Decomposition of the single-machine total tardiness problem with a precedence constraint

The purpose of this study is to extend the well-known decomposition theorem for the single-machine total tardiness problem to the case when a precedence constraint between a pair of jobs is set. Since the decomposition theorem forms the basis of strict solution algorithms for the ordinary single-machine total tardiness problem, it is expected that we can construct efficient solution algorithms by the extended decomposition theorem even for the problem with a precedence constraint. We also extend the elimination rules to restrict the possible job positions in an optimal schedule that are also utilized in strict solution algorithms. Then, we construct a simple strict solution algorithm for our problem and show the effectiveness of our results by numerical experiments.