4OR

pp 1–20 | Cite as

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

Research Paper

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.

Keywords

Assembly line crew scheduling Greedy algorithm Rearrangements k-Partitioning Karmarkar–Karp differencing algorithm 

Mathematics Subject Classification

90B35 90B90 90C27 97M40 90C59 

References

  1. 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
  2. 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
  3. Bernard C, Rüschendorf L, Vanduffel S (2015) Value-at-risk bounds with variance constraints. J Risk Insur. doi:10.1111/jori.12108
  4. Bernard C, Rüschendorf L, Vanduffel S, Yao J (2017) How robust is the value-at-risk of credit risk portfolios? Eur J Financ 23(6):507–534CrossRefGoogle Scholar
  5. Boland PJ, Proschan F (1988) Multivariate arrangement increasing functions with applications in probability and statistics. J Multivar Anal 25(2):286–298CrossRefGoogle Scholar
  6. Chopra S, Rao MR (1993) The partition problem. Math Program 59(1–3):87–115CrossRefGoogle Scholar
  7. Coffman E, Yannakakis M (1984) Permuting elements within columns of a matrix in order to minimize maximum row sum. Math Oper Res 9(3):384–390CrossRefGoogle Scholar
  8. Day PW (1972) Rearrangement inequalities. Can J Math 24(5):930–943CrossRefGoogle Scholar
  9. Dell’Amico M, Martello S (1995) Optimal scheduling of tasks on identical parallel processors. ORSA J Comput 7(2):191–200CrossRefGoogle Scholar
  10. Dell’Amico M, Martello S (2005) A note on exact algorithms for the identical parallel machine scheduling problem. Eur J Oper Res 160(2):576–578CrossRefGoogle Scholar
  11. Dell’Amico M, Iori M, Martello S, Monaci M (2008) Heuristic and exact algorithms for the identical parallel machine scheduling problem. INFORMS J Comput 20(3):333–344CrossRefGoogle Scholar
  12. Embrechts P, Puccetti G, Rüschendorf L (2013) Model uncertainty and VaR aggregation. J Bank Financ 37(8):2750–2764CrossRefGoogle Scholar
  13. Frangioni A, Necciari E, Scutella MG (2004) A multi-exchange neighborhood for minimum makespan parallel machine scheduling problems. J Comb Optim 8(2):195–220CrossRefGoogle Scholar
  14. Gent IP, Walsh T (1998) Analysis of heuristics for number partitioning. Comput Intell 14(3):430–451CrossRefGoogle Scholar
  15. Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell Syst Tech J 45(9):1563–1581CrossRefGoogle Scholar
  16. Hayes B (2002) Computing science: the easiest hard problem. Am Sci 90(2):113–117CrossRefGoogle Scholar
  17. Hsu W-L (1984) Approximation algorithms for the assembly line crew scheduling problem. Math Oper Res 9(3):376–383CrossRefGoogle Scholar
  18. Jakobsons E, Wang R (2016) Negative dependence in matrix arrangement problems. http://ssrn.com/abstract=2756934
  19. 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
  20. Korf RE (1998) A complete anytime algorithm for number partitioning. Artif Intell 106(2):181–203CrossRefGoogle Scholar
  21. Marshall AW, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its applications, 2nd edn. Springer, New YorkGoogle Scholar
  22. Mertens S (1998) Phase transition in the number partitioning problem. Phys Rev Lett 81(20):4281CrossRefGoogle Scholar
  23. Mokotoff E (2004) An exact algorithm for the identical parallel machine scheduling problem. Eur J Oper Res 152(3):758–769CrossRefGoogle Scholar
  24. Puccetti G, Rüschendorf L (2012) Computation of sharp bounds on the distribution of a function of dependent risks. J Comput Appl Math 236(7):1833–1840CrossRefGoogle Scholar
  25. Puccetti G, Wang R (2015) Extremal dependence concepts. Stat Sci 30(4):485–517CrossRefGoogle Scholar
  26. 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
  27. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, BerlinCrossRefGoogle Scholar
  28. Wang B, Wang R (2011) The complete mixability and convex minimization problems with monotone marginal densities. J Multivar Anal 102(10):1344–1360CrossRefGoogle Scholar
  29. Wang B, Wang R (2016) Joint mixability. Math Oper Res 41(3):808–826CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Vrije Universiteit Brussel (VUB)BrusselsBelgium
  2. 2.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  3. 3.ETH ZürichZürichSwitzerland

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