic structures applied to ideals of BCI-algebras. Comput Math Appl 62:3334–3342

    Article  MathSciNet  Google Scholar 

  • Jun YB, Kim CS, Yang KO (2012) Cubic sets. Ann Fuzzy Math Inform 4(1):83–98

    MathSciNet  MATH  Google Scholar 

  • Jun YB, Smarandache F, Kim CS (2017) Neutrosophic cubic sets. New Math Nat Comput 13(01):41–45

    Article  Google Scholar 

  • Liu PD (2016) The aggregation operators based on Archimedean t-conorm and t-norm for the single valued neutrosophic numbers and their application to decision making. Int J Fuzzy Syst 18(5):849–863

    Article  Google Scholar 

  • Liu PD, Tang GL (2016) Some power generalized aggregation operators based on the interval neutrosophic numbers and their application to decision making. J Intell Fuzzy Syst 30:2517–2528

    Article  Google Scholar 

  • Liu PD, Wang YM (2014) Multiple attribute decision-making method based on single valued neutrosophic normalized weighted Bonferroni mean. Neural Comput Appl 25(7–8):2001–2010

    Article  Google Scholar 

  • Liu PD, Wang YM (2016) Interval neutrosophic prioritized OWA operator and its application to multiple attribute decision making. J Syst Sci Complex 29(3):681–697

    Article  Google Scholar 

  • Liu PD, Chu YC, Li YW, Chen YB (2014) Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int J Fuzzy Syst 16(2):242–255

    Google Scholar 

  • Lu Z, Ye J (2017) Cosine measures of neutrosophic cubic sets for multiple attribute decision-making. Symmetry 9(7):121. https://doi.org/10.3390/sym9070121

    Article  Google Scholar 

  • Mondal K, Pramanik S (2015) Neutrosophic tangent similarity measure and its application to multiple attribute decision making. Neutrosophic Sets Syst 9:85–92

    Google Scholar 

  • Pramanik S, Biswas P, Giri BC (2017a) Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment. Neural Comput Appl 28(5):1163–1176

    Article  Google Scholar 

  • Pramanik S, Dalapati S, Alam S, Roy TK, Smarandache F (2017b) Neutrosophic cubic MCGDM method based on similarity measure. Neutrosophic Sets Syst 16:44–56

    Google Scholar 

  • Şahin R (2017) Cross-entropy measure on interval neutrosophic sets and its applications in multicriteria decision making. Neural Comput Appl 28(5):1177–1187

    Article  Google Scholar 

  • Şahin R, Küçük A (2015) Subsethood measure for single valued neutrosophic sets. J Intell Fuzzy Syst 29(2):525–530

    Article  Google Scholar 

  • Şahin R, Liu PD (2016) Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput Appl 27(7):2017–2029

    Article  Google Scholar 

  • Şahin R, Liu PD (2017) Possibility-induced simplified neutrosophic aggregation operators and their application to multicriteria group decision making. J Exp Theor Artif Intell 29(4):769–785

    Article  Google Scholar 

  • Smarandache F (1998) Neutrosophy: neutrosophic probability, set, and logic. American Research Press, Rehoboth

    MATH  Google Scholar 

  • Wang H, Smarandache F, Zhang YQ, Sunderraman R (2005) Interval neutrosophic sets and logic: theory and applications in computin\(g\). Hexis, Phoenix

    MATH  Google Scholar 

  • Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single valued neutrosophic sets. Multispace Multistruct 4:410–413

    MATH  Google Scholar 

  • Ye J (2013) Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int J Gen Syst 42(4):386–394

    Article  MathSciNet  Google Scholar 

  • Ye J (2014a) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26:2459–2466

    MathSciNet  MATH  Google Scholar 

  • Ye J (2014b) Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making. J Intell Fuzzy Syst 26:165–172

    MATH  Google Scholar 

  • Ye J (2014c) Single valued neutrosophic minimum spanning tree and its clustering method. J Intell Syst 23(3):311–324

    Google Scholar 

  • Ye J (2014d) Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. J Intell Syst 23(4):379–389

    Google Scholar 

  • Ye J (2015) Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif Intell Med 63(3):171–179

    Article  Google Scholar 

  • Ye J (2017) Single valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Comput 21(3):817–825

    Article  Google Scholar 

  • Ye J, Fu J (2016) Multi-period medical diagnosis method using a single valued neutrosophic similarity measure based on tangent function. Computer Methods Programs Biomed 123:142–149

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning, part 1. Inf Sci 8:199–249

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, 312000, Zhejiang Province, People’s Republic of China

    Jun Ye

Authors
  1. Jun Ye
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jun Ye.

Ethics declarations

Conflict of interest

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

Additional information

Communicated by A. Genovese, G. Bruno.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, J. Operations and aggregation method of neutrosophic cubic numbers for multiple attribute decision-making. Soft Comput 22, 7435–7444 (2018). https://doi.org/10.1007/s00500-018-3194-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-3194-x

Keywords