Data envelopment analysis (DEA) is a linear programming method for measuring the performance and efficiency of units called decision-making units (DMUs). In many real-world performance measurement problems, the input and output data are not precisely known. Furthermore, the data may include dual-role factors that can be considered an input and output factor simultaneously. We propose a novel DEA model in the presence of imprecise data and imprecise dual-role factors by developing a new pair of mixed binary linear epsilon-based DEA models. The proposed models estimate the lower and upper bound efficiency scores in the presence of interval input, output, and dual-role factors by considering a fixed and unified production frontier for all DMUs. We then extend our models by including the weak ordinal dual-role factors. In contrast to the existing methods that exclude the dual-role factors, we include the dual-role factors and find a strictly positive value for the lower bound of the weights of inputs, outputs, and dual-role factors. We present a case study to demonstrate the applicability and exhibit the superiority of our approach over the existing methods.