Consideration of preference-orders requires the use of an extended rough set model called Dominance-based Rough Set Approach (DRSA). The rough approximations defined within DRSA are based on consistency in the sense of dominance principle. It requires that objects having not-worse evaluation with respect to a set of considered criteria than a referent object cannot be assigned to a worse class than the referent object. However, some inconsistencies may decrease the cardinality of lower approximations to such an extent that it is impossible to discover strong patterns in the data, particularly when data sets are large. Thus, a relaxation of the strict dominance principle is worthwhile. The relaxation introduced in this paper to the DRSA model admits some inconsistent objects to the lower approximations; the range of this relaxation is controlled by an index called consistency level. The resulting model is called variable-consistency model (VC-DRSA). We concentrate on the new definitions of rough approximations and their properties, and we propose a new syntax of decision rules characterized by a confidence degree not less than the consistency level. The use of VC-DRSA is illustrated by an example of customer satisfaction analysis referring to an airline company.