Single machine scheduling with batch set-up times to minimize maximum lateness

This paper considers a problem of scheduling N jobs on a single machine to minimize the maximum lateness. A partitioning of the jobs into F families is given. A set-up time is required at the start of each batch, where a batch is a largest set of contiguously scheduled jobs from the same family. We propose a single-batch heuristic in which all jobs of a family form a batch, and a double-batch heuristic in which each family is partitioned into at most two batches according to the due dates of its jobs. Both heuristics require O(N log N) time. It is shown that the single-batch heuristic has a worst-case performance ratio of 2 -1/F, whereas a composite heuristic which selects the better of the schedules generated by the single- and double-batch heuristics has a worst-case performance ratio of 5/3 for arbitrary F. Lower bounds are derived and are incorporated in a branch and bound algorithm. This algorithm uses a procedure to reduce the size of the problem, and employs a branching rule which forces pairs of jobs to lie in the same batch or in different batches. Computational tests show that the algorithm is effective in solving problems with up to 50 jobs.