Abstract
This paper studies the ordinal and additive inconsistency problems of linguistic preference relations. First, the definition of ordinal consistency of a linguistic preference relation is proposed. Based on the definition of adjacency matrix of a linguistic preference relation, the necessary and sufficient conditions of a linguistic preference relation being ordinally consistent are given. Then, a distance-based nonlinear programming method is developed to identify and adjust the ordinal and additive inconsistencies for linguistic preference relations. The proposed methods can not only solve the ordinal inconsistency, additive inconsistency problems, respectively, but also solve the ordinal and additive inconsistency problems simultaneously. Finally, numerical examples and comparative analysis are provided to show the effectiveness and advantages of the proposed methods.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
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
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 180:4477–4495
Ben-Arieh D, Chen ZF (2006) Linguistic-labels aggregation and consensus measure for autocratic decision making using group recommendations. IEEE Trans Syst Man Cybern Part A Syst Humans 36:558–568
Cabrerizo FJ, Chiclana F, Al-Hmouz R, Morfeq A, Balamash AS, Herrera-Viedma E (2015) Fuzzy decision making and consensus: challenges. J Intell Fuzzy Syst 29:1109–1118
Díaz S, García-Lapresta JL, Montes S (2008) Consistent models of transitivity for reciprocal preferences on a finite ordinal scale. Inf Sci 178:2832–2848
Dong YC, Ding Z, Chiclana F, Herrera-Viedma E (2017) Dynamics of public opinions in an online and offline social network. IEEE Trans Big Data. doi:10.1109/TBDATA.2017.2676810
Dong YC, Hong WC, Xu YF (2013) Measuring consistency of linguistic preference relations: a 2-tuple linguistic approach. Soft Comput 17:2117–2130
Dong YC, Li CC, Chiclana F, Herrera F (2016) Average-case consistency measurement and analysis of interval-valued reciprocal preference relations. Knowledge Based Syst 114:108–117
Dong YC, Li CC, Herrera F (2015) An optimization-based approach to adjusting unbalanced linguistic preference relations to obtain a required consistency level. Inf Sci 292:27–38
Dong YC, Xu YF, Li HY (2008) On consistency measures of linguistic preference relations. Eur J Oper Res 189:430–444
Dong YC, Xu YF, Li HY, Feng B (2010) The OWA-based consensus operator under linguistic representation models using position indexes. Eur J Oper Res 203:455–463
García-Lapresta JL, Meneses LC (2009) Modeling rationality in a linguistic framework. Fuzzy Sets Syst 160:3211–3223
González-Pachón J, Romero C (1999) Distance-based consensus methods: a goal programming approach. Omega Int J Manag Sci 27:341–347
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
Herrera-Viedma E, Martínez L, Mata F, Chiclana F (2005) A consensus support system model for group decision-making problems with multigranular linguistic preference relations. IEEE Trans Fuzzy Syst 13:644–658
Herrera F, Herrera-Viedma E (2000) Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst 115:67–82
Herrera F, Martínez L (2000) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8:746–752
Ju YB, Liu XY, Wang AH (2016) Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making. Soft Comput 20:4037–4053
Lin CS, Kou G, Ergu D (2013) An improved statistical approach for consistency test in AHP. Ann Oper Res 211:289–299
Liu WQ, Dong YC, Chiclana F, Cabrerizo FJ, Herrera-Viedma E (2016a) Group decision-making based on heterogeneous preference relations with self-confidence. Fuzzy Optim Decis Mak. doi:10.1007/s10700-016-9254-8
Liu XW, Pan YW, Xu YJ, Yu S (2012) Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations. Fuzzy Sets Syst 198:1–19
Liu XY, Ju YB, Yang SH (2016b) Some generalized interval-valued hesitant uncertain linguistic aggregation operators and their applications to multiple attribute group decision making. Soft Comput 20:495–510
Liu YJ, Liang CY, Chiclana F, Wu J (2017) A trust induced recommendation mechanism for reaching consensus in group decision making. Knowl Based Syst 119:221–231
Pérez IJ, Cabrerizo FJ, Alonso S, Herrera-Viedma E (2014) A new consensus model for group decision making problems with non homogeneous experts. IEEE Trans Syst Man Cybern Syst 44:494–498
Pérez LG, Mata F, Chiclana F, Kou G, Herrera-Viedma E (2016) Modelling influence in group decision making. Soft Comput 20:1653–1665
Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New York
Siraj S, Mikhailov L, Keane J (2012) A heuristic method to rectify intransitive judgments in pairwise comparison matrices. Eur J Oper Res 216:420–428
Wei CP, Feng XQ, Zhang YZ (2009) Method for measuring the satisfactory consistency of a linguistic judgement matrix. Syst Eng Theory Pract 29:104–110
Wu J, Chiclana F, Fujita H, Herrera-Viedma E (2017) A visual interaction consensus model for social network group decision making with trust propagation. Knowl Based Syst 122:39–50
Wu ZB, Xu JP (2012) Consensus reaching models of linguistic preference relations based on distance functions. Soft Comput 16:577–589
Xia MM, Xu ZS, Wang Z (2014) Multiplicative consistency-based decision support system for incomplete linguistic preference relations. Int J Syst Sci 45:625–636
Xu YJ, Gupta JND, Wang HM (2014a) The ordinal consistency of an incomplete reciprocal preference relation. Fuzzy Sets Syst 246:62–77
Xu YJ, Li KW, Wang HM (2013a) Distance-based consensus models for fuzzy and multiplicative preference relations. Inf Sci 253:56–73
Xu YJ, Li KW, Wang HM (2014b) Consistency test and weight generation for additive interval fuzzy preference relations. Soft Comput 18:1499–1513
Xu YJ, Ma F, Tao FF, Wang HM (2014c) Some methods to deal with unacceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision making. Knowl Based Syst 56:179–190
Xu YJ, Patnayakuni R, Wang HM (2013b) The ordinal consistency of a fuzzy preference relation. Inf Sci 224:152–164
Xu ZS (2005) Deviation measures of linguistic preference relations in group decision making. Omega Int J Manag Sci 33:249–254
Zadeh LA (1975) The concept of a linguistic variable and its applications to approximate reasoning—part I. Inf Sci 8:199–249
Zadeh LA (1975) The concept of a linguistic variable and its applications to approximate reasoning–part II. Inf Sci 9:301–357
Zadeh LA (1975) The concept of a linguistic variable and its applications to approximate reasoning–part III. Inf Sci 9:43–80
Zadeh LA (1996) Fuzzy logic= computing with words. IEEE Trans Fuzzy Syst 4:103–111
Zhang HJ, Dong YC, Herrera-Viedma E (2017) Consensus building for the heterogeneous large-scale GDM with the individual concerns and satisfactions. IEEE Trans Fuzzy Syst. doi:10.1109/TFUZZ.2017.2697403
Acknowledgements
This work was partly supported by the Key Project of National Natural Science Foundation of China (No.71433003), the National Natural Science Foundation of China (NSFC) (No.71471056), the Fundamental Research Funds for the Central Universities (No. 2015B23014), Excellent Innovative Talent Program of Hohai University, sponsored by Qing Lan Project of Jiangsu Province.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by V. Loia.
Rights and permissions
About this article
Cite this article
Xu, Y., Wei, C. & Sun, H. Distance-based nonlinear programming models to identify and adjust inconsistencies for linguistic preference relations. Soft Comput 22, 4833–4849 (2018). https://doi.org/10.1007/s00500-017-2671-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-017-2671-y