In this paper the one-machine scheduling problem with linear earliness and tardiness costs is considered. The7 cost functions are job dependent and asymmetric. The problem consists of two sub-problems. The first one is to find a sequence of jobs and the second one is to find the job completion times that are optimal for the given sequence.9 We consider the second sub-problem and propose an algorithm solving the problem in O(n log n) time. 2005 Published by Elsevier Ltd.11