# Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns

- 1.1k Downloads

## Abstract

Peng *et al* (Int J Uncertain Fuzziness Knowl Based Syst 15:29–41, 2007) introduced, by means of the credibility measure, two dominance relations on fuzzy variables, namely the first- and the second-order dominances.
In this paper, we characterize each of these dominance relations, and we justify that they satisfy six well-known properties of comparison methods. We propose a Game Theory approach for the determination of optimal portfolios when returns are fuzzy by introducing the set of best portfolios with respect to the first- and the second-order dominances. Based on the characterization of the first-order dominance, we numerically display some of the best portfolios of the classical set of portfolios of seven independent assets described by triangular fuzzy numbers.

## Keywords

credibility measure fuzzy variable first-order dominance second-order dominance set of best portfolios## References

- Bilbao-Terol A., Arenas-Parra M., Cañal-Fernández V. and Jiménez M. (2016): A sequential goal programming model with fuzzy hierarchies to sustainable and responsible portfolio selection problem.
*Journal of the Operational Research Society*,**67**(10):1259–1273.CrossRefGoogle Scholar - Brogan J.A. and Stidham S.Jr (2008): Non-separation in the mean-lower-partial-moment portfolio optimization problem,
*European Journal of Operational Research*,**184**(2):701–710.CrossRefGoogle Scholar - Chen I-F. and Tsaur R-C. (2016): Fuzzy portfolio selection using a weighted function of possibilistic mean and variance in business cycles,
*International Journal of Fuzzy Systems*,**18**(2):151–159.CrossRefGoogle Scholar - Cheng S-M, Munif A., Cheng G-S, Liu H-C. and Kuo B-C. (2012): Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights,
*Expert Systems With Applications*,**39**(7):6320–6334.CrossRefGoogle Scholar - Chu T-C and Tsao C-T. (2002): Ranking fuzzy numbers with an area between the centroid point and original point,
*Computers and Mathematics with Applications*,**43**(1–2):111–117.CrossRefGoogle Scholar - Dentcheva D. and Ruszczynski A. (2004): Semi-infinite probabilistic optimization: First-order stochastic dominance constraint,
*Optimization*,**53**(5–6):583–601.CrossRefGoogle Scholar - Detyniecki M. and Yager R. (2001): Ranking fuzzy numbers using α-weighted valuations,
*International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,***8**(5):573.CrossRefGoogle Scholar - Georgescu I. and Kinnunen J. (2013): A Risk Approach by Credibility Theory,
*Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operation Research Society of China*,**5**(4):399–416.Google Scholar - Grauer R. and Best M. (1991): Sensitivity analysis for mean-variance portfolio problems,
*Management Science*,**37**(8):980–989.CrossRefGoogle Scholar - Huang X. (2008): Mean-semivariance models for fuzzy portfolio selection,
*Journal of Computational and Applied Mathematics*,**217**(1):1–8.CrossRefGoogle Scholar - Konno H. and Suzuki K. (1995): A mean-variance-skewness optimization model,
*Journal of the Operations Research Society of Japan*,**38**:137–187.CrossRefGoogle Scholar - Kraus A. and Litzenberger R. (1976): Skewness preference and the valuation of risky assets,
*Journal of Finance*,**31**(4):1085–1100.Google Scholar - Kucukbay F. and Araz C. (2016): Portfolio selection problem: a comparison of fuzzy goal programming and linear physical programming,
*An International Journal of Optimization and Control: Theories and Applications (IJOCTA)*,**6**(2):121–128.Google Scholar - Li X., Qin Z. and Kar S. (2010): Mean-variance-skewness model for portfolio selection with fuzzy returns,
*European Journal of Operational Research*,**202**(1):239–247.CrossRefGoogle Scholar - Liu B. (2004): Uncertainty theory: An introduction to its axiomatics foundations.
*Springer-Verlag, Berlin*.CrossRefGoogle Scholar - Liu B. (2014): Uncertainty theory, 4th ed.,
*Springer-Verlag, Berlin*.Google Scholar - Liu B. and Liu Y.K. (2002): Expected value of fuzzy variable and fuzzy expected value models.
*IEEE Transactions on Fuzzy Systems*,**10**(4):445–450.CrossRefGoogle Scholar - Osuna E. (2012): Crossing points of distribution and a theorem that relates them to second order stochastic dominance.
*Statistics and Probability Letters*,**82**(4):758–764.CrossRefGoogle Scholar - Peng J., Mok H. and Tse W. (2005): Fuzzy dominance based on credibility distributions.
*Springer-Verlag, Berlin*, 295–303.Google Scholar - Peng J., Jiang Q. and Rao C. (2007): Fuzzy dominance: a new approach for ranking fuzzy variables via credibility measure.
*International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems*,**15**(supp02):29–41.CrossRefGoogle Scholar - Saborido R., Ruiz A., Bermúdez J., Vercher E. and Luque M. (2016): Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection.
*Applied Soft Computing*,**39**(c):48–63.CrossRefGoogle Scholar - Sadefo Kamdem J, Tassak DC, Fono LA (2012) Moments and semi-moments for fuzzy portfolio selection.
*Insurance Mathematics and Economics*,**51**(3):517–530Google Scholar - Saeidifar A. (2011): Application of weighting functions to the ranking of fuzzy numbers.
*Computers and Mathematics With Applications*,**62**(5):2246–2258.CrossRefGoogle Scholar - Samuelson P. (1970): The fundamental approximation theorem of portfolio analysis in terms of means, variances and higher moments,
*Reviews of Economics Studies*,**37**(4):537–542.CrossRefGoogle Scholar - Sengupta J. (1989): Portfolio decision as games,
*International Journal of Systems Science*,**20**(8):1323–1334.CrossRefGoogle Scholar - Sharpe W. (1971): A linear programming approximation for the general portfolio analysis problem,
*Journal of Financial and Quantitative Analysis*,**6**(5):1263–1275.CrossRefGoogle Scholar - Tobin J. (1965): The theory of portfolio selection, in F.H. Hahn and F.P.R. Brechling, eds.,
*The Theory of Interest Rates*, LondonGoogle Scholar - Wang X. and Kerre E. (2001): Reasonable properties for the ordering of fuzzy quantities,
*Fuzzy Sets and Systems*,**118**(3):375–385.CrossRefGoogle Scholar - Zadeh L.A. (1978): Fuzzy set as a basis of theory of possibility,
*Fuzzy Sets and Systems*,**1**(1):3–28.CrossRefGoogle Scholar