Multihop relay-based cellular networks are attracting much interest because of their throughput enhancement, coverage extension, and low infrastructure cost. In these networks, relay stations (RSs) between a base station (BS) and mobile stations (MSs) drastically increase the overall spectral efficiency, with improved channel quality for MSs located at the cell edge or in shadow areas, and enhanced throughput of MSs in hot spots. These relay-based networks require an advanced radio resource management scheme because the optimal amount of radio resource for a BS-to-RS link should be allocated according to the MS channel quality and distribution, considering the interference among RSs and neighbor BSs. In this paper, we propose optimal resource planning algorithms that maximize the overall utility of relay-based networks under a proportional fair scheduling policy. In the first phase, we determine an optimal scheduling policy for distributing BS-to-RS link resources to RSs. In the second phase, we determine the optimal amount of the BS-to-RS link resources using the results of the first phase. The proposed algorithms efficiently calculate the optimal amount of resource without exhaustive searches, and their accuracy is verified by comparison with simulation results, in which the algorithms show a perfect match with simulations.