The conventional optimistic data envelopment analysis (DEA) model typically evaluates decision-making units (DMUs) using the best relative efficiency that is derived from the estimated efficient production frontier, while the pessimistic DEA model evaluates the DMUs according to the estimated inefficient production frontier. Although the results of these models vary significantly, both approaches should be incorporated for an overall performance evaluation, and a significant body of literature exists on this topic. This paper contributes to the literature by providing entirely new non-convex optimistic and pessimistic models by applying free disposal hull (FDH) technology, which is important in real-life scenarios. These models may experience a lack of sufficient discrimination power. Accordingly, two improved versions of both approaches are developed. The first version formulates the models in the presence of the slack variables. In the second version, we propose FDH super-efficiency models that may become infeasible. Thus, we propose modified models without infeasibility problem. The paper concludes with a comprehensive empirical study to illustrate the details and applicability of the proposed models.