We present a new dominance rule for the single machine total weighted tardiness problem with job dependent penalties. The proposed dominance rule provides a sufficient condition for local optimality. We show that if any sequence violates the dominance rule, then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. We also develop a new algorithm based on the dominance rule, which is compared to a number of competing heuristics for a set of randomly generated problems. Our computational results of over 40000 problems indicate that the proposed algorithm dominates the competing heuristics in all runs.