Multiple attribute decision making (MADM) problems often consist of various types of quantitative and qualitative attributes. Quantitative attributes can be assessed by accurate numerical values, interval values or fuzzy numbers, while qualitative attributes can be evaluated by belief distributions, linguistic variables or intuitionistic fuzzy sets. However, the determination of attribute weights is still an open issue in MADM problems until now. In the traditional objective weight assignment method, attributes are usually assessed by accurate values. In this paper, an entropy weight assignment method is proposed to dealing with the situation where the assessment of attributes can contain uncertainties, e.g., interval values, or contain both uncertainties and incompleteness, e.g., belief distributions. The advantage of the proposed method lies in that uncertainties and incompleteness contained in the interval numerical values or belief distributions can be preserved in the generated weights. Specifically, several pairs of programming models to generate the weights of attributes are constructed in three different circumstances: (1) quantitative attribute expressed by interval values; (2) incomplete belief distribution with accurate belief degrees; and (3) belief distribution constituted by interval belief degrees. The evidential reasoning approach is then utilized to aggregate the distributions of attributes based on the generated attribute weights. The normalized interval weight vector is defined, and the characteristics of the weight assignment method are discussed. The proposed method has been experimented with real data to illustrate its advantages and the potential in supporting MADM with uncertain and incomplete information.

多属性决策（MADM）问题通常由各种类型的定量和定性属性组成。定量属性可以通过准确的数值，区间值或模糊数来评估，而定性属性可以通过信念分布，语言变量或直觉模糊集来评估。但是，到目前为止，属性权重的确定在MADM问题中仍然是一个未解决的问题。在传统的客观权重分配方法中，通常通过准确的值来评估属性。在本文中，提出了一种熵权分配方法来处理属性评估可能包含不确定性（例如区间值）或同时包含不确定性和不完整性（例如信念分布）的情况。提出的方法的优点在于，可以在生成的权重中保留区间数值或置信度分布中包含的不确定性和不完整性。具体而言，在三种不同情况下构造了几对用于生成属性权重的编程模型：（1）用区间值表示的定量属性；（2）信念分布不准确，信念程度准确；（3）由区间置信度构成的置信分布。然后利用证据推理方法基于生成的属性权重来聚合属性的分布。定义了归一化的区间权重向量，并讨论了权重分配方法的特点。