摘 要

大规模群决策问题近年来发展出了很多新的研究方向，其中基于社会网络和最小成本共识的研究已经成为了群决策领域中的研究热点和前沿问题，具有较高的学术价值和应用前景。但目前各个研究领域之间相互较为独立，尚未对实际决策问题提出一个完整的解决方案。本文旨在将各个领域中的核心模型与算法进行优化整合，形成一套完整的基于网络共识环境下大规模群决策的理论、方法和应用体系。

首先，本文介绍了基础的线性规划、二次规划理论以及最小成本共识模型，将其作为本文研究的预备知识。在经典最小成本共识模型的基础上，通过考虑群体决策对个体决策产生的影响，定义了相关满意度指标，并对原模型进行优化，得到一个标准的二次规划模型。对于优化后的模型，本文运用了拉格朗日乘数法对优化模型进行对偶问题的建模与求解，并对该模型的约束条件和变量进行了经济解释，证明该模型具有一定的学术价值和现实意义。

其次，本文在最小成本共识优化模型的基础上，对基于偏好矩阵的共识达成模型进行了扩展和优化，通过对相关一致性指标的定义，构建偏好矩阵共识优化模型。由于该模型与最小成本共识优化模型之间具有很强的关联性和类比性，所以本文将两个模型进行了各类指标的比较分析，解释了该模型的理论价值和实践意义。

在共识网络演化问题上，本文使用了Louvain算法对整个网络进行聚类，目的是将整个群体划分为若干个社区（即“子群”），保证每个社区之间相互独立而社区内部具有较强的凝聚力。对于每个社区内部的决策者观点，通过构建最小成本共识优化模型得到该社区的群体观点，再将每个社区的群体观点进行整合，得到该问题的最终方案集，随后通过构建偏好矩阵共识优化模型对该方案集中的方案进行分析排序，得到最终方案及决策分析结果。

最后，本文通过对某一地区的政府和所有污染企业为达成统一排污量进行协商谈判这一现实问题进行了共识网络演化以及决策分析，验证了整个共识达成过程的科学性和优化模型的实用性，为今后进一步解决大规模复杂群决策问题提供了一种新的思路。

本文的研究创新与特色：一方面，将大规模群决策中基于共识网络演化以及最小成本共识模型、数学优化等多个领域的相关研究方向进行有机整合，得到了一系列新的理论研究结果；另一方面，运用数学优化方法对各种形式的共识模型进行深化和扩展,同时运用仿真方法对大规模群体决策过程进行仿真模拟，形成了一套较为完整的基于数据、模型、算法三个层面驱动的大规模群决策问题的规范研究框架。

关键词：大规模群决策；二次规划；最小成本共识模型；共识网络演化

Abstract

Many research directions have been developed on the large-scale group decision making (LSGDM) problems in recent years. Among them, research based on social network and the minimum cost consensus model has become a hotspot and frontier issue in the field of group decision making (GDM), and it also has strong research potential. However, each research area is relatively independent of each other, and there is not a complete solution process for real cases in real life at present. This thesis aims to integrate and optimize the core model in each area, and to form a complete framework of theories, methods and applications for LSGDM based on the network consensus.

First of all, this thesis introduces the basic linear programming models, quadratic programming models and minimum cost consensus models, which are used as preliminary knowledge for this study. Based on the original minimum cost consensus model, this thesis defines relevant satisfaction indexes by considering the influence of group on individual decision making and optimizes the original model to obtain a standard quadratic programming model. For the optimization model, this thesis uses the Lagrange multiplier method to obtain the dual problem of the optimization model, and economically explained the constraints and variables of the model, proving that the model has certain practical significance.

Secondly, based on the minimum cost consensus optimization model, this study similarly optimizes the preference matrix consensus model and builds a preference matrix consensus reaching process optimization model by defining relevant consistency indicators. Because the proposed model has a strong analogy with the minimum cost consensus optimization model, this thesis compares the two models with various indicators, explaining the theoretical value and practical significance of the proposed model.

Regarding to the problem of consensus network evolution, this thesis uses the Louvain algorithm to cluster the entire network. It aims to divide the entire network into several communities (i.e. "subgroups"), and to ensure that each community is independent of each other while there is strong cohesion within the community. For the opinions of every decision makers (DMs) within each community, this thesis obtains the group opinion of the community by using a minimum cost consensus optimization model, and integrates the group opinion of each community to obtain the final set of solutions to the problem. Then, we use the preference matrix consensus optimization model to analyze and sort the solutions in the alternative set to obtain the final solution and conclusion.

Finally, this thesis conducts consensus network evolution and decision analysis concerning the problem that the government and all polluting enterprises in a certain area negotiate to reach a uniform pollution emission. It verifies the scientific nature of the entire consensus reaching process and the practicability of the optimization model, and it also provides a new way of thinking for solving LSGDM problems.

There are two research innovations and characteristics in this thesis. On the one hand, this thesis integrates research directions based on consensus evolution network and minimum cost consensus model in LSGDM problems, and obtains a family of new research results. On the other hand, this thesis uses mathematical optimization technique to extend various forms of consensus models. At the same time, this thesis uses the algorithm to simulate the LSGDM process, forming a complete research framework of LSGDM problems based on the three levels like data, model, and algorithm.

Keywords：Large scale group decision making; Quadratic programming; Minimum cost consensus model; Consensus network evolution

目 录

第1章 绪论 1

1.1 研究目的及意义 1

1.2 国内外研究现状 1

1.3 研究内容与框架 6

1.4 本章小结 7

第2章 预备知识 8

2.1 线性规划及其对偶问题 8

2.2 二次规划及其对偶问题 9

2.3 最小成本与最大补偿共识模型 10

2.4 本章小结 11

第3章 最小成本共识优化模型 12

3.1 模型描述 12

3.2 满意度指标的定义 12

3.3 模型的构建 14

3.4 对偶问题的求解 14

3.5 模型的经济解释 16

3.6 模型分析 16

3.6.1 算例分析 16

3.6.2 与原模型的比较分析 17

3.7 本章小结 20

第4章 偏好矩阵共识优化模型 21

4.1 模型描述 21

4.2 一致性指标的定义 21

4.3模型的构建 24

4.4 模型分析 26

4.4.1 算例分析 26

4.4.2 与最小成本共识优化模型的比较分析 27

4.5 本章小结 28

第5章 大规模群决策共识网络演化 29

5.1 Louvain算法描述 29

5.2 网络演化过程 31

5.3 减排共识问题的建模 32

5.3.1 问题背景描述 33

5.3.2 基于Louvain算法的社区网络演化 33