Abstract
In a very recent paper by Xu and Chen (Soft Comput 12:515–521, 2008), a novel procedure for group decision making with incomplete reciprocal relations was developed. In this note, we examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Based on this, we develop a procedure for the decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Theoretical analysis and numerical examples have shown that the function deduced by us is more reasonable and effective than Xu and Chen’s.
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References
Alonso S, Chiclana F, Herrera F, Herrera-Viedma E (2004) A learning procedure to estimate missing values in fuzzy preference relations based on additive consistency. MDAI2004, LNAI vol. 3131, pp 227–238
Alonso S, Chiclana F, Herrera F, Herrera-Viedma E (2008) A consistency-based procedure to estimate missing pairwise preference values. Int J Intell Syst 23:155–175
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(2):201–222
Cabrerizo FJ, Heradio R, Pérez IJ, Herrera-Viedma E (2010a) A selection process based on additive consistency to deal with incomplete fuzzy linguistic information. J Univers Comput Sci 16(1):62–81
Cabrerizo FJ, Pérez IJ, Herrera-Viedma E (2010b) Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information. Knowl Based Syst 23(2):169–181
Chiclana F, Herrera-Viedma E, Alonso S and Herrera F (2008) A note on the estimation of missing pairwise preference values: A U-consistency based method. Int J Uncertainty, Fuzzy Knowl Based Syst 16(2):19–32
Chiclana F, Herrera-Viedma E, Alonso S (2009) A note on two methods for estimating missing pairwise preference values. IEEE Trans Syst Man Cybern B Cybern 39(6):1628–1633
Fedrizzi M, Giove S (2007) Incomplete pairwise comparison and consistency optimization. Eur J Oper Res 183:303–313
Gong ZW (2008) Least-square method to priority of the fuzzy preference relations with incomplete information. Int J Approx Reason 47:258–264
Herrera-Viedma E, Alonso S, Chiclana F, Herrera F (2007a) 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 (2007b) Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Trans Syst Man Cybern B Cybern 37(1):176–189
Lee HS, Chou MT, Fang HH, Tseng WK and Yeh CH (2007) Estimating missing values in incomplete additive fuzzy preference relations. KES 2007/WIRN 2007, LNAI, vol. 4693, pp 1307–1314
Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1:155–167
Shen PD, Chyr WL, Lee HS, and Lin K (2009) Correspondence between incomplete fuzzy preference relation and its priority vector. KES 2009, Part II, LNAI vol. 5712, pp 745–751
Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets Syst 12:117–131
Xu ZS (2004) Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. Int J Approx Reason 36:261–270
Xu ZS (2005) A procedure for decision making based on incomplete fuzzy preference relation. Fuzzy Optim Decis Mak 4:175–189
Xu ZS, Chen J (2008) Group decision-making procedure based on incomplete reciprocal relations. Soft Comput 12:515–521
Xu YJ, Da QL (2008) Weighted least-square method and its improvement for priority of incomplete complementary judgement matrix. Syst Eng Electron 31(7):1273–1276
Xu YJ, Da QL (2009) Methods for priority of incomplete complementary judgement matrices. Syst Eng Electron 31(1):95–99
Xu YJ, Da QL, Liu LH (2009) Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness. Int J Approx Reason 50(8):1287–1297
Zhang JJ (2000) Fuzzy analytical hierarchy process. Fuzzy Syst Math 14(2):80–88
Acknowledgments
The author is very grateful to the two anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. This work was supported by Hohai University “the Fundamental Research Funds for the Central Universities (2009B04514)” and the National Natural Science Foundation of China(No. 50979024,No. 90924027).
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Xu, Y., Da, Q. & Wang, H. A note on group decision-making procedure based on incomplete reciprocal relations. Soft Comput 15, 1289–1300 (2011). https://doi.org/10.1007/s00500-010-0662-3
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DOI: https://doi.org/10.1007/s00500-010-0662-3