In this paper, we propose a customized genetic algorithm (GA) to find the minimum-cost guideway network (GN) of personal rapid transit (PRT) subject to connectivity, reliability, and traffic capacity constraints. PRT is a novel transportation concept, where a number of automated taxi-sized vehicles run on an elevated GN. One of the most important problems regarding PRT is how to design its GN topology for given station locations and the associated inter-station traffic demands. We model the GN as a directed graph, where its cost, connectivity, reliability, and node traffics are formulated. Based on this formulation, we develop the GA with special genetic operators well suited for the GN design problem. Such operators include steady state selection, repair algorithm, and directed mutation. We perform numerical experiments to determine the adequate GA parameters and compare its performance to other optimization algorithms previously reported. The experimental results verify the effectiveness and efficiency of the proposed approach for the GN design problem having up to 210 links.