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

Abstract

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.

Appendices

Appendices

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.

1.1 Appendix A

Proof

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

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

(15)

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 $\mu (x_{0})$

1.2 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 [a, c] 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 [a, c]. 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 $E[\xi ]=\max \{s, 0\}+\min \{t, 0\}+e\cdot (t-s)/N$.

1.3 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 (a, b, c). 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 $|c-b|$.