Abstract: For multi-attribute decision making problems, because of the complexity of human thinking and personal preferences, there often exist some situations with high degree of uncertainty where a decision organization consisting of several experts is not very sure about a value, and is hesitant among several possible values when providing the membership degree of an element to a set. To better describe this decision scenario, hesitant fuzzy sets were originally introduced by Torra and Narukawa in 2009.Hesitation fuzzy sets allow an element to belong to a set with multiple different values of membership, effectively solving the problem of inconsistent preferences of multiple experts. However, the number of membership degree in different hesitant fuzzy elements (HFE) may be different, and the uncertainty of hesitant fuzzy sets includes fuzzy uncertainty and hesitant uncertainty, which directly leads to the complexity of calculating hesitant fuzzy entropy. Existing literature has made significant contributions to the study of hesitant fuzzy entropy, but there are still two shortcomings. One is that it ignores hesitant uncertainty and only considers fuzzy uncertainty, the other is that we must artificially add new membership degrees based on risk preference,when comparing the entropy values of any two hesitant fuzzy elements. Although some literature has addressed one of the issues, both have not yet been addressed simultaneously. The concept of non-probability entropy was proposed by Deluca and Termini to measure the uncertainty of fuzzy sets in 1972. Then, Kosko proposed a concise non-probabilistic fuzzy entropy formula from the perspective of distance, which shows a ratio of the distance: the distances between the fuzzy information and its nearest and farthest non-fuzzy neighbors. Motivated by the principle of the non-probabilistic entropy for fuzzy sets, hesitant fuzzy non-probabilistic entropy measure is studied in this paper.
Firstly, we critically review the existing entropy measures for HFE, and demonstrate that these entropy measures have some shortcomings. Because the multiple membership degrees of hesitant fuzzy element coincide with the multidimensionality of Euclidean space, this paper considers that HFE are seen as some points in Euclidean space, and the membership degrees in HFE could be called coordinates accordingly. In the sequel, we deeply analyze the evolution law of fuzzy uncertainty and hesitant uncertainty in Euclidean space.Based on the hesitancy fuzzy Euclidean distance, the concept of hesitant fuzzy non-probability entropy is proposed creatively. It is a ratio of distance between the hesitant fuzzy elements and the two reference points(the full HFE and the empty HFE).In order to show the superiority of the hesitant fuzzy non-probabilistic entropy, a comparative analysis is carried out with the existing entropy measures. The results of the comparative analysis show that the proposed hesitant fuzzy non-probabilistic entropy has a higher distinguishing ability. And when comparing the entropy sizes of two hesitant fuzzy elements, it makes full use of raw decision information without artificially adding some new membership degrees, avoiding distortion of decision information. In addition, the proposed hesitant fuzzy non-probabilistic entropy can effectively combine the hesitating uncertainty with the fuzzy uncertainty without using a bivariate aggregation function to aggregate the two uncertainties.
On the basis of the above theoretical analysis, a group decision method is developed by applying the proposed hesitant fuzzy non-probabilistic entropy. An investment company wants to invest in tourism projects. There are great risks in tourism project investment, so an in-depth investigation and careful decision-making is necessary. After preliminary investigation, four tourism projects are selected as alternative projects.Renowned experts are invited to conduct a risk assessment of the alternatives, and make a comprehensive risk assessment of each option in order to select the best investment option. Four representative risk assessment indicators are considered in the assessment process: market, policy, facility, and management risk. According to the decision information and proposed decision method, the ranking results of tourism projects are obtained.
The follow-up study will attempt to expand the proposed method to other fuzzy environments. In addition, as the complexity of decision-making problems and number of experts increase, we will study large-scale group decision-making and combine it with complex network theory.

Key words: hesitant fuzzy sets, TOPSIS, non-probabilistic entropy

摘要： 不同的犹豫模糊元包含的隶属度个数往往不同,且犹豫模糊集的不确定性包括模糊不确定性与犹豫不确定性两个方面,这直接导致犹豫模糊熵计算的复杂性。针对此问题,将犹豫模糊元置于欧式空间进行分析,任意一个犹豫模糊元看作欧式空间中的点,而隶属度值则是相应点的坐标,欧式空间的多维性与犹豫模糊元的多值性刚好契合。基于此种思路,首先,基于空间分析,给出了犹豫模糊非概率熵的公理化定义。然后,基于空间距离提出犹豫模糊非概率熵测度公式,即为距离之比,犹豫模糊元分别与最近的(最远的)两个非模糊点之间的距离之比,该犹豫模糊非概率熵能够有效融合犹豫不确定性与模糊不确定性,同时解决了犹豫模糊元中隶属度个数不同带来的计算问题。最后,结合TOPSIS思想,提出了一种基于熵权法的犹豫模糊多属性群决策途径,并用实例分析验证其科学性和有效性。

关键词: 犹豫模糊集, TOPSIS, 非概率熵