Establishing a network of transit routes with satisfactory demand coverage is one of the main goals of transit agencies in moving towards a sustainable urban development. A primary concern in obtaining such a network is reducing operational costs. This paper deals with the problem of minimizing construction costs in a grid transportation network while satisfying a certain level of demand coverage. An algorithm is proposed following the general idea of “constructive algorithms” in related literature. The proposed algorithm, in an iterative approach, selects an origin-destination with maximum demand, generates a basic shortest-path route, and attempts to improve it through a route expansion process. The paper reports the scenarios and further details of the algorithm considered for expanding a transit route in a grid network. A random 6×10 grid network is applied to report the results. The results support that application of the proposed algorithm notably reduces the operational costs for various amounts of demand coverage.