A note on dynamic programming formulations for scheduling job classes with changeover times on a single machine

We consider the problem for scheduling N jobs that are partitioned to F disjoint classes (or families) subject to changeover (or setup) times in order to minimize the total weighted completion time. We first identify a published inefficient forward dynamic programming (DP) procedure due to Potts (1991) that works in O(F²N^F+d), where d is the number of different changeover values (d ≤ F²). We then show how to make a forward DP progress in O(F²N^F) by modifying the evaluation of stage costs. Although the posed scheduling problem is believed to be amenable to backward DP (rather than forward DP) in the literature, the resulting forward DP can work at least as efficiently as an existing (best-known) backward DP procedure. In other words, both forward and backward DPs can be rendered equally efficient when the definitions of states are changed appropriately for DP procedures. This is why DP formulations are often said to be quite an art.