Hybrid backward and forward dynamic programming based Lagrangian relaxation for single machine scheduling

In this paper we consider the single machine scheduling problem with precedence constraints to minimize the total weighted tardiness of jobs. This problem is known to be strongly NP-hard. A solution methodology based on Lagrangian relaxation is developed to solve it. In this approach, a hybrid backward and forward dynamic programming algorithm is designed for the Lagrangian relaxed problem, which can deal with jobs with multiple immediate predecessors or successors as long as the underlying precedence graph representing the precedence relations between jobs does not contain cycles. All precedence constraints are considered in this dynamic programming algorithm, unlike the previous work using forward dynamic programming where some precedence relations were ignored. Computational experiments are carried out to analyze the performance of our algorithm with respect to different problem sizes and to compare the algorithm with a forward dynamic programming based Lagrangian relaxation method. The results show that the proposed algorithm leads to faster convergence and better Lagrangian lower bounds.