Finding representations for an unconstrained bi-objective combinatorial optimization problem

  • Alexandre D. Jesus
  • Luís Paquete
  • José Rui Figueira
Original Paper
  • 46 Downloads

Abstract

Typically, multi-objective optimization problems give rise to a large number of optimal solutions. However, this information can be overwhelming to a decision maker. This article introduces a technique to find a representative subset of optimal solutions, of a given bounded cardinality for an unconstrained bi-objective combinatorial optimization problem in terms of \(\epsilon \)-indicator. This technique extends the Nemhauser–Ullman algorithm for the knapsack problem and allows to find a representative subset in a single run. We present a discussion on the representation quality achieved by this technique, both from a theoretical and numerical perspective, with respect to an optimal representation.

Keywords

Multiobjective combinatorial optimization Representation problem Dynamic programming 

References

  1. 1.
    Bazgan, C., Jamain, F., Vanderpooten, D.: Approximate Pareto sets of minimal size for multi-objective optimization problems. Oper. Res. Lett. 43(1), 1–6 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. In: Larmore, L.L., Goemans, M.X. (eds.) Proceedngs of the 35rd Annual ACM Symposium on Theory of Computing (STOC), pp. 232–241. ACM Press, New york (2003)Google Scholar
  3. 3.
    Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5(2), 266–288 (1957)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Diakonikolas, I., Yannakakis, M.: Small approximate Pareto sets for bi-objective shortest paths and other problems. SIAM J. Comput. 39(4), 1340–1371 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ebem-Chaime, M.: Parametric solution for linear bricriteria knapsack models. Manag. Sci. 42(11), 1565–1575 (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)MATHGoogle Scholar
  7. 7.
    Eusébio, A., Figueira, J.R., Ehrgott, M.: On finding representative non-dominated points for bi-objective integer network flow problems. Comput. Oper. Res. 48, 1–10 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Guerreiro, A.P., Fonseca, C.M., Paquete, L.: Greedy hypervolume subset selection in low dimensions. Evol. Comput. 24(3), 521–544 (2016)CrossRefGoogle Scholar
  9. 9.
    Hamacher, H.W., Pedersen, C.R., Ruzika, S.: Finding representative systems for discrete bicriterion optimization problems. Oper. Res. Lett. 35(3), 336–344 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jesus, A. D.: Implicit enumeration for representation systems in multiobjective optimization. Master’s thesis, University of Coimbra, Portugal (2015)Google Scholar
  11. 11.
    Kuhn, T., Fonseca, C.M., Paquete, L., Ruzika, S., Duarte, M.M., Figueira, J.R.: Hypervolume subset selection in two dimensions: formulations and algorithms. Evol. Comput. 24(3), 411–425 (2016)CrossRefGoogle Scholar
  12. 12.
    Nemhauser, G., Ullman, Z.: Discrete dynamic programming and capital allocation. Manag. Sci. 15(9), 494–505 (1969)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, FOCS ’00, pp. 86–92, Washington, DC, USA, 2000. IEEE Computer SocietyGoogle Scholar
  14. 14.
    Sayın, S.: Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math. Program. 87(3), 543–560 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sayın, S.: A procedure to find discrete representations of the efficient set with specified coverage errors. Oper. Res. 51(3), 427–436 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sayın, S., Kouvelis, P.: The multiobjective discrete optimization problem: a weighted min-max two-stage optimization approach and a bicriteria algorithm. Manag. Sci. 51(10), 1572–1581 (2005)CrossRefMATHGoogle Scholar
  17. 17.
    Serafini, P.: Some considerations about computational complexity for multiobjective combinatorial optimization. In: Recent Advances and Historical Development of Vector Optimization, Lecture Notes in Economics and Mathematics, pp. 221–231, Berlin, Germany. Springer, Berlin (1986)Google Scholar
  18. 18.
    Vaz, D., Paquete, L., Fonseca, C.M., Klamroth, K., Stiglmayr, M.: Representation of the non-dominated set in biobjective discrete optimization. Comput. Oper. Res. 63, 172–186 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Verel, S., Liefooghe, A., Jourdan, L., Dhaenens, C.: Analyzing the effect of objective correlation on the efficient set of MNK-landscapes. In: Proceedings of the 5th Conference on Learning and Intelligent OptimizatioN (LION 5), Lecture Notes in Computer Science, pp. 116–130. Springer, Berlin (2011)Google Scholar
  20. 20.
    Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithmsa comparative case study. In: Proceedings of the International Conference on Parallel Problem Solving from Nature PPSN V, pp. 292–301. Springer, Berlin (1998)Google Scholar
  21. 21.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Alexandre D. Jesus
    • 1
  • Luís Paquete
    • 1
  • José Rui Figueira
    • 2
  1. 1.CISUC, Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal
  2. 2.CEG-IST, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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