IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v86y2023i2d10.1007_s10898-023-01284-x.html
   My bibliography  Save this article

Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems

Author

Listed:
  • Stephan Helfrich

    (RPTU Kaiserslautern-Landau)

  • Tyler Perini

    (RICE University)

  • Pascal Halffmann

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Natashia Boland

    (H. Milton Stewart School of Industrial and Systems Engineering)

  • Stefan Ruzika

    (RPTU Kaiserslautern-Landau)

Abstract

Scalarization is a common technique to transform a multiobjective optimization problem into a scalar-valued optimization problem. This article deals with the weighted Tchebycheff scalarization applied to multiobjective discrete optimization problems. This scalarization consists of minimizing the weighted maximum distance of the image of a feasible solution to some desirable reference point. By choosing a suitable weight, any Pareto optimal image can be obtained. In this article, we provide a comprehensive theory of this set of eligible weights. In particular, we analyze the polyhedral and combinatorial structure of the set of all weights yielding the same Pareto optimal solution as well as the decomposition of the weight set as a whole. The structural insights are linked to properties of the set of Pareto optimal solutions, thus providing a profound understanding of the weighted Tchebycheff scalarization method and, as a consequence, also of all methods for multiobjective optimization problems using this scalarization as a building block.

Suggested Citation

  • Stephan Helfrich & Tyler Perini & Pascal Halffmann & Natashia Boland & Stefan Ruzika, 2023. "Analysis of the weighted Tchebycheff weight set decomposition for multiobjective discrete optimization problems," Journal of Global Optimization, Springer, vol. 86(2), pages 417-440, June.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:2:d:10.1007_s10898-023-01284-x
    DOI: 10.1007/s10898-023-01284-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-023-01284-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-023-01284-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Pascal Halffmann & Tobias Dietz & Anthony Przybylski & Stefan Ruzika, 2020. "An inner approximation method to compute the weight set decomposition of a triobjective mixed-integer problem," Journal of Global Optimization, Springer, vol. 77(4), pages 715-742, August.
    2. Timothy C. Y. Chan & Tim Craig & Taewoo Lee & Michael B. Sharpe, 2014. "Generalized Inverse Multiobjective Optimization with Application to Cancer Therapy," Operations Research, INFORMS, vol. 62(3), pages 680-695, June.
    3. Karakaya, G. & Köksalan, M. & Ahipaşaoğlu, S.D., 2018. "Interactive algorithms for a broad underlying family of preference functions," European Journal of Operational Research, Elsevier, vol. 265(1), pages 248-262.
    4. Hadi Charkhgard & Martin Savelsbergh & Masoud Talebian, 2018. "Nondominated Nash points: application of biobjective mixed integer programming," 4OR, Springer, vol. 16(2), pages 151-171, June.
    5. Alves, Maria João & Costa, João Paulo, 2016. "Graphical exploration of the weight space in three-objective mixed integer linear programs," European Journal of Operational Research, Elsevier, vol. 248(1), pages 72-83.
    6. Botte, Marco & Schöbel, Anita, 2019. "Dominance for multi-objective robust optimization concepts," European Journal of Operational Research, Elsevier, vol. 273(2), pages 430-440.
    7. Anthony Przybylski & Xavier Gandibleux & Matthias Ehrgott, 2010. "A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 371-386, August.
    8. Karakaya, G. & Köksalan, M., 2021. "Evaluating solutions and solution sets under multiple objectives," European Journal of Operational Research, Elsevier, vol. 294(1), pages 16-28.
    9. Bilge Bozkurt & John W. Fowler & Esma S. Gel & Bosun Kim & Murat Köksalan & Jyrki Wallenius, 2010. "Quantitative Comparison of Approximate Solution Sets for Multicriteria Optimization Problems with Weighted Tchebycheff Preference Function," Operations Research, INFORMS, vol. 58(3), pages 650-659, June.
    10. Y. P. Aneja & K. P. K. Nair, 1979. "Bicriteria Transportation Problem," Management Science, INFORMS, vol. 25(1), pages 73-78, January.
    11. Benson, Harold P. & Sun, Erjiang, 2002. "A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program," European Journal of Operational Research, Elsevier, vol. 139(1), pages 26-41, May.
    12. Luque, Mariano & Ruiz, Francisco & Steuer, Ralph E., 2010. "Modified interactive Chebyshev algorithm (MICA) for convex multiobjective programming," European Journal of Operational Research, Elsevier, vol. 204(3), pages 557-564, August.
    13. Pedro Correia & Luís Paquete & José Rui Figueira, 2021. "Finding multi-objective supported efficient spanning trees," Computational Optimization and Applications, Springer, vol. 78(2), pages 491-528, March.
    14. Ted Ralphs & Matthew Saltzman & Margaret Wiecek, 2006. "An improved algorithm for solving biobjective integer programs," Annals of Operations Research, Springer, vol. 147(1), pages 43-70, October.
    15. Arthur M. Geoffrion, 1967. "Solving Bicriterion Mathematical Programs," Operations Research, INFORMS, vol. 15(1), pages 39-54, February.
    16. Alves, Maria Joao & Climaco, Joao, 2000. "An interactive reference point approach for multiobjective mixed-integer programming using branch-and-bound," European Journal of Operational Research, Elsevier, vol. 124(3), pages 478-494, August.
    17. Britta Schulze & Kathrin Klamroth & Michael Stiglmayr, 2019. "Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions," Journal of Global Optimization, Springer, vol. 74(3), pages 495-522, July.
    18. Holzmann, Tim & Smith, J.C., 2018. "Solving discrete multi-objective optimization problems using modified augmented weighted Tchebychev scalarizations," European Journal of Operational Research, Elsevier, vol. 271(2), pages 436-449.
    19. H. P. Benson & E. Sun, 2000. "Outcome Space Partition of the Weight Set in Multiobjective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 17-36, April.
    20. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Stephan Helfrich & Kathrin Prinz & Stefan Ruzika, 2024. "The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1187-1216, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stephan Helfrich & Kathrin Prinz & Stefan Ruzika, 2024. "The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1187-1216, September.
    2. Britta Schulze & Kathrin Klamroth & Michael Stiglmayr, 2019. "Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions," Journal of Global Optimization, Springer, vol. 74(3), pages 495-522, July.
    3. Özgür Özpeynirci & Murat Köksalan, 2010. "An Exact Algorithm for Finding Extreme Supported Nondominated Points of Multiobjective Mixed Integer Programs," Management Science, INFORMS, vol. 56(12), pages 2302-2315, December.
    4. Kerstin Dächert & Tino Fleuren & Kathrin Klamroth, 2024. "A simple, efficient and versatile objective space algorithm for multiobjective integer programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 351-384, August.
    5. Melih Ozlen & Benjamin A. Burton & Cameron A. G. MacRae, 2014. "Multi-Objective Integer Programming: An Improved Recursive Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 470-482, February.
    6. Alves, Maria João & Costa, João Paulo, 2016. "Graphical exploration of the weight space in three-objective mixed integer linear programs," European Journal of Operational Research, Elsevier, vol. 248(1), pages 72-83.
    7. Anthony Przybylski & Xavier Gandibleux & Matthias Ehrgott, 2010. "A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 371-386, August.
    8. Seyyed Amir Babak Rasmi & Ali Fattahi & Metin Türkay, 2021. "SASS: slicing with adaptive steps search method for finding the non-dominated points of tri-objective mixed-integer linear programming problems," Annals of Operations Research, Springer, vol. 296(1), pages 841-876, January.
    9. Karakaya, G. & Köksalan, M., 2021. "Evaluating solutions and solution sets under multiple objectives," European Journal of Operational Research, Elsevier, vol. 294(1), pages 16-28.
    10. Fritz Bökler & Sophie N. Parragh & Markus Sinnl & Fabien Tricoire, 2024. "An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 263-290, August.
    11. Klamroth, Kathrin & Stiglmayr, Michael & Sudhoff, Julia, 2023. "Ordinal optimization through multi-objective reformulation," European Journal of Operational Research, Elsevier, vol. 311(2), pages 427-443.
    12. Pascal Halffmann & Tobias Dietz & Anthony Przybylski & Stefan Ruzika, 2020. "An inner approximation method to compute the weight set decomposition of a triobjective mixed-integer problem," Journal of Global Optimization, Springer, vol. 77(4), pages 715-742, August.
    13. Natashia Boland & Hadi Charkhgard & Martin Savelsbergh, 2015. "A Criterion Space Search Algorithm for Biobjective Integer Programming: The Balanced Box Method," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 735-754, November.
    14. Przybylski, Anthony & Gandibleux, Xavier, 2017. "Multi-objective branch and bound," European Journal of Operational Research, Elsevier, vol. 260(3), pages 856-872.
    15. Bennet Gebken & Sebastian Peitz, 2021. "Inverse multiobjective optimization: Inferring decision criteria from data," Journal of Global Optimization, Springer, vol. 80(1), pages 3-29, May.
    16. Chi-Yo Huang & Jih-Jeng Huang & You-Ning Chang & Yen-Chu Lin, 2021. "A Fuzzy-MOP-Based Competence Set Expansion Method for Technology Roadmap Definitions," Mathematics, MDPI, vol. 9(2), pages 1-26, January.
    17. Julius Bauß & Michael Stiglmayr, 2024. "Augmenting bi-objective branch and bound by scalarization-based information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 85-121, August.
    18. Sedeno-Noda, A. & Gonzalez-Martin, C. & Gutierrez, J., 2005. "The biobjective undirected two-commodity minimum cost flow problem," European Journal of Operational Research, Elsevier, vol. 164(1), pages 89-103, July.
    19. Gholamreza Shojatalab & Seyed Hadi Nasseri & Iraj Mahdavi, 2023. "New multi-objective optimization model for tourism systems with fuzzy data and new approach developed epsilon constraint method," OPSEARCH, Springer;Operational Research Society of India, vol. 60(3), pages 1360-1385, September.
    20. Sophie N. Parragh & Fabien Tricoire & Walter J. Gutjahr, 2022. "A branch-and-Benders-cut algorithm for a bi-objective stochastic facility location problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 44(2), pages 419-459, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:86:y:2023:i:2:d:10.1007_s10898-023-01284-x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.