Restricted accessResearch articleFirst published online 2019-12-23
Consistency analysis and priority weights of multiplicative trapezoidal fuzzy preference relations based on multiplicative consistency and logarithmic least square model
Consistency and the priority vector are two important issues in preference relations. As one of preference relations, multiplicative trapezoidal fuzzy preference relation (MTFPR) is an effective form of vague and imprecise information when decision maker (DM) express his/her opinions by comparing alternatives or criteria with each other in group decision making (GDM). Therefore, it is meaningful to discuss the consistency and the method for deriving priority vector of MTFPRs. In this paper, we define multiplicative consistency of MTFPRs and investigate the necessary and sufficient conditions of multiplicative consistent MTFPRs. Some properties of multiplicative consistent MTFPRs are studied in detail. Based on the necessary and sufficient conditions of multiplicative consistent MTFPRs, two consistent measurement matrices (CMMs) are developed to define an acceptably multiplicative consistent MTFPR. A logarithmic least square model is further constructed for deriving a normalized trapezoidal fuzzy priority vector from a MTFPR. Three numerical examples including a GDM problem are analyzed to demonstrate the validity of the proposed models.
SaatyT.L., The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.
2.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
3.
Van LaarhovenP.J.M. and PedryczW., A fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems11 (1983), 229–241.
4.
BuckleyJ.J., Fuzzy hierarchical analysis, Fuzzy Sets and Systems17 (1985), 233–247.
5.
BuckleyJ.J., FeuringT. and HayashiY., Fuzzy hierarchical analysis revisited, European Journal of Operational Research129 (2001), 48–64.
6.
DongY.C., LiC.C., ChiclanaF. and Herrera-ViedmaE., Average-case consistency measurement and analysis of interval-valued reciprocal preference relations, Knowledgebased Systems114 (2016), 108–117.
7.
WeiY.Q., LiuJ.S. and WangX.Z., Concept of consistence and weights of the judgment matrix in the uncertain type of AHP, System Engineering Theory and Practice7(4) (1994), 16–22.
8.
ChenJ.F., HsiehH.N. and Hung DobQ., Evaluating teaching performance based on fuzzy AHP and comprehensive evaluation approach, Applied Soft Computing28 (2015), 100–108.
9.
KamvysiK., GotzamaniK., AndronikidisA. and GeorgiouA.C., Capturing and prioritizing students’ requirements for course design by embedding Fuzzy-AHP and linear programming in QFD, European Journal of Operational Research237 (2014), 1083–1094.
10.
ZhengG.Z., ZhuN., ZheT., ChenY. and SunB.H., Application of a trapezoidal fuzzy AHP method for work safety evaluation, Safety Science50 (2012), 228–239.
11.
LyP.T.M., LaiW.H., HsuC.W. and ShihF.Y., Fuzzy AHP analysis of Internet of Things (IoT) in enterprises, Technological Forecasting and Social Change136 (2018), 1–13.
12.
GongZ.W., LinY. and YaoT.X., Uncertain fuzzy preference relations and their applications, Springer, Berlin, 2013.
13.
WangT.C. and ChenY.H., Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP, Information Science178(19) (2008), 3755–3765.
14.
GogusO. and BoucherT.O., Strong transitivity, rationality and weak monotonicity in fuzzy pairwise comparisons, Fuzzy Sets and Systems94(1) (1998), 133–144.
15.
DuboisD., The role of fuzzy sets in decision sciences: Old techniques and new directions, Fuzzy Sets and Systems184(1) (2011), 3–28.
16.
XiaM.M. and ChenJ., Consistency and consensus improving methods for pairwise comparison matrices based on Abelian linearly ordered group, Fuzzy Sets and Systems266 (2015), 1–32.
17.
XuY.J. and WangH.M., A comment on “Incomplete fuzzy linguistic preference relations under uncertain environments”, Information Fusion20 (2014), 2–5.
18.
WangZ.J., Consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean, Information Science314 (2015), 169–183.
19.
GongZ.W., Least-square method to priority of the fuzzy preference relations with incomplete information, International Journal of Approximate Reasoning47 (2008), 258–264.
20.
TaninoT., Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems12 (1984), 117–131.
21.
LiuF., Acceptable consistency analysis of interval reciprocal comparison matrices, Fuzzy Sets and Systems160(18) (2009), 2686–2700.
22.
LiuF., ZhangW.G. and ZhangL.H., Consistency analysis of triangular fuzzy reciprocal preference relations, European Journal of Operational Research235 (2014), 718–726.
23.
WangY.M., YangJ.B. and XuD.L., A two-stage logarithmic goal programming method for generating weights from interval comparis on matrices, Fuzzy Sets and Systems152 (2005), 475–498.
24.
WangY.M., FanZ.P. and HuaZ.S., A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations, European Journal of Operational Research182 (2007), 356–366.
25.
CondeE. and PérezM.P.R., A linear optimization problem to derive relative weights using an interval judgement matrix, European Journal of Operational Research201 (2010), 537–544.
26.
LanJ.B., HuM.M., YeX.M. and SunS.Q., Deriving interval weights from an interval multiplicative consistent fuzzy preference relation, Knowledge-Based Systems26 (2012), 128–134.
27.
WangY.M. and ElhagT.M.S., A goal programming method for obtaining interval weights from an interval comparison matrix, European Journal of Operational Research177 (2007), 458–471.
28.
GencS., BoranF.E., AkayD. and XuZ.S., Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations, Information Sciences180 (2010), 4877–4891.
29.
LiuF., ZhangW.G. and FuJ.H., A new method of obtaining the priority weights from an interval fuzzy preference relation, Information Sciences185 (2012), 32–42.
30.
WangZ.J. and ChenY.G., Logarithmic least squares prioritization and completion methods for interval fuzzy preference relations based on geometric transitivity, Information Sciences289 (2014), 59–75.
31.
WangZ.J. and LiK.W., Goal programming approaches to deriving interval weights based on interval fuzzy preference relations, Information Sciences193 (2012), 180–198.
32.
XuZ.S. and ChenJ., Some models for deriving the priority weights from interval fuzzy preference relations, European Journal of Operational Research184 (2008), 266–280.
33.
ZhouY.Y., ChengL.H., ZhouL.G., ChenH.Y. and GeJ.Q., A group decision making approach for trapezoidal fuzzy preference relations with compatibility measure, Soft Computing21 (2017), 2709–2721.
34.
WuP., ZhouL.G., ZhengT. and ChenH.Y., A fuzzy group decision making and its application based on compatibility with multiplicative trapezoidal fuzzy preference relations, International Journal of Fuzzy Systems19(3) (2017), 683–701.
35.
ChenC.T., LinC.T. and HuangS.F., A fuzzy approach for supplier evaluation and selection in supply chain management, International Journal of Production Economics102(2) (2006), 289–301.
36.
SaatyT.L., A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology15(3) (1977), 234–281.
37.
XuY.J., WuD. and WangH.M., A Gower plot-based approach to ascertain and adjust the ordinal and additive inconsistencies for fuzzy linguistic preference relations, International Journal of Fuzzy Systems19(6) (2017), 2003–2019.
38.
QinJ.D., LiuX.W. and PedryczW., An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment, European Journal of Operational Research258(2) (2017), 626–638.
39.
ZhangZ., KouX.Y., YuW.Y. and GuoC.H., On priority weights and consistency for incomplete hesitant fuzzy preference relations, Knowledge-Based Systems143 (2018), 115–126.
40.
ZhangZ. and GuoC.H., Fusion of heterogeneous incomplete hesitant preference relations in group decision making, International Journal of Computational Intelligence Systems9(2) (2016), 245–262.
41.
ZhangZ. and GuoC.H., Deriving priority weights from intuitionistic multiplicative preference relations under group decision-making settings, Journal of the Operational Research Society68(12) (2017), 1582–1599.
42.
ZhangZ., KouX.Y. and DongQ.X., Additive consistency analysis and improvement for hesitant fuzzy preference relations, Expert Systems With Applications98 (2018), 118–128.
43.
LiC.C., RodríguezR.M., HerreraF., MartínezL. and DongY.C., Consistency of hesitant fuzzy linguistic preference relations: An interval consistency index, Information Sciences432 (2018), 347–361.
44.
WuP., ZhouL.G., ChenH.Y. and TaoZ.F., Additive consistency of hesitant fuzzy linguistic preference relation with a new expansion principle for hesitant fuzzy linguistic term sets, IEEE Transaction Fuzzy Systems27(4) (2019), 716–730.
45.
ZhangB.W., DongY.C. and Herrera-ViedmaE., Group decision making with heterogeneous preference structures: An automatic mechanism to support consensus reaching, Group Decision and Negotiation28 (2019), 585–617.
46.
WuP., WuQ., ZhouL.G., ChenH.Y. and ZhouH., A consensus model for group decision making under trapezoidal fuzzy numbers environment, Neural Computing & Applications31(2) (2019), 377–394.
47.
LiuY.T., ZhangH.J., WuY.Z. and DongY.C., Ranking range based approach to MADM under incomplete context and its application in venture investment evaluation, Technological and Economic Development of Economy. DOI: 10.3846/tede.2019.10296
48.
LiC.C., DongY.C., XuY.J., ChiclanaF., Herrera-ViedmaE. and HerreraF., An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and Fusion: Taxonomy and future directions, Information Fusion52 (2019), 143–156.
49.
XiaoJ., WangX.L. and ZhangH.J., Managing personalized individual semantics and consensus in linguistic distribution large-scale group decision making, Information Fusion53 (2020), 20–34.
50.
WuP., ZhouL.G., ChenH.Y. and TaoZ.F., Multi-stage optimization model for hesitant qualitative decision making with hesitant fuzzy linguistic preference relations, Applied Intelligence. DOI: 10.1007/s10489-019-01502-8.
51.
WuP., LiuS.H., ZhouL.G. and ChenH.Y., A fuzzy group decision making model with trapezoidal fuzzy preference relations based on compatibility measure and COWGA operator, Applied Intelligence48(1) (2018), 46–67.
