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Bounding Fronts in Multi-Objective Combinatorial Optimization with Application to Aesthetic Drawing of Business Process Diagrams

  • Julius ŽilinskasEmail author
  • Antanas Žilinskas
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 104)

Abstract

The main concept of branch and bound is to detect subsets of feasible solutions which cannot contain optimal solutions. In multi-objective optimization a bounding front is used—a set of bounding vectors in the objective space dominating all possible objective vectors corresponding to the subset of feasible solutions. The subset cannot contain Pareto optimal (efficient) solutions if each bounding vector in the bounding front corresponding to this subset is dominated by at least one already known decision vector. The simplest bounding front corresponds to a single ideal vector composed of lower bounds for each objective function. However, the bounding fronts with multiple bounding vectors may be tighter and therefore their use may discard more subsets of feasible solutions. In this chapter we investigate the use of bounding vectors and bounding fronts in multi-objective optimization aided to aesthetic drawing of special graphs—business process diagrams. An experimental investigation shows that the use of the bounding front considerably reduces the number of function evaluations and computational time.

Notes

Acknowledgements

The support by Agency for Science, Innovation and Technology (MITA) trough the grant Nr.31V-145 is acknowledged.

References

  1. 1.
    Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1999)zbMATHGoogle Scholar
  2. 2.
    Bennett, C., Ryall, J., Spalteholz, L., Gooch, A.: The aesthetics of graph visualization. In: Cunningham, D.W., Meyer, G., Neumann, L. (eds.) Computational Aesthetics in Graphics, Visualization, and Imaging, pp. 1–8. Elsevier/Morgan Kaufmann, San Francisco (2007)Google Scholar
  3. 3.
    Brusco, M.J., Stahl, S.: Branch-and-Bound Applications in Combinatorial Data Analysis. Springer, New York (2005)zbMATHGoogle Scholar
  4. 4.
    Jančauskas, V., Mackutė-Varoneckienė, A., Varoneckas, A., Žilinskas, A.: On the multi-objective optimization aided drawing of connectors for graphs related to business process management. Comm. Comput. Inform. Sci. 319, 87–100 (2012)CrossRefGoogle Scholar
  5. 5.
    Owen, M., Jog, R.: BPMN and business process management. http://www.bpmn.org (2003)
  6. 6.
    Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds. Optim. Lett. 4(2), 173–183 (2010). doi: 10.1007/s11590-009-0156-3MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Purchase, H.: Metrics for graph drawing aesthetics. J. Visual Lang. Comput. 13(5), 501–516 (2002)CrossRefGoogle Scholar
  8. 8.
    Purchase, H., McGill, M., Colpoys, L., Carrington, D.: Graph drawing aesthetics and the comprehension of UML class diagrams: an empirical study. In: Proceedings of the 2001 Asia-Pacific Symposium on Information Visualisation, vol. 9, pp. 129–137 (2001)Google Scholar
  9. 9.
    Tamassia, R., Battista, G., Batini, C.: Automatic graph drawing and readability of diagrams. IEEE Trans. Syst. Man Cybern. 18(1), 61–79 (1989)CrossRefGoogle Scholar
  10. 10.
    Varoneckas, A., Žilinskas, A., Žilinskas, J.: Multi-objective optimization aided to allocation of vertices in aesthetic drawings of special graphs. Nonlinear Anal. Model. Control 18(4), 476–492 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Žilinskas, A., Žilinskas, J.: Branch and bound algorithm for multidimensional scaling with city-block metric. J. Global Optim. 43(2), 357–372 (2009). doi: 10.1007/s10898-008-9306-xMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Žilinskas, J., Goldengorin, B., Pardalos, P.M.: Pareto-optimal front of cell formation problem in group technology. J. Global Optim. (2014, in press). Doi: 10.1007/s10898-014-0154-6Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Recognition Processes DepartmentInstitute of Mathematics and Informatics, Vilnius UniversityVilniusLithuania
  2. 2.Department of Applied InformaticsInstitute of Mathematics and Informatics, Vilnius UniversityVilniusLithuania

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