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, i.e. if any sequence violates the dominance rule, switching a violating job either lowers the total weighted tardiness or leaves it unchanged. We introduce an 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 over •30000 problems indicate that the proposed algorithm dominates the competing heuristics in all runs. Furthermore, the new dominance rules can be used in reducing the number of alternatives for finding the optimal solution in complete enumeration techniques. We show that the proposed dominance rule increases the number of global dominance relationships generated by the Emmons’ Rule which is used heavily in literature to restrict the search space. We also show that having a better upper bound value usually improves the lower bound value which is obtained from the linear lower bound.