Abstract
Preference relations could be originated from a decision making problem by pairwisely comparing a finite set of alternatives. In order to find an optimal solution, a feasible approach is to elicit the priorities from the derived preference relation. In this paper, we report an optimization-based approach to the priorities elicited from fuzzy preference relations (FPRs). The inherent relation between row/column vectors of FPRs with additive/multiplicative consistency is considered. Under additive consistency, the variance-based additive consistency index (VACI) of FPRs is constructed and some properties are studied. With the knowledge of multiplicative consistency, the concept of transformation-based multiplicative consistency index is proposed. Using numerical simulations, the thresholds of the proposed consistency indexes for FPRs with acceptable additive/multiplicative consistency are determined. A new method for deriving the priority vector from FPRs is proposed by constructing an optimization problem. The optimal solution is studied and some comparisons with the existing methods are made. Finally, numerical examples are carried out to show the effectiveness of the proposed approach.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no data sets were generated or analysed during the current study.
References
Arrow KJ (1963) Social choice and individual values (second version). Wiley, New York
Brunelli M (2017) Studying a set of properties of inconsistency indices for pairwise comparisons. Ann Oper Res 248:143–161
Cabrerizo FJ, Ureña R, Pedrycz W, Herrera-Viedma E (2014) Building consensus in group decision making with an allocation of information granularity. Fuzzy Sets Syst 255(16):115–127
Chen TCT (2020) Guaranteed-consensus posterior-aggregation fuzzy analytic hierarchy process method. Neural Comput Appl 32:7057–7068
Chiclana F, Herrera F, Herrera-Viedma E (1998) Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets Syst 97(1):33–48
Chiclana F, Herrera F, Herrera-Viedma F (2001) Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst 122(2):277–291
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
De Baets B, De Meyer H (2005) Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets Syst 152(2):249–270
De Baets B, De Meyer H (2008) On the cycle-transitive comparison of artificially coupled random variables. Int J Approx Reason 47(3):306–322
Fan ZP, Ma J, Zhang Q (2001) An approach to multiple attribute decision making based on fuzzy preference information on alternatives. Fuzzy Sets Syst 131(1):101–106
Fan ZP, Hu GF, Xiao SH (2002) A method for multiple attribute decision-making with the fuzzy preference relation on alternatives. Comput Ind Eng 46(2):321–327
Fedrizzi M, Brunelli M (2009) On the normalization of a priority vector associated with a reciprocal relation. Int J General Syst 38(5):579–586
Fedrizzi M, Brunelli M (2010) On the priority vector associated with a reciprocal relation and a pairwise comparison matrix. Soft Comput 14(6):639–645
Golany B, Kress M (1993) A multicriteria evaluation of methods for obtaining weights from ratio-scale matrices. Eur J Oper Res 69:210–220
Herrera-Viedma E, Herrera F, Chiclana F, Luque M (2004) Some issues on consistency of fuzzy preference relations. Eur J Oper Res 154(1):98–109
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
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 Syst Man Cybern B Cybern 37:176–189
İç YC, Yurdakul M (2021) Development of a new trapezoidal fuzzy AHP-TOPSIS hybrid approach for manufacturing firm performance measurement. Granul Comput 6:915–929
Koczkodaj WW, Urban R (2018) Axiomatization of inconsistency indicators for pairwise comparisons. Int J Approx Reason 94:18–29
Kou G, Lin CS (2014) A cosine maximization method for the priority vector derivation in AHP. Eur J Oper Res 235:225–232
Lan JB, Chen Z, Liu F (2022) An indirect weak transitivity standard for inconsistent multiplicative reciprocal preference relations. Granul Comput 7:315–322
Li J, Wang JQ (2019) Multi-criteria decision-making with probabilistic hesitant fuzzy information based on expected multiplicative consistency. Neural Comput Appl 31:8897–8915
Li CC, Dong YC, Xu YJ, Chiclana F, Herrera-Viedma E, 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
Li CC, Rodríguez RM, Martínez L, Dong YC, Herrera F (2019) Consensus building with individual consistency control in group decision making. IEEE Trans Fuzzy Syst 27(2):319–332
Lipovetsky S, Conklin WM (2002) Robust estimation of priorities in the AHP. Eur J Oper Res 137:110–122
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
Liu XW, Pan YM, 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 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 246:1–18
Liu F, Zou SC, Li Q (2020) Deriving priorities from pairwise comparison matrices with a novel consistency index. Appl Math Comput 374:125059
Liu F, Yang H, Hu YK (2022) A prioritization approach of non-reciprocal fuzzy preference relations and its extension. Comput Ind Eng 168:108076
Ma J, Fan ZP, Jiang YP, Mao JY, Ma L (2006) A method for repairing the inconsistency of fuzzy preference relations. Fuzzy Sets Syst 157:20–33
Orlovsky SA (1978) Decision making with a fuzzy preference relation. Fuzzy Sets Syst 1(3):155–167
Saaty TL (1980) The analytic hierarchy process. Mcgraw-Hill, New York
Seikh MR, Mandal U (2022) Q-rung orthopair fuzzy Frank aggregation operators and its application in multiple attribute decision-making with unknown attribute weights. Granul Comput 7:709–730
Sun WY, Yuan YX (2006) Optimization theory and methods: nonlinear programming. Springer, New York
Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets Syst 12:117–131
Wang YM, Fan ZP (2007) Fuzzy preference relations: aggregation and weight determination. Comput Ind Eng 53:163–172
Wang YM, Fan ZP, Hua ZS (2007) A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations. Eur J Oper Res 182(1):356–366
Wang J, Lan JB, Ren PY, Luo YY (2012) Some programming models to derive priority weights from additive interval fuzzy preference relation. Knowl-Based Syst 27:69–77
Wu ZB, Xu JP (2012) A concise consensus support model for group decision making with reciprocal preference relations based on deviation measures. Fuzzy Sets Syst 206:58–73
Wu P, Wu Q, Zhou LG, Chen HY, Zhou H (2019) A consensus model for group decision making under trapezoidal fuzzy numbers environment. Neural Comput Appl 31:377–394
Xia MM, Xu ZS (2014) Interval weight generation approaches for reciprocal relations. Appl Math Model 38(3):828–838
Xia MM, Xu ZS, Chen J (2013) Algorithms for improving consistency or consensus of reciprocal [0,1]-valued preference relations. Fuzzy Sets Syst 216:108–133
Xu ZS (2004) Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. Int J Approx Reason 36(3):261–270
Xu ZS (2005) A procedure for decision making based on incomplete fuzzy preference relation. Fuzzy Opt Deci Making 4(3):175–189
Xu ZS, Da QL (2003) An approach to improving consistency of fuzzy preference matrix. Fuzzy Opt Deci Making 2(1):3–12
Xu ZS, Da QL (2005) A least deviation method to obtain a priority vector of a fuzzy preference relation. Eur J Oper Res 164(1):206–216
Xu YJ, Herrera F (2019) Visualizing and rectifying different inconsistencies for fuzzy reciprocal preference relations. Fuzzy Sets Syst 362:85–109
Xu YJ, Da DL, Liu LH (2009) Normalizing rank aggregation method for priority of a fuzzy preference relations and its effectiveness. Int J Approx Reason 50(8):1287–1297
Xu YJ, Patnayakun R, Wang HM (2013) The ordinal consistency of a fuzzy preference relation. Inf Sci 224:152–164
Xu YJ, Li KW, Wang HM (2014) Consistency test and weight generation for additive interval fuzzy preference relations. Soft Comput 18:1499–1513
Xu YJ, Liu XW, Wang HM (2018) The additive consistency measure of fuzzy reciprocal preference relations. Int J Mach Learn Cybern 9(7):1141–1152
Xu YJ, Li MQ, Cabrerizo FJ, Chiclana F, Herrera-Viedma E (2021) Algorithms to detect and rectify multiplicative and ordinal inconsistencies of fuzzy preference relations. IEEE Trans Syst Man Cybern Syst 51(6):3498–3511
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zhang J, Kou G, Peng Y, Zhang Y (2021) Estimating priorities from relative deviations in pairwise comparison matrices. Inf Sci 552:310–327
Acknowledgements
The authors would like to thank the anonymous reviewers for improving the quantity of the paper. The work was supported by the National Natural Science Foundation of China (no. 71871072), the Guangxi Natural Science Foundation (no. 2022GXNSFDA035075), and the Innovation Project of Guangxi Graduate Education (no. YCSW2022110).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
conflict of interest
All authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Luo, Z., Yang, H. & Liu, F. An optimization-based method for eliciting priorities from fuzzy preference relations with a novel consistency index. Granul. Comput. 8, 943–958 (2023). https://doi.org/10.1007/s41066-023-00361-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41066-023-00361-6