Skip to main content
Log in

Interval Complex Neutrosophic Set: Formulation and Applications in Decision-Making

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

Neutrosophic set is a powerful general formal framework which generalizes the concepts of classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, etc. Recent studies have developed systems with complex fuzzy sets, for better designing and modeling real-life applications. The single-valued complex neutrosophic set, which is an extended form of the single-valued complex fuzzy set and of the single-valued complex intuitionistic fuzzy set, presents difficulties to defining a crisp neutrosophic membership degree as in the single-valued neutrosophic set. Therefore, in this paper we propose a new notion, called interval complex neutrosophic set (ICNS), and examine its characteristics. Firstly, we define several set theoretic operations of ICNS, such as union, intersection and complement, and afterward the operational rules. Next, a decision-making procedure in ICNS and its applications to a green supplier selection are investigated. Numerical examples based on real dataset of Thuan Yen JSC, which is a small-size trading service and transportation company, illustrate the efficiency and the applicability of our approach.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Abbreviations

NS:

Neutrosophic set

INS:

Interval neutrosophic set

CFS:

Complex fuzzy set

CIFS:

Complex intuitionistic fuzzy set

IVCFS:

Interval-valued complex fuzzy set

CNS:

Complex neutrosophic set

ICNS:

Interval-valued complex neutrosophic set, or interval complex neutrosophic set

SVCNS:

Single-valued complex neutrosophic set

MCDM:

Multi-criteria decision-making

MCGDM:

Multi-criteria group decision-making

\(\vee\) :

Maximum operator (t-conorm)

\(\wedge\) :

Minimum operator (t-norm)

References

  1. Ali, M., Smarandache, F.: Complex neutrosophic set. Neural Comput. Appl. 28(7), 1817–1834 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Can, M.S., Ozguven, O.F.: PID tuning with neutrosophic similarity measure. Int. J. Fuzzy Syst. 19(2), 489–503 (2017)

    Article  MathSciNet  Google Scholar 

  4. Greenfield, S., Chiclana, F., Dick, S.: Interval-valued complex fuzzy logic. In: IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 2014–2019. IEEE (2016)

  5. Li, Y., Liu, P., Chen, Y.: Some single valued neutrosophic number heronian mean operators and their application in multiple attribute group decision-making. Informatica 27(1), 85–110 (2016)

    Article  Google Scholar 

  6. Liu, P.: The aggregation operators based on archimedean t-Conorm and t-Norm for single-valued neutrosophic numbers and their application to decision-making. Int. J. Fuzzy Syst. 18(5), 849–863 (2016)

    Article  Google Scholar 

  7. Liu, P., Tang, G.: Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and Choquet integral. Cognit. Comput. 8(6), 1036–1056 (2016)

    Article  Google Scholar 

  8. Liu, P., Zhang, L., Liu, X., Wang, P.: Multi-valued neutrosophic number bonferronimean operators and their application in multiple attribute group decision-making. Int. J. Inf. Technol. Decis. Mak. 15(5), 1181–1210 (2016)

    Article  Google Scholar 

  9. Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002)

    Article  Google Scholar 

  10. Ramot, D., Friedman, M., Langholz, G., Kandel, A.: Complex fuzzy logic. IEEE Trans. Fuzzy Syst. 11(4), 450–461 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  12. Smarandache, F.: Neutrosophy. neutrosophic probability, set, and logic, ProQuest information and learning. Ann Arbor, Michigan, USA, 105 p (1998)

  13. Salleh, A.R.: Complex intuitionistic fuzzy sets. In: International Conference on Fundamental and Applied Sciences 2012, vol. 1482(1), pp. 464–470. (2012)

  14. Sahin, R., Yiider, M.: A multi-criteria neutrosophic group decision-making method based TOPSIS for supplier selection. ar Xiv preprint arXiv:1412.5077 (2014)

  15. Son, L.H., Tien, N.D.: Tune up fuzzy C-means for big data: some novel hybrid clustering algorithms based on initial selection and incremental clustering. Int. J. Fuzzy Syst. (2016). doi:10.1007/s40815-016-0260-3

  16. Sahin, R., Yigider, M.: A multi-criteria neutrosophic group decision making metod based TOPSIS for supplier selection. CoRRabs/1412.5077 (2014)

  17. Tian, Z.P., Zhang, H.Y., Wang, J., Wang, J.Q., Chen, X.H.: Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets. Int. J. Syst. Sci. 47(15), 3598–3608 (2016)

    Article  MATH  Google Scholar 

  18. Wang, H., Smarandache, F., Sunderraman, R., Zhang, Y.Q.: Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, vol. 5. Hexis, Arizona (2005)

  19. Wang, H., Smarandache, F., Zhang, Y., Sunderraman, R.: Single valued neutrosophic sets. Rev. Air Force Acad. 17, 10 (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Ye, J.: Similarity measures between interval neutrosophic sets and their applications in multi-criteria decision-making. J. Intell. Fuzzy Syst. 26(1), 165–172 (2014)

    MATH  Google Scholar 

  22. Ye, J.: Single valued neutrosophic cross-entropy for multi-criteria decision-making problems. Appl. Math. Model. 38(3), 1170–1175 (2014)

    Article  MathSciNet  Google Scholar 

  23. Ye, J.: Vector similarity measures of simplified neutrosophic sets and their application in multi-criteria decision-making. Int. J. Fuzzy Syst. 16(2), 204–215 (2014)

    Google Scholar 

  24. Ye, J.: Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single valued neutrosophic environment. J. Intell. Fuzzy Syst. 27(6), 2927–2935 (2014)

    MathSciNet  Google Scholar 

  25. Ye, J., Zhang, Q.S.: Single valued neutrosophic similarity measures for multiple attribute decision-making. Neutrosophic Sets Syst. 2, 48–54 (2014)

    Google Scholar 

  26. Ye, J.: Interval neutrosophic multiple attribute decision-making method with credibility information. Int. J. Fuzzy Syst. 18(5), 914–923 (2016)

    Article  Google Scholar 

  27. Zhang, G., Dillon, T.S., Cai, K.Y., Ma, J., Lu, J.: Operation properties and -equalities of complex fuzzy sets. Int. J. Approx. Reason. 50(8), 1227–1249 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, M., Liu, P., Shi, L.: An extended multiple attribute group decision-making TODIM method based on the neutrosophic numbers. J. Intell. Fuzzy Syst. 30(3), 1773–1781 (2016)

    Article  MATH  Google Scholar 

Download references

Acknowledgement

This research is funded by Graduate University of Science and Technology under grant number GUST.STS.ÐT2017-TT02. The authors are grateful for the support from the Institute of Information Technology, Vietnam Academy of Science and Technology. We received the necessary devices as experiment tools to implement proposed method.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Le Hoang Son.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, M., Dat, L.Q., Son, L.H. et al. Interval Complex Neutrosophic Set: Formulation and Applications in Decision-Making. Int. J. Fuzzy Syst. 20, 986–999 (2018). https://doi.org/10.1007/s40815-017-0380-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-017-0380-4

Keywords

Navigation