The rapid advancement of Urban Air Mobility (UAM) presents new challenges in the strategic placement of vertiports and the efficient allocation of air taxis to these vertiports. In this study, we formulate a mathematical model based on the capacitated set cover model to determine the optimal vertiport locations as well as the number of air taxi allocations while minimizing total infrastructure and operational costs. Our formulation integrates constraints on fleet size, vertiport capacity, and demand coverage to ensure an efficient UAM network which can satisfy the demand in a given region of interest. We demonstrate the applicability of our modeling approach on a case study for the southwest region of Ohio, USA and perform several numerical experiments using real-world population data and varying demand scenarios to analyze the impact of operational range and demand percentage on network configuration. Our findings for the case of Ohio highlight the trade-off between infrastructure investment and operational effectiveness in UAM planning.