统计与数据科学学院知识库

随着社会的不断进步和经济的蓬勃发展，决策活动已遍及生活的各个领域，其中多属性决策作为一种重要的决策方法，得到了广泛应用。然而，传统的以精确数为基础的多属性决策，例如，类似于中小企业信用风险评价、罕见病孤儿药的临床综合评价、企业基于营商环境的投资选择、银行信用债投资选择、以及企业信用债发行时主承销商的选择等决策问题在面对复杂多变的决策环境时面临模糊性和不确定性挑战，这促使模糊多属性决策成为研究的重点。

然而，传统模糊集在表达决策信息时存在犹豫性和偏好信息不足的问题。尽管犹豫模糊集解决了犹豫性问题，却未充分表达决策者偏好。概率犹豫模糊集的提出，通过将概率引入模糊集来表达决策者偏好，有效解决了偏好损失问题，使得基于概率偏好的犹豫模糊信息多属性决策方法成为研究热点。本文主要从信息测度的研究出发，深入探讨了基于概率偏好犹豫模糊信息测度的决策方法及其应用，不仅研究了单一隶属度维度的概率犹豫模糊集，还扩展到综合考虑多维隶属度的概率图犹豫模糊集，对两种不同概率偏好犹豫模糊集的信息测度及基于这些信息测度的多属性决策方法进行了深入研究，并将其应用于实际的决策问题中，旨在为解决诸如具有模糊性和不确定性的决策问题提供新的思路和方法。本文研究的主要工作如下：

针对决策者需通过单一隶属度维度表达决策信息，且需兼顾表达犹豫性与偏好的复杂决策场景。本文研究了概率犹豫模糊环境下基于信息测度的多属性决策方法。主要工作概述如下：

（1）在概率犹豫模糊环境下，本文提出了概率犹豫模糊元素的相似度测度，并在特定条件下证明了这种新提出的相似度实质上是一种概率犹豫模糊元素的熵。基于此，进一步提出了一种在属性权重未知情境下基于相似度与熵的多属性群决策方法。

（2）在概率犹豫模糊环境下，本文提出了一种全新的概率犹豫模糊元素距离测度。在此基础上，依据所提出的新概率犹豫模糊元素距离公式，进一步创新性地提出了一种最大距离偏差求权重的方法。随后，本文将概率犹豫模糊集与ELECTRE II方法相结合，提出了一种在权重未知情况下基于改进距离的ELECTRE II的多属性决策方法。

（3）在概率犹豫模糊环境下，本文将信息能的概念引入到概率犹豫模糊集中，创新性地提出了一种新的概率犹豫模糊集的相关系数及其加权形式。考虑到决策者在决策过程中习惯于使用语言变量进行表达的局限性，本文还提出了一种将决策者的语言变量评估信息转化为概率犹豫模糊信息的方法。最后，基于新提出的相关系数，提出了一种概率犹豫模糊多属性群决策方法。

针对决策者需通过多维隶属度表达决策信息，并同时表达犹豫性和偏好的复杂决策场景。本文研究了概率图犹豫模糊环境下基于信息测度的多属性决策方法。主要工作概述如下：

（1）在概率图犹豫模糊环境下，鉴于关于概率图犹豫模糊集距离测度的研究还很少见或不足，本文提出了两种类型的距离测度：一种考虑了元素个数，另一种则不考虑，以克服前者可能存在的不足。同时，基于距离测度的概念，本文还创新性地提出了一种新的概率图犹豫模糊元素的熵测度。鉴于当前在该模糊环境下多属性决策方法的研究还很少见或不足，结合这些测度和TOPSIS方法，本文进一步提出了一种基于距离和熵测度的TOPSIS多属性决策方法。

（2）在概率图犹豫模糊环境下，鉴于关于概率图犹豫模糊集相似度的研究还很少见或不足，本文提出了概率图犹豫模糊集的信息能概念，并基于此深入研究，提出了两种无需考虑元素个数的Dice相似度，进一步将其推广至广义Dice相似度。随后，鉴于当前在该模糊环境下多属性决策方法的研究还很少见或不足，基于以上研究成果，提出了基于相似度的多属性决策方法。

（3）在概率图犹豫模糊环境下，鉴于关于概率图犹豫模糊集相关系数的研究还很少见或不足，基于对信息能的深入研究，本文进一步提出了概率图犹豫模糊集的关联度测度，并创新性地提出了几个概率图犹豫模糊集的相关系数及其加权形式。鉴于当前在该模糊环境下多属性决策方法的研究还很少见或不足，本文提出了一种基于相关系数的多属性群决策方法。在该方法中，还进一步提出了将投票决策信息转化为概率图犹豫模糊信息的方法。

With the continuous progress of society and the vigorous development of economy, decision-making activities have been widely used in every field of life, among which multi-attribute decision-making, as an important decision-making method, has been widely used. However, traditional multi-attribute decisions based on exact numbers, for example, Problems such as credit risk assessment of small and medium-sized enterprises, clinical comprehensive evaluation of orphan drugs for rare diseases, investment choice of enterprises based on business environment, investment choice of bank credit bonds, and selection of lead underwriter when issuing corporate credit bonds face challenges of fuzziness and uncertainty in the face of complex and changeable decision-making environment, which makes fuzzy multi-attribute decision-making to become a focal point of research.

Nevertheless, traditional fuzzy sets have deficiencies in expressing decision-making information, mainly manifested in hesitancy and insufficient preference information. Although hesitant fuzzy sets resolve the issue of hesitancy, they do not fully express decision-makers' preferences. The introduction of probabilistic hesitant fuzzy sets effectively addresses the problem of preference loss by incorporating probability into fuzzy sets to express decision-makers' preferences, making multi-attribute decision-making methods based on probabilistic hesitant fuzzy information a research hotspot. Starting from the research of information measure, this paper deeply discusses the decision method and its application based on the probabilistic preference hesitant fuzzy information measure. It not only studies the probabilistic hesitant fuzzy set with single membership dimension, but also extends to the probabilistic picture hesitant fuzzy set with comprehensive consideration of multi-dimension membership. This paper studies the information measures of two different probabilistic preference hesitant fuzzy sets and the multi-attribute decision making methods based on these information measures, and applies them to practical decision making problems to provide new ideas and methods for solving fuzzy and uncertain decision making problems such as credit risk assessment of small and medium-sized enterprises. The main work of this paper is as follows:

Addressing the complex decision-making scenarios where decision-makers need to express decision information through a single membership degree dimension while also accommodating hesitation and preference, this paper investigates multi-attribute decision-making methods based on information measures in the probabilistic hesitant fuzzy environment. The main work is summarized as follows:

1. In the probabilistic hesitant fuzzy environment, this paper first proposes the similarity measure of probability hesitant fuzzy elements, and proves in specific conditions that the new proposed similarity measure is actually the entropy of probability hesitant fuzzy elements. Based on this, a multi-attribute group decision-making method using the new proposed similarity measure and entropy in the situation where the attribute weights are unknown is proposed.
2. In the probabilistic hesitant fuzzy environment, a new probabilistic hesitant fuzzy element distance measure is proposed in this paper. On this basis, according to the proposed new probabilistic hesitant fuzzy element distance formula, a new method of calculating the weight by the maximum distance deviation is further innovatively proposed. Then, combining probabilistic hesitation fuzzy set with ELECTRE II method, this paper proposes an ELECTRE II multi-attribute decision making method based on improved distance in the case of unknown weights.
3. In the probabilistic hesitant fuzzy environment, this paper introduces the concept of information energy into probabilistic hesitant fuzzy sets, and innovatively proposes a new correlation coefficient and its weighted form for probabilistic hesitant fuzzy sets. Considering the limitations of decision makers who are accustomed to using language variables for expression in the decision-making process, this paper also proposes a method to transform the decision maker's language variable evaluation information into probabilistic hesitant fuzzy information. Finally, based on the newly proposed correlation coefficient, a probabilistic hesitant fuzzy multi-attribute decision-making method is proposed.

Addressing the complex decision-making scenarios where decision-makers need to express decision information through multi-dimensional membership degrees while also accommodating hesitation and preference, this paper studies multi-attribute decision-making methods based on information measures in the probabilistic picture hesitant fuzzy environment. The main work is summarized as follows:

1. In the probabilistic picture hesitant fuzzy environment, in view of the lack of research on distance measures for probabilistic picture hesitant fuzzy sets, two types of distance measures are proposed in this paper: one takes into account the number of elements, and the other does not, to overcome the possible shortcomings of the former. At the same time, based on the concept of distance measure, a new entropy measure for probabilistic picture hesitant fuzzy elements is proposed. Given that research on multi-attribute decision-making methods in this fuzzy environment is still relatively scarce or insufficient, combining these measures and TOPSIS method, this paper further proposes a TOPSIS multi-attribute decision method based on distance and entropy measures.
2. In the probabilistic picture hesitant fuzzy environment, in view of the lack of research on similarity measures for probabilistic picture hesitant fuzzy sets, this paper proposes the concept of information energy of probabilistic picture hesitant fuzzy sets. Based on this, two Dice similarity measures without considering the number of elements are proposed, which are further extended to generalized Dice similarity measures, aiming to fill the gaps and shortcomings of the existing research. Then, given that research on multi-attribute decision-making methods in this fuzzy environment is still relatively scarce or insufficient, based on the above research results, the multi-attribute decision method based on similarity is constructed.

In the probabilistic picture hesitant fuzzy environment, in view of the lack of research on correlation coefficients for probabilistic picture hesitant fuzzy sets, this paper further puts forward the correlation measure of the probabilistic picture hesitant fuzzy sets, and innovatively puts forward the correlation coefficients and their weighted forms of several probabilistic picture hesitant fuzzy sets. Given that research on multi-attribute decision-making methods in this fuzzy environment is still relatively scarce or insufficient, a multi-attribute group decision making method based on correlation coefficient is proposed in this paper. In this method, a further method is proposed to convert voting decision information into probabilistic picture hesitant fuzzy information.