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Today, Xiaobian brings you the intensive reading of the doctoral dissertation "Multi-attribute Group Decision-making Method Considering Expert Consensus under Generalized Multiplicative Preference Information" 6.1 Standard Hesitant Multiplicative Preference Relationship and Its Distance Measure.
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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）
In this issue, we will introduce the 6.1 standard hesitant multiplicative preference relationship and its distance measure of the intensive doctoral dissertation "Multi-attribute Group Decision-making Method Considering Expert Consensus under Generalized Multiplicative Preference Information" from three aspects: mind map, intensive reading content, and knowledge supplementation.
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）
In the preface to this chapter, the authors first review the research on Hesitant Multiplicative Preference Information (HMPR) and review the research. In actual decision-making, experts may be hesitant about the value of preference information and use hesitant multiplicative preference information to complete the evaluation work, which puts forward higher requirements for the consistency of individual information and the level of group consensus, so this chapter aims to propose a multi-attribute group decision making (MAGDM) method considering the consistency and consensus of HMPR.
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.
Then, the authors introduce the shortcomings of the current MAGDM method under HMPR considering consistency and consensus. Then, the research contribution of this chapter is introduced: this chapter constructs an optimization model based on HMPR consistency and consensus, and proposes the MAGDM method on this basis. Finally, the chapter arrangement of this chapter is introduced.
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）
The concepts related to hesitant multiplicative sets and HMPRs have been introduced in Chapter 2, and this subsection will review the definition of minimized original standardized hesitant multiplicative preference relations (NHMPRs) and construct improved distances between different NHMPRs to refine subsequent research. Due to the increasing complexity of MAGDM problems in practice, experts from different research backgrounds will most likely use the hesitant multiplication of different numbers of elements to evaluate alternatives, which may result in a sharp increase in computational complexity. In order to overcome this shortcoming, that is, to unify the number of elements in the hesitant multiplication number, some researchers have proposed a standardized method for the hesitant multiplication number.
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.
In this subsection, the author first introduces the definition of the standard hesitant multiplication number, as shown in the figure below.
In this subsection, the author initially presents the definition of standard hesitant multiplicative numbers, as shown in the figure below.
Based on the preference coefficient, the authors give the distance measures between different standard hesitant multiplier numbers and prove the satisfying properties of the distance measure according to their definitions.
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.
Similarly, the HMPR composed of hesitant multiplication numbers needs to be standardized to reduce the complexity of the information aggregation process, and the resulting definition of NHMPR is as follows.
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.
Since the final alternative selection stage in this chapter takes the method of transforming the NHMPR into a Multiplicative Preference Relation (MPR) to determine the ranking of alternatives, the concept of an improved scoring function of the standard hesitant multiplicative number will be introduced and the related propositions will be obtained, as shown in the figure below.
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.
According to definition 6.2, the calculation method of the distance measure between different NHMPRs can be obtained, and a simplified calculation formula of the distance measure is obtained.
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）
Hesitant multiplication is a mathematical tool used to deal with ambiguity and uncertainty, and is commonly used in multi-criteria decision problems. It is an extension of the traditional fuzzy number, taking into account the hesitation and uncertainty of decision-makers in making decisions. Compared with the traditional fuzzy number, the hesitant multiplication number has stronger expression ability and more flexible modeling ability, which can better reflect the subjective preference and uncertainty degree of decision makers. In the decision-making process, the hesitant multiplication number can be used to indicate the degree of preference of decision-makers for different decision-making schemes, and then used to calculate weights or rank, so as to assist decision-makers in making more reasonable decisions.
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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参考资料：ChatGPT、百度百科
Bibliography:
WANG Rui. Multi-attribute group decision-making method considering expert consensus under generalized multiplicative preference information[D]. Sichuan: Southwest Jiaotong University, 2022.
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