, Volume 16, Issue 1, pp 31–50 | Cite as

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

Research Paper


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.


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

Mathematics Subject Classification

90B35 90B90 90C27 97M40 90C59 


Compliance with ethical standards

Conflict of interest

The authors declare that there are no conflicts of interest.


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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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