In this paper, we binarize a novel algorithm called the Fox Optimizer using a two-step technique and test its performance against the Set Covering Problem. Additionally, we explore the incorporation of chaotic maps into the binarization process. To benchmark the binary Fox Optimizer, we compare it with two well-known and documented metaheuristics: Particle Swarm Optimization and Grey Wolf Optimizer. Each algorithm is tested with standard, sine chaotic, elitist, and elitist sine chaotic binarization rules. Our findings demonstrate that elitist configurations, especially when combined with sine chaotic binarization, consistently yield superior results, providing robust and reliable performance in obtaining high-quality solutions. Conversely, standard binarization configurations exhibit enhanced convergence capabilities, proving effective for problems with rapid convergence requirements or lower complexity. This study highlights the importance of aligning algorithm configurations with specific problem characteristics to optimize performance in practical applications.