Dual Models and Return Allocations for Consensus Building Under Weighted Average Operators

Recent studies have examined minimum cost consensus models from the moderator’s perspective. However, current dual models aimed at deriving a maximum return have been limited to special cases. A further problem has been the return allocation to the individuals. With this in mind, this article presents two general dual consensus models, the first of which is a minimum cost consensus model, and the second of which allows the different individuals to have different tolerance levels. In both models the individual opinions and the group opinion are connected using a weighted average operator. To understand the economic significance of these models, some properties, such as the weak duality property, the optimality property, and the strong duality property are given. The relationships between the individual opinions, the unit costs, and the unit returns are also proven, and the core return allocation problem is addressed. It was found that in both cases, the costs the moderator needed to pay to a given individual were precisely equal to the returns the individual deserved. Numerical case studies were used to illustrate the proposed models, with the simulation results providing insights into the practical use of these optimization consensus models.