Advertisement

Mathematical Methods of Operations Research

, Volume 68, Issue 2, pp 257–276 | Cite as

Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning

  • Lizhen Shao
  • Matthias Ehrgott
Original Article

Abstract

In this paper, we propose a modification of Benson’s algorithm for solving multiobjective linear programmes in objective space in order to approximate the true nondominated set. We first summarize Benson’s original algorithm and propose some small changes to improve computational performance. We then introduce our approximation version of the algorithm, which computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of \({\varepsilon}\)-nondominated points. This work is motivated by an application, the beam intensity optimization problem of radiotherapy treatment planning. This problem can be formulated as a multiobjective linear programme with three objectives. The constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. With our algorithm we solve the problem approximately within a specified accuracy in objective space. We present results on four clinical cancer cases that clearly illustrate the advantages of our method.

Keywords

Multiobjective linear programming Radiotherapy treatment planning \({\varepsilon}\)-efficient solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Benson HP (1998) Hybrid approach for solving multiple-objective linear programs in outcome space. J Optim Theory Appl 98: 17–35zbMATHCrossRefMathSciNetGoogle Scholar
  2. Benson HP (1998) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J Global Optim 13: 1–24zbMATHCrossRefMathSciNetGoogle Scholar
  3. Chen PC, Hansen P (1991) On-line and off-line vertex enumeration by adjacency lists. Oper Res Lett 10: 403–409zbMATHCrossRefMathSciNetGoogle Scholar
  4. Cotrutz C, Lahanas M, Kappas K, Baltas D (2001) A multiobjective gradient-based dose optimization algorithm for external beam conformal radiotherapy. Phys Med Biol 46: 2161–2175CrossRefGoogle Scholar
  5. Craft DL, Halabi TF, Bortfeld TR (2005) Exploration of tradeoffs in intensity-modulated radiotherapy. Phys Med Biol 50: 5857–68CrossRefGoogle Scholar
  6. Craft DL, Halabi TF, Shih HA, Bortfeld TR (2006) Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med Phys 33: 3399–3407CrossRefGoogle Scholar
  7. Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for pareto set generation in multicriteria optimization problems. Struct Multidiscip Optim 14: 63–69Google Scholar
  8. Das I, Dennis JE (1998) Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8: 631–657zbMATHCrossRefMathSciNetGoogle Scholar
  9. Ehrgott M, Gandibleux X (2007) Bound sets for biobjective combinatorial optimization problems. Comput Oper Res 34: 2674–2694zbMATHCrossRefMathSciNetGoogle Scholar
  10. Ehrgott M, Wiecek M (2005) Multiobjective programming. In: Figueira J, Greco S, Ehrgott M (eds) Multicriteria decision analysis: state of the art surveys. Springer Science + Business Media, New York, pp 667–722CrossRefGoogle Scholar
  11. Hamacher H, Küfer K-H (2002) Inverse radiation therapy planing—A multiple objective optimization approach. Discret Appl Math 118: 145–161zbMATHCrossRefGoogle Scholar
  12. Holder A (2003) Designing radiotherapy plans with elastic constraints and interior point methods. Health Care Management Sci 6(1): 5–16CrossRefGoogle Scholar
  13. Horst R, Thoai NV, Devries J (1988) On finding the new vertices and redundant constraints in cutting plane algorithms for global optimization. Oper Res Lett 7: 85–90zbMATHCrossRefMathSciNetGoogle Scholar
  14. Küfer K-H, Scherrer A, Monz M, Alonso F, Trinkaus H, Bortfeld T, Thieke C (2003) Intensity-modulated radiotherapy—A large scale multi-criteria programming problem. OR Spectr 25: 223–249zbMATHCrossRefGoogle Scholar
  15. Lahanas M, Schreibmann E, Milickovic N, Baltas D (2003) Intensity modulated beam radiation therapy dose optimization with multi-objective evolutionary algorithms. In: Fonseca CM, Fleming PJ, Zitzler E, Deb K, Thiele L (eds) Evolutionary multi-criterion optimization. Lecture Notes in Computer Science, vol 2632. Springer, Berlin, pp 648–661CrossRefGoogle Scholar
  16. Lahanas M, Schreibmann E, Baltas D (2003) Multiobjective inverse planning for intensity modulated radiotherapy with constraint-free gradient-based optimization algorithms. Phys Med Biol 48: 2843–2871CrossRefGoogle Scholar
  17. Lim J, Ferris MC, Wright SJ, Shepard DM, Earl MA (2002) An optimization framework for conformal radiation treatment planning. Technical report, Computer Sciences Department, University of Wisconsin–Madison. Available online at http://pages.cs.wisc.edu/~ferris/papers/conformal.pdf. Accepted for publication, INFORMS Journal on Computing
  18. Loridan P (1984) \({\varepsilon}\)-solutions in vector minimization problems. J Optim Theory Appl 43: 265–276zbMATHCrossRefMathSciNetGoogle Scholar
  19. Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized constraint method for generating the pareto frontier. Struct Multidiscip Optim 25: 86–98CrossRefMathSciNetGoogle Scholar
  20. Nizin P, Kania A, Ayyangar K (2001) Basic concepts of Corvus dose model. Med Dosim 26: 65–69CrossRefGoogle Scholar
  21. Rockafellar RT (1970) Convex Analysis. Princeton University Press, PrincetonzbMATHGoogle Scholar
  22. Romeijn H, Dempsey J, Li J (2004) A unifying framework for multi-criteria fluence map optimization models. Phys Med Biol 49: 1991–2013CrossRefGoogle Scholar
  23. Schlegel W, Mahr A (2002) 3D-Conformal radiation therapy: a multimedia introduction to methods and techniques. Springer, BerlinGoogle Scholar
  24. Shao L (2005) A survey of beam intensity optimization in IMRT. In: Halliburton T (ed) Proceedings of the 40th Annual Conference of the Operational Society of New Zealand, pp 255–265. Available online at https://secure.orsnz.org.nz/conf40/content/papers/Shao.pdf
  25. Verhaegen F (2003) Monte Carlo modelling of external radiotherapy photon beams. Phys Med Biol 48: 107–164CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Engineering ScienceThe University of AucklandAucklandNew Zealand
  2. 2.Laboratoire d’Informatique de Nantes AtlantiqueUniversité de NantesNantesFrance

Personalised recommendations