# 基于犹豫模糊集的养老模式选择决策研究Research on Decision-Making Decision of Pension Model Based on Hesitant Fuzzy Set

DOI: 10.12677/AR.2019.64008, PDF, HTML, XML, 下载: 173  浏览: 261  国家自然科学基金支持

Abstract: To effectively alleviate the burden of pensions brought about by the aging of the population, this paper provides an hesitant fuzzy multi-criteria decision analysis method based on the improved prospect theory, and applies it to the decision of the choice of pension mode. First, we construct an evaluation index system for the old-age model and establish a hesitant fuzzy evaluation matrix. Secondly, based on the irrational behavior of decision makers, the improved prospect theory is introduced to construct an improved foreground decision matrix. For the problem of definite power, an optimization model that comprehensively considers the main and objective factors is established. Then, combined with the improved foreground decision matrix and criterion weights, a hesitant fuzzy multi-criteria decision making method based on improved foreground theory is proposed. Based on this method, the decision-making problem of the pension model is studied, and an effective analysis method is provided for such problems, and the optimization decision of such problems is made through examples.

1. 引言

2. 犹豫模糊集及改进的前景理论

2.1. 犹豫模糊集

${h}_{A}\left(X\right)=\left\{〈x,{h}_{A}\left(x\right)〉|x\in X\right\}$

$s\left(h\left(x\right)\right)=\frac{1}{{l}_{h}}{\sum }_{\gamma \in h\left(x\right)}\gamma$ (1)

2.2. 改进前景理论

2.2.1. 前景理论

1979年，Kahneman和Tversky [16] 提出前景理论(Prospect Theory)，该理论由价值函数和概率权重函数共同组成，即：

$V=\underset{i=1}{\overset{n}{\sum }}w\left({p}_{i}\right)v\left(\Delta {x}_{i}\right)$ (2)

$w\left(p\right)=\left\{\begin{array}{l}\frac{{p}^{\chi }}{{\left({p}^{\chi }+{\left(1-p\right)}^{\chi }\right)}^{1/\chi }},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\Delta x\ge 0\\ \frac{{p}^{\delta }}{{\left({p}^{\delta }+{\left(1-p\right)}^{\delta }\right)}^{1/\delta }},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\Delta x<0\end{array}$ (3)

$v\left(\Delta x\right)=\left\{\begin{array}{l}{\left(\Delta x\right)}^{\alpha },\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\Delta x\ge 0\\ -\theta {\left(-\Delta x\right)}^{\beta },\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\Delta x<0\end{array}$ (4)

Figure 1. Outlook value function

2.2.2. 改进前景理论

$v\left(\Delta x\right)=\left\{\begin{array}{l}\zeta {\left(\Delta x\right)}^{\alpha },\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\Delta x\ge 0\\ -\theta {\left(-\Delta x\right)}^{\beta },\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\Delta x<0\end{array}$ (5)

1) $\alpha ,\beta$ 的取值范围扩大，由前景理论规定的 $0<\alpha ,\beta <1$ 扩展到 $\alpha ,\beta =1$ 以及 $\alpha ,\beta >1$ 参照效用曲线，依据不同的 $\alpha ,\beta$ 的取值范围，将决策者分为保守型、中间型及冒险型三种。

2) 引进新的参数 $\zeta$，若决策者相对损失来说对收益更敏感，则 $\zeta >1,\theta =1$ ；若决策者相对收益来说对损失更敏感，则 $\zeta =1,\theta >1$

3. 基于改进的前景理论的犹豫模糊多准则决策方法

3.1. 问题描述

3.2. 属性权重

1) 主观权重

${W}^{\text{*}}=\left({w}_{1}^{\text{*}},{w}_{2}^{\text{*}},\cdots ,{w}_{n}^{\text{*}}\right)$，且满足 $\underset{j=1}{\overset{n}{\sum }}{w}_{j}=1,{w}_{j}\ge 0,j=1,2,\cdots ,n$$l=\left({l}_{1},{l}_{2},\cdots ,{l}_{n}\right)$ 为专家依据经验判断得到的属性权重，满足 ${l}_{j}\ge 0,j=1,2,\cdots ,n$。依据最小交叉熵原理，构建如下的优化模型：

$\begin{array}{l}\mathrm{min}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}{w}_{j}^{*}\mathrm{ln}\left({w}_{j}^{*}/{l}_{j}\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}{w}_{j}^{*}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{j}^{*}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,2,\cdots ,n\end{array}$ (6)

2) 客观权重

$\begin{array}{l}\mathrm{max}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\underset{i=1}{\overset{m}{\sum }}\underset{k=1}{\overset{m}{\sum }}|{v}_{ij}-{v}_{kj}|{w}_{j}^{-}\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}{\left({w}_{j}^{-}\right)}^{2}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{w}_{j}^{-}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,2,\cdots ,n\end{array}$ (7)

3) 主客观组合权重

${w}_{j}=\frac{{w}_{j}^{*}{w}_{j}^{-}}{\underset{j=1}{\overset{n}{\sum }}{w}_{j}^{*}{w}_{j}^{-}}$ (8)

3.3. 决策方法

4. 养老模式选择的评价指标体系构建

Figure 2. Structural model of pension mode selection

5. 算例分析

Table 1. Expert decision matrix

$S=\left(\begin{array}{rrrrr}\hfill 0.8& \hfill 0.4& \hfill 0.4& \hfill 0.7& \hfill 0.75\\ \hfill 0.55& \hfill 0.6& \hfill 0.6& \hfill 0.6& \hfill 0.55\\ \hfill 0.4& \hfill 0.8& \hfill 0.85& \hfill 0.55& \hfill 0.4\end{array}\right)$ ,

1) 若决策者为冒险型，则 $0<\alpha ,\beta <1$，令 $\alpha =\beta =0.88$$\zeta =1,\theta =2.25$ [14]，得到改进的前景决策矩阵为：

$v=\left(\begin{array}{rrrrr}\hfill 0.2603& \hfill -0.5459& \hfill -0.5857& \hfill 0.1123& \hfill 0.2247\\ \hfill -0.1128& \hfill 0.0000& \hfill -0.0613& \hfill -0.0613& \hfill -0.0613\\ \hfill -0.5056& \hfill 0.2426& \hfill 0.2779& \hfill -0.2076& \hfill -0.4650\end{array}\right)$

2) 若决策者为中间型，则 $\alpha =\beta =1$，令 $\alpha =\beta =1$$\zeta =1,\theta =2.25$ [16]，得到改进的前景决策矩阵为：

$v=\left(\begin{array}{rrrrr}\hfill 0.2167& \hfill -0.4500& \hfill -0.4875& \hfill 0.0833& \hfill 0.1833\\ \hfill -0.0750& \hfill 0.0000& \hfill -0.0375& \hfill -0.0375& \hfill -0.0375\\ \hfill -0.4125& \hfill 0.2000& \hfill 0.2333& \hfill -0.1500& \hfill -0.3750\end{array}\right)$

3) 若决策者为保守型，则 $\alpha ,\beta >1$，令 $\alpha =\beta =1.21$$\zeta =1,\theta =2.25$ [16]，得到改进的前景决策矩阵为：

$v=\left(\begin{array}{rrrrr}\hfill 0.1571& \hfill -0.3209& \hfill -0.3536& \hfill 0.0495& \hfill 0.1284\\ \hfill -0.0367& \hfill 0.0000& \hfill -0.0159& \hfill -0.0159& \hfill -0.0159\\ \hfill -0.2889& \hfill 0.1426& \hfill 0.1719& \hfill -0.0849& \hfill -0.2574\end{array}\right)$

1) 若决策者为冒险型，解得最优准则权重为 $W={\left(0.3400,0.2334,0.1278,0.0947,0.2041\right)}^{\text{T}}$

2) 若决策者为中间型，则最优准则权重为 $W={\left(0.3437,0.2367,0.1313,0.0850,0.2033\right)}^{\text{T}}$

3) 若决策者为保守型，则最优准则权重为 $W={\left(0.3493,0.2420,0.1372,0.0702,0.2014\right)}^{\text{T}}$

Table 2. Sorting results

6. 结语

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