In this letter, we study throughput-optimal scheduling policies in single-path buffer-aided relay networks, where the packets of a user travel one fixed path from the source to the destination and do not require routing. We modify the well-known Max-Weight (MW) policy and propose a version of throughput-optimal algorithms, which we refer to as Modified MW (MMW). In MMW, by defining a suitably large threshold, a link's weight is proportional to just the corresponding local queue size either in the base station or the relay station, in most of the time slots. Moreover, a parameter can be adjusted for improving delay fairness between the relayed and direct users. MMW is suitable for use in both shared and independent channel scenarios and facilitates decentralized network implementations. Numerical results confirm that MMW is better able to provide similar performance for direct and relayed users in terms of queueing delay, compared with the existing algorithms.