We consider a single-machine scheduling problem where penalties occur for jobs that either commence before their target start date or are completed after their due date. The objective is to minimize the maximum penalty subject to certain assumptions on the target start times, due dates, and penalty functions. A more efficient algorithm than the existing one is presented for this problem. The number of computations in the proposed algorithm is of the order of n log n, while in the existing algorithm it is of the order of n2.