Some Properties of the Optimal Decomposition Conditions for the Single Machine Tardiness Problem

We consider the well-known optimal decomposition algorithm for the tardiness problem. The algorithm presents conditions which determine the positions a job could occupy in an optimal sequence. This resulted in optimal solutions for up to 100 jobs. We analyze these conditions and present simplified conditions. We then study a more recent Rule which when combined with these conditions resulted in optimal solutions for up to 500 jobs. We present several properties under which this recent rule is satisfied. We provide mathematical proofs for our properties. We believe that our study will enable more theoretical research in this field and will eventually enable optimal solutions for very large job sets.