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Measuring consistency of interval-valued preference relations: comments and comparison

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Abstract

The concepts of consistency definition and consistency index are usually used to measure the consistency of a preference relation. When interval numbers are used to express the preference information, the consistency of the derived interval-valued preference relations (IVPRs) is worth being investigated. In this study, a comment is provided for the ideas behind consistency definitions and consistency indexes of interval multiplicative reciprocal matrices (IMRMs) and interval additive reciprocal matrices (IARMs), respectively. A comparison is made by considering the two kinds of consistency definitions of IVPRs. It is found that the method of defining the consistency of IVPRs in terms of the imaginary intervals is equivalent to that of defining the approximate consistency. Numerical examples are reported to illustrate the differences of the two consistency definitions of IVPRs. The observations illustrate that the fundamental inconsistency of IVPRs is compatible with the underlying idea of fuzzy sets. It is revealed that a consistent preference relation is only a particular case with a fixed value of the defined consistency index. In general, the consistency index could be used to quantify the deviation degree from a consistent real-valued preference relation.

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References

  1. Brunelli M (2015) Introduction to the analytic hierarchy process. Springer, New York

  2. Chiclana F, Herrera-Viedma E, Alonso S, Herrera F (2009) Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans Fuzzy Syst 17(1):14–23

  3. Dong YC, Chen C, Li CC, Hong WC, Xu YF (2015) Consistency issues of interval pairwise comparison matrices. Soft Comput 19:2321–2335

  4. Dong YC, Li CC, Chiclana F, Herrera-Viedma E (2016) Average-case consistency measurement and analysis of interval-valued reciprocal preference relations. Knowl Based Syst 114:108–117

  5. Dubois D (2011) The role of fuzzy sets in decision sciences: old techniques and new directions. Fuzzy Sets Syst 184:3–28

  6. Golden BL, Wasil EA, Harker PT (1989) The analytic hierarchy process: applications and studies. Springer, Berlin

  7. Herrera F, Martinez L, Sanchez PJ (2005) Managing non-homogeneous information in group decision making. Eur J Oper Res 166(1):115–132

  8. Herrera-Viedma E, Alonso S, Chiclana F, Herrera F (2007) A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Trans Fuzzy Syst 15(5):863–877

  9. Hu BQ, Wang S (2006) A novel approach in uncertain programming part I: new arigthmetic and order relation for interval nunmbers. J Ind Manag Opt 2(4):351–371

  10. Kacprzyk J (1986) Group decision making with a fuzzy linguistic majority. Fuzzy Sets Syst 18(2):105–118

  11. Krejčí J (2017) On additive consistency of interval fuzzy preference relations. Comput Ind Eng 107:128–140

  12. Krejčí J (2019) On extension of multiplicative consistency to interval fuzzy preference relations. Oper Res Int J 19:783–815

  13. Lan JB, Hu MM, Ye XM, Sun SQ (2012) Deriving interval weights from an interval multiplicative consistent fuzzy preference relation. Knowl Based Syst 26:128–134

  14. Li CC, Rodríguez RM, Martínez L, Dong YC, Herrera F (2018) Consistency of hesitant fuzzy linguistic preference relations: an interval consistency index. Inf Sci 432:347–361

  15. Li CC, Dong YC, Xu YJ, Chiclana F, Herrera-Viedma F, Herrera F (2019) An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and fusion: taxonomy and future directions. Inf Fusion 52:143–156

  16. Lin J, Zhang Q (2017) Note on continuous interval-valued intuitionistic fuzzy aggregation operator. Appl Math Model 43:670–677

  17. Liu F (2009) Acceptable consistency analysis of interval reciprocal comparison matrices. Fuzzy Sets Syst 160(18):2686–5700

  18. Liu F, Zhang WG (2014) TOPSIS-based consensus model for group decision-making with incomplete interval fuzzy preference relations. IEEE Trans Cybern 44(8):1283–1294

  19. Liu F, Zhang WG, Zhang LH (2014) A group decision making model based on a generalized ordered weighted geometric average operator with interval preference matrices. Fuzzy Sets Syst 246(1):1–18

  20. Liu F, Pedrycz W, Wang ZX, Zhang WG (2017a) An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations. Fuzzy Sets Syst 322:1–18

  21. Liu F, Pedrycz W, Zhang WG (2017b) Limited rationality and its quantification through the interval number judgments with permutations. IEEE Trans Cybern 47(12):4025–4037

  22. Liu F, Liu ZL, Wu YH (2018a) A group decision making model based on triangular fuzzy additivereciprocal matrices with additive approximation-consistency. Appl Soft Comput 65:349–359

  23. Liu F, Peng YN, Yu Q, Zhao H (2018b) A decision-making model based on interval additive reciprocal matrices with additive approximation-consistency. Inf Sci 422:161–176

  24. Liu F, Yu Q, Pedrycz W, Zhang WG (2018c) A group decision making model based on an inconsistency index of interval multiplicative reciprocal matrices. Knowl Based Syst 145:67–76

  25. Liu F, Yu Q, Huang MJ, Ralescu DA (2020) An inconsistency index of interval additive reciprocal matrices with application to group decision making. J Data Inf Manag. https://doi.org/10.1007/s42488-019-00019-6

  26. Lu J, Zhang GQ, Ruan D, Wu F (2007) Multi-objective group decision making: methods, software and applications with fuzzy set techniques. Sinapore World Scientific Publishing Co., Pte. Ltd, Sinapore

  27. Lu J, Ma J, Zhang GQ, Zhu YJ, Zeng XY, Koehl L (2011) Theme-based comprehensive evaluation in new product development using fuzzy hierarchical criteria group decision-making method. IEEE Trans Ind Elect 58(6):2236–2246

  28. Ma J, Lu J, Zhang GQ (2010) Decider: a fuzzy multi-criteria group decision support system. Knowl Based Syst 23(1):23–31

  29. Meng FY, Tan CQ (2017) A new consistency concept for interval multiplicative preference relations. Appl Soft Comput 52:262–276

  30. Meng FY, Chen XH, Tan CQ (2016) Cooperative fuzzy games with interval characteristic functions. Oper Res Int J 16(01):1–24

  31. Meng FY, An QX, Tan CQ, Chen XH (2017a) An approach for group decision making with interval fuzzy preference relations based on additive consistency and consensus analysis. IEEE Trans Syst Man Cybern Syst 47:2069–2082

  32. Meng FY, Lin J, Tan CQ, Zhang Q (2017b) A new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations. Fuzzy Sets Syst 315:1–25

  33. Meng FY, Tan CQ, Chen XH (2017c) Multiplicative consistency analysis for interval fuzzy preference relations: a comparative study. Omega 68:17–38

  34. Moorse RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs

  35. Nurmi H (1981) Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets Syst 6(3):249–259

  36. Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1(3):155–167

  37. Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 15(3):234–281

  38. Saaty TL (1980) The analytic hierarchy process: planning, priority setting, resource allocation. McGraw-Hill, New York

  39. Saaty TL, Vargas LG (1987) Uncertainty and rank order in the analytic hierarchy process. Eur J Oper Res 32(1):107–117

  40. Su LQ, Li FC, Qiu JQ (1997) Generalized interval numbers and their operation. J Hebei Inst Chem Tech Light Ind 18(1):11–14

  41. Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets Syst 12(2):117–131

  42. Vaidyaab OS, Kumar S (2006) Analytic hierarchy process: an overview of applications. Eur J Oper Res 169(1):1–29

  43. Wan SP, Wang F, Dong JY (2018) A group decision making method with interval valued fuzzy preference relations based on the geometric consistency. Inf Fusion 40:87–100

  44. Wang YM, Yang JB, Xu DL (2005) A two-stage logrithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets Syst 152(3):475–498

  45. Wang ZJ, Lin J, Liu F (2019) Axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations. Inf Sci 491:109–137

  46. Xu ZS (2001) A practical method for priority of interval number complementary judgement matrix. Oper Res Manag Sci 10(1):9–16

  47. Xu ZS (2004) On compatibility of interval fuzzy preference relations. Fuzzy Opt Decis Mak 3(3):217–225

  48. Xu ZS (2011) Consistency of interval fuzzy preference relation in group decision making. Appl Soft Comput 11(5):3898–3909

  49. Xu ZS, Chen J (2008) Some models for deriving the priority weights from interval fuzzy preference relations. Eur J Oper Res 184(1):266–280

  50. Xu ZS, Da QL (2003) An approach to improving consistency of fuzzy preference matrix. Fuzzy Opt Decis Mak 2:3–12

  51. Xu ZS, Wei CP (1999) A consistency improving method in the analytic hierarchy process. Eur J Oper Res 116:443–449

  52. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

  53. Zadeh LA (1996) Fuzzy sets, fuzzy logic, and fuzzy systems. World Scientific Publishing Co., Inc., River Edge

  54. Zhai JR (1998) Generalized interval numbers and their operation. J Hebei Inst Mech Electr Eng 15(02):69–74

  55. Zhou JJ, Zhang ZH, Liu SY (1996) Arithmetic operation of the general interval numbers. J Hebei Min Civil Eng Inst 2:51–55

  56. Zhou LG, He YD, Chen HY, Liu JP (2014) Compatibility of interval fuzzy preference relations with the COWA operator and its application to group decision making. Soft Comput 18(11):2283–2295

  57. Zhou L, Merigó JM, Chen H, Liu J (2016) The optimal group continuous logarithm compatibility measure for interval multiplicative preference relations based on the COWGA operator. Inf Sci 328(20):250–269

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Acknowledgements

The authors would like to thank the anonymous reviewers for the valuable comments and constructive suggestions improving the quality of the paper.

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Correspondence to Fang Liu.

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The work was supported by the National Natural Science Foundation of China (Nos. 71571054, 71871072), the Guangxi Natural Science Foundation for Distinguished Young Scholars (No. 2016GXNSFFA380004), 2017 Guangxi high school innovation team and outstanding scholars plan, and the Innovation Project of Guangxi Graduate Education (No. YCSW2019045).

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Liu, F., Huang, M., Huang, C. et al. Measuring consistency of interval-valued preference relations: comments and comparison. Oper Res Int J (2020). https://doi.org/10.1007/s12351-020-00551-z

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Keywords

  • Interval-valued preference relation (IVPR)
  • Consistency definition
  • Consistency index
  • Deviation degree
  • Equivalence