Robust Versions of the Lower and Upper Possibilistic Mean - Variance Models for the One Period or Two Periods Cases

Öz

It is easy to use possibility theory in modeling incomplete information. Robust optimization is an important tool when there is parameter uncertainty. Thus, in this study, we propose robust versions of the lower and upper possibilistic mean - variance (MV) models when there are multiple possibility distribution scenarios. Here, we use entropy as a diversification constraint. In addition, we reduce these robust versions to concave maximization problems. Furthermore, we generalize them for two periods portfolio selection problem by using fuzzy addition and multiplication. On the other hand, these generalizations are not concave maximization problems. Finally, we give an illustrative example by using different solvers in Gams modeling system.

Anahtar Kelimeler

• Entropy
• Fuzzy Arithmetic
• Portfolio Selection
• Possibility Theory
• Robust Optimization

Bir ya da İki Periyotlu Durumlar için Alt ve Üst Olabilirlik Ortalama - Varyans Modellerinin Dayanıklı Versiyonları

Öz

Tam olmayan bilgiyi modellemede olabilirlik teorisini kullanmak kolaydır. Parametre belirsizliği olduğunda dayanıklı optimizasyon önemli bir araçtır. Bu nedenle bu çalışmada, birden çok olabilirlik dağılımı senaryosu olduğunda alt ve üst olabilirlik ortalama - varyans (OV) modellerinin dayanıklı versiyonları önerilmiştir. Burada entropi çeşitlendirme kısıdı olarak kullanılmıştır. Bununla birlikte bu dayanıklı versiyonlar konkav maksimizasyon problemlerine indirgenmiştir. Üstelik bunlar, iki periyotlu portföy seçimi problemine bulanık toplama ve çarpma kullanılarak genelleştirilmiştir. Öte yandan bu genelleştirmeler, konkav maksimizasyon problemleri değildir. Son olarak, Gams modelleme sisteminde farklı çözücüler kullanılarak açıklayıcı bir örnek verilmiştir.

Anahtar Kelimeler

• Entropi
• Bulanık Aritmetik
• Portföy Seçimi
• Olabilirlik Teorisi
• Dayanıklı Optimizasyon

Kaynakça

1. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353.
2. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3-28.
3. Dubois, D., & Prade, H. (1988). Possibility Theory. Plenum Press, New York.
4. Dubois, D. (2006). Possibility theory and statistical reasoning. Computational Statistics & Data Analysis, 51(1), 47-69.
5. Fullér, R., & Harmati, I. Á. (2018). On possibilistic dependencies: a short survey of recent developments. Soft Computing Based Optimization and Decision Models, 261-273.
6. Carlsson, C., Fullér, R., & Majlender, P. (2002). A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems, 131(1), 13-21.
7. Zhang, W. G. (2007). Possibilistic mean–standard deviation models to portfolio selection for bounded assets. Applied Mathematics and Computation, 189(2), 1614-1623.
8. Zhang, W. G., Wang,Y. L., Chen, Z. P., & Nie, Z. K. (2007). Possibilistic mean-variance models and efficient frontiers for portfolio selection problem. Information Sciences, 177(13), 2787–2801.

9. Zhang, W. G., & Xiao, W. L. (2009). On weighted lower and upper possibilistic means and variances of fuzzy numbers and its application in decision. Knowledge and Information Systems, 18, 311-330.
10. Li, X., Guo, S., & Yu, L. (2015). Skewness of fuzzy numbers and its applications in portfolio selection. IEEE Transactions on Fuzzy Systems, 23(6), 2135-2143.
11. Yang, X. Y., Chen, S. D., Liu, W. L., & Zhang, Y. (2022). A multi-period fuzzy portfolio optimization model with short selling constraints. International Journal of Fuzzy Systems, 24(6), 2798–2812.
12. Gong, X., Min, L., & Yu, C. (2022). Multi-period portfolio selection under the coherent fuzzy environment with dynamic risk-tolerance and expected-return levels. Applied Soft Computing, 114, 108104.
13. Gupta, P., Mehlawat, M. K., Yadav, S., & Kumar, A. (2020). Intuitionistic fuzzy optimistic and pessimistic multi-period portfolio optimization models. Soft Computing, 24(16), 11931-11956.
14. Liu, Y. J., & Zhang, W. G. (2019). Possibilistic moment models for multi-period portfolio selection with fuzzy returns. Computational Economics, 53(4), 1657-1686.
15. Liu, Y. J., & Zhang, W. G. (2018). Fuzzy portfolio selection model with real features and different decision behaviors. Fuzzy Optimization and Decision Making, 17(3), 317-336.
16. Liagkouras, K., & Metaxiotis, K. (2018). Multi-period mean–variance fuzzy portfolio optimization model with transaction costs. Engineering Applications of Artificial Intelligence, 67, 260-269.
17. Liu, Y. J., Zhang, W. G., & Zhao, X. J. (2018). Fuzzy multi-period portfolio selection model with discounted transaction costs. Soft Computing, 22(1), 177-193.
18. Liu, Y. J., & Zhang, W. G. (2015). A multi-period fuzzy portfolio optimization model with minimum transaction lots. European Journal of Operational Research, 242(3), 933-941.
19. Zhang, W. G., Liu, Y. J., & Xu, W. J. (2012). A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs. European Journal of Operational Research, 222(2), 341-349.
20. Liu, Y. J., Zhang, W. G., & Xu, W. J. (2012). Fuzzy multi-period portfolio selection optimization models using multiple criteria. Automatica, 48(12), 3042-3053.
21. Roth, M., Franke, G., & Rinderknecht, S. (2022). A comprehensive approach for an approximative integration of nonlinear-bivariate functions in mixed-integer linear programming models. Mathematics, 10(13), 2226.
22. Göktaş, F. (in press). Mathematical analyses of the upper and lower possibilistic mean – variance models and their extensions to multiple scenarios. Journal of Advanced Research in Natural and Applied Sciences.
23. Corazza, M., & Nardelli, C. (2019). Possibilistic mean–variance portfolios versus probabilistic ones: the winner is... Decisions in Economics and Finance, 42(1), 51-75.
24. Lam, W. S., Lam, W. H., & Jaaman, S. H. (2021). Portfolio Optimization with a Mean–Absolute Deviation–Entropy Multi-Objective Model. Entropy, 23(10), 1266.
25. Ali, M. Y., Sultana, A., & Khan, A. F. M. K. (2016). Comparison of fuzzy multiplication operation on triangular fuzzy number. IOSR Journal of Mathematics, 12(4-I), 35-41.