On the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

We consider the ordinary NP-hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. In [S. S. Panwalkar, C. Koulamas, An O(n) algorithm for the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines, European Journal of Operational Research 221 (2012), 7–13.], the authors presented quadratic algorithm for the problem when each job has its smaller processing time on the first machine. In this note, we improve the running time of the algorithm to O(n log n) using recently introduced modified binary tree data structure. Furthermore, we present O(n) algorithm for the scheduling sequence minimizing makespan for the no-wait variant.