Block rearranging elements within matrix columns to minimize the variability of the row sums
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Several problems in operations research, such as the assembly line crew scheduling problem and the k-partitioning problem can be cast as the problem of finding the intra-column rearrangement (permutation) of a matrix such that the row sums show minimum variability. A necessary condition for optimality of the rearranged matrix is that for every block containing one or more columns it must hold that its row sums are oppositely ordered to the row sums of the remaining columns. We propose the block rearrangement algorithm with variance equalization (BRAVE) as a suitable method to achieve this situation. It uses a carefully motivated heuristic—based on an idea of variance equalization—to find optimal blocks of columns and rearranges them. When applied to the number partitioning problem, we show that BRAVE outperforms the well-known greedy algorithm and the Karmarkar–Karp differencing algorithm.
KeywordsAssembly line crew scheduling Greedy algorithm Rearrangements k-Partitioning Karmarkar–Karp differencing algorithm
Mathematics Subject Classification90B35 90B90 90C27 97M40 90C59
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Conflict of interest
The authors declare that there are no conflicts of interest.
- Alvim AC and Ribeiro CC (2004) A hybrid bin-packing heuristic to multiprocessor scheduling. In: International workshop on experimental and efficient algorithms. Springer, Berlin, pp 1–13Google Scholar
- Bernard C, McLeish D (2016) Algorithms for finding copulas minimizing convex functions of sums. Asia Pac J Oper Res 33(5):1650040. doi: 10.1142/S0217595916500408
- Bernard C, Rüschendorf L, Vanduffel S (2015) Value-at-risk bounds with variance constraints. J Risk Insur. doi: 10.1111/jori.12108
- Jakobsons E, Wang R (2016) Negative dependence in matrix arrangement problems. http://ssrn.com/abstract=2756934
- Karmarkar N, Karp RM (1982) The differencing method of set partitioning. Technical Report UCB/CSD 82/113, Computer Science Division, University of California, BerkeleyGoogle Scholar
- Marshall AW, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its applications, 2nd edn. Springer, New YorkGoogle Scholar
- Rüschendorf L (2013) Mathematical risk analysis. In: Mikosch TV, Resnick SI, Robinson SM (eds) Springer Series in Operations Research and Financial Engineering. Springer, HeidelbergGoogle Scholar