In many real world applications, the performance of entities with undesirable outputs should be assessed while imprecise and vague information is presented. In this study, a fully fuzzy data envelopment analysis (FFDEA) approach is propounded to analyze the relative efficiency of decision making units (DMUs) in which the acceptance degree of decision maker that fuzzy constraints may be breached is incorporated. In order to achieve this purpose, the interval expectation of trapezoidal fuzzy numbers and the order relationship of trapezoidal fuzzy numbers are employed. Also, after converting the FFDEA model into an interval DEA model and then a bi-objective DEA model, parametric examination is applied to aggregate objectives. Therefore, input- and output-oriented FFDEA approaches with undesirable outputs are provided to estimate the relative efficiency of processes whilst the acceptance degree of the violated fuzzy constraints is considered. Datasets from the existing studies are used to clarify the introduced technique and to describe its applicability.