We propose a mixed integer programming formulation for the single machine scheduling problem with release times and the objective of minimizing the weighted sum of the start times. The basic formulation involves start time and sequence determining variables, and lower bounds on the start times. Its linear programming relaxation solves problems in which all release times are equal. For the general problem, good lower bounds are obtained by adding additional valid inequalities that are violated by the solution to the linear programming relaxation. We report computational results and suggest some modifications based on including additional variables that are likely to give even better results.