Advertisement

Soft Computing

, Volume 14, Issue 7, pp 773–782 | Cite as

Ranking of fuzzy numbers by a new metric

  • Tofigh AllahviranlooEmail author
  • M. Adabitabar Firozja
Original Paper

Abstract

In this paper, a new approach for comparison among fuzzy numbers based on new metric distance (D TM) is proposed. All reasonable properties of ranking function are proved. At first, the distance on the interval numbers based on convex hall of endpoints is proposed. The existing distance measures for interval numbers, (Bardossy and Duckstein in Fuzzy rule-based modeling with applications to geophysical, biological and engineering systems. CRC press, Boca Raton, 1995; Diamond in Info Sci 46:141–157, 1988; Diamond and Korner in Comput Math Appl 33:15–32, 1997; Tran and Duckstein in Fuzzy Set Syst 130:331–341, 2002; Diamond and Tanaka Fuzzy regression analysis. In: Slowinski R (ed) Fuzzy sets in decision analysis, operations research and statistics. Kluwer, Boston, pp 349–387, 1998) do not satisfy the properties of a metric distance, while the proposed distance does. It is extended to fuzzy numbers and its properties are proved in detail. Finally, we compare the proposed definition with some of the known ones.

Keywords

Metric distance Fuzzy numbers Ranking 

References

  1. Abbasbandy S, Asady B (2006) Ranking of fuzzy numbers by sign distance. Inf Sci 176:2405–2416zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bardossy A, Duckstein L (1995) Fuzzy rule-based modeling with applications to geophysical, biological and engineering systems. CRC press, Boca RatonzbMATHGoogle Scholar
  3. Bardossy A, Hagaman R, Duckstein L, Bogardi I (1992) Fuzzy least squares regression: theory and application. In: Kacprzyk J, Fedrizi M (eds) Fuzzy l. Omnitech Press, Warsaw and Physica-Verlag, Heidelberg, pp 181–193Google Scholar
  4. Bortolan G, Degan R (2006) A review of some method for ranking fuzzy sets. Fuzzy Sets Syst 15:1–19CrossRefGoogle Scholar
  5. Caldas M, Jafari S (2005) θ-Compact fuzzy topological spaces. Chaos Solitons Fractals 25:229–232Google Scholar
  6. Chang SL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cybern 2:30–34zbMATHMathSciNetGoogle Scholar
  7. Cheng CH (1998) A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst 95:307–317zbMATHCrossRefGoogle Scholar
  8. Li DF, Yang JB (2004) Fuzzy linear programming technique for multiattribute group decision making in fuzzy environments. Inf Sci 158:263–275zbMATHCrossRefGoogle Scholar
  9. Diamond P (1998) Fuzzy least squares. Inf Sci 46:141–157CrossRefMathSciNetGoogle Scholar
  10. Diamond P, Korner R (1997) Extended fuzzy linear models and least squares estimates. Comput Math Appl 33:15–32zbMATHCrossRefMathSciNetGoogle Scholar
  11. Diamond P, Tanaka H (1998) Fuzzy regression analysis. In: Slowinski R (ed) Fuzzy sets in decision analysis, operations research and statistics. Kluwer, Boston, pp 349–387Google Scholar
  12. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkzbMATHGoogle Scholar
  13. Elnaschie MS (2004) A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos Solitons Fractals 19:209–236CrossRefGoogle Scholar
  14. Elnaschie MS (2006a) Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics. Chaos Solitons Fractals 27:297–330CrossRefGoogle Scholar
  15. Elnaschie MS (2006b) Superstrings, entropy and the elementary particles content of the standard model. Chaos Solitons Fractals 29:48–54CrossRefGoogle Scholar
  16. Feng G, Chen G (2005) Adaptive control of discrete-time chaotic systems: a fuzzy control approach. Chaos Solitons Fractals 23:459–467Google Scholar
  17. Fortemps P, Roubens M (1996) Ranking and defuzzification methods based on area compaensation. Fuzzy Sets Syst 82:319–330zbMATHCrossRefMathSciNetGoogle Scholar
  18. Goetschel R, Vaxman W (1981) A pseudometric for fuzzy sets and certain related result. J Math Anal Appl 81:507–523zbMATHCrossRefMathSciNetGoogle Scholar
  19. Goetschel R, Vaxman W (1983) Topological properties of fuzzy numbers. Fuzzy Sets Syst 10:87–99zbMATHCrossRefGoogle Scholar
  20. Huang H, Wu CH (2009) On the triangle inequalities in fuzzy metric spaces. Inf Sci (in press)Google Scholar
  21. Jiang W, Guo-Dong Q, Bin D (2005) H variable universe adaptive fuzzy control for chaotic system. Chaos Solitons Fractals 24:1075–1086zbMATHCrossRefMathSciNetGoogle Scholar
  22. Ma M, Kandel A, Friedman M (2000a) A new approach for defuzzification. Fuzzy Sets Syst 111:351–356zbMATHCrossRefMathSciNetGoogle Scholar
  23. Ma M, Kandel A, Friedman M (2000b) Correction to “a new approach for defuzzification”. Fuzzy Sets Syst 128:133–134CrossRefMathSciNetGoogle Scholar
  24. Modarres M, Nezhad SS (2001) Ranking fuzzy numbers by preference ratio. Fuzzy Sets Syst 118:429–439zbMATHCrossRefGoogle Scholar
  25. Negoita CV, Ralescu DA (1975) Applications of fuzzy sets to systems analysis. Wiley, Now YorkzbMATHGoogle Scholar
  26. Ekel PY, Fernando H, Schuffner Neto (2006) Algorithms of discrete optimization and their application to problems with fuzzy cofficients. Inf Sci 176:2846–2868Google Scholar
  27. Slavka B (2003) Alpa-bounds of fuzzy numbers, Inf Sci 152:237–266Google Scholar
  28. Tanaka Y, Mizuno Y, Kado T (2005) Chaotic dynamics in the Friedman equation. Chaos Solitons Fractals 24:407–422Google Scholar
  29. Tran L, Duckstein L (2002) Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy Sets Syst 130:331–341zbMATHCrossRefMathSciNetGoogle Scholar
  30. Wang X, Kree EE (2001) Reasonable properties for the ordering of fuzzy quantities I. Fuzzy Sets Syst 118:375–385zbMATHCrossRefGoogle Scholar
  31. Xu R, Li C (2001) Multidimensional least-squares fitting with a fuzzy model. Fuzzy Sets Syst 119:215–223zbMATHCrossRefGoogle Scholar
  32. Yang MS, Ko (1997) On cluster-wise fuzzy regression analysis. IEEE Transaction on Systems. Man Cybern B 27:1–13Google Scholar
  33. Yao JS, Wu K (2000) Ranking fuzzy numbers based on decomposition principle and signed distance. Fuzzy Sets Syst 116:275–288zbMATHCrossRefMathSciNetGoogle Scholar
  34. Xu ZS, Chen J (2009) An iteractive metchod for fuzzy multiple attribute group decision making. Inf Sci (in press)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of mathematicsIslamic Azad UniversityQaemshahrIran

Personalised recommendations