﻿ 基于信任关系和互补判断矩阵的群决策方法
 控制与决策  2020, Vol. 35 Issue (5): 1240-1246 0

### 引用本文 [复制中英文]

[复制中文]
LI Sheng-li, WEI Cui-ping, SONG Yan-hong. Group decision making method for fuzzy complementary judgement matrices based on trust relationships[J]. Control and Decision, 2020, 35(5): 1240-1246. DOI: 10.13195/j.kzyjc.2018.1288.
[复制英文]

### 文章历史

1. 扬州大学 数学学院，江苏 扬州 225002;
2. 太原师范学院 数学系，山西 晋中 030619;
3. 北京电子科技学院 计算机科学与技术系，北京 100070

Group decision making method for fuzzy complementary judgement matrices based on trust relationships
LI Sheng-li 1,2, WEI Cui-ping 1, SONG Yan-hong 3
1. Department of Mathematics, Yangzhou University, Yangzhou 225002, China;
2. Department of Mathematics, Taiyuan Normal University, Jinzhong 030619, China;
3. Department of Computer Science and Technology, Beijing Electronic Science and Technology Institute, Beijing 100070, China
Abstract: With the development of information and network technology, group decision making problems based on social network have attracted the attention of more and more researchers. This paper focuses on a consensus adjustment process and an alternative selection approach for the group decision making problems with fuzzy complementary judgement matrices under the social network environment. Firstly, we build a trust relationship among decision makers by fusing the information from three aspects: social relation among decision makers, the decision maker' status, and his/her knowledge ability. Then a consensus index is proposed to minimize consensus compensation between elements as far as possible, a consensus model based on the trust relationship is established, and the feasibility of the model is proved theoretically. Finally, the eigenvector centrality of the trust matrix is used to calculate the weights of experts, which is incorporated into aggregating the individual preference values and ranking alternatives. The effectiveness of the proposed method is further verified by a numerical example.
Keywords: group decision making    consensus    consistency    social network    trust relationship    fuzzy complementary judgement matrices
0 引言

1 预备知识 1.1 模糊互补判断矩阵

 (1)

1.2 社会网络关系

 (2)

aij=f(ei, ej), 则称A=(aij)n×n为邻接矩阵.矩阵A刻画了两两节点之间是否存在某种关系, 这种邻接矩阵的元素不是1就是0, 但是在许多情况下, 需要细化这一关系, 用[0, 1]之间的数字来刻画两两之间的关系强度, 因此邻接矩阵拓展为加权的邻接矩阵.

2 基于信任关系的共识调整模型

m个专家给出m个不同的互补判断矩阵, 当专家们的意见达成一定程度的共识时, 便可对每个专家的意见进行集结得到群体互补判断矩阵, 从而对方案进行排序.当群体共识水平没有达到决策者要求时, 需要引导个别专家对一些意见进行修正.下面分别讨论如何进行共识度测量, 如何建立专家之间信任关系, 以及如何利用专家之间的信任关系对不满足共识性要求的一部分元素进行修正, 以达到群体共识.

2.1 共识测度

 (3)

 (4)

 (5)

ACDh(0≤ ACDh≤1)的值越大, 代表专家eh与群体的共识度越高.设定阈值γ∈[0.5, 1), 当所有专家共识度都大于等于γ时, 群体达成共识.当专家的共识水平没有达到要求时, 需要根据下面的共识模型进行调整.

2.2 信任关系的建立

 (6)

2.3 共识调整模型

 (7)

step 1:设置l=0, R0k=(rij, 0k)n×n=(rijk)n×n, k=1, 2, …, m.

step 2:计算每个决策者经过第l次调整后共识指标ACDlk和一致性指标CIlk, k=1, 2, …, m.如果对于任意k都有ACDlkγ或者llmax, 则转到step 4;否则执行下一步.

step 3:当存在ACDlkγ时, 识别出共识度最低的专家eh, 并结合专家的一致性指标, 共识性指标以及社会关系矩阵求得专家之间的信任矩阵TDl.根据矩阵TDl得到专家eh信任的专家的集合TSh={eh1, eh2, …, e_hp}, 然后对(i, j)∈ APSh中元素根据式(7)进行修正.修正后的矩阵记为Rl+1h=(rij, l+1h)n×n, 令l=l+1, 转到step 2.

step 4:令Rk=Rlk, 输出Rk, k=1, 2, …, m.

 (8)

 (9)

 (10)

 (11)

 (12)

Rm中相应APS位置上元素的一致性测度得到了提高, 由式(5)可知ACDm>ACDm.

3 方案选择过程

step 1:确定专家的重要性指标向量.将专家之间的信任矩阵TD=(TDij)m×m行归一化, 即令.专家ei的重要性指标ui可根据文献[21]中的思想给出, 即

 (13)

U=(u1, u2, …, um)T可由如下线性方程组确定：

 (14)

step 2:根据各个专家的重要性指标计算专家的相关权重, 然后对R1, R2, …, Rm进行集结得到群体决策矩阵Rc=(rijc)n×n, 其中

 (15)

step 3:由文献[14]中方法可计算各个方案的得分向量

 (16)

step 4:根据得分值dp对方案进行排序, dp越大方案越优.

4 应用实例

step 1:建立决策者之间的信任关系.

step 1.1:利用式(1)得决策者的初始个体一致性指标分别为

step1.2:根据2.3节给出的共识性度量, 由式(4)计算每个决策者在元素层面的平均共识度分别为

q=6, 则每个决策者的共识性指标为

 (17)

step1.3:结合专家的共识性水平, 一致性水平以及表示决策者之间相互关系的矩阵SL, 由式(6)得到信任关系矩阵

α=0.70, 则信任关系可建立为

step2:对于给定共识阈值γ=0.75, 由于ACD1 < 0.75, 需要根据共识模型对e1的偏好信息进行修正.由矩阵T得TS1={e3, e4, e5}.令σ=0.5, 根据式(7)得到关于专家e1的建议为r12=0.55, r21=0.45, r16=0.862 5, r61=0.137 5, r36=0.725, r63=0.275.

step 3:将TD行归一化后得TD, 求解线性方程组(14), 得到专家的重要性指标u=(0.391 8, 0.454 2, 0.424 2, 0.465 1, 0.493 8), 进而求得专家相关权重w1=0.175 8, w2=0.203 8, w3= 0.190 3, w4=0.207 8, w5=0.222 3.

5 结论

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