Block Rearranging Elements within Matrix Columns to Minimize the Variability of the Row Sums

20 Pages Posted: 24 Jul 2015 Last revised: 2 Mar 2017

Kris Boudt

Vrije Universiteit Brussel (VUB); VU University Amsterdam

Edgars Jakobsons

ETH Zürich - Department of Mathematics

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Date Written: January 18, 2017

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 scheduling, Rearrangements, Karmarkar-Karp difference algorithm

JEL Classification: C61, C63

Suggested Citation

Boudt, Kris and Jakobsons, Edgars and Vanduffel, Steven, Block Rearranging Elements within Matrix Columns to Minimize the Variability of the Row Sums (January 18, 2017). Available at SSRN: https://ssrn.com/abstract=2634669 or http://dx.doi.org/10.2139/ssrn.2634669

Kris Boudt

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
http://www.vub.ac.be/
Brussels, 1050
Belgium

VU University Amsterdam ( email )

De Boelelaan 1105
Amsterdam, ND North Holland 1081 HV
Netherlands

Edgars Jakobsons

ETH Zürich - Department of Mathematics ( email )

ETH Zentrum HG-F 42.1
Raemistr. 101
CH-8092 Zurich, 8092
Switzerland

Steven Vanduffel (Contact Author)

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

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