Skip to main content
SpringerLink
Log in

The Aggregation Operators Based on Archimedean t-Conorm and t-Norm for Single-Valued Neutrosophic Numbers and their Application to Decision Making

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

Abstract

The single-valued neutrosophic set (SVNS) can be easier to describe the incomplete, indeterminate and inconsistent information, and Archimedean t-conorm and t-norm (ATT) can generalize most of the existing t-conorms and t-norms, including Algebraic, Einstein, and Hamacher Frank t-conorms and t-norms. In this paper, we extended ATT to the single-valued neutrosophic numbers (SVNNs), and proposed a single-valued neutrosophic number-weighted averaging (SVNNWA) operator and a single-valued neutrosophic number-weighted geometric (SVNNWG) operator based on ATT. First, we presented some new operational laws for SVNNs based on ATT, and discussed some special cases and properties of them. Then we proposed the SVNNWA and SVNNWG operators, and studied some properties and special cases of them. Further, we gave the decision-making methods for multiple attribute decision-making (MADM) and multiple attribute group decision-making (MAGDM) problems based on these operators. Finally, two examples are given to verify the developed approaches.

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

Access this article

Log in via an institution

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

References

  1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–356 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Atanassov, K.T.: More on intuitionistic fuzzy sets. Fuzzy Sets Syst. 33, 37–46 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Atanassov, K.T.: Operators over interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 64, 159–174 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hu, S.K., Lu, M.T., Tzeng, G.H.: Improving mobile commerce adoption using a new hybrid fuzzy MADM model. Int. J. Fuzzy Syst. 17(3), 399–413 (2015)

    Article  Google Scholar 

  7. Wei, G.W.: Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information. Int. J. Fuzzy Syst. 17(3), 484–489 (2015)

    Article  MathSciNet  Google Scholar 

  8. Khalid, A., Abbas, M.: Distance measures and operations in intuitionistic and interval-valued intuitionistic fuzzy soft set theory. Int. J. Fuzzy Syst. 17(3), 490–497 (2015)

    Article  MathSciNet  Google Scholar 

  9. Wang, X.F., Wang, J.Q., Deng, S.Y.: Some geometric operators for aggregating intuitionistic linguistic information. Int. J. Fuzzy Syst. 17(2), 268–278 (2015)

    Article  MathSciNet  Google Scholar 

  10. Merigó, J.M., Casanovas, M., Liu, P.D.: Decision making with fuzzy induced heavy ordered weighted averaging operators. Int. J. Fuzzy Syst. 16(3), 277–289 (2014)

    Google Scholar 

  11. Wang, H., Smarandache, F., Zhang, Y., Sunderraman, R., Single valued neutrosophic sets. In: Proceedings of 10th 476 International Conference on Fuzzy Theory and Technology, Salt Lake City, p. 477 (2005)

  12. Smarandache, F.: A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic. American Research Press, Rehoboth (1999).

    MATH  Google Scholar 

  13. Guo, Y., Cheng, H.D., Zhang, Y., Zhao, W.: A new neutrosophic approach to image denoising. New Math. Nat. Comput. 5, 653–662 (2009)

    Article  MATH  Google Scholar 

  14. Guo, Y., Cheng, H.D.: New neutrosophic approach to image segmentation. Pattern Recogn. 42, 587–595 (2009)

    Article  MATH  Google Scholar 

  15. Wang, H., Smarandache, F., Zhang, Y.Q., et al.: Interval Neutrosophic Sets and Logic: Theory and Applications in Computing. Hexis, Phoenix (2005)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Beliakov, G., Bustince, H., James, S., Calvo, T., Fernandez, J.: Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: the median operator. IEEE Trans. Fuzzy Syst. 20, 487–498 (2012)

    Article  Google Scholar 

  19. Chen, S.M., Lee, L.W.: Autocratic decision making using group recommendations based on the ILLOWA operator and likelihood-based comparison relations. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 42, 115–129 (2012)

    Article  Google Scholar 

  20. Huang, Y.S., Chang, W.C., Li, W.H.: Aggregation of utility-based individual preferences for group decision-making. Eur. J. Oper. Res. 229, 462–469 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, P.D.: Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making. IEEE Trans. Fuzzy Syst. 22, 83–97 (2014)

    Article  Google Scholar 

  22. Liu, P.D.: Some generalized dependent aggregation operators with intuitionistic linguistic numbers and their application to group decision making. J. Comput. Syst. Sci. 79, 131–143 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, P.D.: A weighted aggregation operators multi-attribute group decision-making method based on interval-valued trapezoidal fuzzy numbers. Expert Syst. Appl. 38, 1053–1060 (2011)

    Article  Google Scholar 

  24. Liu, P.D., Jin, F.: Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making. Inf. Sci. 205, 58–71 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, P.D., Zhang, X., Jin, F.: A multi-attribute group decision-making method based on interval-valued trapezoidal fuzzy numbers hybrid harmonic averaging operators. J. Intell. Fuzzy Syst. 23, 159–168 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Merigó, J.M.: The probabilistic weighted average and its application in multi-person decision making. Int. J. Intell. Syst. 27, 457–476 (2012)

    Article  Google Scholar 

  27. Merigó, J.M., Gil-Lafuente, A.M.: The induced generalized OWA operator. Inf. Sci. 179, 729–741 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Merigó, J.M., Gil-Lafuente, A.M.: Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making. Inf. Sci. 236, 1–16 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Merigó, J.M., Yager, R.R.: Generalized moving averages, distance measures and OWA operators. Int. J. Uncertain Fuzziness Knowl Based Syst. 21, 533–559 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, Z.S., Yager, R.R.: Power-geometric operators and their use in group decision making. IEEE Trans. Fuzzy Syst. 18, 94–105 (2010)

    Article  Google Scholar 

  31. Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man Cybern. B 18, 183–190 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zarghami, M., Szidarovszky, F.: Revising the OWA operator for multi criteria decision making problems under uncertainty. Eur. J. Oper. Res. 198, 259–265 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu, Z.S.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15, 1179–1187 (2007)

    Article  Google Scholar 

  34. Wang, W.Z., Liu, X.W.: Intuitionistic fuzzy geometric aggregation operators based on Einstein operations. Int. J. Intell. Syst. 26, 1049–1075 (2011)

    Article  Google Scholar 

  35. Liu, P.D., Shi, L.L.: The generalized hybrid weighted average operator based on interval neutrosophic hesitant set and its application to multiple attribute decision making. Neural Comput. Appl. 26(2), 457–471 (2015)

    Article  Google Scholar 

  36. Liu, P.D., Chu, Y.C., Li, Y.W., Chen, Y.B.: Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int. J. Fuzzy Syst. 16(2), 242–255 (2014)

    Google Scholar 

  37. Liu, P.D., Wang, Y.M.: Multiple attribute decision making method based on single valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl. 25, 2001–2010 (2014)

    Article  Google Scholar 

  38. Xia, M.M., Xu, Z.S., Zhu, B.: Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm. Knowl. Based Syst. 31, 78–88 (2012)

    Article  Google Scholar 

  39. Alkhazaleh, S.M.R., Salleh, A.R.: Generating theorems for s-norms and t-norms. J. Qual. Meas. Anal. 4(1), 231–242 (2008)

    Google Scholar 

  40. Alkhazaleh, S.: More on neutrosophic norms and conorms. Neutrosophic Sets Syst. 9, 23–30 (2015)

    Google Scholar 

  41. Ye, J.: A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst. (2013). doi:10.3233/IFS-130916

    Google Scholar 

  42. Roychowdhury, S., Wang, B.H.: On generalized Hamacher families of triangular operators. Int. J. Approx.Reason. 19, 419–439 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Dombi, J.: A general class of fuzzy operators, the De-Morgan class of fuzzy operators and fuzziness induced by fuzzy operators. Fuzzy Sets Syst. 8, 149–163 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  44. Klement, E.P., Mesiar, R. (eds.): Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier, New York (2005)

    MATH  Google Scholar 

  45. Smarandache, F., Vladareanu, L.: Applications of neutrosophic logic to robotics-an introduction. In: IEEE International Conference on Granular Computing, pp. 607–612. (2011)

  46. Wang, T.C., Lee, H.D.: Developing a fuzzy TOPSIS approach based on subjective weights and objective weights. Expert Syst. Appl. 36, 8980–8985 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province, National Soft Science Project of China (2014GXQ4D192), Shandong Provincial Social Science Planning Project (No. 15BGLJ06), and The Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (No. 2015Z057). The authors also would like to express appreciation to the anonymous reviewers and Editors for their very helpful comments that improved the paper.

Author information

Authors and Affiliations

  1. School of Management Science and Engineering, Shandong University of Finance and Economics, Ji’nan, 250014, China

    Peide Liu

Authors
  1. Peide Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peide Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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, 849–863 (2016). https://doi.org/10.1007/s40815-016-0195-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-016-0195-8

Keywords

Search

Navigation