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On product of positive L-R fuzzy numbers and its application to multi-period portfolio selection problems

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With the wide applications of fuzzy theory in optimization, fuzzy arithmetic attracts great attention due to its inevitability in solution process. However, the complexity of the Zadeh extension principle significantly reduces the practicability of fuzzy optimization technology. In this paper, we prove some important properties on positive L-R fuzzy numbers, and propose a new calculation method for the product of multiple positive L-R fuzzy numbers. Furthermore, a numerical integral-based simulation algorithm (NISA) is proposed to approximate the expected value, variance and skewness of the product of positive L-R fuzzy numbers. As applications, a fuzzy multi-period utility maximization model for portfolio selection problem is considered. For handling the large number of multiplications on L-R fuzzy numbers during the optimization process, a genetic algorithm integrating NISA is designed. Finally, some numerical experiments are presented to demonstrate the advantages of NISA. The results greatly enrich the fuzzy arithmetic methods and promote the practicability of fuzzy optimization technology.

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  1. Abbasbandy, S., & Amirfakhrian, M. (2006). The nearest trapezoidal form of a generalized left right fuzzy number. International Journal of Approximate Reasoning, 43(2), 166–178.

  2. Abbasbandy, S., & Asady, B. (2004). The nearest trapezoidal fuzzy number to a fuzzy quantity. Applied Mathematics and Computation, 156(2), 381–386.

  3. Ahmed, F., & Kilic, K. (2019). Fuzzy analytic hierarchy process: A performance analysis of various algorithms. Fuzzy Sets and Systems, 362, 110–128.

  4. Anile, A. M., Deodato, S., & Privitera, G. (1995). Implementing fuzzy arithmetic. Fuzzy Sets and Systems, 72(4), 239–250.

  5. Chanas, S. (2001). On the interval approximation of a fuzzy number. Fuzzy Sets and System, 122(2), 353–356.

  6. Chang, P. T., & Hung, K. C. (2006). $\alpha $-cut fuzzy arithmetic: simplifying rules and a fuzzy function optimization with a decision variable. IEEE Transactions on Fuzzy Systems, 14(4), 331–338.

  7. Chou, C. C. (2003). The canonical representation of multiplication operation on triangular fuzzy numbers. Computers and Mathematics with Applications, 45(10), 1601–1610.

  8. Dimitar, P. F., & Ronald, R. Y. (1997). Operations on fuzzy numbers via fuzzy reasoning. Fuzzy Sets and Systems, 91(2), 137–142.

  9. Dubois, D., & Prade, H. (1980). Fuzzy sets and systems: Theory and applications. New York: Academic Press.

  10. Grzegorzewski, P. (2002). Nearest interval approximation of a fuzzy number. Fuzzy Sets and System, 130(3), 321–330.

  11. Guo, S. N., Yu, L. A., Li, X., & Kar, S. (2016). Fuzzy multi-period portfolio selection with different investment horizons. European Journal of Operational Research, 254(3), 1026–1035.

  12. Holčapek, M., & Štěpnička, M. (2014). MI-algebras: A new framework for arithmetics of (extensional) fuzzy numbers. Fuzzy Sets and Systems, 257, 102–131.

  13. Huang, X. X. (2011). Mean-risk model for uncertain portfolio selection. Fuzzy Optimization and Decision Making, 10(1), 71–89.

  14. Lai, K. K., Yu, L. A., & Wang, S. Y. (2006). Mean–variance–skewness–kurtosis-based portfolio optimization. In Proceedings of IEEE international multi-symposium in computer and computational sciences (IMSCCS 2006) (Vol. 2, pp. 292–297).

  15. Li, D., & Peter, S. (2011). A portfolio selection model using fuzzy returns. Fuzzy Optimization and Decision Making, 10(2), 167–191.

  16. Li, X., Guo, S. N., & Yu, L. A. (2015). Skewness of fuzzy numbers and its applications in portfolio selection. IEEE Tansactions on Fuzzy Systems, 23(6), 2135–2143.

  17. Liu, B. D., & Iwamura, K. (1998). Chance constrained programming with fuzzy parameters. Fuzzy Sets and Systems, 94(2), 227–237.

  18. Liu, B. D., & Liu, Y. K. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 10(4), 445–450.

  19. Liu, S., Wang, S. Y., & Qiu, W. (2003). Mean–variance–skewness model for portfolio selection with transaction costs. International Journal of Systems Science, 34(4), 255–262.

  20. Ma, M., Kandel, A., & Friedman, M. (2000). A new approach for defuzzication. Fuzzy Sets and Systems, 111(3), 351–356.

  21. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.

  22. Triesch, E. (1993). On the convergence of product-sum series of $L$-$R$ fuzzy numbers. Fuzzy Sets and Systems, 53(2), 189–192.

  23. Wang, S. Y., & Zhu, S. S. (2002). On fuzzy portfolio selection problems. Fuzzy Optimization and Decision Making, 1(4), 361–377.

  24. Wu, X. L., & Liu, Y. K. (2012). Optimizing fuzzy portfolio selection problems by parametric quadratic programming. Fuzzy Optimization and Decision Making, 11(4), 411–449.

  25. Yu, L. A., Wang, S. Y., Wen, F. H., & Lai, K. K. (2012). Genetic algorithm-based multi-criteria project portfolio selection. Annals of Operations Research, 197(1), 71–86.

  26. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.

  27. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3–28.

  28. Zeng, W., & Li, H. (2007). Weighted triangular approximation of fuzzy numbers. International Journal of Approximate Reasoning, 46(1), 137–150.

  29. Zhang, Y. Y., Li, X., & Guo, S. N. (2017). Portfolio selection problems with Markowitz’s mean–variance framework: A review of literature. Fuzzy Optimization and Decision Making, 17(2), 125–158.

  30. Zhou, J. D., Li, X., Kar, S., Zhang, G. Q., & Yu, H. T. (2017). Time consistent fuzzy multi-period rolling portfolio optimization with adaptive risk aversion factor. Journal of Ambient Intelligence and Humanized Computing, 8(5), 651–666.

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The authors would like to thank the two anonymous referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Nos. 71722007, 71433001, 71931001), the Fundamental Research Funds for the Central Universities (No. XK1802-5), the Research Grants Council of Hong Kong (Nos. 15210815, 17301519), IMR and RAE Research Fund from Faculty of Science, the University of Hong Kong.

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Correspondence to Sini Guo.

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In this section, we present a proof on properties (i) and (ii), give the computation procedure of fuzzy simulation, and introduce a granular computing method to derive fuzzy returns based on historical data.

Appendix A


\(\mathbf (i) \): If \(x_{0}\in (a_{1}a_{2}, b_{1}b_{2})\), according to the Zadeh extension principle, we have

$$\begin{aligned} \mu (x_{0})=\sup \limits _{(x_{1}, x_{2}) \in L_{1}\cup L_{2}\cup R_{2}}\{\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}|x_{1}x_{2}=x_{0}\}. \end{aligned}$$

For any \((x_{1}, x_{2})\in L_{2}\) with \(x_{1}x_{2}=x_{0}\), take \(x^{*}_{1}=x_{0}/b_{2}\) and \(x^{*}_{2}=b_{2}\). It is obvious that \(x_{1}\le x^{*}_{1}\le b_{1}\) and \((x^{*}_{1}, x^{*}_{2})\in L_{1}\). Since \(\xi _{1}\) is a L-R fuzzy number, \(\mu _{1}(x)\) is increasing when \(x\in (a_{1}, b_{1})\) with \(\mu _{1}(x^{*}_{1})\ge \mu _{1}(x_{1})\) and \(\mu _{2}(x^{*}_{2})\ge \mu _{2}(x_{2})\), which implies that

$$\begin{aligned} \min \{\mu _{1}(x^{*}_{1}), \mu _{2}(x^{*}_{2})\}\ge \min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}. \end{aligned}$$

Taking all \((x_{1}, x_{2})\in L_{2}\) into consideration, we have

$$\begin{aligned}&\sup \limits _{(x_{1}, x_{2}) \in L_{1}}\{\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}|x_{1}x_{2}=x_{0}\}\ge \\&\quad \sup \limits _{(x_{1}, x_{2}) \in L_{2}}\{\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}|x_{1}x_{2}=x_{0}\}. \end{aligned}$$

Similarly, we can prove that

$$\begin{aligned}&\sup \limits _{(x_{1}, x_{2}) \in L_{1}}\{\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}|x_{1}x_{2}=x_{0}\}\ge \\&\quad \sup \limits _{(x_{1}, x_{2}) \in R_{2}}\{\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}|x_{1}x_{2}=x_{0}\}. \end{aligned}$$

According to Eq. (15), we have

$$\begin{aligned} \mu (x_{0})=\sup \limits _{(x_{1}, x_{2}) \in L_{1}}\{\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}|x_{1}x_{2}=x_{0}\}. \end{aligned}$$

Now, we employ the reduction to absurdity to prove that there exist \(x_1 \in [a_{1}, b_{1}]\) and \(x_2 \in [a_{2}, b_{2}]\) such that \(x_{1}x_{2}=x_{0}\) and \(\mu _{1}(x_{1})=\mu _{2}(x_{2})=\mu (x_{0})\). Denote the \(\gamma \)-level sets of \(\xi _{i}\) as \([a_{i1}(\gamma ), a_{i2}(\gamma )]\), \(i=1, 2\). Suppose that \((x^{*}_{1}, x^{*}_{2})\in L_{1}\) satisfying \(x^{*}_{1}x^{*}_{2}=x_{0}\), and \(\min \{\mu _{1}(x^{*}_{1}), \mu _{2}(x^{*}_{2})\}\)\(=\mu (x_{0})\). If \(\mu _{1}(x^{*}_{1})\ne \mu _{2}(x^{*}_{2})\), without loss of generalization, assume \(\mu _{1}(x^{*}_{1})=\gamma _{1}\), \(\mu _{2}(x^{*}_{2})=\gamma _{2}\) and \(\gamma _{1}<\gamma _{2}\). Then take \(x^{**}_{1}=(1+\varepsilon )x^{*}_{1}\) and \(x^{**}_{2}=x^{*}_{2}/(1+\varepsilon )\), where \(\varepsilon \) is a small enough positive number satisfying

$$\begin{aligned} 0<\varepsilon <\min \{-1+a_{11}((\gamma _{1}+\gamma _{2})/2)/x^{*}_{1}, -1+x^{*}_{2}/a_{21}((\gamma _{1}+\gamma _{2})/2)\} \end{aligned}$$

which ensures \((x^{**}_{1}, x^{**}_{2})\in L_{1}\) and \(\mu _{2}(x^{**}_{2})>(\gamma _{1}+\gamma _{2})/2>\)\(\mu _{1}(x^{*}_{1})\). Since \(\mu _{i}(x)\) is increasing in interval \((a_{i}, b_{i})\), \(i=1, 2\), then we have \(\min \{\mu _{1}(x^{**}_{1}), \mu _{2}(x^{**}_{2})\}>\min \{\mu _{1}(x^{*}_{1}),\)\( \mu _{2}(x^{*}_{2})\}\), which contradicts with \(\min \{\mu _{1}(x^{*}_{1}), \mu _{2}(x^{*}_{2})\}=\mu (x_{0})\) for the reason that \(\mu (x_{0})\) is the supremum of \(\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}\) over \(L_{1}\). This process illustrates that the value of \(\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}\) can be increased by getting \(\mu _{1}(x_{1})\) close to \(\mu _{2}(x_{2})\) with the above operations (See Fig. 5). If and only if \(\mu _{1}(x_{1})=\mu _{2}(x_{2})\), \(\min \{\mu _{1}(x_{1}), \mu _{2}(x_{2})\}\) arrives at its supremum over \(L_{1}\). Hence, for any \(x_{0}\in (a_{1}a_{2}, b_{1}b_{2})\), \(\mu (x_{0})=\mu _{1}(x_{1})=\mu _{2}(x_{2})\).

\(\mathbf (ii) \) If \(x_{0}\in (b_{1}b_{2}, c_{1}c_{2})\), the conclusion can be proved in the similar way. The proof is complete. \(\square \)

Fig. 5

Iteration process for \(\min (\mu _{1}(x_{1}), \mu _{2}(x_{2}))\) approaching \(\mu (x_{0})\)

Appendix B

This appendix gives a basic computation procedure of fuzzy simulation (Guo et al. 2016). Suppose that \(\xi =(a, b, c)\) is a triangular fuzzy number with credibility function \(\nu (x)\), where \(\nu (x)=\mu (x)/2\). The steps for computing \(E[\xi ]\) is shown as follows. Firstly, randomly select N points \(y_{1}, y_{2}, \ldots , y_{N}\) in [ac] and calculate their credibilities \(\nu _{1}, \nu _{2}, \ldots , \nu _{N}\). Then, set \(e=0\), \(s=\min \{y_{1}, y_{2}, \ldots , y_{N}\}\) and \(t=\max \{y_{1}, y_{2}, \ldots , y_{N}\}\). Secondly, randomly select a number r from [ac]. If \(r>0\), set \(e\rightarrow e+\mathbf{Cr }\{\xi \ge \textit{r}\}\). Otherwise, set \(e\rightarrow e-\mathbf{Cr }\{\xi \le \textit{r}\}\). Here \(\mathbf{Cr }\{\xi \ge \textit{r}\}\) and \(\mathbf{Cr }\{\xi \le \textit{r}\}\) are credibility measure given by

$$\begin{aligned} \mathbf{Cr }\{\xi \ge \textit{r}\}=\left\{ \begin{array}{ll} \max \{\nu _{k}|y_{k}\ge r\}, &{}\quad \mathrm{if}~~\max \{\nu _{k}|y_{k}\ge r\}< 0.5 \\ 1-\max \{\nu _{k}|y_{k}< r\}, &{} \quad \mathrm{if}~~\max \{\nu _{k}|y_{k}\ge r\}\ge 0.5, \end{array} \right. \end{aligned}$$
$$\begin{aligned} \mathbf{Cr }\{\xi \le \textit{r}\}=\left\{ \begin{array}{ll} \max \{\nu _{k}|y_{k}\le r\}, &{} \quad \mathrm{if}~~\max \{\nu _{k}|y_{k}\le r\}< 0.5 \\ 1-\max \{\nu _{k}|y_{k}> r\}, &{} \quad \mathrm{if}~~\max \{\nu _{k}|y_{k}\le r\}\ge 0.5. \end{array} \right. \end{aligned}$$

Thirdly, repeat the second operation N times to update e constantly and finally output \(E[\xi ]=\max \{s, 0\}+\min \{t, 0\}+e\cdot (t-s)/N\).

Appendix C

This appendix introduces a granular computing method to derive fuzzy returns from historical data (Zhou et al. 2017). Denote \(r_{1},r_{2},\ldots ,r_{N}\) as the historical returns. We employ the following methods to generate the triangular fuzzy number (abc). Firstly, set \(b=\sum _{i=1}^{N}r_{i}/N \), and calculate the membership degree of \( r_{i} \) by \(\mu (r_{i})=\mathbf 1 _{(-\infty , b)}(r_{i})\cdot (r_{i}-a)/(b-a)+(1-\mathbf 1 _{(-\infty , b)}(r_{i}))\cdot (b-r_{i})/(c-b)\), where \(\mathbf 1 _{(-\infty , b)}(r_{i})=1\) if \(r_{i}<b\), \(\mathbf 1 _{(-\infty , b)}(r_{i})=0\) otherwise. Secondly, assume \( \alpha \) is a given positive number, then determine a by maximizing the value of \(\sum _{a\le r_{i}< b}\mu (r_{i})\cdot \exp (-\alpha |b-a|),\) where \( \sum _{a\le r_{i}< b}\mu (r_{i}) \) is intended for covering most of the data points with \(r_{i}< b\), while \( \exp (-\alpha |b-a|) \) is applied to minimize the support length \( |b-a| \). Finally, determine c by maximizing \(\sum _{b\le r_{i}\le c}\mu (r_{i})\cdot \exp (-\alpha |c-b|),\) where \(\sum _{b\le r_{i}\le c}\mu (r_{i})\) is used to cover most of the data points with \(r_{i}> b\) and \( \exp (-\alpha |c-b|) \) is intended for minimizing \(|c-b|\).

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Li, X., Jiang, H., Guo, S. et al. On product of positive L-R fuzzy numbers and its application to multi-period portfolio selection problems. Fuzzy Optim Decis Making 19, 53–79 (2020). https://doi.org/10.1007/s10700-019-09308-6

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  • Fuzzy sets
  • Fuzzy arithmetic
  • Positive L-R fuzzy number
  • Multi-period portfolio selection