Negative Dependence in Matrix Arrangement Problems
26 Pages Posted: 5 Apr 2016
Date Written: March 11, 2016
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: Suggested Citation