Some input-output classifier data envelopment analysis (DEA) models in multiplier and envelopment forms were developed to designate the status of flexible measures, playing either input or output roles. These models ignore the role of non-Archimedean epsilon in the input-output classification process. We show that these epsilon-free models may ignore some flexible measures in the performance evaluation process and hence the status of such flexible measure(s) can be randomly and inappropriately identified. To fill this gap, we develop a pair of epsilon-based multiplier and envelopment classifier models. We also develop an approach to find a suitable epsilon value for our developed classifier models. A case study of the supplier selection problem in the Iranian Space Research Center (ISRC) is provided to illustrate the potential application of our new epsilon-based approach.