Neural Computing and Applications

, Volume 30, Issue 1, pp 241–260 | Cite as

Multi-criteria group decision-making method based on interdependent inputs of single-valued trapezoidal neutrosophic information

  • Ru-xia Liang
  • Jian-qiang WangEmail author
  • Lin Li
Original Article


Single-valued trapezoidal neutrosophic numbers (SVTNNs) are very useful tools for describing complex information, because they are able to maintain the completeness of the information and describe it accurately and comprehensively. This paper develops a method based on the single-valued trapezoidal neutrosophic normalized weighted Bonferroni mean (SVTNNWBM) operator to address multi-criteria group decision-making (MCGDM) problems. First, the limitations of existing operations for SVTNNs are discussed, after which improved operations are defined. Second, a new comparison method based on score function is proposed. Then, the entropy-weighted method is established in order to obtain objective expert weights, and the SVTNNWBM operator is proposed based on the new operations of SVTNNs. Furthermore, a single-valued trapezoidal neutrosophic MCGDM method is developed. Finally, a numerical example and comparison analysis are conducted to verify the practicality and effectiveness of the proposed approach.


Multi-criteria group decision-making Single-valued trapezoidal neutrosophic number Single-valued trapezoidal neutrosophic weighted Bonferroni mean operator Entropy-weighted method 



This work was supported by the National Natural Science Foundation of China (Nos. 71571193).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.


  1. 1.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353CrossRefzbMATHGoogle Scholar
  2. 2.
    Derrac J, Chiclana F, Garcia S, Herrera F (2016) Evolutionary fuzzy k-nearest neighbors algorithm using interval-valued fuzzy sets. Inf Sci 329:144–163CrossRefGoogle Scholar
  3. 3.
    Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96CrossRefzbMATHGoogle Scholar
  4. 4.
    Atanassov KT, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Wan S, Lin L-L, Dong J (2016) MAGDM based on triangular Atanassov’s intuitionistic fuzzy information aggregation. Neural Comput Appl. doi: 10.1007/s00521-016-2196-9 Google Scholar
  6. 6.
    Zhou H, Wang J-Q, Zhang H-Y (2016) Multi-criteria decision-making approaches based on distance measures for linguistic hesitant fuzzy sets. J Oper Res Soc. doi: 10.1057/jors.2016.41 Google Scholar
  7. 7.
    Beg I, Rashid T (2014) Group decision making using intuitionistic hesitant fuzzy sets. Int J Fuzzy Log Intell Syst 14(3):181–187CrossRefGoogle Scholar
  8. 8.
    Liu H-W, Wang G-J (2007) Multi-criteria decision making methods based on intuitionistic fuzzy sets. Eur J Oper Res 179(1):220–233CrossRefzbMATHGoogle Scholar
  9. 9.
    Xu Z-S (2012) Intuitionistic fuzzy multi-attribute decision making: an interactive method. IEEE Trans Fuzzy Syst 20(3):514–525CrossRefGoogle Scholar
  10. 10.
    Wang J-Q, Han Z-Q, Zhang H-Y (2014) Multi-criteria group decision-making method based on intuitionistic interval fuzzy information. Group Decis Negot 23(4):715–733CrossRefGoogle Scholar
  11. 11.
    Amorim P, Curcio E, Almada-Lobo B, Barbosa-Póvoa APFD, Grossmann IE (2016) Supplier selection in the processed food industry under uncertainty. Eur J Oper Res 252(3):801–814MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen S-M, Cheng S-H, Chiou C-H (2016) Fuzzy multi-attribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology. Inf Fusion 27:215–227CrossRefGoogle Scholar
  13. 13.
    Liu P-D, Liu X (2016) The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making. Int J Mach Learn Cybern. doi: 10.1007/s13042-016-0508-0 Google Scholar
  14. 14.
    Wu J, Xiong R, Chiclana F (2016) Uninorm trust propagation and aggregation methods for group decision making in social network with four tuple information. Knowl Based Syst 96(2):29–39Google Scholar
  15. 15.
    Wang J-Q, Nie R-R, Zhang H-Y (2013) New operators on triangular intuitionistic fuzzy numbers and their applications in system fault analysis. Inf Sci 251:79–95MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang J-Q (2008) Overview on fuzzy multi-criteria decision-making approach. Control Decis 23(6):002MathSciNetGoogle Scholar
  17. 17.
    Wan S-P (2013) Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl Math Model 37(6):4112–4126MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Smarandache F (1998) Neutrosophy: neutrosophic probability, set, and logic. American Research Press, Rehoboth, pp 1–105zbMATHGoogle Scholar
  19. 19.
    Smarandache F (1999) A unifying field in logics: neutrosophic logic neutrosophy, neutrosophic set, neutrosophic probability. American Research Press, Rehoboth, pp 1–141zbMATHGoogle Scholar
  20. 20.
    Smarandache F (2008) Neutrosophic set—a generalization of the intuitionistic fuzzy set. Int J Pure Appl Math 24(3):38–42MathSciNetGoogle Scholar
  21. 21.
    Deli I, Şubaş Y (2015) Some weighted geometric operators with SVTrN-numbers and their application to multi-criteria decision making problems. J Intell Fuzzy Syst. doi: 10.3233/jifs-151677 zbMATHGoogle Scholar
  22. 22.
    El-Hefenawy N, Metwally M-A, Ahmed Z-M, El-Henawy I-M (2016) A review on the applications of neutrosophic sets. J Comput Theor Nanosci 13(1):936–944CrossRefGoogle Scholar
  23. 23.
    Şubaş Y (2015) Neutrosophic numbers and their application to multi-attribute decision making problems. Unpublished Masters Thesis, 7 Aralık University, Graduate School of Natural and Applied ScienceGoogle Scholar
  24. 24.
    Liu C, Luo Y (2016) Correlated aggregation operators for simplified neutrosophic set and their application in multi-attribute group decision making. J Intell Fuzzy Syst 30(3):1755–1761CrossRefzbMATHGoogle Scholar
  25. 25.
    Wu X-H, Wang J, Peng J-J, Chen X-H (2016) Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. Int J Fuzzy Syst. doi: 10.1007/s40815-016-0180-2 Google Scholar
  26. 26.
    Ji P, Zhang H-Y, Wang J-Q (2016) A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput Appl. doi: 10.1007/s00521-016-2436-z Google Scholar
  27. 27.
    Liu P-D, Li H (2015) Multiple attribute decision-making method based on some normal neutrosophic Bonferroni mean operators. Neural Comput Appl 25(7–8):1–16Google Scholar
  28. 28.
    Broumi S, Talea M, Bakali A, Smarandache F (2016) Single valued neutrosophic graphs. J New Theory 10:86–101Google Scholar
  29. 29.
    Broumi S, Talea M, Bakali A, Smarandache F (2016) Interval valued neutrosophic graphs. Publ Soc Math Uncertain 10:5Google Scholar
  30. 30.
    Broumi S, Deli I, Smarandache F (2014) Interval valued neutrosophic parameterized soft set theory and its decision making. Appl Soft Comput 28(4):109–113zbMATHGoogle Scholar
  31. 31.
    Şahin R, Liu P (2016) Correlation coefficient of single-valued neutrosophic hesitant fuzzy sets and its applications in decision making. Neural Comput Appl. doi: 10.1007/s00521-015-2163-x Google Scholar
  32. 32.
    Tian Z-P, Wang J, Wang J-Q, Zhang H-Y (2016) An improved MULTIMOORA approach for multi-criteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information. Neural Comput Appl. doi: 10.1007/s00521-016-2378-5 Google Scholar
  33. 33.
    Tian Z-P, Wang J, Wang J-Q, Zhang H-Y (2016) Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decis Negot. doi: 10.1007/s10726-016-9479-5 Google Scholar
  34. 34.
    Tian Z-P, Wang J, Zhang H-Y, Wang J-Q (2016) Multi-criteria decision-making based on generalized prioritized aggregation operators under simplified neutrosophic uncertain linguistic environment. Int J Mach Learn Cybern. doi: 10.1007/s13042-016-0552-9 Google Scholar
  35. 35.
    Ma Y-X, Wang J-Q, Wang J, Wu X-H (2016) An interval neutrosophic linguistic multi-criteria group decision-making method and its application in selecting medical treatment options. Neural Comput Appl. doi: 10.1007/s00521-016-2203-1 Google Scholar
  36. 36.
    Chan H-K, Wang X-J, Raffoni A (2014) An integrated approach for green design: life-cycle, fuzzy AHP and environmental management accounting. Br Account Rev 46(4):344–360CrossRefGoogle Scholar
  37. 37.
    Chan H-K, Wang X-J, White GRT, Yip N (2013) An extended fuzzy-AHP approach for the evaluation of green product designs. IEEE Trans Eng Manag 60(2):327–339CrossRefGoogle Scholar
  38. 38.
    Ye J (2015) Some weighted aggregation operators of trapezoidal neutrosophic numbers and their multiple attribute decision making method.
  39. 39.
    Deli I, Şubaş Y (2016) A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems. Int J Mach Learn Cybern. doi: 10.1007/s13042-016-0505-3 Google Scholar
  40. 40.
    Broumi S, Smarandache F (2014) Single valued neutrosophic trapezoid linguistic aggregation operators based multi-attribute decision making. Bull Pure Appl Sci Math Stat 33(2):135–155CrossRefGoogle Scholar
  41. 41.
    Said B, Smarandache F (2016) Multi-attribute decision making based on interval neutrosophic trapezoid linguistic aggregation operators. Handb Res Gen Hybrid Set Struct Appl Soft Comput. doi: 10.5281/zenodo.49136 Google Scholar
  42. 42.
    Tian Z-P, Wang J, Wang J-Q, Chen X-H (2015) Multi-criteria decision-making approach based on gray linguistic weighted Bonferroni mean operator. Int Trans Oper Res. doi: 10.1111/itor.12220 Google Scholar
  43. 43.
    Ye J (2015) Multiple attribute group decision making based on interval neutrosophic uncertain linguistic variables. Int J Mach Learn Cybern. doi: 10.1007/s13042-015-0382-1 Google Scholar
  44. 44.
    Bonferroni C (1950) Sulle medie multiple di potenze. Bolletino dell`Unione Matematica Italiana 5:267–270MathSciNetzbMATHGoogle Scholar
  45. 45.
    Li D, Zeng W, Li J (2016) Geometric Bonferroni mean operators. Int J Intell Syst. doi: 10.1002/int.21822 Google Scholar
  46. 46.
    Liu P, Zhang L, Liu X, Wang P (2016) Multi-valued neutrosophic number Bonferroni mean operators with their applications in multiple attribute group decision making. Int J Inf Technol Decis Mak. doi: 10.1142/s0219622016500346 Google Scholar
  47. 47.
    Liu P-D, Jin F (2012) The trapezoid fuzzy linguistic Bonferroni mean operators and their application to multiple attribute decision making. Sci Iran 19(6):1947–1959MathSciNetCrossRefGoogle Scholar
  48. 48.
    Zhu W-Q, Liang P, Wang L-J, Hou Y-R (2015) Triangular fuzzy Bonferroni mean operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 29(4):1265–1272MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Chen Z-S, Chin K-S, Li Y-L, Yang Y (2016) On generalized extended Bonferroni means for decision making. IEEE Trans Fuzzy Syst. doi: 10.1109/tfuzz.2016.2540066 Google Scholar
  50. 50.
    Zhang H-Y, Ji P, Wang J, Chen X-H (2017) A novel decision support model for satisfactory restaurants utilizing social information: a case study of Tour Manag 59: 281–297CrossRefGoogle Scholar
  51. 51.
    Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30(3):183–224MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wang H-B, Smarandache F, Zhang Y, Sunderraman R (2010) Single valued neutrosophic sets. Multispace Multistruct 4:410–413zbMATHGoogle Scholar
  53. 53.
    Deli I, Şubaş Y (2014) Single valued neutrosophic numbers and their applications to multi-criteria decision making problem. viXra preprint viXra 1412.0012Google Scholar
  54. 54.
    Xu Z-S, Yager R-R (2011) Intuitionistic fuzzy Bonferroni means. IEEE Trans Syst Man Cybern B (Cybern) 41(2):568–578CrossRefGoogle Scholar
  55. 55.
    Zhou W, He J-M (2012) Intuitionistic fuzzy normalized weighted Bonferroni mean and its application in multi-criteria decision making. J Appl Math 1110-757x:1–16Google Scholar
  56. 56.
    Shannon C-E (2001) A mathematical theory of communication. ACM SIGMOBILE Mob Comput Commun Rev 5(1):3–55MathSciNetCrossRefGoogle Scholar
  57. 57.
    López-de-Ipiña K, Solé-Casals J, Faundez-Zanuy M, Calvo P-M, Sesa E, Martinez de Lizarduy U, Bergareche A (2016) Selection of entropy based features for automatic analysis of essential tremor. Entropy 18(5):184CrossRefzbMATHGoogle Scholar
  58. 58.
    Wei C, Yan F, Rodríguez R-M Entropy measures for hesitant fuzzy sets and their application in multi-criteria decision making. J Intell Fuzzy Syst (Preprint) 1–13Google Scholar
  59. 59.
    Verma R, Maheshwari S (2016) A new measure of divergence with its application to multi-criteria decision making under fuzzy environment. Neural Comput Appl. doi: 10.1007/s00521-016-2311-y Google Scholar
  60. 60.
    Kumar A, Peeta S (2015) Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem. Transp Res B Methodol 71(4):213–229CrossRefGoogle Scholar
  61. 61.
    Yue Z (2014) TOPSIS-based group decision making methodology in intuitionistic fuzzy setting. Inf Sci 277(2):141–153MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Wang J-H, Hao J-Y (2006) A new version of 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 14(3):435–445CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.School of BusinessCentral South UniversityChangshaPeople’s Republic of China
  2. 2.School of BusinessHunan UniversityChangshaPeople’s Republic of China

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