We present a neuro-dominance rule for single machine total weighted tardiness problem. To obtain the neuro-dominance rule (NDR), backpropagation artificial neural network (BPANN) has been trained using 5000 data and also tested using 5000 another data. The proposed neurodominance rule provides a sufficient condition for local optimality. It has been proved that if any sequence violates the neuro-dominance rule then violating jobs are switched according to the total weighted tardiness criterion. The proposed neuro-dominance rule is compared to a number of competing heuristics and meta heuristics for a set of randomly generated problems. Our computational results indicate that the neuro-dominance rule dominates the heuristics and meta heuristics in all runs. Therefore, the neuro-dominance rule can improve the upper and lower bounding schemes.