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We address the problem of scheduling a single batching machine to minimize the maximum lateness with a constraint restricting the batch size. A solution for this NP-hard problem is defined by a selection of jobs for each batch and an ordering of those batches. As an alternative, we choose to represent a solution as a sequence of jobs. This approach is justified by our development of a dynamic program to find a schedule that minimizes the maximum lateness while preserving the underlying job order. Given this solution representation, we are able to define and evaluate various job-insert and job-swap neighborhood searches. Furthermore we introduce a new neighborhood, named split-merge, that allows multiple job inserts in a single move. The split-merge neighborhood is of exponential size, but can be searched in polynomial time by dynamic programming. Computational results with an iterated descent algorithm that employs the split-merge neighborhood show that it compares favorably with corresponding iterated descent algorithms based on the job-insert and job-swap neighborhoods.

This paper presents a two dimensional convex irregular bin packing problem with guillotine cuts. The problem combines the challenges of tackling the complexity of packing irregular pieces, guaranteeing guillotine cuts that are not always orthogonal to the edges of the bin, and allocating pieces to bins that are not necessarily of the same size. This problem is known as a two-dimensional multi bin size bin packing problem with convexirregular pieces and guillotine cuts. Since pieces are separated by means of guillotine cuts, our study is restricted to convex pieces. A beam search algorithm is described, which is successfully applied to both the multi and single bin size instances. The algorithm is competitive with the results reported in the literature for the single bin size problem and provides the first results for the multi bin size problem.

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