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Block rearranging elements within matrix columns to minimize the variability of the row sums

Author

Listed:
  • Kris Boudt

    (Vrije Universiteit Brussel (VUB)
    Vrije Universiteit Amsterdam)

  • Edgars Jakobsons

    (ETH Zürich)

  • Steven Vanduffel

    (Vrije Universiteit Brussel (VUB))

Abstract

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.

Suggested Citation

  • Kris Boudt & Edgars Jakobsons & Steven Vanduffel, 2018. "Block rearranging elements within matrix columns to minimize the variability of the row sums," 4OR, Springer, vol. 16(1), pages 31-50, March.
    • Handle: RePEc:spr:aqjoor:v:16:y:2018:i:1:d:10.1007_s10288-017-0344-4
    DOI: 10.1007/s10288-017-0344-4
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    References listed on IDEAS

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    1. E. G. Coffman & M. Yannakakis, 1984. "Permuting Elements Within Columns of a Matrix in Order to Minimize Maximum Row Sum," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 384-390, August.
    2. Embrechts, Paul & Puccetti, Giovanni & Rüschendorf, Ludger, 2013. "Model uncertainty and VaR aggregation," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 2750-2764.
    3. Wen-Lian Hsu, 1984. "Approximation Algorithms for the Assembly Line Crew Scheduling Problem," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 376-383, August.
    4. Wang, Bin & Wang, Ruodu, 2011. "The complete mixability and convex minimization problems with monotone marginal densities," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1344-1360, November.
    5. Mokotoff, Ethel, 2004. "An exact algorithm for the identical parallel machine scheduling problem," European Journal of Operational Research, Elsevier, vol. 152(3), pages 758-769, February.
    6. Carole Bernard & Don McLeish, 2016. "Algorithms for Finding Copulas Minimizing Convex Functions of Sums," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(05), pages 1-26, October.
    7. Mauro Dell'Amico & Manuel Iori & Silvano Martello & Michele Monaci, 2008. "Heuristic and Exact Algorithms for the Identical Parallel Machine Scheduling Problem," INFORMS Journal on Computing, INFORMS, vol. 20(3), pages 333-344, August.
    8. Boland, Philip J. & Proschan, Frank, 1988. "Multivariate arrangement increasing functions with applications in probability and statistics," Journal of Multivariate Analysis, Elsevier, vol. 25(2), pages 286-298, May.
    9. Mauro Dell’Amico & Silvano Martello, 1995. "Optimal Scheduling of Tasks on Identical Parallel Processors," INFORMS Journal on Computing, INFORMS, vol. 7(2), pages 191-200, May.
    10. Dell'Amico, Mauro & Martello, Silvano, 2005. "A note on exact algorithms for the identical parallel machine scheduling problem," European Journal of Operational Research, Elsevier, vol. 160(2), pages 576-578, January.
    11. Antonio Frangioni & Emiliano Necciari & Maria Grazia Scutellà, 2004. "A Multi-Exchange Neighborhood for Minimum Makespan Parallel Machine Scheduling Problems," Journal of Combinatorial Optimization, Springer, vol. 8(2), pages 195-220, June.
    12. Carole Bernard & Ludger Rüschendorf & Steven Vanduffel & Jing Yao, 2017. "How robust is the value-at-risk of credit risk portfolios?," The European Journal of Finance, Taylor & Francis Journals, vol. 23(6), pages 507-534, May.
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    Cited by:

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    4. Nabil Bouamara & Kris Boudt & S'ebastien Laurent & Christopher J. Neely, 2023. "Sluggish news reactions: A combinatorial approach for synchronizing stock jumps," Papers 2309.15705, arXiv.org.

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