Modeling the minimum cost consensus problem in an asymmetric costs context

The unit costs of up- and down-adjustments are not equal in some real-life minimum cost consensus (MCC) problems, such as when each individual has two different cost coefficients that depend on the adjustment direction of his/her opinion. To solve these problems, the MCC model with directional constraints (MCCM-DC) is constructed on the basis of goal programming theory and the rectilinear distance function. To analyse the impact of individuals' limited compromises and tolerance behaviors on the consensus modeling, we further develop the ε-MCCM-DC and the threshold-based (TB)-MCCM-DC. Then, the relationships and transformation conditions of these models are investigated. Furthermore, the validity of the proposed models is demonstrated by the case of trans-boundary pollution control negotiations in China's Taihu Lake Basin. The analysis results show the following: First, the consensus opinion obtained from MCCM-DC is more inclined to the lower cost direction, and its total consensus costs will no longer ascend after reaching a critical point with the increase of unit adjustment costs. Second, the optimal solution of MCCM-DC is the lower bound of ε-MCCM-DC and the upper bound of TB-MCCM-DC. Compared with consensus models without directional constraints, the proposed models can obtain a better consensus opinion at lower costs due to the flexibility in adjusting individual opinions and can also characterize the MCC problems in a more realistic way.