Additive and Multiplicative Consistency Modeling for Incomplete Linear Uncertain Preference Relations and Its Weight Acquisition

The use of consistency methods to supplement or generate missing information so as to obtain a rational and logical ranking list has been a key point in studies of the incomplete preference relation. Existing works on incomplete interval fuzzy preference relations (IFPRs) utilize only endpoints of the original intervals and generate missing values using interval operations, resulting in the loss and distortion of information. This article considers IFPRs as linear uncertain preference relations (LUPRs), and introduces belief degree and inverse uncertainty distributions to explore the consistency and ranking problems with LUPRs. Definitions related to uncertain preference relations (UPRs) as well as their additive and multiplicative consistency are first given. Then, models to solve weight vectors are proposed under both consistency scenarios. Finally, algorithms that generate missing information in incomplete LUPRs are provided. By building mathematical relationships between the minimum deviation and the belief degree, analytic formulas for missing values are obtained when the minimum deviation is achieved. It is verified that consistent UPRs and the proposed weight-vector solving models under incomplete preference relations are extensions of the traditional IFPRs.