Abstract
In this paper, we first analyze several existing ranking approaches for a set of interval numbers which are based on the possibility degree matrix. We show by counterexamples that these methods are not appropriate to rank a set of interval numbers from two viewpoints. Furthermore, we investigate the relationship between the possibility degree matrix and the reciprocal fuzzy preference matrix and analyze the reason that leads to the irrational ranking result. Finally, we propose an improved ranking algorithm for a set of interval numbers based on the Boolean matrix. Two numerical examples and a practical application are provided to illustrate the feasibility and effectiveness of our proposed method.
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Ahn BS (2006) The uncertain OWA aggregation with weighting functions having a constant level of orness. Int J Intell Syst 21:469–483
Ahn BS (2007) The OWA aggregation with uncertain descriptions on weights and input arguments. IEEE Trans Fuzzy Syst 15:1130–1134
Baas S, Kwakernaak H (1977) Rating and ranking of multiple-aspect alternatives using fuzzy sets. Automatica 13:47–58
Chiclana F, Herrera F, Herrera-Viedma E (2001) Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst 122:277–291
Facchinetti G, Ricci RG, Muzzioli S (1998) Note on ranking fuzzy triangular numbers. Int J Intell Syst 13:613–622
Genc S, Boran FE, Akay D, Xu ZS (2010) Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations. Inf Sci 180:4877–4891
Herrera F, Herrera-Viedma E (2000) Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst 115:67–82
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
Kao C, Liu ST (2001) Fractional programming approach to fuzzy weighted average. Fuzzy Sets Syst 120:435–444
Kundu S (1997) Min-transitivity of fuzzy leftness relationship and its application to decision making. Fuzzy Sets Syst 86:357–367
Lan JB, Sun Q, Chen QM, Wang ZX (2013) Group decision making based on induced uncertain linguistic OWA operators. Decis Support Syst 55:296–303
Li DQ, Wang JY, Li HX (2009) Note on “On the normalization of interval and fuzzy weights”. Fuzzy Sets Syst 160:2722–2725
Mitchell HB, Schaefer PA (2000) On ordering fuzzy numbers. Int J Intell Syst 15:981–993
Nakahara Y, Sasaki M, Gen M (1992) On the linear programming problems with interval coefficients. Int J Comput Ind Eng 23:301–304
Nakahara Y (1998) User oriented ranking criteria and its application to fuzzy mathematical programming problems. Fuzzy Sets Syst 94:275–286
Park JH, Gwak MG, Kwun YC (2011) Uncertain linguistic harmonic mean operators and their applications to multiple attribute group decision making. Computing 93:47–64
Pavlačka O (2014) On various approaches to normalization of interval and fuzzy weights. Fuzzy Sets Syst 243:110–130
Peng B, Ye CM, Zeng SZ (2012) Uncertain pure linguistic hybrid harmonic averaging operator and generalized interval aggregation operator based approach to group decision making. Knowl-Based Syst 36:175–181
Qiu JB, Wei YL, Wu LG (2016) A novel approach to reliable control of piecewise affine systems with actuator faults. IEEE Trans Circuits Syst II: Express Briefs. doi:10.1109/TCSII.2016.2629663
Sengupta A, Pal TK (2000) On comparing interval numbers. Eur J Oper Res 127:28–43
Wan SP, Dong JY (2014) A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J Comput Syst Sci 80:237–256
Wang YM, Yang JB, Xu DL (2005) A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets Syst 152:475–498
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
Wei GW (2008) Dependent uncertain linguistic OWA operator. Lecture Notes in Computer Science 5009:156–163
Wei GW, Zhao XF, Lin R, Wang HJ (2013) Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Appl Math Model 37:5277–5285
Wei YL, Qiu JB, Lam HK, Wu LG (2016a) Approaches to T–S fuzzy-affine-model-based reliable output feedback control for nonlinear itô stochastic systems. IEEE Trans Fuzzy Syst. doi:10.1109/TFUZZ.2016.2566810
Wei YL, Qiu JB, Shi P, Lam HK (2016b) A new design of H-infinity piecewise filtering for discrete-time nonlinear time-varying delay systems via T–S fuzzy affine models. IEEE Trans Syst Man Cybern: Syst. doi:10.1109/TSMC.2016.2598785
Xu ZS, Da QL (2002) The uncertain OWA operator. Int J Intell Syst 17:569–575
Xu ZS (2004) Uncertain lingistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Inf Sci 168:171–184
Xu ZS (2006) Induced uncertain linguistic OWA operators applied to group decision making. Inf Fusion 7:231–238
Xu ZS (2008) On multi-period multi-attribute decision making. Knowl-Based Syst 21:164–171
Xu ZS, Chen J (2008) Some models for deriving the priority weights from interval fuzzy preference relations. Eur J Oper Res 184:266–280
Xu ZS, Yager RR (2010) Power-geometric operators and their use in group decision making. IEEE Trans Fuzzy Syst 18:94–105
Zhou LG, Chen HY, Merigó JM, Gil-Lafuente AM (2012) Uncertain generalized aggregation operators. Expert Syst Appl 39:1105–1117
Acknowledgements
The authors wish to express their gratitude to the anonymous referees and the Editor-in-Chief, Professor Antonio Di Nola, for their kind suggestions and helpful comments in revising the paper.
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This article dees not contain any studies with human participants performed by any of the authors.
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Communicated by V. Loia.
This work is supported by Grants from the National Natural Science Foundation of China (10971243, 61472043).
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Li, D., Zeng, W. & Yin, Q. Novel ranking method of interval numbers based on the Boolean matrix. Soft Comput 22, 4113–4122 (2018). https://doi.org/10.1007/s00500-017-2625-4
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DOI: https://doi.org/10.1007/s00500-017-2625-4