Note—A Note on Shwimer's Theorem A

In 1972 Joel Shwimer published some interesting research “on the n-job, one-machine, sequence-independent scheduling problem with tardiness penalties … where the goal is to minimize the total penalty costs resulting from the tardiness of the jobs” (Shwimer, J. 1972. On the n-job, one-machine, sequence-independent scheduling problem with tardiness penalties: a branch-bound solution. Management Sci.18 (6) B-301–B313.). The proposed algorithm is essentially based on his Theorem A. Unfortunately the given proof of Theorem A contains an inconsistency. In this note we present a different proof of the mentioned theorem.