Variable neighborhood search for the linear ordering problem

  • Authors:

  • Carlos G. Garcia;Dionisio Pérez-Brito;Vicente Campos;Rafael Martí

  • Affiliations:

  • Departamento de Economía de las Instituciones, Estadística Económica, Universidad de La Laguna, Spain;Departamento de Economía de las Instituciones, Estadística Económica, Universidad de La Laguna, Spain;Departamento de Estadistica e Investigacion Operativa, Universitat de València, Burjassot, Valencia, Spain;Departamento de Estadistica e Investigacion Operativa, Universitat de València, Burjassot, Valencia, Spain

  • Venue:
  • Computers and Operations Research
  • Year:
  • 2006

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Abstract

Given a matrix of weights, the linear ordering problem (LOP) consists of finding a permutation of the columns and rows in order to maximize the sum of the weights in the upper triangle. This NP-complete problem can also be formulated in terms of graphs, as finding an acyclic tournament with a maximal sum of arc weights in a complete weighted graph. In this paper, we first review the previous methods for the LOP and then propose a heuristic algorithm based on the variable neighborhood search (VNS) methodology. The method combines different neighborhoods for an efficient exploration of the search space. We explore different search strategies and propose a hybrid method in which the VNS is coupled with a short-term tabu search for improved outcomes. Our extensive experimentation with both real and random instances shows that the proposed procedure competes with the best-known algorithms in terms of solution quality, and has reasonable computing-time requirements.Variable neighborhood search (VNS) is a metaheuristic method that has recently been shown to yield promising outcomes for solving combinatorial optimization problems. Based on a systematic change of neighborhood in a local search procedure, VNS uses both deterministic and random strategies in search for the global optimum.In this paper, we present a VNS implementation designed to find high quality solutions for the NP-hard LOP, which has a significant number of applications in practice. The LOP, for example, is equivalent to the so-called triangulation problem for input-output tables in economics. Our implementation incorporates innovative mechanisms to include memory structures within the VNS methodology. Moreover we study the hybridization with other methodologies such as tabu search.