Consistency and consensus modeling of linear uncertain preference relations

Interval operations, as currently defined, suffer from the problem of not satisfying the conditions of global complementarity and consistency of interval fuzzy preference relations (IFPRs). In this paper, we resolve this difficulty by constructing linear uncertain preference relations (LUPRs). By considering all the information and the uncertain distribution of an interval, we propose the concept of uncertain preference relations (UPRs) for the first time. Then we apply uncertainty distributions to characterize interval judgments that are considered as a whole to participate in the uncertain operation to achieve the desired conditions of global complementarity and consistency. Based on this, we prove that IFPRs and the definitions of their additive consistency are special cases of those of LUPRs. Moreover, we investigate two types of consensus models developed based on LUPRs between the minimum deviation and belief degree. We prove that the minimum deviation is a linear, increasing function of the belief degree, and then establish sufficient and necessary conditions for the consensus model to satisfy additive consistency. Finally, the LUPRs models presented in this paper is applied, incorporating with expert assistance in decision-making, to the sensitivity assessment of the meteorological industry in a region of China, and the LUPRs models can be utilized to obtain results with smaller deviations.