Assignment of attribute weights with belief distributions for MADM under uncertainties

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.