Improved Dynamic Programs for Some Batching Problems involving the Maximum Lateness Criterion

We study a class of scheduling problems involving the maximumlateness criterion and an element of batching. For all the problemsthat we examine, algorithms appear in the literature which consistof a sorting step to determine an optimal job sequence, followedby a dynamic programming step which determines the optimal batches.In each case, the dynamic program is based on a backward recursionof which a straightfoward implementation requires O(n^2) time, wheren is the number of jobs. We present improved implementations of thesedynamic programs that are based on monotonicity properties of theobjective expressed as a function of the length of the first batch.These properties and the use of efficient data structures enable usto exclude partial schedules that cannot lead to an overall optimumearly on in the enumeration process. The four problems that weconsider are solved in O(n log n) time; in two occasions, thebatching step is actually performed in linear time and the overallcomplexity is determined by the sorting step.