多标记学习和协同过滤算法（基于模糊评价信息的多属性决策方法双犹豫模糊集多属性决策2）

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本期推文小编将继续分享博士论文《基于模糊评价信息的多属性决策方法研究》第二章的内容，接下来我们开始今天的学习吧！

This issue of the tweet editor will continue to share the intensive reading series of the doctoral dissertation "Research on Multi-attribute Decision Making Method Based on Fuzzy Evaluation Information", bringing you the content of the second chapter of the dissertation, let's start today's learning!

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思维导图

本节的内容主要是分别将 PBM 算子和 PGBM 算子应用于 q 阶双犹豫模糊集中，依次提出 q 阶双犹豫模糊 PBM 算子、q 阶双犹豫模糊 PGBM 算子，以及它们的加权形式 q 阶双犹豫模糊加权PBM 算子以及 q 阶双犹豫模糊加权 PGBM 算子的具体内容。

This part is a preface to the content of the second chapter, which introduces us to the background and significance of the research questions in this chapter, as well as the specific content of the research.

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精读内容

2.2.1q阶双犹豫PBM算子

作者根据 q 阶双犹豫模糊元的运算法则，可以得到以下定理2.2和定理2.3，并分别进行了证明。

The authors can obtain the following Theorem 2.2 and Theorem 2.3 based on the algorithm of q-order double hesitant fuzzy elements, and prove them respectively.

接着，作者强调了 q-RDHFPBM 算子具有的两个参数 s 和 t，当 s 和 t取值不同时，q-RDHFPBM 算子可以有不同的形式，针对 s 和 t 不同的取值对 q-RDHFPBM 算子的特殊性进行讨论，分为几种情况，如下所示：

Then, the authors highlight the two parameters s and t that the q-RDHFPBM operator has. q-RDHFPBM operator can take different forms when s and t take different values. The special features of the q-RDHFPBM operator are discussed for the different values of s and t and are divided into several cases as follows.

2.2.2q阶双犹豫模糊加权PBM算子

为了解决q-RDHFPBM 算子最主要的缺点：没有考虑 q 阶双犹豫模糊元的权重信息，作者提出了 q-RDHFPBM算子的加权形式，根据q 阶双犹豫模糊元的运算法则，得到定理2.4，其证明有界性的过程与上文类似。

To address the most important drawback of the q-RDHFPBM operator: the weight information of the q-order double hesitant fuzzy element is not considered, the authors propose a weighted form of the q-RDHFPBM operator and obtain Theorem 2.4 according to the operator of the q-order double hesitant fuzzy element, which proves boundedness in a similar way as above.

2.2.3 q阶双犹豫模糊PGBM算子

首先，作者给出了q阶双犹模糊PGBM算子的定义：

First, the authors give the definition of the q-order double Judas fuzzy PGBM operator.

接着，根据定义 2.3 中给出的 q 阶双犹豫模糊元的运算法则，可以得到定理 2.5、2.7，与 q-RDHFPBM 算子一样，参数 s 和 t 的一些特殊取值决定了 q-RDHFPGBM算子存在一些特殊形式，针对 q-RDHPGBM 算子的特殊形式进行如下讨论。

Then, according to the operators of q-order double hesitant fuzzy elements given in Definition 2.3, Theorems 2.5 and 2.7 can be obtained. As with the q-RDHFPBM operator, some special values of the parameters s and t determine the existence of some special forms of the q-RDHFPGBM operator, and the special forms of the q-RDHPGBM operator are discussed as follows.

2.2.4q阶双犹豫模糊加权PGBM算子

与 q-RDHFPBM 类似，q-RDHFPGBM 算子也不具有处理输入变量权重的能力，作者给出了 q-RDHFPGBM 算子的加权形式，即q-RDHFWPGBM算子。

Similar to q-RDHFPBM, the q-RDHFPGBM operator does not have the ability to handle the weights of the input variables, and the authors give a weighted form of the q-RDHFPGBM operator, the q-RDHFWPGBM operator.

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补充知识

在本节内容中提到的q 阶双犹豫模糊集和PBM算子是什么意思呢？

What do the q-order double hesitant fuzzy sets and the PBM operator mentioned in this section mean?

q 阶双犹豫模糊集：q阶双犹豫模糊集是经典的q阶正交模糊集的有效拓展形式。与双犹豫模糊集类似，q阶双犹豫模糊集由几个独立的介于闭区间0到1在的离散值组成，分别表示可能的隶属度和非隶属度，并且满足的约束条件为隶属度最大值的q次方与非隶属度最大的q次方之和不超过1。

q-order double hesitant fuzzy set: q-order double hesitant fuzzy set is an efficient extended form of the classical q-order orthogonal fuzzy set. Similar to the double hesitant fuzzy set, the q-order double hesitant fuzzy set consists of several independent discrete values between the closed interval 0 to 1 in, representing the possible affiliation and non-affiliation degrees respectively, and satisfying the constraint that the sum of the qth power of the maximum of the affiliation degree and the qth power of the maximum of the non-affiliation degree does not exceed 1.

PBM算子：PBM算子是将PA算子和BM算子相结合的道路一种新的复合型算子，该算子充分吸收了PA算子和BM算子的优点，适合解决复杂环境下的多属性决策问题。

PBM operator: the PBM operator is a new composite operator of the road combining the PA operator and the BM operator by He et al. (2015), which fully absorbs the advantages of the PA operator and the BM operator and is suitable for solving multi-attribute decision problems in complex environments.

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参考资料：DeepL翻译

参考文献：[1]赵红梅. 基于模糊评价信息的多属性决策方法研究 [D]. 北京交通大学, 2021.

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文案 |Yuan

排版 |Yuan

审核 |Qian

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