Maximum expert consensus models with both type- $$\alpha $$ and type- $$\varepsilon $$ constraints

The maximum expert consensus model (MECM) aims to maximize the number of consensual decision-makers (DMs) within a limited budget. However, it may fail to achieve high group satisfaction or even cannot reach an acceptable consensus due to its neglect of the group consensus level, resulting in type- $$\alpha $$ constraints not being satisfied. To address this issue, we extend the existing MECM by considering both type- $$\alpha $$ and type- $$\varepsilon $$ consensus constraints to enable the group consensus level and the number of consensual DMs as large as possible. Firstly, we construct a dual-MECM that considers the above two constraints. Secondly, we further develop a dual-MECM considering compromise limits (dual-MECM-CL). To provide a reference for budgeting, a dual minimum cost consensus model (dual-MCCM) is established to determine the upper and lower bounds of the budget. Subsequently, we explore the relationships between the two proposed MECMs and the existing MECM. Finally, the effectiveness of the proposed models is illustrated by numerical examples. The results show that: (1) The dual-MECM can ensure that the majority of DMs reach consensus while maintaining a high group consensus level. (2) With a limited budget, the improvement of the overall consensus level will lead to the reduction in the number of consensual DMs. (3) Consideration of individual compromise limits may reduce the number of consensual DMs within the same budget. Therefore, the proposed models can derive a more reasonable consensus result due to full consideration of consensus measurements and DMs’ behaviors.