The consensus models with interval preference opinions and their economic interpretation

•We construct two consensus models based on minimum cost and maximum return.•Interval preferences are applied to denote individuals׳ opinions.•Economic meaning and relations among variables of the consensus models are explored.•Correlations exist between unit return, cost, individual׳s and moderator׳s opinions.•We consider the interests of both moderator and individuals.

This paper aims to explore the case when an individual opinion is interval preference in consensus decision making. And for this purpose, we construct two multi-objective optimization models: one based on the minimum cost from the perspective of the moderator, the other the maximum return from the perspective of the individuals. On the basis of multi-objective programming theories, these multi-objective programming models are then transformed into two single-objective linear programming models, i.e., the primal model and the dual model. The primal model focuses on how to obtain a consensus with the minimum cost, while the dual model is concerned with how to get the maximum return. With the help of dual linear programming theories, we have revealed the following economic significance of the primal-dual consensus models: the primal-dual consensus models can not only help us probe into the relations between the minimum cost paid by the moderator and the maximum return expected by individuals who changed their opinions before, but also help us explore the relations between the unit cost that the moderator pays each individual, unit return that each individual receives, each individual opinion and the consensus opinion. This paper with the aid of theoretical analysis and an illustrative example indicates that once the consensus is obtained, the optimal unit return and optimal consensus opinion value are also solved. This paper also points out that the amount of the total return acquired by all the individuals who have abandoned their original opinions before is equivalent to that of the total cost paid by the moderator to reach the consensus. This paper also argues that there exists compact correlations between the individual׳s unit return, the consensus opinion, the individual׳s interval opinion, and the moderator׳s unit cost.