In this research, the problem of scheduling a set of jobs on a single machine to minimize total weighted tardiness with unequal release dates is considered. We present a new dominance rule by considering the time depending orderings between each pair of jobs. The proposed rule provides a sufficient condition for local optimality. Therefore, if any sequence violates the dominance rule then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. Based on the dominance rule, an algorithm is developed which is compared to a number of heuristics in the literature. Our computational results indicate that the proposed algorithm dominates the competing algorithms in all runs, therefore it can improve the upper bounding scheme and can be used in reducing the number of alternatives in any enumerative algorithm. Furthermore, the proposed dominance rule is incorporated in a branch and bound algorithm in conjunction with lower bounding scheme, branching condition and search strcitegy. To the best of our knowledge, author know of no other published exact approach for l|rj|
problem. This enhances contribution of our study in the literature.