Abstract: Ecological civilization construction theory plays a very important role in the theoretical system of Chinese style socialism. While focusing on economic and social development, China pays more attention to the protection of natural resources and realize the harmonious coexistence of humans and nature. On December 12, 2016, the National Development and Reform Commission and other four departments released the Evaluation Index System for Ecological Civilization Construction. The public eco-environment satisfaction has been clearly defined as the main indicator to reflect public satisfaction with the ecological environment quality. Because of the complexity of evaluation and difficulty of quantifying information, probabilistic linguistic information becomes an effective form applied in ecological environment evaluation. Constructing new evaluation models and methods of probabilistic linguistic information turns to be a key scientific problem with both theoretical and practical significance in the evaluation decision-making.
When hesitancy fuzzy sets are used for decision evaluation, the importance degree of each membership degree is the same. However, in reality, the importance degree of each membership degree should be different due to the personal preference of decision makers and the difference in the number of decision makers in group decision making. In order to solve this problem,ZHU(2014) combined probabilistic information with hesitant fuzzy set and introduced probabilistic hesitant fuzzy set which consists of several possible membership values associated with their corresponding probabilistic information. To expand information representation dimensions, HAO et al.(2017) proposed the probabilistic dual hesitation fuzzy set, which not only considered multiple membership values of elements and their corresponding probabilistic information, but also paid attention to the non-membership of elements and their respective importance degrees.
Operational laws are the core problem of multi-attribute decision making. From the existing literature, the information operational laws of the probability dual hesitation fuzzy sets are all based on Archimedean norm, such as the probability dual hesitation fuzzy information operational laws based on Algebra norm and the probability dual hesitation fuzzy information operational laws based on Einstein norm. However, after the operation of the above two algorithms, the number of membership degree and non-membership degree elements of the integration result will mostly increase, thus greatly increasing the computation amount. At the same time, the corresponding probability of membership degree and non-membership degree of integration results will decrease with the increase of the number of operations, resulting in the distortion of probability information. Therefore, it is necessary to further explore the operation rules of the probabilistic dual hesitation fuzzy information and construct related integration operators.
In this paper, some operational laws of probabilistic dual hesitant fuzzy information are defined based on Archimedean copula and corresponding co-copula, which provide a new idea for the fusion of probabilistic dual hesitant fuzzy information. These operational laws can not only keep the closure of the operation, but also reflect the relationship between probabilistic dual hesitant fuzzy information. Based on the further study of the properties of these operational laws, the Archimedean copula weighted probabilistic dual hesitant fuzzy arithmetic average aggregation operator and the Archimedean copula weighted probabilistic dual hesitant fuzzy geometric average aggregation operator are proposed, and their properties are also discussed. Finally, the multiple attribute group decision making approach of probabilistic dual hesitant fuzzy information based on these operators is developed. By this approach, a formula can be used to determine the weighting vector of attributes. An instance of public eco-environment satisfaction evaluation is used to illustrate the effectiveness of the proposed approach. In addition, the comparison with existing methods is discussed.

Key words: multiple attribute group decision making, probabilistic dual hesitant fuzzy information, Archimedean copula, weighted aggregation operator, information entropy

摘要： 本文基于阿基米德copula和相应的协copula定义了概率对偶犹豫模糊信息的一些运算法则,为概率对偶犹豫模糊信息的融合提供了一种新思路。这些运算法则不仅可以保持运算的封闭性,而且可以反映出概率对偶犹豫模糊信息之间的关系。在进一步研究运算法则性质的基础上,提出了阿基米德copula加权概率对偶犹豫模糊算术平均算子和几何平均算子,并讨论了其性质。针对概率对偶犹豫模糊信息环境下属性权重信息未知的多属性群决策问题,给出了求解属性权重的公式,并利用上述算子进行信息集结,获得了群决策方法。最后,通过公众生态环境满意度评价的实例说明所提方法的有效性,利用参数对评价结果的影响说明决策方法的灵活性。此外,还将本文方法与现有方法进行了对比分析。

关键词: 多属性群决策, 概率对偶犹豫模糊信息, 阿基米德copula, 加权集成算子, 信息熵