Abstract Imperfect data have to be processed in a different way when they are involved in classification or regression tasks. For instance, in the case of supervised classification, predicting a subset of candidate classes when imperfect data are present is preferable to predicting a single class using point prediction methods which offer less guarantee. This is even more necessary when machine learning algorithms are involved in sensitive domains, e.g., medical field. Of course, the corresponding classifier should be cautious, i.e., the predicted subset of candidate classes contains the true class, but also relevant or precise, i.e., its size is not too large. The well known Strong Dominance based classification algorithm (SD) is a good candidate as a robust method but some times the predicted subsets are too large. This paper proposes two cautious classification algorithms within the framework of belief functions that aim to make flexible the SD classifier. Indeed, these classifiers control the granularity of the partial order that is returned by Strong Dominance relation. Ideally, an output which is a compromise between the robustness, i.e., cautiousness, offered by the SD and the precision offered by point prediction classifiers must be guaranteed for the decision-maker regarding his preferences. The first classifier is based on a interval criterion that is built from the pignistic criterion to which is associated a threshold while the second classifier is based on a convex mixture between the pignistic criterion and the Strong Dominance binary relation. The outputs of these three classifiers are theoretically and experimentaly compared.