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Exact Methods for Computing All Lorenz Optimal Solutions to Biobjective Problems

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Algorithmic Decision Theory (ADT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9346))

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Abstract

This paper deals with biobjective combinatorial optimization problems where both objectives are required to be well-balanced. Lorenz dominance is a refinement of the Pareto dominance that has been proposed in economics to measure the inequalities in income distributions. We consider in this work the problem of computing the Lorenz optimal solutions to combinatorial optimization problems where solutions are evaluated by a two-component vector. This setting can encompass fair optimization or robust optimization. The computation of Lorenz optimal solutions in biobjective combinatorial optimization is however challenging (it has been shown intractable and NP-hard on certain problems). Nevertheless, to our knowledge, very few works address this problem. We propose thus in this work new methods to generate Lorenz optimal solutions. More precisely, we consider the adaptation of the well-known two-phase method proposed in biobjective optimization for computing Pareto optimal solutions to the direct computing of Lorenz optimal solutions. We show that some properties of the Lorenz dominance can provide a more efficient variant of the two-phase method. The results of the new method are compared to state-of-the-art methods on various biobjective combinatorial optimization problems and we show that the new method is more efficient in a majority of cases.

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References

  1. Kostreva, M.M., Ogryczak, W.: Linear optimization with multiple equitable criteria. RAIRO - Operations Research 33 (7 1999) 275–297

    Google Scholar 

  2. Kostreva, M., Ogryczak, W., Wierzbicki, A.: Equitable aggregations and multiple criteria analysis. Eur. J. Oper. Res. 158(2), 362–377 (2004)

    Article  MathSciNet  Google Scholar 

  3. Perny, P., Spanjaard, O., Storme, L.X.: A decision-theoretic approach to robust optimization in multivalued graphs. Annals OR 147(1), 317–341 (2006)

    Article  MathSciNet  Google Scholar 

  4. Yu, P.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14(3), 319–377 (1974)

    Article  MathSciNet  Google Scholar 

  5. Baatar, D., Wiecek, M.: Advancing equitability in multiobjective programming. Computational&. Applied Mathematics 52(1–2), 225–234 (2006)

    MATH  Google Scholar 

  6. Moghaddam, A., Yalaoui, F., Amodeo, L.: Lorenz versus Pareto dominance in a single machine scheduling problem with rejection. In: Takahashi, R., Deb, K., Wanner, E., Greco, S. (eds.) Evolutionary Multi-Criterion Optimization. Lecture Notes in Computer Science, vol. 6576, pp. 520–534. Springer, Berlin Heidelberg (2011)

    Chapter  Google Scholar 

  7. Endriss, U.: Reduction of economic inequality in combinatorial domains. In: AAMAS. (2013) 175–182

    Google Scholar 

  8. Ulungu, E., Teghem, J.: The two-phases method: An efficient procedure to solve biobjective combinatorial optimization problems. Foundation of Computing and Decision Science 20, 149–156 (1995)

    MATH  Google Scholar 

  9. Visée, M., Teghem, J., Pirlot, M., Ulungu, E.: Two-phases method and branch and bound procedures to solve the bi-objective knapsack problem. J. Global Optim. 12, 139–155 (1998)

    Article  MathSciNet  Google Scholar 

  10. Ehrgott, M., Skriver, A.: Solving biobjective combinatorial max-ordering problems by ranking methods and a two-phases approach. Eur. J. Oper. Res. 147(3), 657–664 (2003)

    Article  MathSciNet  Google Scholar 

  11. Przybylski, A., Gandibleux, X., Ehrgott, M.: Two-phase algorithms for the biobjective assignement problem. Eur. J. Oper. Res. 185(2), 509–533 (2008)

    Article  Google Scholar 

  12. Raith, A., Ehrgott, M.: A two-phase algorithm for the biobjective integer minimum cost flow problem. Computers&. Oper. Res. 36(6), 1945–1954 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Hardy, G., Littlewood, J., Pólya, G.: Inequalities. Cambridge University Press, Cambridge Mathematical Library (1952)

    MATH  Google Scholar 

  14. Shorrocks, A.F.: Ranking income distributions. Economica 50(197), 3–17 (1983)

    Article  Google Scholar 

  15. Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  16. Perny, P., Weng, P., Goldsmith, J., Hanna, J.: Approximation of Lorenz-Optimal Solutions in Multiobjective Markov Decision Processes. In: Conference on Uncertainty in Artificial Intelligence. (2013)

    Google Scholar 

  17. Perny, P., Spanjaard, O.: An Axiomatic Approach to Robustness in Search Problems with Multiple Scenarios. In: Proceedings of the 19th conference on Uncertainty in Artificial Intelligence. (2003) 469–476

    Google Scholar 

  18. Laumanns, M., Thiele, L., Zitzler, E.: An adaptive scheme to generate the Pareto front based on the epsilon-constraint method. In Branke, J., Deb, K., Miettinen, K., Steuer, R., eds.: Practical Approaches to Multi-Objective Optimization. Number 04461 in Dagstuhl Seminar Proceedings (2005)

    Google Scholar 

  19. Cohon, J.: Multiobjective Programming and Planning. Academic Press, New York (1978)

    MATH  Google Scholar 

  20. Aneja, Y., Nair, K.: Bicriteria transportation problem. Manage. Sci. 25, 73–78 (1979)

    Article  MathSciNet  Google Scholar 

  21. Yager, R.: On ordered weighted averaging aggregation operators in multicriteria decision making. In: IEEE Trans. Systems, Man and Cybern. Volume 18. (1998) 183–190

    Google Scholar 

  22. Ogryczak, W.: Inequality measures and equitable approaches to location problems. Eur. J. Oper. Res. 122(2), 374–391 (2000)

    Article  MathSciNet  Google Scholar 

  23. Bazgan, C., Hugot, H., Vanderpooten, D.: Solving efficiently the 0–1 multi-objective knapsack problem. Computers&. Oper. Res. 36(1), 260–279 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Eppstein, D.: Finding the \(k\) shortest paths. SIAM J. Computing 28(2), 652–673 (1998)

    Article  MathSciNet  Google Scholar 

  25. Jiménez, V.M., Marzal, A.: A lazy version of Eppstein’s k shortest paths algorithm. In: Proceedings of the 2Nd International Conference on Experimental and Efficient Algorithms. WEA’03, Berlin, Heidelberg, Springer-Verlag (2003) 179–191

    Google Scholar 

  26. Stewart, B.S., White III, C.C.: Multiobjective A*. J. ACM 38(4), 775–814 (1991). October

    Article  Google Scholar 

  27. Mandow, L., Pérez-De-la Cruz, J.L.: A new approach to multiobjective A* search. In: Proceedings of the 19th International Joint Conference on Artificial Intelligence. IJCAI’05, San Francisco, CA, USA, Morgan Kaufmann Publishers Inc. (2005) 218–223

    Google Scholar 

  28. Gandibleux, X., Vancoppenolle, D., Tuyttens, D.: A first making use of GRASP for solving MOCO problems. In: 14th International Conference in Multiple Criteria Decision-Making, Charlottesville (1998)

    Google Scholar 

  29. Florios, K., Mavrotas, G.: Generation of the exact Pareto set in multi-objective traveling salesman and set covering problems. Appl. Math. Comput. 237, 1–19 (2014)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. PSL, Université Paris-Dauphine LAMSADE, CNRS UMR 7243, 75775, Paris Cedex 16, France

    Lucie Galand

  2. Sorbonne Universités, UPMC Université Paris 06, CNRS, LIP6, UMR 7606, 75005, Paris, France

    Thibaut Lust

Authors
  1. Lucie Galand
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  2. Thibaut Lust
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Correspondence to Thibaut Lust .

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Editors and Affiliations

  1. University of New South Wales, Kensington, Australia

    Toby Walsh

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Galand, L., Lust, T. (2015). Exact Methods for Computing All Lorenz Optimal Solutions to Biobjective Problems. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_19

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  • DOI: https://doi.org/10.1007/978-3-319-23114-3_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23113-6

  • Online ISBN: 978-3-319-23114-3

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