﻿ 基于前景理论和证据推理的区间灰数多属性决策
 计算机系统应用  2019, Vol. 28 Issue (9): 33-40 PDF

Multi-Attribute Decision-Making Method of Interval Grey Number Based on Prospect Theory and Evidential Reasoning
XIONG Ning-Xin, WANG Ying-Ming
School of Economics and Management, Fuzhou University, Fuzhou 350108, China
Foundation item: National Natural Science Foundation of China (61773123)
Abstract: In terms of the problem of Multi-Attribute Decision-Making (MADM) under the uncertain background, the interval grey number MADM method based on the prospect theory and Evidential Reasoning (ER) is put forward. Firstly, considering the property of interval grey number, an improved distance formula of interval grey number is developed. On this basis, taking the positive and negative ideal solutions as the reference point, the distance set between the attribute value and the positive and negative ideal solution is calculated. Secondly, the decision makers’ psychological risk factors are introduced into the interval grey number MADM to develop a prospect value function based on the improved distance formula. Thirdly, the alternatives are selected by ER and the comparison rule of interval numbers. Finally, an illustrative example shows that the proposed method has rationality and feasibility.
Key words: interval grey number     prospect theory     evidential reasoning     distance measure     multi-attribute decision-making

1 预备知识 1.1 区间灰数

 $\forall \otimes \Rightarrow {d^ * } \in D,\theta \in D,D = [a,b],p\left( \theta \right)$

1)若 $D$ 为一个离散集合, 称灰数 $\otimes$ 为离散型灰数,记为 $\forall \otimes \Rightarrow {d^ * } \in D,D = [{d_1},{d_2}, \cdots ,{d_n}],\theta \in D,p\left( \theta \right)$ .

2)既有下界又有上界的灰数称为连续型灰数,也称为区间灰数. 若 $D$ 为一个连续集合(即区间集合),记 $\forall \otimes \Rightarrow {d^ * } \in D,D = [a,b],\theta \in D,p\left( \theta \right)$ , $a$ $b$ 分别称为区间灰数 $\otimes$ 的下界和上界.

 $\tilde \otimes = \alpha a + (1 - \alpha )b,\alpha \in [0,1]$

 $\hat \otimes = \frac{1}{2}\left( {a + b} \right)$ (1)

 $d\left( {{ \otimes _1},{ \otimes _2}} \right) = 2\sqrt[{ - 1/2}]{{{{({a_1} - {a_2})}^2} + {{({b_1} - {b_2})}^2}}}$ (2)

 $\frac{{{{\hat \otimes }_1}}}{{1 + l\left( {{ \otimes _1}} \right)}} > \frac{{{{\hat \otimes }_{\rm{2}}}}}{{1 + l\left( {{ \otimes _2}} \right)}}$ (3)

${ \otimes _1} \succ { \otimes _2}$ , 否则 ${ \otimes _1} \prec { \otimes _2}$ .

 ${ \otimes _1} \cup { \otimes _2} = \left\{ {\zeta \left| {\zeta \in \left[ {{a_1},{b_1}} \right]{\text{或}}\zeta \in \left[ {{a_2},{b_2}} \right]} \right.} \right\}.$

 ${ \otimes _1} \cap { \otimes _2} = \left\{ {\zeta \left| {\zeta \in \left[ {{a_1},{b_1}} \right]{\text{且}}\zeta \in \left[ {{a_2},{b_2}} \right]} \right.} \right\}.$
1.2 前景理论

 $\upsilon (\Delta x) = \left\{ \begin{gathered} \;\;\;\;\;\Delta {x^\alpha }\;\;\;\;\;\;\;\;\Delta x \ge 0 \\ - \sigma {( - \Delta x)^\beta }\;\;\;\;\Delta x < 0\; \\ \end{gathered} \right.$ (4)

1.3 证据推理理论

Step 1.在证据推理框架中, 方案 ${a_i}\left( {i = 1,2, \cdots ,m} \right)$ 在属性 ${c_j}\left( {j = 1,2, \cdots ,n} \right)$ 下的属性值表示证据, 在使用证据推理算法前需要将证据转换为统一的信度结构.定义一组评价集 $H = \left\{ {{H_n}|{H_n} \prec {H_{n + 1}},n = 1,2, \cdots ,N} \right\}$ , 假设 ${\beta _{n,i}}$ 表示方案在属性 ${c_j}$ 下被评价为等级 ${H_n}$ 的信任度, 方案在属性 ${c_j}$ 下的置信度分布评价表示为:

 $S({e_i}) = \left\{ {({H_n},{\beta _{n,i}})|n = 1,2, \cdots ,N} \right\}\\ 0 \le {\beta _{n,i}} \le 1,\sum\limits_{n = 1}^N {{\beta _{n,i}}} \le 1$ (5)

Step 2. 计算基本概率分配及未分配概率.

 ${m_{n,i}} = {w_i}{\beta _{n,i}},n = 1,2, \cdots ,N$ (6)

 ${m_{H,i}} = 1 - \sum\limits_{n = 1}^N {{m_{n,i}}} = 1 - {w_i}\sum\limits_{n = 1}^N {{\beta _{n,i}}}$ (7)
 ${m_{H,i}} = {\bar m_{H,i}} + {\tilde m_{H,i}}$ (8)

 ${\bar m_{H,i}} = 1 - {w_i}$ (9)
 ${\tilde m_{H,i}} = {w_i}\left(1 - \sum\limits_{n = 1}^N {{\beta _{n,i}}} \right)$ (10)

Step 3. 采用解析合成算法[17]对基本概率分配函数实现证据集成. 具体算法如下所示:

 $\left\{ \begin{split} &\left\{ {{H_n}} \right\}:{m_n} = k\left[ {\prod\limits_{i = 1}^L {\left( {{m_{n,i}} + {{\bar m}_{H,i}} + {{\tilde m}_{H,i}}} \right) - \prod\limits_{i = 1}^L {\left( {{{\bar m}_{H,i}} + {{\tilde m}_{H,i}}} \right)} } } \right], \\ & n = 1,2, \cdots ,N \\ &\left\{ H \right\}:{\tilde m_H} = k\left[ {\prod\limits_{i = 1}^L {\left( {{{\bar m}_{H,i}} + {{\tilde m}_{H,i}}} \right) - \prod\limits_{i = 1}^L {{{\bar m}_{H,i}}} } } \right]\\ &\left\{ H \right\}:{\bar m_H} = k\left[ {\prod\limits_{i = 1}^L {{{\bar m}_{H,i}}} } \right]\\ &k = {\left[ {\sum\limits_{n = 1}^N {\prod\limits_{i = 1}^L {\left( {{m_{n,i}} + {{\bar m}_{H,i}} + {{\tilde m}_{H,i}}} \right) - \left( {N - 1} \right)\prod\limits_{i = 1}^L {\left( {{{\bar m}_{H,i}} + {{\tilde m}_{H,i}}} \right)} } } } \right]^{ - 1}}\\ &\left\{ {{H_n}} \right\}:{\beta _n} = \frac{{{m_n}}}{{1 - {{\bar m}_H}}},n = 1,2, \cdots ,N\\ &\left\{ H \right\}:{\beta _H} = \frac{{{{\tilde m}_H}}}{{1 - {{\bar m}_H}}} \end{split}\right.$ (11)

${\beta _n}$ 表示方案被评价为等级 ${H_n}$ 的置信度, ${\beta _H}$ 表示分配到确定评价等级的不确定性, 综合置信度为:

 $S(y) = \left\{ {\left( {{H_n},{\beta _n}} \right),n = 1,2, \cdots ,N} \right\}$

2 决策模型 2.1 问题描述

2.2 改进的区间灰数距离公式

 $d({ \otimes _1},{ \otimes _2}) = {2^{ - 1/2}}{[{( - 3 - 0)^2} + {(2 - 1)^2}]^{1/2}} = \sqrt 5$
 $d({ \otimes _3},{ \otimes _2}) = {2^{ - 1/2}}{[{(1 - 0)^2} + {(4 - 1)^2}]^{1/2}} = \sqrt 5$

 $d({ \otimes _1},{ \otimes _2}) = |{\hat \otimes _1} - {\hat \otimes _{\rm{2}}}| + \frac{1}{2}|l({ \otimes _1}) - l({ \otimes _{\rm{2}}})|$ (12)

 $\begin{split} &{d^2}\left( {{ \otimes _1},{ \otimes _2}} \right) \\ & = {\int_0^1 {\int_0^1 {\left\{ {\left[ {{{\hat \otimes }_{\rm{1}}} + \alpha \left( {l\left( {{ \otimes _1}} \right)} \right)} \right]} \right. - \left. {\left[ {{{\hat \otimes }_{\rm{2}}} + \beta \left( {l\left( {{ \otimes _2}} \right)} \right)} \right]} \right\}} } ^2}d\alpha d\beta \\ & - \int_0^1 {\int_0^1 {\left\{ {\left[ {{{\hat \zeta }_1} + \alpha l\left( {{\zeta _1}} \right)} \right]} \right.} } - {\left. {\left[ {{{\hat \zeta }_1} + \beta l\left( {{\zeta _1}} \right)} \right]} \right\}^2}d\alpha d\beta \\ & = {\left[ {\left( {{{\hat \otimes }_{\rm{1}}}} \right) - \left( {{{\hat \otimes }_{\rm{2}}}} \right)} \right]^2} + \frac{1}{{12}}\left\{ {{{[l\left( {{ \otimes _1}} \right)]}^2} + {{[l\left( {{ \otimes _2}} \right)]}^2}} \right\} - \frac{1}{6}{[l\left( {{\zeta _1}} \right)]^2} \\ \end{split}$ (13)

(1)非负性: $d\left( {{ \otimes _1},{ \otimes _2}} \right) \ge {\rm{0}}$ .

(2) 对称性: $d\left( {{ \otimes _1},{ \otimes _2}} \right){\rm{ = }}d\left( {{ \otimes _{\rm{2}}},{ \otimes _{\rm{1}}}} \right)$ .

(3) 三角不等式: $d\left( {{ \otimes _1},{ \otimes _2}} \right) \le d\left( {{ \otimes _1},{ \otimes _{\rm{3}}}} \right){\rm{ + }}d\left( {{ \otimes _{\rm{2}}},{ \otimes _{\rm{3}}}} \right)$ .

2.3 决策模型构建

Step 1. 数据规范化

 ${\bar x_{ij}}\left( {\bar \otimes } \right) = \frac{{{{\bar x}_{ij}}}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_{ij}}} \right)}^2}} } }},\;\;i = 1,2, \cdots ,n$ (14)
 ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} _{ij}}\left( {\bar \otimes } \right) = \frac{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_{ij}}}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{{\bar x}_{ij}}} \right)}^2}} } }},\;\;i = 1,2, \cdots ,n$ (15)

 ${\bar x_{ij}}\left( {\bar \otimes } \right) = \frac{{\left( {1/{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_{ij}}} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {1/{{\bar x}_{ij}}} \right)}^2}} } }},\;\;i = 1,2, \cdots ,n$ (16)
 ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} _{ij}}\left( {\bar \otimes } \right) = \frac{{\left( {1/{{\bar x}_{ij}}} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {1/{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} }_{ij}}} \right)}^2}} } }},\;\;i = 1,2, \cdots ,n$ (17)

Step 2. 选取决策参考点

 $\begin{split} {X^ + }( \otimes ) &= \left\{ {\mathop {\max }\limits_i {x_{i1}}(\bar \otimes ),\mathop {\max }\limits_i {x_{i2}}(\bar \otimes ), \cdots ,\mathop {\max }\limits_i {x_{in}}(\bar \otimes )} \right\} \\ &= \left\{ {x_1^ + (\bar \otimes ),x_2^ + (\bar \otimes ), \cdots ,x_n^ + (\bar \otimes )} \right\} \\ \end{split}$ (18)
 $\begin{split} {X^ - }( \otimes ) &= \left\{ {\mathop {\min }\limits_i {x_{i1}}(\bar \otimes ),\mathop {\min }\limits_i {x_{i2}}(\bar \otimes ), \cdots ,\mathop {\min }\limits_i {x_{in}}(\bar \otimes )} \right\} \\ & = \left\{ {x_1^ - (\bar \otimes ),x_2^ - (\bar \otimes ), \cdots ,x_n^ - (\bar \otimes )} \right\} \\ \end{split}$ (19)

Step 3. 计算距离测度

$D({x_i},x_i^ + )$ $D({x_i},x_i^ - )$ 分别表示方案 ${x_{ij}}$ 与正理想解 ${X^ + }( \otimes )$ 和负理想解 ${X^ - }( \otimes )$ 之间的距离集, 即:

 $\begin{gathered} D\left( {{x_i},x_i^ + } \right) = \left\{ {d\left( {{x_{i1}}\left( {\bar \otimes } \right),x(\bar \otimes )} \right),d\left( {{x_{i2}}\left( {\bar \otimes } \right),x_2^ + \left( \otimes \right)} \right),\cdots ,} \right. \\ \left. { d\left( {{x_{in}}\left( {\bar \otimes } \right),x_n^ + \left( {\bar \otimes } \right)} \right)} \right\} \\ \end{gathered}$ (20)
 $\begin{gathered} D\left( {{x_i},x_i^ - } \right) = \left\{ {d\left( {{x_{i1}}\left( {\bar \otimes } \right),x_1^ - (\bar \otimes )} \right),d\left( {{x_{i2}}\left( {\bar \otimes } \right),x_2^ - \left( \otimes \right)} \right),\cdots ,} \right. \\ \left. { d\left( {{x_{in}}\left( {\bar \otimes } \right),x_{in}^ - \left( {\bar \otimes } \right)} \right)} \right\} \\ \end{gathered}$ (21)

Step 4. 构建前景价值矩阵

 ${v_{ij}} = \left\{ \begin{gathered} {(D({x_{ij}}(\bar \otimes ),x_j^ - (\bar \otimes )))^\alpha } \\ - \sigma {(D({x_{ij}}(\bar \otimes ),x_j^ + (\bar \otimes )))^\beta } \\ \end{gathered} \right.$ (22)

 ${v^ - } = - \sigma {(D({x_{ij}}(\bar \otimes ),x_j^ + (\bar \otimes )))^\beta }$ (23)

 ${v^ + } = {(D({x_{ij}}(\bar \otimes ),x_j^ - (\bar \otimes )))^\alpha }$ (24)

Step 5. 规范化前景决策矩阵

 $\bar v_{ij}^ - = \frac{{v_{ij}^ - - \min v_{ij}^ - }}{{\max v_{ij}^ + - \min v_{ij}^ - }}$ (25)
 $\bar v_{ij}^ + = \frac{{v_{ij}^ + - \min v_{ij}^ - }}{{\max v_{ij}^ + - \min v_{ij}^ - }}$ (26)

Step 6. 融合前景值

 $S({e_i}) = \left\{ {({H_n},{\beta _{n,ij}})|n = 1,2} \right\}$

${m_{n,ij}}$ 为基本概率分配, 表示 ${x_{ij}}(\bar \otimes )$ 支持 ${a_i}$ 接近正理想解的程度, ${m_{H,ij}}$ 为未分配概率, 根据公式(6-10)可得:

 ${m_{1,ij}} = {w_j}\bar v_{ij}^ - ,{m_{2,ij}} = {w_i}(1 - \bar v_{ij}^ + )$ (27)
 ${m_{H,ij}} = 1 - \sum\limits_{n = 1}^N {{m_{n,ij}}} = {w_j}(\bar v_{ij}^ + - \bar v_{ij}^ - )$ (28)

Step 7. 根据文献[18]提出的区间数大小可能度比较公式确定综合前景值的大小, 并对方案做出排序.

 $p\left( {V\left( a \right) \succ V\left( b \right)} \right) = \frac{{\min \{ l(a) + l(b),\max ({a^R} - {b^L},0)\} }}{{l(a) + l(b)}}$ (29)

$p\left( {V\left( a \right) \succ V\left( b \right)} \right) \ge \dfrac{1}{2}$ , 则称 $V\left( a \right) \succ V\left( b \right)$ , 否则反之.

3 算例分析 3.1 问题描述

3.2 决策过程

Step 1. 数据规范化

Step 2. 选取决策参考点

 $\begin{split} {X^ + }\left( \otimes \right) = &\left\{ {\left[ {0.554,0.765} \right],\left[ {0.447,0.943} \right],\left[ {0.670,0.782} \right]} \right., \\ &\left. {\left[ {0.538,0.721} \right],\left[ {0.571,0.663} \right]} \right\} \\ \end{split}$
 $\begin{split} {X^ - }\left( \otimes \right) =& \left\{ {\left[ {0.240,0.295} \right],\left[ {0.149,0.471} \right],\left[ {0.359,0.401} \right]} \right., \\ &\left. {\left[ {0.231,0.361} \right],\left[ {0.326,0.368} \right]} \right\} \end{split}$

Step 3. 计算距离测度

 $D\left( {{x_i},x_i^ + } \right) = \left[ {\begin{array}{*{20}{c}} {0.397}&{0.140}&{0.274}&0&0 \\ 0&0&{0.328}&{0.340}&{0.272} \\ {0.123}&{0.245}&{0.354}&0&{0.237} \\ {0.217}&{0.421}&0&{0.259}&{0.030} \end{array}} \right]$
 $D({x_i},x_i^ - ) = \left[ {\begin{array}{*{20}{c}} 0&{0.367}&{0.076}&{0.340}&{0.272} \\ {0.397}&{0.421}&{0.029}&0&0 \\ {0.297}&{0.234}&0&{0.340}&{0.041} \\ {0.197}&0&{0.348}&{0.098}&{0.256} \end{array}} \right]$

Step 4. 构建前景价值矩阵

Step 5. 综合前景值

 $V\left( {{a_1}} \right) = \left[ {0.516,0.902} \right]$
 $V\left( {{a_2}} \right) = \left[ {0.341,0.799} \right]$
 $V\left( {{a_3}} \right) = \left[ {0.383,0.843} \right]$
 $V\left( {{a_4}} \right) = \left[ {0.404,0.850} \right]$

Step 6. 方案排序

3.3 分析比较

(1) 未考虑决策面对损失和收益时的风险态度, 文献[19]提出基于正负靶心的灰靶决策模型, 定义最优和最劣方案分别作为灰靶的正负靶心, 在综合考虑方案与正负靶心的距离基础上, 建立单目标优化方程, 再根据综合靶心距 ${\varepsilon _i}$ 对方案排序. 采用文献[19]对4种火炮方案进行综合靶心距求解的结果为 ${\varepsilon _1} = 0.287\;005$ , ${\varepsilon _2} = 0.334\;964$ , ${\varepsilon _3} = 0.325\;629$ , ${\varepsilon _4} = 0.404\;164$ .该方法未考虑决策者的心理因素, 可与本文考虑决策者心理因素下的结果进行比较.

(2)考虑决策的心理行为, 文献[20]提出一种基于改进的TODIM方法的区间灰数多属性决策模型, TODIM 方法在前景理论的基础上提出的一种多属性决策方法, 根据两两方案相比较时的收益和损失求解优势度 $\Phi ({a_i})$ , 并对方案做出选择, 该方法与前景理论方法有类似之处, 因此具有可比性. 采用文献[20]的方法得到4种火炮方案的总优势度为 $\Phi ({a_1}) = 0.5665$ , $\Phi ({a_2}) =$ $- 0.2791$ , $\Phi ({a_3}) = - 0.1107$ , $\Phi ({a_4}) = - 0.1767$ .

4 结论与展望

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