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今天小编为您带来精读博士论文《广义乘性偏好信息下考虑专家共识的多属性群决策方法》6.1标准犹豫乘性偏好关系及其距离测度。
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Today, the editor brings the " 6.1 standard hesitation multiplicative preference relation and its distance measure of the intensively read PhD thesis 'A Multi-Attribute Group Decision-Making approach considering expert consensus under generalised nultiplicative preference information'".
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一、内容摘要（Content Summary）
本期推文将从思维导图、精读内容、知识补充三个方面介绍精读博士论文《广义乘性偏好信息下考虑专家共识的多属性群决策方法》的6.1标准犹豫乘性偏好关系及其距离测度。
This issue of the article will introduce the 6.1 standard hesitation multiplicative preference relation and its distance measure from three aspects: mind mapping, in-depth reading content, and supplementary knowledge, focusing on the doctoral dissertation "Multi-attribute Group Decision-making Method Considering Expert Consensus under Generalized Multiplicative Preference Information".
二、思维导图（Mind Mapping）
三、精读内容（Detailed Reading Content）
（一）章节前言（Chapter Introduction）
在该章节的前言，作者首先对犹豫乘性偏好信息（HMPR）进行了相应的研究综述，并进行了研究评述。而实际决策中，专家可能对于偏好信息的取值犹豫不决而运用犹豫乘性偏好信息完成评估工作，此时对于个体信息的一致性与群体共识水平提出了更高的要求，因此，本章旨在提出考虑HMPR的一致性与共识的多属性群决策（MAGDM）方法。
In the introduction of this chapter, the author provides a comprehensive review and evaluation of the research on hesitant multiplicative preference relations (HMPR). In practical decision-making, experts may exhibit hesitancy in determining preference values and utilize hesitant multiplicative preference information to carry out evaluations. Under such circumstances, higher requirements are placed on the consistency of individual information and the level of consensus within a group. Therefore, the aim of this chapter is to propose a multi-attribute group decision-making (MAGDM) approach that considers the consistency and consensus of HMPR.
然后，作者介绍了目前考虑一致性与共识的HMPR下的MAGDM方法的不足。接着，介绍了本章节的研究贡献：本章构建了一种基于HMPR的一致性与共识的优化模型，并在此基础上提出了MAGDM方法。最后，介绍了本章的章节安排。
Subsequently, the author discusses the limitations of existing MAGDM methods that consider consistency and consensus under HMPR. The research contributions of this chapter are then presented, which include the construction of an optimization model based on HMPR that considers consistency and consensus, and the development of the MAGDM method based on this model. Finally, the chapter concludes with an overview of its structure.
（二）标准犹豫乘性偏好关系及其距离测度（Standard Hesitant Multiplicative Preference Relations and Their Distance Measures）
在第2章中已经介绍了有关犹豫乘性集与HMPR的相关概念，本小节将在此基础上回顾最小化原始标准化犹豫乘性偏好关系（NHMPR）的定义，并构建不同 NHMPR之间的改进距离以完善后续研究。由于实际中MAGDM问题的复杂程度不断提高，来自不同研究背景的专家将极有可能运用不同元素数量的犹豫乘性数来评估备选方案，因此可能造成的后果是运算复杂程度急剧增加。为了克服这个不足，即统一犹豫乘性数中的元素数量，有研究者提出了犹豫乘性数的标准化方法。
In Chapter 2, relevant concepts pertaining to hesitant multiplicative sets and HMPR have already been introduced. This subsection builds upon the previous knowledge by revisiting the definition of minimizing normalized hesitant multiplicative preference relations (NHMPR) and constructing improved distance measures between different NHMPR to enhance further research. As the complexity of MAGDM problems in practice continues to increase, experts from different research backgrounds may employ hesitant multiplicative numbers with different element quantities for evaluating alternative solutions. Consequently, this can lead to a significant increase in computational complexity. To overcome this limitation and achieve a unified element quantity in hesitant multiplicative numbers, researchers have proposed a standardization method for hesitant multiplicative numbers.
在该小节，作者首先介绍了标准犹豫乘性数的定义，如下图所示。
In this subsection, the author initially presents the definition of standard hesitant multiplicative numbers, as shown in the figure below.
基于偏好系数构建的标准犹豫乘性数，作者给出了不同标准犹豫乘性数之间的距离测度并且根据其定义证明了距离测度满足的性质。
The author then provides the distance measures between different standard hesitant multiplicative numbers based on preference coefficients and proves the properties of these distance measures according to their definitions.
同样地，由犹豫乘性数组成的HMPR也需要标准化操作以减少信息集结过程运算的复杂程度，得到的NHMPR的定义如下。
Similarly, HMPR composed of hesitant multiplicative numbers also requires a standardization operation to reduce the computational complexity during the information aggregation process. The definition of NHMPR obtained is as follows.
由于在后续的研究中，本章在最终的备选方案选择阶段采取将NHMPR转化为一个乘性偏好关系（MPR）的方式以确定备选方案的排序，因此将引入标准犹豫乘性数的改进记分函数的概念并且得到了相关的命题，如下图所示。
Considering that in subsequent research, this chapter adopts the approach of transforming NHMPR into a multiplicative preference relation (MPR) during the final stage of selecting alternative solutions, the concept of an improved scoring function using standard hesitant multiplicative numbers is introduced and related propositions are derived, as shown in the figure below.
根据定义6.2可以得到不同NHMPR之间的距离测度的计算方法，并得到了该距离测度的简化计算式。
Based on Definition 6.2, the calculation method for the distance measures between different NHMPR is obtained, along with a simplified formula for this distance measure.
四、知识补充——犹豫乘性数（ Supplementary Knowledge - Hesitant Multiplicative Numbers）
犹豫乘性数是一种用于处理模糊性和不确定性的数学工具，常用于多准则决策问题中。它是对传统的模糊数进行了扩展，考虑了决策者在做决策时的犹豫和不确定性因素。与传统的模糊数相比，犹豫乘性数具有更强的表达能力和更灵活的建模能力，能够更好地反映决策者的主观偏好和不确定性程度。在决策过程中，犹豫乘性数可以用来表示决策者对不同决策方案的偏好程度，进而用于计算权重或进行排序，从而辅助决策者做出更合理的决策。
Hesitant multiplicative numbers are mathematical tools used to handle ambiguity and uncertainty, commonly employed in multi-criteria decision-making problems. They extend traditional fuzzy numbers by considering the hesitation and uncertainty factors that decision-makers encounter when making decisions. Compared to traditional fuzzy numbers, hesitant multiplicative numbers possess stronger expressive power and more flexible modeling capabilities, allowing them to better reflect decision-makers' subjective preferences and the degree of uncertainty. In the decision-making process, hesitant multiplicative numbers can be used to represent decision-makers' preferences for different decision alternatives, enabling the calculation of weights or rankings to assist decision-makers in making more informed decisions.
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参考文献：
王睿. 广义乘性偏好信息下考虑专家共识的多属性群决策方法 [D]. 四川: 西南交通大学, 2022.
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