This paper proposes an exact exponential algorithm for the single machine total tardiness problem. It exploits the structure of a basic branchand-reduce framework based on the well known Lawler's decomposition property that solves the problem with worst-case complexity O * (3 n) in time and polynomial space. The proposed algorithm, called branch-and-merge, is an improvement of the branch-and-reduce technique with the embedding of a node merging operation. Its time complexity converges to O * (2 n) keeping the space complexity polynomial. This improves upon the best-known complexity result for this problem provided by dynamic programming across the subsets with O * (2 n) worst-case time and space complexity. The branch-and-merge technique is likely to be generalized to other sequencing problems with similar decomposition properties.
The paper presents a new generalization of the one-dimensional cutting stock problem (1D-CSP) that considers cut losses dependent on the items' cutting sequence. It is shown that this generalization can still be solved approximately by standard 1D-CSP approaches. Furthermore, a pattern-based heuristic (denoted by HSD) is presented that specifically considers sequence-dependent cut losses (SDCL). A computational study shows that whenever some variability in SDCL occurs, consideration of SDCL in the HSD heuristic is beneficial. Finally, two case studies illustrate the relevance of this new generalization.
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