﻿ 考虑风险偏好的概率犹豫模糊多属性决策方法
 计算机系统应用  2020, Vol. 29 Issue (10): 36-43 PDF

Probabilistic Hesitant Fuzzy Multi-Attribute Decision-Making Method Considering Risk Preference
LUO Hua, WANG Ying-Ming
School of Economics and Management, Fuzhou University, Fuzhou 350108, China
Foundation item: National Natural Science Foundation of China (61773123)
Abstract: Aiming at the problem that the attribute value is a probabilistic hesitant fuzzy number and the decision maker’s attitude to risk is different, a probabilistic fuzzy multi-attribute decision-making method considering risk preference is proposed. First, considering the decision maker’s hesitation may affect the decision-making effect, a hesitation formula expressed by the difference between the number of elements in the probability hesitant fuzzy element is given. The extended Hamming distance and the extended Euclidean distance are defined based on the hesitation and the difference of element values. Then, a foreground decision matrix is established based on the expected values given by decision makers, and the maximum weight method is used to calculate the attribute weights. Based on this, the comprehensive foreground value of each plan is calculated and ranked. At last, the example analysis of purchasing ERP system software verifies the validity and rationality of the proposed method.
Key words: multi-attribute decision-making     probabilistic hesitant fuzzy set     foreground theory     expected value     maximum deviation

1 预备知识 1.1 概率犹豫模糊集相关知识

 $H = \left\{ {\left\langle {x,h(x)} \right\rangle |x \in X} \right\}$

X上的一个犹豫模糊集. 其中, $h(x) = \left\{ {{\gamma ^\lambda }|\lambda =1,2, \cdots , } \right.$ $\left. {l} \right\}$ 为其中一个犹豫模糊元, $l$ 为概率犹豫模糊元 $h(x)$ 中元素的个数, ${\gamma ^\lambda } \in \left[ {0,1} \right]$ 为非空集合X中的元素 $x$ 属于犹豫模糊集H的隶属度.

 ${H_P}{\rm{ = }}\left\{ {\left\langle {x,h({P_x})} \right\rangle |x \in X} \right\}$

 ${h^c}({P_x}) = \left\{ {[1 - {\gamma ^\lambda }]({P^\lambda })|\lambda = 1,2, \cdots ,l} \right\}$

 $\begin{split} &{E(h({P_x})) = \displaystyle\sum\limits_{\lambda = 1}^l {{\gamma ^\lambda }{P^\lambda }}}\\ &{D(h({P_x})) = \displaystyle\sum\limits_{\lambda = 1}^l {{{({\gamma ^\lambda } - E(h({P_x})))}^2}} {P^\lambda }} \end{split}$

① 若 $E({h_1}({P_x})) > E({h_2}({P_x}))$ , 则 ${h_1}({P_x}) > {h_2}({P_x})$ .

② 若 $E({h_1}({P_x})) < E({h_2}({P_x}))$ , 则 ${h_1}({P_x}) < {h_2}({P_x})$ .

③ 若 $E({h_1}({P_x})) = E({h_2}({P_x}))$ $D({h_1}({P_x})) < D({h_2}({P_x}))$ ${h_1}({P_x}) > {h_2}({P_x})$ ;

$E({h_1}({P_x})) = E({h_2}({P_x}))$ $D({h_1}({P_x})) = D({h_2}({P_x}))$ ${h_1}({P_x}) = {h_2}({P_x})$ ;

$E({h_1}({P_x})) = E({h_2}({P_x}))$ $D({h_1}({P_x})) > D({h_2}({P_x}))$ ${h_1}({P_x}) < {h_2}({P_x})$ .

 $D\left( {{h_1}(P),{h_2}(P)} \right) = \displaystyle\sum\limits_{\lambda = 1}^l {\left| {\gamma _1^\lambda P_1^\lambda - \gamma _2^\lambda P_2^\lambda } \right|}$ (1)

1.2 前景理论相关知识

 $U = \sum\limits_{i = 1}^n {w({P_i})v({x_i})}$

Tversky等[16]给出了价值函数为幂函数:

 $v(\Delta x) = \left\{ \begin{array}{l} \Delta {x^\alpha }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Delta x \ge {\rm{0}}{\kern 1pt} \\ - \theta {( - \Delta x)^\beta }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Delta x < {\rm{0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \end{array} \right.$

 $v({h_2}) = \left\{ \begin{array}{l} {({d_H}({h_1},{h_2}))^\alpha }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {h_2} \ge {h_1} \\ - \theta {({d_H}({h_1},{h_2}))^\beta }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {h_2} < {h_1} \\ \end{array} \right.$
2 确定前景决策矩阵及属性权重

2.1 距离公式

 $u(h({P_x})) = \sqrt {1 - \frac{1}{{1 + \ln l}}} ,\;\;u({H_P}) = \frac{1}{n}\sum\limits_{i = 1}^n {u({h_i}({P_x}))}$

$u(h({P_x}))$ 为概率犹豫模糊元 $h({P_x})$ 的犹豫度, $u({H_P})$ 为概率犹豫模糊集 ${H_P}$ 的犹豫度. 概率犹豫模糊元中元素的个数越多, 犹豫程度越大. 当且仅当概率犹豫模糊元中元素个数为1时, 其犹豫度为0.

(1) 概率犹豫模糊元 ${h_1}({P_x})$ ${h_2}({P_x})$ 之间拓展的海明距离为:

 $\begin{split} {D_H}({h_1}({P_x}),{h_2}({P_x})) =& \frac{1}{2}(\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right| \\ &+\dfrac{1}{l}\displaystyle\sum\limits_{\lambda = 1}^l {\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|} ) \end{split}$ (2)

(2) 概率犹豫模糊元 ${h_1}({P_x})$ ${h_2}({P_x})$ 之间拓展的标准欧式距离为:

 $\begin{split} {D_H}({h_1}({P_x}),{h_2}({P_x})) =& \left( {\frac{1}{2}\left( {{{\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right|}^2}} \right.} \right.\\ &+{\left. {\left. {\frac{1}{l}\sum\limits_{\lambda = 1}^l {{{\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|}^2}} } \right)} \right)^{\frac{1}{2}}} \end{split}$ (3)

(3) 概率犹豫模糊元 ${h_1}({P_x})$ ${h_2}({P_x})$ 之间拓展的一般欧式距离为:

 $\begin{split} {D_H}({h_1}({P_x}),{h_2}({P_x})) = &\left[ {\frac{1}{2}\left( {{{\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right|}^\varphi }} \right.} \right.\\ &+{\left. {\left. { \frac{1}{l}\sum\limits_{\lambda = 1}^l {{{\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|}^\varphi }} } \right)} \right]^{\frac{1}{\varphi }}} \end{split}$ (4)

(4) 概率犹豫模糊元 ${h_1}({P_x})$ ${h_2}({P_x})$ 之间的拓展的加权海明距离为:

 $\begin{split} {D_H}({h_1}({P_x}),{h_2}({P_x})) =& \mu \left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right| \\ &+\nu \frac{1}{l}\sum\limits_{\lambda = 1}^l {\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|} \end{split}$ (5)

(5) 概率犹豫模糊元 ${h_1}({P_x})$ ${h_2}({P_x})$ 之间拓展的加权标准欧式距离为:

 $\begin{split} {D_H}({h_1}({P_x}),{h_2}({P_x})) =& \left( {\mu {{\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right|}^2}} \right.\\ &+{\left. { \nu \frac{1}{l}\sum\limits_{\lambda = 1}^l {{{\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|}^2}} } \right)^{\frac{1}{2}}} \end{split}$ (6)

(6) 概率犹豫模糊元 ${h_1}({P_x})$ ${h_2}({P_x})$ 之间拓展的加权一般欧式距离为:

 $\begin{split} {D_H}({h_1}({P_x}),{h_2}({P_x})) =& \left( {\mu {{\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right|}^\varphi }} \right.\\ &+{\left. { \nu \frac{1}{l}\sum\limits_{\lambda = 1}^l {{{\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|}^\varphi }} } \right)^{\frac{1}{\varphi }}} \end{split}$ (7)

$0 \le D({h_1}({P_x}),{h_2}({P_x})) \le 1$ ;

$D({h_1}({P_x}),{h_2}({P_x})) = 0$ , 当且仅当 ${h_1}({P_x})$ = ${h_2}({P_x})$ ;

$D({h_1}({P_x}),{h_2}({P_x}))$ = $D({h_2}({P_x}),{h_1}({P_x}))$ .

① 因为 $0 \le u({h_1}({P_x})) \le 1$ , $0 \le u({h_2}({P_x})) \le 1$ , 则 $0 \le \left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right| \le 1$ ; 又因为 $0 \le {\gamma _1}^\lambda \cdot {P_1}^\lambda \le 1$ , $0 \le {\gamma _2}^\lambda \cdot {P_2}^\lambda \le 1$ , 因此 $0 \le \left| {{\gamma _1}^\lambda \cdot {P_1}^\lambda - {\gamma _2}^\lambda \cdot {P_2}^\lambda } \right| \le 1$ , 故 $0 \le D({h_1}({P_x}),{h_2}({P_x})) \le 1$ .

②若 ${h_1}({P_x})$ = ${h_2}({P_x})$ , 则 $u({h_1}({P_x})) = u({h_2}({P_x}))$ ${\gamma _1}^\lambda \cdot {P_1}^\lambda$ = ${\gamma _2}^\lambda \cdot {P_2}^\lambda$ , 因此 $\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right| = 0$ $\left| {{\gamma _1}^\lambda \cdot {P_1}^\lambda - {\gamma _2}^\lambda \cdot {P_2}^\lambda } \right| = 0$ , 故 $D({h_1}({P_x}),{h_2}({P_x})) = 0$ ;

$D({h_1}({P_x}),{h_2}({P_x})) = 0$ , 则 $\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right| = 0$ $\left| {{\gamma _1}^\lambda \cdot {P_1}^\lambda - {\gamma _2}^\lambda \cdot {P_2}^\lambda } \right| = 0$ , 因此 $u({h_1}({P_x})) = u({h_2}({P_x}))$ ${\gamma _1}^\lambda \cdot {P_1}^\lambda$ = ${\gamma _2}^\lambda \cdot {P_2}^\lambda$ , 故 ${h_1}({P_x})$ = ${h_2}({P_x})$ .

${D_H}({h_1}({P_x}),{h_2}({P_x})) = \dfrac{1}{2}\left( {\left| {u({h_1}({P_x})) - u({h_2}({P_x}))} \right|} \right.$

 $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +\left. {\frac{1}{l}\sum\limits_{\lambda = 1}^l {\left| {\gamma _1^\lambda p_1^\lambda - \gamma _2^\lambda p_2^\lambda } \right|} } \right)$
 ${D_H}({h_2}({P_x}),{h_1}({P_x})) = \frac{1}{2}\left( {\left| {u({h_2}({P_x})) - u({h_1}({P_x}))} \right|} \right.$
 $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; +\left. {\frac{1}{l}\sum\limits_{\lambda = 1}^l {\left| {\gamma _2^\lambda p_2^\lambda - \gamma _1^\lambda p_1^\lambda } \right|} } \right)$

2.2 建立前景决策矩阵

${E_j}$ 为参考点, 则 ${B_{ij}}$ 的前景价值函数为:

 $v({B_{ij}}) = \left\{ \begin{array}{l} {({D_H}({E_j},{B_{ij}}))^\alpha }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {B_{ij}} \ge {E_j} \\ - \theta {({D_H}({E_j},{B_{ij}}))^\beta }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {B_{ij}} < {E_j} \\ \end{array} \right.$ (8)

2.3 确定属性权重

 $\max {\kern 1pt} {\kern 1pt} {\kern 1pt} f(w) = \sum\limits_{i = 1}^m {\sum\limits_{k = 1}^m {\sum\limits_{j = 1}^n {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } } {w_j}$
 ${\rm {s.t.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ \begin{array}{l} \displaystyle\sum\limits_{j = 1}^n {{{({w_j})}^2} = 1} \\ 0 \le {w_j} \le 1 \\ \end{array} \right.$

 $\begin{split} L(w,\eta ) = &\sum\limits_{i = 1}^m {\sum\limits_{k = 1}^m {\sum\limits_{j = 1}^n {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } } {w_j} \\ &+\frac{\eta }{2}{\left (\sum\limits_{j = 1}^n {{w_j}} ^2 - 1\right)} \end{split}$ (9)

 $\left\{ \begin{split} &\frac{{\partial L({w_j},\eta )}}{{\partial {w_j}}} = \sum\limits_{i = 1}^m {\sum\limits_{k = 1}^m {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } + \eta {w_j} = 0 \\ & \frac{{\partial L({w_j},\eta )}}{{\partial \eta }} = \sum\limits_{j = 1}^n {w_j^2 - 1 = 0} \\ \end{split} \right.$ (10)

 ${w_j} = \frac{{\displaystyle\sum\limits_{i = 1}^m {\displaystyle\sum\limits_{k = 1}^m {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } }}{{\sqrt {\displaystyle\sum\limits_{j = 1}^n {{{\left( {\displaystyle\sum\limits_{i = 1}^m {\displaystyle\sum\limits_{k = 1}^m {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } } \right)}^2}} } }}$ (11)

${w_j}$ 进行单位化得属性权重为:

 $w_j^* = \dfrac{{\displaystyle\sum\limits_{i = 1}^m {\displaystyle\sum\limits_{k = 1}^m {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } }}{{\displaystyle\sum\limits_{j = 1}^n {\left( {\displaystyle\sum\limits_{i = 1}^m {\displaystyle\sum\limits_{k = 1}^m {\left| {{D_H}({E_j},{B_{ij}}) - {D_H}({E_j},{B_{kj}})} \right|} } } \right)} }}$ (12)
3 决策步骤

 ${B'_{ij}} = \left\{ {\begin{array}{*{20}{l}} {{B_{ij}} = \left\{ {\gamma _{ij}^\lambda (P_{ij}^\lambda )|\lambda = 1,2, \cdots ,l} \right\}}&{{C_j} \in {\text{效益型}}}\\ {B_{ij}^C = \left\{ {[1 - \gamma _{ij}^\lambda ](P_{ij}^\lambda )|\lambda = 1,2, \cdots ,l} \right\}}&{{C_j} \in {\text{成本型}}} \end{array}} \right.$

 ${V_i}^* = \sum\limits_{j = 1}^n {v({{B'}_{ij}})w_j^*}$ (13)

4 算例分析 4.1 算例分析

 $\begin{split} Q =& \left( {\{ 0.6(0.5),0.8(0.5)\} ,{\kern 1pt} {\kern 1pt} \{ 0.5(0.3),0.6(0.5),0.7(0.2)\} ,{\kern 1pt} {\kern 1pt} } \right.\\ &\left. {\{ 0.6(0.6),0.7(0.4)\} ,{\kern 1pt} {\kern 1pt} \{ 0.6(0.4),0.8(0.4),0.9(0.2)\} } \right) \end{split}$

 ${{{W}}^*} = \left( {0.2307,{\rm{ }}0.2785,{\rm{ }}0.2195,{\rm{ }}0.2713} \right)$

 ${{{V}}_1} \!=\! {\rm{ }} - 0.5904;\;\;\;{{{V}}_2} \!=\! {\rm{ }} - 0.3765;\;\;\;{{{V}}_3} = {\rm{ }}0.2437;\;\;\;{{{V}}_4} = {\rm{ }}0.1633.$

4.2 比较分析

 ${{{V}}_1} \!=\! {\rm{ }} - 0.3429;\;\;\;{{{V}}_2} \!=\! {\rm{ }} - 0.4119;\;\;\;{{{V}}_3} = {\rm{ }}0.2620;\;\;\;{{{V}}_4} = {\rm{ }}0.1984.$

 ${{\rm{V}}_1} \!=\! {\rm{ }} - 0.4949;\;\;\;{{\rm{V}}_2} \!=\! {\rm{ }} - 0.6450;\;\;\;{{\rm{V}}_3} = {\rm{ }}0.0975;\;\;\;{{\rm{V}}_4} = {\rm{ }}0.0642.$

5 结论

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