This paper considers bandwidth allocation and scheduling problems on Ethernet Passive Optical Networks (EPON). EPON is one of the good candidates for the optical access network. This paper formulates the bandwidth allocation problem as a nonlinear mathematical one and characterizes the optimal bandwidth allocation which maximizes weighted sum of throughput and fairness. Based upon the characterization, two heuristic algorithms are suggested with various numerical tests. The test results show that our algorithms can be used for efficient bandwidth allocation on the EPON. This paper also shows that the WSPT (Weighted Shortest Processing Time) rule is optimal for minimization the total delay time in transmitting the traffic of the given allocated bandwidth.