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

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.

Sluggish news reactions: A combinatorial approach for synchronizing stock jumps

“…The combinatorial optimization problem ( 17) is rooted in the pioneering work of Puccetti and Rüschendorf (2012) and Embrechts et al (2013) on rearrangements and the Rearrangement Algorithm; looping over each column in a matrix to order it oppositely to the sum of the other columns (see Appendix B for a run-through). This algorithm is best known as an actuarial tool to bound portfolio risk, but also has applications in other disciplines, such as such as option pricing (e.g., Bondarenko and Bernard, 2023) and operations research (e.g., Boudt et al, 2018).…”

“…We extend the pioneering work of Puccetti and Rüschendorf (2012) and Embrechts et al (2013) on rearrangements. Their algorithm is best known for actuarial applications, but can also be applied to other disciplines, such as option pricing (e.g., Bondarenko and Bernard, 2023) and operations research (e.g., Boudt et al, 2018). Rearrangements can also synchronize stock jumps on a fine sampling grid, provided we penalize economically implausible rearrangements.…”