Premature convergence is one of the best-known drawbacks that affects the performance of evolutionary algorithms. An alternative for dealing with this problem is to explicitly try to maintain proper diversity. In this paper, a new replacement strategy that preserves useful diversity is presented. The novelty of our method is that it combines the idea of transforming a single-objective problem into a multiobjective one, by considering diversity as an explicit objective, with the idea of adapting the balance induced between exploration and exploitation to the various optimization stages. Specifically, in the initial phases, larger amounts of diversity are accepted. The diversity measure considered in this paper is based on calculating distances to the closest surviving individual. Analyses with a multimodal function better justify the design decisions and provide greater insight into the working operation of the proposal. Computational results with a packing problem that was proposed in a popular contest illustrate the usefulness of the proposal. The new method significantly improves on the best results known to date for this problem and compares favorably against a large number of state-of-the-art schemes.