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Multiple linear ordering problem

Author

Listed:
  • Benito-Marimon, Marc
  • Martí, Rafael
  • Martínez-Gavara, Anna

Abstract

The linear ordering problem (LOP) is a well-known NP-hard combinatorial optimization problem. Given a set of elements and its dominance matrix, the objective is to find a simultaneous ordering of rows and columns of the matrix, to maximize the sum of entries above the main diagonal. This linear ordering returns a rank of the elements maximizing the dominance, which provides a practical tool in many applications from economy to social networks. However, the LOP may have many alternate optima with significantly different orderings, which makes it difficult to provide an optimal representative ranking. In this paper, we study the rankability of the input matrix in terms of the similarities among alternate optimal rankings. We propose new mathematical models to formulate the rankability as an optimization problem. We also present a heuristic algorithm, based on the VNS methodology, to target large instances. Our computational testing shows that the analysis of several alternate optimal solutions may provide the user with practical information to overcome the limitations of a single optimal ranking.

Suggested Citation

  • Benito-Marimon, Marc & Martí, Rafael & Martínez-Gavara, Anna, 2026. "Multiple linear ordering problem," European Journal of Operational Research, Elsevier, vol. 331(2), pages 351-364.
    • Handle: RePEc:eee:ejores:v:331:y:2026:i:2:p:351-364
    DOI: 10.1016/j.ejor.2025.10.040
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    References listed on IDEAS

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    1. Pierre Hansen & Nenad Mladenović & Raca Todosijević & Saïd Hanafi, 2017. "Variable neighborhood search: basics and variants," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(3), pages 423-454, September.
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