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

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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.

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Notes

  1. Rearranging refers to the act of permuting (swapping) elements.

  2. In Bernard et al. (2017) it is shown that for some particular type of matrices this approach makes it possible to obtain after d − 1 steps (rearrangements) the situation in which all columns are oppositely ordered to the sum of all others. This development allows to obtain approximations for the minimum variance of a sum of (scaled) Bernoulli distributions in a fast way.

  3. In Graham (1966), the so-called parallel machine scheduling problem is solved by iteratively assigning subsequent jobs to the machine whose current completion time is minimum.

  4. The use of block rearrangements is consistent with the notion of \(\varSigma \)-countermonotonic matrices, as developed in Section 3.4 of Puccetti and Wang (2015); see in particular Definition 3.8 and Theorem 3.8 herein.

  5. See also Section 2.3 in Bernard et al. (2015).

  6. In probabilistic terms, this specific situation corresponds to mixability of random variables, a concept that has recently been studied in Wang and Wang (2011, 2016).

  7. Vectors \(W,Z\in \mathbb {R}^n\) are said to be oppositely ordered if \( (w_j-w_i)(z_j-z_i)\le 0\) for all \(1\le i < j\le n\).

  8. It is a direct consequence of the fact that unless W and Z have the same distribution, \(W\preccurlyeq _{\mathrm {cx}}Z\) implies \({E}[f(W)]<{E}[f(Z)]\) for any strictly convex function f; See Theorem 3.A43 of Shaked and Shanthikumar (2007).

  9. Note that this shuffling step already leads to a substantial reduction in the variability among the row sums, compared to a sorted (co-monotonic) arrangement.

  10. Sorting of the columns is done to render our paper more transparent with respect to reproducibility of the numerical results and also to enable a more fair comparison between block rearrangement algorithms and the RA of Puccetti and Rüschendorf (2012) in the case of heterogeneous columns. In fact, the RA goes through the columns sequentially (starting with the first column and so on). If the first few columns show little variability, rearranging them would not have much effect on the variability of row sums. For the block rearrangement algorithms, this issue does not exist, as they do not apply rearrangements sequentially starting with the first column.

  11. To verify this condition, one needs to consider \((2^{d}-2)/2=2^{d-1}-1\) different partitions of the matrix into two blocks. The division by two is because for each partition \(\delta \), the corresponding \(1-\delta \) does not need to be considered. Furthermore, the cases in which all elements of \(\delta \) are zero (resp. one), do not require consideration.

  12. Recall that \({E}[S]=\frac{1}{n}\sum _{j=1}^{n}s_{j}\) is invariant under intra-column rearrangements.

  13. The greedy algorithm has a running time of \(O(d\log d)\) and is flexible enough to deal with real numbers. Most partitioning algorithms are designed to work with integer numbers or positive real numbers. See Korf (1998) for a review of alternative solutions.

  14. In fact, from Proposition A.1 in Bernard et al. (2017) it follows that the greedy algorithm yields an arrangement in which each column is oppositely ordered to the sum of the other columns and is thus—in the particular context of the partition problem—as good as the RA.

  15. The results for the BRAVE algorithms are based on the greedy algorithm for finding the blocks. Using the KK algorithm, we obtained sightly better results, but the computation times were longer. We also tested the Block RA1 algorithm of Bernard and McLeish (2016) with block selection as in (2). These simulations showed that block selection based on maximal Spearman correlation of the row sums in a few iterations leads to a rearranged matrix with maximal Spearman correlation that is close to −1, but the improvement in the objective function V is relatively low compared to the RA, BRAVE and BRAVE \(+\) RA algorithms. Moreover, the Spearman correlation approach requires sorting the partial row sums for all considered partitions at every iteration, and is time-consuming. For the sake of brevity, we do not report these results.

  16. This is to be expected. The columns in this experiment are similar and on average BRA chooses blocks of equal size so that the variances of their row are typically close to each other.

  17. To save space we do not report the running times for the Pareto and the Bernoulli–Beta set-ups, since the obtained results for the running times are qualitatively similar. This is expected, since the nature of the randomized matrix entries has little impact on the computer time spent on the sorting and summation operations done in the rearrangement algorithms.

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Authors and Affiliations

  1. Vrije Universiteit Brussel (VUB), Brussels, Belgium

    Kris Boudt & Steven Vanduffel

  2. Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Kris Boudt

  3. ETH Zürich, Zürich, Switzerland

    Edgars Jakobsons

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  1. Kris Boudt
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  2. Edgars Jakobsons
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  3. Steven Vanduffel
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Correspondence to Steven Vanduffel.

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Boudt, K., Jakobsons, E. & Vanduffel, S. Block rearranging elements within matrix columns to minimize the variability of the row sums. 4OR-Q J Oper Res 16, 31–50 (2018). https://doi.org/10.1007/s10288-017-0344-4

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