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本期推文将通过思维导图，精读内容，知识补充三个板块，复刻来自于南京航空航天大学的博士论文《后悔规避视角下的灰色风险型决策方法及其应用研究》，为大家带来《博士论文犹豫模糊元距离测度》，一起来看看吧！

Share interest, spread happiness, increase knowledge, leave beautiful! Dear, this is LearningYard Academy.

In this issue, we will use mind mapping, intensive reading and knowledge supplement to copy the doctoral thesis of Nanjing University of Aeronautics and Astronautics titled "Grey Risk-based Decision Making Method and Its Application from the Perspective of Regret Avoidance", and bring you "Doctoral thesis Hesitation fuzzy element distance Measure". Let's take a look!

【1】思维导图

论文研究内容版块内容的思维导图如下所示：

The mind map of the research content section of the paper is as follows:

【2】精读内容

犹豫模糊集在决策背景下反映了群体间意见难于形成一致意见的犹豫不确定性，因而成为描述和处理现实复杂群决策问题的利器和国内外不确定决策领域的最新研究热点。

Hesitancy fuzzy sets reflect the hesitancy uncertainty which is difficult to form consensus among groups in the context of decision making, so they become a sharp tool to describe and deal with complex group decision making problems and a new research hotspot in uncertain decision making field at home and abroad.

距离测度向来是模糊集理论中一个基础而又十分重要的课题，也是决策理论中构造决策方法的主要途径之一。该论文的这个章节围绕研究所涉及到的相关理论和知识进行梳理，对现有的犹豫模糊元的距离构造思想进行了简要回顾并分析了各自的优缺点。该章节具体的逻辑顺序是按照距离测度理论提出时间来的，在介绍基础测度的同时引入了其他思想对距离测度公式进行了改进。

Distance measure is always a basic and very important subject in fuzzy set theory and also one of the main ways to construct decision method in decision theory. This chapter of the thesis is based on the relevant theories and knowledge involved in the research, and briefly reviews the existing hesitant fuzzy element distance construction ideas and analyzes their advantages and disadvantages. The specific logical sequence of this chapter is based on the distance measure theory. While introducing the basic measure, other ideas are introduced to improve the distance measure formula.

下面本文对Hausdorff距离测度和基于集论的距离测度进行一个简要回顾。

The following is a brief review of the Hausdorff distance measure and the distance measure based on set theory.

基于平均距离的思想与Hausdorff和平均距离的思想如出一辙，它们避免了复杂的附加运算且对犹豫模糊信息实施直接计算，体现了犹豫模糊集的精髓，只不过基于Hausdorff距离的存在计算过程信息损失严重的问题，而基于平均距离思想的不能满足距离公理中自反性条件。

The idea based on average distance is similar to the idea of Hausdorff and average distance. They avoid complex additional operations and directly calculate hesitant fuzzy information, which reflects the essence of hesitant fuzzy set. However, based on the existence of Hausdorff distance, there is a serious problem of information loss in the calculation process. However, the one based on the mean distance idea cannot satisfy the reflexivity condition in the distance axiom.

基于集论(Set-Theoretic)的距离测度其本质是通过基本的集合运算诸如交与并等基础运算刻画两向量间的相近与否的程度。基于集论的方法的距离研究广泛存在于人类知识的表达、行为分析以及实际问题的解决等领域。

The essence of distance measure based on Set theory is to describe the degree of similarity between two vectors by basic set operations such as intersection () and union (). Distance research based on set theory is widely used in the fields of human knowledge expression behavior analysis and practical problem solving.

【3】知识补充

本文介绍了HFE的距离测度，在这里再简要介绍犹豫模糊熵的距离测度。

The distance measure of HFE is introduced in this paper, and then the distance measure of hesitancy fuzzy entropy is briefly introduced here.

以下是犹豫模糊熵的定义。

The following is the definition of hesitation fuzzy entropy.

第一种是基于犹豫模糊值与补集差异程度的熵测度。Xu和Xia从模糊集的熵测度出发，提出了适用于犹豫模糊集的熵度量公式：

The first is the entropy measure based on hesitancy fuzzy value and complement difference degree. Starting from the entropy measure of fuzzy sets, Xu and Xia proposed an entropy measure formula suitable for hesitant fuzzy sets:

其次是基于得分函数和偏差函数的熵测度，Wei等提出的熵测度方式基于得分函数和偏差函数。

Secondly, entropy measure is based on score function and deviation function. The entropy measure proposed by Wei et al is based on score function and deviation function.

最后是以犹豫模糊元各元素与0.5距离为基准的熵测度。Farhadinia以0.5为参照，将犹豫模糊元中各元素与0.5之间的标准Hamming距离纳入犹豫模糊熵测度因素。这种计算方式较为简便，十分容易上手。

Finally, the entropy measure is based on the distance between each element and 0.5. Taking 0.5 as a reference, Farhadinia included the standard Hamming distance between each element and 0.5 in the hesitating fuzzy element into the measure factor of hesitating fuzzy entropy. This method is simple and easy to use.

本期的分享就到这里，如果您对今天的文章有独特的想法，欢迎给我们留言，让我们相约明天，祝您今天过得开心快乐！

本文由LearningYard学苑原创，仅代表作者个人观点，如有侵权请联系删除。

翻译参考来源：有道翻译。

内容参考来源：

[1]王铁旦. 犹豫模糊环境下基于TODIM的可信性FMEA改进方法研究[D].昆明理工大学, 2021.