Skip to main content
SpringerLink
Log in

Checking and adjusting order-consistency of linguistic pairwise comparison matrices for getting transitive preference relations

  • Regular Article
  • Published:
OR Spectrum Aims and scope Submit manuscript

Abstract

Linguistic pairwise comparison matrices for decision-making need to be (strongly) order-consistent, which means the judgments should be (strongly) transitive. We introduce the equivalent conditions of (strong) transitivity by using route matrices and digraphs and develop an adjustment procedure to help decision makers correcting the inconsistency. The conclusion can extend to other ordinal-scaled or numerical comparison matrices as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Aguaron J, Escobar MT, Moreno-Jiménez JM (2003) Consistency stability intervals for a judgement in AHP decision support systems. Eur J Oper Res 145:382–393

    Article  Google Scholar 

  • Aguaron J, Moreno-Jiménez JM (2003) The geometric consistency index: approximated thresholds. Eur J Oper Res 147:45–137

    Article  Google Scholar 

  • Alonso S, Cabrerizo FJ, Chiclana F, Herrera F, Herrera-Viedma E (2009) Group decision making with incomplete fuzzy linguistic preference relations. Int J Intell Syst 24:201–222

    Article  Google Scholar 

  • Alonso S, Pérez IJ, Cabrerizo FJ, Herrera-Viedma E (2013) A linguistic consensus model for Web 2.0 communities. Appl Soft Comput 13:57–149

    Article  Google Scholar 

  • Ashikhmin I, Furems E (2005) UniComBOS—intelligent decision support system for multi-criteria comparison and choice. J Multi-Criteria Decis Anal 13:147–157

    Article  Google Scholar 

  • Bana e Costa CA, Vansnick J-C (2008) A critical analysis of the eigenvalue method used to derive priorities in AHP. Eur J Oper Res 187:1422–1428

    Article  Google Scholar 

  • Bozóki S (2008) Solution of the least squares method problem of pairwise comparison matrices. Cent Eur J Oper Res 16:345–358

    Article  Google Scholar 

  • Bozóki S, Rapcsák T (2008) On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. J Glob Optim 42:157–175

    Article  Google Scholar 

  • 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 17:14–23

    Article  Google Scholar 

  • de Soto AR, Trillas E (1999) On antonym and negate in fuzzy logic. Int J Intell Syst 14:295–303

    Article  Google Scholar 

  • Dodd F, Donegan H, McMaster T (1993) A statistical approach to consistency in AHP. Math Comput Model 18:19–22

    Article  Google Scholar 

  • Dong Y, Hong W-C, Xu Y (2013) Measuring consistency of linguistic preference relations: a 2-tuple linguistic approach. Soft Comput 17:2117–2130

    Article  Google Scholar 

  • Doukas H (2013) Modelling of linguistic variables in multicriteria energy policy support. Eur J Oper Res 227:227–238

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Escobar MT, Aguarón J, Moreno-Jiménez JM (2004) A note on AHP group consistency for the row geometric mean priorization procedure. Eur J Oper Res 153:318–322

    Article  Google Scholar 

  • Fülöp J (2008) A method for approximating pairwise comparison matrices by consistent matrices. J Glob Optim 42:423–442

    Article  Google Scholar 

  • Furems E, Gnedenko L (1992) Interactive procedure for non-transitivity revealing and correcting under pairwise comparisons. INFOR: Inf Syst Oper Res 30:118–126

    Google Scholar 

  • Gass S (1998) Tournaments, transitivity and pairwise comparison matrices. J Oper Res Soc 49:616–624

    Article  Google Scholar 

  • Herrera F, Herrera-Viedma E, Martínez L (2000) A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst 114:43–58

    Article  Google Scholar 

  • Herrera F, Alonso S, Chiclana F, Herrera-Viedma E (2009) Computing with words in decision making: foundations, trends and prospects. Fuzzy Optim Decis Making 8:337–364

    Article  Google Scholar 

  • Herrera F, Herrera-Viedma E (2000) Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst 115:67–82

    Article  Google Scholar 

  • Herrera-Viedma E, Herrera F, Chiclana F, Luque M (2004) Some issues on consistency of fuzzy preference relations. Eur J Oper Res 154:98–109

    Article  Google Scholar 

  • Herrera-Viedma E, Chiclana F, Herrera F, Alonso S (2007) Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Trans Cybern 37:176–189

    Article  Google Scholar 

  • Ishizaka A, Lusti M (2004) An expert module to improve the consistency of AHP matrices. Int Trans Oper Res 11:97–105

    Article  Google Scholar 

  • Jensen RE, Hicks TE (1993) Ordinal data AHP analysis: a proposed coefficient of consistency and a nonparametric test. Math Comput Model 17:135–150

    Article  Google Scholar 

  • Ju Y, Wang A (2013) Extension of VIKOR method for multi-criteria group decision making problem with linguistic information. Appl Math Model 37:3112–3125

    Article  Google Scholar 

  • Kéri G (2011) On qualitatively consistent, transitive and contradictory judgment matrices emerging from multiattribute decision procedures. Cent Eur J Oper Res 19:215–224

    Article  Google Scholar 

  • Koczkodaj W (1993) A new definition of consistency of pairwise comparisons. Math Comput Model 18:79–84

    Article  Google Scholar 

  • Kou G, Ergu D, Shang J (2014) Enhancing data consistency in decision matrix: adapting Hadamard model to mitigate judgment contradiction. Eur J Oper Res 236:261–271

    Article  Google Scholar 

  • Kwiesielewicz M, van Uden E (2004) Inconsistent and contradictory judgements in pairwise comparison method in the AHP. Comput Oper Res 31:713–719

    Article  Google Scholar 

  • Larichev OI, Moshkovich HM (1995) ZAPROS-LM—a method and system for ordering multiattribute alternatives. Eur J Oper Res 82:503–521

    Article  Google Scholar 

  • Leung L, Cao D (2000) On consistency and ranking of alternatives in fuzzy AHP. Eur J Oper Res 124:102–113

    Article  Google Scholar 

  • Martínez L, Ruan D, Herrera F (2010) Computing with words in decision support systems: an overview on models and applications. Int J Comput Intell Syst 3:382–395

    Article  Google Scholar 

  • Martínez L, Herrera F (2012) An overview on the 2-tuple linguistic model for computing with words in decision making: extensions, applications and challenges. Inf Sci 207:1–18

    Article  Google Scholar 

  • Millet I (1998) The effectiveness of alternative preference elicitation methods in the analytic hierarchy process. J Multi-Criteria Decis Anal 6:41–51

    Article  Google Scholar 

  • Moshkovich HM, Mechitov AI (2013) Verbal decision analysis: foundations and trends. Adv Decis Sci Article ID 697072, p 9. Doi:10.1155/2013/697072

  • Pedrycz W, Song M (2014) A granulation of linguistic information in AHP decision-making problems. Inf Fusion 17:93–101

    Article  Google Scholar 

  • Rodríguez RM, Martínez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20:109–119

    Article  Google Scholar 

  • Rodríguez RM, Martínez L (2012) An analysis of symbolic linguistic computing models in decision making. Int J Gen Syst 42:121–136

    Article  Google Scholar 

  • Saaty TL (1980) The analytic hierarchy process. McGraw Hill, New York

    Google Scholar 

  • Stein WE, Mizzi PJ (2007) The harmonic consistency index for the analytic hierarchy process. Eur J Oper Res 177:488–497

    Article  Google Scholar 

  • Tapia GJM, del Moral MJ, Martínez MA, Herrera-Viedma E (2012) A consensus model for group decision making problems with linguistic interval fuzzy preference relations. Expert Syst Appl 39:10022–10030

    Article  Google Scholar 

  • Trillas E, Moraga C, Guadarrama S, Cubillo S, Castiñeira E (2007) Computing with antonyms. In: Nikravesh M, Kacprzyk J, Zadeh L (eds) Forging new frontiers: fuzzy pioneers I. Springer, Berlin, pp 133–153

    Chapter  Google Scholar 

  • Von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

  • Wang Y-M, Chin K-S, Luo Y (2009) Aggregation of direct and indirect judgments in pairwise comparison matrices with a re-examination of the criticisms by Bana e Costa and Vansnick. Inf Sci 179:329–337

    Article  Google Scholar 

  • Wang J-Q, Wang P, Wang J, Zhang H-Y, Chen X-H (2015) Atanassov’s interval-valued intuitionistic linguistic multi-criteria group decision-making method based on trapezium cloud model. IEEE Trans Fuzzy Syst 23:542–554

    Article  Google Scholar 

  • Wang J-Q, Peng J-J, Zhang H-Y, Liu T, Chen X-H (2015) An uncertain linguistic multi-criteria group decision-making method based on a cloud model. Group Decis Negot 24:171–192

    Article  Google Scholar 

  • Xu ZS (2004) Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. Int J Approx Reason 36:70–261

    Article  Google Scholar 

  • Yang W-E, Wang J-Q, Wang X-F (2012) An outranking method for multi-criteria decision making with duplex linguistic information. Fuzzy Set Syst 198:20–33

    Article  Google Scholar 

  • Yang W-E (2014) Semantic dominance analysis for multicriteria decision-making problems with unbalanced linguistic scale. Int Trans Oper Res 21:627–647

    Article  Google Scholar 

  • Yang W-E, Wang J (2013) Multi-criteria semantic dominance: a linguistic decision aiding technique based on incomplete preference information. Eur J Oper Res 231:81–171

    Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8:199–249

    Article  Google Scholar 

  • Zhu J, Hipel KW (2012) Multiple stages grey target decision making method with incomplete weight based on multi-granularity linguistic label. Inf Sci 212:15–32

    Article  Google Scholar 

Download references

Acknowledgments

We sincerely thank the editors and four anonymous referees. Their careful and detailed comments helped us improve the paper considerably. This work was supported by the National Natural Science Foundation of China (Nos. 71401185, 71431008); MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 13YJC630200); China Postdoctoral Science Foundation (Nos. 2015T80869, 2014M552135); and Hunan Provincial Natural Science Foundation of China (No. 14JJ4050).

Author information

Authors and Affiliations

  1. School of Business Administration, Hunan University, Changsha, 410082, China

    Wu-E Yang & Chao-Qun Ma

  2. School of Business, Central South University, Changsha, 410083, China

    Zhi-Qiu Han

  3. School of Economics, Central South University of Forestry and Technology, Changsha, 410004, China

    Wu-E Yang & Wen-Jun Chen

  4. Hunan Provincial Engineering Laboratory of Risk Management and Control Techniques for the Innovative Enterprises, Changsha, 410004, China

    Wu-E Yang & Wen-Jun Chen

Authors
  1. Wu-E Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao-Qun Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi-Qiu Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wen-Jun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wu-E Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, WE., Ma, CQ., Han, ZQ. et al. Checking and adjusting order-consistency of linguistic pairwise comparison matrices for getting transitive preference relations. OR Spectrum 38, 769–787 (2016). https://doi.org/10.1007/s00291-015-0417-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00291-015-0417-8

Keywords

Search

Navigation