Skip to main content
SpringerLink
Log in

On possibility-degree formulae for ranking interval numbers

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Since interval numbers are used to evaluate the opinions of decision makers and express the weights of alternatives in various decision making problems, it is requisite to give a feasible method to rank them. In the present paper, the two existing possibility degree formulae for ranking interval numbers are proved to be equivalent. A generalized possibility degree formula is proposed by considering the attitude of decision makers with a prescribed function. Some known possibility degree formulae for ranking interval numbers can be recovered by choosing a special function. The proposed method is applied to uniformly define the weak transitivity of interval multiplicative and additive reciprocal preference relations. Numerical examples are carried out to illustrate the new formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Baas SM, Kwakernaak H (1977) Rating and ranking multiple-aspect alternatives using fuzzy sets. Automatica 13:47–58

    Article  MathSciNet  MATH  Google Scholar 

  • Brunelli M (2015) Introduction to the analytic hierarchy process. Springer, New York

    Book  MATH  Google Scholar 

  • Dubois D, Prade H (1983) Ranking of fuzzy numbers in the setting of possibility theory. Inf Sci 30:183–224

    Article  MathSciNet  MATH  Google Scholar 

  • Da QL, Liu XW (1999) Interval number linear programming and its satisfactory solution. Syst Eng Theory Pract 19:3–7

    Google Scholar 

  • Facchinetti G, Ricci RG, Muzzioli S (1998) Note on ranking fuzzy triangular numbers. Int J Intell Syst 13:613–622

    Article  Google Scholar 

  • Guo KH, Mou YJ (2010) Multiple attribute decision-making method with intervals based on possibility degree matrix. J Comput Appl 32:218–222

    Google Scholar 

  • Herrera F, Martinez L, Sanchez PJ (2005) Managing non-homogenous information in group decision making. Eur J Oper Res 166:115–132

    Article  MATH  Google Scholar 

  • Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48:219–225

    Article  MATH  Google Scholar 

  • Kundu S (1997) Min-transitivity of fuzzy leftness relationship and its application to decision making. Fuzzy Sets Syst 86:357–367

    Article  MathSciNet  MATH  Google Scholar 

  • Lipovetsky S, Tishler A (1999) Interval estimation of priorities in the AHP. Eur J Oper Res 114:153–164

    Article  MATH  Google Scholar 

  • Liu F (2009) Acceptable consistency analysis of interval reciprocal multiplicative preference relations. Fuzzy Sets Syst 160:2686–2700

    Article  MATH  Google Scholar 

  • Li KW, Wang ZJ, Tong XY (2016) Acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices. Eur J Oper Res 250:628–638

    Article  MathSciNet  MATH  Google Scholar 

  • Moorse RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Moorse RE (1995) Methods and applications of interval analysis. SIAM, Philadelphia (Second Printing)

    Google Scholar 

  • Nakahara Y, Sasaki M, Gen M (1992) On the linear programming problems with interval coefficients. Comput Ind Eng 23(1–4):301–304

    Article  Google Scholar 

  • Nakahara Y (1998) User oriented ranking criteria and its application to fuzzy mathematical programming problems. Fuzzy Sets Syst 94:275–286

    Article  MathSciNet  MATH  Google Scholar 

  • Saaty TL, Vargas LG (1987) Uncertainty and rank order in the analytic hierarchy process. Eur J Oper Res 32(1):107–117

    Article  MathSciNet  MATH  Google Scholar 

  • Sengupta A, Pal TK (2000) On comparing interval numbers. Eur J Oper Res 127:28–43

    Article  MathSciNet  MATH  Google Scholar 

  • Sengupta A, Pal TK (2009) Fuzzy preference ordering of interval numbers in decision problems. Springer-Verlag, Berlin Heidelberg

    Book  MATH  Google Scholar 

  • Sengupta A, Pal TK, Chakraborty D (2001) Interpretation of inequality constraints involving interval coecients and a solution to interval programming. Fuzzy Sets Syst 119:129–138

    Article  MATH  Google Scholar 

  • Sugihara K, Ishii H, Tanaka H (2004) Interval priorities in AHP by interval regression analysis. Eur J Oper Res 158:745–754

    Article  MathSciNet  MATH  Google Scholar 

  • Sevastianov P (2007) Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory. Inf Sci 177:4645–4661

    Article  MathSciNet  MATH  Google Scholar 

  • Sun HB, Yao WX (2010) Comments on methods for ranking interval numbers. J Syst Eng 25:305–312

    Google Scholar 

  • Tanino T (1988) Fuzzy preferene orderings in group decision making. In: Kacprzyk J, Roubens M (eds) Non-conventional preference relations in decision making. Springer-Verlag, Berlin, pp 54–71

    Chapter  Google Scholar 

  • Wang X, Kerre EE (2001) Reasonable properties for the ordering of fuzzy quantities (I) (II). Fuzzy Sets Syst 112:387–405

    Article  MathSciNet  MATH  Google Scholar 

  • Wang YM, Yang JB, Xu DL (2005a) Interval weight generation approaches based on consistency test and interval comparison matrices. Appl Math Comput 167:252–273

    MathSciNet  MATH  Google Scholar 

  • Wang YM, Yang JB, Xu DL (2005b) A two-stage logarithmic goal programming method for generating weights from interval multiplicative preference relations. Fuzzy Sets Syst 152:475–498

    Article  MATH  Google Scholar 

  • Wang ZJ, Li KW (2012) Goal programming approaches to deriving interval weights based on interval fuzzy preference relations. Inf Sci 193:180–198

    Article  MathSciNet  MATH  Google Scholar 

  • Wang ZJ (2016) A two-stage linear goal programming approach to eliciting interval weights from additive interval fuzzy preference relations. Soft Comput 20:2721–2732

    Article  MATH  Google Scholar 

  • Xu ZS, Da QL (2003) Possibility degree method for ranking interval numbers and its application. J Syst Eng 18:67–70

    Google Scholar 

  • Xu ZS (2004) On compatibility of interval fuzzy preference matrices. Fuzzy Optim Decis Mak 3:217–225

  • Xu ZS, Chen J (2008) Some models for deriving the priority weights from interval fuzzy preference relations. Eur J Oper Res 184:266–280

    Article  MathSciNet  MATH  Google Scholar 

  • Xia MM, Chen J (2015) Studies on interval multiplicative preference relations and their application to group decision making. Group Decis Negot 24:115–144

    Article  Google Scholar 

  • Yager RR (1999) On ranking fuzzy numbers using valuations. Int J Int Syst 14:1249–1268

    Article  MATH  Google Scholar 

  • Yager RR, Detyniecki M, Bouchon-Meunier B (2001) A context-dependent method for ordering fuzzy numbers using probabilities. Inf Sci 138:237–255

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Q, Fan ZP, Pan DH (1999) A ranking approach with possibilities for multiple attribute decision making problems with interval. Control Decis 14(6):703–706

    Google Scholar 

  • Zhang Z (2016) Logarithmic least squares approaches to deriving interval weights, rectifying inconsistency and estimating missing values for interval multiplicative preference relations. Soft Comput. doi:10.1007/s00500-016-2049-6

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Nos. 71571054, 71201037), the Guangxi Natural Science Foundation (No. 2014GXNSFAA118013), and the Guangxi Natural Science Foundation for Distinguished Young Scholars (No. 2016GXNSFFA380004).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Guangxi University, Nanning, 530004, Guangxi, People’s Republic of China

    Fang Liu, Li-Hua Pan, Zu-Lin Liu & Ya-Nan Peng

Authors
  1. Fang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-Hua Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zu-Lin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ya-Nan Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fang Liu.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Additional information

Communicated by V. Loia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, F., Pan, LH., Liu, ZL. et al. On possibility-degree formulae for ranking interval numbers. Soft Comput 22, 2557–2565 (2018). https://doi.org/10.1007/s00500-017-2509-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-017-2509-7

Keywords

Search

Navigation