Soft Computing

, Volume 22, Issue 22, pp 7463–7477 | Cite as

Acceptably consistent incomplete interval-valued intuitionistic multiplicative preference relations

  • Mamata Sahu
  • Anjana GuptaEmail author
  • Aparna Mehra


We study the consistency property, and especially the acceptably consistent property, for incomplete interval-valued intuitionistic multiplicative preference relations. We propose a technique which first estimates the initial values for all missing entries in an incomplete interval-valued intuitionistic multiplicative preference relation and then improves them by a local optimization method. Two examples are presented in order to illustrate applications of the proposed method in group decision-making problems.


Multiplicative preference relation Interval-valued intuitionistic fuzzy number Acceptably consistent preference relation Multi-criteria group decision-making problems 



The authors are thankful to the esteemed referees for their valuable comments which help to improve the presentation of the paper substantially. The authors acknowledge the editor-in-chief for being considerate and supportive.

Compliance with ethical standards

Conflict of interest

All three authors declare that they have no conflict of interest regarding the publication of this manuscript.


  1. Aczél J, Saaty TL (1983) Procedures for synthesizing ratio judgments. J Math-Mat Psychol 27:93–102CrossRefGoogle Scholar
  2. Alonso S, Chiclana F, Herrera F, Herrera-viedma E, Alcala-Fernadez J, Porcel C (2008) A consistency based procedure to estimate missing pairwise preference values. Int J Intell Syst 23:155–175CrossRefGoogle Scholar
  3. Alonso S, Herrera-Viedma E, Chiclana F, Herrera F (2010) A web based consensus support system for group decision making problems and incomplete preferences. Inf Sci 180:4477–4495MathSciNetCrossRefGoogle Scholar
  4. Atanassov, K. (2012). On intuitionistic fuzzy sets theory. Springer, BerlinCrossRefGoogle Scholar
  5. Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349MathSciNetCrossRefGoogle Scholar
  6. Brans JP, Vincke P (1985) A preference ranking organization method. Manag Sci 31:647–656CrossRefGoogle Scholar
  7. Chen SM, Lin TE, Lee LW (2014) Group decision making using incomplete fuzzy preference relations based on the additive consistency and the order consistency. Inf Sci 259:1–15MathSciNetCrossRefGoogle Scholar
  8. Chiclana F, Herrera F, Herrera Viedma E (2001) Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst 122:277–291MathSciNetCrossRefGoogle Scholar
  9. Deschrijver G, Kerre EE (2003) On the composition of intuitionistic fuzzy relations. Fuzzy Sets Syst 136:333–361MathSciNetCrossRefGoogle Scholar
  10. Elsner L, Johnson CR, Dias Da Silva JA (1988) The Perron root of a weighted geometric mean of nonnegative matrices. Linear and Multilinear Algebra 24:1–13MathSciNetCrossRefGoogle Scholar
  11. Fan ZP, Ma J, Jiang YP, Sun YH, Ma L (2006) A goal programming approach to group decision making based on multiplicative preference relation and fuzzy preference relations. Eur J Oper Res 174:311–321CrossRefGoogle Scholar
  12. Fedrizzi M, Giove S (2007) Incomplete pair wise comparison and consistency optimization. Eur J Oper Res 183:303–313CrossRefGoogle Scholar
  13. Figueira J, Greco S, Ehrgott M (eds) (2005) Multiple criteria decision analysis: state of the art surveys. International series in operations research and management science. Springer, New York, p 78Google Scholar
  14. Gong ZW, Li LS, Forrest J, Zhao Y (2011) The optimal priority models of the intuitionis tic fuzzy preference relation and their application in selecting industries with higher meteorological sensitivity. Expert Syst Appl 38:4394–4402CrossRefGoogle Scholar
  15. Harker PT (1987a) Alternative modes of questioning in the analytic hierarchy process. Math Modell 9:353–360MathSciNetCrossRefGoogle Scholar
  16. Harker PT (1987b) Incomplete pairwise comparisons in the analytic hierarchy process. Math Modell 9:837–848MathSciNetCrossRefGoogle Scholar
  17. Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer, New YorkCrossRefGoogle Scholar
  18. Jiang Y, Xu ZS, Xu JP (2014) Interval-valued intuitionistic multiplicative sets. Int J Uncertain, Fuzziness Knowl-Based Syst 22(3):385–406MathSciNetCrossRefGoogle Scholar
  19. Jiang Y, Xu ZS, Yu XH (2013) Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations. Appl Soft Comput 13:2075–2086CrossRefGoogle Scholar
  20. Jiang Y, Xu ZS, Yu XH (2015) Group decision making based on incomplete intuitinistic multiplicative preference relations. Inf Sci 295:33–52CrossRefGoogle Scholar
  21. Kou G, Peng Y, Chen Z, Shi Y (2009) Multiple criteria mathematical programming for multi-class classification and application in network intrusion detection. Inf Sci 179:371–381CrossRefGoogle Scholar
  22. Liao HC, Xu ZS, Xia MM (2014) Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation. J Intell Fuzzy Syst 27:2969–2985MathSciNetzbMATHGoogle Scholar
  23. Lio H, Xu Z (2014) Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency. IEEE Trans Fuzzy Syst 22:1669–1689CrossRefGoogle Scholar
  24. Liu F (2009) Acceptable consistency analysis of interval reciprocal comparison matrices. Fuzzy Sets Syst 160:2686–2700MathSciNetCrossRefGoogle Scholar
  25. Liu F, Zhang WG, Wang ZX (2012) A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making. Eur J Oper Res 218:747–754MathSciNetCrossRefGoogle Scholar
  26. Meng F, Chen X (2015) An approach to incomplete multiplicative preference relations and its application in group decision making. Inf Sci 309:119–137CrossRefGoogle Scholar
  27. Meng F, Tan C (2017) A new consistency concept for interval multiplicative preference relations. Appl Soft Comput 52:262–276CrossRefGoogle Scholar
  28. Meng F, Tan C, Chen X (2017) Multiplicative consistency analysis for interval fuzzy preference relations: a comparative study. Omega 68:17–38CrossRefGoogle Scholar
  29. Nishizawa K (1997) A method to find elements of cycles in an incomplete directed graph and its applications-binary AHP and petri nets. Comput Math Appl 33:33–46MathSciNetCrossRefGoogle Scholar
  30. Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1:155–167MathSciNetCrossRefGoogle Scholar
  31. Petra G, Lidija ZS (2012) Acceptably consistency of aggregated comparison matrices in analytic hierarchy process. Eur J Oper Res 223:417–420CrossRefGoogle Scholar
  32. Rezaei J (2015) Best-worst multi-criteria decision-making method. Omega 53:49–57CrossRefGoogle Scholar
  33. Roy B (1991) The outranking approach and the foundations of ELECTRE methods. Theor Decis 31:49–73MathSciNetCrossRefGoogle Scholar
  34. Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 15:234–281MathSciNetCrossRefGoogle Scholar
  35. Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New YorkzbMATHGoogle Scholar
  36. Saaty TL (2001) Fundamentals of decision making and priority theory with the analytic hierarchy process. RWS Publications, PittsburghGoogle Scholar
  37. Saaty TL, Vargas LG (1987) Uncertainty and rank order in the analytic hierarchy process. Eur J Oper Res 32:107–117MathSciNetCrossRefGoogle Scholar
  38. Szmidt E, Kacprzyk J (2002) Using intuitinistic fuzzy sets in group decision making. Control Cybernet 31:1037–1053zbMATHGoogle Scholar
  39. Triantaphyllou E (2000) Multi-criteria decision making methods: a comparative study, applied optimization series 44. Springer, NewYorkCrossRefGoogle Scholar
  40. Ureña R, Chiclana F, Morente-Molinera JA, Herrera-Viedma E (2015) Managing incomplete preference relations in decision making: a review and future trends. Inf Sci 302:14–32MathSciNetCrossRefGoogle Scholar
  41. Wan S, Wang F, Dong J (2017) Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making. Eur J Oper Res 263:571–582MathSciNetCrossRefGoogle Scholar
  42. Wan S, Wang F, Dong J (2018) A threephase method for group decision making with interval-valued intuitionistic fuzzy preference relations. IEEE Trans Fuzzy Syst 26(2):998–1010CrossRefGoogle Scholar
  43. Wan S, Xu GL, Dong J (2016) A novel method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations. Inf Sci 372:53–71CrossRefGoogle Scholar
  44. Wang YM, Yang JB, Xu DL (2005) A twostage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets Syst 152:475–498CrossRefGoogle Scholar
  45. Xia MM, Xu ZS (2013) Group decision making based on intuitionistic multiplicative aggregation operators. Appl Math Model 37:5120–5133MathSciNetCrossRefGoogle Scholar
  46. Xia MM, Xu ZS, Liao HC (2013) Preference relations based on intuitionistic multiplicative information. IEEE Trans Fuzzy Syst 21:113–133CrossRefGoogle Scholar
  47. Xu ZS (2004a) Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. Int J Approx Reason 36:261–270MathSciNetCrossRefGoogle Scholar
  48. Xu ZS (2004b) On compatibility of interval fuzzy preference relations. Fuzzy Optim Decis Making 3:217–225MathSciNetCrossRefGoogle Scholar
  49. Xu ZS (2007a) A survey of preference relations. Int J Gen Syst 36:179–203MathSciNetCrossRefGoogle Scholar
  50. Xu ZS (2007b) Intuitionistic preference relations and their application in group decision making. Inf Sci 177:2363–2379MathSciNetCrossRefGoogle Scholar
  51. Xu ZS (2012) A consensus reaching process under incomplete multiplicative preference relations. Int J Gen Syst 41:333–351MathSciNetCrossRefGoogle Scholar
  52. Xu ZS (2013) Priority weight intervals derived from intuitionistic multiplicative preference. IEEE Trans Fuzzy Syst 21:642–654CrossRefGoogle Scholar
  53. Xu ZS, Cai XQ (2009) Incomplete interval-valued intuitionistic preference relations. Int J Gen Syst 38:871–886MathSciNetCrossRefGoogle Scholar
  54. Xu ZS, Cai XQ (2012) Intuitionistic fuzzy information aggregation: theory and applications. Science Press, BerlinCrossRefGoogle Scholar
  55. Xu GL, Wan S, Wang F, Dong J, Zeng YF (2016) Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations. Knowl-Based Syst 98:30–43CrossRefGoogle Scholar
  56. Yager RR, Xu ZS (2006) The continuous ordered weighted geometric operator and its application to decision making. Fuzzy Sets Syst 157:1393–1402MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsDelhi Technological UniversityDelhiIndia
  2. 2.Department of MathematicsIndian Institute of Technology DelhiHauz Khas, DelhiIndia

Personalised recommendations