﻿ 基于前景理论的犹豫二元语义灰关联群决策法
 计算机系统应用  2019, Vol. 28 Issue (3): 152-157 PDF

1. 西安理工大学 理学院, 西安 710054;
2. 西安理工大学 高科学院, 西安 710109

Hesitant 2-Tuple Linguistic Grey Relational Group Decision-Making Approach Based on Prospect Theory
LIU Rui1, WANG Qiu-Ping1, WANG Xiao-Feng1, YAN Hai-Xia2
1. Faculty of Sciences, Xi’an University of Technology, Xi’an 710054, China;
2. Hi-Tech College of Xi’an University of Technology, Xi’an 710109, China
Foundation item: National Natural Science Foundation of China (61772416); 2015 Scientific Research Program of Education Bureau, Shaanxi Province (15JK2068)
Abstract: For the multi-attribute group decision-making problems, where the information of the attribute weights and the expert weights is completely unknown and the preference information is in the form of hesitant 2-tuple linguistic, a multi-attribute group decision-making method based on the prospect theory and the grey relation analysis is proposed. Firstly, the weights of the experts are determined by the matrix vec operator and the grey relation analysis, and the weights of attributes are calculated by the maximizing deviation method. Subsequently, a comparison method of the hesitant 2-tuple linguistic elements is given, and the positive and negative ideal solutions based on that are determined and used as the decision reference point. Then the hesitant 2-tuple linguistic prospect value function according to the prospect theory and the grey relational coefficient is acquired, and then the ratio of the gains to losses of the alternatives is obtained, and the alternatives are ranked accordingly. Finally, the proposed method is applied to a numerical example of investment decision, and the results show the rationality and effectiveness of the method.
Key words: hesitant 2-tuple linguistic element     prospect theory     matrix vec operator     grey relational analysis     multi-attribute group decision-making

1 基础概念 1.1 犹豫二元语义术语集

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

(1) 若 ${s_i} > {s_l}$ , 则 $\left( {{s_i},{\alpha _{ij}}} \right) \succ \left( {{s_l},{\alpha _{lm}}} \right)$

(2) 若 ${s_i} = {s_l}$ , 则:

1) 若 $e\left( {{\alpha _{ij}}} \right) > e\left( {{\alpha _{lm}}} \right)$ , 则 $\left( {{s_i},{\alpha _{ij}}} \right) \succ \left( {{s_l},{\alpha _{lm}}} \right)$

2) 若 $e\left( {{\alpha _{ij}}} \right) = e\left( {{\alpha _{lm}}} \right)$ , 则 $\left( {{s_i},{\alpha _{ij}}} \right) \sim \left( {{s_l},{\alpha _{lm}}} \right)$

 $\begin{array}{l} d\left( {\left( {{s_i},{\alpha _{ij}}} \right),\left( {{s_l},{\alpha _{lm}}} \right)} \right) = \left| {i - l} \right| + \\ \max \left\{ {\mathop {\max }\limits_{{a_k} \in {\alpha _{ij}}} \left\{ {\mathop {\min }\limits_{{b_n} \in {\alpha _{lm}}} \left( {\left| {{a_k} - {b_n}} \right|} \right)} \right\},\mathop {\max }\limits_{{b_n} \in {\alpha _{lm}}} \left\{ {\mathop {\min }\limits_{{a_k} \in {\alpha _{ij}}} \left( {\left| {{a_k} - {b_n}} \right|} \right)} \right\}} \right\} \end{array}$ (1)
1.2 前景理论

 $v\left( {\Delta x} \right) = \left\{ \begin{array}{l} {\left( {\Delta x} \right)^\alpha },\;\;\;\;\;\;\;\;\Delta x \ge 0\\ - \theta {\left( { - \Delta x} \right)^\beta }{\rm{,}}\;\;\Delta x < 0 \end{array} \right.$ (2)

2 基于前景理论的犹豫二元语义灰关联群决策法

2.1 决策者权重的确定

(1) 将决策者 ${e_k}\left( {k = 1,2, \cdots ,t} \right)$ 所给决策矩阵 ${{{X}}^k} = {\left( {h_{ij}^k} \right)_{m \times n}}$ 按行依次拉直成一个长向量 ${\bar {{X}}^k}$ , 即:

 $\begin{array}{l} {{\bar {{X}}}^k} = {\left( {{h^k}\left( s \right)} \right)_{1 \times mn}} = \left( {h_{11}^k,h_{12}^k, \cdots ,h_{1n}^k,h_{21}^k,h_{22}^k, \cdots ,h_{2n}^k, \cdots ,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left. {h_{m1}^k,}{h_{m2}^k, \cdots ,h_{mn}^k} \right)_{1 \times mn}},s = 1,2, \cdots ,mn. \end{array}$

(2) 取决策者 ${e_k}\left( {k = 1,2, \cdots ,t} \right)$ 的决策信息 ${\bar {{X}}^k}$ 为参考序列, 决策群体的决策信息 ${\bar {{X}}^1},{\bar {{X}}^2}, \cdots ,{\bar {{X}}^t}$ 为比较序列, 则决策者 ${e_k}$ 与第 $l$ 个决策者 ${e_l}$ 的关联系数为:

 $r_l^k\left( s \right) = \frac{{\mathop {\min }\limits_l \mathop {\min }\limits_s d\left( {{h^k}\left( s \right),{h^l}\left( s \right)} \right) + \rho \mathop {\max }\limits_l \mathop {\max }\limits_s d\left( {{h^k}\left( s \right),{h^l}\left( s \right)} \right)}}{{d\left( {{h^k}\left( s \right),{h^l}\left( s \right)} \right) + \rho \mathop {\max }\limits_l \mathop {\max }\limits_s d\left( {{h^k}\left( s \right),{h^l}\left( s \right)} \right)}}$ (3)

 $r_l^k = \frac{1}{{mn}}\sum\limits_{s = 1}^{mn} {r_l^k\left( s \right)}$ (4)

(3) 确定决策者 ${e_k}\left( {k = 1,2, \cdots ,t} \right)$ 与决策群体的平均关联度为:

 ${r^k} = \frac{1}{t}\sum\limits_{l = 1}^t {r_l^k}$ (5)

(4) 确定决策者 ${e_k}\left( {k = 1,2, \cdots ,t} \right)$ 的权重, 即:

 ${\lambda _k} = {{{r^k}} / {\sum\limits_{l = 1}^t {{r^l}} }}$ (6)
2.2 属性权重的确定

 $\left\{ \begin{array}{l} \max D\left( {{\omega ^k}} \right) = \displaystyle\sum\limits_{j = 1}^n {\displaystyle\sum\limits_{i = 1}^m {\displaystyle\sum\limits_{l = 1}^m {\omega _j^kd\left( {h_{ij}^k,h_{lj}^k} \right)} } } \\ {\rm {s.t.}} \; \displaystyle\sum\limits_{j = 1}^n {{{\left( {\omega _j^k} \right)}^2}} = 1,\omega _j^k > 0, \; j = 1,2, \cdots ,n \end{array} \right.$ (7)

 $\omega _j^k = \sum\limits_{i = 1}^m {\sum\limits_{l = 1}^m {{{d\left( {h_{ij}^k,h_{lj}^k} \right)} / {\sqrt {\sum\nolimits_{j = 1}^n {{{\left[ {\sum\nolimits_{i = 1}^m {\sum\nolimits_{l = 1}^m {d\left( {h_{ij}^k,h_{lj}^k} \right)} } } \right]}^2}} } }}} }$ (8)

 ${\left( {\omega _j^k} \right)^*} = {{\omega _j^k} / {\sum\limits_{j = 1}^n {\omega _j^k} }} \; \left( {j = 1,2, \cdots ,n} \right)$ (9)
2.3 正、负前景值

 ${\left( {h_j^ + } \right)^k} = \left\{ \begin{array}{l} \mathop {\max }\limits_i h_{ij}^k,\;\;\;\;\;{\text{效益型属性}}{C_j}\\ \mathop {\min }\limits_i h_{ij}^k,\;\;\;\;\;\;{\text{成本型属性}}{C_j} \end{array} \right.$ (10)
 ${\left( {h_j^ - } \right)^k} = \left\{ \begin{array}{l} \mathop {\min }\limits_i h_{ij}^k,\;\;\;\;\;{\text{效益型属性}}{C_j}\\ \mathop {\max }\limits_i h_{ij}^k,\;\;\;\;\;{\text{成本型属性}}{C_j} \end{array} \right.$ (11)

 $v_{ij}^k = \left\{ \begin{array}{l} {\left( {1 - {{\left( {\xi _{ij}^ - } \right)}^k}} \right)^\alpha },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{以负理想方案为参考点}}\\ - \theta \cdot {\left[ { - \left( {{{\left( {\xi _{ij}^ + } \right)}^k} - 1} \right)} \right]^\beta }{\rm{,}}\;\;\;\;{\text{以正理想方案为参考点}} \end{array} \right.$ (12)

 ${\left( {v_i^{\rm{ + }}} \right)^k} = \sum\limits_{j = 1}^n {{{\left( {\omega _j^k} \right)}^*}{{\left( {v_{ij}^{\rm{ + }}} \right)}^k}} , \; i = 1,2, \cdots ,m,k = 1,2, \cdots ,t$ (13)
 ${\left( {v_i^ - } \right)^k} = \sum\limits_{j = 1}^n {{{\left( {\omega _j^k} \right)}^{\rm{*}}}{{\left( {v_{ij}^ - } \right)}^k}} , \; i = 1,2, \cdots ,m,k = 1,2, \cdots ,t$ (14)

 $v_i^ + = \sum\limits_{k = 1}^t {{\lambda _k}{{\left( {v_i^{\rm{ + }}} \right)}^k}} , \; i = 1,2, \cdots ,m$ (15)
 $v_i^ - = \sum\limits_{k = 1}^t {{\lambda _k}{{\left( {v_i^ - } \right)}^k}} , \; i = 1,2, \cdots ,m$ (16)

 ${R_i} = \left| {{{v_i^ + } / {v_i^ - }}} \right|, \; i = 1,2, \cdots ,m$ (17)
2.4 决策步骤

Step 1. 根据各决策者给出的决策信息构造犹豫二元语义决策矩阵 ${{{X}}^k} = {\left( {h_{ij}^k} \right)_{m \times n}}\left( {k = 1,2, \cdots ,t} \right)$ ;

Step 2. 根据2.1节确定决策者权重;

Step 3. 根据2.2节确定各犹豫二元语义决策矩阵中的属性权重;

Step 4. 根据定义2及式(10)、(11)确定各决策矩阵的犹豫二元语义正、负理想方案;

Step 5. 根据式(13)、(14)分别计算决策者 ${e_k}\left( {k = } \right.1, \left. {2, \cdots ,t} \right)$ 关于各方案的正、负前景值;

Step 6. 根据式(15)、(16)分别计算决策群体关于各方案的正、负前景值;

Step 7. 根据式(17)计算各方案的收益损失比值 ${R_i}\left( {i = 1,2, \cdots ,m} \right)$ ;

Step 8. 将 ${R_i}\left( {i = 1,2, \cdots ,m} \right)$ 按降序排列, 便可得到整个方案集由优到劣的排序.

2.5 区分度

 $\eta = \dfrac{{{\delta _{\max }} - {\delta _{\sec }}}}{{{\delta _{\max }}}} \times 100\%$ (18)

3 算例分析

Step 1. 建立犹豫二元语义决策矩阵 ${{{X}}^k} = {\left( {h_{ij}^k} \right)_{5 \times 4}} \;$ $\left( {k = 1,2,3} \right)$ .

Step 2. 根据2.1节可得决策者的权重向量为:

 ${{\lambda }} = {\left( {0.329 \; 9,0.332 \; {\rm{ 6}},0.337 \; 5} \right)^{\rm{T}}}.$

Step 3. 根据2.2节可得犹豫二元语义决策矩阵Xk (k=1, 2, 3)的属性权重向量分别为:

 $\begin{array}{l} {{{W}}^1} = {\left( {0.305\;2,\;0.181\;2,\;0.314\;8,\;0.198\;8} \right)^{\rm{T}}}\\ {{{W}}^2} = {\left( {0.321\;0,\;0.194\;1,\;0.215\;5,\;0.269\;4} \right)^{\rm{T}}}\\ {{{W}}^3} = {\left( {0.164\;5,\;0.271\;1,\;0.222\;4,\;0.342\;1} \right)^{\rm{T}}} \end{array}$

Step 4. 确定各决策矩阵的犹豫二元语义正、负理想方案:

 $\begin{array}{l} {\left( {{A^ + }} \right)^1} = (({s_6},\;( - 0.4, - 0.3,0.1))\;({s_2},( - 0.1,0.2,0.3)),\\ \;\;\;\;\;\;\;\;\;\;\;({s_1},( - 0.45, - 0.2)),({s_4},( - 0.2,0.1,0.2))) \end{array}$
 $\begin{array}{l} {\left( {{A^{\rm{ - }}}} \right)^1} = (({s_2},(0,0.1,0.2)),({s_4},(0.2,0.32,0.45)),\\ \;\;\;\;\;\;\;\;\;\;\;({s_5},( - 0.2,0,0.4)),({s_2},( - 0.3,0.1))), \end{array}$
 $\begin{array}{l} {\left( {{A^ + }} \right)^2} = (({s_6},(0.2,0.4)),({s_2},( - 0.1,0.2)),({s_1},( - 0.2,0.3)),\\ \;\;\;\;\;\;\;\;\;\;\;\;({s_5},( - 0.1,0.3)) \end{array}$
 $\begin{array}{l} {\left( {{A^{\rm{ - }}}} \right)^2} = (({s_1},(0.4)),({s_5},( - 0.1,0,0.1)),({s_4},(0.3,0.4)),\\ \;\;\;\;\;\;\;\;\;\;\;({s_1},( - 0.3, - 0.2,0))) \end{array}$
 $\begin{array}{l} {\left( {{A^ + }} \right)^3} = (({s_4},( - 0.5,0.1,0.2)),({s_2},( - 0.2, - 0.1,0)),\\ \;\;\;\;\;\;\;\;\;\;\;({s_2},(0.1,0.2,0.3)),({s_6},( - 0.05,0.25))) \end{array}$
 $\begin{array}{l} {\left( {{A^ - }} \right)^3} = (({s_2},( - 0.2,0,0.1)),({s_5},(0.2,0.3)),({s_5},( - 0.3, - 0.2)),\\ \;\;\;\;\;\;\;\;\;\;\;({s_1},( - 0.3, - 0.2,0))) \end{array}$

Step 5. 根据式(13), (14)分别计算决策者 ${e_k}\left( {k =}\right.$ ${\left. 1, \right.}2, \left. 3 \right)$ 关于方案 ${A_i}\left( {i = 1,2, \cdots ,5} \right)$ 的正、负前景值( $\alpha =$ $\beta =0.88,\theta = 2.25$ [14]), 具体结果如表4表5所示.

Step 6. 根据式(15)、(16)分别计算决策群体关于方案Ai的正、负前景值: $v_1^ + = 0.328 \; 3$ , $v_2^ + = 0.346{\rm{ 3}}$ , $v_3^ + = 0.223{\rm{ 7}}$ , $v_4^ + = 0.551{\rm{ 5}}$ , $v_5^ + = 0.601{\rm{ 7}}$ , $v_1^ - = - 1.107{\rm{ 7}}$ , $v_2^ - = - 1.000{\rm{ 6}}$ , $v_3^ - = - 1.228{\rm{ 7}}$ , $v_4^ - = - 0.714{\rm{ 6}}$ , $v_5^ - = - 0.356{\rm{ 0}}$ .

Step 7. 根据式(17)得各方案的收益损失比值分别为: R1 = 0.296 3, R2 = 0.3461, R3 = 0.1820, R4 = 0.7718, R5 = 1.6904.

Step 8. 将 ${R_i}\left( {i = 1,2, \cdots ,5} \right)$ 按降序排列, 即 ${R_5} > {R_4} >$ ${R_2} > {R_1} > {R_3}$ , 则 ${A_5} \succ {A_4} \succ {A_2} \succ {A_1} \succ {A_3}$ , 故方案 ${A_5}$ 为最佳候选企业. 优选结果与文献[13]相同.

4 结束语

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