Applied Intelligence

, Volume 49, Issue 7, pp 2582–2602 | Cite as

Nested probabilistic-numerical linguistic term sets in two-stage multi-attribute group decision making

  • Xinxin Wang
  • Zeshui XuEmail author
  • Xunjie Gou


Fuzzy linguistic approach is considered as an effective solution to accommodate uncertainties in qualitative decision making. In the face of increasingly complicated environment, we usually face the fact that multi-attribute group decision making contains nested information, and the whole process needs to be evaluated twice so that the experts can make full use of decision information to get more accurate result, and we call this kind of problem as two-stage multi-attribute group decision making (TSMAGDM). However, the existing linguistic approaches cannot represent such nested evaluation information to deal with the above situation. In this paper, a novel linguistic expression tool called nested probabilistic-numerical linguistic term set (NPNLTS) which considers both quantitative and qualitative information, is proposed to handle TSMAGDM. Based on which, some basic operational laws and aggregation operators are put forward. Then, an aggregation method and an extended TOPSIS method are developed respectively in TSMAGDM with NPNLTSs. Finally, an application case about strategy initiatives of HBIS GROUP on Supply-side Structural Reform is presented, and some analyses and comparisons are provided to validate the proposed methods.


Nested probabilistic-numerical linguistic term sets Two-stage multi-attribute group decision making The aggregation method The extended TOPSIS method 



The work was supported by the National Natural Science Foundation of China (Nos. 71571123, 71771155 and 71801174).


  1. 1.
    Parsons S (1996) Current approaches to handling imperfect information in data and knowledge bases. IEEE Trans Knowl Data Eng 8(3):353–372MathSciNetCrossRefGoogle Scholar
  2. 2.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  3. 3.
    Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Kluwer, New YorkzbMATHGoogle Scholar
  4. 4.
    Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type 2. Inf Control 31:312–340MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Garibaldi JM, Jaroszewski M, Musikasuwan S (2008) Nonstationary fuzzy sets. IEEE Trans Fuzzy Syst 16(4):1072–1086CrossRefGoogle Scholar
  6. 6.
    Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefzbMATHGoogle Scholar
  7. 7.
    Yager RR (1986) On the theory of bags. Int J Gener Syst 13:23–37MathSciNetCrossRefGoogle Scholar
  8. 8.
    Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539zbMATHGoogle Scholar
  9. 9.
    Pedrycz W (2013) Granular computing: analysis and design of intelligen systems. CRC Press/Francis Taylor, Boca RatonCrossRefGoogle Scholar
  10. 10.
    Zadeh LA (1996) Fuzzy logic = computing with words. IEEE Trans Fuzzy Syst 4:103–111CrossRefGoogle Scholar
  11. 11.
    Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8(3):199–249MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wang H, Xu ZS (2016) Multi-groups decision making using intuitionistic-valued hesitant fuzzy information. Int J Comput Int Sys 9(3):468–482CrossRefGoogle Scholar
  13. 13.
    García-Lapresta JL, Pérez-Román D (2016) Consensus-based clustering under hesitant qualitative assessments. Fuzzy Sets Syst 292:261–273MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dong YC, Li CC, Herrera F (2016) Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its use to deal with hesitant unbalanced linguistic information. Inf Sci 367:259–278CrossRefGoogle Scholar
  15. 15.
    Wang JQ, Wang J, Chen QH, Zhang HY, Chen XH (2014) An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets. Inf Sci 280:338–351MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang YX, Xu ZS, Wang H, Liao HC (2016) Consistency-based risk assessment with probabilistic linguistic preference relation. Appl Soft Comput 49:817–833CrossRefGoogle Scholar
  17. 17.
    Lahdelma R, Salminen P (2006) Stochastic multicriteria acceptability analysis using the dataenvelopment model. Eur J Oper Res 170(1):241–252CrossRefzbMATHGoogle Scholar
  18. 18.
    Xian SD, Jing N, Xue WT (2017) A new intuitionistic fuzzy linguistic hybrid aggregation operator and its application for linguistic group decision making. Int J Intell Syst 32(12):1332–1352CrossRefGoogle Scholar
  19. 19.
    Liao HC, Jiang LS, Xu ZS (2017) A linear programming method for multiple criteria decision making with probabilistic linguistic information. Inf Sci 415:341–355CrossRefGoogle Scholar
  20. 20.
    Zhang YX, Xu ZS, Liao HC (2017) A consensus process for group decision making with probabilistic linguistic preference relations. Inf Sci 414:260–275CrossRefGoogle Scholar
  21. 21.
    Dong YC, Chen X, Herrera F (2015) Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group 492 decision making. Inf Sci 297:95–117CrossRefGoogle Scholar
  22. 22.
    Parreiras R, Ekel PY, Martini J, Palhares RM (2010) A flexible consensus scheme for multicriteria group decision making under linguistic assessments. Inf Sci 180:1075–1089CrossRefGoogle Scholar
  23. 23.
    Tao ZF, Chen HY, Zhou LG, Liu JP (2014) 2-tuple linguistic soft set and its application to group decision making. Soft comp 19:1201–1213CrossRefzbMATHGoogle Scholar
  24. 24.
    Garg H, Kumar K (2018) Distance measures for connection number sets based on set pair analysis and its applications to decision-making process. Appl Intell 48:3346–3359CrossRefGoogle Scholar
  25. 25.
    Mendel JM (2002) An architecture for making judgement using computing with words, Int. J ApplMath Comput Sci 12(3):325–335MathSciNetzbMATHGoogle Scholar
  26. 26.
    Zhou SM, John RI, Chiclana F, Garibaldi JM (2010) On aggregating uncertain information by type-2 OWA operators for soft decisionmaking. Int J Intell Syst 25(6):540–558zbMATHGoogle Scholar
  27. 27.
    Herrera F, Martínez L (2000) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752CrossRefGoogle Scholar
  28. 28.
    Xu ZS (2004) A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf Sci 166(1–4):19–30MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xu ZS (2005) Deviation measures of linguistic preference relations in group decision making. Omega 33(3):249–254CrossRefGoogle Scholar
  30. 30.
    Xu ZS, Wang H (2017) On the syntax and semantics of virtual linguistic terms for information fusion in decision making. Inf Fusion 34:43–48CrossRefGoogle Scholar
  31. 31.
    Rodriguez RM, Martinez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20:109–119CrossRefGoogle Scholar
  32. 32.
    Pang Q, Wang H, Xu ZS (2016) Probabilistic linguistic term sets in multi-attribute group decision making. Inf Sci 369:128–143CrossRefGoogle Scholar
  33. 33.
    Wei CP, Zhao N, Tang XJ (2014) Operatorsand comparisons ofhesitantfuzzy linguisticterm sets. IEEE Trans Fuzzy Syst 22(3):575–585CrossRefGoogle Scholar
  34. 34.
    Zhu B, Xu ZS (2014) Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Trans Fuzzy Syst 22(1):35–45MathSciNetCrossRefGoogle Scholar
  35. 35.
    Gou XJ, Xu ZS (2016) Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets. Inf Sci 372:407–427CrossRefGoogle Scholar
  36. 36.
    Liao HC, Xu ZS, Zeng XJ (2014) Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inf Sci 271:125–142MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liao HC, Xu ZS, Zeng XJ, Merigó JM (2015) Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets. Knowl Based Syst 76:127–138CrossRefGoogle Scholar
  38. 38.
    Gou XJ, Xu ZS, Liao HC (2017) Hesitant fuzzy linguistic entropy and cross-entropy measures and alternative queuing method for multiple criteria decision making. Inf Sci 388:225–246CrossRefGoogle Scholar
  39. 39.
    Beg I, Rashid T (2013) Topsis for hesitant fuzzy linguistic term sets. Int J Intell Syst 28(12):1162–1171CrossRefGoogle Scholar
  40. 40.
    Bai CZ, Zhang R, Qian LX, Wu YN (2017) Comparisons of probabilistic linguistic term sets for multi-criteria decision making. Knowl Based Syst. 119:284–291CrossRefGoogle Scholar
  41. 41.
    Herrera F, Herrera-Viedma E, Verdegay JL (1995) A sequential selection process in group decision making with a linguistic assessment approach. Inf Sci 85:223–239CrossRefzbMATHGoogle Scholar
  42. 42.
    Yang JB (2001) Rule and utility based evidential reasoning approach for multiattribute decision analysis under uncertainties. Eur J Oper Res 131:31–61MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kumar K, Garg H (2018) TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. Comput Appl Math 37:1319–1329MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Garg H (2017) A new improved score function of an interval-valued Pythagorean fuzzy set based TOPSIS method. Int J Uncertain Quan 7:463–474CrossRefGoogle Scholar
  45. 45.
    Xu ZS, Xia MM (2011) On distance and correlation measures of hesitant fuzzy information. Int J Intell Syst 26:410–425CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Business SchoolSichuan UniversityChengduChina

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