In many applications it can be advantageous for the decision maker to have multiple options available for a possible realization of the project. One way to increase the number of interesting choices is in certain cases to consider in addition to the optimal solution x* also nearly optimal or approximate solutions which differ in the design space from x* by a certain value. In this paper we address the efficient computation and discretization of the set E of epsilon-approximate solutions for scalar optimization problems. For this we will suggest two strategies to archive and update the data coming from the generation process of the search procedure, and will use Differential Evolution coupled with the new archivers for the computation of E. Finally, we will demonstrate the behavior of the archiver empirically on some academic functions as well as on two models related to space mission design.