Abstract
This article describes a heuristic for scheduling so-called ‘modular’ projects. Exact solutions to this NP-hard problem can be obtained with existing branch-and-bound and dynamic-programming algorithms, but only for small to medium-size instances. The proposed heuristic, by contrast, can be used even for large instances, or when instances are particularly difficult because of their characteristics, such as a low network density. The proposed heuristic draws from existing results in the literature on sequential testing, which will be briefly reviewed. The performance of the heuristic is assessed over a dataset of 360 instances.
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Notes
A strict (partial) order \(O\subset V\times V\) over a set \(V\) is a relation defined on \(V\) that is both asymmetric (\((i,j)\in O\) implies \((j,i)\not \in O\)) and transitive (\((i,j)\in O\) and \((j,l)\in O\) implies \((i,l)\in O\)).
To compute the transitive reduction of a poset \((V,O)\) we remove all elements \((i,l)\) of \(O\) for which there is an element \(j\in V\) such that \((i,j)\in O\) and \((j,l)\in O\).
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The authors thank the two anonymous referees for their time and for the thoughtful suggestions, which have helped us improve the paper.
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Appendix: An instance with \(n=20\)
Appendix: An instance with \(n=20\)
In this appendix, for illustration purposes, we describe the outputs of the algorithms proposed in this text for the instance depicted in Fig. 6. The numerical data for this instance can be found in Table 5. The instance is part of the dataset (instance name \(g\_n20\_os6\_4\)).
Precedence network
The initial ordering generated by Greedy 1 is
which is easy to verify manually with the data from Table 5 and the pseudocode of Greedy 1. Greedy 2 removes jobs 1, 3 and 5 and the order of the modules is redetermined. In this case, the heuristic order of the modules does not change (\(L' = L''\) in lines 13-16 of the pseudocode of Greedy 2), and
The expected profit from \(L\) and \(L'\) is approximately 14.72 and 15.05, respectively, so Greedy 2 will return \(L'\). An optimal EMS policy found via B&B turns out to be slightly better, with an expected profit of 15.32. An optimal ordering is given by
this list is also found by the two implementations of Greedy 3.
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Huysmans, M., Coolen, K., Talla Nobibon, F. et al. A fast greedy heuristic for scheduling modular projects. J Heuristics 21, 47–72 (2015). https://doi.org/10.1007/s10732-014-9272-z
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DOI: https://doi.org/10.1007/s10732-014-9272-z