Negative Dependence in Matrix Arrangement Problems

26 Pages Posted: 5 Apr 2016

See all articles by Edgars Jakobsons

Edgars Jakobsons

ETH Zürich - Department of Mathematics

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: March 11, 2016

Abstract

Minimizing an arrangement increasing (AI) function with a matrix input over intra-column permutations is a difficult optimization problem of a combinatorial nature. Unlike maximization of AI functions (which is achieved by perfect positive dependence, namely, arranging all columns in an increasing order), minimization is a much more challenging problem due to the lack of a universal definition and construction of compensating arrangements in more than two dimensions. We consider AI functions with a special structure, which facilitates finding close-to-optimal solutions by employing the concept of Sigma-countermonotonicity and the (Block) Rearrangement Algorithm. We show that many classical optimization problems, including stochastic crew scheduling and assembly of reliable systems, have objective functions with this structure, and illustrate with a numerical case study. This paves a path to obtaining approximate solutions for problems that have so far been considered intractable.

Keywords: Schur-Convexity, Negative Dependence, Scheduling, Systems Assembly, Archimedean Copulas, Rearrangement Algorithm

JEL Classification: C61

Suggested Citation

Jakobsons, Edgars and Wang, Ruodu, Negative Dependence in Matrix Arrangement Problems (March 11, 2016). Available at SSRN: https://ssrn.com/abstract=2756934 or http://dx.doi.org/10.2139/ssrn.2756934

Edgars Jakobsons

ETH Zürich - Department of Mathematics ( email )

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

Ruodu Wang (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

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