Advertisement

Soft Computing

, Volume 23, Issue 15, pp 5975–5990 | Cite as

Comparing the magnitude of fuzzy intervals and fuzzy random variables from the standpoint of gradual numbers

  • Farid Aiche
  • Didier DuboisEmail author
Foundations
  • 110 Downloads

Abstract

Gradual numbers have been introduced to separate fuzziness (understood as gradation) from uncertainty in so-called fuzzy numbers. Then a fuzzy number can be viewed as a standard interval of functions, each interpreted as a gradual number. Gradual numbers are naturally met when representing probabilities of fuzzy events, midpoints of fuzzy intervals, etc. They can be viewed as a non-monotonic generalization of cumulative probability distributions. This paper presents three methods for comparing gradual numbers that generalize stochastic orderings to such non-monotonic functions. Then it proposes joint extensions of stochastic dominance and statistical preference to random fuzzy intervals when the fuzzy intervals are understood as intervals of gradual numbers. This approach, which combines known probabilistic orderings with known forms of interval orderings, can be viewed as a systematic way of constructing methods for ranking fuzzy random variables.

Keywords

Gradual numbers Stochastic dominance Statistical preference Probability Interval orderings Fuzzy intervals Fuzzy random variables 

Notes

Acknowledgements

The first author benefited from visit scholarships granted by University of Tizi-Ouzou, and E.N.P.E.I., Algiers.

Compliance with ethical standards

Conflict of interest

The authors declare that none of them has any conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Aiche F, Dubois D (2010) An extension of stochastic dominance to fuzzy random variables. In: Hüllermeier E et al (eds) Proceedings of international conference on information processing and management of uncertainty in knowledge-based systems (IPMU 2010), LNAI, vol 6178. Springer, pp 159–168Google Scholar
  2. Aiche F, Abbas M, Dubois D (2013) Chance-constrained programming with fuzzy stochastic coefficients. Fuzzy Optim Decis Mak 12(2):125–152MathSciNetzbMATHCrossRefGoogle Scholar
  3. Arrow KJ, Hurwicz L (1977) An optimality criterion for decision making under ignorance. In: Arrow KJ, Hurwicz L (eds) Studies in resource allocation processes. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  4. Boukezzoula R, Galichet S, Foulloy L, Elmasry M (2014) Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages. Fuzzy Sets Syst 257:67–84MathSciNetzbMATHCrossRefGoogle Scholar
  5. Campos L, Munoz A (1989) A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst 29:145–153MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chanas S, Nowakowski M (1988) Single value simulation of fuzzy variable. Fuzzy Sets Syst 25:43–57MathSciNetzbMATHCrossRefGoogle Scholar
  7. Chanas S, Zielinski P (1999) Ranking fuzzy real intervals in the setting of random sets—further results. Inf Sci 117:191–200zbMATHCrossRefGoogle Scholar
  8. Chanas S, Delgado M, Verdegay JL, Vila MA (1993) Ranking fuzzy real intervals in the setting of random sets. Inf Sci 69:201–217zbMATHCrossRefGoogle Scholar
  9. Chateauneuf A, Cohen M, Tallon J-M (2009) Decision under risk: the classical expected utility model. In: Bouyssou D (ed) Decision-making process, chap 8. ISTE and Wiley, London, pp 363–382Google Scholar
  10. Couso I, Dubois D (2009) On the variability of the concept of variance for fuzzy random variables. IEEE Trans Fuzzy Syst 17:1070–1080CrossRefGoogle Scholar
  11. Couso I, Dubois D (2012) An imprecise probability approach to joint extensions of stochastic and interval orderings. In: Proceedings of international conference on information processing and management of uncertainty in knowledge-based systems, IPMU 2012, vol 3, pp 388–399Google Scholar
  12. Couso I, Sánchez L (2011) Upper and lower probabilities induced by a fuzzy random variable. Fuzzy Sets Syst 165(1):1–23MathSciNetzbMATHCrossRefGoogle Scholar
  13. Couso I, Dubois D, Sánchez L (2014) Random sets and random fuzzy sets as ill-perceived random variables. Springer briefs in computational intelligence. Springer, BerlinzbMATHGoogle Scholar
  14. Couso I, Moral S, Sánchez L (2015) The behavioral meaning of the median. Inf Sci 294:127–138MathSciNetzbMATHCrossRefGoogle Scholar
  15. David H (1963) The method of paired comparisons. Griffin’s statistical monographs & courses, vol 12. Griffin, LondonGoogle Scholar
  16. De Baets B, De Meyer H (2008) On the cycle-transitive comparison of artificially coupled random variables. Int J Approx Reason 47:306–322MathSciNetzbMATHCrossRefGoogle Scholar
  17. Delgado M, Martín-Bautista MJ, Sánchez D, Vila MA (2002) A probabilistic definition of a non convex fuzzy cardinality. Fuzzy Sets Syst 126(2):41–54zbMATHCrossRefGoogle Scholar
  18. Denoeux T (2009) Extending stochastic order to belief functions on the real line. Inf Sci 179:1362–1376zbMATHCrossRefGoogle Scholar
  19. Destercke S, Couso I (2015) Ranking of fuzzy intervals seen through the imprecise probabilistic lens. Fuzzy Sets Syst 278:20–39MathSciNetzbMATHCrossRefGoogle Scholar
  20. Dubois D (2006) Possibility theory and statistical reasoning. Comput Stat Data Anal 51:47–69MathSciNetzbMATHCrossRefGoogle Scholar
  21. Dubois D (2011) The role of fuzzy sets in decision sciences: old techniques and new directions. Fuzzy Sets Syst 184(1):3–28MathSciNetzbMATHCrossRefGoogle Scholar
  22. Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30:183–224MathSciNetzbMATHCrossRefGoogle Scholar
  23. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24:279–300MathSciNetzbMATHCrossRefGoogle Scholar
  24. Dubois D, Prade H (1988) Possibility theory. Plenum Press, New YorkzbMATHCrossRefGoogle Scholar
  25. Dubois D, Prade H (1991) Random sets and fuzzy interval analysis. Fuzzy Sets Syst 42:87–101MathSciNetzbMATHCrossRefGoogle Scholar
  26. Dubois D, Prade H (2008) Gradual elements in a fuzzy set. Soft Comput 12:165–175zbMATHCrossRefGoogle Scholar
  27. Dubois D, Kerre E, Mesiar R, Prade H (2000) Fuzzy interval analysis. In: Dubois D, Prade H (eds) Fundamentals of fuzzy sets, The handbooks of fuzzy sets series. Kluwer, Boston, pp 483–581zbMATHCrossRefGoogle Scholar
  28. Ferson S, Ginzburg LR (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54:133–144CrossRefGoogle Scholar
  29. Ferson S, Hajagos JG (2004) Arithmetic with uncertain numbers: rigorous and (often) best possible answers. Reliab Eng Syst Saf 85:135–152CrossRefGoogle Scholar
  30. Ferson S, Ginzburg L, Kreinovich V, Myers DM, Sentz K (2003) Constructing probability boxes and Dempster–Shafer structures. Technical report, Sandia National Laboratories, USAGoogle Scholar
  31. Fishburn P (1987) Interval orderings. Wiley, New YorkGoogle Scholar
  32. Fortemps P, Roubens M (1996) Ranking and defuzzification methods based on area compensation. Fuzzy Sets Syst 82:319–330MathSciNetzbMATHCrossRefGoogle Scholar
  33. Fortin J, Dubois D, Fargier H (2008) Gradual numbers and their application to fuzzy interval analysis. IEEE Trans Fuzzy Syst 16:388–402CrossRefGoogle Scholar
  34. Friedman M, Ma M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets Syst 96:201–209MathSciNetzbMATHCrossRefGoogle Scholar
  35. Gil MA, López-Díaz M, Ralescu DA (2006) Overview on the development of fuzzy random variables. Fuzzy Sets Syst 157(19):2546–2557MathSciNetzbMATHCrossRefGoogle Scholar
  36. Herencia JA (1996) Graded sets and points: a stratified approach to fuzzy sets and points. Fuzzy Sets Syst 77:191–202MathSciNetzbMATHCrossRefGoogle Scholar
  37. Kruse R, Meyer KD (1987) Statistics with vague data. D. Reidel, DordrechtzbMATHCrossRefGoogle Scholar
  38. Kwakernaak H (1978) Fuzzy random variables I. Definitions and theorems. Inf Sci 15:1–29MathSciNetzbMATHCrossRefGoogle Scholar
  39. Martin TP, Azvine B (2013) The X-mu approach: fuzzy quantities, fuzzy arithmetic and fuzzy association rules. In: IEEE symposium on foundations of computational intelligence (FOCI), Singapore, pp 24–29Google Scholar
  40. Montes I, Destercke S (2017) Comonotonicity for sets of probabilities. Fuzzy Sets Syst 328:1–34MathSciNetzbMATHCrossRefGoogle Scholar
  41. Montes I, Miranda E, Montes S (2017) Imprecise stochastic orders and fuzzy rankings. Fuzzy Optim Decis Mak 16(3):297–327MathSciNetzbMATHCrossRefGoogle Scholar
  42. Molodtsov D (1999) Soft set theory—first results. Comput Math Appl 37(4/5):19–31MathSciNetzbMATHCrossRefGoogle Scholar
  43. Negoita CV, Ralescu DA (1975) Applications of fuzzy sets to systems analysis. Birkhauser, BaselzbMATHCrossRefGoogle Scholar
  44. Ogura Y, Li S-M, Ralescu DA (2001) Set defuzzification and Choquet integral. Int J Uncertain Fuzziness Knowl Based Syst 9(1):1–12MathSciNetzbMATHCrossRefGoogle Scholar
  45. Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114:409–420MathSciNetzbMATHCrossRefGoogle Scholar
  46. Rocacher D, Bosc P (2005) The set of fuzzy rational numbers and flexible querying. Fuzzy Sets Syst 155(3):317–339MathSciNetzbMATHCrossRefGoogle Scholar
  47. Sánchez D, Delgado M, Vila MA, Chamorro-Martinez J (2012) On a non-nested level-based representation of fuzziness. Fuzzy Sets Syst 192:159–175MathSciNetzbMATHCrossRefGoogle Scholar
  48. Smets P (2005) Belief functions on real numbers. Int J of Approx Reason 40:181–223MathSciNetzbMATHCrossRefGoogle Scholar
  49. Wang X, Kerre E (2001) Reasonable properties for the ordering of fuzzy quantities (2 parts). Fuzzy Sets Syst 118:375–406zbMATHCrossRefGoogle Scholar
  50. Williamson RC, Downs T (1990) Probabilistic arithmetic I. Numerical methods for calculating convolutions and dependency bounds. Int J Approx Reason 4(2):89–158MathSciNetzbMATHCrossRefGoogle Scholar
  51. Yager RR (1978) Ranking fuzzy subsets over the unit interval. In: Proceedings of IEEE international conference on decision and control, pp 1435–1437Google Scholar
  52. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Ecole Nationale Préparatoire aux Etudes d’Ingéniorat (E.N.P.E.I.) Badji MokhtarRouiba, AlgiersAlgeria
  2. 2.Université Mouloud Mammeri de Tizi-OuzouTizi-OuzouAlgeria
  3. 3.IRITCNRS and Université de ToulouseToulouse Cedex 9France

Personalised recommendations