Skip to main content
SpringerLink
Log in

Finding representations for an unconstrained bi-objective combinatorial optimization problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  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)

    Article  MathSciNet  Google Scholar 

  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. Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5(2), 266–288 (1957)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ebem-Chaime, M.: Parametric solution for linear bricriteria knapsack models. Manag. Sci. 42(11), 1565–1575 (1996)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guerreiro, A.P., Fonseca, C.M., Paquete, L.: Greedy hypervolume subset selection in low dimensions. Evol. Comput. 24(3), 521–544 (2016)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jesus, A. D.: Implicit enumeration for representation systems in multiobjective optimization. Master’s thesis, University of Coimbra, Portugal (2015)

  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)

    Article  Google Scholar 

  12. Nemhauser, G., Ullman, Z.: Discrete dynamic programming and capital allocation. Manag. Sci. 15(9), 494–505 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  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 Society

  14. Sayın, S.: Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math. Program. 87(3), 543–560 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sayın, S.: A procedure to find discrete representations of the efficient set with specified coverage errors. Oper. Res. 51(3), 427–436 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the European Regional Development Fund (FEDER), through the COMPETE 2020—Operational Program for Competitiveness and Internationalization (POCI).

Author information

Authors and Affiliations

  1. CISUC, Department of Informatics Engineering, University of Coimbra, Pólo II, 3030-290, Coimbra, Portugal

    Alexandre D. Jesus & Luís Paquete

  2. CEG-IST, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

    José Rui Figueira

Authors
  1. Alexandre D. Jesus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luís Paquete
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José Rui Figueira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís Paquete.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jesus, A.D., Paquete, L. & Figueira, J.R. Finding representations for an unconstrained bi-objective combinatorial optimization problem. Optim Lett 12, 321–334 (2018). https://doi.org/10.1007/s11590-017-1129-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1129-6

Keywords

Search

Navigation