Supporting the objective of reaching consensus is a notable research area in the context of group decision making. Recently, several approaches based on an allocation of information granularity have been proposed for supporting it. Those approaches improve the consensus among the individuals participating in the decision problem by means of the required flexibility provided by the information granularity level. However, they do not take into account that the adjusted preferences could be very different from the preferences communicated by the individuals. Furthermore, they cannot be applied in decision problems where various criteria are considered to evaluate the choices. To overcome these shortcomings, we introduce a new consensus approach based on minimum adjustment for multi-criteria group decision making with an allocation of information granularity. An experimental study is also provided to illustrate this consensus approach and analyze its performance.