A Quadratic Programming Method for Ranking Alternatives Based on Multiplicative and Fuzzy Preference Relations

Document Type: Research Paper

Authors

1 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, No.1 Xikang Road, Nanjing, 210098, Jiangsu,China and Business School, Hohai University, Jiangning Campus, No.8 Focheng West Road, Jiangning, Nanjing, 211100, Jiangsu, China

2 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, No.1 Xikang Road, Nanjing, 210098, Jiangsu, China and Business School, Hohai University, Jiangning Campus, No.8 Focheng West Road, Jiangning, Nanjing, 211100, Jiangsu, China

Abstract

This paper proposes a quadratic programming method (QPM) for ranking alternatives based on multiplicative preference relations (MPRs) and fuzzy preference relations (FPRs). The proposed QPM can be used for deriving a ranking from either a MPR or a FPR, or a group of MPRs, or a group of FPRs, or their mixtures. The proposed approach is tested and examined with two numerical examples, and comparative analyses with the existing methods are provided to show the effectiveness and advantages of the QPM.

Keywords


[1] S. Alonso, F. Cabrerizo, F. Chiclana, F. Herrera and E. Herrera-Viedma, Group decision mak-
ing with incomplete fuzzy linguistic preference relations, International Journal of Intelligent
Systems, 24 (2009), 201{222.
[2] F. Chiclana, F. Herrera and E. Herrera-Viedma, Integrating multiplicative preference relations
in a multipurpose decision-making model based on fuzzy preference relations, Fuzzy Sets and
Systems, 122 (2001), 277{291.
[3] F. Chiclana, F. Herrera and E. Herrera-Viedma, Integrating three representation models in
fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Sys-
tems, 97 (1998), 33{48.
[4] K. O. Cogger and P. L. Yu, Eigenweight vectors and least-distance approximation for revealed
preference in pairwise weight ratios, Journal of Optimization Theory and Applications, 46
(1985), 483{491.
[5] G. Crawford and C. Williams, A note on the analysis of subjective judgement matrices,
Journal of Mathematical Psychology, 29 (1985), 387{405.
[6] Z. P. Fan, J. Ma, Y. P. Jiang, Y. H. Sun and L. Ma, A goal programming approach to group
decision making based on multiplicative preference relations and fuzzy preference relations,
European Journal of Operational Research, 174 (2006), 311{321.
[7] Z. P. Fan, J. Ma and Q. Zhang, An approach to multiple attribute decision making based on
fuzzy preference information on alternatives, Fuzzy Sets and Systems, 131 (2002), 101{106.
[8] Z. P. Fan, S. H. Xiao and G. F. Hu, An optimization method for integrating two kinds of
preference information in group decision-making, Computers & Industrial Engineering, 46
(2004), 329{335.
[9] Z. P. Fan and Y. Zhang, A goal programming approach to group decision-making with three
formats of incomplete preference relations, Soft Computing, 14 (2010), 1083{1090.
[10] E. Fernandez and J. C. Leyva, A method based on multiobjective optimization for deriving
a ranking from a fuzzy preference relation, European Journal of Operational Research, 154
(2004), 110{124.

[11] P. T. Harker, Alternative modes of questioning in the analytic hierarchy process, Mathemat-
ical Modelling, 9 (1987), 353{360.
[12] F. Herrera, E. Herrera-Viedma and F. Chiclana, Multiperson decision-making based on multi-
plicative preference relations, European Journal of Operational Research,129(2001),372{385.
[13] R. E. Jensen, An alternative scaling method for priorities in hierarchy structures, Journal of
Mathematical Psychology, 28 (1984), 317{332.
[14] J. Kacprzyk, Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems,
18 (1986), 105{118.
[15] J. Kacprzyk and M. Roubens, Non-Conventional Preference Relations in Decision-Making,
Springer, Berlin, 1988.
[16] S. Lipovetsky, The synthetic hierarchy method: An optimizing approach to obtaining priori-
ties in the AHP, European Journal of Operational Research, 93 (1996), 550{564.
[17] S. Lipovetsky and A. Tishler, Interval estimation of priorities in the AHP, European Journal
of Operational Research, 114 (1999), 153{164.
[18] H. Nurmi, Approaches to collective decision making with fuzzy preference relations, Fuzzy
Sets and Systems, 6 (1981), 249{259.
[19] S. A. Orlovsky, Decision-making with a fuzzy preference relation, Fuzzy Sets and Systems, 1
(1978), 155{167.
[20] T. L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.
[21] T. Tanino, Fuzzy preference orderings in group decision-making, Fuzzy Sets and Systems, 12
(1984), 117{131.
[22] T. Tanino, Fuzzy preference relation in group decision making, in: J. Kacprzyk, M. Roubens
(Eds.) Non-Conventional Preference Relation in Decision Making, Springer, Berlin, (1988),
54{71.
[23] T. Tanino, On group decision making under fuzzy preferences, in: J. Kacprzyk, M. Fedrizzi
(Eds.) Multiperson Decision Making Using Fuzzy Sets and Possibility Theory, Kluwer,
Netherlands, 1990, 172{185.
[24] L. G. Vargas, An overview of the analytic process and its applications, European Journal of
Operational Research, 48 (1990), 2{8.
[25] L. F.Wang and S. B. Xu, The Introduction to the Analytic Hierarchy Process, Chinese People
University Press, Beijing, 1990.
[26] Y. M.Wang and Z. P. Fan, Fuzzy preference relations: Aggregation and weight determination,
Computers & Industrial Engineering, 53 (2007), 163{172.
[27] Y. M. Wang and Z. P. Fan, Group decision analysis based on fuzzy preference relations:
Logarithmic and geometric least squares methods, Applied Mathematics and Computation,
194 (2007), 108{119.
[28] Y. M. Wang, Z. P. Fan and Z. S. Hua, A chi-square method for obtaining a priority vector
from multiplicative and fuzzy preference relations, European Journal of Operational Research,
182 (2007), 356{366.
[29] Y. M. Wang and C. Parkan, Multiple attribute decision making based on fuzzy preference
information on alternatives: Ranking and weighting, Fuzzy Sets and Systems, 153 (2005),
331{346.
[30] D. A. Wismer, Introduction to Nonlinear Optimization: A Problem Solving Approach, North-
Holland Company, New York, 1978.
[31] Y. J. Xu, L. Chen, K. W. Li and H. M. Wang, A chi-square method for priority derivation in
group decision making with incomplete reciprocal preference relations, Information Sciences,
36 (2015), 166{179.
[32] Y. J. Xu, L. Chen and H. M. Wang, A least deviation method for priority derivation in
group decision making with incomplete reciprocal preference relations, International Journal
of Approximate Reasoning, 66 (2015), 91{102.
[33] Y. J. Xu, Q. L. Da and L. H. Liu, Normalizing rank aggregation method for priority of a fuzzy
preference relation and its e ectiveness, International Journal of Approximate Reasoning, 50
(2009), 1287{1297.

[34] Y. J. Xu, Q. L. Da and H. M. Wang, A note on group decision-making procedure based on
incomplete reciprocal relations, Soft Computing, 15 (2011), 1289{1300.
[35] Y. J. Xu, J. N. D. Gupta and H. M.Wang, The ordinal consistency of an incomplete reciprocal
preference relation, Fuzzy Sets and Systems, 246(2014), 62{77.
[36] Y.J. Xu, K.W. Li and H.M. Wang, Incomplete interval fuzzy preference relations and their
applications, Computers & Industrial Engineering, 67 (2014), 93{103.
[37] Y. J. Xu, F. Ma, F. F. Tao and H. M. Wang, Some methods to deal with unacceptable
incomplete 2-tuple fuzzy linguistic preference relations in group decision making, Knowledge-
Based Systems, 56 (2014), 179{190.
[38] Y.J. Xu, R. Patnayakuni and H. M. Wang, Logarithmic least squares method to priority
for group decision making with incomplete fuzzy preference relations, Applied Mathematical
Modelling, 37 (2013), 2139{2152.
[39] Y. J. Xu, R. Patnayakuni and H. M. Wang, A method based on mean deviation for weight
determination from fuzzy preference relations and multiplicative preference relations, Inter-
national Journal of Information Technology & Decision Making, 11 (2012), 627{641.
[40] Y. J. Xu, R. Patnayakuni and H. M. Wang, The ordinal consistency of a fuzzy preference
relation, Information Sciences, 224 (2013), 152{164.
[41] Y. J. Xu and H. M. Wang, Eigenvector method, consistency test and inconsistency repairing
for an incomplete fuzzy preference relation, Applied Mathematical Modelling, 37 (2013),
5171{5183.
[42] Z. S. Xu, Generalized chi square method for the estimation of weights, Journal of Optimiza-
tion Theory and Applications, 107 (2000), 183{192.
[43] Z. S. Xu, Goal programming models for obtaining the priority vector of incomplete fuzzy
preference relation, International Journal of Approximate Reasoning, 36 (2004), 261{270.
[44] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transations on Fuzzy Systems, 15
(2007), 1179{1187.
[45] Z. S. Xu and Q. L. Da, A least deviation method to obtain a priority vector of a fuzzy
preference relation, European Journal of Operational Research, 164 (2005), 206{216.