Skip to main content
SpringerLink
Log in

Modified Distance Calculation in Generational Distance and Inverted Generational Distance

  • Conference paper
  • First Online:
Evolutionary Multi-Criterion Optimization (EMO 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9019))

Included in the following conference series:

Abstract

In this paper, we propose the use of modified distance calculation in generational distance (GD) and inverted generational distance (IGD). These performance indicators evaluate the quality of an obtained solution set in comparison with a pre-specified reference point set. Both indicators are based on the distance between a solution and a reference point. The Euclidean distance in an objective space is usually used for distance calculation. Our idea is to take into account the dominance relation between a solution and a reference point when we calculate their distance. If a solution is dominated by a reference point, the Euclidean distance is used for their distance calculation with no modification. However, if they are non-dominated with each other, we calculate the minimum distance from the reference point to the dominated region by the solution. This distance can be viewed as an amount of the inferiority of the solution (i.e., the insufficiency of its objective values) in comparison with the reference point. We demonstrate using simple examples that some Pareto non-compliant results of GD and IGD are resolved by the modified distance calculation. We also show that IGD with the modified distance calculation is weakly Pareto compliant whereas the original IGD is Pareto non-compliant.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bader, J., Zitzler, E.: HypE: An algorithm for fast hypervolume-based many-objective optimization. Evolutionary Computation 19, 45–76 (2011)

    Article  Google Scholar 

  2. Bosman, P.A.N., Thierens, D.: The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Trans. on Evolutionary Computation 7, 174–188 (2003)

    Article  Google Scholar 

  3. Coello, C.A.C., Lamont, G.B.: Applications of Multi-Objective Evolutionary Algorithms. World Scientific, Singapore (2004)

    Book  MATH  Google Scholar 

  4. Coello, C.A.C., Reyes Sierra, M.: A study of the parallelization of a coevolutionary multi-objective evolutionary algorithm. In: Monroy, R., Arroyo-Figueroa, G., Sucar, L.E., Sossa, H. (eds.) MICAI 2004. LNCS (LNAI), vol. 2972, pp. 688–697. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  5. Czyzak, P., Jaszkiewicz, A.: Pareto simulated annealing - A metaheuristic technique for multiple-objective combinatorial optimization. J. Multi-Criteria Decision Analysis 7, 34–47 (1998)

    Article  MATH  Google Scholar 

  6. Deb, K.: Multi-objective optimization using evolutionary algorithms. John Wiley & Sons, Chichester (2001)

    MATH  Google Scholar 

  7. Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, Part I: Solving problems with box constraints. IEEE Trans. on Evolutionary Computation 18, 577–601 (2014)

    Article  Google Scholar 

  8. Hansen, M.P., Jaszkiewicz, A.: Evaluating the quality of approximations of the non-dominated set, Technical Report IMM-REP-1998-7, Technical Univ. of Denmark (1998)

    Google Scholar 

  9. He, Z., Yen, G.G., Zhang, J.: Fuzzy-based Pareto optimality for many-objective evolutionary algorithms. IEEE Trans. on Evolutionary Computation 18, 269–285 (2014)

    Article  Google Scholar 

  10. Ishibuchi, H., Akedo, N., Nojima, Y.: Behavior of multi-objective evolutionary algorithms on many-objective knapsack problems. IEEE Trans. on Evolutionary Computation (in press)

    Google Scholar 

  11. Ishibuchi, H., Masuda, H., Tanigaki, Y., Nojima, Y.: Difficulties in specifying reference points to calculate the inverted generational distance for many-objective optimization problems. In: Proc. of MCDM 2014 (under IEEE SSCI 2014), pp. 170–177 (2014) http://www.cs.osakafu-u.ac.jp/ci/Papers/DownloadablePapers.php

  12. Ishibuchi, H., Masuda, H., Tanigaki, Y., Nojima, Y.: Evolutionary many-objective optimization: A short review. In: Proc. of IEEE CEC 2008, pp. 2424–2431 (2008)

    Google Scholar 

  13. Ishibuchi, H., Yoshida, T., Murata, T.: Balance between genetic search and local search in memetic algorithms for multiobjective permutation flowshop scheduling. IEEE Trans. on Evolutionary Computation 7, 204–223 (2003)

    Article  Google Scholar 

  14. Knowles, J.D., Corne, D.W.: On metrics for comparing non-dominated sets. In: Proc. of CEC 2002, pp. 711–716 (2002)

    Google Scholar 

  15. Knowles, J.D., Thiele, L., Zitzler, E.: A tutorial on the performance assessment of stochastic multiobjective optimizers. TIK Report No. 214, ETH Zurich (2006)

    Google Scholar 

  16. Martínez, S.Z., Hernández, V.A.S., Aguirre, H., Tanaka, K., Coello, C.A.C.: Using a family of curves to approximate the pareto front of a multi-objective optimization problem. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds.) PPSN 2014. LNCS, vol. 8672, pp. 682–691. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  17. Russo, L.M.S., Francisco, A.P.: Quick hypervolume. IEEE Trans. on Evolutionary Computation 18, 481–502 (2014)

    Article  Google Scholar 

  18. Schütze, O., Esquivel, X., Lara, A., Coello, C.A.C.: Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization. IEEE Trans. on Evolutionary Computation 16, 504–522 (2012)

    Article  Google Scholar 

  19. Sierra, M.R., Coello, C.A.C.: A new multi-objective particle swarm optimizer with improved selection and diversity mechanisms, Technical Report, CINVESTAV-IPN (2004)

    Google Scholar 

  20. Tan, K.C., Khor, E.F., Lee, T.H.: Multiobjective Evolutionary Algorithms and Applications. Springer, London (2005)

    MATH  Google Scholar 

  21. Van Veldhuizen D.A.: Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations. Ph.D. Thesis, Graduate School of Engineering, Air Force Institute of Technology, Wright-Patterson AFB, Ohio, USA (1999)

    Google Scholar 

  22. While, L., Bradstreet, L., Barone, L.: A fast way of calculating exact hypervolumes. IEEE Trans. on Evolutionary Computation 16, 86–95 (2012)

    Article  Google Scholar 

  23. Yang, S., Li, M., Liu, X., Zheng, J.: A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans. on Evolutionary Computation 17, 721–736 (2013)

    Article  Google Scholar 

  24. Zitzler, E., Brockhoff, D., Thiele, L.: The hypervolume indicator revisited: on the design of pareto-compliant indicators via weighted integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  25. Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms - a comparative case study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  26. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Fonseca, V.G.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Trans. on Evolutionary Computation 7, 117–132 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Intelligent Systems, Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka, 599-8531, Japan

    Hisao Ishibuchi, Hiroyuki Masuda, Yuki Tanigaki & Yusuke Nojima

Authors
  1. Hisao Ishibuchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroyuki Masuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuki Tanigaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yusuke Nojima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yusuke Nojima .

Editor information

Editors and Affiliations

  1. Institute for Polymers and Composites/I3N University of Minho Guimarães Portugal, Guimaraes, Portugal

    António Gaspar-Cunha

  2. Dept. of Electrical and Computer Engg, University of Coimbra, Coimbra, Portugal

    Carlos Henggeler Antunes

  3. CINVESTAV-IPN Depto. de Computacíon, Col. San Pedro Zacatenco, México DF, Mexico

    Carlos Coello Coello

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Ishibuchi, H., Masuda, H., Tanigaki, Y., Nojima, Y. (2015). Modified Distance Calculation in Generational Distance and Inverted Generational Distance. In: Gaspar-Cunha, A., Henggeler Antunes, C., Coello, C. (eds) Evolutionary Multi-Criterion Optimization. EMO 2015. Lecture Notes in Computer Science(), vol 9019. Springer, Cham. https://doi.org/10.1007/978-3-319-15892-1_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-15892-1_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15891-4

  • Online ISBN: 978-3-319-15892-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation