Cognitive Computation

, Volume 8, Issue 6, pp 1036–1056 | Cite as

Multi-criteria Group Decision-Making Based on Interval Neutrosophic Uncertain Linguistic Variables and Choquet Integral

  • Peide Liu
  • Guolin Tang



The interval neutrosophic uncertain linguistic variables (INULVs) can be better at handling the uncertainty of the decision-makers’ cognition in multi-criteria group decision-making (MCGDM) problems. Most MCGDM methods with INULVs are based on the supposition that all criteria are independent; however, they may be correlative in real decisions. The Choquet integral can process MCGDM problems with correlated criteria, which fail to aggregate INULVs. So, it is necessary and meaningful to propose the MCGDM method with INULVs based on the Choquet integral by considering the correlations between the attributes.


By combining INULVs with the Choquet integral, we propose the interval neutrosophic uncertain linguistic Choquet averaging (INULCA) operator and the interval neutrosophic uncertain linguistic Choquet geometric (INULCG) operator. These two operators reflect the existing correlation between two adjacent coalitions. In order to globally consider the correlations between elements or their ordered positions, the generalized Shapley INULCA (GS-INULCA) operator and the generalized Shapley INULCG (GS-INULCG) operator are further proposed. Furthermore, some models based on the grey relational analysis (GRA) method for determining the optimal fuzzy measures on the expert set and the criteria set are respectively built.


Based on the proposed operators and built models, a method is developed to cope with the MCGDM problems with INULVs, and the validity and advantages of the proposed method are analyzed by comparison with some existing approaches.


The method proposed in this paper can effectively handle the MCGDM problems in which the attribute information is expressed by INULVs, the attributes’ and experts’ weights are partly known, and the experts and attributes are interactive.


Multiple-criteria group decision-making (MCGDM) Interval neutrosophic uncertain linguistic set Grey relational analysis Choquet integral Generalized Shapley function Fuzzy measure 



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 (15BGLJ06), and the science and technology project of colleges and universities in Shandong Province (J13LN19 and J16LN25).

Compliance with Ethical Standards

Conflict of interest

Peide Liu and Guolin Tang declare that they have no conflict of interest.

Informed Consent

Informed Consent was not required as no human or animals were involved.

Human and Animals Rights

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Akusok A, Miche Y, Hegedus J, Nian R, Lendasse A. A two stage methodology using K-NN and false-positive minimizing ELM for nominal data classification. Cogn Comput. 2014;6(3):432–45.CrossRefGoogle Scholar
  2. 2.
    Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20(1):87–96.CrossRefGoogle Scholar
  3. 3.
    Bausys R, Zavadskas EK, Kaklauskas A. Application of neutrosophic set to multicriteria decision-making by COPRAS. Econ Comput Econ Cybern Stud Res. 2015;40(2):91–105.Google Scholar
  4. 4.
    Baušys R, Zavadskas EK. Multicriteria decision-making approach by VIKOR under interval neutrosophic set environment. Econ Comput Econ Cybern Stud Res. 2015;49(4):33–48.Google Scholar
  5. 5.
    Belles-Sampera J, Merigó JM, Guillén M, Santolino M. Indicators for the characterization of discrete Choquet integrals. Inf Sci. 2014;267:201–16.CrossRefGoogle Scholar
  6. 6.
    Broumi S, Smarandache F. New distance and similarity measures of interval neutrosophic sets. Int Conf Inf Fus. 2014;17:1–7.CrossRefGoogle Scholar
  7. 7.
    Broumi S, Ye J, Smarandache F. An extended TOPSIS method for multiple attribute decision-making based on interval neutrosophic uncertain linguistic variables. Neutrosophic Sets Syst. 2015;8:22–31.Google Scholar
  8. 8.
    Chen LH, Xu ZS, Yu XH. Prioritized measure-guided aggregation operators. IEEE Trans Fuzzy Syst. 2014;22(5):1127–38.CrossRefGoogle Scholar
  9. 9.
    Chi PP, Liu PD. An extended TOPSIS method for the multiple attribute decision-making problems based on interval neutrosophic set. Neutrosophic Sets Syst. 2013;1:63–70.Google Scholar
  10. 10.
    Choquet G. Theory of capacities. Annales del Institut Fourier. 1953;5:131–295.CrossRefGoogle Scholar
  11. 11.
    Guo Y, Sengur A. A novel color image segmentation approach based on neutrosophic set and modified fuzzy c-means. Circ Syst Signal Process. 2013;32(4):1699–723.CrossRefGoogle Scholar
  12. 12.
    Laurent PA. A neural mechanism for reward discounting: insights from modeling hippocampal–striatal interactions. Cogn Comput. 2013;5(1):152–60.CrossRefGoogle Scholar
  13. 13.
    Liu PD, Chu YC, Li YW, Chen YB. Some generalized neutrosophic number hamacher aggregation operators and their application to group decision-making. Int J Fuzzy Syst. 2014;16(2):242–55.Google Scholar
  14. 14.
    Liu PD, Jin F. A multi-attribute group decision-making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers. Appl Math Model. 2012;36(6):2498–509.CrossRefGoogle Scholar
  15. 15.
    Liu PD, Jin F, Zhang X, Su Y, Wang MH. Research on the multi-attribute decision-making under risk with interval probability based on prospect theory and the uncertain linguistic variables. Knowl-Based Syst. 2011;24(4):554–61.CrossRefGoogle Scholar
  16. 16.
    Liu PD, Liu Y. An approach to multiple attribute group decision-making based on intuitionistic trapezoidal fuzzy power generalized aggregation operator. Int J Comput Intell Syst. 2014;7(2):291–304.CrossRefGoogle Scholar
  17. 17.
    Liu PD, Shi LL. The generalized hybrid weighted average operator based on interval neutrosophic hesitant set and its application to multiple attribute decision-making. Neural Comput Appl. 2015;26(2):457–71.CrossRefGoogle Scholar
  18. 18.
    Liu PD, Wang YM. Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput Appl. 2014;25(7–8):2001–10.CrossRefGoogle Scholar
  19. 19.
    Liu PD, Yu XC. 2-dimension uncertain linguistic power generalized weighted aggregation operator and its application for multiple attribute group decision-making. Knowl-Based Syst. 2014;57(1):69–80.CrossRefGoogle Scholar
  20. 20.
    Liu PD, Zhang X. An approach to group decision-making based on 2-dimension uncertain linguistic assessment information. Technol Econ Dev Econ. 2012;18(3):424–37.CrossRefGoogle Scholar
  21. 21.
    Liu PD, 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. 2012;23(5):159–68.Google Scholar
  22. 22.
    Llamazares B. Constructing Choquet integral-based operators that generalize weighted means and OWA operators. Inform Fus. 2015;23:131–8.CrossRefGoogle Scholar
  23. 23.
    Marichal JL. The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision-making. Disc Appl Math. 2000;107(1–3):139–64.CrossRefGoogle Scholar
  24. 24.
    Mathew JM, Simon P. Color texture image segmentation based on neutrosophic set and nonsubsampled contourlet transformation. Appl Algorithms. 2014;8321:164–73.CrossRefGoogle Scholar
  25. 25.
    Meng FY, Chen XH. Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures. Cogn Comput. 2015;7:445–63.CrossRefGoogle Scholar
  26. 26.
    Meng FY, Chen XH, Zhang Q. Some interval-valued intuitionistic uncertain linguistic Choquet operators and their application to multi-attribute group decision-making. Appl Math Model. 2014;38(9–10):2543–57.CrossRefGoogle Scholar
  27. 27.
    Meng FY, Wang C, Chen XH. Linguistic interval hesitant fuzzy sets and their application in decision-making. Cogn Comput. 2016;8:52–68.CrossRefGoogle Scholar
  28. 28.
    Mukherjee A, Sarkar S. A new method of measuring similarity between two neutrosophic soft sets and its application in pattern recognition problems. Neutrosophic Sets Syst. 2015;8:63–8.Google Scholar
  29. 29.
    Patryk A, Laurent A. Neural mechanism for reward discounting: insights from modeling hippocampal–striatal interactions. Cogn Comput. 2013;5(1):152–60.CrossRefGoogle Scholar
  30. 30.
    Rodríguez LF, Ramos F. Development of computational models of emotions for autonomous agents: a review. Cogn Comput. 2014;6(3):351–75.CrossRefGoogle Scholar
  31. 31.
    Smarandache F. Neutrosophy. Neutrosophic probability, set, and logic. Rehoboth: American Research Press; 1998.Google Scholar
  32. 32.
    Smarandache F. N-norm and N-conorm in neutrosophic logic and set, and the neutrosophic topologies. Crit Rev Creighton Univ. 2009;3:73–83.Google Scholar
  33. 33.
    Sugeno M. Theory of fuzzy integral and its application. Doctorial dissertation, Tokyo Institute of Technology, 1974.Google Scholar
  34. 34.
    Wang H, Smarandache F, Zhang YQ, Sunderraman R. Interval neutrosophic sets and logic: theory and applications in computing. Phoenix: Hexis; 2005.Google Scholar
  35. 35.
    Wang H, Smarandache F, Zhang YQ, Sunderraman R. Single valued neutrosophic sets. Multispace Multistruct. 2010;4:410–3.Google Scholar
  36. 36.
    Wei GW. Grey relational analysis method for 2-tuple linguistic multiple attribute group decision-making with incomplete weight information. Expert Syst Appl. 2011;38:4824–8.CrossRefGoogle Scholar
  37. 37.
    Xu ZS. Asymmetric fuzzy preference relations based on the generalized sigmoid scale and their application in decision-making involving risk appetites. IEEE Trans Fuzzy Syst. 2015;24(3):741–56.Google Scholar
  38. 38.
    Xu ZS. Choquet integrals of weighted intuitionistic fuzzy information. Inf Sci. 2010;180:726–36.CrossRefGoogle Scholar
  39. 39.
    Xu ZS. Induced uncertain linguistic OWA operators applied to group decision-making. Inform Fus. 2006;7:231–8.CrossRefGoogle Scholar
  40. 40.
    Ye J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst. 2014;26:2459–66.Google Scholar
  41. 41.
    Ye J. Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif Intell Med. 2015;63(3):171–9.CrossRefPubMedGoogle Scholar
  42. 42.
    Ye J. Multiple attribute group decision-making based on interval neutrosophic uncertain linguistic variables. Int J Mach Learn Cybern. (2015). doi: 10.1007/s13042-015-0382-1
  43. 43.
    Ye J. Similarity measures between interval neutrosophic sets and their multicriteria decision-making method. J Intell Fuzzy Syst. 2014;26(1):165–72.Google Scholar
  44. 44.
    Ye J. Single valued neutrosophic cross-entropy for multicriteria decision-making problems. Appl Math Model. 2014;38:1170–5.CrossRefGoogle Scholar
  45. 45.
    Ye J. Some aggregation operators of interval neutrosophic linguistic numbers for multiple attribute decision-making. J Intell Fuzzy Syst. 2014;27:2231–41.Google Scholar
  46. 46.
    Ye J. Trapezoidal neutrosophic set and its application to multiple attribute decision-making. Neural Comput Appl. 2015;26(5):1157–66.CrossRefGoogle Scholar
  47. 47.
    Ye J. Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision-making. Int J Fuzzy Syst. 2014;16(2):204–10.Google Scholar
  48. 48.
    Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–53.CrossRefGoogle Scholar
  49. 49.
    Zavadskas EK, Baušys R, Lazauskas M. Sustainable assessment of alternative sites for the construction of a waste incineration plant by applying WASPAS method with single-valued neutrosophic set. Sustainability. 2015;7(12):15923–36.CrossRefGoogle Scholar
  50. 50.
    Zhang HY, Ji P, Wang JQ, Chen XH. A neutrosophic normal cloud and its application in decision-making. Cogn Comput. 2016. doi: 10.1007/s12559-016-9394-8.
  51. 51.
    Zhang HY, Wang JQ, Chen XH. An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput Appl. 2016;27:615–27.CrossRefGoogle Scholar
  52. 52.
    Zhang HY, Wang JQ, Chen XH. Interval neutrosophic sets and their application in multicriteria decision-making problems. Sci World J. 2014;2014:1–15. doi: 10.1155/2014-943645953.Google Scholar
  53. 53.
    Zhang ZM, Wu C. A novel method for single-valued neutrosophic multi-criteria decision-making with incomplete weight information. Neutrosophic Sets Syst. 2014;4:35–64.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Management Science and EngineeringShandong University of Finance and EconomicsJinanChina
  2. 2.School of Economics and ManagementBeijing University of TechnologyBeijingChina

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