基于指数加权的区间直觉模糊熵及其应用

Interval-valued Intuitionistic Fuzzy Entropy Based on Exponential Weighting and Its Application

熵是刻画模糊集不确定性程度的一个重要手段。为刻画区间直觉模糊集的不确定性,首先基于区间数的Hukuhara差(简称H-差)提出区间直觉模糊集的核区间的概念,其能够有效反映区间直觉模糊集中隶属度与非隶属度的力量对比所产生的模糊性。考虑到区间直觉模糊集的不确定性由模糊性和犹豫性共同决定,提出了更符合人们直觉的区间直觉模糊集不确定度量的基本准则,由于区间直觉模糊集的模糊程度和犹豫程度所占比重并不能完全确定,因此为更好地描述两者对区间直觉模糊集不确定性程度的影响,利用指数函数加权的方法构造出一种新的区间直觉模糊熵。通过性质讨论和不同方法下区间直觉模糊熵的对比实例分析可知,在犹豫度区间长度相同的情况下,区间直觉模糊熵随着核区间的左右区间数的增大而减小;在核区间相同的情况下,区间直觉模糊熵随着犹豫度区间的左右区间数的增大而增大,符合其不确定性度量的基本准则。所提方法能充分反映不确定性随模糊性和犹豫性的增加而增加,这符合人们的直觉。其次,分析并验证了当区间直觉模糊集退化为直觉模糊集时,该方法构造的直觉模糊熵也能够有效度量直觉模糊集的不确定性程度。最后,将新的熵公式有效地应用到属性权重完全未知的多属性决策分析中,并通过实例验证了其合理性,为解决多属性决策问题提供了一种新的思路。

Entropy is an important means to describe the uncertainty degree of fuzzy sets.To depict the uncertainty of interval-valued intuitionistic fuzzy sets,this paper first put forward the definition of core interval of interval- valued intui- tionistic fuzzy sets based on Hukuhara difference ( H- difference) of interval numbers,which can effectively reflect the fuzziness generated by the force comparison between membership degree and non-membership degree of interval-valued intuitionistic fuzzy sets.Considering the uncertainty of interval-valued intuitionistic fuzzy sets is codetermined by the fuzziness and hesitancy,this paper proposed the basic criterion of the uncertainty measurement of interval- valued intui- tionistic fuzzy sets,which more accords with human intuition.Due to the difficulty for completely determining the proportion of fuzziness and hesitancy,in order to better describe the influence of the fuzziness and hesitancy on the uncertainty degree of interval-valued intuitionistic fuzzy sets,this paper presented a new interval-valued intuitionistic fuzzy entropy based on the exponential weighted method.Comparison example analysis under properties discussion and diffe- rent for interval-valued intuitionistic fuzzy entropy demonstrates that when hesitancy degree interval is the same,the interval -valued intuitionistic fuzzy entropy decreases with the increase of the number of left and right intervals of the core interval,and when core interval is the same,the new fuzzy entropy increases with the increase of the number of left and right intervals of the hesitancy degree interval,which are accord with basic principle of uncertainty measurement.The proposed method completely shows that the uncertainty can increase with the increase of the fuzziness and hesitancy,which is accordance with human intuition.Secondly,this paper analyzed and verified that when the interval- valued intui- tionistic fuzzy sets degenerate into the intuitionistic fuzzy sets,the new fuzzy entropy constructed by the proposed me- thod can measure the degree of uncertainty of the intuitionistic fuzzy sets effectively.Finally,the proposed new entropy formula is applied effectively in multiple attributes decision-making analysis with unknown attribute weights and the rationality of the method is verified by an example,which provides a new way to solve multi-attribute decision-making problem.